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 THE LIFE, LETTERS, AND LABOURS OF FRANCIS 
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 BIOMETRIKA. A Journal for the Statistical Study of Biological 
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TABLES FOR STATISTICIANS 
 AND BIOMETRICIANS 
 
 EDITED BY 
 
 KARL PEARSON, F.R.S. 
 
 GALTON PROFESSOR, UNIVERSITY OF LONDON 
 
 ISSUED WITH ASSISTANCE FROM THE GRANT MADE BY 
 
 THE WORSHIPFUL COMPANY OF DRAPERS TO THE 
 
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PREFACE 
 
 |~ AM very conscious of the delay which has intervened between the announce- 
 ment of the publication of these Tables and their appearance. This delay has 
 been chiefly due to two causes." First the great labour necessary, which largely 
 fell on those otherwise occupied, and secondly the great expense involved (a) in 
 the calculation of the Tables, and (b) in their publication. This matter of expense 
 is one which my somewhat urgent correspondents, I venture to think, have entirely 
 overlooked. It is perfectly true that only one single Table in this volume has 
 been directly paid for, but a very large part of the labour of calculation has been 
 done by the Staff of the Biometric Laboratory, whose very existence depends on 
 the generous grant made to that laboratory by the Worshipful Company of 
 Drapers. Our staff is not a large one and it has many duties, so that the progress 
 of calculation has of necessity been slow. Even now I am omitting projected 
 Tables, which I can only hope may be incorporated in a later edition of this 
 work, e.g. Tables of the Incomplete B- and T-functions, and the Table needed to 
 complete Everitt's work on High Values of Tetrachoric r when r lies between 
 — - 80 and — TOO. It would only satisfy my ideal of what these Tables should be, 
 had I been able to throw into one volume with the present special tables, 
 extensive tables of squares, of square roots, of reciprocals and of the natural 
 trigonometric functions tabled to decimals of a degree. Logarithmic tables are 
 relatively little used by the statistician to-day, which is the age of mechanical 
 calculators, and he is perfectly ready to throw aside the fiction that there is any 
 gain in the cumbersome notation of minutes and seconds of angle — a system 
 which would have disappeared long ago, but for the appalling 'scrapping' of 
 astronomical apparatus it would involve. But the ideal of one handy book for 
 the statistician cannot be realised until we have a body of scientific statisticians 
 far more numerous than at present. Statisticians must for the time being carry 
 about with them not only this volume but a copy of Barlow's Tables, and a 
 set of Tables of the Trigonometrical Functions. 
 
vi Tables for Statisticians and Biometricians 
 
 Beside the cost of calculating these Tables, to which I have referred, must be 
 added the cost of printing them. I had to do this slowly as opportunity offered 
 in my Journal Biometrika, and the Tables as printed were moulded, in order 
 that stereos might be taken for reproduction. Even as it is, there are a number 
 of Tables in this volume, either printed here for the first time (e.g. Tables of the 
 Logarithm of the Factorial and of the Fourth Moment), or published here for the 
 first time (e.g. Tables of the G(r, v) Integrals), the setting up of which has 
 naturally been very expensive. 
 
 From the beginning of this work in 1901* when the first of these Tables was 
 published and moulded, I have had one end in view, the publication, as funds 
 would permit, of as full a series of Tables as possible. It is needless to say that 
 no anticipation of profit was ever made, the contributors worked for the sake 
 of science, and the aim was to provide what was possible at the lowest rate we 
 could. The issue may appear to many as even now costly ; let me assure those 
 inclined to cavil, that to pay its way with our existing public double or treble 
 the present price would not have availed; we are able to publish because of the 
 direct aid provided by initial publication in Biometrika and by direct assistance 
 from the Drapers' Company Grant. Yet a few years ago when a reprint of these 
 Tables in America was only stopped by the threat to prevent the circulation of 
 the book in which they were to appear entering any country with which we had 
 a reasonable copyright law, I was vigorously charged with checking the progress 
 of science and acting solely from commercial ends ! Meanwhile without any leave, 
 large portions of these tables have been reprinted, sometimes without even citing 
 the originals, in American psychological text-books. Two Russian subjects have 
 reissued many of these Tables in Russian and Polish versions, and copies of their 
 works in contravention of copyright are carried into other European countries. 
 It does not seem to have occurred to these men of science that there was any- 
 thing blameworthy in depriving Biometrika of such increased circulation as it 
 obtained from being the sole locus of these Tables, nor did they see in their 
 actions any injury to science as a whole resulting from lessening my power to 
 publish other work of a similar character. It is a singular phase of modern science 
 that it steals with a plagiaristic right hand while it stabs with a critical left. 
 
 The Introduction gives a brief description of each individual table ; it is by no 
 means intended to replace actual instruction in the use of the tables such as 
 
 * When issuing their prospectus in the spring of 1901 the Editors of Biometrika promised to 
 provide " numerical tables tending to reduce the labour of statistical arithmetic" 
 
Preface vii 
 
 is given in a statistical laboratory, nor does it profess to provide an account 
 of the innumerable uses to which they may be put, or to warn the reader of the 
 many difficulties which may arise from inept handling of them. Additional aid 
 may be found in the text which usually accompanies the original publication of 
 the tables. 
 
 In conclusion here I wish to thank the loyal friends and colleagues — Dr W. F. 
 Sheppard, Mr W. Palin Elderton, Dr Alice Lee, Mr P. F. Everitt, Miss Julia Bell, 
 Miss Winifred Gibson, Mr A. Rhind, Mr H. E. Soper and others — whose un- 
 remitting exertions have enabled so much to be accomplished, if that much is 
 indeed not the whole we need. I have further to acknowledge the courtesy 
 of the Council of the British Association, who have permitted the republication 
 of the Tables of the G (r, v) Integrals, originally published in their Transactions. 
 
 To the Syndics of the Cambridge Press I owe a deep debt of gratitude for 
 allowing me the services of their staff in the preparation of this work. Pages and 
 pages of these Tables' were originally set up for Biometrika, or were set up afresh 
 here, without the appearance of a single error. To those who have had experience 
 of numerical tables prepared elsewhere, the excellence of the Cambridge first proof 
 of columns of figures is a joy, which deserves the fullest acknowledgement. 
 
 Should this work ever reach a second edition I will promise two things, 
 rendered possible by the stereotyping of the tables : it shall not only appear 
 at a much reduced price, but it shall be largely increased in extent. 
 
 KARL PEARSON. 
 
 Biometric Laboratory, 
 February 7, 1914. 
 
 Errata 
 
 The reader is requested to make before using these Tables the following corrections on 
 pp. 82, 83, 84 and 85 : 
 
 For 177 VF2i and 177 sfFs 2 at the top of the Tables read 1-177 ViVSi and 1-177 V#2 a . 
 
When you can measure what you are speaking about and express it in 
 numbers, you know something about it, but when you cannot measure it, when 
 you cannot express it in numbers, your knowledge is of a meagre and unsatis- 
 factory kind. 
 
 Lord Kelvin. 
 
 La theorie des probabilites n'est au fond que le bon sens reduit au calcul ; 
 elle fait appr£cier avec exactitude ce que les esprits justes sentent par une sorte 
 d'instinct, sans qu'ils puissent souvent s'en rendre compte. 
 
 Laplace. 
 
ERRATA, ANTE USUM DILIGENTER CORRIGENDA. 
 
 Introduction. 
 
 p. xiii. Equation (i) cancel the + sign which follows A 3 w > or replace by - sign, 
 p. xiv. For Equation (vii) bis 
 
 ff i i(u -u 1 + u_ l + u i ) + 6%(5u 1 -5u -u_ l -u i )+u a -u (6) = 0, 
 read 2 i (-2« -Mi + «-i+»2) + <5i( 5 "i- 3M o- U-i-u 2 )+u -u (d) = O, 
 and add : " This equation is most effectively dealt with by finding the value of 
 
 %-«o(d) 
 
 <?o= 
 
 and then calculating : 
 
 kiZuo+U-i + Uz-bua) 
 
 p. xxxiv. Formula (xxxi), For A 7 (ab — cdf read N(ad—bcf- 
 
 p. xxxv. Table (3) was taken from Biometrika, Vol. IX. p. 292. Unfortunately it was not 
 there noted that Mr Yule's unit was 1000 houses : see his Theory of Statistics, p. 61. He 
 has drawn the Editor's attention to this regrettable omission. The table for the statistical 
 constants at the centre of the page should read : 
 
 (3) Houses x»- 1439-2998, P=8-730/10 312 , while # 2 , <j> and C 2 remain unchanged. 
 
 On p. xxxvi. Lines 3 — 9 while correct for the illustration actually given as table (3) 
 on p. xxv, are of course incorrect for the true unit of 1000 houses. The statement in 
 Lines 19 — 23 with regard to the houses building or built is incorrect ; there is very marked 
 positive association. We must now include the house-data, and Lines 26 — 27 should 
 read : " If we regard these four tables the order of ascending association judged by either 
 <j> or <7 2 is (3), (4), (5), (2) as against Mr Yule's (2), (3), (4), (5)." 
 
 p. xlvii. Line 6. For 4th -071,162 read 4th -073,116. 
 
 p. xlviii. Table column (i), 22nd Line of figures. For 45 read 5-0, and for S(x) at foot 
 read S(y). 
 
 p. xlix. Line 1. For &i = 10e read 6 1 = 10c l . 
 
 p. lv. Formula (xlii). 
 
 For log CJft/ ) = -0399,0899 + etc., 
 read log ( F Jft^ ) = 0-399,0899 +etc. 
 
 p. lx. Table, 4 = 2, B = ". For (21-556) read (31-556). 
 „ A=2, 5 = 8. For (12-202) read (13-202). 
 
 p. lxii. Line 11. 
 
 „ -67449^ , -67449^ 
 
 For _^. 2ftMld _^z fc , 
 
 read -67449 2,3, and -674492,3,. 
 
p. ixiii. The two solidi have been dropped in the biquadratic : 
 
 For ft (8ft- 9ft - 12) (4ft - 3ft) = (10ft- 12ft - 18) 2 (ft +3)*, 
 read ft (8ft - 9ft - 12)/(4ft - 3ft) = (10ft - 12ft - 18) 2 /(ft + 3) 2 . 
 
 p. lxv. Formula (lxxvi) 
 
 For S ft 20,^ = 205 -etc., 
 
 read Ftp, 2s 2 ^/s, & = 2ft - etc. 
 p. lxxv. Table, column Nvi, 3rd line. For 38 read 36. 
 
 p. lxxvii. Line 2. For " We look out 5-8 in Table L. " read " We look out 5-8 in Table LI." 
 p. lxxx. Line 5. For e -<-«*i g rea d e -4.560c e_ 
 p. lxxxiii. Line 13 from bottom. For 2-371,76665 read 2-371,6665. 
 
 Text of Tables. 
 p. 13. Table V, H-S41. For X 2 = "03172 read -03072. 
 
 pp. 82, 83, 84 and 85. For Vll-JN^ and f*77«/Fj, at the top of the Tables read 1-177nA^2, 
 
 and 1-177 ViV : 2 2 . 
 p. 92. Table XLVIII. 
 
 >r m = 20\ 
 ?rc = 20J 1 
 
 51-2195 
 
 read ra = 20\ 
 m = 20J 1 
 
 51-2195 
 
 25-6098 
 
 25-6098 
 
 2 
 
 12-4765 
 
 2 
 
 12-4765 
 
 3 
 
 5-9099 
 
 3 
 
 5-9099 
 
 4 
 
 2-7154 
 
 4 
 
 2-7154 
 
 5 
 
 1-2068 
 
 5 
 
 1-2068 
 
 6 
 
 •5172 
 
 6 
 
 •51 72 
 
 7 
 
 ■0839 
 
 7 
 
 •2130 
 
 8 
 
 •0315 
 
 8 
 
 •0839 
 
 9 
 
 •0112 
 
 9 
 
 •0315 
 
 10 
 
 •0037 
 
 10 
 
 ■0112 
 
 11 
 
 •0012 
 
 11 
 
 •0037 
 
 12 
 
 •0003 
 
 12 
 
 •0012 
 
 IS 
 
 •oooi 
 
 13 
 
 •0003 
 
 U 
 
 •0000 
 
 U 
 
 •0001 
 
 15 
 
 — 
 
 15 
 
 •0000 
 
 p. 126. Table LI V. For r = 2, <£° = 5, log H(r, ») = -106,5985, 
 read log H(r, »)- -196,5985. 
 
 p. 141. „ For r = 45, <£°=44, log F(r, v ) = -483,7836, 
 
 read log F(r, v) = 7-483,7836. 
 
 p. 142. „ For r = 50, <£° = 4, \ogH{r, v) = -932,5457, 
 
 read log H{r, v) = -392,5457. 
 
 p. 142. „ For r= 50, </>° =31, log ff(r, v)= 933,2995, 
 
 read log H(r, v) = -393,2995. 
 
 p. 143. For log log e -637 7799 16. 
 read log loge T-637 7843 11. 
 
 The issue of this list of Errata has been intentionally delayed in order to make it as 
 complete as a wider use of the volume would render possible. The Editor will be as grateful 
 for further emendations, as he has been for the above. 
 
CONTENTS 
 
 Preface 
 
 Introduction to the Use of the Tables . 
 
 PAGE 
 V 
 
 xiii 
 
 TABLES * 
 
 TABLE PAGE 
 
 I. Table of Deviates of the Normal Curve for each Permille of 
 
 Frequency xv 1 
 
 II. Tables of the Probability Integral : Area and Ordinate of 
 
 the Normal Curve in terms of the Abscissa . . xvii 2-7 
 
 III. Tables of the Probability Integral : Abscissa and Ordinate 
 
 in terms of difference of Areas ..... xvii 9-10 
 
 IV. Tables of the Probability Integral : Logarithms of Areas 
 
 for high Values of Deviate xxi 11 
 
 V. Probable Errors of Means and Standard Deviations . xxii 12-18 
 
 VI. Probable Errors of Coefficients of Variation . . . xxii 18 
 
 VII. Abac for Probable Error of a Coefficient of Correlation r xxiii 19 
 
 VIII. Probable Error of a Coefficient of Correlation : Table to 
 
 facilitate the calculation of 1 — r 2 . . . . xxiii 20-21 
 
 IX. Values of the Incomplete Normal Moment Functions, First 
 
 to Tenth Moments xxiv 22-23 
 
 X. Numerical Values and Graph of Generalised Probable Error xxv 24 
 
 XL Values of the Functions ^r„ i|r 2 , and ^ 3 required to determine 
 the constants of a Normal Frequency Distribution from 
 the Moments of its Truncated Tail .... xxviii 25 
 
 XII. Tables for Testing Goodness of Fit : 3 to 30 frequency 
 
 groupings ......... xxxi 26-28 
 
 * The Roman figures to the pages refer to the Introduction, where the Table is discussed, the 
 Arabic to the Table itself. 
 
 B. 6 
 
Tables for Statisticians and Biometriciam 
 
 TABLE 
 
 XIII. Tables for Testing Goodness of Fit: Auxiliary Table A, 
 
 to assist in determining P for high values of x> . 
 
 XIV— XVI. Tables for Testing Goodness of Fit : Auxiliary Tables 
 
 B — D. Numerical values needed in the calculation 
 
 and extension of Tables of P . 
 
 XVII. Tables for Testing Goodness of Fit: Value of (-log P) for 
 
 high values of ^ 2 when the frequency is in a fourfold 
 Grouping ......... 
 
 XVIII. Probability of Association on a Correlation Scale : Values 
 
 of (— log P) for an observed value of the Tetrachoric 
 Correlation r and the Standard Deviation of r for 
 zero Association ....... 
 
 XIX. Probability of Association on a Correlation Scale : Values 
 
 of x 2 corresponding to Values of (- log P) 
 
 XX. Probability of Association on a Correlation Scale : Values 
 
 of log^; 2 corresponding to Values of Tetrachoric r 
 and a r ......... 
 
 XXI. Probability of Association on a Correlation Scale : Abac to 
 
 determine <r r for a Population of given size, divided 
 into a fourfold Table with given marginal Frequencies 
 
 XXII. Probability of Association on a Correlation Scale: Abac to 
 
 determine the Equiprobable Tetrachoric Correla- 
 tion r,. from a Knowledge of log^ 3 and a cr r . 
 
 XXIII. Approximate Values of Probable Error of r from a four- 
 
 fold Correlation Table : Values of % r for values of r 
 
 XXIV. Approximate Values of Probable Error of r from a four- 
 
 fold Correlation Table : Values of ^ a for Values of 
 
 marginal |(1 + a) 
 
 XXV. Table to determine the Probability that the mean of a 
 very small sample n (4 to 10), drawn from a normal 
 Population will not exceed (in algebraic sense) the 
 Mean of the Population by more than z times the 
 Standard Deviation of the Sample .... 
 
 XXVI. Table to assist the Calculation of the Ordinates of the 
 
 Frequency Curve y = i/ e~ px,a (1 + xja) p 
 
 XXVII. Tables of Powers of Natural Numbers, 1 to 100 
 XXVIII. Tables of Sums of Powers of Natural Numbers 1 to 100 . 
 
 XXIX. Tables to facilitate the Determination of Tetrachoric Corre- 
 lation : Tables of the Tetrachoric Functions t, to t 6 
 for given marginal Frequencies .... 
 
 PAGE PAGE 
 
 xxxiii 29 
 
 xxxiii 30 
 
 xxxiv 31 
 
 xxxvi 31 
 
 xxxvi 32 
 
 xxxvi 32 
 
 xlii 33 
 
 xlii 34 
 
 xl 35 
 
 xl 35 
 
 xliii 36 
 
 xlv 37 
 
 xlvi 38-39 
 
 xlvi 40-41 
 
 1 42-51 
 
Contents 
 
 XI 
 
 TABLE PAOE PAGE 
 
 XXX. Tables to facilitate the Determination of Tetrachoric 
 Correlation : Supplementary Tables for determining 
 high Tetrachoric Correlations (r = "80 to TOO) for 
 given positions of the dichotomic lines . . liii 52-57 
 
 XXXI. Table of the Logarithms of the Gamma Function, 
 
 logF(p) from p-1 to p = 2 . . . . lv 58-61 
 
 XXXII. Table to deduce the Subtense from a knowledge of Arc 
 
 and Chord in the Case of the Common Catenary lvi 62-63 
 
 XXXIII. A, Extension of Table XXXII for very flat Catenaries; 
 
 B, Extension of Table XXXII for very narrow 
 
 Catenaries lvi 64* 
 
 XXXIV. Diagram to find the Correlation Coefficient r from the 
 
 Mean Positive Contingency on the Hypothesis of 
 
 a Normal Distribution ...... lvii 65 
 
 XXXV. Diagram to determine the Type of a Frequency Distri- 
 
 bution from a Knowledge of the Constants fa and 
 
 fa. Customary Values of fa and fa . . . lxiii 66 
 
 XXXVI. Diagram showing Distribution of Frequency Types for 
 
 High or Unusual Values of fa and fa, . . . lxiii 67 
 
 XXXVII. Probable Errors of Frequency Constants: Table for the 
 
 Probable Error of fa for given Values of fa and fa . lxii 68-69 
 
 XXXVIII. Probable Errors of Frequency Constants : Table for the 
 
 Probable Error of fa 2 for given values of fa and fa . lxii 70-71 
 XXXIX. Probable Errors of Frequency Constants : Values of the 
 Correlation of Deviations in fa and fa(Rp v ^) for 
 given values of fa and fa lxii 72-73 
 
 XL. Probable Errors of Frequency Constants : Probable Error 
 of the Distance between Mean and Mode for given 
 Values of fa and fa lxii 74-75 
 
 XLI. Probable Errors of Frequency Constants : Probable Error 
 
 of the Skewness for given Values of fa and fa . lxii 76-77 
 
 XLII. Values of the Frequency Constants fa, fa, fa and fa 
 for given Values of fa and fa on the Assumption that 
 the Frequency can be described by one or other of 
 Pearson's Frequency Types ..... lxii 78-79 
 XLI 1 1. Probable Errors of the Frequency Constants : Probable 
 Error of the Criterion of Type, /c 2 , for given Values 
 
 of fa and fa lxii 80-81 
 
 XLIV. Probable Error of the Determination of Frequency Type : 
 Value of Semi-minor Axis of Probability Ellipse for 
 
 given Values of fa and fa lxiv 82-83 
 
 6—2 
 
Xll 
 
 Tables for Statisticians and Biometricians 
 
 TABLE 
 
 XLV. Probable Error of the Determination of Frequency Type : 
 Value of Semi-major Axis of Probability Ellipse for 
 given Values of /3i and /3 2 
 
 XLVI. Probable Error of the Determination of Frequency Type : 
 Value of Angle between Major Axis of Probability 
 Ellipse and Axis of fi 2 for given Values of /9, and /3 2 
 
 XLVII. Probable Error of the Determination of Frequency Type : 
 Diagram to determine for given Values of & and /9 2 , 
 a given Frequency Distribution belongs definitely 
 to a given Type of Frequency .... 
 
 XLVIII. Probable Occurrences in Second Small Samples. Per- 
 centage Frequency of each number of Successes in 
 a Second Small Sample of m after p Successes in 
 a First Small Sample of n . 
 
 XLIX. Logarithm of Factorial |_n from n = 1 to 1000 . 
 
 L. Table of Fourth Moments of Subgroup Frequencies, i.e. 
 n x x*, for n = 1 to 400, x = 1 to 19, for Verification 
 of the Calculation of Raw Moments 
 
 LI. General Term of Poisson's Exponential Limit to the 
 Binomial, i.e. of the so-called " Law of Small 
 Numbers." Value of e~ m m x j\x for m = - l to 15'0, 
 and x = up to such figure as makes the Function 
 significant in the sixth decimal Place . 
 
 LII. Table of Poisson's Exponential Limit to the Binomial to 
 be used in the Determination of the Probable Errors 
 of Cell Frequencies n = 1 to 30. Percentage of 
 Occurrences in random Samples when the Population 
 proportionately sampled would give actually n to 
 the Cell 
 
 LIII. Angles, Arcs and Decimals of Degrees. Table giving 
 (a) the Arc for each Degree from 1° to 180°, (b) the 
 Arc for each Minute and Second of Angle and (c) 
 the Value in Decimals of a Degree of each Arc and 
 Second of Angle 
 
 LIV. Table of the G(r,v) Integrals. Values of log.F(r, v) 
 and H(r, v) for r = 1 to 50 and <£ = tan -1 v from 
 0° to 45° 
 
 LV. Miscellaneous Constants in Frequent Use by Statisticians 
 and Biometricians ....... 
 
 PAGE PAGE 
 
 lxiv 84-85 
 
 lxiv 86-87 
 
 lxv 
 
 88 
 
 lxx 89-97 
 
 lxxiii 98-101 
 
 lxxiv 102-112 
 
 lxxvi 113-121 
 
 lxxvii 122-124 
 
 lxxix 125 
 
 lxxxi 126-142 
 lxxxiii 143 
 
INTRODUCTION TO THE USE OF THE TABLES 
 
 For this introduction to the use of the Tables I have largely drawn on the 
 prefaces to the original papers in Biometrika, and record here my acknowledge- 
 ments to the authors of the same. 
 
 Interpolation. 
 
 (1) A word must first be said as to interpolation. Let a function u be tabled 
 for the argument x proceeding by differences Ax = h. Then the scheme of such 
 a table with the differences of u is : 
 
 #_3 
 
 «-» 
 
 #—3 
 
 2t_ 2 
 
 %-\ 
 
 U-, 
 
 Xq 
 
 «. 
 
 .r, 
 
 «1 
 
 X., 
 
 It., 
 
 *, 
 
 «1 
 
 x t 
 
 u 4 
 
 a - 5 
 
 M, 
 
 Am_ 3 
 
 Att_„ 
 
 A«_, 
 
 A«„ 
 
 A?/, 
 
 Aw, 
 
 Au., 
 
 Au 4 
 
 A*u_., 
 A 2 i/_ , 
 
 A 2 «, 
 A 2 w., 
 
 A 2 M, 
 
 A 3 »_, 
 
 A 3 «_. 2 
 
 A 3 «_, 
 
 A 3 «„ 
 
 A 3 «, 
 
 A 3 u„ 
 
 A 4 «_., 
 A 4 '/_., 
 A 4 «_, etc., etc. 
 
 A 4 M, 
 
 vhere : 
 
 A« g =»f s+1 - M„ 
 
 A 2 */, = A« s+1 - Am,, 
 
 A : 7/ s = AX+i - A 2 » s etc., etc. 
 Now there are three interpolation formulae which it is desirable to remember 
 If the function be required for the value x + dh and this value be termed u o (0) 
 then we have : ' 
 
 «.(*)-«. + ^-^ AX+ ^Li^^^ + (i)> 
 
 m^^^-Max-^5^., (ii) , 
 
 where <£= 1-0. This is Everett's formula*. And lastly : 
 
 «. (0) = «. + ^ | (A«„ + Aw.,) + t ^ _ ?Ojl^ | ( A 3 M _, + A 3 u_ 2 ) . . . (iii), 
 
 where we work with the differences on or adjacent to the horizontal through x a . 
 * Journal of the Institute of Actuaries, Vol. xxxv, p. 452. 
 
xiv Tables for Statisticians and Biometricians [Interpolation 
 
 It is very rarely indeed that we need go beyond second differences, often the 
 first will suffice. Not infrequently the inverse problem arises, namely we are 
 given u o (0) and have to determine from it. If we only go as far as second 
 differences, either (i) or (iii) gives us a quadratic to find and the root will 
 generally be obvious without ambiguity. Usually it suffices to find 
 
 and then determine from 
 
 0=( ( , O (0)- ? , O )/A"o+^ (1 2 7 0,) AV^'«. (iv); 
 
 or to find & = (h„(0) - «„)/£ (Ah„ + Am_,) 
 
 and then 
 
 6 - («. w - a.vi (a* + *o - f; HA ffXj (v) - 
 
 Very often good results are readily obtained by. applying Lagrange's inter- 
 polation formula which for three values of u reduces to 
 
 u o (0) = (l-0 1 )n v -^0(l-0)u^ l + ^0(l+0)u 1 (vi). 
 
 Or, we may use the mean of two such formulae and take 
 
 u o (0) = (l - 0)(1 ~ i0)r, o + \0 (o - 0) Ul -l0(l - 0)(u_ i + u 2 ) ... (vii). 
 The resulting quadratics are respectively: 
 
 0-- (J («, + m_,) - «„) + 1 \ («, - u_,) + U, - u, (5) = (vi) bis , 
 
 and s ^ («, - «, + '/_, + m 2 ) + \ (5;*! - 5w — it_, - M,) + u, - m (0) = . . .(vii) bis . 
 
 (2) There are some tables in this book which are of double entry, e.g. those 
 for the Tetrachoric Functions and for the G (r, v) Integrals. The simplest solid 
 interpolation formula, using second differences, is : 
 
 «*,!, = «So + a*A«M + Z/A'wo,o 
 
 + 4 {#(* - 1) AX,o + 2ay A A'w„,n + y(y-l) A%„) (viii>, 
 
 where A denotes a difference with regard to x, and A' with regard to y. But if 
 we consider u x<y to be the ordinate of a surface, and the figure, p. xv, to represent 
 the xy plane of such a surface, then it is clear that, if P be the point x, y, and 
 A, B, G, D, &c. the adjacent points at which the ordinates are known from the 
 table of double entry, only the points A, B, C, D, J, and N are used by the above 
 formula; and of these points, not equal weight is given to the fundamental points 
 A, B, G, D, for G only appears in a second difference. If another point of 
 the fundamental square other than A be taken as origin, we get a divergent, 
 occasionally a widely divergent result. If we use only four points — A, B, C, D — 
 to determine the value of the function at P, then we might take the ordinate 
 
I] 
 
 Introduction 
 
 xv 
 
 at P of the plane which (by the method of least squares) most nearly passes 
 through the four points of the surface vertically above A, B, G, D. We have then 
 
 u x,y = \ («o,o + «M + "0,1 + «i,i) + i («M - M o,o + «M - «o,i) (x ~ '5) 
 
 + i O'o.i - «'o,o + Mi, 1 - «i,o) (y-'o) (ix), 
 
 but by trial it has been found that this formula gives occasionally worse results 
 than that for first differences, using only three points. To find by the methods 
 of simple interpolation (with first or first and second differences) the points 
 a and b, and then interpolate P between them, generally gives a fairly good 
 result; but this result usually differs somewhat from that obtained by first simply 
 
 F 
 
 © 
 (-1, -1) 
 
 (-1,0) 
 E 
 
 © 
 
 O 
 
 © 
 (0, -i) 
 
 (0, 0) 
 A — x 
 ® 
 
 a 
 
 x .... 
 
 : t 
 
 i .'/ 
 
 : I 
 
 ,® p. 
 
 H 
 
 & 
 (1, -1) 
 
 0,0) 
 B 
 
 ,...® 
 
 I 
 
 © 
 (2, -1) 
 
 (2,0) 
 J 
 
 ® 
 
 x/ 
 
 © 
 
 S 
 
 (-1, 1) 
 
 © 
 
 D 
 
 (0, 1) 
 
 X © 
 
 b C 
 
 (1, 1) 
 
 © 
 K 
 
 I 
 
 © © © © 
 
 R N ML 
 
 (-1,2) (0,2) (1,2) (2,2) 
 
 interpolating e and / and then interpolating between e and /*. Various other 
 methods for interpolation in ^-dimensioned space will be found discussed by Palin 
 Elderton in Biometrikaf. The ideal method can hardly yet be said to be known, 
 and it may well vary from table to table and from one part of the same table 
 to auother. One or other of the above methods will, however, suffice in practice 
 for most statistical purposes. 
 
 I consider now the individual tables. 
 
 Table I (p. 1) 
 
 Table of Deviates of the Normal Curve for each Permille of Frequency. (Calcu- 
 lated by Sheppard and published by Galton in Biometrika, Vol. v. p. 405.) 
 
 If N be the total number in a population, zhx the frequency between x and 
 x + Bx, a- the standard-deviation, then the frequency curve of the population 
 assuming its distribution to be Gaussian or normal will be : 
 
 iV 
 
 V27TI 
 
 .(ix) 
 
 * B. A. Report, Dover 1899. Tables of G (r, »)-Integrals, Beport of the Committee (Drawn up 
 by K. Pearson). -f Vol. vi. p. 94. 
 
XVI 
 
 Tables for Statisticians ami Biometricians 
 
 [I 
 
 the origin being the mean. Table I. gives the value of x\a for each thousandth 
 of the area of this curve, — each ' permille ' — reckoned from left to right. 
 
 In entering the table we enter from the left-hand column and top row if the 
 permille be less than 500. For example, if the frequency below a particular value 
 were 387 per thousand, the corresponding deviate would be -0'287l, the number 
 placed at the intersection of the *38 row from left and 007 column from top. 
 The negative sign is always to be given when reading permilles below 500, 
 because the deviate will be in defect of the mean, supposing increasing variates 
 to be plotted as usual from left to right. 
 
 On the other hand if the permille be greater than 500 we enter the table from 
 the right-hand column and bottom row. For example, if the permille be 748, the 
 deviate is 4- 06682, the number placed at the intersection of the - 74 row from 
 right and '008 column from bottom of the table. The plus sign must be given, as 
 the deviation is in excess of the mean, if the convention as to plotting variables 
 has been observed. 
 
 Illustration: The following observations were made on the nature of the 
 degree taken by 1011 Cambridge undergraduates measured at the Anthropological 
 Society's Laboratory : 
 
 Poll 
 
 Third Class 
 
 487 
 189 
 
 Second Class 
 First Class 
 
 182 
 153 
 
 Find the deviates of these on a normal or Gaussian scale. 
 
 The sums from the lowest to each class top are 487, 676, 858 and 1011 
 respectively. If we term with Francis Galton the one man in a thousand of 
 surpassing intelligence or special ability a "genius," we have on multiplying by 
 •0009891197 the reciprocal of 1011, the series for entering Table I. Thus we 
 find: 
 
 Hence : (-481) 
 
 Deviates : 
 
 A 
 Ax -7 
 
 ■4817 
 
 •6680 
 
 •8487 
 
 and -9990 
 
 •0476 
 •0451 
 ■0025 
 •00175 
 - -0458 
 
 (•668) '4344 
 
 (•669) -4372 
 
 A -0028 
 
 A x -6 -00168 
 
 + •4361 
 
 (-848)1-0279 
 
 (-849)1-0322 
 
 A -0043 
 
 A x -7 -00301 
 
 + 1-0309 
 
 (-999) 3-0902 
 + 3-0902 
 
 Supposing with Pearson* that 100 units of intelligence ("mentaces") separate 
 the lowest man of the First Class from the highest man of the Poll, we have 
 + 1'0309 — (— '0458) = 100/ a, where tr is the standard deviation of intelligence. 
 Thus o-= 100/1"0767 = 92 - 88 mentaces. Hence we conclude that the range of 
 Third Class men is from - 4-25 (i.e. 9288 x (- -0458)) below to + 40-50 
 
 Biometrika, Vol. v. p. 109. 
 
I — III] Introduction xvii 
 
 (i.e. 92 - 88 x (+ -4361)) above the average undergraduate. The range of Second 
 Class men is from + 40"50 to +9575 mentaces above the average undergraduate, 
 and the range of First Class men all those with more than 95"75 mentaces above 
 the average. The "genius" corresponds to an excess of no less than 287 - 02 
 mentaces. If we suppose that one individual in 1000 is completely feeble- 
 minded or practically wanting in all intelligence, we should credit roughly the 
 average man with 300 mentaces, and we should then have our range of intel- 
 ligence on a Gaussian scale : 
 
 Poll : below 29G mentaces ; 
 
 Third Class : above 296 and below 340 mentaces ; 
 
 Second Class : above 340 and below 396 mentaces ; 
 
 First Class : above 396 mentaces ; 
 
 " Genius " : above 587 mentaces. 
 
 In rough numbers: Poll, below 300; Third Class, 300 to 350; Second Class, 
 350 to 400 ; First Class, over 400 ; " Genius," over 600. 
 
 Of course there is much that is hypothetical here, but the numbers give us 
 some appreciation of the distribution of ability, and they serve to illustrate the 
 construction of a Gaussian or normal scale. When more than three or four 
 significant figures are needed Tables II and III must be used. 
 
 Tables II and III (pp. 2—10) 
 
 Tables of the Probability Integral : Area and Ordinate of the Normal Curve in 
 terms of the Abscissa; and Abscissa and Ordinate in Terms of Difference of Areas. 
 (Calculated by Dr W. F. Sheppard, and published in Biometrika, Vol. n. pp. 
 174—190.) 
 
 " Sheppard's Tables " were the first to express the Gaussian * or normal 
 probability integral in terms of tlie standard deviation ; they are so familiar to 
 statisticians that it would almost seem a work of supererogation to explain their 
 ' use,' which is further too manifold for full description. We can only give a few 
 sample illustrations. 
 
 It is most important when using these tables to pay attention to the signs of 
 the differences recorded at the tops of the columns. 
 
 Illustration (i). The mean length of cubit in 1063 adult English males is 
 recorded as 18"-31 ± "019 and of their 1063 adult sons as 18"52 + -021. Determine 
 the odds against these two measurements being really identical, i.e. random 
 samples from the same population. We assume that the deviation of means and 
 their differences follow the normal law. The difference is 0""21 and the probable 
 error of this difference = V(-019> + ('021) s = 0"0283. Since the probable error 
 
 * The term is usual, but inaccurate. Laplace had reached the probability integral and suggested 
 its tabulation several years before Gauss. 
 
 B. e 
 
xviii Tables for Statisticians and Biometricians [II — III 
 
 = '67449 x standard deviation, we have the standard deviation of the difference 
 = 0"'04196. Hence the deviation in terms of the standard deviation 
 
 = 0-21/(0-04196) = 5-0048. 
 
 Table II, p. 8, gives the area |(1 + a) of the normal curve up to the abscissa x/a. 
 Noting the remark at the foot of the table, we have 
 
 */<t=5-00, ^(1 -fa) = -999,999,7133, 
 xja = 5-01 , \{1 + a) = "999,999,7278, 
 
 A = 145, 
 
 A x 48 70, 
 
 x/a = 5-0048, | (1 + a) = -999,999,7203. 
 Hence } (1 - «)- -000,000,2797. 
 
 Accordingly if we suppose the deviation as likely to be in defect as in 
 excess, the probability that we shall reach the observed deviation, or exceed it, is 
 2x|(l-«), and that we shall not is | (1 + a) — £ (1 — a), or the odds against the 
 result on a pure random sampling chance are -999,999,4406 to -000,000,5594, or 
 1,787,629 to 1, i.e. overwhelming odds. Thus we may reasonably argue that sons 
 in the professional classes in 1900 were substantially differentiated from their 
 fathers by a longer forearm of about £". 
 
 Illustration (ii). Find the value in mentaces of the mean intelligence of Poll- 
 men, First, Second and Third Class men as given by the numbers in the Illustration 
 to Table I. 
 
 The equation to the normal or Gaussian curve being 
 
 N 
 
 z = 
 
 ,-iW') 1 
 
 V2tt<t 
 
 we easily find that if there be * tabled ' ordinates z 1 and z 2 * at the abscissae 
 #! and x 2 , which cut off an area n 12 , then the mean x 12 of this area is given bj' 
 
 x 12 = a-.(z 1 -z 2 )l(n 12 IN) (x). 
 
 It will be sufficient to take the values of the abscissae already found, i.e. 
 ^/o- = - -0458, x 2 /<r = + -4361, 
 x 3 /<r = + 1 -0309, xja = + 3"0902. 
 We require the z's for these. For example : 
 
 x = -04, z= -398,6233 
 
 •05, z = -398,4439 
 
 = -58, A, -1793 
 
 A 2 - 397. 
 
 * The symbol z here used is that of the Tables, i.e. —. — c -i (*/")'. 
 
II— III] 
 
 Introduction 
 
 xix 
 
 Therefore by formula (i) p. xiii : 
 
 *, = -398,6233 --58 [1793] + -^! — [397] , 
 
 = -398,6233] 
 
 - 1040 > = -398,5241. 
 + 48j 
 
 Or, we might proceed as follows: for the Poll-men £(1 - a) - -4817, hence 
 a = -0366. But from Table III, p. 9, which gives z for a : 
 
 « = -03, z = -398,6603 
 
 a = 04, z = -398,4408 
 
 = -66, A,= -2194 
 
 A,= -627. 
 Hence by formula (i) : 
 
 ■RR y -«J4 
 
 *, = -398,6603 - -66 [2194] + — ~- [627] 
 
 - -398,6603' 
 
 - 1448 [• =-398,5225. 
 
 + 70 
 
 We conclude therefore that z would be correct to five figures with second differ- 
 ences, and that for four figures, first differences from either Table II or Table III 
 will suffice. 
 
 If we use formula (ii) p. xiii — Everitt's formula — we find from Table II: 
 4 = 398,6233 - -58 [1793] + *§ *«" [397] + ^ * fl 8236 [398] 
 
 6 
 
 = -398,6233 
 
 -1040 
 
 + 13 
 
 + 5 
 
 and from Table III : 
 
 •398,5211, 
 
 *, = -398,6603 - -66 [2194] + I^IL**** [62 7] + 34 * ~ 8844 [627] 
 
 = 398,6603 
 -1448 
 
 + 
 + 
 
 37 
 31 ) 
 
 •398,5223. 
 
 Working with formula (iii), Table II gives us z 1 m 398,5242 and Table III 
 z x = -398,5225 with second differences. We shall not therefore without higher 
 differences get from any of our formulae closer than -398,522 with a possible error 
 
 c2 
 
xx Tables for Statisticians and Biometricians [II — III 
 
 of 1 or 2 in the last place. This is, of course, amply sufficient for statistical 
 purposes, where four figures as a rule would be sufficient. 
 
 Using formula (i) p. xiii we obtain : 
 
 z 1 = -39852, z 2 = -23450, 
 
 e, = -36275, e t - -00337. 
 Whence : 
 
 »»i = + ~t -,v~ o- = - '8273o- = - 76-84 mentaces, 
 "4817 
 
 •39852 - -36275 _„. 1I700 
 
 x lt = H — ^ <r= + '192oo- = + 17'88 mentaces, 
 
 ■18o9 
 
 •36275 - -23450 „,_, " . 
 
 CCJ3 = H — — o-= + -7121a- = + 66-14 mentaces, 
 
 '1801 
 
 •23450 - -00337 , „„_ „„. 
 
 x u = H o-= + l - o378o- = + 142-83 mentaces, 
 
 "1503 
 
 •00337 — 
 x 4x = + ™*T' o- = + 3-3700O- - + 31301 mentaces. 
 
 Assuming as before the average man to have 300 mentaces of intelligence 
 we find : 
 
 Average Poll-man has 223 mentaces. 
 Average Third Class man has 318 mentaces. 
 Average Second Class man has 366 mentaces. 
 Average First Class man has 443 mentaces. 
 Average man of "genius" has 613 mentaces. 
 
 Thus the average First Class Honours man is twice as able as the average Poll- 
 man, and the average "genius" has not quite twice the ability of the average 
 Third Class Honours man. 
 
 Illustration (iii). It is required to determine normal curve frequencies corre- 
 sponding to the following frequencies of the cephalic index in Bavarian skulls. 
 
 Here the mean and standard deviation found by moments in the usual way are 
 
 wi = 83-069, o- = 3-432. 
 
 The deviations from the mean were next expressed in terms of the standard 
 deviation, i.e. these deviations are 
 
 -13569, -12569, ... -0-569, + 431, +1-431, +2431, ... + 14431, 
 
 and they are multiplied on a calculator by the reciprocal of the standard deviation, 
 whence the column xja is found. Table II gives us £(l+a) knowing xj a ; this 
 has been calculated by first differences only. We shall consider as an illustration 
 to Table XII, whether the normal distribution thus reached is to be considered 
 a good fit to the observations. 
 
IV] 
 
 Introduction 
 
 xxi 
 
 Index 
 
 Observed 
 
 xja 
 
 *(! + «) 
 
 Calculated 
 Frequency 
 
 C9-5— 70S 
 
 1 
 
 -3-9539 
 
 •99996 
 
 Under 70'5 -1 
 
 70-5— 71 -5 
 
 1 
 
 -3-6625 
 
 •99988 
 
 •2 
 
 71-5— 72-5 
 
 — 
 
 -3-3711 
 
 •99963 
 
 •6 
 
 72-5—73-5 
 
 2-5 
 
 -3-0797 
 
 •99896 
 
 1-5 
 
 73-5— 74S 
 
 1-5 
 
 -2-7883 
 
 •99735 
 
 3 3 
 
 74-5—75-5 
 
 3-5 
 
 -2-4969 
 
 •99374 
 
 6-7 
 
 75-5— 76-5 
 
 125 
 
 -2-2055 
 
 •98629 
 
 12-7 
 
 76-5—77-5 
 
 17 
 
 -1-9141 
 
 •97219 
 
 22-1 
 
 77-5— 78 -5 
 
 37 
 
 - 1 -6228 
 
 •94768 
 
 35-3 
 
 78-5—79-5 
 
 55 
 
 -1-3314 
 
 •90846 
 
 51-9 
 
 79-5—80-5 
 
 71-5 
 
 - 1 -0400 
 
 •85082 
 
 70-1 
 
 80-5—81-5 
 
 82 
 
 - -7486 
 
 •77294 
 
 87-0 
 
 8V5—82-5 
 
 116 
 
 - -4572 
 
 ■67623 
 
 99-4 
 
 82-5— 8S-5 
 
 98 
 
 - -1658 
 
 •56584 
 
 104-2 
 
 83-5—84-5 
 
 107 
 
 •1256 
 
 •54997 
 
 100-5 
 
 84-5—85-5 
 
 82 
 
 •4170 
 
 •66165 
 
 89-1 
 
 85-5—86-5 
 
 74 
 
 •7084 
 
 •76064 
 
 72-6 
 
 86-5—87-5 
 
 58 
 
 •9998 
 
 •84129 
 
 54-3 
 
 87-5—88-5 
 
 34 5 
 
 1-2912 
 
 •90167 
 
 37 4 
 
 88-5—89-5 
 
 19 
 
 1 -5825 
 
 •94324 
 
 23-7 
 
 89-5—90-5 
 
 10 
 
 1 -8739 
 
 •96953 
 
 13-8 
 
 90-5—91-5 
 
 8 
 
 2-1653 
 
 •98482 
 
 7-4 
 
 91-5—92-5 
 
 3 
 
 2-4567 
 
 •99299 
 
 3 6 
 
 92-5—93-5 
 
 1-5 
 
 2-7481 
 
 •99700 
 
 1-6 
 
 93-5—94-5 
 
 2 
 
 3-0395 
 
 •99882 
 
 •7 
 
 94-5—95-5 
 
 1-5 
 
 3-3309 
 
 •99957 
 
 •3 
 
 95-5—96-5 
 
 — 
 
 3-6223 
 
 •99985 
 
 Over 95-5 -1 
 
 96-5—97-5 
 
 — 
 
 3-9137 
 
 •99995 
 
 — 
 
 97-5—98-5 
 
 1 
 
 4-2050 
 
 •99999 
 
 — 
 
 Totals ... 
 
 900 
 
 — 
 
 
 900-2 
 
 Table IV (p. 11) 
 
 Extension of the Table of the Probability Integral F=$(l —a). (Calculated 
 by Julia Bell, M.A., Drapers' Research Memoirs, Biometric Series, vm, p. 27.) 
 
 It has been found needful occasionally to determine probabilities for deviations 
 exceeding considerably the limit #/<7 = 6 of Sheppard's Table II. 
 
 Illustration. If x/tr = 3431, determine to two significant figures the probability 
 of a deviation occurring as large or larger than this. 
 
 The table gives us : 
 
 
 
 
 33 
 
 238-39135 
 
 14-56180 
 
 
 34 
 
 252-95315 
 
 1499573 
 
 •43393 
 
 35 
 
 267-94888 
 
 15-42967 
 
 •43394 
 
 36 
 
 283-37855 
 
 
 
xxii Tables for Statisticians and Biometricians [V — VI 
 
 Hence using formula (ii) p. xiii : 
 (- log F) = 25295315 + -31 [14-99573] 
 
 •31 x -9039 ,„„„„ -69 x -5239 AnnnA 
 
 x -43394 
 
 o 
 
 
 6 
 
 
 = 252-95315} 
 
 
 + 4-64868 
 
 ► = 257-55542. 
 
 
 - -04641 J 
 
 
 
 Hence log F = - 257-55542 = 
 
 = 258-44458, 
 
 J F , = 2-7834/10 258 , 
 
 
 which measures the improbability required. 
 
 
 Table V (pp. 12—18) and Table VI (p. 18) 
 
 Probable Errors of Means, Standard Deviations and Coefficients of Variation. 
 (Table V calculated by Winifred Gibson, B.Sc; Table VI by Dr Raymond Pearl 
 and T. Blakeman, M.A. Biometrika, Vol. IV. pp. 385 — 393.) 
 
 If m be a mean, o- a standard deviation and V = lOOo-fin a coefficient of 
 variation, for a population of n, we have 
 
 Probable Error of Mean 
 
 = -G744898fl-/VH = XiO- ( xi ). 
 
 Probable Error of Standard Deviation 
 
 = -6744898o-/V2n = xsff ( xii ). 
 
 Probable Error of the Coefficient of Variation 
 
 = -6744898 F x jl + 2 (-^Yj /V2n (xiii), 
 
 = -6744898/^271 x -f 
 
 -J&x* ( xiv )- 
 
 Table V gives % and ^ 2 for each value of n up to 1000, Table VI gives ■x/r for 
 each value of V proceeding by units from to 50. 
 
 When the frequency n is greater than 1000, the tables may still be used by 
 taking out a square factor, which can be divided out at sight. 
 
 Illustration (i). n = 2834 = 4 x 708-5. 
 
 n = 708, xi = -02535 ; n = 709, *, = -02533. 
 .-. n = 708-5, x ,= 02534, and .'. for n = 2834, 
 we have Xi = "01267. 
 
 Illustration (ii). In the case of the 900 Bavarian crania of the Illustration (iii) 
 
 to Table II the values 
 
 mi = 83-069, a = 3-432, 
 
V — VIII] Introduction 
 
 xxm 
 
 and therefore F= 41315 were found. It is required to find the probable errors 
 of these values. 
 
 For 900, ^1= 02248 and % 2 = '01590, hence the probable errors of m and <r are 
 
 p.e. of ?» = ^ 1 o- = - 077, 
 
 p.e. of a = fta = 055. 
 Next for V= 41315, 
 
 ^ = 4-00639 + -1315 [1-00609] - A. (-1315) (-8685) x [299] 
 = 413852. 
 
 p.e. of V = X2 x ^ = -01590 x 4-13852 
 = -0658. 
 Hence our results should be recorded as 
 
 m = 83-069 + -077, 
 a = 3432 + -055, 
 7=4-1315 + -0658. 
 
 Table VII (p. 19) 
 
 Abac for Probable Errors of r. (Calculated by Dr David Heron, drawn by 
 H Gertrude Jones, Biometrika, Vol. vn. p. 411.) 
 
 The probable error of a coefficient of correlation 
 
 = - 67 ^ 9 (l -,->). 
 \n 
 
 To ascertain the value of this function approximately, turn the page horizontal, 
 
 enter with the proper frequency on the scale at base, follow the corresponding 
 
 vertical until the sloping line with the given correlation is reached, then move 
 
 along the horizontal to the left until the scale of probable error is reached, which 
 
 will give the required approximate probable error of the correlation in a population 
 
 of the given size. 
 
 Illustration. r = '67l and n = 415. 
 
 Probable error of r = -018. The actual value is -0182. 
 
 Table VIII (pp. 20—21) 
 Values of I- r". (Calculated by H. E. Soper, M.A.) 
 Illustration. As in the last example let 
 
 r = -671 and n = 415. 
 Then probable error of r = ^, x (1 — r 3 ) 
 
 = -549,759 (from Table VIII) 
 x -03311 (from Table V) 
 = 0182. 
 
 The value of 1 - r 2 for r to four instead of to three figures can be obtained by 
 interpolation. 
 
XXIV 
 
 Tables for Statisticians and Biometricians 
 
 [IX 
 
 Table IX (pp. 22—23) 
 
 Values of the Incomplete Normal Moment Functions. (Calculated by Dr Alice 
 Lee, Biometrika, Vol. vi. p. 59.) 
 
 The nth incomplete normal moment function is defined to be 
 
 fin («) 
 
 \/2tt 
 
 r x n e 
 
 Jo 
 
 ■Jx2 
 
 dx 
 
 .(XV). 
 
 We take 
 
 m n {x) = ii n {x)j{{n— l)(ft — 3) (» - 5) ... lj if n be even) 
 = fi n (*)/{(« - 1) (h - 3) (« - 5) ... 2) if n be odd I 
 and m n (x) is the function tabled. 
 
 In multiple correlation (supposed normal), the frequency surface is 
 
 -4x 2 \ 
 
 z = 
 
 {2Tr)i' l <r 1 (T.,...a- n \ r R 
 
 e 
 
 where 
 and 
 
 X s = ^ {SiRgpXjl**) + 28' (RpgXpXg/apag)} j 
 
 1, 
 
 J *12 > **13 • • • ?'l 
 1 , ?'23 ... 7"o 
 
 ' 111 > ' M 
 
 I'n-i > M 
 
 ..(xvii), 
 .(xviii), 
 
 while jRpp and R pq are the usual minors. 
 
 ^ 2 = constant is the "ellipsoid" of equal frequency in n-dimensional space. 
 The total frequency, i.e. the volume of the surface, inside any ellipsoid ^ is 
 
 I x =( X zdV 
 
 Jo 
 
 and 
 
 r ,,, V27rit„_i (y) c i 
 
 IJN = cT-rjr i o\ lf n be even 
 Xl 2. 4.6 ... (n — 2) 
 
 W.(x) 
 
 .(xix). 
 
 if « be odd 
 
 ] .3. 5.. .(ft -2) 
 
 Thus a knowledge of the incomplete normal moment functions enables us 
 to predict for multiple variables whether an outlying observation consisting of 
 a system of n variate values is or is not reasonably probable. 
 
 If I X JN = \, we obtain the 'ellipsoidal' contour y,o within which half the 
 frequency lies. This x<> 1% fcne " generalised probable error " of Pearson and Lee. 
 
IX— X] 
 
 Introduction 
 
 xxv 
 
 Values of the "generalised probable error" coefficients are given in Table X 
 for n = l to 11, and by means of a smooth curve the results may probably be 
 extended to ?! = 1 5. The values found for this extension are : 
 
 
 « = 12 
 
 n = 13 
 
 « = 14 
 
 n = 15 
 
 Xo 
 
 3-367 
 
 3-513 
 
 3-654 
 
 3-791 
 
 Illustration (i). Let us consider long bone data for Frenchmen. 1 = F= femur, 
 2 = H = humerus, 3 = T= tibia, 4 = R = radius*, then by formula (xviii) p. xxiv : 
 
 R = 
 
 Further in cms : 
 
 1, -8421, -8058, -7439 
 
 •8421, 1, -8601, -8451 
 
 •8058, -8601, 1, -7804 
 
 ■7439, -8451, -7804, 1 
 
 to, = 45-23, o-, = 2-372, 
 to 2 = 33 01, a 2 = 1-538, 
 m, -36-81, 0-3 = 1-799, 
 to 4 = 24-39, o- 4 =l-170. 
 What is the chance that the following individual may be considered French ? 
 
 J" = 36-97, #' = 26-82, T'= 3056, i?' = 20-68. 
 The deviations in terms of their standard deviations are : 
 
 x x = (F' - to,)/o-, = - 3-482, x, = (//' - ?« 2 )/o- 2 = - 4-059, 
 x 3 = (T - mj/tr, = - 3-474, g> 4 = (R' - to 4 )/o- 4 = - 3-171. 
 
 Further : 
 
 R a 
 
 s =3-7810, -g? = 6-5496, 
 
 R 
 
 R. 2l 
 R 
 
 I! 
 
 R 
 ft 
 
 4-3406, 
 
 ^ = 3-6508, 
 
 ^23 
 
 = 20231, ^- 3 = 11404, ^ = 02130, ^=21946, 
 
 = 2-3175, 
 
 -74 
 R 
 
 = 0-6842. 
 
 Whence 
 
 X 2 = 16741,035 and x = 40916, 
 
 n is even, hence: I JN = ^iM 4 ^! 6 ) = ^2 
 
 ■n-.m,, 
 
 * For particulars of these length measurements the reader must consult R. S. Proc. Vol. 61, 
 pp. 343 et seq. and Phil. Trans. Vol. 192, A, p. 180. 
 
 B. d 
 
XXVI 
 
 Tables for Statisticians and Biometricians [IX — X 
 
 and from the Table, p. 22, we have by formula (i), p. xiii : 
 
 m, (4-0916) = -397,7378 + -916[3650] - \ (-916) (-084) [1043] 
 = 398,0682. 
 Hence I X I N = ^tt x '398,0682 = -9978. 
 
 Thus the odds are 9978 to 22, say 454 to 1 against a deviation-complex as 
 great as or greater than this occurring in a French male skeleton, i.e. the bones very 
 improbably were those of a Frenchman. Actually they were those of a male of 
 the Aino race. 
 
 Illustration (ii). The following are the ordinates of a frequency distribution 
 for the speed of American trotting horses*. It is assumed that they form a 
 truncated normal curve, and we require to determine (i) the mean of the whole 
 population, (ii) its standard deviation, and (iii) what fraction the ' tail ' is of the 
 whole population. 
 
 The values of frequency in an arbitrary scale are : 
 
 Seconds 
 
 Frequency 
 
 Seconds 
 
 Frequency 
 
 29—28 
 
 92-8 
 
 W—19 
 
 45-8 
 
 28—27 
 
 100-4 
 
 19—18 
 
 38-4 
 
 27—26 
 
 95-0 
 
 18—17 
 
 27-8 
 
 26—25 
 
 71-2 
 
 17—16 
 
 19-8 
 
 25—24 
 
 67-6 
 
 16—15 
 
 10-7 
 
 24—23 
 
 613 
 
 15—14 
 
 15-8 
 
 28—22 
 
 61-4 
 
 14— 13 
 
 7-9 
 
 22—21 
 
 44-8 
 
 13—12 
 
 5-0 
 
 21—20 
 
 44-5 
 
 12—11 
 
 2-1 
 
 
 
 11—10 
 
 5-6 
 
 Taking the working origin at 20 — 19 seconds, we find 
 
 sV = - 3-9214, ^' = 32-545,666 
 
 for raw moment coefficients. Hence, if d be the distance from 29 seconds, i.e. the 
 stump of the tail from the mean, and 2 the standard deviation of the tail about its 
 
 mean : 
 
 d = 95 - 3-9214 = 5-5786 sees., 
 
 2* = „ 2 ' _ „/* = 17-168,288, 
 
 and accordingly 2 2 /e£ l = "5517. 
 
 If this value be compared with those for fa in Table XT, p. 25, it will be seen 
 that we have got slightly more than the half of a normal curve, i.e. not a true 
 tail. We cannot therefore use Table XI, but must fall back on Table IX. 
 
 * Galton, R. S. Proc. Vol. 62, p. 310. See for another method of fitting, Pearson, Biometrika, 
 Vol. n. p. 3. 
 
IX — X] Introduction xxvii 
 
 Let x be the distance from stump to centre of curve, n equal the area of 
 truncated portion, and N be whole population. Then 
 
 njN= f + f -j== e'^dx' = \ + m {xj<r) (xx); 
 
 Jo Jo v27r 
 
 (Jo J -xi**j2ir J 
 
 (\/2tt Jo v2tt J 
 
 = N, 
 
 a 
 
 ==- m,(a;/o-)| (xxi); 
 
 nu' = N*-\\ +| =6-**"^ 
 
 roo rO x 'i 
 
 1 + 1^ 
 
 I Jo J -i!a N'ltr 
 
 = TVcr 2 {£ + TO, (a'/o-)} (xxii). 
 
 Now d = x + x, and S 2 = /i/ — a?. 
 
 1 a; 
 
 j ~7S=* ~ ^ W°") + - (i + w (x/o-)} 
 „ c* v2tt <^ , .... 
 
 Hence — = ■ , , . (xxm), 
 
 a i + m (x a) 
 
 .(xxiv), 
 
 {J + m, (a/or)) {4 + to (a?/o-)} - j-^== - m, (ar/<r) 
 ^ = l4 + 7»,(«/a)}» 
 
 £2 ( i + m2 )(« +mo )_(^=- Wl )' 
 
 and d? = ~T~i F~ (xxv) ' 
 
 say, for brevity. 
 
 Here m x and m? are given by Table IX and \ + m„ is the £ + i<* of Table II. 
 
 Formula (xxv) has not yet been tabled for different values of x', as it occurs 
 much more rarely than the corresponding function for a true tail. 
 
 If we take three values x' = 0, Ol and 0"2, we have, from Tables II and IX, 
 x'= 0, 4 + w„ = -500,0000, \ + to 2 = -500,0000, — L -to 1 = -398,9423, 
 
 V27T 
 
 x'=-l, „ =-539,8278, „ =-500,1325, „ =-396,9526, 
 
 x'=% „ =-579,2597, „ =-501,0512, „ =-391,0427. 
 
 Whence from formula (xxv) for the three values of x 
 
 2 2 /d 2 =-5708, -5528 and -534.5. 
 
 di 
 
xxviii Tables for Statisticians and Biometricians [DL — XI 
 
 But our value of S a /^ 2 is '5517. Thus we find by interpolation 
 
 «'=-10G0. 
 It remains to determine m , «ij and m 2 for this value of x, or simpler £ + m , 
 
 \ + m 2 and — m, from the above values for x = "1 and x' = "2. We find 
 
 V27T 
 
 \ + m, = -542,194, £ + m, = -500,184, --L - m, = -396,598. 
 
 V27T 
 
 Whence d/a = |||^ + '1060 = '8375. 
 
 Thus <r = 5-5786/(-8375) = 6-6610 sees., 
 
 x = x x a = '7061 sees., 
 njN = -5422. 
 
 This gives a mean of 28'29 sees, with a variability measured by 666 sees. 
 Those actually registered as trotters are 54 °/ of the population. 
 
 Table X (p. 24) 
 See under Table IX, p. xxv. 
 
 Table XI (p. 25) 
 
 Constants of Normal Curve from Moments of Tail about Stump. (Pearson and 
 Lee, Biometrika, Vol. vi. pp. 65 and 68.) 
 
 This Table may be of service in cases of the following kind : 
 
 (a) In some cases a record is actually truncated as in the case of the 
 American Trotters dealt with on p. xxvi. Or, again, we may take a record of 
 stature obtained by measuring all men who exceed 69", or a record of mental 
 capacity found by measuring all persons with low intelligence in a community. 
 
 (b) When we recognise heterogeneity, e.g. when we have a mixture of male 
 and female bones, or two strains of trypanosomes, we can occasionally get a rough 
 approximate analysis by supposing the tails to represent homogeneous material 
 and then fitting them with normal curves. From the tails we get two components, 
 and if their compound agrees fairly well with the observed total we have 
 performed an analysis far more rapidly than by using the nonic equation*. 
 Difficulties arise owing to deviations from Gaussian frequency being not infrequent; 
 different dichotomic lines may give different results, and owing to paucity of 
 material in the 'tails' and corresponding irregularity there will be large probable 
 errors. 
 
 Cases under (a) and (b) will be treated by exactly the same process, but our 
 Table supposes that the distribution considered is less than half the normal distribu- 
 tion. The rules for determining the mean, standard deviation and total frequency 
 of the untruncated population are given on p. 25 under Table XI itself. 
 
 * Pearson, Phil. Trans. Vol. 185, A, p. 84. 
 
XI] 
 
 Introduction 
 
 xxix 
 
 Illustration (i). The following distribution represents the ' tail ' of a group of 
 .301 mentally defective children measured by G. Jaederholm using Binnet-Simon 
 test methods* : 
 
 Mental Defect in Years. 
 
 
 
 * 
 
 $5 
 
 $ 
 
 
 $ 
 
 03 
 
 * 
 
 
 
 SO 
 
 •* 
 
 7 
 
 >c 
 
 >-0 
 
 SO 
 
 so 
 
 i~ 
 
 
 
 4 
 
 1 
 I 
 
 i 
 * 
 
 1 
 
 i 
 3 
 
 1 
 -=t" 
 
 1 
 
 i 
 
 1 
 
 i 
 
 1 
 
 ,1 
 
 03 
 
 
 
 
 S3 
 1 
 
 1 
 
 t 
 
 1 
 
 1 
 
 1 
 
 to 
 
 1 
 
 to 
 
 1 
 
 
 Frequency ... 
 
 34 
 
 18 
 
 13 
 
 3 
 
 4 
 
 4 
 
 1 
 
 1 
 
 78 
 
 We find, with origin at — 3"45, 
 
 d = 1-8077 in } years and 2 2 = 27258. 
 
 Hence ^ 1 = 2 a /d a = -834. 
 
 The Table (p. 25) gives us /t' = 2186 and ^- 2 = 2*833. Hence the mean is at 
 distance from — 3'45 on left= 2 - 186 x <r, and for the standard deviation 
 
 <r = yjr 2 x d = 2833 x 1-8077 = 5-1212 £ years = 2-5606 years, 
 
 whence mean is at distance h = 5'597 years from — 3'45. 
 
 Now h' = 2186 corresponds by Table IT to £ (1 + a) = -98559. 
 
 .-. njN= -01441, or N= 78/(-01441) = 5413. 
 
 Thus the distribution is the tail of a population of 5413 individuals with 
 a mean at 2*15 years of mental excess, and a standard deviation of 256 years. 
 
 The example is merely illustrative and of no importance in itself. 
 
 Illustration (ii). Assuming that the correlation data follow the normal law, 
 determine indirectly the 'plural' partial correlation-)- of habits of mother and 
 health of baby for the limited universe of unclean homes. 
 
 The correlations between the various characters found by tetrachoric tables are 
 Habits of mother and health of baby: r 12 = -3060, 
 
 Habits of mother and cleanliness of home : r 13 = -7958, 
 Health of baby and cleanliness of home : r, a = '2578. 
 There are 947 out of 2931 homes which are not clean. Data from Bradford. 
 Here n/N= 947/2931 = -3231 = ^ (1 -a); 
 
 hence a = 3538, and by Table III 
 
 x ' = x j a = -459,049. 
 
 * Mendelism and the Problem of Mental Defect. II. On the Continuity of Mental Defect. Dulau&Co., 
 37, Soho Square, W. 
 
 + Biometrika, Vol. ix. p. 289. 
 
xxx Tables for Statisticians and Biometricians [XI 
 
 Now on the assumption of a normal distribution : 
 
 Jx v2tt<t 
 
 m JV<7 S | <**«" ***•<&/ 
 
 J x' 
 
 = iW {/x s (so )-/x s (*')}• 
 Here Table IX (p. 22) shows that if s be odd, m, (oo ) = '398,9423, i.e. 1/V2T, 
 and (p. 23) if s be even, m, (00 ) = -500,0000. 
 
 Hence in obtaining the moment coefficients of the tail, about the mean of the 
 whole population, m s (x') should be subtracted from '398,9423 or from '500,0000 
 before the results are multiplied by (s — 1) (s — 3) ... 2 or (s — 1) (s — 3) ... 1, when 
 s is odd or even respectively. It is convenient to term fi s ( 00 ) — ft, (x') the com- 
 plementary incomplete moment function of order s*. 
 
 For s = 1 and s = 2, we have 
 
 W/ij' = criV {?»! (00 ) — mj («')}, 
 
 «/f/ = o- 2 iV [m 2 (00 ) — ma (#')], 
 
 for in this case the multiplying factors to proceed from m s {x') to /x,(* - ') are both 
 unity. 
 
 Now cc = x/a can be found when n is known from Tables II or III. Hence we 
 have for the distance of centroid of tail from its stump, and for the square of its 
 standard-deviation about its centroid : 
 
 "IT, 
 
 d = /*/ — A'o" = 
 
 a 
 
 {OTj (00 ) — Ml] (x)) — ti 
 
 .(xxvi). 
 
 N 
 2* = <r> — {m 2 (00 ) - m, (a;')} - ft? 
 
 Of course m, (oo ) — m l (x') is the z of Sheppard's Tables II and III. 
 
 Returning to our numerical example, we have from Table IX (p. 22) : 
 
 m, (-45905) = '030,6721 + '5905 [162049] - \ (-5905) (-4095) [26358] 
 
 = -039,9222, 
 
 m, (00 ) - m, (-45905) = -359,02. 
 
 Found directly from Sheppard's Tables, it equals '35905. 
 
 Similarly from Table IX (p. 23) : 
 
 m, (-45905) = '008,1136 + 5905 [73,162] - 1 ('5905) ('4095) [30661] 
 
 = 012,0630, 
 
 and m 2 (00 ) - m, ('45905) = '487,9370. 
 
 * It is the function used by Dr Alice Lee and myself, Biometrika, Vol. vi. p. 65 to form Table XI, 
 p. 25, but by an oversight not adequately distinguished in symbol from /», (x') of p. GO of the same memoir. 
 
XI — XII] Introduction xxxi 
 
 We thus reach d = <r (-35902/-3231 - -45905) 
 
 = -65212a-, 
 
 and S 2 = o- 2 {-48794/-3231 - (1-11117) 2 ) 
 
 = -274,786c- 2 . 
 
 Or the distance of centroid from stump, and the standard deviation of the tail are 
 respectively 
 
 d = -65212a- and 2 = -52426a-. 
 
 We now find X = Vl -Wfa* = -85155. 
 
 The formula for the 'plural' correlation coefficient is* 
 
 ,r M - . r »- r »' r » ( XX vii), 
 
 '" Vl-r 13 ' 2 Vl-r B ' 2 V h 
 
 where r )3 ' = Ar I3 , ?v/ = Xr m 
 
 = •2195, =-6777. 
 
 Thus . 3 r 12 = -2191. 
 
 Actually taking out the universe of not clean homes, the correlation of habits of 
 mother and health of child is -1615. The difference is considerable, but 3 r 12 is 
 deduced from the entire population of 2931 homes, while -1615 depends on only a 
 third of this number. The 'singular' partial correlation is 3 r 12 =-1723, i.e. the 
 average relation between habits of mother and health of child for each individual 
 grade of cleanliness of home. 
 
 Table XII (p. 26) 
 
 Tables for testing Goodness of Fit. (W. Palin Elderton, Biometrika, Vol. I. 
 pp. 155—163.) 
 
 The theory of testing frequency distributions for goodness of fit was first given 
 by Pearson f and may be summed up as follows : 
 
 If a frequency distribution or table contains n 'cells' and the contents of these 
 cells be m/, m^', m 3 ' ... m n ' in number, while m 1; m^, m 3 ... m n be the numbers that 
 would occur in these cells on any theory ; then calculate 
 
 I m r j 
 
 /square of difference of theoretical and observed frequenciesN 
 \ theoretical frequency / 
 
 and the probability that random sampling would lead to as large or larger deviation 
 
 between theory and observation is 
 
 P-4/v^ + y|.-*(^& + ^ + ... +00 ^ 
 
 *..„.„ 1 . 3. 5 ... (n — 3). 
 
 if n be even 
 
 »'— 3 
 
 / H 1+ l + 2V 2 T4T6 + - + 2.4.6 *..(»'- 8) ) if "' be odd 
 
 (n' - 3) 
 
 (xxix), 
 
 Pearson, Phil. Trans. Vol. 200, A, p. 25. t Phil. Mag. Vol. l. pp. 157—175, 1900. 
 
XXX11 
 
 Tables for Statisticians and Biometricians 
 
 [XII 
 
 Short provisional tables of P were given in the article referred to and were 
 replaced in the following year by the present standard tables of Palin Elderton. 
 
 In using the test for goodness of fit, due regard should be paid to the 
 conditions under which it is deduced. It is assumed that the frequencies form 
 a normal system of variates. This is legitimate only when in the binomial (p + q) n , 
 q is not very small as compared with p. If q be not very small as compared with p, 
 even for n finite, the binomial approaches closely to the normal curve. Accordingly 
 in using the test it is desirable to club together small frequencies at the tails of 
 curves or margins of surfaces. The difficulty becomes very obvious when theory 
 can go by fractions, but observations only by units. 
 
 The theory can be extended to cover much ground in all sorts of sampling*. 
 
 Illustration. The following data for observed frequencies of cephalic index in 
 Bavarian crania and for corresponding frequencies of a fitted Gaussian curve have 
 already been considered on p. xx. Test the goodness of fit. 
 
 
 Observed 
 
 Gaussian 
 (■») 
 
 m' - m 
 
 (n' - mf 
 m 
 
 Under 75-5 
 
 9-5 
 
 12-4 
 
 - 2-9 
 
 
 68 
 
 75-5—76-5 
 
 12-5 
 
 12-7 
 
 - 0-2 
 
 
 00 
 
 76-5—77-5 
 
 17 
 
 22-1 
 
 - 5-1 
 
 1 
 
 18 
 
 77-5—78-5 
 
 37 
 
 35-3 
 
 + 1-7 
 
 
 08 
 
 78-5—79-5 
 
 55 
 
 51-9 
 
 + 3-1 
 
 
 19 
 
 79-5—80-5 
 
 71-5 
 
 70-1 
 
 + 1-4 
 
 
 03 
 
 80-5—81 -5 
 
 82 
 
 87-0 
 
 - 5-0 
 
 
 29 
 
 81-5—83-5 
 
 116 
 
 99-4 
 
 + 16-6 
 
 2 
 
 77 
 
 82-5—83-5 
 
 98 
 
 104 -2 
 
 - 6-2 
 
 
 37 
 
 83-5—84-5 
 
 107 
 
 100-5 
 
 4- 6-5 
 
 
 42 
 
 84-5—85-5 
 
 82 
 
 89-1 
 
 - 7-1 
 
 
 57 
 
 85-5—86-5 
 
 74 
 
 72-6 
 
 + 1-4 
 
 
 03 
 
 86-5—87-5 
 
 58 
 
 54-3 
 
 + 3-7 
 
 
 25 
 
 87-5—88-5 
 
 34-5 
 
 37-4 
 
 - 2-9 
 
 
 22 
 
 88-5—89-5 
 
 19 
 
 23-7 
 
 - 4-7 
 
 
 93 
 
 89-5—90-5 
 
 10 
 
 13-8 
 
 - 3-8 
 
 1 
 
 05 
 
 90-5—91-5 
 
 8 
 
 7-4 
 
 + 0-6 
 
 
 05 
 
 Over 91 -5 
 
 9 
 
 6 3 
 
 + 2-7 
 
 1-16 
 
 Totals ... 
 
 900 
 
 900-2 
 
 18 Groups 
 
 X s = 10-27 
 
 * A word of caution must be given about a recent extension by Slutsky (see Journal of Royal 
 Statistical Society, Vol. Lxxvn. pp. 78—84) who has applied it to test the goodness of fit of regression 
 curves. In such cases the means and standard deviations of each array should, I think, be deduced 
 from the theoretical surface, and the method would then agree with that illustrated on pp. xxiv— xxvi, 
 i.e. on the probability of a given complex of variates differing from the run of values of a given 
 population significantly. Slutsky after assuming that the observed frequencies and standard deviations 
 of the arrays may replace the theoretical values, deduces his P from Elderton's Tables instead of from 
 the incomplete normal moment tables. He finds for the fit of a straight regression line, used to predict 
 the probable price of rye at Samara from the price a month previously, x 2 = 22-2, giving P = -02, a bad 
 fit. Had he, however, used the theoretical standard-deviation of an array, i.e. oVl -?•'-', instead of the 
 very irregular observed standard deviations of individual arrays, he would have found x 2 =8-84, leading 
 to P=-64 an excellent fit, which would probably have been still further improved by the use of 
 theoretical total frequencies for the arrays, based, say, on a Gaussian distribution. 
 
XIII — XVI] Introduction xxxiii 
 
 Taking froin the column for n = 18 (p. 27) the values for x 2 = 10 and 11, we 
 interpolate P = -891 for ^ 2 =1027 by first differences, and conclude that in 89 out 
 of 100 trials we should get in random sampling a fit as bad or worse than that 
 observed, if the real distribution were Gaussian. Accordingly we say that a 
 Gaussian curve describes excellently the distribution of Bavarian cephalic iudices. 
 
 Tables XIII— XVI (pp. 29—30) 
 
 Auxiliary Tables provided by W. Palin Elderton (Biometrika, Vol. I. pp. 162 — 
 163), useful for calculating values of P for x 2 outside the range of the existing table. 
 
 For such cases we must turn back to the fundamental formulae (xxix) of p. xxxi, 
 and the numerical values of considerable portions of these formulae will be found 
 evaluated in these auxiliary tables. 
 
 Illustration. Find P for n' — 11 and x 2 = 78, ^ 2 = 39, hence by formula 
 we have 
 
 = e- 39 (40 + 760-5 + 9886 5 + 96393375) 
 = e- 39 x 107080-375, 
 where the powers of 39 are taken out of Table XXVII (p. 38). 
 Hence using Table XIII, 
 
 logP= 170625,1520 + 50297,0988 = 12-0922,2508, 
 which gives us P = T23659/10 12 . 
 
 As a rule we can select n to be odd, but, if it is necessarily even, there is more 
 trouble, not in the determination of the series, but in the evaluation of the 
 integral 
 
 e' hy? d X . 
 
 -vi£ 
 
 A table of the values of F=^I for % = ^ fco 500 has been given as Table IV 
 (p. 11). This gives X 2 = 25 to 250000 but the intervals are large. 
 If greater accuracy be required then Schlomilch's formula* 
 
 1 
 
 v ttJ x * v tt* 6 y x \ x ^ 
 
 + 
 
 ! +2) X 2 (X 2 + 2 )(X 2 + 4) 
 5 9 
 
 X 2 (X 2 + 2) (x 2 + 4) (x* + 6) x* it + 2) (X 2 + 4) ( X 2 + 6) <tf + 8) 
 
 129 \ 
 
 X 2 (X 2 + 2)(x 2 + 4)(x 2 +6)(x 2 + 8)(x 2 + 10) + "-[ (***) 
 
 must be used. 
 
 Here a/ — X e ~ w '" be found in Table XIII, and the series converges fairly 
 
 rapidly. 
 
 * Compendium der h'dheren Analysis, Bd. n. S. 270, Braunschweig, 1879. 
 
XXXIV 
 
 Tables for Statisticians and Biometricians 
 
 [XVII 
 
 Table XVII (p. 31) 
 
 Values of (— log P) corresponding to given values of ^ in a fourfold table. 
 (K. Pearson : On a Novel Method of regarding the Association of two Variates 
 classed solely in Alternate Categories. Drapers' Company Research Memoirs, 
 Biometric Series, viii. Dulau & Co.) 
 
 If individuals be classed by the characters into A and not-.4, B and not-5, we 
 form a tetrachoric table of the form 
 
 
 A | Not-4 
 
 Totals 
 
 B ... 
 Not-£ 
 
 a b 
 
 c d 
 
 a + h 
 c+d 
 
 Totals 
 
 a+c 
 
 b + d 
 
 N 
 
 For such a table : 
 
 % 2 = 
 
 N(ab-cdf 
 
 (a + b) (c + d)(b + d)(a + c)' 
 
 .(xxxi), 
 
 gives a measure of the probability of independence, and, if the two attributes are 
 highly associated, %" will be large and P the probability of independence very 
 small and largely outside Palin Elderton's Table XII. Tabic XVII provides for 
 such cases. 
 
 Illustrations. The following tables are given by Mr G. U. Yule in his Theory 
 of Statistics*. His conclusions with regard to them are: 
 
 1. Datura : " No Association." 
 
 2. Eye Colour in Father and Son : " Shows the tendency to resemblance." 
 
 3. Houses in course of erection, Urban and Rural : " Distinct Positive 
 Association." 
 
 4. Imbecility and Deaf-Mutism : " High Degree of Association." 
 
 5. Developmental Defects and Dullness : " Very high indeed." 
 
 It is required to measure the degree of probability that the variates in these 
 five cases are independent. 
 
 (1) Datura. (2) Eye-Colour in Father and Son. 
 
 Colour of Flower. Father. 
 
 
 
 Violet 
 
 White 
 
 Totals 
 
 Prickly ... 
 Smooth... 
 
 47 
 12 
 
 21 
 3 
 
 68 
 15 
 
 Totals ... 
 
 59 
 
 24 
 
 83 
 
 a 
 o 
 
 
 Light 
 
 Not Light 
 
 Totals 
 
 Light 
 
 Not Light ... 
 
 471 148 
 151 230 
 
 619 
 381 
 
 Totals ... 
 
 622 
 
 378 
 
 1000 
 
 * Pp. 37, 34, 62, 33, 34 and 45 respectively. 
 
XVII] 
 
 (3) Houses in course of erection in 
 Urban and Rural Districts. 
 
 Introduction xxxv 
 
 (4) Imbecility and Deaf-Mutism. 
 
 
 Built 
 
 Building 
 
 Totals 
 
 Urban . . . 
 Rural ... 
 
 4960 
 1749 
 
 50 
 12 
 
 5010 
 1761 
 
 Totals ... 
 
 6709 62 
 
 6771 
 
 
 Imbecile 
 
 Non-Imbecile 
 
 Totals ... 
 
 Deaf Mute 
 Non-Deaf Mute... 
 
 451 
 48,431 
 
 14,795 
 32,464,323 
 
 15,246 
 32,512,754 
 
 Totals 
 
 48,882 
 
 32,479,118 
 
 32,528,000 
 
 (5) Developmental Defects and Dullness. 
 
 
 With Defects 
 
 Without 
 
 Totals 
 
 Dull 
 Not-Dull ... 
 
 888 1186 
 1420 22,793 
 
 2074 
 24,213 
 
 Totals 
 
 2308 23,979 
 
 26,287 
 
 (f>- the mean square contingency = x 2 /iV an< ^ */> i s the product-moment coefficient 
 on the assumption that the 'presence of the character' is to be considered as 
 a concrete unit*. The coefficient of mean square contingency C. 2 = ^<p-/(l +4>°). 
 
 The following table gives the values of %-, <f>-, j> and C' 2 , and the values of P 
 deduced. 
 
 
 X 2 
 
 2 
 
 * 
 
 c 2 
 
 P 
 
 (1) Datura 
 
 (2) Eye-Colour 
 
 | (3) Houses 
 
 (4) Imbecility and Deaf-Mutism 
 ] (5) Defects and Dullness 
 
 •7080 
 133-3265 
 1-4393 
 8014-62 
 3256-797 
 
 •085,301 
 •133,327 
 
 •000,2125 
 •000,2464 
 •123,894 
 
 •2921 
 •3651 
 •0146 
 •0157 
 •3519 
 
 •2803 
 •3430 
 •0146 
 •0157 
 •3320 
 
 •8713 
 
 1 -035/1028 
 
 •6948 
 3-179/10 1739 
 2-846/10 700 
 
 For example, from Table XVII by Formula (i) we have for x"= 1333265 : 
 (- log P) = 20-809 + ™™* [10-770] - J (^f 65 ) ( 1G ^ 5 ) [-026] 
 
 = 27-985, 
 and therefore log P = 28-015, 
 
 which leads to P = 1 -035/1 28 . 
 
 Or again, for x~= 801462 : 
 
 -V,p-i»«. + !*-p»«n- t @«)(^ [ « fl 
 
 = 1738-498, 
 
 * Pearson and Heron, Biometrika, Vol. ix. p. 167. 
 
 (2 
 
xxxvi Tables for Statisticians and Biometricians [XVII — XX 
 
 and therefore log P = 1739602, 
 
 and P = 3-179/10™. 
 
 In the first and third cases a different treatment must be used. For ^ = 14393 
 we use Table XII. 
 
 We have for n = 4 : 
 
 P = -801 253 + -4393 [- 228846] - $ (-4393) (-5607) [+ 48064] 
 = -6948. 
 Had we worked from Table XVII by Formula (i), we should have had 
 
 P = -6950. 
 For x 2 = -7080, we can use Table XII, remembering that for ^ 2 = 0, P = l. 
 We have 
 
 P = 1-000,000 + -708 [- 198,747] - } ('708) (-292) [- 30,099] 
 
 = -8624. 
 Had we worked from Table XVII by Formula (i), we should have had P = *865, 
 close enough for practical purposes. 
 
 The true value of P worked from 
 
 IV27T iv 
 
 •~ W * + ?S? # ~***) 
 
 by using Table II is P = "8713. See p. xxxviii. 
 
 Examining the values of P we see that having regard to the errors of random 
 sampling we can only say that there is no relation between rural and urban 
 districts and houses building or built ; there is clearly no ' distinct association,' 
 for in 69 out of 100 cases in sampling from independent material we should get 
 more highly associated results. There is likewise no association on the given 
 material in the Datura characters. The other three cases have clearly very marked 
 association, quite independent of any influence of random sampling. If we regard 
 these three tables the order of ascending association judged by either <f> or C 2 is 
 (4), (5), (2), as against Mr Yule's (2), (4), (5). If we disregard the non-significance 
 and take merely intensity of association, without regard to random sampling, the 
 order is (3), (4), (1), (5), (2), as against Mr Yule's order (1), (2), (3), (4), (5). 
 
 The best method of inquiry at present for relative association in the case of 
 four-fold tables is, I hold, first to investigate P and throw out as not associated 
 those cases like the ' Houses, built and building ' above. Then to use either 
 " tetrachoric r t " or C 2 according as we are justified in considering the variates 
 as continuous or not. r P (see p. xxxvii) may be used as control. 
 
 Tables XVIII— XX (pp. 31—32) 
 
 Tables for determining the Equiprobable Tetrachoric Correlation r P . (Pearson 
 and Bell : On a Novel Method of regarding the Association of two Variates classed 
 
XVIII— XX] 
 
 Introduction 
 
 xxxvu 
 
 solely in Alternate Categories. Draper's' Company Research Memoirs, Biometric 
 Series, vm. Dulau & Co.) 
 
 We have seen under the discussion of the previous Table how to find a measure 
 of the improbability of two variates being independent, when they are classed in 
 alternate categories. The difficulty in such cases is to appreciate the relative 
 importance of very large inverse powers of 10. The object of the present tables is 
 to enable us to deduce a tetrachoric correlation, r t , of which the improbability 
 is the same as that of the given system supposing it to arise, when the two 
 variates have the same marginal frequencies but are really independent. In order 
 to do this we have to determine <r r for the given marginal frequencies, i.e. the 
 standard deviation of r t on the assumption that r is really zero. This may be easily 
 found from Abac Diagram XXI or from Table XXIV (see below). Table XVIII 
 then gives us the value of (— log P) for each value of r t and a>. If we now turn 
 to our original table and calculate its rf, this as we have seen will correspond 
 to a given (— log P). We now make the (— log P) from our ^ 2 correspond to the 
 (— logP) from our r t and „a,., this gives us a value of r t which has the same degree 
 of improbability as our observed table. In other words, instead of trying to 
 appreciate the meaning of inverse high powers of 10, we say that a table of the 
 same marginal frequency would be as improbable if it had a tetrachoric correlation 
 r t arising from random sampling of independent variates. Thus we read our 
 improbability on a scale of tetrachoric correlation. We use our correlation merely 
 as a scale to measure probability on. 
 
 As log% 2 provides a more satisfactory basis for interpolation, and as many 
 readers use logarithm tables and not calculators, log^ 2 will be the form in which 
 X 1 will be often presented. Table XX provides the value of r t corresponding to 
 given a r and given log % 2 . 
 
 We will assume for the present that „cr,. can be readily found from the marginal 
 totals : see p. xli below. 
 
 Illustration. Obtain the values of r P for the five tables given above on 
 pp. xxxiv — v. 
 
 The values of log^ 2 and cr r are as follows: 
 
 
 log x" 
 
 a"r 
 
 (1) Datura 
 
 (2) Eye-Colour, Father and Son 
 
 (3) Houses in Course of Erection 
 
 (4) Imbecility and Deaf-Mutism 
 
 (5) Developmental Defects and Dullness 
 
 T-8500 
 2-1249 
 •1582 
 3-9039 
 3-5128 
 
 •1941 
 •0514 
 •0634 
 •0175 
 •0201 
 
 Of the values here recorded for log ^ 2 and <r r , those for the first and third 
 cases lie beyond the limits of our Table XX. But if the point <r r = - 0634 and 
 log x 2 = '1582 be noted on the Abac XXII, p. 34, it will be seen that, having due 
 
xxxviii Tables for Statisticians and Biometriciaus [XVIII — XX 
 
 regard to the spacings of the correlation curves, the value of the equiprobable 
 correlation is under "0:3, say '027. In other words no significant association can be 
 asserted. 
 
 In the case of O ov = 1941 we are thrown back on the original formulae*. In 
 the first place we must find P for the given value of ^ 2 , i.e. - 7080 (see p. xxxv). 
 But for w'= 4 from formula (xxix), 
 
 = 2 (-200,0578 + -280,0088 x "84142} 
 = -871,3256. 
 To obtain r we have to use the formula below, where O ov= "1941, and 
 
 m— A ( • — 8), the /*„, fit, /i s being the normal moment functions of Table IX. 
 
 W2irJ; 
 
 e 
 
 X 
 
 P = 
 
 ■ -^5= [j Mo (V2«0 - & (V2//H-)} - lj ^ (V2m) - to (^2mr)j 
 
 - sss- !/ " c (V ^" } " ^ (V2Vir)1 + m& !ms (V2 ~' 7t) ~ * ( vi ™ r >5j 
 
 (xxxii). 
 
 Substituting the values of n a,. = -1941 and V2wt = 4-852,107, we have for 
 
 r =03, P= -90550, 
 r = -04, P= -86501. 
 Whence for P = -87133, we have r = -038. 
 
 We now turn to the three cases which fall inside Table XX. 
 
 (2) Eye-colour, Father and Son. 
 
 log x 2 = 2-1249 „<7, = -0514, 
 
 r = 0-5 log x" - 2-0942 
 n0 -, = 0o ^ = o f) j^ ^ = 2 2748 
 
 r-06 log x 2 = 2-1239 
 ^,-•06 5=0 . 7 i og%2= 2-2935. 
 
 Linear differences will suffice 
 
 „<r, = -05 r = 0-5 +.^[1] = 0-517, 
 
 o£r) . = . OG r = 0-G+;^[-l] = 0G01. 
 
 Hence O o- r = -0514 gives 
 
 r = -517 + ^x-084 
 
 = •517 + -012 = -529. 
 
 * Drapers' Company Research Memoirs. Biometrie Series VII. "A Novel Method," etc.: see 
 pp. 12, 13. 
 
XVIII— XX] 
 
 Introduction 
 
 XXXIX 
 
 Interpolating for <7,. first, 
 
 ,• = •5 „<r,.= -05U log x" = 2-0737, 
 r = -6 O 0V = -051-1- log x' = 2-2537. 
 
 Hence for log x~ = 2-1249 : 
 
 •0512 ri , .__ 
 r " = ' O + -1800 [ ' 1] = "° 28 - 
 We conclude that the equiprobable correlation is - 53. 
 
 (4) Imbecility and Deaf-mutism. 
 
 log x" = 3-9039 cr,. = -Ol75, 
 r = 0-95, O o-, = -01 , log x 2 = *3673 ; „°v = "02, log x 2 = 37660. 
 Hence: r = 0'J5, O o-, = -0175, log x 2 = 39163. 
 
 Again : 
 
 r = 090, „ct,. = -01, log x 2 = -i-2207 ; „o-,. = -02, log % 2 = 3-6197. 
 Hence: r-090, <r,. = -0175, log X * = 3-7699. 
 
 Interpolating log x'- = 39039 between 39163 and 37699, we find 
 
 r P = 0-946. 
 
 (5) Developmental Defects and Dullness. 
 
 log x * = 3-5128, o-,= 0201. 
 r = 0-8, <r,. - -02, log X 2 = 3'4097 ; <x,. = 03, log tf = 3-0598. 
 Hence : „o-, = 0201, log X 2 = 3"4062. 
 
 r = 0<), ,«r-"02, log x 2 = 3-6197; O o-,. = -03, log %= = 3-2690. 
 Hence : log tf = 3-6162, for „cr,. = -0201. 
 Thus, by interpolating logx s = 3-5128 between 3'4062 and 3-6162, we find 
 
 r>=-851. 
 We have accordingly the following results : 
 
 
 •2803 
 •3430 
 •0146 
 •0157 
 •3320 
 
 P 
 
 r P 
 
 r t 
 
 Q 
 
 (1) Datura 
 
 (2) Eye-Oolour 
 
 (3) Houses 
 
 (4) Imbecility and Deaf-Mutism 
 
 (5) Defects and Dullness 
 
 •8713 
 1 -035/10 28 
 
 •6948 
 3-179/10" 39 
 2-846/10 706 
 
 •038 
 ■529 
 •027 
 •946 
 •851 
 
 -•188+ -140 
 •550+ -027 
 
 - -081 + -043 
 •330+ -012 
 ■652+ -009 
 
 -•282 
 •581 
 
 -•190 
 •907 
 •846 
 1 
 
 It will be seen that equiprobable r P confirms generally the results from P, i.e. 
 the tables for 'Datura' and ' Houses' give no sensible association. r t also confirms 
 this view and shows that ' Houses ' is even lower in the scale than ' Datura.' The 
 order of r P is the same as that of Yule's coefficient of association Q, but neither 
 r r , r t , G lt P or Q support the conclusions stated to flow from the percentages on 
 
xl Tables for Statisticians and Biometrieians [XXIII — XXIV 
 
 p. xxxiv. Both r P and Q give very high results for (4) and (5), and this is in accord- 
 ance with the view elsewhere expressed that for extreme dichotomies Q is not to 
 be trusted. It may further be doubted, whether for such dichotomies the theory 
 of the distribution of deviations on which r P is based can in its turn be accepted. 
 On the whole r t seems to me the most satisfactory coefficient of association, to be 
 controlled by results for r P in the cases where neither the dichotomies are extreme, 
 nor the numbers so large or so small as to fall outside the moderate range of 
 Tables XVIII— XX or Abacs XXI and XXII. 
 
 Abacs XXI and XXII (pp. 33—34). 
 Sec after Tables XXIII and XXIV. 
 
 Tables XXIII and XXIV 
 
 Tables for determining approximately the probable error of a tetrachoric 
 correlation. (Pearson, Biometrika, Vol. IX. pp. 22 — 27. Tables calculated by 
 Julia Bell, M.A.) 
 
 Given a tetrachoric table 
 
 a 
 
 b 
 
 a+b 
 
 c 
 
 d 
 
 c + d 
 
 a + c 
 
 b + d, 
 
 N 
 
 so arranged that a + c > b + d and a i- b > c + d, 
 
 then if |(1 + *) - (o + b)/HT, 4(l+a,) = (o + c)/JV, 
 
 and r t be the correlation, we have approximately : 
 
 Probable error of »•«->#. %r t • Xo, ■ Xa 2 > 
 where Xi = '67449/ViV 
 
 and is tabled in Table V, p. 12, 
 
 _ vni+» 1 )i(i-« 1 ) Y „ y*g+*)i(i-Q / xxxiii x 
 
 H and K being found from the z column of Table II, p. 2, and 
 
 Xr t 
 
 ■n 
 
 z V 1 -( ! lr ! )' «-«* 
 
 sin _1 rj being read in degrees. % a and ^ a are tabled in Table XXIV and Xr in 
 Table XXIII (p. 35). 
 
 This value of the probable error is only approximate aud may diverge con- 
 siderably from the true value* for extreme dichotomies. In such cases the full 
 formula must be used. 
 * Phil. Trans. Vol. 195, p. 14. Xo i° formula (') should of course not be included under the radical. 
 
XXIII— XXIV] 
 
 Introduction 
 
 xli 
 
 When r t is zero in the population and not in the sample, the standard deviation 
 tr r of r = is given accurately by -= % a % ai> . 
 
 Illustration (i). Tetrachoric r t for the Table 
 
 22,793 
 1,186 
 
 1,420 
 
 888 
 
 24,213 
 
 2,074 
 
 23,979 
 
 2,308 
 
 26,287 
 
 is '652. Find approximately its probable error. 
 From Table XXIII : 
 
 r = 6.5, X.- = '6785 ; r = "66, X >- = '6675. 
 .-. x,- = '6785- 0110 x -2 = -6763. 
 
 Now 4(1+ a,) = '9211, 4(1 +«2> = '9122. 
 
 Hence from Table XXIII, 
 
 Xa, - l'»249 + 11 [754] = 1-8332, 
 
 Xa, = 1'7623 + -22 [626] = 17761, 
 
 X ai X ai = 3'2559. 
 
 Xi cannot be found from Table V in this case as N is beyond its range. But it 
 equals 
 
 •67449/V26287 = -67449/162-13 = -00416. 
 
 Thus finally p.e. of r t = 00416 x -6763 x 32559 
 
 = 009. 
 
 Illustration (ii). Find the value of <r r for the table : 
 
 471 
 151 
 
 148 
 230 
 
 619 
 381 
 
 622 
 
 378 
 
 1000 
 
 Here 4(1 + a,) = '619 and 4 (1 + a 2 ) = -622. 
 
 X 0i = 1-2712 + -9 [36] = 1-2744, 
 Xa , = 1-2748 + -2 [39] = 1 -2756. 
 0^ = ^X^1000 = -0514. 
 
 In a similar manner the values for all the O o>'s in the table on p. xxxvii were 
 found. 
 
 B. / 
 
xlii Tables for Statisticians and Biometricians [XXI, XXII 
 
 Abac XXI (p. 33) 
 
 For determination of the standard-deviation of the correlation coefficients obtained 
 by random sampling from a four-fold table in which the correlation is zero. 
 (Drapers' Company Research Memoirs, Biometiic Series, vm. G. H. Soper's Abac.) 
 
 Method of use : Enter with the total frequency of the sample on the left-hand 
 scale, and with the first value of \ (1 + o) on the bottom scale. The horizontal 
 through the former and the vertical through the latter meet at a point. At this 
 point pass up the diagonal to the left-hand scale again. Where you meet that 
 scale pass along the horizontal until you meet the vertical through the second 
 value of | (1 + a). Then from this point pass along the diagonal again to the left- 
 hand scale, whence traverse the horizontal to the right-hand scale and there the 
 required value of cr r may be read off. 
 
 Illustration (i). Find the value of cr r for the case just given of 
 JV=1000, ^(1+« 1 ) = -619, |(1 + a„) = -622. 
 The vertical through '619 meets the 1000 horizontal in a point whose diagonal 
 reaches the left-hand scale almost exactly in 620. Whence passing horizontally 
 we reach the vertical through - 622 in a point about midway between two diagonal 
 lines. Passing up midway between these two diagonals, we reach almost exactly 
 the 380 line on the left-hand scale. Passing across to the right-hand scale along 
 this line, we see that we are slightly above the middle of the division between 
 •050 and '052, say '0512. The actual value of a r is "0514. 
 
 Illustration (ii). Let 
 
 JV=-6771, |(1 + «0 = -7399, |(1 + a 2 ) = '9908. 
 A similar process gives first 450 ou left-hand scale and then about 248, whence 
 crossing to right-hand scale we find cr r = - 0635 instead of "0634 actual. 
 
 Abac XXII (p. 34) 
 
 Abac to determine from log %' 2 and a r the value of the equiprobable correlation 
 r P ,for a fourfold table. (Drapers' Company Research Memoirs, Biometric Series, 
 vm. G. H. Soper's Abac.) 
 
 The rule is very simple : Enter the Abac with the proper value of a r on the 
 scale at the foot and rise on the vertical till the horizontal through the proper 
 value of log^ 2 on the left-hand scale is reached. Then follow the curve through 
 the meet of these two lines to the right-hand scale, where the requisite correlation 
 will be found inscribed. 
 
 Illustration. Take the Table for Eye Colour in Father and Son given on 
 p. xxxiv. Here, as just shewn, ov="0514 and (p. xxxvii) log^ 2 =2'1249. If we enter 
 with the vertical through - 0514 on the scale at the bottom, and the horizontal 
 through 2 - 1249 on the left-hand scale, the curve through their point of intersection 
 reaches the right-hand scale just below the "53 mark, say "529. This agrees with 
 the correlation found above (p. xxxviii) by interpolation from Table XX. 
 
XXV] 
 
 Introduction 
 
 xliii 
 
 Table XXV (p. 36) 
 
 Value of the probability that the mean of a small sample of n, draiun at random 
 from a population following the normal law, will not exceed (in the algebraic sense) 
 the mean of that population by more than z times the standard deviation of the 
 sample. ("Student": Biometrika, Vol. vi. p. 19.) 
 
 When n is greater than 10, it will be sufficient as a rule to use the approximate 
 result 
 
 v^^3 r - <»-**■ 
 
 P= . e 2 dx (xxxv) 
 
 V27T J -a° 
 
 as a measure of the probability. This may be found from Table II. 
 
 Illustration (i). Experiments of A. R. Cushney and A. R. Peebles on the 
 difference in effect of Dextro-hyoscyamine hydrobromide and Laevo-hyoscyamine 
 hydrobromide*. 
 
 
 Additional Hours of 
 
 Patient 
 
 Sleep 
 
 
 (Laevo - Dextro) 
 
 1 
 
 + 1-2 
 
 2 
 
 + 2-4 
 
 3 
 
 + 1-3 
 
 It 
 
 + 1-3 
 
 5 
 
 
 
 6 
 
 + 1-0 
 
 7 
 
 + 1-8 
 
 8 
 
 +0-8 
 
 9 
 
 +4-6 
 
 10 
 
 + 1-4 
 
 Mean 
 
 + 1-58 
 
 Standard Deviation 
 
 1-17 
 
 + 158 
 
 Table XXV shows that for ^=135: 
 
 P = -99854, 
 
 or the odds are 666 to 1 that leavo- is a better soporific than dextro-hyoscyamine 
 hydrobromide. 
 
 Illustration (ii). Difference in weight of crops of potatoes grown by Dr Voelcker 
 with (i) sulphate of potash and (ii) kainite as artificial manure. 
 
 * Journal of Phytiology, 1904. 
 
 /2 
 
xliv 
 
 Tables for Statisticians and Biometricians [XXV 
 
 Gain by sulphate of potash. 
 
 1904 (a) 10 cwt. 8 qr. 20 lbs. 
 
 (b) 1 ton 10 cwt. 1 qr. 26 lbs. 
 
 1905 (a) 6 cwt. qr. 3 lbs. 
 (b) 13 cwt. 2 qr. 8 lbs. 
 
 Average gain=15'25 cwt., and the standard deviation = 9 cwt.,2=15'25/9 = 1'694. 
 
 Here n = 4, and Table XXV gives us 
 
 P = -9653 + 0-94 x [46] = -9696, 
 
 or the odds are about 32 to 1 that the sulphate of potash is a better dressing than 
 kainite for potatoes. 
 
 Illustration (iii). Test whether it is of advantage to kiln -dry barley seed 
 before sowing. The following table gives price of head corn in shillings per 
 quarter for 11 sowings, the first seven in 1899 and the last four in 1900. 
 
 Not Kiln-dried 
 
 Kiln-dried 
 
 A 
 
 1899 ■ 
 
 1900 • 
 
 26-5 
 28 
 29-5 
 30 
 27-5 
 26 
 l 29 
 29-5 
 28-5 
 30 
 28-5 
 
 26-5 
 
 26-5 
 
 28-5 
 
 29 
 
 27 
 
 26 
 
 26 
 
 28-5 
 
 28 
 
 29 
 
 28 
 
 
 
 1-5 
 
 1 
 
 1 
 
 0-5 
 
 
 
 3 
 
 1 
 
 0-5 
 
 1 
 
 0-5 
 
 Mean 
 
 Standard Deviation 
 
 — 
 
 •91 
 •79 
 
 * x dec. 
 
 Here 
 and if 
 
 The Gaussian curve gives 
 
 *-JZL 
 
 a; = -91/-79 = 1-1519, 
 x = »/VI/8 = 1-1519 x 2-8284 = 3-258, 
 
 1 /-S-258 , , 2 
 
 P = ^L[ e-** da?, 
 
 which evaluated by Table II, p. 6, gives 
 
 P = -99944, 
 or the odds are 2845 to 1 in favour of not kiln-drying seed barley. 
 
XXVI] 
 
 Introduction 
 
 xlv 
 
 If we had actually worked with the non-approximate formula, we should have 
 
 found 
 
 P=-9976, 
 
 or odds of 416 to 1, considerably less than the approximate formula provide, but 
 not enough difference to vitiate any conclusion likely to be drawn in practice*. 
 
 Table XXVI (p. 37) 
 
 Table for use in plotting Type III Curves, i.e. 
 
 -v-(, as\ p 
 
 .(xxxvi) 
 
 (W. P. Elderton, Biometrika, Vol. II. p. 270.) 
 
 Rule : Taking p for the curve, multiply the values in the Table by p in 
 succession on the machine with p on as multiplier. Then subtract the results from 
 the logarithm of y , and we have the logarithms of the ordinates of the curve at 
 the abscissae found by multiplying X in the first column of the Table by a of the 
 curve. The curve can then be plotted. Its origin will be the mode. It is usually 
 quite unnecessary to use the whole series of ordinates, either alternate ordinates 
 will suffice, or we cut off one or both tails at a considerable distance from their 
 tabulated values. 
 
 Illustration. The frequency curve of barometric heights at Dunrobin Castle is 
 given by the curve 
 
 on.Qqoq X I r \ 22-9323 
 
 The range X = — '65 to +"90 is easily seen to be sufficient. Column (i) of 
 the accompanying table gives aX for these values, the second gives 
 22-9323 x (log 10 (1+X)-X log* e) ; 
 
 * The three illustrations above are drawn from "Student's " original paper. He gives (I. c. p. 19) 
 the values for P as drawn from the Gaussian for n = 10 to compare with those obtained from the full 
 formula. They are, — corrected for slips : 
 
 z 
 
 Full Formula 
 
 Gaussian 
 
 z 
 
 Full Formula 
 
 Gaussian 
 
 ■1 
 
 •61462 
 
 •60411 
 
 1-1 
 
 •99539 
 
 •99819 
 
 ■2 
 
 •71846 
 
 •70159 
 
 1-2 
 
 •99713 
 
 ■99925 
 
 ■3 
 
 •80423 
 
 •78641 
 
 1-3 
 
 •99819 
 
 •99971 
 
 ■4 
 
 •86970 
 
 •85520 
 
 1-4 
 
 •99885 
 
 •99989 
 
 ■5 
 
 •91609 
 
 •90691 
 
 1-5 
 
 ■99926 
 
 •99996 
 
 ■6 
 
 •94732 
 
 •94375 
 
 1-6 
 
 •99951 
 
 •99999 
 
 ■7 
 
 •96747 
 
 •96799 
 
 1-7 
 
 •99968 
 
 — 
 
 ■8 
 
 •98007 
 
 •98285 
 
 1-8 
 
 •99978 
 
 
 
 ■9 
 
 •98780 
 
 •99137 
 
 1-9 
 
 •99985 
 
 ■ 
 
 1-0 
 
 •99252 
 
 •99592 
 
 20 
 
 •99990 
 
 — 
 
 Clearly even for n = 10, the Gaussian ascends too rapidly in P, and this must be borne in mind in 
 deducing conclusions for z = l and upwards when n = ll to 20, say. 
 
xlvi Tables for Statisticians and Biometricians [XXVI — XXVIII 
 
 actually these values are negative and must be subtracted from log y„, i.e. T.592,621; 
 the resulting values are given in the third column. In column (iv) are given 
 the autilogarithms of the numbers in column (iii), and these must be plotted to 
 the values in column (i) to obtain the graph of the curve which is a good fit. 
 
 (i) (ii) (iii) (iv) 
 
 x=aX l>[log 10 (l 
 
 + .Y)-jnoge]* 
 
 log;/ 
 
 y 
 
 - 9-77 2 
 
 474,991 
 
 - -882,370 
 
 •13 
 
 - 8-88 1 
 
 923,630 
 
 - -331,009 
 
 •47 
 
 - 7-99 1 
 
 472,368 
 
 •120,253 
 
 1-32 
 
 - 7-11 1 
 
 103,755 
 
 •488,866 
 
 3-08 
 
 - 6-22 
 
 804,557 
 
 •788,064 
 
 6-14 
 
 - 5-33 
 
 564,456 
 
 1-028,165 
 
 10-67 
 
 - 4-44 
 
 375,287 
 
 1-217,334 
 
 16-49 
 
 - 3-55 
 
 230,493 
 
 1-362,128 
 
 23-02 
 
 - 2-67 
 
 124,683 
 
 1 -467,938 
 
 29-37 
 
 - 1-78 
 
 053,386 
 
 1 -539,235 
 
 34-61 
 
 - 0-89 
 
 012,888 
 
 1 579,733 
 
 38-00 
 
 o-oo 
 
 000,000 
 
 1-592,621 
 
 39-14 
 
 0-89 ! 
 
 012,039 
 
 1 -580,582 
 
 38-07 
 
 1-78 ! 
 
 046,713 
 
 1 -545,908 
 
 35-15 
 
 2-67 
 
 101,957 
 
 1 -490,664 
 
 30-95 
 
 3-55 
 
 176,074 
 
 1-416,547 
 
 26-09 
 
 4-44 
 
 267,482 
 
 1-325,139 
 
 21-14 
 
 5-33 
 
 374,828 
 
 1-217,793 
 
 16-51 
 
 6-22 
 
 496,920 
 
 1-095,701 
 
 12-47 
 
 711 
 
 632,702 
 
 ■959,919 
 
 9-12 
 
 7-99 
 
 781,189 
 
 •811,432 
 
 6-48 
 
 8-88 
 
 941,509 
 
 •651,112 
 
 4-48 
 
 9-77 1 
 
 112,905 
 
 •479,716 
 
 3-02 
 
 10-66 1 
 
 294,689 
 
 •297,932 
 
 1-99 
 
 11-55 1 
 
 486,196 
 
 •106,425 
 
 1-28 
 
 12-44 1 
 
 686,831 
 
 - -094,210 
 
 •80 
 
 13-32 1 
 
 896,111 
 
 - -303,490 
 
 •50 
 
 14-21 2 
 
 113,510 
 
 - -520,889 
 
 •30 
 
 15-10 2 
 
 338,613 
 
 - -745,992 
 
 •18 
 
 15-99 2 
 
 570,963 
 
 - -978,342 
 
 •11 
 
 Once the reader is used to the process it will be found to work readily, and the 
 same multipliers are kept on the mechanical calculator throughout. 
 
 Tables XXVII and XXVIII (pp. 38—41) 
 
 Tables of the Poivers and Sums of the Powers of the natural numbers from 1 to 
 100. (W. Palin Elderton, Biometrika, Vol. n. p. 474.) 
 
 These tables can be used in a great variety of ways, for example in finding the 
 roots of equations, or in fitting parabolae of various orders to curves. 
 Illustration (i). Find the positive root of the equation : 
 </> (r) = 002,726r' + -057,149^ + •0l7,192»- 5 
 
 + -083,578^ + -OSS^l?- 3 + •134,717»- 2 + r - "560,386 = 0. 
 * Actually these values are negative, and are therefore subtracted from log y to give (iii). 
 
XXVII— XXVIII] 
 
 Introduction 
 
 xlvii 
 
 The positive root is less thau '56, but the term in r 2 shows that it must be less 
 than -.52. Take "52 and '50 as trials. From Table XXVII we have 
 
 1st -520,000 and 500,000, 
 
 2nd -270,400 „ -250,000, 
 
 3rd -140,608 „ -125,000, 
 
 4th -071,162 „ -062,500, 
 
 5th -038,020 „ -031,250, 
 
 6th 019,771 „ -015,625, 
 
 7th -010,281 „ -007,813. 
 
 Multiply out by the coefficients of <f> (r), retaining the products always on the 
 arithmometer. We find 
 
 <j> (-52) = + -016,384. 
 
 $ (-50) = - -008,990. 
 
 Interpolating r = "52 — Hff| x 2 = '5071, 
 
 which is correct to last figure. 
 
 Illustration (ii). Fit a cubic parabola to the data below, giving the average 
 age of husband to each age of wife in Italy (see Biometrika, Vol. II. p. 20). We 
 will suppose each observation to be of equal weight, — this is of course not the fact, 
 but it will illustrate the general method of fitting parabolic curves. In the paper 
 just cited illustrations are given up to parabolae of the sixth order. The object 
 here is to show the use of Table XXVII. 
 
 Age of 
 
 Probable Age 
 
 Age of 
 
 Probable Age 
 
 Age of 
 
 Probable Age 
 
 Bride 
 
 of Groom 
 
 Bride 
 
 of Groom 
 
 Bride 
 
 of Groom 
 
 15-5 
 
 25-0 
 
 25-5 
 
 27-0 
 
 35-5 
 
 36 
 
 16-5 
 
 25-2 
 
 26-5 
 
 27-5 
 
 36-5 
 
 37-0 
 
 17-5 
 
 25-4 
 
 27-5 
 
 28-0 
 
 37-5 
 
 38-5 
 
 18-5 
 
 25-5 
 
 28-5 
 
 29-0 
 
 38-5 
 
 39-5 
 
 19-5 
 
 25-5 
 
 29-5 
 
 30-0 
 
 39 5 
 
 41-5 
 
 20-5 
 
 25-5 
 
 30-5 
 
 32-0 
 
 40-5 
 
 41-5 
 
 21-5 
 
 25-75 
 
 31-5 
 
 33 
 
 41-5 
 
 42-5 
 
 22-5 
 
 26-0 
 
 32-5 
 
 33-5 
 
 42-5 
 
 43-5 
 
 23-5 
 
 26-0 
 
 335 
 
 34-0 
 
 43 5 
 
 43-5 
 
 24-5 
 
 26-8 
 
 34 5 
 
 34-5 
 
 44-5 
 
 43-5 
 
 ■ — 
 
 — 
 
 — 
 
 ■ 
 
 45-5 
 
 43-5 
 
 The ages of groom have been taken as approximate means. Now we can take 
 our axis of x, the age of bride through 30'5, and the age of groom to be measured 
 from 32'0. x will accordingly range from —15 to +15, and the age 32 + y of 
 groom will range from y = - 7 to y=ll"5. We can now re-arrange the above 
 table in a form suitable for working on the following table. Then the squares, 
 cubes, and if necessary, higher powers of x are taken from Table XXVII, p. 38, 
 and are given as Columns (iii) and (iv) below. The entries in Column (i) are then 
 multiplied by those in (ii), (iii) and (iv) by continuous process on the machine, and 
 
xlviii 
 
 Tables for Statisticians and Biometricians [XXVIII 
 
 it is not needful to enter separate products, the sums being reached which are 
 placed at the foot. Next from Table XXVIII we read off 
 
 8(x) = 0, £(0 = 2(S(15 2 )), S(tf) = 0, #<y) = 2(S(15 4 )), 
 
 S(x>) = 0, S(x«) = 2(S(U% 
 These give us : 
 
 8 (a?) = 2480, 8 (a?) = 356,624, S (x°) = 6096,5840. 
 We have now all the numerical data for a solution. Let the required cubic be 
 
 y — c + c 1 x + c 2 x* + c z x a . 
 Then we must make u = S (y — c — c v x — c.x* — c^x?)- a minimum. The resulting 
 equations are 
 
 S(y) - c,8 (1) + 0, S (x) + c,S (O + c 3 S («•), 
 
 8 (xy) = c„S (x) + Cl S (x*) + 0,8 (*») + c 3 S (x*), 
 S (x"-y) = c 8 (x*) + Cl S{o?) + c,S (of) + c,S (x>), 
 S (afy) = cS (x 3 ) + 'oJS-W + cS (^) + 0,8 («•> 
 
 (i) 
 
 (ii) 
 
 (hi) 
 
 (iv) 
 
 (v) 
 
 (vi) 
 
 (vii) 
 
 V 
 
 X 
 
 x- 
 
 .r 3 
 - 3375 
 
 xy 
 
 x-y 
 
 x'y 
 
 - 7-0 
 
 -15 
 
 225 
 
 
 
 ■ _ 
 
 - 6-8 
 
 -14 
 
 196 
 
 -2744 
 
 — 
 
 — 
 
 — 
 
 - 6-6 
 
 -13 
 
 169 
 
 -2197 
 
 — 
 
 — 
 
 — 
 
 - 6-5 
 
 -12 
 
 144 
 
 -1728 
 
 — 
 
 — 
 
 — 
 
 - (i-5 
 
 -11 
 
 121 
 
 - 1331 
 
 — 
 
 — 
 
 — 
 
 - 6-5 
 
 -10 
 
 100 
 
 -1000 
 
 — 
 
 — 
 
 — 
 
 - 6-25 
 
 - 9 
 
 81 
 
 - 729 
 
 — 
 
 — 
 
 — 
 
 - 6-0 
 
 - 8 
 
 64 
 
 - 512 
 
 — 
 
 — 
 
 — 
 
 - 6-0 
 
 - 7 
 
 49 
 
 - 343 
 
 — 
 
 — 
 
 — 
 
 - 5-2 
 
 - 6 
 
 36 
 
 - 216 
 
 — 
 
 — 
 
 — 
 
 - 5-0 
 
 - S 
 
 25 
 
 - 125 
 
 — 
 
 — 
 
 — 
 
 - 4-5 
 
 - 4 
 
 16 
 
 - 64 
 
 — 
 
 — 
 
 — 
 
 - 4-0 
 
 - 3 
 
 9 
 
 - 27 
 
 — 
 
 — 
 
 — 
 
 - 3-0 
 
 - 2 
 
 4 
 
 - 8 
 
 — 
 
 — 
 
 — 
 
 - 2-0 
 
 - 1 
 
 1 
 
 1 
 
 — 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 1-0 
 
 1 
 
 1 
 
 1 
 
 — 
 
 — 
 
 — 
 
 1-5 
 
 2 
 
 4 
 
 8 
 
 — 
 
 — 
 
 — 
 
 2-0 
 
 3 
 
 9 
 
 27 
 
 — 
 
 — 
 
 — 
 
 2-5 
 
 4 
 
 16 
 
 64 
 
 — 
 
 — 
 
 — 
 
 4-0 
 
 5 
 
 25 
 
 125 
 
 — 
 
 — 
 
 — 
 
 4-5 
 
 6 
 
 36 
 
 216 
 
 — 
 
 — 
 
 — 
 
 6-5 
 
 7 
 
 49 
 
 343 
 
 — 
 
 — 
 
 — 
 
 7-5 
 
 8 
 
 64 
 
 512 
 
 — 
 
 — 
 
 — 
 
 9-5 
 
 9 
 
 81 
 
 729 
 
 — 
 
 — 
 
 — 
 
 9-5 
 
 10 
 
 100 
 
 1000 
 
 — 
 
 — 
 
 — 
 
 10-5 
 
 11 
 
 121 
 
 1331 
 
 — 
 
 — 
 
 — 
 
 11-5 
 
 12 
 
 144 
 
 1728 
 
 — 
 
 — 
 
 — 
 
 11 "6 
 
 13 
 
 169 
 
 2197 
 
 — 
 
 — 
 
 — 
 
 11-5 
 
 14 ' 
 
 196 
 
 2744 
 
 — 
 
 — 
 
 — . 
 
 11-5 
 
 15 
 
 225 
 
 3375 
 
 
 — 
 
 
 8(x) = 23 •(!.-> 
 
 — 
 
 — 
 
 — ' 
 
 S(xy) = 1833 -45 
 
 S(x 2 y) = 4560-35 
 
 S (xh/) = 248,807-85 
 
XXVIII] Introduction 
 
 Write b = c , b, = 10c„, 6 2 =100c 2 , 6 3 =1000c 3 . 
 
 Then our equations are 
 
 •23650 = b„ x -31000 + b, x -24800, 
 
 1-83345 = b, x -24800 + 6 3 x -35662, 
 
 •45603 = b u x -24800 + b. 2 x -35662, 
 
 248808= 6, x -35662 + 6 3 x -60966; 
 
 giving b„ = - -58626, .-. c„ = - "58626, 
 
 6, = 1-686453, c 2 = -016,8645, 
 
 6, = 9-59613, c, = -959,613, 
 
 6 3 = -1-532,144, c 3 = - -001,532,144, 
 
 and the required cubic is 
 
 y= - -58626 + -959,613a; + -016,8645a? - 001,532,144^. 
 
 xlix 
 
 30 
 
 Age of Bride. 
 
 The graph of the cubic and the observations are given in the accompanying 
 diagram. If X and Y be the actual ages of bride and groom, then 
 
 Y = 61-30457 - 4-344.941Z + -157,0553Z 2 - -001,53214X 3 . 
 
 For higher parabolic curves fitted to the same data, see Biometrika, Vol. n. 
 pp. 21—22. 
 
 B. g 
 
Tables for Statisticians and Biometricians [XXIX 
 
 Table XXIX (pp. 42—51) 
 
 Tables of the Tetrachoric Functions. (P. F. Everitt, Biometrika, Vol. VII. 
 pp. 437—451.) 
 
 The purpose of these tables is to expedite the calculation of tetrachoric r t , the 
 correlation coefficient from a four-fold table, when we suppose the variates to be 
 Gaussian in the law of their frequency. 
 
 Let the table be 
 
 a 
 
 b 
 
 a + b 
 
 c 
 
 d 
 
 c + d 
 
 a + c 
 
 b + d 
 
 lV' 
 
 where a is the quadrant in which the mean falls, then b + d and c + d are clearly 
 each less than ±N. Let 
 
 t = (b + d)jN = i (1 - aO, < - (o + d)/N = A ( 1 . - «,), 
 
 then d/N= t t ' + Tit/?- + t 2 t./?-' j + . . . + TnT„'r" + (xxxvii) 
 
 is the equation to determine r the tetrachoric correlation, and Table XXIX gives 
 the values for given t , i.e. ^(1— a) of the following six tetrachoric functions 
 ii, To ... t 6 , and further of h, the ratio of the abscissa of the dichotomic line to the 
 standard deviation of the corresponding variate. 
 
 It is occasionally needful to go beyond the first six tetrachoric functions. In 
 this case the following finite difference formula is available : 
 
 t„ = /tp„T„_, - f] fl T M (xxxviii), 
 
 where p n =l/'Jn, q n = (n - 2)/Vn(rc — 1) (xxxix). 
 
 The following table gives the values of p n and q n from n = 7 to 24. 
 
 11 
 
 Vn 
 
 fn 
 
 •77152 
 
 n 
 
 H 
 
 In 
 
 7 
 
 •37796 
 
 16 
 
 ■25000 
 
 •90370 
 
 S 
 
 ■35355 
 
 •80178 
 
 17 
 
 •24254 
 
 •90951 
 
 .9 
 
 •33333 
 
 •82496 
 
 18 
 
 •23570 
 
 •91466 
 
 10 
 
 •31623 
 
 •84327 
 
 19 
 
 •22942 
 
 •91925 
 
 11 
 
 ■30151 
 
 •85812 
 
 20 
 
 ■22361 
 
 •92338 
 
 13 
 
 •28868 
 
 •87039 
 
 n 
 
 •21822 
 
 •92711 
 
 IS 
 
 •27735 
 
 •88070 
 
 22 
 
 •21320 
 
 •93048 
 
 U 
 
 ■26726 
 
 ■88950 
 
 2S 
 
 •20851 
 
 •93356 
 
 15 
 
 •25820 
 
 •89709 
 
 «l 
 
 •20412 
 
 •93638 
 
XXIX] 
 
 Introduction 
 
 li 
 
 Illustration (i). Find the correlation between dullness and developmental 
 defects as indicated in the following table for 26,287 children. 
 
 
 Without Defects 
 
 With Defects 
 
 Totals 
 
 Not Dull ... 
 Dull 
 
 22,793 
 1,186 
 
 1,420 
 888 
 
 24,213 
 2,074 
 
 Totals 
 
 28,979 
 
 2,308 
 
 26,287 
 
 Here 
 
 mtr = -078,898, 
 
 •in-jsr 
 
 / 
 T\ = 
 
 •1594,5, 
 
 / 
 
 T 2 = 
 
 •15268, 
 
 / 
 T 3 = 
 
 05431, 
 
 / 
 
 T 4 = 
 
 - 05137, 
 
 r 5 = 
 
 - -06755, 
 
 T, = 
 
 •00017, 
 
 £ = 
 
 1-35442. 
 
 mula 
 
 (xxxviii) fc 
 
 
 •05221, 
 
 / 
 
 T 8 = 
 
 •02486, 
 
 / 
 
 T 9 = 
 
 - 03185, 
 
 Tiu' = 
 
 - -03460. 
 
 t»' = «7 = -087,800, 
 Whence by interpolation from Table, p. 43 : 
 t, = -14712, 
 t„ = -14694, 
 t 3 = -05977, 
 r 4 =- 04262, 
 T , = - -00702, 
 To = - -00752, 
 h = 1-41253, 
 Proceeding to apply the difference formula (xxxviii) for four further functions 
 we have 
 
 Tr = -04770, 
 t 8 = -02985, 
 t, = - -02530, 
 t„, = - -03647, 
 Hence the equation for r is 
 
 •026,854 = -023,458r + -022,435r 2 + -003,246r 3 
 
 + •002,189r 4 + -004,527r 5 - -OOO.OOl?- 6 
 + •002.490?- 7 + 000,742r 8 + -OOO^Oer 9 
 + -0()l,262r 10 . 
 Whence we find r = 652 ± -009. 
 
 Illustration (ii). Find the tetrachoric correlation for the four-fold table given 
 for Houses in course of Erection on p. xxxv. Here 
 
 | (1 - a,) = t = Jfff = -260,080 ; f(I - a,) = t 8 ' -jffr" -009,157. 
 By simple linear interpolation, 
 
 t, = '32442, t,' = -02468, 
 t 2 = -14753, t/ = -04116, 
 t 3 =- -07766, t : ,' = -04599, 
 t 4 =- 11015, t 4 '= -03048. 
 
 J/2 
 
lii 
 
 Tables for Statisticians and Biometricians [XXIX 
 
 Hence the equation for r: 
 
 - 000,6093 = -008,007?- + -006,072?-- - -003,572?-" 
 - •003.357?- 4 . 
 Whence r=- 081 + 043. 
 Or, the association is not definitely significant. 
 
 Illustration (iii). Find the tetrachoric r for the Table of Bradford Parents : 
 
 Mother's Habits. 
 
 GO 
 
 • 1-4 
 
 M 
 
 
 Good 
 
 Bad 
 
 Totals 
 
 Good 
 
 994 
 
 67 
 
 1061 
 
 GO 
 
 Bad 
 
 159 
 
 476 
 
 635 
 
 fa 
 
 Totals ... 
 
 1153 
 
 543 
 
 1696 
 
 Here a brief experience will show the reader that to proceed by tetrachoric 
 functions will require a very large amount of labour. 
 
 We have 
 
 1(1- a,) = t = 543/1696 = '32017 ; \ (1 - a 2 ) = T „' = 635/1696 = -37441, 
 d/N= 476/1696 = -28066. 
 
 We have accordingly the following series of tetrachoric functions — the first 
 6 from the table, the remaining 18 from the difference formula. 
 
 djN 
 
 i(i-a) 
 
 n 
 
 T-2 
 
 T 3 
 
 U 
 
 Tj 
 
 ** 
 
 •28066 
 
 •32017 
 •37441 
 
 •35769 
 •37901 
 
 ■11817 
 
 •08581 
 
 -•11415 
 
 - -13887 
 
 - -09489 
 - -07178 
 
 ■05674 
 
 •08288 
 
 •08012 
 •06325 
 
 rj 
 
 *-8 
 
 r» 
 
 TlO 
 
 Til 
 
 n« 
 
 ra 
 
 nt 
 
 - -02963 
 
 - -05629 
 
 - -06913 
 - -05709 
 
 •01368 
 •04034 
 
 •06032 
 •05223 
 
 - -00324 
 
 - -02957 
 
 - -05294 
 
 - -04819 
 
 - -00401 
 ■02176 
 
 •04659 
 •04558 
 
 m 
 
 tig 
 
 *il 
 
 1-18 
 
 n» 
 
 Tao 
 
 m 
 
 T ti 
 
 •00922 
 - -01575 
 
 - -04103 
 
 - -04245 
 
 -•01304 
 •01103 
 
 ■03609 
 •03966 
 
 •01586 
 - -00723 
 
 - -03167 
 
 - -03714 
 
 - -01793 
 •00411 
 
 •02768 
 •03484 
 
 T 23 
 
 T 24 
 
 h 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •01944 
 - -00151 
 
 - -02407 
 
 - -03272 
 
 •46732 
 •32020 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
XXIX — XXX] Introduction liii 
 
 Considering only the equation as far as Everitt's Tables extend, we have 
 ( r ) = - -16079 + 13557c + -01014r 2 + 01585r»+ , 0068lr* + -00470?- 5 + -00507?'" = 0. 
 This leads to r = 93G5, but the series indicates that the terms are far from 
 converging rapidly. 
 
 The first 12 tetrachoric functions were then used, the last six being found 
 by the table of p n and q n above, and the value of r was found to be - 9152. 
 Then 18 functions were used and gave r = -9114. 
 
 Lastly 24 tetrachoric functions were used, and the equation below obtained, 
 which led to r = '9105. 
 
 $ ( r ) = - -16079 + -13557r + 01014r 2 + -01585^ + -00681?- 4 + -00470c 5 
 + OOSO?;- 11 + -00167r + -00395^ + O0055r« + 00315c 10 
 + OOOlOr" + -00255c 12 - -00009c 13 + -00212-c" - -00015 c 13 
 + 00174c 111 - 00014c 17 + -00143r 18 - -00011?- 19 + -00118?- 20 
 - -00057r 21 + -00096c 5 - - -00003c- 3 + -00079c 24 . 
 It will be seen that even with this very large amount of labour we cannot be 
 sure of having reached a final result*. To obviate this the following table was 
 constructed by Everitt, and there is no doubt that the extension of this table to 
 the whole range of correlation would much simplify the discovery of tetrachoric r t . 
 At present the calculation of high values of r t , for negative correlations is in hand. 
 
 Table XXX (pp. 52—27) 
 
 Supplementary Tables for determining High Correlations from Tetrachoric 
 Groupings. (P. F. Everitt, Biometrika, Vol. vm. pp. 385 — 395.) 
 
 Using the notation of p. 1, 
 
 d 1 /•»/•=>= - j —1— (a;2 + j, 2 _ 2raj ,) 
 
 "= == e 'i-H v * »' dxdy (xl) 
 
 in the case of a tetrachoric table, or 
 
 d _ J_ /""--to 1 
 
 N 
 
 = -Lf e'^Ydy 
 \l-lirh y 
 
 x. rr 1 f -hXdX ., , h-yr 
 where Y= — = e 2 , if t «- 
 ^2-rrJt 
 
 .(xli). 
 
 Vl -r 2 
 
 * Mr H. E. Soper working out this example draws my attention to the fact that convergence is closely 
 given by a form : r n =r„ (1 + n .c"), where n is the number of terms used and a and c are constants. 
 Hence (r„ - r„ ) (r n+2m - r„ ) = (r n+m - r K ) 2 , 
 
 r n + r n+2m _ 2)-„ t m 
 
 In our ease take n = 6, m = 6, and we find 
 
 °° r r , + r M -2r n 
 The value j- 24 is -9105. In this case « = -1567 and c = -7574, but we cannot assert that these would 
 be constants for all tables. If we use r V i, l"u and fjj, we find )•„ = -9102. 
 
liv 
 
 Tables for Statisticians and Biometricians [XXX 
 
 Hence r, h being known, Fis a tabled integral for each value of Y. Accordingly 
 
 by aid of Table II we know 
 
 w 
 
 V2 7 
 
 , and using a quadrature formula, d/N can 
 
 be found for each value of h, k and r. 
 
 Table XXX gives, for r = -80, -85, "90, -95 and 100, and values of h and k 
 proceeding by •], the values of d/N. For given values of h, k and djN, we can 
 then find r by interpolation from these tables. The process is far shorter than 
 that required by Table XXIX when we have to proceed to many terms. Un- 
 fortunately opportunity has not yet arisen for fully completing similar tables 
 for r negative and over "80. 
 
 Illustration. Determine the correlation in habits between Mother and Father 
 in Bradford. The data are 
 
 Mother. 
 
 - 
 Pm 
 
 
 Habits Good 
 
 Habits Bad 
 
 Totals 
 
 Habits Good 
 Habits Bad 
 
 994 
 159 
 
 67 
 ■176 
 
 1061 
 635 
 
 Totals 
 
 1153 543 
 
 1696 
 
 Here (b +d)/N= -32017, (c+d)/N= -37441, and therefore h = -46722, /„■ = -32020 
 from Table II. Also djN= 476/1696 = -28066. 
 
 Inspection of Table XXX shows that r will be likely to lie between -90 and -95. 
 
 We extract from the Table for d/N : 
 
 r=-90 
 
 h=i 
 
 ft = -5 
 
 *«=-3 
 
 •2943 
 
 •2784 
 
 •2728 
 •2602 
 
 r = -95 
 
 ft =-4 
 
 ft = -5 
 
 *=-3 
 
 k=-4 
 
 •3135 
 
 •2980 
 
 •2898 
 •2787 
 
 Hence : 
 
 »•= -90 
 
 ft =-4 
 
 ! 
 ft= -5 | 
 
 tm -32020 
 
 •2911 
 
 •2703 i 
 
 r= -95 
 
 ft = -4 
 
 ft=-o 
 
 k= -32020 
 
 •3104 
 
 •2876 
 
 Tin 
 
 r=-90 
 
 ft = -46722 
 
 /•= -32020 
 
 •2771 
 
 r = -95 
 
 ft =-46722 
 
 £= -32020 
 
 •2951 
 
XXXI] Introduction lv 
 
 We have now the desired h and k and have to interpolate djN = '280(36 between 
 •2771 and 2951. There results /•= -9099. 
 
 This is in excellent agreement with the value 9105 deduced from 24 terms, or 
 from the final value "9102, which can bo deduced from the 12, 18 and 24 term 
 values on the logarithmic rate of decrease hypothesis : see footnote p. liii. 
 
 Table XXXI (pp. 58—61) 
 
 The Y-Function. (J. H. Duffell : Biometrika, Vol. vn. pp. 43—47.) 
 
 It is well known that Y (x + 1 ) = *T (&•), and this property enables us to raise 
 or lower the argument of the T-function at will. As a rule in most statistical 
 investigations we require Y (x + \)jafe~ x . The following formula due to Pearson 
 will then be found to give Y(x + l)jx x e~ x with great exactness : 
 
 log (-^~) = -0399,0899 + i log x + -080,929 sin ^J 6 ^ . . .( x iii). 
 
 For values of x + 1 less than 6 and often for values less than 10, we find 
 log r (x + 1) or \ogY(p) from Table XXXI by reduction to^j between 1 and 2. 
 
 The reader's attention must be especially drawn as to the rules, given on 
 the Table itself, as to (i) characteristic, (ii) change of third figure of mantissa 
 at a bar, and (iii) the sign of the differences on the facing pages of the tables. 
 The difference tabled under 1144, say, is the drop from 1*144 to 1145. 
 
 Illustration (i). Find Y (2346). 
 
 By the reduction formula T (*2346) = r(l*2346)/*2346. 
 
 Hence log Y (*2346) = log Y (1*2346) - 1*370,3280. 
 
 log T (1*234) = 1-958,9685 A = - 1069, 
 
 log Y (1*235) = 1-958,8616 -6A = - [641*4]. 
 .*. log T (1*2346) = 1-958,9685 - [641] 
 = T-958,9044. 
 log T (2346)= 1*958,9044 
 -1*370,3280 
 
 •588,5764 
 Or F (*2346) = 3*87772. 
 
log r (87614) = 
 
 lvi Tables/or Statisticians and Biometricians [XXXII — XXXIII 
 
 Illustration (ii). Find r (8-7614). 
 
 T (8-7614) = 7-7614 x 6'7614 x 57614 x 47614 x 3-7614 x 2-7614 
 
 x 1-7614 T(l-7614). 
 
 •889,9401 + log T (1-7614) 
 
 •830,0366 
 
 •760,5280 
 
 •677,7347 
 
 •575,3495 
 
 •441,1293 
 
 •245,8580 
 
 Hence 
 
 = 4-420,5762 + log V (1-7614). 
 log T (1-7614) = 1-964,5473 + -4 [1113] 
 = 1-964,5918. 
 .-. log T (8-7614) = 4-385,1680. 
 
 T (8-7614) = 24275-49. 
 
 Table XXXII (pp. 62—63) 
 
 Table XXXIII, A and B (p. 64). 
 
 Subtense from Arc and Chord in the case of the Common Catenary. (Julia Bell 
 and H. E. Soper: see Biometrika, Vol. vni. pp. 316, 338, and Vol. ix. pp. 401—2.) 
 
 If c be the parameter of the common catenary, then we know that 
 
 y = c cosh u (xliii), 
 
 where u — xjc is its equation. 
 If the chord be 2x, then 
 
 subtense/chord = (y — c)/(2x)\ 
 
 = (sinh frt ) 3 > (xliv), 
 
 u J 
 
 arc/chord = — ( x l v )> 
 
 arc — chord _ sinh u — u _ /3 / l - \ 
 
 chord = u = 100 { } ' 
 
 subtense _ (sinh \iif _ a / \ "\ 
 
 chord ■■ "T~~~l00 ( ; " 
 
 Corresponding values of a and /3 are given in the Tables XXXII and XXXIII. 
 
XXXIII A and B] Introduction lvii 
 
 Illustration (i). A cable of 132.5 is suspended over the gap between two 
 towers of the same height, 115 feet apart. What will be the droop of the cable ? 
 
 g-100< 18 ^* 18) -ll-52. 
 
 llo 
 
 Table XXXIII A, gives us a = 21-62 = 100 subtense/chord. 
 .-. subtense = 2162 x 115 
 
 = 24-86. 
 Thus the droop is 24-86 ft. 
 
 Illustration (ii). A catenary arch is to have a rise of 50 ft., centre line 
 measurement, and a span of 200. What is the length of the centre line ? 
 
 a =100 x 50/200 = 25-0, 
 but a = 25 by Table XXXII gives /3 = 151. 
 
 100 (arc — chord)/chord = lo'l. 
 .-. arc = 2302 ft.* 
 
 Illustration (iii). For some races the shape of the nasal bridge is very ap- 
 proximately a catenary. Thus if the nasal chord from dacryon to dacryon be 
 measured and also the tape measure from dacryon to dacryon, we obtain the 
 mesodacryal index @. The tables enable us to pass to the mesodacryal index a, 
 and thus ascertain the nasal subtense, which is slightly harder of direct measure- 
 ment than the arcual or tape measure. 
 
 In the skull of a male gorilla the mesodacryal chord was 22'6 mm., and the 
 mesodacryal arc 30 mm. Determine the mesodacryal subtense 
 
 S - 100 3 °- 22 ' 6 - 1Q0 * 7 " 4 - 32-74 
 / *" 100 22-6 _ 226 - d27 *- 
 
 Hence, from Table XXXII : 
 
 a = 38-84 = 100 subtense/22'6. 
 
 .-. subtense = 22'6 x "3884 = 8-8 mm. 
 
 The actual value of the mesodacryal subtense measured ou the skull was 
 
 8'7 mm. 
 
 Abac XXXIV (p. 65) 
 
 Diagram to find the Correlation Coefficient r from Mean Contingency on the 
 Hypothesis of a Normal Frequency Distribution. (Pearson : Drapers' Company 
 Research Memoirs, No. 1, "On the Theory of Contingency.") 
 
 If n plJ be the frequency in the cell of the pth column and qth row of a correlation 
 or contingency table, and m p be the total frequency in the pth column, n q the 
 
 * Should there be any use for this table for constructional purposes, which there ought to be when the 
 value of the catenary arch is more fully recognised, I will in a later edition of this work give the value 
 of u corresponding to each f>, so that the parameter c can be at once read off and the form of the arch 
 readily plotted. It might also be desirable to give the values of a and /3 to two decimal places. We 
 have these data in our MS. copies. 
 
 B. ft 
 
lviii Tables for Statisticians and Biometricians [XXXIV 
 
 total frequency in the gth row, and N the whole population, then if the two 
 variates are independent, the frequency to be expected in the p, qth. cell will be 
 
 A7" v 1l i v mp — n 1 m P 
 N N ~ N ' 
 
 and the observed excess over this, i.e. n pq — 9 „ p , is termed the 'contingency' in 
 
 this cell. The total contingency must be of course zero, i.e. the sum of all the 
 cell contingencies. If, however, we take only the positive excess contingencies and 
 
 divide them by N, i.e. yjr = -== 2+ I n pg ? w £ ) > we obtain the so-called f mean 
 
 contingency.' On the assumption of normal frequency distribution it is possible 
 to deduce the actual correlation from yfr, provided that the cells are sufficiently 
 small for summation to replace integration. As in practice our cells are hardly 
 likely to exceed 8x8, and may be smaller and unequal in area, we shall generally 
 find a value below that of the true correlation, even if the system be accurately 
 normal. A corrective factor corresponding to the class-index correlation has not 
 yet been theoretically deduced. But experience seems to show that to add half the 
 correction due to class-index correlations gives good results. That is to say, that, 
 if r$ be the correlation found from the Abac, p. 65, and r xC and r xC be the class- 
 index correlations for x and y, we should take for the true correlation: 
 
 r = r+ + i 
 
 -i 
 
 r, 
 
 + 
 
 r xCjyC y 
 
 * 
 
 .(xlviii). 
 
 I'xCjyCyl 
 
 It is clear that this is the same thing as taking the mean of the crude mean 
 contingency correlation and its value as corrected for the class-index correlations. 
 The following illustrations may indicate the method of procedure. 
 
 Illustration (i). Find the correlation from the table on p. lix by mean 
 contingency. The first number in each cell is the frequency reduced to 1000, the 
 second number is that to be expected on the basis of independent probability, and 
 the third is the mean contingency of the cell. 
 
 The sum of the positive contingencies is 94136, hence the mean contingency 
 is '094. Entering the diagram with -094 on the base scale, we pass up the vertical 
 to the curve, and then along the horizontal to the left hand scale and find r^ = - 285. 
 
 The class-index correlation for the vertical marginal frequency is r yC ='9645, 
 and that for the horizontal marginal frequency is - 9624*. Hence 
 
 r*l(r xCx r yCy ) = -m, 
 
 and r = £ (-307 + -285) = -296. 
 
 The table is actually a true Gaussian distribution with correlation equal 
 to '300. 
 
 * Biometrika, Vol. ix. p. 218. 
 
XXXIV] 
 
 Introduction 
 
 First Variate A. 
 
 lix 
 
 
 J 2 
 
 3 
 
 4 
 
 5+6 
 
 7 
 
 8 
 
 Totals 
 
 
 i 
 
 404 1716 
 
 (1-224) (10-948) 
 2-816 6-2n 
 
 7-55 
 
 (8-976) 
 -1-426 
 
 3-30 
 
 (6-120) 
 -2-820 
 
 0-91 0-92 
 
 (2-346) (3-434) 
 -1-436 -2-514 
 
 12 
 
 (0-952) 
 -0-832 
 
 34 
 
 
 % 
 
 17-41 123-59 
 
 (10-836) (96-922) 
 6-574 26-668 
 
 79-76 
 
 (79-464) 
 0-296 
 
 44-64 
 
 (54-180) 
 -9-540 
 
 14-61 
 
 (20-769) 
 -6-159 
 
 17-67 
 
 (30-401) 
 - 12-731 
 
 3-32 
 
 (8-428) 
 - 5-108 
 
 301 
 
 »1 
 
 01 
 
 8-86 93 00 
 
 3 (10-224) (91-448) 
 i -1S64 1'552 
 
 78-31 
 
 (74-976) 
 
 52 04 
 
 (51-120) 
 0-920 
 
 19-20 
 
 (19-596) 
 -0-396 
 
 26-40 
 
 (28-684) 
 -2-284 
 
 619 
 
 (7-952) 
 -1-762 
 
 284 
 
 •g 
 
 o 
 
 c 
 
 4 
 
 2-83 
 
 (4-932) 
 -2-102 
 
 37-73 
 
 (44-114) 
 
 37-24 
 
 (36-168) 
 1-072 
 
 27-51 
 
 (24-660) 
 2-850 
 
 10-95 
 C9-453) 
 1-497 
 
 16-31 
 
 (13-837) 
 2-473 
 
 4-43 
 
 (3-836) 
 0-594 
 
 137 
 
 GO 
 
 5 + 6 
 
 1-62 
 
 (3-780) 
 -2-160 
 
 25-21 
 
 (33-810) 
 -8-600 
 
 27-75 
 
 (27-720) 
 0-030 
 
 22 09 
 
 (18-900) 
 3-190 
 
 9-26 
 
 (7-245) 
 2-015 
 
 14-64 
 
 (10-605) 
 4-035 
 
 4-43 
 
 (2-940) 
 1-490 
 
 105 
 
 
 7 
 
 102 
 
 (3-528) 
 -2-508 
 
 19-50 
 
 (31-506) 
 -12-056 
 
 24-47 
 
 (25-872) 
 -2 -402 
 
 21-39 
 
 (17-640) 
 3-750 
 
 9-58 
 
 (6-762) 
 2-818 
 
 16-36 
 
 (9-898) 
 6-462 
 
 5-68 
 
 (2-744) 
 2-936 
 
 98 
 
 
 8 
 
 0-22 5-81 
 
 (1-476) ; (13-202) 
 -1-256 1 -7-S92 
 
 8-92 
 
 (10-824) 
 -1-904 
 
 903 
 
 (7-380) 
 1-650 
 
 4-49 
 
 (2-829) 
 1-661 
 
 8-70 
 
 (4-141) 
 4-559 
 
 3-83 
 
 (1-148) 
 2-682 
 
 41 
 
 
 Totals t 36 
 
 322 
 
 264 
 
 180 
 
 69 
 
 101 
 
 28 
 
 1000 
 
 Illustration (ii). Find r^ by mean contingency for the table on p. lx: 
 
 The sum of the positive contingencies is 169'846, or we have mean contingency 
 ■v|r = -170, whence the diagram leads us to rj, = '480. The marginal frequencies are 
 the same as in Illustration (i). Thus we have 
 
 r = i (-517 + -480) = -499. 
 
 The table gives actually a true Gaussian distribution with correlation -500. 
 It will be seen from Illustrations (i) and (ii), that if the distribution be Gaussian, 
 even if the marginal frequencies are in fairly irregular groupings, ?-^ will be 
 reasonably close to the true contingency, and corrected as suggested above will 
 give excellent results. 
 
 hi 
 
lx Tables for Statisticians and Biometricians [XXXV — XLVI 
 
 First Variate A. 
 
 
 
 i 
 
 2 
 
 3 
 
 4 
 
 5 + 6 
 
 7 
 
 8 
 
 Totals 
 
 
 i 
 
 7 38 
 
 (1-224) 
 6-156 
 
 19-85 
 
 (10-948) 
 8-902 
 
 ► 4-94 
 
 (8-976) 
 -4-036 
 
 1-38 
 
 (6-120) 
 -4-740 
 
 0-26 
 
 (2-346) 
 -2-086 
 
 018 
 
 (3-434) 
 -3-254 
 
 001 
 
 (0-952) 
 -0-951 
 
 34 
 
 
 2 
 
 20 58 
 
 (10-836) 
 9-744 
 
 145-47 
 
 (96-922) 
 48-548 
 
 78-94 
 
 (79-464) 
 -0-524 
 
 35-98 
 
 (54-180) 
 -18-200 
 
 9'72 
 
 (20-769) 
 -11-049 
 
 9-27 
 
 (30-401) 
 -21-131 
 
 104 
 
 (8-428) 
 -7-388 
 
 301 
 
 0) 
 
 a 
 
 6 01 
 
 (10-224) 
 
 -4-214 
 
 93-63 
 
 (91-182) 
 2-182 
 
 85-41 
 
 (74-976) 
 10-4S4 
 
 54-34 
 
 (51-120) 
 8-220 
 
 18-59 
 
 (19-596) 
 -1-006 
 
 22-33 
 
 (28-684) 
 -6-854 
 
 3-69 
 
 (7-952) 
 -4-262 
 
 284 
 
 a 
 
 4 
 
 1-26 
 
 (4-932) 
 -3-672 
 
 31-81 
 
 (44-114) 
 -12-304 
 
 39-49 
 
 (36-168) 
 3-822 
 
 3103 
 
 (24-660) 
 6-370 
 
 12-29 
 
 (9-453) 
 2-837 
 
 17-36 
 
 (13-837) 
 3-523 
 
 3-76 
 
 (3-836) 
 - -076 
 
 137 
 
 o 
 
 0) 
 
 02 
 
 5 + 
 
 0-53 
 
 (3-780) 
 - 3-250 
 
 1811 
 
 (33-810) 
 -15-700 
 
 27-79 
 
 (27-720) 
 0-070 
 
 2514 
 
 (18-90) 
 6-240 
 
 1109 
 
 (7-245) 
 3-845 
 
 17-62 
 
 (10-605) 
 7-015 
 
 4-72 
 
 (2-940) 
 1-780 
 
 105 
 
 
 7 
 
 0-22 
 
 (3-528) 
 -S-308 
 
 1102 
 
 (21-556) 
 -20-536 
 
 21-59 
 
 (25-872) 
 -4-282 
 
 23-66 
 
 (17-640) 
 6-020 
 
 11-86 
 
 (6-762) 
 5-098 
 
 21-89 
 
 (9-898) 
 11-992 
 
 7-76 
 
 (2-744) 
 5-016 
 
 98 
 
 
 S 
 
 002 
 
 (1-476) 
 -1-456 
 
 2-11 
 
 (12-202) 
 -11-092 
 
 5-84 
 
 (10-824) 
 '-4-984 
 
 8-47 
 
 (7-380) 
 1-090 
 
 519 
 
 (2-829) 
 2-361 
 
 12-35 
 
 (4-141) 
 8-209 
 
 7 02 
 
 (1-148) 
 5-872 
 
 41 
 
 
 Totals 
 
 36 
 
 322 
 
 264 
 
 180 
 
 69 
 
 101 
 
 28 
 
 1000 
 
 Tables XXXV— XLVI (pp. 66—87) 
 
 Criteria for Frequency Types and Probable Errors of Frequency Constants. 
 (A. J. Rhind: Biometrika, Vol. vil. pp. 127—147 and pp. 386—397.) 
 
 It is desirable to consider all these tables under one heading, namely the 
 general investigation of frequency type and of the probable errors of frequency 
 constants. 
 
 The main lines of Pearson's theory of frequency are involved in the following 
 statements: 
 
XXXV— XLVI] Introduction lxi 
 
 If the differential equation to the uni-modal frequency distribution be 
 
 1 di/ x — a , . . . 
 
 ydx f{x) 
 
 we may suppose f(x) expanded in a series of powers of x, and so 
 
 1 dy _ x — a ... 
 
 y dx c + c x x + c 2 x 2 + ... + c n x n + 
 
 then a, c , c lt c 2 , ... c H ... can be uniquely determined from the 'moment co- 
 efficients' of the frequency distribution. These constants are functions of certain 
 other constants /3 1; /3 2 — 3, /3 3 , /3 4 — 15, ... which vanish for the Gaussian curve, and 
 are small for any distribution not widely divergent from the Gaussian. Further 
 c , Ci, c 2 ...c n ... converge, if, as usual, these constants are less than unity, the 
 factors of convergence being of the order V/3-constant. As a matter of fact c„ 
 involves the (n + 2)th moment coefficient, and thus we obtain values of the 
 c-constants subject to very large errors, if we retain terms beyond C,. If we stop 
 at c 2 then our differential equation is of the form 
 
 ldy = x-a y . 
 
 ydx Co + dx+CvX* 
 
 and we need only & = (ifff** and /3 2 = /J. 4 /fi«-, where fu, /li 3 , /j. 4 are the second, third 
 and fourth moment coefficients about the mean. 
 
 1 dij oc ■■ • ct 
 
 If we take the form — r- — . we reach the Gaussian, in which each con- 
 
 ydx c 
 
 tributory cause-group is independent, and if the number of groups be not very 
 
 large, each cause-group is of equal valency and contributes with equal frequency 
 
 results in excess and defect of its mean contribution. If we take — ^ = — , 
 
 y dx c + CiX 
 
 then each contributory cause-group is still of equal valency and independent, but 
 
 does not give contributions in excess and defect of equal frequency. 
 
 Finally if we take - -M- ■ ■ : , then contributory cause-groups are 
 
 ' ydx c + CiX + dx* J -. 
 
 not of equal valency, they are not independent, but their results correlated, and 
 
 further contributions in excess and defect are not equally probable. The use of this 
 
 1 di] sc — a 
 
 form - -~= was adopted to allow of this wide generalisation of the 
 
 y dx c + CiX + c 2 ar r ° 
 
 Gaussian hypothesis. 
 
 If we adopt it, every /3-constant is expressible by means of the formulae : 
 
 A,(even) = ( M +l){i / S n _ 1 + (l+ia) y 8„_ 2 j/(l-H«-l)«) 0"). 
 
 £„ (odd) =(n+ 1){£& A l -. + (l+i«)/S«- 2 }/(l-H»-l) a ) 0»i). 
 
 where a = (2ft-3ft-6)/08,'+3) (liv), 
 
 in terms of lower /S-constants. 
 
lxii Tables for Statisticians and Biometricians [XXXV — XLVI 
 
 Table XLII, (a) — (d) gives the values of /3 8 , /3 4 , /3 6 and /3„ in terms of /?, and 
 /3». Hence as soon as /3, and /3 2 are calculated we can find the numerical values of 
 
 & = WsIh-2, &"*/%//*»*. 0, = ftuslfif, /3 = fi s /fi 2 4 (lv), 
 
 theoretically. Although these values will not be those which would be absolutely 
 deduced from the data themselves, they will, considering the large probable errors 
 of /i 5 , fi e , ft? and fi s be reasonable approximations to them. The values of the 
 probable errors of /3, and /3 2 are determinable by formulae involving /3,, /3 2 ... /3 8 . 
 
 From these formulae, Tables XXXVII and XXXVIII, giving the values of 
 V-ATXp, and ViVSp, have been constructed. Hence multiplying by ^ from Table V, 
 we obtain 
 
 •67449 - , -67449^ 
 
 -—2^ and — j=r S ft 
 
 ViV Pl >JN 
 
 the probable errors of /3, and y8 2 . 
 
 If we add to the standard deviations of /8, and /3 2 , the correlation between 
 deviations in /3, and /3 2 , namely R^p,, which correlation is given in Table XXXIX, 
 we can find the probable errors of any functions of /3, and /3„. Two such important 
 functions are the distance d from mean to mode and the skewness sk of the 
 distribution. The probable errors of d and sk can be found from Tables XL and 
 XLI respectively, the former by multiplying the tabulated value viVSd/cr by crx^ 
 (from Table V), and the latter by multiplying the tabulated value \/NS S k by %i 
 (from Table V). 
 
 Thus far we have only been concerned with the constants which describe 
 certain physical characters of the frequency distribution without regard to the 
 type of curve suited to the distribution. We now turn to the latter subject. 
 
 It is known that the type of frequency depends upon a certain criterion k. 2 . 
 Hence near the critical values of « 2 more than one type of curve may describe the 
 frequency witliin the limit of the probable error of « 2 . Table XLIII gives the 
 probable error of * 2 , if the entries in that table be multiplied by the ■& of 
 Table V. 
 
 The following are the series of Type curves which arise according to the value 
 of the criteria 
 
 «,-2&-8A-6 (Ivi), 
 
 , ft (ft + a? ( ivii) 
 
 /9 2 is by necessity >f&. Hence for our curves all possible values of f3 u /3 2 lie in 
 the positive quadrant between the lines /3 2 = J/3, and & = -^-/3, + 1, the latter being 
 if we go to /3 8 the limit of failure of Type IV, for its fi 6 becomes infinite. Beyond 
 the latter line distributions are heterotypic. 
 
XXXV — XLVI] Introduction lxiii 
 
 Criterion Type Equation to Curve 
 
 *» = ft-0, &>S VII y = y * 
 
 ( i+ 3 < iviii )- 
 
 K ., = Q ft = 0, ft = 3 Normal 2/ = 3/ e 2ff2 (lix). 
 
 «, = o ft = o, ft<3 n E y-8>(i-5) (*)• 
 
 * 2 = ft = 0, A<1'8 II b 2/ = yo _J_ i (ki). 
 
 - 1/ tan -1 - 
 S>0<1 IV ,j- s ,'- — ~ (Ixii). 
 
 *«!• V y=y e-yl x x-P (Ixiii). 
 
 a;,, > 1 < ao VI y = y (<c-a) mi /x'" ! (lxiv). 
 
 *,= », i.e. 2/8,-3/8,-6 = III y -^''-(l +-)* (hcv). 
 
 k 2 < 0, i.e. negative. Below /=0 I t y=y h-|- — ) (1 ) (lxvi). 
 
 K ,<0. Inside /=0 J T y = y (l + ^J' 1 /(l-~)'"\..(\xyii). 
 
 k, < 0. Above /= U x y = y„ -— — . . .(lx viii). 
 
 K) K) 
 
 For « 2 < 0,/=0 represents the biquadratic 
 
 ft (8ft - 9ft - 12) (4ft - 3ft) = (10ft - 12ft - 18V (ft + 3)f . . .(lxix). 
 
 Type I is thus divided into three subclasses, limited range curves, J-shaped 
 curves and U-shaped curves. 
 
 Diagram XXXV enables the reader at once to find the type appropriate to his 
 distribution, and Diagram XXXVI gives the same figure on a much larger scale to 
 indicate the changes that occur with large values of ft and ft. 
 
 Knowing the values of ft and ft the computer can fix his point on the 
 Diagram XXXV, but he may come so near a critical point or line, that one curve 
 may appear as reasonable as another. It is clear, for example, that in the 
 neighbourhood of the Gaussian point 0, he might possibly use II, l t , III, VI, V, 
 
 * The branch of the cubic k 2 = 1 with which we are concerned passes through the Gaussian point, 
 at which p = oo , and along this branch p is always >5. 
 
lxiv Tables for Statisticians and Biometricians [XXXV — XLVI 
 
 IV or VIII, and as all these types at that point transform into each other, the 
 forms actually deduced will be almost identical, however different their equations. 
 But there will be other occasions when doubt as to the use of the simpler of two 
 curves may arise; for example if /Sj = '8, /3 2 = 4-15, are we justified in using 
 Type III as simpler than Type I ? 
 
 Now we have to remember that the variates &, /3 2 form a frequency surface, of 
 which the equation is 
 
 M - x * ( ftl . H. _ afiftftftflA 
 Z = - e 2(1-R%f>j\2tf + Xp* 2^ s ft J (ha) 
 
 and that the contours of this surface projected onto the j3 1 , /3 2 plane of 
 Diagram XXXV form a series of similar and similarly placed ellipses. Within 
 any one of these ellipses a certain amount of the volume of the &, /3 2 - frequency 
 lies, and therefore if this system of contours were properly placed round the /Si, /S 2 
 point on Diagram XXXV we could tell at once the probability that the given point, 
 owing to random sampling, should fall outside a given elliptic contour. 
 
 The ellipse which has for principal semi-axes 11772! and ri772 2 , where £, and 
 2 2 are the principal axes of the ellipse : 
 
 i - 1 (§L+£L- ?^A&\ rl n 
 
 R\p t VSft 2 2ft 2 2ft 2ft / 
 
 covers an area on which stands just one half the frequency, i.e. it is the ellipse 
 determined by the generalised probable error of two variates (see Table X, p. 24). 
 
 The semi-minor axis l'1772!and the semi-major axis l'1772 2 of this "Probability 
 Ellipse " multiplied by \/N are given in Tables XLIV and XLV respectively, and 
 Table XLVI gives the angle in degrees between the major axis of this ellipse and 
 the axis of /9 a . It is thus possible to construct from Tables XLIV — XLVI the 
 " probability ellipse" round a given point /3,, /3 2 > and to test the area within which 
 half the frequency lies. If the probability required be not \, but much less, then 
 we note that the probability, that a point will lie outside the ellipse with semi- 
 axes X2j and \2 2 is P = e ~ ^ x . 
 
 ,_ -67449 
 Let \2 2 =1177v / iV r 2 2 x ^p (lxxii), 
 
 or 
 and 
 
 Hence 
 
 Accordingly 
 
 tu 2 *■ *■ 
 
 "*" */Njq 
 
 \ 2 = 
 
 2 x -630,672, 
 
 p 
 
 — g-gX-315,336 
 
 logP = 
 
 ---q x -136,949. 
 
 9 = 10 
 
 P = -0427, 
 
 g = 12 
 
 P = 0227, 
 
 2=15 
 
 P = -0088, 
 
 2 = 20 
 
 P = -0018. 
 
XXXV— XLVI] Introduction lxv 
 
 Hence we select the grade of working probability we require, roughly 1 in 23, 
 1 in 44, 1 in 114 or 1 in 555, and this determines q. Divide iVthe total frequency 
 by q and look up in Table V, Xi f° r ^ll> multiply this by the 1177 ViV2 2 of 
 Table XLV, p. 84, and we obtain the semi-major axis of the required ellipse- 
 Multiply the same Xi by 1177 ViV2i of Table XLIV and we have the semi-minor 
 axis. We can then construct round the point ft, ft this ellipse and ascertain if 
 it cuts critical boundaries on Diagram XXXV, p. 66, the orientation being given by 
 Table XLVI, p. 86. Less accurately, but for practical purposes effectively, we may 
 work on Diagram XL VII, p. 88. We proceed just as before, to select our q and so 
 determine our \2 2 and \2i. Then we take the ratio of 2i/2 2 . We now pick out 
 of the ellipses on p. 88 the set having the nearest 2,/2 2 value and out of this set 
 the ellipse with the nearest \2 2 value of its semi-major axis. This ellipse or if 
 necessary an interpolated one is transferred to tracing paper and placed with its 
 centre at the given point (ft, ft), and its major axis touching the dotted curve. If 
 this ellipse does not cut a critical line, we can be certain that to the given degree of 
 probability the curve is of the type into the area of which its (3 U ft point falls. 
 
 It would be impossible in an Introduction to these tables to give the whole 
 theory of frequency curves*. But one or two formulae may be usefully placed 
 here for reference. 
 
 Distance d from mode to mean = , .. _ * ttt (lxxiii), 
 
 2 (oft, - 6ft -9) 
 
 Skewness sk = g '_^ & + _ (lxxiv), 
 
 JVV = ft (4ft - 24ft + 36 + 9ftft - 12ft + 35ft) (lxxv), 
 
 NZtf = ft - 4ft ft + 4ft 3 - ft/ + 16ft ft - 8ft + 16ft) (lxxv Us), 
 
 SftSfl.Eftp, = 2ft - 3ftft - 4ftft + 6ft 2 ft + 3ftft- 6ft + 12ft 2 + 24ft (lxxvi). 
 
 It is from the above formulae that the Tables now under discussion have been 
 calculated. 
 
 Illustration. The following percentages of black measured with a colour top 
 are stated to occur with the recorded frequencies in the skin colour of white 
 and negro crosses f. 
 
 Discuss the type of frequency curve suited to the data and determine the chief 
 physical constants of the distribution and their probable errors. 
 
 * The general theory is given in " Skew Variation in Homogeneous Material," Phil. Trans. Vol. 186 
 (1895), A, pp. 343—414: Supplement, Vol. 197 (1901), A, pp. 443—459; "On the Mathematical Theory 
 of Errors of Judgment," Phil. Trans. Vol. 198 (1902), pp. 274—279 ; "Das Fehlergesetz und seine 
 Verallgemeinerungen durch Fechner und Pearson," A Rejoinder, Biometrika, Vol. iv. pp. 169—212. 
 " Skew Frequency Curves," A Rejoinder to Professor Kapteyn, Ibid. Vol. v. pp. 168—171, and " On the 
 curves which are most suitable for describing the frequency of Random Samples of a Population," 
 Ibid. Vol. v. pp. 172—175. 
 
 t Extracted from C. B. Davenport, Heredity of Skin Color in Negro-White Crosses, Carnegie 
 Institution of Washington, 1913. 
 
 B. * 
 
lxvi Tables for Statisticians and Biometricians [XXXV — XLVI 
 
 The working origin was taken at 20, the centre of the group 18 — 22. The 
 centre of the first group at 1*47 % is i(20 - ]-47) = 3706 on the negative side of 
 the working origin and may be taken to contribute — 56, + 206, — 763, + 2830 
 
 Percentage 
 
 Frequency 
 
 Percentage 
 
 Frequency 
 
 0— 2* 
 
 15 
 
 43-47 
 
 45 
 
 3— 7 
 
 120 
 
 48—-,.' 
 
 24 
 
 8— 111 
 
 139 
 
 68—57 
 
 14 
 
 13—17 
 
 157 
 
 58—0..' 
 
 fi 
 
 18—32 
 
 158 
 
 63—67 
 
 3 
 
 23—27 
 
 139 
 
 68- ; .' 
 
 3 
 
 28--:: 
 
 117 
 
 78 — 77 
 
 2 
 
 33—37 
 
 92 
 
 78- 8$ 
 
 2 
 
 38—42 
 
 50 
 
 88 -87 
 
 — 
 
 to the first, second, third and fourth moments respectively. The working unit 
 being 5 °/ o , the raw moment coefficients arc : 
 
 vfm -567,2191, ■/,'« 7703,4990, 
 
 v,' = 28-982,5042, vl = 253-268,8730. 
 
 Whence transferring to mean and correcting, we have 
 
 Mean = 228361, a = 270156, 
 
 ^,-7*298,428, fi 3 = 16-238,780,. ^=198-909,921. 
 
 These lead to 
 
 ft - -678,295, ft = 37 34,202, 
 
 «, = 2ft - 3/3, - 6 = - -566,483, k, = - 1-052,180, 
 
 tk = 495,087, Distance from Mean to Mode = d = 1-337,508. 
 
 These values, except the mean, are all in working units. Therefore in per- 
 centages of black : 
 
 a= 13-5078 and d = 6-6875. 
 
 We can now find the probable errors of these constants. We first want ^, 
 from Table V, but 1086 is outside the limit of n. We therefore take ^ 2 for 543 
 and have ^ = -02047, and we find ^ 2 = -014,47. We can repeat our constants with 
 their probable errors 
 
 Mean = 228361 ± 2765,- a = 135078 ± -1955. 
 Then from Table XXXVII, 
 
 ft = 3-7: ^2,, = 4-70 + !$$[2] = 4-71, 
 
 ft = 3-8 : ViV2 ft = 5-05 + f§§ [2] = 5-06. 
 
 Hence for ft = 37342 : >JN2 $< = 4-71 + f$& [35], 
 
 V J r 2 ft = 4-83. 
 * 4 at and 11 at 2, giving a mean at 1-47 %. 
 
XXXV — XLVI] Introduction lxvii 
 
 Similarly from Table XXXVIII : 
 
 ft - 3-7 : \/Nl^ = 1202 - §{$[66] = 11-65. 
 ft = 3-8 : VFSfl, = 1360 - }$ [72] = 13-19. 
 Hence for ft = 37342: V^S^ = 11-6.5 + ^^[1-54], 
 
 \fNSp, - 12-18. 
 Thus we find, multiplying by ^ : 
 
 ft= -6783 + -0989, 
 ft = 3-7342 ±-2493. 
 
 It is clear that the ft and ft are significantly different from the Gaussian 
 ft = and ft = 3. 
 
 We next turn to the skewness, using Table XLI : 
 
 ft = 3-7: \/NZ sk = 1-98 + fff [21] = 2-10, 
 ft =3-8: VF2 8i = 1-88 + ffj [16] =1-97. 
 Hence for ft - 37342 : </N2* = 210 - flfa [13] 
 
 = 206. 
 Thus the skewness = -4951 ± -0422, or the distribution is significantly skew. 
 Passing to Table XL for the probable error of d, we have 
 
 ft = 3-7 : ViV tajo = 214 + fgf [20] = 2-25, 
 ft = 3-8: *JNZ d l* = 2-03 + $$ [17] = 213. 
 Hence for ft=3-7342: ViV2 rf /<x = 2-25- T ^[12] 
 
 = 2-21. 
 Thus Probable Error ofrf = X ,x<rx 221 = 6111, 
 
 and d = 6-6875 ±6111. 
 
 The probable error of K t is to be found from the relation : 
 
 (VF2,,) 2 = 4 (Viv% a ) 2 + 9 (V^2 ft ) 2 - 12 (V^) (Vtf 2 fc ) x ii ft A . (Ixxvii) 
 Thus we require ii PlP . Table XXXIX, p. 72, will provide this : 
 ft = 3-7 : iJ ftft = -892 + m [5] = -895, 
 ft = 3-8 : R„ lh = -893 + ffflf [5] = -896. 
 Hence for ft = 3'7342 we may take R Plh = -895. Accordingly 
 (V^S»,) 2 = 593-4096 + 2099601 - 631-8278 
 = 171-5419. 
 Or, VFS,, = 130974. 
 
 Hence p.e. of «, = X i x ^iVX, = 2681 , 
 
 or, k, = - -566,483 ± "2681. 
 
 ii 
 
lxviii Tables for Statisticians and Biometricians [XXXV — XL VI 
 
 It would look therefore as if Kj were significantly negative, but it is just possible 
 that «, might be zero. Such a big probable error for k, suggests our being 
 in the neighbourhood of a critical limit. This is verified on examining Table 
 XLIII to find the value of ViV2„. 2 . We see that it is over 80, and we thus 
 conclude that the probable error of «- 2 may lie between 1 and 2. Thus we cannot 
 be definitely certain of the sign or magnitude of k 2 , when we are even relatively in 
 the neighbourhood of k 2 = 0. 
 
 If we turn to Diagram XXXV (p. 66) we see that the point y8i = -678 and 
 /3 2 =3734 is not very close but is approaching the line along which Type III is 
 applicable, and this is the source of the disturbance noted. 
 
 We accordingly try to measure the probability that Type III would be as 
 satisfactory as Type I within the area of which our &, /3 2 values actually lie. 
 We must take M77V.V2, from Table XLIV and ll77v / iV ; 2 2 from Table XLV. 
 We have 
 
 A = '6789, & = 3-7 then M77VF2, = 23 
 „ A-3'8 M n =2 -5. 
 
 Hence for & = 37342, MttVES, = 2"3 + &% [2] = 2-37. 
 
 Again: ft =3-7 l-177VF2 a = 16 -$$[1] = 15 ' 43 > 
 
 /3 2 = 3-8 1-mVyS, -18- {|f [1]- 17-48, 
 & = 3-7342 M77VT2 2 = 15-43 + T 3 ^ [2] - 1611. 
 
 Thus: 2,/2 2 = 2-37/1611 = -147 = -15, say. Or, if we turn to Diagram XLVII 
 (p. 88), our system of ellipses is half-way between the 3rd (2 1 /2 2 = , 14) and the 
 4th (2i/2 2 = -16). Now if such a system of ellipses be traced off and centred at 
 the point /S, = '678, fi 2 = 3 - 734 on Diagram XLVII to the right and then the 
 major-axis be brought into parallelism with the dotted lines, we find that the 
 biggest of these ellipses \2 2 = 5 fails to reach the critical line III. But the 
 semi-major axis of the probability-ellipse is 11772 2 = 16-ll/v / iV= '493. Hence 
 we must conclude that it is more probable that the curve is of Type I than of 
 Type III. This is readily determined and is usually sufficient guide. Actually 
 the value of \2 2 must be about '6 before we get an ellipse to approximately touch 
 the Type III line. But 2 2 = -493/1-177 = -419, and accordingly \ = -6/-419 = 1-432, 
 which gives P = e~* x ='36 nearly, or the odds are 16 to 9 that the point would 
 not lie outside this contour. But if it did lie outside this contour, the chance 
 of its being on or over the Type III line corresponds to only a very small section 
 of the total frequency outside this contour. If we invert the problem and put the 
 system of ellipses on the nearest point of the Type III line we find that the odds 
 are very much in favour of the point ft = "678, ft = 3-734 lying outside such a 
 system. On the whole it is reasonable to conclude that Type I is properly used 
 although we should probably not get bad results from a Type III curve. In 
 some respects a suitable fit would be obtained by using Type I, and fixing its 
 
XXXV — XLVI] Introduction lxix 
 
 start at zero*, but the vagueness of what is meant by ' percentage of black ' as a 
 factor, when the entire pigmentation of the skin probably arises from a single 
 melanin pigment, only varying in concentration in the pigment granules and in 
 the density of granules themselves. We have therefore contented ourselves by 
 fitting a Type I curve, as further illustration of the use of the tables in the 
 present work. The theory of fitting is given in the paper cited below f . Following 
 the usual notation we find : 
 
 r = 6 (& - /?, - l)/(3ft - 2ft + 6) = 21-7755, 
 e = r 2 /{4 + J/3, (r + 2) 2 /(r + 1)} = 57-764,468, 
 & 2 = A M- 2 (r+l)/e = (36-9391) 2 . 
 
 Hence: wi, = 2-0917, m 2 = 176838, 
 
 o, = 3-9071, a, = 33-0320, 
 
 an 
 
 (1 
 
 (g. \ 2-0917 / g. \17.683S 
 
 1 + 3^9071/ I 1 " 3¥032()J ' 
 
 To find y since m* is large, we use the approximation to the formula: 
 
 f r(» tl +w 2 +l ) | 
 
 N(rn 1 + m i + 1) 1e _ < Ml+m °->(mi + inS m ^ lll *\ 
 
 ro»._+i). (lxxvm >' 
 
 V* = 7, r, m , iu x 
 
 6 
 
 r (m, + i)i 
 
 1 to," 
 
 ...(lxxix) 
 
 1/ 1 l\ 
 
 namelv -^ (m 1 +m 2 + 1) /to, + to 2 12 \m, + m 2 raj 
 
 " y ° 6 T (to, + I )/(«-"' m,"') V ^ ~ e 
 
 the evaluation of the two T-functious for to 2 + in* + I and to 2 + 1 following easily 
 by Stirling's Theorem. If we write Z = r(3-0917)/{e- M917 (20917) 20917 } we have 
 
 \ogZ= log2-0917 -2-0917 log 2 0917, 
 + log 1-0917, 
 + log T (1-0917), 
 + 2-0917 loge. 
 
 From Table XXXI (p. 58) we find log T (1-0917) = T'979,8897 and log e is 
 given by Table LV (p. 143). Hence we determine, log Z = -576,5176. Evaluating 
 the rest of the expression for log y we have : 
 
 log 2/ = 2-233,3936, 
 2/„=171157. 
 
 Thus 
 
 our curve is 
 
 (m \ 20917 / <r \ 17-6838 
 
 1 + 3-90-71 i l ~ 33^320 
 
 * For method, see Phil. Trans. Vol. 186, A, pp. 370, 371. 
 
 t Phil. Tram. Vol. 186, A, pp. 367—370. See also Palin Elderton : Frequency Curves and Corre- 
 lation, Layton Brothers. 
 
lxx Tables for Statisticians and Biometricians [XXXV — XLVI 
 
 with origin at the mode =161486 in actual percentages, = 3 - 2297 in working 
 units. To calculate y we take the origin at and have 
 
 log y = 26-134,8705 + 20917 log (* + -6774) + 17-6838 log (362617 -x), 
 where x may be put 0, 1, 2, 3, 4, 5... working units corresponding to 0, 5, 10, 15, 
 20, 25... actual percentages. The curve is shewn in the accompanying diagram, 
 and considering the nature of the data is a reasonable graduation. 
 
 200 
 
 "fe. 
 
 10 15 
 
 20 25 30 35 40 45 50 55 
 Per cent, of Black in Skin Colour. 
 
 60 65 70 
 
 80 
 
 Table XLVIII (pp. 89—97). 
 
 Percentage frequency of Occurrences in a Second Sample of m after p Occur- 
 rences in a First Sample n. (M. Greenwood, Biometrika, Vol. ix. pp. 69 — 90.) 
 
 If we assume the truth of Bayes' Theorem then an event having occurred 
 p times and failed q times in n trials, the chance that it will occur s times and fail 
 m — s times in a second series of m trials is : 
 
 C' = 
 
 [ af> +s {l-x) 
 
 J 
 
 i +m ~ s dx 
 
 \s \m — 
 
 [ X* (1 - x)i dx 
 
 J 
 
 These results can be evaluated as all the indices are integers and the series 
 C + C i + Cj+ ... + C s + ... expressed in the usual hypergeometrical form : 
 
 \ q + m \ n+ 1 j m p + l m(m- 
 \q~]n+m+'l \ + |1 q + m + |2 
 
 -1) (p + l)(p + 2) 
 
 (q + m)(q +m- 1) 
 
 2)(p+3) 
 
 in (m — 1) (m — 2) 
 
 ft 
 
 (q + m) (q + m-l)(q+m-2) 
 
 + 
 
 .(lxxx). 
 
XLVIII] 
 
 Introduction 
 
 lxxi 
 
 If n be large and p not widely different from q, then results may be obtained from 
 the Gaussian curve, using as S. D. a/to™ -, but if either p/n or q/a be very 
 
 small and m or n are commensurable, this no longer holds*. The case, however, 
 of p and q widely different and n and m commensurable and themselves small 
 numbers frequently arises, especially in laboratory work or in the treatment of 
 rare diseases - !-. Tlie present table gives the evaluation of the hypergeometrical 
 series, formula (lxxx) above, for a series of values of m, n, p and q. It is not 
 sufficiently comprehensive to allow of very accurate interpolation in certain of its 
 ranges, but it has involved a large amount of work, and will undoubtedly be 
 of help till a more complete table can be calculated. Meanwhile if the reader 
 feels in doubt as to any interpolation, it is not a very arduous task to calculate 
 the result required from formula (lxxx) by aid of Table XLIX. 
 
 Illustration (i). In a batch of 79 recruits for a certain regiment four were 
 found to be syphilitic. What number of syphilitics may be anticipated in a 
 further batch of 40 recruits ? 
 
 Here n = 79, p = 4, q = 75 and m = 40. We must first interpolate in the 
 2> = 4 column on p. 97 between n= 100, m = 25, and n= 100, m = 50 for m = 40, 
 i.e. we must go ^| towards the m = 50 series, or we must add 0"4 times the first 
 series to 0'6 times the second series. We then repeat the same process for the 
 series for p = 4 and n = 50, in = 25 and n = 50, m — 50 on p. 95. There results : 
 
 Occurrences 
 
 »i=50, ?m = 40, p = 4 
 
 n=100, m = 40, p = i 
 
 
 
 6-8654 
 
 20-7406 
 
 1 
 
 13-5880 
 
 26 
 
 7023 
 
 2 
 
 16-3802 
 
 21 
 
 5249 
 
 S 
 
 15-7066 
 
 14 
 
 1138 
 
 4 
 
 13-2702 
 
 8 
 
 2460 
 
 5 
 
 10-3867 
 
 4 
 
 4566 
 
 6 
 
 7-7259 
 
 2 
 
 2637 
 
 7 
 
 5-5275 
 
 1 
 
 0886 
 
 8 
 
 3-8221 
 
 
 4977 
 
 9 
 
 2-5583 
 
 
 2171 
 
 10 
 
 1-6581 
 
 
 0906 
 
 11 
 
 1-0409 
 
 
 0362 
 
 12 
 
 •6332 
 
 
 0139 
 
 IS 
 
 •3734 
 
 
 0052 
 
 U 
 
 •2137 
 
 
 0019 
 
 15 
 
 •1188 
 
 
 0007 
 
 16 arid over 
 
 •1311 
 
 •0003 
 
 * For a full discussion of the subject : see Pearson, " On the Influence of Past Experience on Future 
 Expectation," Philosophical Magazine, 1907, p. 365. 
 
 t Tables recently published by Ross and Stott ("Tables of Statistical Error," Annals of Tropical 
 Medicine and Parasitology, Vol. v. No. 3, 1911), appear to be designed to meet such cases, but being 
 based on the Gaussian curve are, I think, very likely to lead the user to fallacious conclusions. 
 
lxxii 
 
 Tables for Statisticians and Biometricians [XLVIII 
 
 We must interpolate between these two series for n = 79, that is we must take 
 0'42 times the first series and 0'58 times the second series. The results are 
 given below, and set against the direct calculation from formula (lxxx), using 
 Table XLIX. 
 
 
 By Interpolation. 
 
 Direct Calculation. 
 
 
 ;i = 79, i> = i, »i = 40 
 
 n = 79, ;i = 4, m = 40 
 
 
 
 14-9130 
 
 12-6143 
 
 1 
 
 21-1943 
 
 21-9379 
 
 2 
 
 19-3642 
 
 22-5152 
 
 3 
 
 14-7828 
 
 17-6667 
 
 k 
 
 10-3562 
 
 11-6727 
 
 5 
 
 6-9472 
 
 6-8143 
 
 6 
 
 4-5578 
 
 3-6137 
 
 7 
 
 2-9529 
 
 1-7713 
 
 8 
 
 1 8940 
 
 ■8118 
 
 9 
 
 1-2004 
 
 •3507 
 
 10 
 
 •7489 
 
 1436 
 
 11 
 
 •4582 
 
 •0560 
 
 12 ■ 
 
 •2740 
 
 •0208 
 
 IS 
 
 ■1598 
 
 •0074 
 
 u 
 
 •0908 
 
 0025 
 
 15 
 
 •0503 
 
 •0008 
 
 16 
 
 •0271 
 
 ) 
 
 17 
 
 ■0142 
 
 V0003 
 
 18 and over 
 
 i 
 
 •0140 
 
 ) 
 
 The interpolation does not give a result very close to the actual series. For 
 example, not more than three syphilitics might be anticipated in 70 °/ of samples 
 of 40 by the interpolated series ; actually not more than 3 are to be expected 
 in 75 % °f samples. At the same time the result is much better than the 
 normal curve theory provides. In the latter case we have 
 
 Hence 
 
 Mean = 40 x 7 4 5 = 2 025, 
 Standard-Deviation = V40 x ^ x J| 
 (3-5 -2-025)/l-387 = 1-064 
 
 1-387. 
 
 and by Table II this value of x corresponds to |(1 + a) = -86, i.e. in 86 % per cent, 
 of samples of 40, we should have not more than 3 cases. It will be seen therefore 
 that (i) the values at the latter end of the Table are not close enough to obtain 
 very accurate results by interpolation, but (ii) that the Gaussian gives a still 
 poorer approximation. 
 
 Illustration (ii). Of 10 patients subjected by a first surgeon to a given 
 operation only one dies. A second surgeon in performing the same operation 
 on 7 patients, presumably equally affected, loses 4 cases. Would it be reason- 
 able to assume the second surgeon had inferior operative skill ? 
 
XLVIII— XLIX] 
 
 Introduction 
 
 lxxiii 
 
 On p. 91, we have the series for p = 1 when n = 10 for the values m = 5 and 
 m = 10. Taking '6 of the first series and "4 of the second we have : 
 
 
 Interpolation j 
 
 rom 
 
 Actual Value from 
 
 
 Table. formula. 
 
 
 « = 10, m = 7, p = l 
 
 m = 10, m = 7, p = l 
 
 
 
 37-9762 
 
 35-9477 
 
 1 
 
 30-6704 
 
 31 -4542 
 
 2 
 
 17-3366 
 
 18-8725 
 
 S 
 
 8-2114 
 
 8-9869 
 
 4 
 
 3-5248 
 
 3-4565 
 
 5 
 
 1 -4446 
 
 1-0370 
 
 6 
 
 •5676 
 
 •2200 
 
 7 
 
 •1996 
 
 ■0251 
 
 8 
 
 
 •0561' 
 
 
 — 
 
 9 
 
 
 •■0114 
 
 
 — 
 
 10 
 
 
 ;-ooi2; 
 
 
 — 
 
 The chance that if the two surgeons are of equal skill 4 or more patients will 
 die out of the second surgeon's 7 operations is '058 by interpolation and '047 
 actually. Hence the odds against the occurrence are 16 to 1 by the table and 
 20 to 1 actually. It will be observed that interpolation gives small values at 
 impossible numbers of deaths, but these have to be reckoned in to obtain the 
 total number 100. That all seven patients should die under the second surgeon, 
 if of equal skill, involves odds of 500 to 1 about in the interpolation result, but 
 4000 to 1 about actually. On the Gaussian hypothesis in the original problem the 
 mean - 7 x & = -7 and the S. D. = V7 x ^ x ■& = ' 7 937, and (3-5 - -7)/-7937 = 3-52 
 roughly, or this corresponds to odds of about 4545 to 1 — which are wholly un- 
 reasonable. Thus the Table gives by interpolation odds of approximately the 
 right value, which may serve many useful purposes, for those who are unable to 
 work out the values required from formula (lxxx). At the same time it is clear 
 that a much larger Table with closer values of the quantities involved is desirable. 
 
 Table XLIX (pp. 98—101) 
 
 The Logarithms of Factorials. (Calculated by Julia Bell, published here for 
 the first time.) 
 
 This table was obtained by adding up in succession consecutive logarithms in 
 a table of logarithms to 12 figures. Not until the work was completed did we 
 realise the existence of the splendid table of C. F. Degen*, which was then used 
 to confirm our own results. De Morgan in his Treatise on the Theory of Probabilities 
 of 1837 published an abridgement to six decimals of Degen's Table of Factorials. 
 His values cannot, however, be trusted to the sixth figure of the mantissa. The 
 
 * Tabularum ad faciliorem et breviorem probabilitatis computationem utilum. Havniae, mdcccxxiv. 
 This gives the logarithms of the factorials up to 1200 with 18 figures in the mantissa. 
 
 B. * 
 
lxxiv Tables for Statistician* and Biometricians [XLIX — L 
 
 use of a factorial table is extremely varied, especially in problems in probability 
 involving high numbers. 
 
 Illustration. In a certain district the number of children born per month 
 is 662 and the chance of a birth being male is '51 and of its being female "49. 
 Evaluate the chance that in a given month there should be an equal number of 
 boys and girls born, and compare it with the chance of the most probable numbers 
 (338 boys and 324 girls) being born. 
 
 The chance of equal numbers of boys and girls being born is : 
 
 1662 
 
 Therefore 
 
 1 a 331 i 1' 707 > 5702 1 + 1581714,6166 
 
 ° g * " |+ 1-690,1961] -1383-941,4114 
 
 where the logs of the factorials are found from Table XLIX. Hence 
 
 logC' c = 200-660,64o3) - 
 
 + 197-773,2042[ - i4Jd >»* ao 
 
 or G e = -027155, or once in about 36'8 months, say once in three years the records 
 may be expected to show equal numbers of boys and girls born in the month*. 
 
 The chance of the most probable number of boys and girls is given by 
 
 1662 
 
 o,„=(-5ir(-49r^24, 
 
 log G m = 338 x 1-707,5702 + 1581-714,6156 
 + 324x1-690,1961- 709-645.9652 
 - 674-359,6453 
 = 200-782,2640) 
 + 197-709,0051) SFW1 » awi - 
 
 Or C m = "030993, or the most probable numbers will only be born once in 
 32'3 months, or say once in two years and eight months. 
 
 We have C e /C m = "876, or the chance of equal boys and girls is 88% of the 
 chance of the most probable numbers of boys and girls. 
 
 Table L (pp. 102—112) 
 
 Tables of Fourth- Moments of Subgroup Frequencies. (Calculated by Alice Lee 
 and P. F. Everitt ; published here for the first time.) 
 
 In the usual method of determining the raw moments of a frequency, we take 
 moments about an arbitrary origin, which is towards the apparent mode and 
 
 * Actually of course the problem is more complex, because the number of children born per month 
 is not constant. 
 
L] 
 
 Introduction 
 
 lxxv 
 
 multiply by plus and minus abscissae increasing by units — the 'working unit.' 
 Thus an error made in an early moment may be carried on to the later moments. 
 To control the results Table L was calculated a number of years ago, and from it 
 the fourth moments for such frequencies as most usually occur can be read off at 
 sight, and the raw fourth moment column thus tested before proceeding further. 
 
 (i) 
 
 (ii) 
 
 (iii) 
 
 (iv) 
 
 (v) 
 
 (vi) 
 
 (vii) 
 
 Head 
 Length 
 
 Frequency 
 
 Abscissa 
 
 AV,' 
 
 AW 
 
 AW 
 
 AV 
 
 Table L 
 
 171 
 
 1 
 
 -20 
 
 - 20 
 
 + 400 
 
 - 8,000 
 
 + 160,000 
 
 160,000 
 
 2 
 
 1 
 
 19 
 
 19 
 
 361 
 
 6,859 130,321 
 
 130,321 
 
 3 
 
 2 
 
 18 
 
 38 
 
 648 
 
 11,664 209,952 
 
 209,952 
 
 4 
 
 
 
 17 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 5 
 
 3 
 
 16 
 
 48 
 
 768 
 
 12,288 
 
 196,608 
 
 196,608 
 
 G 
 
 3 
 
 15 
 
 4:> 
 
 675 
 
 10,125 
 
 151,875 
 
 151,875 
 
 7 
 
 5 
 
 14 
 
 70 
 
 980 
 
 13,720 ' 192,080 
 
 192.080 
 
 8 
 
 7 
 
 13 
 
 91 
 
 1,183 
 
 15,379 i 199,927 199,927 
 
 9 
 
 12 
 
 12 
 
 144 
 
 1,728 
 
 20,736 '■ 248,832 j 248,832 
 
 180 
 
 13 
 
 11 
 
 143 
 
 1,573 
 
 17,303 , 190,333 
 
 190,333 
 
 1 
 
 17 
 
 10 
 
 170 
 
 1,700 
 
 17,000 i 170,000 
 
 170,000 
 
 2 
 
 28 
 
 9 
 
 252 
 
 2,268 
 
 20,412 183,708 
 
 183,708 
 
 3 
 
 24 
 
 8 
 
 192 
 
 1,536 
 
 12,288 
 
 98,304 
 
 98,304 
 
 4 
 
 43 
 
 7 
 
 301 
 
 2,107 
 
 14,749 
 
 103,243 
 
 103,243 
 
 5 
 
 53 
 
 6 
 
 318 
 
 1,908 
 
 11,448 
 
 68,688 
 
 68,688 
 
 G 
 
 57 
 
 5 
 
 285 
 
 1,425 
 
 7,125 
 
 35,625 
 
 35,625 
 
 7 
 
 55 
 
 4 
 
 220 
 
 880 
 
 3,520 
 
 14,080 
 
 14,080 
 
 8 
 
 68 
 
 3 
 
 204 
 
 612 
 
 1,836 
 
 5,508 
 
 5,508 
 
 9 
 
 83 
 
 2 
 
 166 
 
 332 
 
 664 1,328 
 
 1,328 
 
 190 
 1 
 
 2 
 
 85 
 
 96 
 
 102 
 
 - 1 
 
 
 
 + 1 
 
 - 85 
 
 + 85 
 + 102 
 
 85 
 
 + 85 
 + 102 
 
 85 
 102 
 
 
 + 102 
 
 + 102 
 
 3 
 
 79 
 
 2 
 
 158 
 
 316 
 
 632 
 
 1,264 
 
 1,264 
 
 4 
 
 83 
 
 3 
 
 249 
 
 747 
 
 2,241 
 
 6,723 
 
 6,723 
 
 5 
 G 
 
 7 
 
 66 
 66 
 56 
 
 4 
 5 
 6 
 
 264 
 330 
 336 
 
 1,056 
 1,650 
 2,016 
 
 4,224 
 8,250 
 
 
 16,896 
 
 16,896 
 41,250 
 72,576 
 
 
 42,250 
 
 12,096 
 
 72,576 
 
 8 
 
 43 
 
 7 
 
 301 
 
 2,107 
 
 14,749 
 
 103,243 
 
 103,243 
 
 9 
 
 35 
 
 8 
 
 280 
 
 2,240 
 
 17,920 
 
 143,360 143,360 
 
 200 
 
 30 
 
 9 
 
 270 
 
 2,430 
 
 21,870 
 
 196,830 | 196,830 
 
 1 
 
 20 
 
 10 
 
 200 
 
 2,000 
 
 20,000 
 
 200,000 200,000 
 
 2 
 
 24 
 
 11 
 
 264 
 
 2,904 
 
 31,944 
 
 351,384 351,384 
 
 3 
 
 14 
 
 12 
 
 168 
 
 2,016 
 
 24,192 
 
 290,304 290,304 
 
 4 
 
 13 
 
 13 
 
 169 
 
 2,197 
 
 28,561 
 
 371,293 i 371,293 
 
 5 
 
 8 
 
 14 
 
 112 
 
 1,568 
 
 21,952 
 
 307,328 , 307,328 
 
 6 
 
 3 
 
 15 
 
 45 
 
 675 
 
 10,125 
 
 151,875 , 151,875 
 
 7 
 
 6 
 
 16 
 
 96 
 
 1,536 
 
 24,576 
 
 393,216 
 
 393,216 
 
 8 
 
 
 
 17 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 9 
 
 1 
 
 18 
 
 18 
 
 324 
 
 5,832 
 
 104,976 
 
 104,976 
 
 210 
 
 1 
 
 + 19 
 
 + 19 
 
 + 361 
 
 + 6,859 
 
 + 130,321 
 
 130,321 
 
 Totals 
 
 1306 
 
 — 
 
 + 
 
 + 
 
 + 
 
 + 
 
 — 
 
 *2 
 
lxxvi Tables for Statisticians and Biometricians [L — LI 
 
 The multiplication can therefore be done very rapidly and it suffices to re-examine 
 not the whole of the arithmetic but only those rows which do not agree with the 
 table. 
 
 Illustration. Calculate the first four raw moments of the distribution of head 
 lengths in 1306 non-habitual criminals on the previous page and test whether 
 they are correct. 
 
 This was an actually worked out case, and it will be seen that in this instance 
 only one slip was made — that of a wrong multiplication by 5 in the contribution 
 to the fourth moment of the frequency of head lengths 196. Often far more 
 serious blunders are found. Correction would be made and the columns then 
 added up on the adding machine. Two points should be noticed. First it is not 
 in practice necessary to copy out the results from Table L, — they are merely 
 compared on the table itself with the items in column (vii) and any divergence 
 noted. Secondly in actual practice, it would be quite sufficient to take 20 instead 
 of 40 sub-groups in this case. Sheppard's corrections would fully adjust for the 
 difference. 
 
 Table LI (pp. 113—121) 
 
 Tables of the General Term of Poisson's Exponential Expansion (" Law of 
 Small Numbers"). (H. E. Soper, Biometrika, Vol. x. p. 25.) 
 
 The limit to the binomial series 
 
 p n + np n-, q + "ft*" 1 ) p»-Y + "^"^t^V -Y + Oxxxi), 
 
 when q is very small, but nq = m is finite, was first shewn by Poisson to be 
 
 m' m? m 
 
 X 
 
 .. lit no ill/ \ ,i ..\ 
 
 1+w + i.2 + T7273 + - + .n + "J ( lxxxll >- 
 
 The present table provides the value of the terms of this series, i.e. e~ m m x jx ! 
 to six decimals for m = O'l to m = 15 by tenths. 
 
 A previous table for m = O'l to m = 10 to four decimals has been published by 
 Bortkewitsch*, but his values are not always correct to the fourth decimal. 
 Poisson's exponential limit to the binomial has been termed the " Law of Small 
 Numbers " by Bortkewitsch, but there are objections to the term. The approxi- 
 mation depends on the smallness of q (or, of course, p) and the largeness of n, so 
 that the mean m is finite. Thus 100 murders per annum might be quite a "small 
 number," if they occurred in a population of 40,000,000, for n would be large and 
 q would be small. It is therefore space and time which has limited the present 
 table to m = 15, not the idea of m being small of necessity. 
 
 Illustration (i). The number of monthly births in the Canton Vaud being 
 taken as 662, and one birth in 114 being that of an imbecile, find the chance of 12 
 or more imbeciles being born in a month. 
 
 * Das Gesetz der kleinen Zahlen, Leipzig, 1898. 
 
LI— LII] 
 
 Introduction 
 
 lxxvii 
 
 /113 1 \ m 
 The binomial is I —^ + -— J . n is accordingly large and q small, while 
 
 nq = 58 nearly. We look out 5"8 in Table L and sum the terms for 12 and beyond. 
 We find the chance of 12 or more = -01595. Actually worked from the binomial, 
 it is '01564. Or about once in five years, we might expect in Canton Vaud a 
 month with 12 imbecile births*. 
 
 Illustration (ii). Bortkewitsch (loc. cit. p. 25) gives the following deaths from 
 kicks of a horse in ten Prussian Army Corps during 20 years, reached after 
 excluding four corps for special reasons: 
 
 Annual 
 
 Frequency 
 
 Frequency 
 
 Deaths 
 
 Observed 
 
 Poisson's Series 
 
 
 
 109 
 
 108-72 
 
 ; 
 
 65 
 
 66-22 
 
 2 
 
 22 
 
 20-22 
 
 3 
 
 3 
 
 4-12 
 
 4 
 
 1 
 
 ■63 
 
 5 
 
 — 
 
 •08 
 
 and over 
 
 — 
 
 •01 
 
 Totals ... 
 
 200 
 
 200 
 
 The mean m of the observed frequency is '01, whence using Table LI (p. 113) 
 and taking - 9 the series for 0'6 and "1 times the series for 0*7, we reach figures, 
 which multiplied by 200 give us the column headed " Frequency, Poisson's Series " 
 above. Such good agreement, however, is very rare. A good fit to actual data 
 with the Exponential Binomial Limit is not often found. Its chief use lies in 
 theoretical investigations of chance and probable error : see Whitaker, Biometrika, 
 Vol. x. p. 36. 
 
 Table LII (pp. 122—124) 
 
 Table of Poisson's Exponential for Cell Frequencies 1 to 30. (Lucy Whitaker, 
 Biometrika, Vol. x. pp. 36—71.) 
 
 Given a cell in which the frequency is n s corresponding to the population N. 
 Then if n, anil N are very large (or we suppose, without this, the individual to be 
 returned before a second draw), the number in this sth cell will be distributed in 
 M samples of in according to the binomial law 
 
 * See Eugenics Laboratory Memoirs, XIII. ' ' A Second Study of the Influence of Parental Alcoholism," 
 p. 22. 
 
Ixxviii Tables for Statisticians and Biometricians [LII 
 
 The mean will be mn s /N and the standard deviation a / m ~ ( 1 — -X\ . If 
 
 we 
 
 only have a single sample of m and do not know the distribution in the actual 
 population we are compelled to give n s /N the value »i s /to, where m, is the number 
 found in the sth cell of the sample. If n s jN or m s /m be very small and m large« 
 the binomial will approach Poisson's Exponential Limit, and in such cases the 
 deviations in the samples for the sth cell will be distributed very differently from 
 those following a Gaussian law, and the usual rule for deducing the probability of 
 deviations of a given size by means of the probability integral fails markedly. 
 It is not till we get something like 30 out of 1000 in a cell that we can trust the 
 Gaussian to give us at all a reasonable approach. The present table endeavours 
 to provide material in the case of cell frequencies 1 to 30, which will supply the 
 place of the probability integral. 
 
 Illustration (i). Suppose the actual number to be expected in a cell is 17, 
 what is the probability that the observed number will deviate by more than 5 
 from this result? Looking at p. 123 we see that in 8-467 °/ Q of cases there will be 
 a deviation in defect of G or more and in 9 - 526 °/ of cases a deviation in excess of 
 6 or more. Hence in 17'993 °/ say 18 °/ o of cases we should get values less than 
 12 or greater than 22. Thus once in every 5 or 6 trials we should get values 
 which differ as widely as G or more from the true value. 
 
 Now look at the matter from the Gaussian standpoint. The standard 
 deviation is 
 
 y»"(>-")V"> : 
 
 a. 
 
 ■ml 
 
 Here m is supposed large compared with 17, so that the S. D. = V 17 = 4 - 123 
 nearly. But suppose m = 800, we should have 
 
 S. D. = Vl7 (1 --02125) = Vl7 x -97875 = 4-079. 
 
 Now we want deviations in excess of 5, i.e. we must take 5'5/4079 = 1348. 
 If we turn to Table II we find for this argument 
 
 | (1 + a) = -9102 or \ (1 - a) = '0898. 
 
 Hence we should conclude that in not more than 17"96 % of cases would deviations 
 exceed ± 5. Actually such occur in 17'99 % °f cases. Thus the actual per- 
 centages are very close, but the Poisson series tells us that 8 - 47 °/ of cases will 
 be in defect and 9'53 % m excess, while the Gaussian gives 8-98 % in both excess 
 and defect. We may further ask the percentage of times that 17 itself would 
 occur; according to the Gaussian it will occur in 9*76 % of trials, actually it will 
 occur in 9"63 °/ . With values of cell-frequency less than 17, say in the single 
 digits, far greater divergences will be encountered. 
 
LII— LIII] 
 
 Introduction, 
 
 lxxix 
 
 Illustration (ii). Consider the fourfold Table below and discuss the relative 
 probabilities that it has arisen from a population which shews 0, 1, 2, 3, etc. indi- 
 
 
 A Not-J 
 
 Totals 
 
 n 
 
 Not-Z? ... 
 
 127-5 
 863-5 87 
 
 127-5 
 950-5 
 
 
 991 
 
 87 
 
 1078 
 
 viduals for this size of sample in the cell B, not-A. On the assumption that 
 is really the population of this cell, the probability is unity. Hence we have the 
 following result. 
 
 Population) 
 of cell ( 
 
 
 1 
 
 1 
 
 2 
 
 3 • 
 
 4 
 
 5 
 
 6 
 
 7 8 
 
 .9 
 
 to 
 
 U 
 
 12 
 
 13 & 
 over 
 
 Probability ) 
 
 ofO [ 
 
 occurring ) 
 
 •36788 
 
 •13534-04979 
 
 •01832 
 
 •00674 
 
 •00248 
 
 •00091 -00034 
 
 •00012 
 
 •00005 
 
 •00002 
 
 •00001 
 
 •00000 
 
 
 
 
 
 
 
 Sum 
 
 = 1-58200. 
 
 
 
 
 
 
 Whence taking the a priori probabilities proportional to the probability of 
 occurring on the separate possibilities we have : 
 
 Probabilities that the Table arose from a population 
 ivith x in the B, not-A cell. 
 
 X 
 
 Probability 
 
 j; P 
 
 robability 
 
 
 
 •632,110 
 
 7 
 
 000,575 
 
 1 
 
 •232,541 
 
 8 
 
 000,215 
 
 2 
 
 •085,550 
 
 9 
 
 000,076 
 
 8 
 
 •031,473 
 
 to 
 
 000,032 
 
 4 
 
 •011,580 
 
 11 
 
 000,013 
 
 S 
 
 •004,260 
 
 12 
 
 000,006 
 
 6 
 
 •001,568 
 
 IS and over '■ 
 
 000,000 
 
 The " association " of such a Table cannot therefore be considered " perfect," for 
 in 37 °/ of cases it would arise from a Table with a unit or more in the B, not- A 
 cell. The above is actually a Table of the correlation of stature in father and son. 
 Grave caution is therefore needful in discussing such "perfect association" tables. 
 
 Table LIII (p. 125) 
 
 Angles, Arcs and Decimals of Degrees. (Based on Hutton's Mathematical 
 Tables.) 
 
 This Table gives degrees in radians for the first two quadrants; it then gives 
 minutes and seconds from 1 to 60 in fractions of a degree and in radians. The 
 
lxxx Table* for Statisticians and Biometriciam [LIII 
 
 need of such a table is very obvious, and arises in too great a variety of circum- 
 stances to be specified. 
 
 Illustration. It is required to plot the curve * : 
 
 x = 14-9917 tan 0, 
 
 y = 23.5-323 cos 32 ' 8023 ftr 4 ' 56 ' 16 0. 
 
 Here log y = log 235323 + 328023 log cos - 4-5696 log ex0. 
 
 To cover the whole range of observations we must proceed from 0= — 45° to 
 = + 45° roughly. It will be found sufficient to take 6 by steps of 3° and 
 ultimately perhaps of 4°. Hence 14-9917 is put on the arithmometer and multi- 
 plied in succession by the natural tangents of 3°, 6°, 9°.., etc. Plus and minus 
 signs are given to these values of x. The corresponding values of y are found 
 in three columns. The first is obtained by putting 32-8023 on the arithmometer 
 and multiplying by the logarithmic cosines of 3°, 6°, 9°, etc. The second is 
 obtained by multiplying (taking the third factor from Table LIII) 
 
 4-5696 x log e x -017,4533 = -216,7955 
 
 on the machine and multiplying the result in succession by 3, 6, 9, etc. 
 
 The first column is added to log 235323 = 2-371,6644 and the second column 
 first subtracted aud then added to the result to obtaiu the value of logy for 
 positive and negative abscissae respectively. The antilogarithms give the ordinates. 
 
 Another problem sometimes arises given x to find y. For example : In the 
 above curve the mode is at 117'9998 cms. of stature and the origin at 113'8228, 
 thus the distance between them =4-1770 cms. or since the working unit of 
 x = 2 cms. and the positive direction of x is towards dwarfs, the mode is at 
 x = — 2-0885. Required to find the maximum ordinate y mo . We have 
 
 tan = - 20885/14-9917 = - -139,3104, 
 whence by a table of natural tangents 
 
 = -V 55' -851265, 
 = - V 55' 51". 
 The log cosine of this value of is 
 
 1-995,8962. 
 Table LIII gives us : 
 
 7°= -122,1730 in arc 
 
 55'= -015,9989 „ „ 
 
 •851,265' = -851,265 x -000,2909= '000,2476,, „ 
 Hence =--138,4195 „ „ 
 
 * See Phil. Trans. Vol. 186, A, p. 387. Pearson's Type IV frequency curve fitted to the stature 
 of 2192 St Louis School Girls aged 8. 
 
LIU— LIV] Introduction lxxxi 
 
 Hence \ gy m0 = 2-371,6644 
 
 + 32-8023 (- -004,1038) 
 
 + 1-984,5521 x -138,4195 
 2-511,7510. 
 Hence y mo = 324901. 
 
 Table LIV (pp. 126—142) 
 
 Tables of the G (r, v) Integrals. (Calculated by Alice Lee, D.Sc. Transactions 
 British Association Report, Dover, 1899, pp. 65 — 120.) 
 
 The purpose of this table is to obtain the value of the integral 
 
 Q 
 
 (r,v)=j sm r 0e*°dd (lxxxiii). 
 
 In order to obtain small differences in tabulated values two additional 
 functions F(r, v) and H (r, v) are introduced. 
 
 The relations between the three functions are then expressed by the following 
 series of equations : 
 
 F{r,v) = e-^Q(r,v) (Ixxxiv), 
 
 F{r,v) = e * { *^ +1 H{r,v) (hxxv), 
 
 G(r,v) = e i '"F(r,v) (lxxxvi), 
 
 e"* +} " r (cos6) r+1 
 
 G(r,v) = ,—- Y S{r,v) (lxxxvii), 
 
 vr — 1 ' 
 
 g(r,y) " \ <£l£* r < r - w '> < lxxxviii )' 
 
 H {r ' V)= (cos </>)-+' G{r ' ») -(Ixxxix), 
 
 where tan <f> = v/r. 
 
 Pearson's Type IV Skew Frequency Curve is of the form 
 
 - v tan -1 - 
 
 y ~ y, {i + (?yi* (r+2) ' (xc) ' 
 
 Hence if N be its total area, i.e. the entire population under discussion, 
 
 ^ = y ae _i '" r [" sin r 6e^d0, 
 Jo 
 
 /o 
 )Y 1 
 a F (r, v) 
 B. , 
 
 *— r-frfcrrs (™)- 
 
liixii Tables for Statisticians and Biometricians [LI V 
 
 The function H (r, v) is introduced because, as a rule, its logarithms have far 
 smaller differences and it is thus capable of more exact determination from a table 
 of double entry. Its physical relation to the curve may be expressed as follows ; 
 let the origin be transferred to the mean, then if y 1 be the ordinate at the mean, 
 
 *-!*(b (xci)bis - 
 
 where a is the standard-deviation of the curve 
 
 a . ... 
 
 = , (xcii). 
 
 vr — 1 cos <f> 
 
 The distance of the mean from the origin is given by 
 
 /*/ = — a tan <f> (xciii). 
 
 When r is fairly large : 
 
 cos- 1 
 
 / ~ 5 T7T- 0'' tan 
 
 1 / r e 3r Vir . 
 
 jr(r,»)"V 2tt (cos0) r+1 " (XC1V) - 
 
 TT 1 / V 1 -Aff2 / N 
 
 Hence =a/ x -i—e ^ (xcv), 
 
 H(r,v) Vr-1 V2tt 
 
 , /l-4cos a d> 
 
 where g=I ^/ _____£, 
 
 and thus the evaluation if <£ be > 60° may be made by aid of Table II*. 
 
 Illustration. In the curve fitted to the statures of St Louis School Girls, 
 aged 8 (p. lxxx), we have 
 
 if- 2192, a =149917, 
 
 r = 30-8023, v = 4-56967. 
 Find y„. 
 
 We have tan <j ) = v /r= -148,3548. 
 
 Hence <j> = 8° 26'-31315 = 8°-43855. 
 
 Turning to the Tables, p. 136, we see the large differences of log.F(r, v) at 
 this value of <f>, and accordingly settle to work with log H(r, v). 
 We have for log H (r, v ), 
 
 r = 30 r = 31 
 
 $ = 8° -388,2032 -388,5583, 
 
 <f> = 9° -388,2278 "388,5822, 
 
 4> = 8°-4386, r = 30: 
 
 log H (r, 1/) = -388,2032 + (-4386) [246] - £ (-4386) x (-5614) [28] 
 
 = •388,2137. 
 
 * For a fuller discussion of these integrals see Phil. Trans. Vol. 186, A, pp. 376—381, B. A. Trans. 
 Report, Liverpool, 1896, Preliminary Keport of Committee... , and the B.A. Trans. Report, Dover, 1899, 
 already cited. 
 
LIV — LV] Introduction Ixxxiii 
 
 <£ = 8°-4386, r-Sl: 
 
 log S(r, v) = -388,558a + (-4386) [238] - } (-438(5) (-5614) [27] 
 = 388,5684. 
 <f> = 8°-4386, r = 32 : 
 
 log H(r, v) = -388,8910 + (-4386) [231] - \ (4386) (-5614) [26] 
 = -388,9008. 
 Hence </> = 8°-4386, r = 30 8023 : 
 
 log H (r, *) = -388,2137 + 8023 [3547] - £ (-8023) (1977) [- 223] 
 = -388-5001. 
 Hence by formula (lxxxv) : 
 
 log F (r, v) = i>4> log e + r + 1 log cos <£ - £ log (r - 1) + log ZZ" (r, i/). 
 Or, using Tables LIII and LV, we have 
 
 log F(r, v) = 292,2901 - -737,1249 
 
 + 1-849,6578 
 + -388,5001 
 
 •530,4480 
 - -737,1249 
 
 log F(r, v) = 1-793,3231 
 Finally from formula (xci) : 
 
 log y = log N - log a - 1793,3231 
 
 = 3-340,8405 - 1-175,8509 
 
 - 1-793,3231 
 
 - -969,1740 
 = 2371,76665. 
 Or y = 235324*. 
 
 Table LV. 
 
 This table contains some miscellaneous constants in frequent statistical or 
 biometric use and requires no illustration. It lias already been used in the 
 illustrations to previous tables. 
 
 I have had the generous assistance of my colleagues Miss E. M. Elderton and 
 Mr H. E. Soper in the preparation of the Illustrations to these Tables. I can 
 hardly hope that arithmetical slips have wholly escaped us in a first edition, 
 and I shall be grateful for the communication of any corrections that my readers 
 may discover are necessary. 
 
 * The value 235-323 obtained in Phil Trans. Vol. 186, A, p. 387, was found by the approximate 
 formula (xciv) before tables were calculated. 
 
Every reader may now see in what way the higher branches of mathematics 
 are concerned in our present subject. They are the abbreviators of long and 
 tedious operations, and it would be perfectly possible, with sufficient time and 
 industry, to do without their use When both the ordinary and the mathe- 
 matical result are derived from the same hypothesis, the latter must be the more 
 correct : and in those numerous cases in which the difficulty lies in reducing the 
 original circumstances to a mathematical form, there is nothing to show that we 
 are less liable to error in deducing a common sense result from principles too 
 indefinite for calculation, than we should be in attempting to define more closely, 
 and to apply numerical reasoning. — De Morgan. 
 
Tables of the Probability Integral 
 
 TABLE I. 
 
 Table of Deviates of the Normal Curve for each Perinille of Frequency. 
 
 Permille 
 
 ■000 
 
 ■001 
 
 •002 
 
 ■003 
 
 ■004 
 
 ■005 
 
 •006 
 
 ■007 
 
 ■008 
 
 ■009 
 
 ■010 
 
 
 •00 
 
 00 
 
 3-0902 
 
 2-8782 
 
 2-7478 
 
 2 6521 
 
 2-5758 
 
 2-5121 
 
 2-4573 
 
 2-4089 
 
 2-3656 
 
 2-3263 
 
 ■99 
 
 ■01 
 
 2-3203 
 
 2-2904 
 
 2-2571 
 
 2-2262 
 
 2-1973 
 
 2-1701 
 
 2-1444 
 
 2-1201 
 
 2-0969 
 
 2-0749 
 
 2-0537 
 
 ■98 
 
 •02 
 
 2-0537 
 
 2-0335 
 
 2-0141 
 
 1-9954 
 
 1-9774 
 
 1-9600 
 
 1-9431 
 
 1-9268 
 
 1-9110 
 
 1-8957 
 
 1-8808 
 
 ■97 
 
 ■OS 
 
 1 -8808 
 
 1-8663 
 
 1-8522 
 
 1-8384 
 
 1-8250 
 
 1-8119 
 
 1-7991 
 
 1 -7866 
 
 1-7744 
 
 1-7624 
 
 1-7507 
 
 ■96 
 
 ■04 
 
 1-7507 
 
 1-7392 
 
 1-7279 
 
 1-7169 
 
 1-7060 
 
 1-6954 
 
 1-6849 
 
 1-6747 
 
 1-6646 
 
 1-6546 
 
 1-6449 
 
 ■95 
 
 •05 
 
 1-6449 
 
 1-6352 
 
 1-6258 
 
 1-6164 
 
 1-6072 
 
 1-5982 
 
 1-5893 
 
 1 -5805 
 
 1-5718 
 
 1-5632 
 
 1-5548 
 
 ■94 
 
 ■06 
 
 1-5548 
 
 1 -5464 
 
 1-5382 
 
 1-5301 
 
 1-5220 
 
 1-5141 
 
 1-5063 
 
 1-4985 
 
 1-4909 
 
 1-4833 
 
 1-4758 
 
 ■93 
 
 ■07 
 
 1-4758 
 
 1-4684 
 
 1-4611 
 
 1 -4538 
 
 1-4466 
 
 1 -4395 
 
 1-4325 
 
 1-4255 
 
 1-4187 
 
 1-4118 
 
 1 -4051 
 
 ■92 
 
 ■08 
 
 1-4051 
 
 1-3984 
 
 1-3917 
 
 1-3852 
 
 1-3787 
 
 1 -3722 
 
 1-3658 
 
 1-3595 
 
 1-3532 
 
 1-3469 
 
 1 -3408 
 
 •91 
 
 ■00 
 
 1-3408 
 
 1-3346 
 
 1 -3285 
 
 1-3225 
 
 1-3165 
 
 1-3106 
 
 1-3047 
 
 1-2988 
 
 1-2930 
 
 1-2873 
 
 1-2816 
 
 ■90 
 
 •10 
 
 1-2816 
 
 1-2759 
 
 1-2702 
 
 1-2646 
 
 1-2591 
 
 1-2536 
 
 1-2481 
 
 1-2426 
 
 1-2372 
 
 1-2319 
 
 1 -2265 
 
 ■89 
 
 •11 
 
 1-2265 
 
 1-2212 
 
 1-2160 
 
 1-2107 
 
 1-2055 
 
 1 -2004 
 
 1-1952 
 
 1-1901 
 
 1-1850 
 
 1-1800 
 
 1-1750 
 
 ■88 
 
 •12 
 
 1-1750 
 
 1-1700 
 
 1-1650 
 
 1-1601 
 
 1-1552 
 
 1-1503 
 
 1-1468 
 
 1-1407 
 
 1-1359 
 
 1-1311 
 
 1-1264 
 
 ■87 
 
 •13 
 
 1-1264 
 
 1-1217 
 
 1U170 
 
 1-1123 
 
 1-1077 
 
 1-1031 
 
 1-0985 
 
 1 -0939 
 
 1-0893 
 
 1 -08 18 
 
 1-0803 
 
 •86 
 
 ■n 
 
 1-0803 
 
 1 -0758 
 
 1-0714 
 
 1-0669 
 
 1 -0625 
 
 1-0581 
 
 1-0537 
 
 1-0494 
 
 1-0450 
 
 1-0407 
 
 1-0364 
 
 ■85 
 
 •IB 
 
 1 0364 
 
 1 -0322 
 
 1-0279 
 
 1-0237 
 
 1-0194 
 
 1-0152 
 
 1-0110 
 
 1-0069 
 
 1-0027 
 
 0-9986 
 
 0-9945 
 
 ■84 
 
 •16 
 
 0-0945 
 
 0-9904 
 
 0-9863 
 
 0-9822 
 
 0-9782 
 
 0-9741 
 
 0-9701 
 
 0-9661 
 
 0-9621 
 
 0-9581 
 
 0-9542 
 
 •83 
 
 •17 
 
 0-9542 
 
 0-9502 
 
 0-9463 
 
 0-9424 
 
 0-9385 
 
 9346 
 
 0-9307 
 
 0-9269 
 
 0-9230 
 
 0-9192 
 
 09154 
 
 ■82 
 
 •18 
 
 0-9154 
 
 0-9116 
 
 0-9078 
 
 0-9040 
 
 0-9002 
 
 0-8965 
 
 0-8927 
 
 0-8890 
 
 0-8853 
 
 0-8816 
 
 0-8779 
 
 ■81 
 
 ■19 
 
 0-8779 
 
 0-8742 
 
 0-8705 
 
 0-8669 
 
 8633 
 
 0-8596 
 
 0-8560 
 
 0-8524 
 
 0-8488 
 
 0-8452 
 
 0-8416 
 
 ■80 
 
 •20 
 
 0-8416 
 
 0-8381 
 
 0-8345 
 
 0-8310 
 
 0-8274 
 
 0-8239 
 
 0-8204 
 
 0-8169 
 
 0-8134 
 
 0-8099 
 
 0-8064 
 
 ■79 
 
 •21 
 
 0-8064 
 
 0-8030 
 
 0-7995 
 
 0-7961 
 
 0-7926 
 
 0-7892 
 
 0-7858 
 
 0-7824 
 
 0-7790 
 
 0-7756 
 
 0-7722 
 
 ■78 
 
 
 0-7722 
 
 0-7688 
 
 0-7655 
 
 0-7621 
 
 0-7588 
 
 0-7554 
 
 0-7521 
 
 0-7488 
 
 0-7454 
 
 0-7421 
 
 0-7388 
 
 ■77 
 
 '..'■! 
 
 0-7388 
 
 0-7356 
 
 0-7323 
 
 0-7290 
 
 0-7257 
 
 0-7225 
 
 0-7192 
 
 0-7160 
 
 0-7128 
 
 0-7095 
 
 0-7063 
 
 ■76 
 
 ■-'.', 
 
 0-7063 
 
 0-7031 
 
 0-6999 
 
 0-6967 
 
 0-6935 
 
 0-6903 
 
 0-6871 
 
 0-6840 
 
 0-6808 
 
 0-6776 
 
 0-6745 
 
 ■75 
 
 •25 
 
 0-6745 
 
 0-6713 
 
 0-6682 
 
 0-6651 
 
 0-6620 
 
 0-6588 
 
 0-6557 
 
 0-6526 
 
 0-6495 
 
 0-6464 
 
 0-6433 
 
 ■74 
 
 ■26 
 
 06433 
 
 0-6403 
 
 0-6372 
 
 0-6341 
 
 0-6311 
 
 0-6280 
 
 0-6250 
 
 0-6219 
 
 0-6189 
 
 0-6158 
 
 0-6128 
 
 •73 
 
 ■27 
 
 0-6128 
 
 0-6098 
 
 0-6068 
 
 0-6038 
 
 0-6008 
 
 0-5978 
 
 0-5948 
 
 0-5918 
 
 0-5888 
 
 0-5858 
 
 0-5828 
 
 '72 
 
 •28 
 
 0-5828 
 
 0-5799 
 
 5769 
 
 0-5740 
 
 0-5710 
 
 0-5681 
 
 0-5651 
 
 0-5622 
 
 0-5592 
 
 0-5563 
 
 0-5534 
 
 ■71 
 
 ■29 
 
 0-5534 
 
 0-5505 
 
 05476 
 
 0-5446 
 
 0-5417 
 
 0-5388 
 
 5359 
 
 0-5330 
 
 0-5302 
 
 0-5273 
 
 0-5244 
 
 •70 
 
 ■30 
 
 0-5244 
 
 0-5215 
 
 0-5187 
 
 0-5158 
 
 0-5129 
 
 0-5101 
 
 0-5072 
 
 0-5044 
 
 0-5015 
 
 0-4987 
 
 0-4959 
 
 •69 
 
 •31 
 
 0-4959 
 
 0-4930 
 
 0-4902 
 
 0-4874 
 
 0-4845 
 
 0-4817 
 
 0-4789 
 
 0-4761 
 
 0-4733 
 
 0-4705 
 
 0-4677 
 
 ■68 
 
 ■32 
 
 0-4677 
 
 0-4649 
 
 0-4621 
 
 0-4593 
 
 0-4565 
 
 0-4538 
 
 0-4510 
 
 0-4482 
 
 0-4454 
 
 0-4427 
 
 0-4399 
 
 ■67 
 
 •S3 
 
 0-4399 
 
 0-4372 
 
 0-4344 
 
 0-4316 
 
 0-4289 
 
 0-4261 
 
 0-4234 
 
 0-4207 
 
 0-4179 
 
 0-4152 
 
 0-4125 
 
 •66 
 
 ■3/ t 
 
 0-4125 
 
 0-4097 
 
 0-4070 
 
 0-4043 
 
 0-4016 
 
 0-3989 
 
 0-3961 
 
 0-3934 
 
 0-3907 
 
 0-3880 
 
 0-3853 
 
 ■65 
 
 ■35 
 
 0-3853 
 
 0-3826 
 
 0-3799 
 
 0-3772 
 
 0-3745 
 
 0-3719 
 
 0-3692 
 
 0-3665 
 
 0-3638 
 
 0-3611 
 
 0-3585 
 
 ■64 
 
 ■36 
 
 0-3585 
 
 0-3558 
 
 0-3531 
 
 0-3505 
 
 0-3478 
 
 0-3451 
 
 0-3425 
 
 0-3398 
 
 0-3372 
 
 0-3345 
 
 03319 
 
 •63 
 
 ■37 
 
 0-3319 
 
 0-3292 
 
 0-3266 
 
 0-3239 
 
 0-3213 
 
 0-3186 
 
 0-3160 
 
 0-3134 
 
 0-3107 
 
 0-3081 
 
 0-3055 
 
 ■62 
 
 ■38 
 
 0-3055 
 
 0-3029 
 
 0-3002 
 
 0-2976 
 
 0-2950 
 
 0-2924 
 
 0-2898 
 
 0-2871 
 
 0-2845 
 
 0-2819 
 
 0-2793 
 
 ■61 
 
 •SO 
 
 0-2793 
 
 0-2767 
 
 0-2741 
 
 0-2715 
 
 0-2689 
 
 0-2663 
 
 0-2637 
 
 0-2611 
 
 0-2585 
 
 0-2559 
 
 02533 
 
 ■60 
 
 ■40 
 
 0-2533 
 
 0-2508 
 
 0-2482 
 
 0-2456 
 
 0-2430 
 
 0-2404 
 
 0-2378 
 
 0-2353 
 
 0-2327 
 
 0-2301 
 
 0-2275 
 
 ■59 
 
 ■41 
 
 0-2275 
 
 0-2250 
 
 0-2224 
 
 0-2198 
 
 0-2173 
 
 0-2147 
 
 0-2121 
 
 0-2096 
 
 0-2070 
 
 0-2045 
 
 0-2019 
 
 •58 
 
 •42 
 
 0-2019 
 
 1993 
 
 0-1968 
 
 0-1942 
 
 0-1917 
 
 0-1891 
 
 0-1866 
 
 0-1840 
 
 0-1815 
 
 0-1789 
 
 0-1764 
 
 ■57 
 
 •43 
 
 0-1764 
 
 0-1738 
 
 0-1713 
 
 0-1687 
 
 0-1662 
 
 0-1637 
 
 0-1611 
 
 0-1586 
 
 0-1560 
 
 0-1535 
 
 0-1510 
 
 ■56 
 
 ■44 
 
 0-1510 
 
 0-1484 
 
 0-1459 
 
 0-1434 
 
 0-1408 
 
 0-1383 
 
 0-1358 
 
 0-1332 
 
 0-1307 
 
 0-1282 
 
 0-1257 
 
 ■55 
 
 •45 
 
 0-1257 
 
 01231 
 
 0-1206 
 
 0-1181 
 
 0-1156 
 
 0-1130 
 
 0-1105 
 
 0-1080 
 
 0-1055 
 
 0-1030 
 
 0-1004 
 
 ■54 
 
 ■46 
 
 0-1004 
 
 0-0979 
 
 0-0954 
 
 0-0929 
 
 0-0904 
 
 0-0878 
 
 0-0853 
 
 0-0828 
 
 0-0803 
 
 0-0778 
 
 0-0753 
 
 ■53 
 
 ■47 
 
 0-0753 
 
 0-0728 
 
 0-0702 
 
 0-0677 
 
 0-0652. 
 
 0-0627 
 
 0-0602 
 
 0-0577 
 
 0-0552 
 
 0-0527 
 
 0502 
 
 ■52 
 
 •48 
 
 0-0502 
 
 0476 
 
 0-0451 
 
 0-0426 
 
 0-0401 
 
 0-0376 
 
 0-0351 
 
 0-0326 
 
 0-0301 
 
 0-0276 
 
 0-0251 
 
 ■51 
 
 ■49 
 
 0-0251 
 
 0-0226 
 
 0-0201 
 
 0-0175 
 
 0-0150 
 
 0-0125 
 
 o-oioo 
 
 0-0075 
 
 0-0050 
 
 0-0025 
 
 0-0000 
 
 ■50 
 
 
 •010 
 
 ■009 
 
 ■008 
 
 •007 
 
 ■006 
 
 ■005 
 
 ■004 
 
 ■003 
 
 ■002 
 
 •001 
 
 •000 
 
 Permille 
 
 B. 
 
Tables for Statisticians and Biometricians 
 TABLE II. Area and Ordinate in terms of Abscissa. 
 
 i(l+«) 
 
 ■00 
 01 
 ■02 
 ■03 
 •Ob 
 ■05 
 
 ■06 
 ■07 
 •08 
 ■00 
 ■10 
 
 ■11 
 ■12 
 •18 
 ■14 
 ■15 
 
 ■16 
 ■17 
 ■18 
 ■19 
 
 ■21 
 
 •25 
 
 •26 
 ■27 
 ■28 
 ■29 
 •SO 
 
 •SI 
 •32 
 ■S3 
 ■34 
 •35 
 
 ■36 
 ■37 
 •38 
 •39 
 •40 
 
 41 
 
 42 
 ■43 
 ■44 
 ■46 
 
 ■46 
 ■47 
 ■48 
 ■49 
 
 ■50 
 
 •5000000 
 •5039894 
 ■5079783 
 •5119665 
 •5159534 
 •5199388 
 
 •5239222 
 ■5279032 
 •5318814 
 •5358564 
 
 •5398278 
 
 •5437953 
 
 •5477584 
 •5517168 
 •5556700 
 •5596177 
 
 •5635595 
 •5674949 
 •5714237 
 •5753454 
 •5792597 
 
 •5831662 
 •5870644 
 •5909541 
 ■5948349 
 •5987063 
 
 ■6025681 
 •6064199 
 •6102612 
 •6140919 
 •6179114 
 
 •6217195 
 •6255158 
 •6293000 
 •6330717 
 ■6368307 
 
 •6405764 
 •6443088 
 •6480273 
 •6517317 
 •6554217 
 
 •6590970 
 •6627573 
 •6664022 
 •6700314 
 ■6736448 
 
 •6772419 
 •6808225 
 •6843863 
 •6879331 
 •6914625 
 
 A 
 
 + 
 
 39894 
 39890 
 39882 
 39870 
 39854 
 39834 
 
 39810 
 39782 
 39750 
 39714 
 39675 
 
 39631 
 39584 
 39532 
 39477 
 39418 
 
 39355 
 39288 
 39217 
 39143 
 39065 
 
 38983 
 38897 
 38808 
 38715 
 38618 
 
 38518 
 38414 
 38306 
 38195 
 38081 
 
 37963 
 
 37842 
 37717 
 37589 
 37458 
 
 37323 
 37185 
 37044 
 36900 
 36753 
 
 36602 
 36449 
 36293 
 36133 
 35971 
 
 35806 
 35638 
 35467 
 35294 
 
 A 2 
 
 
 
 4 
 
 8 
 
 12 
 
 16 
 
 20 
 
 24 
 28 
 32 
 36 
 40 
 
 44 
 48 
 51 
 55 
 59 
 
 63 
 67 
 71 
 
 74 
 
 82 
 86 
 89 
 93 
 97 
 
 100 
 104 
 107 
 111 
 114 
 
 118 
 121 
 125 
 128 
 131 
 
 135 
 
 138 
 141 
 144 
 147 
 
 150 
 153 
 156 
 159 
 162 
 
 165 
 168 
 171 
 173 
 176 
 
 •3989423 
 •3989223 
 •3988625 
 •3987628 
 •3986233 
 •3984439 
 
 •3982248 
 •3979661 
 ■3976677 
 •3973298 
 •3969525 
 
 •3965360 
 •3960802 
 •3955854 
 ■3950517 
 •3944793 
 
 •3938684 
 •3932190 
 •392531 5 
 •3918060 
 •3910427 
 
 •3902419 
 •3894038 
 •3885286 
 •3876166 
 •3866681 
 
 •3856834 
 •3846627 
 •3836063 
 •3825146 
 
 •3813878 
 
 •3802264 
 •3790305 
 •3778007 
 •3765372 
 •3752403 
 
 •3739106 
 •3725483 
 •3711539 
 •3697277 
 •3682701 
 
 ■3667817 
 ■3652627 
 •3637136 
 •3621349 
 •3605270 
 
 •3588903 
 •3572253 
 •3555325 
 •3538124 
 •3520653 
 
 199 
 
 598 
 
 997 
 
 1395 
 
 1793 
 
 2191 
 
 2588 
 2984 
 3379 
 3773 
 4166 
 
 4558 
 4948 
 5337 
 5724 
 6110 
 
 6493 
 
 6875 
 7255 
 7633 
 
 8008 
 
 8381 
 8752 
 9120 
 9485 
 9847 
 
 10207 
 10564 
 10917 
 11268 
 11615 
 
 11958 
 12298 
 12635 
 12968 
 13297 
 
 13623 
 13944 
 
 14262 
 14575 
 14885 
 
 15190 
 15491 
 15787 
 16079 
 16367 
 
 16650 
 16928 
 17202 
 17470 
 
 A 2 
 
 399 
 399 
 399 
 398 
 398 
 397 
 
 397 
 396 
 395 
 394 
 393 
 
 392 
 390 
 389 
 387 
 386 
 
 384 
 382 
 380 
 378 
 375 
 
 373 
 
 371 
 368 
 365 
 362 
 
 360 
 357 
 354 
 350 
 347 
 
 344 
 340 
 337 
 333 
 329 
 
 325 
 322 
 318 
 313 
 309 
 
 305 
 301 
 806 
 
 292 
 
 288 
 
 283 
 278 
 274 
 269 
 264 
 
 50 
 51 
 52 
 63 
 
 54 
 
 55 
 
 ■56 
 57 
 58 
 69 
 
 GO 
 
 61 
 62 
 63 
 64 
 65 
 
 66 
 67 
 68 
 69 
 70 
 
 71 
 72 
 73 
 
 75 
 
 76 
 77 
 78 
 79 
 SO 
 
 81 
 82 
 88 
 84 
 85 
 
 ■86 
 ■87 
 ■8S 
 89 
 
 90 
 
 91 
 92 
 a.; 
 94 
 95 
 
 96 
 
 or 
 
 98 
 
 99 
 
 100 
 
 4(l+«) 
 
 •6914625 
 •6949743 
 ■6984682 
 •7019440 
 •7054015 
 ■7088403 
 
 •7122603 
 •7156612 
 
 •7190427 
 •7224047 
 ■7257469 
 
 •7290691 
 •7323711 
 •7356527 
 ■7389137 
 •7421539 
 
 •7453731 
 ■7485711 
 •7517478 
 •7549029 
 •7580363 
 
 •7611479 
 •7642375 
 •7673049 
 •7703500 
 •7733726 
 
 •7763727 
 •7793501 
 •7823046 
 •7852361 
 •7881446 
 
 •7910299 
 •7938919 
 •7967306 
 ■7906468 
 •8023375 
 
 •8051055 
 •8078498 
 •8105703 
 •8132671 
 •8159399 
 
 •8185887 
 •8212136 
 •8238145 
 •8263912 
 •8289439 
 
 •8314724 
 •8339768 
 •8364569 
 •8389129 
 •8413447 
 
 + 
 
 35118 
 34939 
 34758 
 34574 
 34388 
 34200 
 
 34009 
 33815 
 33620 
 33422 
 33222 
 
 33020 
 32816 
 32610 
 32402 
 32192 
 
 31980 
 31767 
 31551 
 31331 
 31116 
 
 30896 
 30674 
 30451 
 30226 
 30001 
 
 29773 
 29545 
 29316 
 29085 
 28853 
 
 28620 
 28387 
 28152 
 27917 
 27680 
 
 27443 
 27205 
 26967 
 26728 
 26489 
 
 26249 
 26008 
 25768 
 25527 
 25285 
 
 25044 
 24802 
 24560 
 24318 
 
 A 2 
 
 176 
 179 
 181 
 184 
 186 
 189 
 
 191 
 193 
 196 
 198 
 200 
 
 202 
 204 
 206 
 208 
 210 
 
 212 
 214 
 215 
 217 
 219 
 
 220 
 222 
 223 
 225 
 226 
 
 227 
 228 
 230 
 231 
 232 
 
 233 
 234 
 235 
 235 
 236 
 
 237 
 
 238 
 238 
 239 
 239 
 
 240 
 240 
 241 
 241 
 241 
 
 242 
 242 
 242 
 242 
 242 
 
Tables of the Probability Integral 
 
 TABLE 11.— {continued). 
 
 •3520653 
 •3502919 
 ■3484925 
 •3466G77 
 •3448180 
 •3429439 
 
 •3410458 
 •3391243 
 ■3371799 
 •3352132 
 •3332246 
 
 •3312147 
 •3291840 
 •3271330 
 ■3250623 
 ■3229724 
 
 •3208638 
 •3187371 
 •3165929 
 •3144317 
 •3122539 
 
 •3100603 
 •3078513 
 •3056274 
 •3033893 
 ■3011374 
 
 ■2988724 
 •2965948 
 •2943050 
 ■2920038 
 •2896916 
 
 •2873689 
 •2850364 
 •2826945 
 •2803438 
 •2779849 
 
 ■2756182 
 •2732444 
 •2708640 
 •2684774 
 •2660852 
 
 •2636880 
 •2612863 
 •2588805 
 •2564713 
 •2540591 
 
 •2516443 
 •2492277 
 •2468095 
 •2443904 
 •2419707 
 
 17734 
 
 17994 
 18248 
 18497 
 18741 
 18981 
 
 19215 
 19444 
 19667 
 19886 
 20099 
 
 20307 
 20510 
 20707 
 20899 
 21086 
 
 21267 
 21442 
 21613 
 21777 
 21936 
 
 22090 
 22239 
 22381 
 
 22519 
 22650 
 
 22777 
 22897 
 23013 
 23122 
 23227 
 
 23325 
 23419 
 23507 
 23589 
 23666 
 
 23738 
 23805 
 23866 
 23922 
 23972 
 
 24017 
 24058 
 24093 
 24122 
 24147 
 
 24167 
 24182 
 24191 
 24196 
 
 A 2 
 
 264 
 259 
 254 
 249 
 244 
 239 
 
 234 
 
 229 
 224 
 219 
 213 
 
 208 
 203 
 197 
 192 
 
 187 
 
 181 
 176 
 170 
 165 
 159 
 
 154 
 148 
 143 
 137 
 132 
 
 126 
 121 
 115 
 110 
 104 
 
 99 
 93 
 
 88 
 83 
 
 77 
 
 72 
 66 
 61 
 56 
 51 
 
 45 
 40 
 35 
 30 
 
 25 
 
 20 
 
 15 
 
 10 
 
 5 
 
 
 
 ■00 
 ■01 
 ■02 
 ■03 
 
 ■04 
 
 ■05 
 
 10G 
 1-07 
 108 
 V09 
 1-10 
 
 1-11 
 1-12 
 VIS 
 1-14 
 1-15 
 
 1-16. 
 
 V17 
 
 1-18 
 
 1-19 
 
 1-20 
 
 1-21 
 
 1-22 
 1-23 
 1-2J, 
 1-25 
 
 1-26 
 1-27 
 1-28 
 1-29 
 ISO 
 
 131 
 
 1-32 
 1-33 
 
 1*4 
 
 1-35 
 
 1-36 
 1-37 
 1-38 
 1-39 
 1-1,0 
 
 1-41 
 1-1,2 
 1-P 
 
 1-44 
 
 i-i,r, 
 
 VJf6 
 1-1,7 
 11,8 
 1-1,9 
 1-50 
 
 *(l+n) 
 
 8413447 
 8437524 
 8401358 
 8484960 
 508300 
 8531409 
 
 8554277 
 8576903 
 8599289 
 8621434 
 8643339 
 
 8665005 
 8686431 
 
 8707619 
 
 8728568 
 8749281 
 
 8769756 
 8789995 
 8809999 
 8829768 
 8849303 
 
 8868606 
 8887676 
 8906514 
 8925123 
 8943502 
 
 8961653 
 8979577 
 8997274 
 9014747 
 9031995 
 
 9049021 
 9065825 
 9082409 
 9098773 
 9114920 
 
 9130850 
 9146565 
 9162067 
 
 91773:><; 
 9192433 
 
 9207302 
 9221962 
 9236415 
 9250663 
 
 9204707 
 
 9278550 
 9292191 
 9305634 
 9318879 
 •9331928 
 
 + 
 
 24076 
 23834 
 23592 
 23351 
 23109 
 22868 
 
 22626 
 22386 
 22145 
 21905 
 21665 
 
 21426 
 21188 
 20950 
 20712 
 20475 
 
 20239 
 20004 
 19769 
 19535 
 19302 
 
 19070 
 18839 
 18609 
 18379 
 18151 
 
 17924 
 
 17697 
 17472 
 17248 
 17026 
 
 16804 
 16584 
 16365 
 16147 
 15930 
 
 15715 
 15501 
 15289 
 15078 
 
 14868 
 
 14660 
 14453 
 14248 
 14044 
 13842 
 
 13642 
 13443 
 13245 
 13049 
 
 A 2 
 
 242 
 242 
 242 
 242 
 242 
 241 
 
 241 
 241 
 240 
 240 
 240 
 
 239 
 239 
 238 
 237 
 237 
 
 236 
 235 
 235 
 234 
 233 
 
 232 
 231 
 230 
 
 229 
 228 
 
 227 
 226 
 225 
 224 
 223 
 
 222 
 220 
 219 
 218 
 217 
 
 215 
 214 
 212 
 211 
 210 
 
 208 
 207 
 205 
 204 
 202 
 
 201 
 199 
 197 
 196 
 194 
 
 2419707 
 2395511 
 2371320 
 2347138 
 2322970 
 2298821 
 
 2274696 
 2250599 
 2226535 
 2202508 
 2178522 
 
 2154582 
 2130691 
 2106856 
 2083078 
 2059363 
 
 2035714 
 2012135 
 1988631 
 1966208 
 
 1941861 
 
 1918602 
 1895432 
 1872354 
 1849373 
 1826491 
 
 1803712 
 1781038 
 1758474 
 1736022 
 1713086 
 
 1691468 
 1609370 
 1047397 
 1025551 
 1003833 
 
 1582248 
 1500797 
 1539483 
 1518308 
 1497275 
 
 1470385 
 1455041 
 1435046 
 1414600 
 1394300 
 
 1374165 
 1354181 
 1334353 
 1314684 
 1295176 
 
 24196 
 24191 
 24182 
 24168 
 24149 
 24125 
 
 24097 
 24064 
 24027 
 23986 
 23940 
 
 2389(1 
 23830 
 23778 
 23715 
 23649 
 
 23578 
 83604 
 
 23426 
 23344 
 23259 
 
 23170 
 23077 
 22981 
 22882 
 22779 
 
 22673 
 22564 
 22452 
 22337 
 
 22218 
 
 22097 
 21973 
 21847 
 21717 
 21585 
 
 21451 
 21314 
 21175 
 21033 
 20890 
 
 20744 
 20596 
 20446 
 20294 
 20140 
 
 19985 
 19828 
 19669 
 19508 
 
 1—2 
 
 A 2 
 + 
 
 10 
 14 
 19 
 24 
 
 28 
 33 
 37 
 41 
 
 46 
 
 50 
 54 
 58 
 62 
 66 
 
 70 
 
 74 
 78 
 82 
 85 
 
 89 
 93 
 96 
 99 
 103 
 
 106 
 109 
 112 
 115 
 118 
 
 121 
 124 
 127 
 129 
 132 
 
 134 
 137 
 139 
 142 
 144 
 
 146 
 
 148 
 150 
 152 
 154 
 
 155 
 157 
 159 
 160 
 162 
 
Tables for Statisticians and Biometricians 
 TABLE II. Area and Oidinate in terms of Abscissa. 
 
 iO+«) 
 
 1-50 
 1-51 
 V52 
 1-53 
 1-54 
 1-55 
 
 1-66 
 V57 
 1-58 
 1-59 
 1-00 
 
 1-61 
 1-62 
 1-63 
 1-64 
 1-65 
 
 166 
 167 
 1-68 
 1-C9 
 1-70 
 
 1-71 
 1-72 
 1-73 
 1-7 k 
 V75 
 
 1-76 
 V77 
 1-78 
 1-79 
 1-80 
 
 1-81 
 1-82 
 1-88 
 1-84 
 1-85 
 
 1-86 
 1-87 
 1-88 
 1-89 
 1-90 
 
 1-91 
 1-92 
 V93 
 1-94 
 1-95 
 
 1-96 
 1-97 
 1-98 
 1-99 
 2-00 
 
 •9331928 
 •9344783 
 •9357445 
 ■9369916 
 •9382198 
 •9394292 
 
 •9406201 
 •9417924 
 •9429466 
 •9440826 
 •9452007 
 
 •9463011 
 •9473839 
 •9484493 
 •9494974 
 •9505285 
 
 •9515428 
 •9525403 
 •9535213 
 •9544860 
 •9554345 
 
 •9563671 
 •9572838 
 ■9581849 
 •9590705 
 •9599408 
 
 •9607961 
 •9616364 
 •9624620 
 •9632730 
 •9640697 
 
 •9648521 
 •9656205 
 •9663750 
 •9671159 
 •9678432 
 
 •9685572 
 •9692581 
 •9699460 
 •9706210 
 •9712834 
 
 •9719334 
 •9725711 
 •9731966 
 •9738102 
 •9744119 
 
 •9750021 
 ■9755808 
 •9761482 
 •9767045 
 •9772499 
 
 A 
 
 + 
 
 12855 
 12662 
 12471 
 12282 
 12094 
 11908 
 
 11724 
 11541 
 11360 
 11181 
 11004 
 
 10828 
 10654 
 10482 
 10311 
 10142 
 
 9975 
 9810 
 9647 
 9485 
 9325 
 
 9167 
 9011 
 8856 
 8704 
 8553 
 
 8403 
 8256 
 8110 
 7966 
 
 7824 
 
 7684 
 7545 
 7409 
 7273 
 7140 
 
 7009 
 6879 
 6751 
 6624 
 6500 
 
 6377 
 6255 
 6136 
 6018 
 5902 
 
 5787 
 5674 
 5563 
 5453 
 
 194 
 193 
 191 
 
 189 
 188 
 186 
 
 184 
 183 
 181 
 179 
 177 
 
 176 
 174 
 172 
 170 
 169 
 
 1G7 
 165 
 163 
 162 
 160 
 
 158 
 156 
 155 
 153 
 151 
 
 149 
 147 
 146 
 144 
 142 
 
 140 
 139 
 137 
 135 
 133 
 
 132 
 
 130 
 128 
 126 
 125 
 
 123 
 121 
 120 
 118 
 116 
 
 115 
 113 
 111 
 110 
 
 108 
 
 •1295176 
 •1275830 
 •1256646 
 •1237628 
 ■1218775 
 •1200090 
 
 •1181573 
 •1163228 
 
 •1145048 
 •1127042 
 •1109208 
 
 •1091548 
 •1074061 
 •1056748 
 •1039611 
 •1022649 
 
 •1005864 
 ■0989255 
 ■0972823 
 0956568 
 0940491 
 
 •0924591 
 •0908870 
 •0893326 
 •0877961 
 ■0862773 
 
 •0847764 
 •0832932 
 •0818278 
 •0803801 
 •0789502 
 
 •0775379 
 •0761433 
 ■0747663 
 •0734068 
 •0720649 
 
 •0707404 
 •0694333 
 •0681436 
 •0668711 
 •0656158 
 
 •0643777 
 •0631566 
 •0619524 
 •0607652 
 ■0595947 
 
 •0584409 
 •0573038 
 •0561831 
 •0550789 
 •0539910 
 
 19346 
 19183 
 19018 
 18853 
 18685 
 18517 
 
 18348 
 18177 
 18006 
 17834 
 17661 
 
 17487 
 17312 
 17137 
 16962 
 16786 
 
 16609 
 16432 
 16255 
 16077 
 15899 
 
 15722 
 15544 
 15366 
 15188 
 15010 
 
 14832 
 14654 
 14477 
 14300 
 14123 
 
 13946 
 13770 
 13594 
 13419 
 13245 
 
 13071 
 12897 
 12725 
 12553 
 12382 
 
 12211 
 12041 
 11873 
 11705 
 11538 
 
 11372 
 11206 
 11042 
 10879 
 
 A 2 
 + 
 
 162 
 163 
 165 
 166 
 167 
 168 
 
 169 
 170 
 171 
 172 
 173 
 
 174 
 174 
 175 
 176 
 176 
 
 177 
 177 
 177 
 
 178 
 178 
 
 178 
 178 
 178 
 178 
 178 
 
 178 
 178 
 177 
 177 
 177 
 
 176 
 
 176 
 176 
 175 
 175 
 
 174 
 173 
 173 
 172 
 171 
 
 170 
 170 
 169 
 168 
 167 
 
 166 
 165 
 164 
 163 
 162 
 
 2-00 
 2-01 
 2-02 
 2-03 
 2-04 
 2-05 
 
 2-06 
 2-07 
 g-OS 
 
 i-09 
 2-10 
 
 2-11 
 
 2-12 
 2-13 
 2-14 
 2-15 
 
 2-16 
 2-17 
 2-18 
 2-19 
 2-20 
 
 2-21 
 
 2-22 
 2-23 
 2-24 
 2-25 
 
 2-26 
 2-27 
 2-28 
 2-29 
 2-30 
 
 2-31 
 2S2 
 2-33 
 2-34 
 2-35 
 
 2-36 
 2-37 
 2-38 
 2-39 
 
 2-40 
 
 2-41 
 2-42 
 2-43 
 
 2-44 
 2-45 
 
 2-46 
 2-47 
 2-48 
 2-49 
 2-50 
 
 !(! + «) 
 
 ■9772499 
 •9777844 
 •9783083 
 •9788217 
 •9793248 
 •9798178 
 
 •9803007 
 
 •9807738 
 •9812372 
 •9816911 
 •9821356 
 
 •9S25708 
 •9829970 
 •9834142 
 •9838226 
 •9842224 
 
 •9846137 
 •9849966 
 •9853713 
 •9857379 
 •9860966 
 
 •9864474 
 •9867906 
 •9871263 
 
 •9874545 
 •9877755 
 
 •9880894 
 ■9883962 
 •9886962 
 •9889893 
 •9892759 
 
 •9895559 
 •9898296 
 •9900969 
 •9903581 
 •9906133 
 
 •9908625 
 •9911060 
 •9913437 
 •9915758 
 •9918025 
 
 •9920237 
 •9922397 
 •9924506 
 •9926564 
 •9928572 
 
 •9930531 
 •9932443 
 ■9934309 
 •9936128 
 •9937903 
 
 A 
 
 + 
 
 5345 
 5239 
 5134 
 5031 
 4929 
 4>29 
 
 4731 
 
 4634 
 4539 
 4445 
 4352 
 
 4262 
 4172 
 4084 
 3998 
 3913 
 
 3829 
 3747 
 3666 
 3587 
 3509 
 
 3432 
 
 3357 
 3283 
 3210 
 3138 
 
 3068 
 2999 
 2932 
 2865 
 2800 
 
 2736 
 2674 
 2612 
 2552 
 
 2492 
 
 2434 
 2377 
 2321 
 2267 
 2213 
 
 2160 
 2108 
 2058 
 2008 
 1960 
 
 1912 
 1865 
 
 1820 
 1775 
 
 A 2 
 
 108 
 106 
 105 
 103 
 102 
 100 
 
 98 
 97 
 95 
 94 
 92 
 
 91 
 89 
 
 88 
 86 
 85 
 
 84 
 82 
 81 
 79 
 
 78 
 
 77 
 75 
 74 
 73 
 71 
 
 70 
 69 
 68 
 66 
 65 
 
 64 
 63 
 62 
 60 
 59 
 
 58 
 57 
 56 
 55 
 54 
 
 53 
 52 
 51 
 50 
 49 
 
 48 
 47 
 46 
 45 
 44 
 
Tables of the Probability Integral 
 TABLE II.— {continued). 
 
 •0539910 
 0529192 
 •0518636 
 ■0508239 
 •0498001 
 ■0487920 
 
 0477996 
 
 •0468226 
 •0458611 
 •0449148 
 •0439836 
 
 •0430674 
 •0421661 
 
 •0412795 
 •0404076 
 •0395500 
 
 •0387069 
 •0378779 
 •0370629 
 •0362619 
 •0354746 
 
 •0347009 
 •0339408 
 •0331 939 
 •032460:! 
 •0317397 
 
 •0310319 
 •0303370 
 •0296546 
 •0289847 
 •0283270 
 
 ■0276816 
 •0270481 
 ■0264265 
 ■0258166 
 •0252182 
 
 •0246313 
 •0240556 
 •0234910 
 ■0229374 
 ■0223945 
 
 •0218624 
 0213407 
 ■0208294 
 •0203284 
 •0198374 
 
 •0193563 
 •0188850 
 •0184233 
 ■0179711 
 •0175283 
 
 10717 
 10557 
 10397 
 10238 
 10081 
 9924 
 
 9769 
 9616 
 9463 
 9312 
 9162 
 
 9013 
 8866 
 8720 
 8575 
 8432 
 
 8290 
 8149 
 8010 
 7873 
 7737 
 
 7602 
 7468 
 7337 
 7206 
 7077 
 
 6950 
 6824 
 6699 
 6576 
 6455 
 
 6335 
 6216 
 6099 
 5984 
 5870 
 
 5757 
 5646 
 5536 
 5428 
 5322 
 
 5217 
 5113 
 5011 
 4910 
 4811 
 
 4713 
 
 4617 
 4522 
 
 4428 
 
 A 2 
 
 + 
 
 
 X 
 
 1(1 +«) 
 
 A 
 
 + 
 
 A 2 
 
 z 
 
 162 
 
 2-50 
 
 •9937903 
 
 1731 
 
 1688 
 1646 
 1605 
 1565 
 1525 
 
 44 
 
 •0175283 
 
 161 
 
 
 '2-51 
 
 •9939634 
 
 43 
 
 •0170947 
 
 160 
 
 
 2S2 
 
 •9941323 
 
 42 
 
 •0166701 
 
 159 
 
 
 2-53 
 
 •9942969 
 
 41 
 
 •0162545 
 
 157 
 156 
 
 
 2-54 
 2-55 
 
 •9944574 
 ■9946139 
 
 40 
 39 
 
 •0158476 
 •0154493 
 
 155 
 
 
 2-G6 
 
 •9947664 
 
 1487 
 1449 
 1412 
 1376 
 1341 
 
 39 
 
 •0150596 
 
 154 
 
 
 2-H7 
 
 •9949151 
 
 38 
 
 •0146782 
 
 153 
 
 
 2-58 
 
 •9950600 
 
 37 
 
 •0143051 
 
 161 
 
 
 2S9 
 
 •9952012 
 
 36 
 
 •0139401 
 
 150 
 
 
 2-60 
 
 •9953388 
 
 35 
 
 •0135830 
 
 149 
 
 
 2-61 
 
 •9954729 
 
 1306 
 1272 
 1239 
 1207 
 1176 
 
 35 
 
 •0132337 
 
 147 
 
 
 2-62 
 
 •9956035 
 
 34 
 
 •0128921 
 
 146 
 
 
 2-6S 
 
 •9957308 
 
 33 
 
 •0125581 
 
 145 
 
 
 2-6J t 
 
 •9958547 
 
 32 
 
 •0122315 
 
 143 
 
 
 2-65 
 
 •9959754 
 
 32 
 
 ■0119122 
 
 142 
 
 
 2-66 
 
 •9960930 
 
 1145 
 1115 
 1085 
 1056 
 1028 
 
 31 
 
 •0116001 
 
 140 
 
 
 2-67 
 
 •9962074 
 
 30 
 
 •0112951 
 
 139 
 
 
 2-68 
 
 ■9963189 
 
 29 
 
 •0109969 
 
 138 
 
 
 2-69 
 
 •9964274 
 
 29 
 
 •0107056 
 
 136 
 
 
 2-70 
 
 •9965330 
 
 28 
 
 •0104209 
 
 135 
 
 
 2-71 
 
 •9966358 
 
 1001 
 974 
 948 
 922 
 897 
 
 27 
 
 •0101428 
 
 133 
 
 
 2-72 
 
 •9967359 
 
 27 
 
 •0098712 
 
 132 
 
 
 2-73 
 
 •9968333 
 
 26 
 
 •0096058 
 
 130 
 
 
 2-74 
 
 ■9969280 
 
 26 
 
 •0093406 
 
 129 
 
 
 2-75 
 
 •9970202 
 
 25 
 
 •0090936 
 
 127 
 
 
 2-76 
 
 •9971099 
 
 873 
 849 
 825 
 803 
 781 
 
 24 
 
 •0088465 
 
 126 
 
 
 2-77 
 
 ■9971972 
 
 24 
 
 •0086052 
 
 125 
 
 
 2-78 
 
 •9972821 
 
 23 
 
 •0083697 
 
 123 
 
 
 2-79 
 
 •9973646 
 
 23 
 
 •0081398 
 
 122 
 
 
 2-80 
 
 •9974449 
 
 22 
 
 •0079155 
 
 120 
 
 
 2-81 
 
 •9975229 
 
 759 
 738 
 717 
 697 
 678 
 
 22 
 
 •0076965 
 
 119 
 
 
 2-82 
 
 •9975988 
 
 21 
 
 •0074829 
 
 117 
 
 
 2-83 
 
 •9976726 
 
 21 
 
 •0072744 
 
 116 
 
 
 284 
 
 •9977443 
 
 20 
 
 •0070711 
 
 114 
 
 
 2-85 
 
 •9978140 
 
 20 
 
 •0068728 
 
 113 
 
 
 2-86 
 
 •9978818 
 
 658 
 640 
 622 
 604 
 
 587 
 
 19 
 
 •0066793 
 
 111 
 
 
 2-87 
 
 •9979476 
 
 19 
 
 •0064907 
 
 110 
 
 
 2-88 
 
 ■99801 16 
 
 18 
 
 •0063067 
 
 108 
 
 
 2-89 
 
 •9980738 
 
 18 
 
 •0061274 
 
 107 
 
 
 2-90 
 
 •9981342 
 
 17 
 
 •0059525 
 
 105 
 
 
 2-91 
 
 •9981929 
 
 570 
 553 
 537 
 522 
 507 
 
 17 
 
 •0057821 
 
 104 
 
 
 2-92 
 
 •9982 198 
 
 16 
 
 •0056160 
 
 102 
 
 
 2-93 
 
 •9983052 
 
 16 
 
 •0054541 
 
 101 
 
 
 2-9J t 
 
 •9983589 
 
 16 
 
 •0052963 
 
 99 
 
 
 2-95 
 
 •9984111 
 
 15 
 
 •0051426 
 
 98 
 
 
 2-96 
 
 •9984618 
 
 492 
 
 478 
 464 
 450 . 
 
 15 
 
 •0049929 
 
 96 
 
 
 2-97 
 
 •9985110 
 
 14 
 
 •0048470 
 
 95 
 
 
 298 
 
 •9985588 
 
 14 
 
 •0047050 
 
 93 
 
 
 2-99 
 
 •9986051 
 
 14 
 
 ■0045666 
 
 92 
 
 
 3-00 
 
 •9986501 
 
 13 
 
 •0044318 
 
 A 2 
 + 
 
 4336 
 
 4246 
 4157 
 4069 
 3982 
 3897 
 
 3814 
 3731 
 3650 
 3571 
 3493 
 
 3416 
 3340 
 3266 
 3193 
 3121 
 
 3051 
 8881 
 
 2913 
 2847 
 2781 
 
 2717 
 2654 
 2592 
 2531 
 2471 
 
 2413 
 2355 
 
 2299 
 2244 
 2189 
 
 2136 
 2084 
 2033 
 1983 
 1934 
 
 1886 
 1839 
 1793 
 1748 
 1704 
 
 1661 
 1619 
 1578 
 1537 
 1497 
 
 1459 
 1421 
 1384 
 1347 
 
 92 
 91 
 
 89 
 88 
 86 
 85 
 
 84 
 82 
 81 
 80 
 
 78 
 
 77 
 76 
 74 
 73 
 
 72 
 
 70 
 69 
 68 
 67 
 66 
 
 64 
 63 
 62 
 61 
 60 
 
 59 
 57 
 56 
 55 
 54 
 
 53 
 52 
 51 
 50 
 49 
 
 48 
 47 
 46 
 46 
 44 
 
 43 
 42 
 41 
 40 
 40 
 
 39 
 38 
 37 
 36 
 35 
 
Tables for Statisticians and Biometricians 
 
 TABLE II. Area and Ordinate in terms of Abscissa. 
 
 X 
 
 i(l+<0 
 
 A 
 + 
 
 A 2 
 
 z 
 
 A 
 
 A' 2 
 + 
 
 3-00 
 
 s-oi 
 
 •9986501 
 •9986938 
 
 437 
 424 
 411 
 
 39!) 
 387 
 375 
 
 13 
 13 
 
 •0014318 
 •0043007 
 
 1312 
 1277 
 1243 
 1210 
 1178 
 1146 
 
 35 
 35 
 
 S-02 
 
 •9987361 
 
 13 
 
 •00417;?.) 
 
 34 
 
 s-os 
 
 •9987772 
 
 12 
 
 •00404MJ 
 
 33 
 
 .",■04 
 
 •9988171 
 
 12 
 
 •0039276 
 
 32 
 
 3-05 
 
 •9988558 
 
 12 
 
 •0038098 
 
 32 
 
 S-06 
 $■07 
 
 •9988933 
 ■9989297 
 
 364 
 363 
 
 342 
 332 
 322 
 
 11 
 
 11 
 
 •0036951 
 •0035836 
 
 1115 
 
 1085 
 
 1056 
 
 1027 
 
 999 
 
 31 
 30 
 
 $■08 
 
 •9989650 
 
 11 
 
 •0034751 
 
 29 
 
 $■09 
 
 •9989992 
 
 10 
 
 •0033695 
 
 29 
 
 $■10 
 
 •9990324 
 
 10 
 
 •0032668 
 
 28 
 
 $■11 
 
 •9990046 
 
 312 
 302 
 293 
 
 284 
 275 
 
 10 
 
 •0031669 
 
 971 
 944 
 918 
 
 893 
 868 
 
 27 
 
 $■12 
 
 •9990957 
 
 10 
 
 •0030698 
 
 27 
 
 $■13 
 
 •9991260 
 
 9 
 
 •0029754 
 
 20 
 
 $■14 
 
 •9991553 
 
 9 
 
 •0028835 
 
 20 
 
 $■15 
 
 •9991836 
 
 9 
 
 0027943 
 
 25 
 
 .:-ir, 
 
 ■9992112 
 
 267 
 258 
 250 
 242 
 235 
 
 9 
 
 •0027075 
 
 843 
 820 
 797 
 774 
 752 
 
 24 
 
 $■17 
 
 •9992378 
 
 8 
 
 •0026231 
 
 24 
 
 $■18 
 
 •9992636 
 
 8 
 
 •0025412 
 
 23 
 
 $■19 
 
 •9992886 
 
 8 
 
 •0024615 
 
 23 
 
 $-20 
 
 •9993129 
 
 8 
 
 •0023841 
 
 22 
 
 $-21 
 
 •9993363 
 
 227 
 220 
 213 
 206 
 200 
 
 7 
 
 •0023089 . 
 
 731 
 710 
 689 
 669 
 650 
 
 21 
 
 $■22 
 
 ■9993590 
 
 7 
 
 •0022358 
 
 21 
 
 $■23 
 
 •9993810 
 
 7 
 
 •0021649 
 
 20 
 
 $■21, 
 
 •9994024 
 
 7 
 
 •0020960 
 
 20 
 
 $■25 
 
 •9994230 
 
 7 
 
 •0020290 
 
 19 
 
 326 
 
 •9994429 
 
 193 
 187 
 181 
 175 
 169 
 
 6 
 
 ■0019641 
 
 631 
 612 
 595 
 577 
 500 
 
 19 
 
 3-27 
 
 •9994623 
 
 6 
 
 •0019010 
 
 18 
 
 3-28 
 
 •9994810 
 
 6 
 
 •0018397 
 
 18 
 
 3-29 
 
 •9994991 
 
 6 
 
 •0017803 
 
 17 
 
 3-30 
 
 •9995166 
 
 6 
 
 •0017226 
 
 17 
 
 3-31 
 
 •9995335 
 
 164 
 159 
 153 
 
 148 
 143 
 
 6 
 
 •0016666 
 
 543 
 
 527 
 512 
 496 
 481 
 
 17 
 
 3-32 
 
 •9995499 
 
 5 
 
 •0016122 
 
 10 
 
 3-33 
 
 •9995658 
 
 5 
 
 0015595 
 
 16 
 
 .-:::.', 
 
 •9995811 
 
 5 
 
 •0015084 
 
 15 
 
 3-35 
 
 •9995959 
 
 5 
 
 •0014587 
 
 15 
 
 $■$6 
 
 •9996103 
 
 139 
 134 
 130 
 125 
 121 
 
 5 
 
 •0014106 
 
 407 
 453 
 439 
 420 
 413 
 
 15 
 
 $■$7 
 
 •9996242 
 
 5 
 
 •0013639 
 
 14 
 
 $■38 
 
 •9996376 
 
 4 
 
 •0013187 
 
 14 
 
 $■$9 
 
 •9996505 
 
 4 
 
 •0012748 
 
 13 
 
 S-Jfi 
 
 •9996631 
 
 4 
 
 •0012322 
 
 13 
 
 3-1,1 
 
 •9996752 
 
 117 
 113 
 109 
 106 
 102 
 
 4 
 
 •0011910 
 
 400 
 388 
 370 
 304 
 353 
 
 13 
 
 3-42 
 
 •9996869 
 
 4 
 
 •0011510 
 
 12 
 
 3-4$ 
 
 •9996982 
 
 4 
 
 •0011122 
 
 12 
 
 3-U 
 
 •9997091 
 
 4 
 
 ■0010747 
 
 12 
 
 3-1,5 
 
 •9997197 
 
 4 
 
 •0010383 
 
 11 
 
 $-lfi 
 
 •9997299 
 
 99 
 95 
 92 
 
 89 
 
 3 
 
 •0010030 
 
 342 
 331 
 320 
 310- 
 
 11 
 
 3-1,7 
 
 •9997398 
 
 3 
 
 •0009689 
 
 11 
 
 8-1,8 
 
 •9997493 
 
 3 
 
 •0009358 
 
 10 
 
 8-1,9 
 
 •9997585 
 
 3 
 
 •0009037 
 
 10 
 
 $-50 
 
 •9997674 
 
 3 
 
 •0008727 
 
 10 
 
 $-50 
 351 
 3-52 
 3-53 
 3-51, 
 3-55 
 
 $■56 
 8-57 
 $■58 
 3-59 
 8-60 
 
 361 
 
 .:■>;.> 
 8-63 
 3'Gl, 
 3-65 
 
 3-G6 
 3-67 
 S-G8' 
 $■(!!) 
 
 li-;u 
 
 8-71 
 3-72 
 $■7$ 
 
 $■74 
 $■75 
 
 3-76 
 $■77 
 3-78 
 8-79 
 3-80 
 
 881 
 
 8-82 
 3-83 
 
 O'oJj. 
 
 3-85 
 
 3-86 
 3-87 
 8-88 
 8-89 
 8-90 
 
 8-91 
 
 ,;■>.>; 
 .;■:>.; 
 ,:■'.», 
 8-95 
 
 8-96 
 $■97 
 
 ,:-us 
 899 
 4'00 
 
 *(!+«) 
 
 •9997074 
 •9997759 
 •9997842 
 •9997922 
 •9997999 
 •9998074 
 
 •9998140 
 •9998215 
 •9998282 
 •9998347 
 9998409 
 
 9998409 
 •9998527 
 •9998583 
 •9998037 
 •9998089 
 
 •9998739 
 
 •9998787 
 •9998834 
 •9998879 
 ■9998922 
 
 •9998904 
 •9999004 
 •9999043 
 •9999080 
 •9999110 
 
 •9999150 
 •9999184 
 •9999216 
 •9999247 
 •9999277 
 
 •9999305 
 •9999333 
 •9999359 
 •9999386 
 
 •9999409 
 
 •9999433 
 ■9999456 
 •9999478 
 •9999499 
 •9999519 
 
 •9999539 
 •9999557 
 ■9999575 
 •9999593 
 •9999609 
 
 •9999625 
 •9999641 
 •9999655 
 •9999670 
 •9999083 
 
 A 
 + 
 
 86 
 83 
 
 80 
 77 
 74 
 72 
 
 69 
 67 
 65 
 62 
 
 00 
 
 58 
 56 
 54 
 
 52 
 50 
 
 48 
 47 
 45 
 43 
 42 
 
 40 
 39 
 37 
 36 
 35 
 
 33 
 32 
 
 31 
 
 30 
 29 
 
 28 
 27 
 20 
 25 
 
 24 
 
 23 
 22 
 21 
 20 
 19 
 
 19 
 18 
 17 
 
 17 
 10 
 
 15 
 15 
 14 
 14 
 
 A 2 
 
 3 
 
 2 
 2 
 
 2 
 2 
 
 2 
 2 
 2 
 2 
 2 
 
 2 
 2 
 2 
 2 
 
 2 
 
Tables of the Probability Integral 
 
 TABLE II— (continued). 
 
 z 
 
 A 
 
 A 2 
 
 + 
 
 
 X 
 
 J(l + ") 
 
 A 
 + 
 
 A 2 
 
 ~ 
 
 A 
 
 A 3 
 
 + 
 
 2 
 
 •0008727 
 
 301 
 
 291 
 282 
 273 
 264 
 256 
 
 10 
 
 4-oo 
 
 •9999683 
 
 13 
 13 
 
 12 
 12 
 11 
 11 
 
 1 
 
 •0001338 
 
 53 
 51 
 49 
 47 
 45 
 43 
 
 ■0008426 
 
 10 
 
 
 lfOl 
 
 •9999696 
 
 1 
 
 •0001286 
 
 2 
 
 •0008135 
 
 9 
 
 
 4-02 
 
 •9999709 
 
 
 
 •0001235 
 
 2 
 
 ■0007853 
 
 9 
 
 
 4-03 
 
 •9999721 
 
 
 •0001186 
 
 2 
 
 •0007581 
 •0007317 
 
 9 
 
 8 
 
 
 4-04 
 4-05 
 
 •9999733 
 •9999744 
 
 
 ■0001140 
 •0001094 
 
 2 
 2 
 
 •0007061 
 
 247 
 239 
 232 
 224 
 217 
 
 8 
 
 
 4-06 
 
 •9999755 
 
 10 
 
 10 
 
 9 
 
 9 
 
 9 
 
 
 •0001051 
 
 42 
 40 
 39 
 37 
 36 
 
 2 
 
 •0006814 
 
 8 
 
 
 407 
 
 •9999765 
 
 
 •0001009 
 
 2 
 
 •0006575 
 
 8 
 
 
 4-os 
 
 •9999775 
 
 
 •0000969 
 
 2 
 
 •0006343 
 
 8 
 
 
 4'09 
 
 •9999784 
 
 
 ■0000930 
 
 
 0006119 
 
 7 
 
 
 4-10 
 
 ■9999793 
 
 
 •0000893 
 
 
 •0005902 
 
 210 
 203 
 196 
 189 
 183 
 
 7 
 
 
 4-n 
 
 •9999802 
 
 8 
 8 
 8 
 
 7 
 7 
 
 
 •0000857 
 
 35 
 33 
 
 32 
 31 
 
 30 
 
 
 •0005693 
 
 7 
 
 
 4-12 
 
 •999981 1 
 
 
 •0000822 
 
 
 •0005490 
 
 7 
 
 
 4-13 
 
 •9999819 
 
 
 •0000789 
 
 
 •0005294 
 
 6 
 
 
 4-n 
 
 •9999826 
 
 
 •0000757 
 
 
 •0005105 
 
 6 
 
 
 4-15 
 
 •9999834 
 
 
 •0000726 
 
 
 •0004921 
 
 177 
 171 
 165 
 160 
 155 
 
 6 
 
 
 tie 
 
 •9999841 
 
 
 
 •0000697 
 
 28 
 
 
 •0004744 
 
 6 
 
 
 4-n 
 
 •9999848 
 
 7 
 
 
 ■0000668 
 
 
 •0004673 
 
 6 
 
 
 4-is 
 
 •9999854 
 
 i 
 
 6 
 6 
 6 
 
 
 •0000641 
 
 27 
 26 
 25 
 24 
 
 
 •0004408 
 
 6 
 
 
 J,- J!) 
 
 •9999861 
 
 
 •0000615 
 
 
 •0004248 
 
 5 
 
 
 4-20 
 
 •9999867 
 
 
 •0000589 
 
 
 •0004093 
 
 149 
 144 
 139 
 135 
 130 
 
 5 
 
 
 4-21 
 
 •9999872 
 
 6 
 
 
 ■0000565 
 
 23 
 
 22 
 22 
 21 
 20 
 
 
 •0003944 
 
 5 
 
 
 4-22 
 
 •9999878 
 
 
 •0000542 
 
 
 •0003800 
 
 5 
 
 
 4*s 
 
 •9999883 
 
 
 
 
 •0000519 
 
 
 •0003661 
 •0003526 
 
 5 
 5 
 
 
 4*4 
 
 4-25 
 
 ■0999888 
 •9999893 
 
 
 
 5 
 5 
 
 
 •0000498 
 •0000477 
 
 
 •0003396 
 •0003271 
 
 125 
 121 
 117 
 113 
 
 109 
 
 4 
 4 
 
 
 4-26 
 4-27 
 
 •9999898 
 •9999902 
 
 4 
 4 
 4 
 4 
 4 
 
 
 •0000457 
 •0(XK)438 
 
 19 
 18 
 18 
 17 
 16 
 
 
 •0003149 
 
 4 
 
 
 4-2S 
 
 •9999907 
 
 
 •0000420 
 
 
 •0003032 
 
 4 
 
 
 4-29 
 
 •9999911 
 
 
 •0000402 
 
 
 •0002919 
 
 4 
 
 
 4'30 
 
 ■9999915 
 
 
 •0000385 
 
 
 •0002810 
 
 105 
 
 102 
 
 98 
 
 95 
 
 91 
 
 4 
 
 
 4-si 
 
 •9999918 
 
 4 
 3 
 3 
 3 
 3 
 
 
 •0000369 
 
 16 
 15 
 14 
 14 
 13 
 
 
 •0002705 
 
 4 
 
 
 4-32 
 
 •9999922 
 
 
 •0000354 
 
 
 ■0008604 
 
 4 
 
 
 4-33 
 
 '8998920 
 
 
 •0000339 
 
 
 0002606 
 
 3 
 
 
 4-34 
 
 •9999929 
 
 
 ■0000324 
 
 
 •0002411 
 
 3 
 
 
 4-35 
 
 •9999932 
 
 
 •0000310 
 
 
 0002320 
 
 88 
 85 
 82 
 79 
 
 76 
 
 3 
 
 
 4'36 
 
 •9999935 
 
 3 
 
 
 •0000297 
 
 13 
 12 
 12 
 11 
 11 
 
 
 •0002232 
 
 3 
 
 
 4-37 
 
 •9999938 
 
 
 •0000284 
 
 
 •0002147 
 
 3 
 
 
 4-38 
 
 •9999941 
 
 
 
 3 
 3 
 
 2 
 
 
 •0000272 
 
 
 
 •0002065 
 
 3 
 
 
 4-39 
 
 •9999943 
 
 
 ■0(100261 
 
 
 •0001987 
 
 3 
 
 
 4-40 
 
 •9999946 
 
 
 •0000249 
 
 
 •0001910 
 
 73 
 71 
 68 
 66 
 63 
 
 3 
 
 
 4-41 
 
 •9999948 
 
 9 
 
 
 •0000239 
 
 10 
 
 10 
 
 9 
 
 9 
 
 9 
 
 
 •0001837 
 
 3 
 
 
 4-42 
 
 •9999951 
 
 2 
 2 
 2 
 
 2 
 
 
 •0000228 
 
 
 •0001766 
 
 3 
 
 
 4-4S 
 
 •9999953 
 
 
 ■0000218 
 
 
 ■0001698 
 
 2 
 
 
 4-44 
 
 •9999955 
 
 
 ■0000209 
 
 
 •0001633 
 
 2 
 
 
 4-45 
 
 •9999957 
 
 
 •0000200 
 
 
 •0001569 
 
 61 
 59 
 
 57 
 55 
 
 2 
 
 
 4-4fl 
 
 •9999959 
 
 2 
 
 B 
 
 
 •0000191 
 
 8 
 8 
 8 
 
 7 
 
 
 ■0001508 
 
 2 
 
 
 4-47 
 
 •9999961 
 
 
 •0000183 
 
 
 •0001449 
 •0001393 
 
 2 
 
 2 
 
 
 4-48 
 4-49 
 4-50 
 
 •9999963 
 •9999964 
 
 2 
 2 
 
 
 •0000175 
 •0000167 
 
 
 0001338 
 
 
 2 
 
 
 •9999960 
 
 
 ■0000160 
 
 
Tables for Statisticians and Biometricians 
 TABLE II. Area and Ordinate in terms of Abscissa*. 
 
 X 
 
 i(i+o) 
 
 z 
 
 4-50 
 
 66023 
 
 159837 
 
 Jf-Bl 
 
 67586 
 
 152797 
 
 4~52 
 
 69080 
 
 146051 
 
 4'53 
 
 70508 
 
 139590 
 
 m 
 
 71873 
 
 133401 
 
 4-55 
 
 73177 
 
 127473 
 
 4-56 
 
 74423 
 
 121797 
 
 4-57 
 
 75614 
 
 116362 
 
 4-58 
 
 76751 
 
 111159 
 
 4-50 
 
 77838 
 
 106177 
 
 4-60 
 
 78875 
 
 101409 
 
 4-61 
 
 79867 
 
 96845 
 
 4-62 
 
 80813 
 
 92477 
 
 4-63 
 
 81717 
 
 88297 
 
 4-64 
 
 82580 
 
 84298 
 
 4-65 
 
 83403 
 
 80472 
 
 4-66 
 
 84190 
 
 76812 
 
 4-67 
 
 84940 
 
 73311 
 
 4-68 
 
 85056 
 
 69962 
 
 4-69 
 
 86340 
 
 66760 
 
 4-70 
 
 86992 
 
 63698 
 
 4-71 
 
 87614 
 
 60771 
 
 4'72 
 
 88208 
 
 57972 
 
 4'7S 
 
 88774 
 
 55296 
 
 7-74 
 
 89314 
 
 52739 
 
 4-75 
 
 89829 
 
 50295 
 
 476 
 
 90320 
 
 47960 
 
 4'77 
 
 90789 
 
 45728 
 
 4-78 
 
 91235 
 
 43596 
 
 4-70 
 
 91661 
 
 41559 
 
 4-80 
 
 92067 
 
 39613 
 
 4-81 
 
 92453 
 
 37755 
 
 4-82 
 
 92822 
 
 35980 
 
 4-83 
 
 93173 
 
 34285 
 
 4-84 
 
 93508 
 
 32667 
 
 4-85 
 
 93827 
 
 31122 
 
 4'86 
 
 94131 
 
 29647 
 
 .>,-s; 
 
 94420 
 
 28239 
 
 4-88 
 
 94696 
 
 26895 
 
 4-89 
 
 94958 
 
 25613 
 
 4-90 
 
 95208 
 
 24390 
 
 4-91 
 
 95446 
 
 23222 
 
 4-92 
 
 95673 
 
 22108 
 
 4-93 
 
 95889 
 
 21046 
 
 4-94 
 
 96094 
 
 20033 
 
 4-95 
 
 96289 
 
 19066 
 
 4'9(i 
 
 96475 
 
 18144 
 
 4-97 
 
 96652 
 
 17265 
 
 4-98 
 
 96821 
 
 16428 
 
 499 
 
 96981 
 
 15629 
 
 X 
 
 i(l+n) 
 
 z 
 
 o-oo 
 
 97133 
 
 14867 
 
 5-01 
 
 97278 
 
 14141 
 
 5-02 
 
 97416 
 
 13450 
 
 5-03 
 
 97548 
 
 12791 
 
 5-04 
 
 97672 
 
 12162 
 
 5V5 
 
 97791 
 
 11564 
 
 5-Uti 
 
 97904 
 
 10994 
 
 5V7 
 
 98011 
 
 10451 
 
 5V8 
 
 98113 
 
 9934 
 
 0V9 
 
 98210 
 
 9441 
 
 5-10 
 
 98302 
 
 8972 
 
 5-11 
 
 98389 
 
 8526 
 
 5-12 
 
 98472 
 
 8101 
 
 5-13 
 
 98551 
 
 7696 
 
 5-14 
 
 98626 
 
 7311 
 
 5-15 
 
 98698 
 
 6944 
 
 5-16 
 
 98765 
 
 6595 
 
 5-17 
 
 98830 
 
 6263 
 
 5-18 
 
 98891 
 
 5947 
 
 5-19 
 
 98949 
 
 5647 
 
 5-m 
 
 99004 
 
 5361 
 
 5-21 
 
 99056 
 
 5089 
 
 5-22 
 
 99105 
 
 4831 
 
 6-2$ 
 
 99152 
 
 4585 
 
 5-24 
 
 99197 
 
 4351 
 
 5-25 
 
 99240 
 
 4128 
 
 6-26 
 
 99280 
 
 3917 
 
 5-27 
 
 99318 
 
 3716 
 
 5-28 
 
 99354 
 
 3525 
 
 5-29 
 
 99388 
 
 3344 
 
 530 
 
 99421 
 
 3171 
 
 6-31 
 
 99452 
 
 3007 
 
 5-32 
 
 99481 
 
 2852 
 
 5-33 
 
 99509 
 
 2704 
 
 5-34 
 
 99535 
 
 2563 
 
 5-35 
 
 99560 
 
 2430 
 
 5-36 
 
 99584 
 
 2303 
 
 5-37 
 
 99606 
 
 2183 
 
 5-38 
 
 99628 
 
 2069 
 
 5-39 
 
 99648 
 
 1960 
 
 5'40 
 
 99667 
 
 1857 
 
 5-41 
 
 99685 
 
 1760 
 
 5-42 
 
 99702 
 
 1667 
 
 0-43 
 
 99718 
 
 1579 
 
 5-44 
 
 99734 
 
 1495 
 
 5-45 
 
 99748 
 
 1416 
 
 5-46 
 
 99762 
 
 1341 
 
 5'47 
 
 99775 
 
 1270 
 
 5-48 
 
 99787 
 
 1202 
 
 5-49 
 
 99799 
 
 1138 
 
 X 
 
 *(!+«) 
 
 z 
 
 5-50 
 
 99810 
 
 1077 
 
 G-51 
 
 99821 
 
 1019 
 
 5-52 
 
 99831 
 
 965 
 
 5-53 
 
 99840 
 
 913 
 
 5-54 
 
 99849 
 
 864 
 
 5-55 
 
 99857 
 
 817 
 
 5-56 
 
 99865 
 
 773 
 
 5-57 
 
 99873 
 
 731 
 
 5-58 
 
 99880 
 
 691 
 
 5-59 
 
 99886 
 
 654 
 
 5-60 
 
 99893 
 
 618 
 
 561 
 
 99899 
 
 585 
 
 5-62 
 
 99905 
 
 553 
 
 5-63 
 
 99910 
 
 522 
 
 5-64 
 
 99915 
 
 494 
 
 5-65 
 
 99920 
 
 467 
 
 5-66 
 
 99924 
 
 441 
 
 5-67 
 
 99929 
 
 417 
 
 5-68 
 
 99933 
 
 394 
 
 6-69 
 
 99936 
 
 372 
 
 5-70 
 
 99940 
 
 351 
 
 5-71 
 
 99944 
 
 332 
 
 5-72 
 
 99947 
 
 313 
 
 5-7S 
 
 99950 
 
 296 
 
 5-74 
 
 99953 
 
 280 
 
 5-75 
 
 99955 
 
 264 
 
 6-76 
 
 99958 
 
 249 
 
 5-77 
 
 99960 
 
 235 
 
 5-78 
 
 99963 
 
 222 
 
 5-79 
 
 99965 
 
 210 
 
 5-SU 
 
 99967 
 
 198 
 
 5-81 
 
 99969 
 
 187 
 
 5-82 
 
 99971 
 
 176 
 
 5-83 
 
 99972 
 
 166 
 
 5-84 
 
 99974 
 
 157 
 
 B-8o 
 
 99975 
 
 148 
 
 6-86 
 
 99977 
 
 139 
 
 6-87 
 
 99978 
 
 131 
 
 5-88 
 
 99979 
 
 124 
 
 5-89 
 
 99981 
 
 117 
 
 5-90 
 
 99982 
 
 110 
 
 5-91 
 
 99983 
 
 104 
 
 r>-<)2 
 
 99984 
 
 98 
 
 6-98 
 
 99985 
 
 92 
 
 5-94 
 
 99986 
 
 87 
 
 5-95 
 
 99987 
 
 82 
 
 5-96 
 
 99987 
 
 77 
 
 6-97 
 
 99988 
 
 73 
 
 5-98 
 
 99989 
 
 68 
 
 699 
 
 99990 
 
 65 
 
 6V0 
 
 89890 
 
 61 
 
 * PreBx -99999 to each entry. 
 
°f71 
 
 o oa 
 
 Tables of the Probability Integral 9 
 
 TABLE III. Abscissa and Ordinate in terms of difference of Areas. 
 
 ■00 
 ■01 
 •OS 
 ■03 
 
 ■04 
 
 ■05 
 
 ■06 
 ■07 
 ■08 
 ■00 
 ■10 
 
 ■11 
 ■12 
 •IS 
 ■H 
 
 •16 
 ■17 
 ■18 
 ■19 
 ■20 
 
 ■21 
 
 •so 
 ■si 
 
 •32 
 ■S3 
 ■Sit 
 •SB 
 
 •86 
 ■37 
 ■38 
 ■SO 
 ■40 
 
 ■41 
 
 ■44 
 •45 
 
 ■46 
 
 ■47 
 
 ■48 
 ■49 
 ■50 
 
 •ooooooo 
 
 •0125335 
 •0250689 
 •0376083 
 •0501536 
 •0627068 
 
 •0752699 
 •0878448 
 •1004337 
 •1130385 
 •1256613 
 
 •1383042 
 •1509692 
 •1636585 
 •1763742 
 •1891184 
 
 •2018935 
 ■2147016 
 •2275450 
 •2404260 
 •2533471 
 
 •2663106 
 •2793190 
 •2923749 
 •3054808 
 •3186394 
 
 •3318533 
 •3451255 
 •3584588 
 •3718561 
 •3853205 
 
 •3988551 
 •4124631 
 •4261480 
 •4399132 
 •4537622 
 
 •4676988 
 •4817268 
 •4958503 
 •5100735 
 •5244005 
 
 •5388360 
 •5533847 
 •5680515 
 •5828415 
 •5977601 
 
 •6128130 
 •6280060 
 •6433454 
 
 •6588377 
 •6744898 
 
 A 
 
 + 
 
 125335 
 125354 
 125394 
 125453 
 125532 
 125631 
 
 125750 
 125889 
 126048 
 126228 
 126429 
 
 126650 
 126893 
 127157 
 127443 
 127751 
 
 128081 
 128434 
 128811 
 129211 
 129635 
 
 130084 
 130659 
 131059 
 131586 
 132140 
 
 132722 
 133333 
 133973 
 134644 
 135346 
 
 136081 
 136849 
 137652 
 138490 
 139366 
 
 140281 
 141235 
 142231 
 143271 
 144355 
 
 145487 
 146668 
 147900 
 149186 
 150529 
 
 151930 
 153394 
 154923 
 156521 
 
 A a 
 
 + 
 
 
 20 
 39 
 59 
 79 
 99 
 
 119 
 
 139 
 159 
 180 
 201 
 
 221 
 243 
 264 
 
 286 
 308 
 
 330 
 353 
 
 376 
 
 400 
 424 
 
 449 
 474 
 500 
 527 
 554 
 
 582 
 611 
 640 
 671 
 702 
 
 735 
 
 768 
 803 
 839 
 876 
 
 914 
 
 954 
 
 996 
 
 1039 
 
 1085 
 
 1132 
 1181 
 1232 
 1286 
 1342 
 
 1402 
 1464 
 1529 
 1598 
 1670 
 
 A 3 
 
 + 
 
 20 
 20 
 20 
 20 
 20 
 20 
 
 20 
 20 
 20 
 21 
 21 
 
 21 
 21 
 22 
 22 
 22 
 
 23 
 23 
 24 
 24 
 25 
 
 25 
 26 
 
 27 
 27 
 28 
 
 29 
 30 
 30 
 31 
 32 
 
 34 
 35 
 36 
 37 
 39 
 
 40 
 42 
 43 
 45 
 47 
 
 49 
 51 
 54 
 56 
 59 
 
 62 
 65 
 69 
 
 72 
 
 •3989423 
 •3989109 
 •3988169 
 ■3986603 
 •3984408 
 •3981587 
 
 •3978138 
 •3974060 
 •3969353 
 ■3964016 
 •3958049 
 
 •3951450 
 •3944218 
 •3936352 
 •3927852 
 •3918715 
 
 •3908939 
 •3898525 
 •3887469 
 •3875769 
 •3863425 
 
 •3850434 
 ■3836794 
 •3822501 
 ■3807555 
 •3791952 
 
 •3775690 
 •3758766 
 •3741177 
 •3722919 
 •3703990 
 
 •3684386 
 •3664103 
 •3643138 
 •3621487 
 •3599146 
 
 •3576109 
 •3552374 
 •3527935 
 •3502788 
 •3476926 
 
 •3450346 
 •3423041 
 •3395005 
 •3366233 
 •3336719 
 
 •3306455 
 •3275435 
 •3243652 
 •3211098 
 •3177766 
 
 313 
 
 940 
 1567 
 2194 
 2821 
 3449 
 
 4078 
 4707 
 5337 
 5967 
 6599 
 
 7232 
 7866 
 8501 
 9137 
 9775 
 
 10415 
 11056 
 11699 
 12344 
 12991 
 
 13641 
 14292 
 14946 
 15603 
 16262 
 
 16924 
 17589 
 18258 
 18929 
 19604 
 
 20283 
 20965 
 21651 
 22342 
 23036 
 
 23735 
 24439 
 25148 
 25861 
 26580 
 
 27305 
 28035 
 28772 
 29514 
 30264 
 
 31020 
 31783 
 32554 
 33333 
 
 627 
 627 
 627 
 627 
 627 
 628 
 
 628 
 629 
 630 
 631 
 632 
 
 633 
 634 
 635 
 636 
 638 
 
 640 
 641 
 643 
 645 
 647 
 
 649 
 652 
 654 
 657 
 659 
 
 662 
 665 
 668 
 672 
 675 
 
 679 
 682 
 686 
 690 
 695 
 
 699 
 704 
 709 
 714 
 719 
 
 725 
 730 
 736 
 743 
 749 
 
 756 
 763 
 771 
 
 779 
 
 787 
 
 A 3 
 
 2 
 2 
 
 2 
 2 
 2 
 
 2 
 2 
 3 
 3 
 3 
 
 3 
 3 
 3 
 3 
 
 4 
 
 4 
 4 
 4 
 4 
 4 
 
 5 
 5 
 5 
 5 
 6 
 
 6 
 6 
 6 
 
 7 
 7 
 
 7 
 7 
 8 
 8 
 
10 
 
 Tables for Statisticians and Biometricians 
 
 TABLE III. Abscissa and Ordinate in terms of difference of Areas. 
 
 ■50 
 ■51 
 
 1 
 •( 
 
 •52 
 
 
 ■53 
 
 , 
 
 n 
 
 
 •55 
 
 . 
 
 ■56 
 
 
 ■57 
 
 
 ■58 
 
 •1 
 
 ■59 
 
 • 
 
 ■60 
 
 •1 
 
 •61 
 
 •1 
 
 ■62 
 
 •1 
 
 ■63 
 
 •( 
 
 ■64. 
 
 • 
 
 ■65 
 
 ■ 
 
 ■66 
 
 • 
 
 •67 
 
 .; 
 
 ■68 
 
 * 
 
 ■69 
 
 11 
 
 ■70 
 
 11 
 
 ■71 
 
 11 
 
 •72 
 
 I' 
 
 •73 
 
 1- 
 
 •7k 
 
 1- 
 
 ■75 
 
 1- 
 
 •76 
 
 1- 
 
 ■77 
 
 1- 
 
 ■78 
 
 I' 
 
 •79 
 
 1- 
 
 •80 
 
 I' 
 
 •6744898 
 •6903088 
 •7063026 
 •7224791 
 •7388468 
 •7554150 
 
 •7721932 
 •7891917 
 •8064212 
 •8238936 
 •8416212 
 
 •8596174 
 •8778963 
 ■8964734 
 •9153651 
 •9345893 
 
 •9541653 
 •9741139 
 •9944579 
 •0152220 
 •0364334 
 
 •0581216 
 •0803193 
 •1030626 
 •1263911 
 •1503494 
 
 •1749868 
 •2003589 
 •2265281 
 •2535654 
 •2815516 
 
 A 
 + 
 
 158191 
 159937 
 161765 
 163678 
 165682 
 167782 
 
 169984 
 172296 
 174724 
 177276 
 179961 
 
 182789 
 185771 
 188917 
 192242 
 195760 
 
 199486 
 203440 
 207641 
 212114 
 216882 
 
 221977 
 227432 
 233286 
 239583 
 246374 
 
 253721 
 261693 
 270373 
 279861 
 
 A 2 
 
 + 
 
 1670 
 1747 
 1828 
 1913 
 2004 
 2100 
 
 2203 
 2312 
 2428 
 2552 
 2685 
 
 2828 
 2981 
 3147 
 3325 
 3518 
 
 3727 
 3954 
 4201 
 4472 
 4769 
 
 5095 
 5455 
 5854 
 6297 
 6792 
 
 7347 
 7972 
 8681 
 9488 
 10414 
 
 A 3 
 
 + 
 
 76 
 81 
 86 
 91 
 96 
 102 
 
 109 
 116 
 124 
 133 
 143 
 
 153 
 165 
 178 
 193 
 209 
 
 227 
 248 
 271 
 297 
 326 
 
 360 
 399 
 443 
 495 
 555 
 
 625 
 709 
 808 
 926 
 
 3177766 
 143646 
 3108732 
 3073013 
 3036481 
 2999125 
 
 2960936 
 2921902 
 2882013 
 2841256 
 2799619 
 
 2757089 
 2713653 
 2669295 
 2624000 
 2577753 
 
 2530535 
 2482330 
 2433117 
 2382877 
 2331588 
 
 2279226 
 2225767 
 2171185 
 2115451 
 2058535 
 
 2000405 
 1941024 
 1880356 
 1818357 
 1754983 
 
 34119 
 34915 
 35719 
 36532 
 37356 
 38189 
 
 39034 
 39889 
 40757 
 41637 
 42530 
 
 43437 
 44358 
 
 45295 
 46247 
 47217 
 
 48205 
 49213 
 50240 
 51289 
 52362 
 
 53459 
 54582 
 55734 
 56916 
 58130 
 
 59380 
 60669 
 61999 
 63374 
 
 A 3 
 
 787 
 795 
 804 
 814 
 823 
 834 
 
 844 
 856 
 867 
 880 
 893 
 
 907 
 921 
 937 
 953 
 970 
 
 988 
 1007 
 1028 
 1049 
 1072 
 
 1097 
 1123 
 1152 
 1182 
 1215 
 
 1250 
 1288 
 1330 
 1375 
 1425 
 
 A 3 
 
 9 
 
 9 
 
 9 
 
 10 
 
 10 
 
 11 
 
 11 
 12 
 12 
 13 
 14 
 
 15 
 15 
 16 
 17 
 18 
 
 19 
 20 
 22 
 23 
 25 
 
 26 
 28 
 30 
 33 
 35 
 
 38 
 42 
 45 
 50 
 
Tables of the Probability Integral 
 
 11 
 
 TABLE IV. 
 
 Extension of Table of the Probability Integral F=^(l — a). 
 
 1 f°° 
 F=-f=\ e~te*dx. The table gives (— logJP) for x. 
 
 v2ttj x 
 
 X 
 
 -logF 
 
 X 
 
 -log J? 
 
 X 
 
 -log J 1 
 
 5 
 
 6-54265 
 
 SO 
 
 197-30921 
 
 50 
 
 544-96634 
 
 6 
 
 9-00586 
 
 SI 
 
 210-56940 
 
 60 
 
 783-90743 
 
 7 
 
 11-89285 
 
 82 
 
 224-26344 
 
 70 
 
 1066-26576 
 
 8 
 
 15-20614 
 
 S3 
 
 238-39135 
 
 80 
 
 1392-04459 
 
 9 
 
 18-94746 
 
 Sit 
 
 252-95315 
 
 90 
 
 1761-24604 
 
 10 
 
 23-11805 
 
 85 
 
 267-94888 
 
 100 
 
 2173-87154 
 
 11 
 
 27-71882 
 
 SO 
 
 283-37855 
 
 150 
 
 4888-38812 
 
 12 
 
 32-75044 
 
 vi 
 
 299-24218 
 
 200 
 
 8688-58977 
 
 13 
 
 38-21345 
 
 88 
 
 315-53979 
 
 250 
 
 13574-49960 
 
 H 
 
 44-10827 
 
 39 
 
 332-27139 
 
 300 
 
 19546-12790 
 
 15 
 
 50-43522 
 
 40 
 
 349-43701 
 
 350 
 
 26603-48018 
 
 10 
 
 57-19458 
 
 41 
 
 367-03664 
 
 400 
 
 34746-55970 
 
 n 
 
 64-38658 
 
 4* 
 
 385-07032 
 
 450 
 
 43975-36860 
 
 18 
 
 72-01140 
 
 43 
 
 403-53804 
 
 500 
 
 54289-40830 
 
 19 
 
 80-06919 
 
 44 
 
 422-43983 
 
 
 
 20 
 
 88-56010 
 
 45 
 
 441-77568 
 
 
 N.B. Toobtain anything 
 
 n 
 
 97-48422 
 
 46 
 
 461-54561 
 
 
 but a rough apprecia- 
 
 22 
 
 106-84167 
 
 47 
 
 481-74964 
 
 
 tion after x = 50, the 
 
 23 
 
 116-63253 
 
 48 
 
 502-38776 
 
 
 table would require 
 
 ft* 
 
 126-85686 
 
 49 
 
 523-45999 
 
 
 much extension, but 
 for many practical 
 
 25 
 
 137-51475 
 
 50 
 
 544-96634 
 
 
 problems it suffices to 
 
 26 
 
 148-60624 
 
 
 
 
 take after #=50: 
 
 27 
 
 16013139 
 
 
 
 
 
 28 
 
 172-09024 
 
 
 
 
 F= J_Vi*\ 
 
 29 
 
 184-48283 
 
 
 
 
 n/2tt* 
 
 SO 
 
 197-30921 
 
 
 
 
 
 From each of the values in this table -30103 must be subtracted, if we wish to 
 obtain the probability 2F, then given by ( - log 2F), that the value is greater than .v. 
 without regard to sign. 
 
 2—2 
 
12 Tables for Statisticians and Biometricians 
 
 TABLE V. Probable Errors of Means and Standard Deviations. 
 
 n 
 
 Xi 
 
 X* 
 
 1 
 
 •67449 
 
 •47694 
 
 2 
 
 •47694 
 
 •33724 
 
 3 
 
 •38942 
 
 •27536 
 
 4 
 
 •33724 
 
 •23847 
 
 5 
 
 •30164 
 
 •21329 
 
 6 
 
 •27536 
 
 •19471 
 
 7 
 
 •25493 
 
 •18026 
 
 8 
 
 •23847 
 
 •16862 
 
 9 
 
 •22483 
 
 •15898 
 
 10 
 
 •21329 
 
 •15082 
 
 11 
 
 •20337 
 
 •14380 
 
 12 
 
 •19471 
 
 •13768 
 
 13 
 
 •18707 
 
 •13228 
 
 U 
 
 •18026 
 
 •12747 
 
 15 
 
 •17415 
 
 •12314 
 
 16 
 
 •16862 
 
 •11923 
 
 17 
 
 •16359 
 
 •11567 
 
 18 
 
 •15898 
 
 •11241 
 
 19 
 
 •15474 
 
 •10942 
 
 20 
 
 •1508-2 
 
 •10665 
 
 21 
 
 •14719 
 
 •10408 
 
 22 
 
 •14380 
 
 •10168 
 
 23 
 
 •14064 
 
 •09945 
 
 2J t 
 
 •13768 
 
 •09735 
 
 25 
 
 •13490 
 
 •09539 
 
 26 
 
 •13228 
 
 •09353 
 
 27 
 
 •12981 
 
 •09179 
 
 28 
 
 •12747 
 
 •09013 
 
 29 
 
 •12525 
 
 •08856 
 
 SO 
 
 •12314 
 
 •08708 
 
 31 
 
 •12114 
 
 •08566 
 
 32 
 
 •11923 
 
 •08431 
 
 S3 
 
 •11741 
 
 •08302 
 
 34 
 
 •11567 
 
 •08179 
 
 35 
 
 •11401 
 
 •08062 
 
 36 
 
 •11241 
 
 •07949 
 
 S7 
 
 •11088 
 
 •07841 
 
 S8 
 
 •10942 
 
 •07737 
 
 39 
 
 •10800 
 
 •07637 
 
 40 
 
 •10665 
 
 •07541 
 
 41 
 
 •10534 
 
 •07448 
 
 42 
 
 •10408 
 
 •07359 
 
 43 
 
 •10286 
 
 •07273 
 
 44 
 
 •10168 
 
 •07190 
 
 45 
 
 •10055 
 
 •07110 
 
 46 
 
 •09946 
 
 •07032 
 
 47 
 
 •09838 
 
 •06957 
 
 48 
 
 •09735 
 
 •06884 
 
 49 
 
 •09636 
 
 •06813 
 
 50 
 
 •09539 
 
 •06745 
 
 n 
 
 51 
 
 *i 
 
 x 2 
 
 •09445 
 
 •06678 
 
 52 
 
 •09353 
 
 •06614 
 
 53 
 
 •09265 
 
 •06551 
 
 54 
 
 •09179 
 
 •06490 
 
 55 
 
 •09095 
 
 •06431 
 
 56 
 
 •09013 
 
 •06373 
 
 57 
 
 •08934 
 
 •06317 
 
 58 
 
 •08856 
 
 •06262 
 
 59 
 
 •08781 
 
 •06209 
 
 60 
 
 •08708 
 
 •06157 
 
 61 
 
 •08C36 
 
 •06107 
 
 68 
 
 •08566 
 
 •06057 
 
 63 
 
 •08198 
 
 •06009 
 
 64 
 
 •08431 
 
 •05962 
 
 65 
 
 •08366 
 
 ■05916 
 
 66 
 
 •08302 
 
 •05871 
 
 67 
 
 •08240 
 
 •05827 
 
 68 
 
 •08179 
 
 •05784 
 
 69 
 
 •08120 
 
 •05742 
 
 70 
 
 •08062 
 
 •05700 
 
 71 
 
 •08005 
 
 •05660 
 
 72 
 
 •07949 
 
 •05621 
 
 73 
 
 •07894 
 
 •05582 
 
 U 
 
 ■07841 
 
 •05544 
 
 75 
 
 •07788 
 
 •05507 
 
 76 
 
 •07737 
 
 •05471 
 
 77 
 
 •07687 
 
 ■05435 
 
 78 
 
 •07637 
 
 •05400 
 
 79 
 
 •07589 
 
 •05366 
 
 80 
 
 •07541 
 
 •05332 
 
 81 
 
 •07494 
 
 •05299 
 
 82 
 
 •07448 
 
 •05267 
 
 8S 
 
 •07403 
 
 •05235 
 
 84 
 
 •07359 
 
 •05204 
 
 85 
 
 •07316 
 
 •05173 
 
 86 
 
 •07273 
 
 •05143 
 
 87 
 
 •07231 
 
 ■05113 
 
 88 
 
 •07190 
 
 •05084 
 
 89 
 
 •07150 
 
 •05056 
 
 90 
 
 •07110 
 
 •05027 
 
 91 
 
 •07071 
 
 •05000 
 
 92 
 
 •07032 
 
 •04972 
 
 93 
 
 •06994 
 
 •04946 
 
 94 
 
 •06957 
 
 •04919 
 
 95 
 
 •06920 
 
 •04893 
 
 96 
 
 •06884 
 
 •04868 
 
 97 
 
 •06848 
 
 •04843 
 
 98 
 
 •06813 
 
 •04818 
 
 99 
 
 •06779 
 
 •04793 
 
 100 
 
 •06745 
 
 •04769 
 
 n 
 
 Xi 
 
 X 2 
 
 101 
 
 •06711 
 
 •04746 
 
 102 
 
 •06678 
 
 •04722 
 
 103 
 
 •06646 
 
 •04699 
 
 104 
 
 ■06614 
 
 •04677 
 
 105 
 
 •06582 
 
 •04654 
 
 106 
 
 •06551 
 
 •04632 
 
 107 
 
 •06521 
 
 •04611 
 
 108 
 
 •06490 
 
 •04589 
 
 109 
 
 ■06460 
 
 •04568 
 
 110 
 
 •06431 
 
 •04547 
 
 111 
 
 •06402 
 
 •04527 
 
 112 
 
 •06373 
 
 •04507 
 
 113 
 
 •06345 
 
 ■04487 
 
 114 
 
 •06317 
 
 •04467 
 
 115 
 
 •06290 
 
 •04447 
 
 116 
 
 ■06262 
 
 •04428 
 
 117 
 
 •06236 
 
 •04409 
 
 118 
 
 •06209 
 
 •04391 
 
 119 
 
 •06183 
 
 •04372 
 
 120 
 
 ■06157 
 
 •04354 
 
 121 
 
 •06132 
 
 •04336 
 
 122 
 
 •06107 
 
 •04318 
 
 123 
 
 •06082 
 
 •04300 
 
 124 
 
 •06057 
 
 •04283 
 
 125 
 
 •06033 
 
 •04266 
 
 126 
 
 •06009 
 
 •04249 
 
 127 
 
 •05985 
 
 •04232 
 
 128 
 
 •05962 
 
 •04216 
 
 129 
 
 •05939 
 
 ■04199 
 
 130 
 
 •05916 
 
 •04183 
 
 1S1 
 
 ■05893 
 
 •04167 
 
 132 
 
 •05871 
 
 •04151 
 
 133 
 
 •05849 
 
 •04136 
 
 134 
 
 •05827 
 
 •04120 
 
 135 
 
 •05805 
 
 •04105 
 
 136 
 
 •05784 
 
 •04090 
 
 1S7 
 
 •05763 
 
 ■04075 
 
 188 
 
 •05742 
 
 •04060 
 
 139 
 
 •05721 
 
 •04045 
 
 140 
 
 •05700 
 
 •04031 
 
 141 
 
 ■05680 
 
 ■04017 
 
 142 
 
 •05660 
 
 •04002 
 
 143 
 
 •05640 
 
 •03988 
 
 144 
 
 •05621 
 
 ■03974 
 
 11,5 
 
 •05601 
 
 ■03961 
 
 146 
 
 •05582 
 
 •03947 
 
 W 
 
 •05563 
 
 ■03934 
 
 148 
 
 •05544 
 
 ■03920 
 
 149 
 
 •05526 
 
 •03907 
 
 150 
 
 •05507 
 
 •03894 
 
Tables for Facilitating the Computation of Probable Errors Yd 
 TABLE V. Probable Errors of Means and Standard Deviations. 
 
 n 
 
 x % 
 
 Xl 
 
 151 
 
 •05489 
 
 •03881 
 
 152 
 
 •05471 
 
 •03868 
 
 153 
 
 •05453 
 
 •03856 
 
 154 
 
 •05435 
 
 •03843 
 
 155 
 
 •05418 
 
 •03831 
 
 156 
 
 •05400 
 
 •03819 
 
 157 
 
 •05383 
 
 •03806 
 
 158 
 
 •05366 
 
 •03794 
 
 159 
 
 •05349 
 
 •03782 
 
 1G0 
 
 •05332 
 
 •03771 
 
 161 
 
 •05316 
 
 •03759 
 
 162 
 
 •05299 
 
 •03747 
 
 16S 
 
 •05283 
 
 •03736 
 
 164 
 
 •05267 
 
 •03724 
 
 165 
 
 •05251 
 
 •03713 
 
 166 
 
 •05235 
 
 •03702 
 
 167 
 
 •05219 
 
 •03691 
 
 168 
 
 •05204 
 
 •03680 
 
 169 
 
 •05188 
 
 •03669 
 
 170 
 
 •05173 
 
 •03658 
 
 171 
 
 •05158 
 
 •03647 
 
 172 
 
 ■05143 
 
 •03637 
 
 173 
 
 •05128 
 
 •03626 
 
 174 
 
 •05113 
 
 •03616 
 
 175 
 
 •05099 
 
 •03605 
 
 176 
 
 •05084 
 
 •03595 
 
 177 
 
 •05070 
 
 •03585 
 
 178 
 
 •05056 
 
 ■03575 
 
 179 
 
 •05041 
 
 •03565 
 
 180 
 
 •05027 
 
 •03555 
 
 181 
 
 •05013 
 
 03545 
 
 182 
 
 •05000 
 
 •03535 
 
 183 
 
 •04986 
 
 •03526 
 
 184 
 
 •04972 
 
 03516 
 
 185 
 
 •04959 
 
 •03507 
 
 186 
 
 •04946 
 
 •03497 
 
 187 
 
 •04932 
 
 •03488 
 
 188 
 
 •04919 
 
 •03478 
 
 189 
 
 •04906 
 
 •03469 
 
 190 
 
 •04893 
 
 •03460 
 
 191 
 
 •04880 
 
 •03451 
 
 192 
 
 •04868 
 
 •03442 
 
 193 
 
 •04855 
 
 •03433 
 
 194 
 
 •04843 
 
 •03424 
 
 195 
 
 ■04830 
 
 •03415 
 
 196 
 
 •04818 
 
 •03407 
 
 197 
 
 •04806 
 
 •03398 
 
 198 
 
 •04793 
 
 •03389 
 
 199 
 
 •04781 
 
 •03381 
 
 200 
 
 •04769 
 
 •03372 
 
 n 
 
 X| 
 
 *2 
 
 201 
 
 •04757 
 
 •03364 
 
 202 
 
 •04746 
 
 •03356 
 
 203 
 
 •04734 
 
 •03347 
 
 204 
 
 •04722 
 
 •03339 
 
 205 
 
 •04711 
 
 •03331 
 
 206 
 
 •04699 
 
 •03323 
 
 207 
 
 •04688 
 
 •03315 
 
 ao8 
 
 •04677 
 
 •03307 
 
 209 
 
 •04666 
 
 •03299 
 
 210 
 
 •04654 
 
 •03291 
 
 211 
 
 •04643 
 
 •03283 
 
 212 
 
 •04632 
 
 •03276 
 
 213 
 
 •04622 
 
 •03268 
 
 214 
 
 •04611 
 
 •03260 
 
 215 
 
 •04600 
 
 •03253 
 
 216 
 
 •04589 
 
 •03245 
 
 217 
 
 •04579 
 
 •03238 
 
 218 
 
 •04568 
 
 •03230 
 
 219 
 
 •04558 
 
 •03223 
 
 220 
 
 •04547 
 
 •03216 
 
 221 
 
 •04537 
 
 •03208 
 
 222 
 
 •04527 
 
 •03201 
 
 223 
 
 •04517 
 
 •03194 
 
 224 
 
 •04507 
 
 •03187 
 
 225 
 
 •04497 
 
 •03180 
 
 226 
 
 •04487 
 
 •03173 
 
 227 
 
 •04477 
 
 •03166 
 
 228 
 
 •04467 
 
 ■03159 
 
 229 
 
 •04457 
 
 •03152 
 
 230 
 
 •04447 
 
 •03145 
 
 231 
 
 •04438 
 
 •03138 
 
 232 
 
 •04428 
 
 •03131 
 
 233 
 
 •04419 
 
 •03125 
 
 234 
 
 •04409 
 
 •03118 
 
 235 
 
 •04400 
 
 •03111 
 
 236 
 
 •04391 
 
 •03105 
 
 237 
 
 •04381 
 
 ■03098 
 
 238 
 
 •04372 
 
 •03092 
 
 239 
 
 •04363 
 
 •03085 
 
 240 
 
 •04354 
 
 •03079 
 
 241 
 
 •04345 
 
 •03172 
 
 242 
 
 •04336 
 
 •03066 
 
 243 
 
 •04327 
 
 •03060 
 
 244 
 
 •04318 
 
 •03053 
 
 245 
 
 •04309 
 
 •03047 
 
 246 
 
 •04300 
 
 •03041 
 
 247 
 
 •04292 
 
 •03035 
 
 248 
 
 •04283 
 
 •03029 
 
 249 
 
 •04274 
 
 •03022 
 
 250 
 
 •04266 
 
 •03016 
 
 251 
 252 
 253 
 254 
 255 
 
 256 
 
 257 
 258 
 259 
 
 261 
 262 
 263 
 264 
 265 
 
 266 
 267 
 268 
 269 
 
 271 
 272 
 273 
 274 
 275 
 
 276 
 277 
 278 
 279 
 280 
 
 281 
 
 282 
 283 
 284 
 285 
 
 286 
 287 
 288 
 289 
 290 
 
 291 
 
 292 
 293 
 294 
 295 
 
 296 
 297 
 298 
 299 
 300 
 
 •04257 
 •04249 
 •04240 
 •04232 
 •04224 
 
 •04216 
 •04207 
 •04199 
 •04191 
 •04183 
 
 •04175 
 •04167 
 •04159 
 •04151 
 •04143 
 
 •04136 
 •04128 
 •04120 
 •04112 
 •04105 
 
 •04097 
 •04090 
 •04082 
 •04075 
 •04067 
 
 •04060 
 •04053 
 ■04045 
 •04038 
 •04031 
 
 •04024 
 •04017 
 •04009 
 •04002 
 •03995 
 
 •03988 
 •03981 
 •03974 
 •03968 
 •03961 
 
 •03954 
 •03947 
 •03940 
 •03934 
 •03927 
 
 •03920 
 •03913 
 •03907 
 •03901 
 ■03894 
 
 •03010 
 •03004 
 •02998 
 •02993 
 •02987 
 
 •02981 
 •02975 
 •02969 
 •02964 
 •02958 
 
 •02952 
 •02947 
 •02941 
 •02935 
 •02930 
 
 •02924 
 •02919 
 •02913 
 •02908 
 •02903 
 
 •02897 
 •02892 
 •02887 
 •02881 
 •02876 
 
 •02871 
 •02866 
 •02860 
 •02855 
 •02850 
 
 •02845 
 •02840 
 •02835 
 •02830 
 •02825 
 
 •02820 
 •02815 
 •02810 
 •02806 
 •02801 
 
 •02796 
 •02791 
 ■08786 
 
 •02782 
 •02777 
 
 •02772 
 •02767 
 •02763 
 •02758 
 •02754 
 
14 Tables for Statisticians and Biometricians 
 
 TABLE V. Probable Errors of Meam and Standard Deviations. 
 
 X, 
 
 301 
 302 
 SOS 
 
 304 
 
 sun 
 
 see 
 
 807 
 
 sos 
 
 S09 
 
 ■•no 
 ■111 
 
 SIS 
 
 SIS 
 
 314 
 
 315 
 
 816 
 
 317 
 318 
 319 
 3.20 
 
 ■:>.' 
 323 
 324 
 325 
 
 SM 
 
 327 
 
 331 
 
 334 
 385 
 
 836 
 
 339 
 
 340 
 341 
 
 842 
 
 .;>,.; 
 344 
 
 345 
 
 347 
 348 
 849 
 850 
 
 •03888 
 •03881 
 •03875 
 •038C8 
 •03862 
 
 •0385G 
 •03850 
 •03843 
 ■03837 
 •03831 
 
 •03825 
 ■0381!) 
 •03812 
 •03800 
 •03800 
 
 •03794 
 •03788 
 •03782 
 •03776 
 •03771 
 
 •03765 
 •03759 
 03753 
 ■03747 
 ■03741 
 
 •03736 
 ■03730 
 •03724 
 •03719 
 •03713 
 
 •03707 
 •03702 
 ■03696 
 •03691 
 ■03685 
 
 •03680 
 •03674 
 •03669 
 ■03663 
 •03658 
 
 •03653 
 •03647 
 •03642 
 •03637 
 ■03631 
 
 •03626 
 •03621 
 •03616 
 •03610 
 ■03605 
 
 ■02749 
 •02744 
 •02740 
 •02735 
 •02731 
 
 •02726 
 
 •02722 
 ■02718 
 •02713 
 •02709 
 
 •02704 
 •02700 
 ■02696 
 •02692 
 •02687 
 
 •02683 
 •02679 
 •02675 
 •02670 
 ■02600 
 
 •02662 
 •02658 
 •02651 
 •02650 
 •02646 
 
 ■02642 
 •02637 
 •02633 
 •02629 
 •02625 
 
 •02621 
 •02618 
 •02614 
 •02610 
 •02606 
 
 •02602 
 •02598 
 ■02594 
 •02590 
 •02587 
 
 •02583 
 •02579 
 •02575 
 •02571 
 •02568 
 
 •02564 
 •02560 
 •02557 
 •02553 
 •02549 
 
 n 
 
 *i 
 
 X-2 
 
 SSI 
 
 •03600 
 
 •02546 
 
 sss 
 
 •03595 
 
 •02542 
 
 SS3 
 
 •03590 
 
 •02538 
 
 864 
 
 •03585 
 
 •02535 
 
 SSG 
 
 •03580 
 
 •02531 
 
 sss 
 
 •03575 
 
 •02528 
 
 557 
 
 •03570 
 
 •02524 
 
 358 
 
 •03505 
 
 •02521 
 
 SSS 
 
 •03560 
 
 •02517 
 
 see 
 
 •03555 
 
 •02514 
 
 301 
 
 •03550 
 
 •02510 
 
 sen 
 
 •03545 
 
 •02507 
 
 sea 
 
 •03540 
 
 •02503 
 
 304 
 
 •03535 
 
 •02500 
 
 305 
 
 •03530 
 
 •02496 
 
 366 
 
 •03526 
 
 •02493 
 
 367 
 
 •03521 
 
 •02490 
 
 868 
 
 •03516 
 
 •02486 
 
 809 
 
 •03511 
 
 •02483 
 
 370 
 
 •03507 
 
 •02479 
 
 371 
 
 •03502 
 
 •02476 
 
 372 
 
 •03497 
 
 •02473 
 
 373 
 
 •03492 
 
 •02469 
 
 874 
 
 •03488 
 
 •02466 
 
 875 
 
 •03483 
 
 •02463 
 
 876 
 
 •03478 
 
 •02460 
 
 377 
 
 •03474 
 
 •02456 
 
 878 
 
 •03469 
 
 •02453 
 
 379 
 
 •03465 
 
 •02450 
 
 880 
 
 •03460 
 
 •02447 
 
 381 
 
 •03456 
 
 •02443 
 
 882 
 
 •03451 
 
 •02440 
 
 883 
 
 •03446 
 
 •02437 
 
 384 
 
 •03442 
 
 •02434 
 
 385 
 
 •03438 
 
 •02431 
 
 886 
 
 03433 
 
 •02428 
 
 387 
 
 •03429 
 
 ■02424 
 
 388 
 
 •03424 
 
 •02421 
 
 889 
 
 •03420 
 
 •02418 
 
 390 
 
 •03415 
 
 •02415 
 
 891 
 
 •03411 
 
 •02412 
 
 392 
 
 •03407 
 
 •02409 
 
 393 
 
 •03402 
 
 •02406 
 
 894 
 
 •03398 
 
 •02403 
 
 395 
 
 •03394 
 
 •02400 
 
 396 
 
 •03389 
 
 •02397 
 
 397 
 
 •03385 
 
 •02394 
 
 398 
 
 •03381 
 
 •02391 
 
 399 
 
 •03377 
 
 •02388 
 
 400 
 
 •03372 
 
 •02385 
 
 401 
 
 4<)2 
 403 
 
 404 
 
 405 
 
 406 
 
 407 
 408 
 409 
 410 
 
 411 
 
 412 
 413 
 
 414 
 
 415 
 
 416 
 417 
 418 
 419 
 420 
 
 421 
 
 422 
 
 42s 
 
 W 
 
 425 
 
 427 
 428 
 429 
 430 
 
 431 
 432 
 433 
 434 
 435 
 
 436 
 437 
 
 440 
 
 441 
 442 
 443 
 
 444 
 445 
 
 446 
 447 
 448 
 449 
 450 
 
 *i 
 
 •03368 
 •03364 
 •03360 
 •03356 
 •03352 
 
 •03347 
 •03343 
 •03339 
 •03335 
 •03331 
 
 •03327 
 •03323 
 •03319 
 •03315 
 •03311 
 
 •03307 
 •03303 
 •03299 
 •03295 
 •03291 
 
 •03287 
 •03283 
 •03279 
 •03276 
 •03272 
 
 •03268 
 •03204 
 •03200 
 •03256 
 •03253 
 
 ■03249 
 ■03245 
 •03241 
 •03238 
 •03234 
 
 •03230 
 •03227 
 •03223 
 •03219 
 •03216 
 
 •03212 
 •03208 
 •03205 
 •03201 
 •03197 
 
 •03194 
 •03190 
 •03187 
 •03183 
 •03180 
 
 •02382 
 •02379 
 •02376 
 •02373 
 
 •02370 
 
 •02307 
 •02364 
 •02361 
 •02358 
 •02355 
 
 •02353 
 
 •02350 
 •02347 
 •02344 
 •02341 
 
 •02338 
 •02330 
 •02333 
 •02330 
 •02327 
 
 ■02324 
 ■02322 
 •02319 
 •02316 
 •02313 
 
 •02311 
 •02308 
 •02305 
 •02303 
 •02300 
 
 •02297 
 ■02295 
 •02292 
 •02289 
 •02287 
 
 •02284 
 •02281 
 •02279 
 •02276 
 •02274 
 
 •02271 
 •02269 
 •02266 
 •02263 
 •02261 
 
 •02258 
 •02256 
 •02253 
 •02251 
 •02248 
 
Tables for Facilitating the Computation of Probable Errors 15 
 TABLE V. Probable Errors of Means and Standard Deviations. 
 
 11 
 
 Xl 
 
 *2 
 
 451 
 
 •03176 
 
 •02246 
 
 452 
 
 •031 73 
 
 •02243 
 
 453 
 
 •03169 
 
 •02241 
 
 454 
 
 •03166 
 
 •02238 
 
 455 
 
 •03162 
 
 •02236 
 
 456 
 
 •03159 
 
 ■02233 
 
 457 
 
 •03155 
 
 •02231 
 
 458 
 
 •03152 
 
 •02229 
 
 459 
 
 •03148 
 
 •02226 
 
 400 
 
 •03145 
 
 •02224 
 
 461 
 
 ■03141 
 
 •02221 
 
 462 
 
 •03138 
 
 •02219 
 
 463 
 
 •03135 
 
 •02217 
 
 464 
 
 •03131 
 
 •02214 
 
 465 
 
 •03128 
 
 •02212 
 
 466 
 
 •03125 
 
 •02209 
 
 467 
 
 •03121 
 
 •02207 
 
 468 
 
 •03118 
 
 •02205 
 
 469 
 
 •03115 
 
 •02202 
 
 470 
 
 •03111 
 
 •02200 
 
 471 
 
 •03108 
 
 •02198 
 
 472 
 
 •03105 
 
 •02195 
 
 473 
 
 •03101 
 
 •02193 
 
 474 
 
 •03098 
 
 ■02191 
 
 475 
 
 •03095 
 
 •02188 
 
 4.76 
 
 03092 
 
 •02186 
 
 477 
 
 •03088 
 
 •02184 
 
 478 
 
 •03085 
 
 •02181 
 
 479 
 
 •03082 
 
 •02179 
 
 480 
 
 •03079 
 
 •02177 
 
 481 
 
 •03075 
 
 •02175 
 
 482 
 
 ■03072 
 
 •02172 
 
 483 
 
 •03069 
 
 •02170 
 
 484 
 
 •03066 
 
 •02168 
 
 485 
 
 •03063 
 
 •02166 
 
 486 
 
 •03060 
 
 •02163 
 
 487 
 
 •03056 
 
 •02161 
 
 488 
 
 •03053 
 
 •02159 
 
 489 
 
 •03050 
 
 •02157 
 
 400 
 
 •03047 
 
 •02155 
 
 491 
 
 •03044 
 
 •02152- 
 
 492 
 
 •03041 
 
 •02150 
 
 403 
 
 •03038 
 
 •02148 
 
 494 
 
 •03035 
 
 •02146 
 
 495 
 
 •03032 
 
 •02144 
 
 496 
 
 •03029 
 
 •02142 
 
 497 
 
 •03026 
 
 •02139 
 
 498 
 
 •03022 
 
 •02137 
 
 499 
 
 •03019 
 
 •02135 
 
 500 
 
 •03010 
 
 •02133 
 
 n 
 
 Xi 
 
 *2 
 
 501 
 
 •03013 
 
 •02131 
 
 502 
 
 •03010 
 
 •02129 
 
 503 
 
 •03007 
 
 •02127 
 
 504 
 
 •03004 
 
 •02124 
 
 505 
 
 •03001 
 
 •02122 
 
 506 
 
 •02998 
 
 •02120 
 
 507 
 
 •02996 
 
 •02118 
 
 508 
 
 •02993 
 
 •02116 
 
 509 
 
 •02990 
 
 •02114 
 
 510 
 
 •02987 
 
 •02112 
 
 511 
 
 •02984 
 
 •02110 
 
 512 
 
 ■02981 
 
 •02108 
 
 513 
 
 •02978 
 
 •02106 
 
 514 
 
 •02975 
 
 •02104 
 
 515 
 
 •02972 
 
 •02102 
 
 516 
 
 •02969 
 
 •02100 
 
 517 
 
 •02966 
 
 •02098 
 
 518 
 
 •02964 
 
 •02096 
 
 519 
 
 •02961 
 
 •02094 
 
 520 
 
 •02958 
 
 •02092 
 
 521 
 
 •02955 
 
 •02089 
 
 522 
 
 •02952 
 
 •02087 
 
 523 
 
 •02949 
 
 •02085 
 
 524 
 
 •02947 
 
 •02084 
 
 525 
 
 •02944 
 
 ■02082 
 
 526 
 
 •02941 
 
 •02080 
 
 527 
 
 •02938 
 
 •02078 
 
 528 
 
 •02935 
 
 •02076 
 
 529 
 
 •02933 
 
 •02074 
 
 530 
 
 •02930 
 
 •02072 
 
 681 
 
 •02927 
 
 •02070 
 
 532 
 
 •02924 
 
 •02068 
 
 5-13 
 
 •02922 
 
 ■02066 
 
 684 
 
 •02919 
 
 •02064 
 
 586 
 
 •02916 
 
 •02062 
 
 536 
 
 •02913 
 
 •02060 
 
 537 
 
 •02911 
 
 •02058 
 
 538 
 
 •02908 
 
 ■02056 
 
 539 
 
 •02905 
 
 •02054 
 
 540 
 
 •02903 
 
 •02052 
 
 541 
 
 •02900 
 
 •02051 
 
 542 
 
 ■02897 
 
 •02049 
 
 543 
 
 •02895 
 
 •02047 
 
 544 
 
 •02892 
 
 •02045 
 
 545 
 
 •02889 
 
 •02043 
 
 546 
 
 •02887 
 
 •02041 
 
 547 
 
 •02884 
 
 •02039 
 
 548 
 
 •02881 
 
 •02037 
 
 549 
 
 •02879 
 
 •02036 
 
 550 
 
 •02876 
 
 •02034 
 
 n 
 
 X, 
 
 x 2 
 
 551 
 
 •02873 
 
 •02032 
 
 552 
 
 •02871 
 
 •02030 
 
 553 
 
 •02868 
 
 •02028 
 
 554 
 
 •02866 
 
 •02026 
 
 555 
 
 •02863 
 
 •02024 
 
 556 
 
 •02860 
 
 •02023 
 
 557 
 
 •02858 
 
 •02021 
 
 558 
 
 •02855 
 
 •02019 
 
 559 
 
 •02853 
 
 •02017 
 
 560 
 
 •02850 
 
 •02015 
 
 561 
 
 •02848 
 
 •02014 
 
 562 
 
 •02845 
 
 ■02012 
 
 688 
 
 •02843 
 
 •02010 
 
 564 
 
 •02840 
 
 •02008 
 
 565 
 
 •02838 
 
 •02006 
 
 566 
 
 •02835 
 
 ■02005 
 
 567 
 
 •02833 
 
 •02003 
 
 568 
 
 •02830 
 
 •02001 
 
 569 
 
 •02828 
 
 •01999 
 
 570 
 
 •02825 
 
 •01998 
 
 571 
 
 •02823 
 
 •01996 
 
 572 
 
 •02820 
 
 •01994 
 
 573 
 
 •02818 
 
 •01992 
 
 574 
 
 ■02815 
 
 •01991 
 
 575 
 
 •02813 
 
 •01990 
 
 576 
 
 •02810 
 
 •01987 
 
 577 
 
 •02808 
 
 •01986 
 
 578 
 
 •02806 
 
 •01984 
 
 579 
 
 •02803 
 
 •01982 
 
 580 
 
 •02801 
 
 •01980 
 
 581 
 
 •02798 
 
 •01978 
 
 582 
 
 •02796 
 
 •01977 
 
 583 
 
 •02793 
 
 •01975 
 
 584 
 
 •02791 
 
 •01974 
 
 585 
 
 •02789 
 
 •01972 
 
 586 
 
 •02786 
 
 •01970 
 
 587 
 
 •02784 
 
 •01969 
 
 588 
 
 •02782 
 
 •01967 
 
 589 
 
 •02779 
 
 •01965 
 
 590 
 
 •02777 
 
 •01964 
 
 591 
 
 •02774 
 
 •01962 
 
 592 
 
 •02772 
 
 •01960 
 
 593 
 
 •02770 
 
 •01959 
 
 594 
 
 •02767 
 
 •01957 
 
 595 
 
 •02765 
 
 •01955 
 
 596 
 
 •02763 
 
 •01954 
 
 597 
 
 •02761 
 
 •01952 
 
 598 
 
 •02758 
 
 •01950 
 
 599 
 
 •02756 
 
 •01949 
 
 600 
 
 •02754 
 
 •01947 
 
16 Tables for Statisticians and Biometricians 
 
 TABLE V. Probable Errors of Means and Standard Deviations. 
 
 n 
 
 *i 
 
 X, 
 
 601 
 
 •02751 
 
 •01945 
 
 602 
 
 •02749 
 
 •01944 
 
 60S 
 
 •02747 
 
 •01942 
 
 604 
 
 •02744 
 
 •01941 
 
 605 
 
 •02742 
 
 •01939 
 
 606 
 
 •02740 
 
 •01937 
 
 607 
 
 •02738 
 
 •01936 
 
 60S 
 
 •02735 
 
 •01934 
 
 600 
 
 •02733 
 
 •01933 
 
 610 
 
 •02731 
 
 •01931 
 
 611 
 
 •02729 
 
 •01929 
 
 612 
 
 •02726 
 
 •01928 
 
 613 
 
 •02724 
 
 •01926 
 
 614 
 
 •02722 
 
 •01925 
 
 615 
 
 •02720 
 
 •01923 
 
 616 
 
 •02718 
 
 •01922 
 
 617 
 
 •02715 
 
 ■01920 
 
 618 
 
 •02713 
 
 •01919 
 
 619 
 
 •02711 
 
 •01917 
 
 620 
 
 •02709 
 
 •01915 
 
 621 
 
 •02707 
 
 •01914 
 
 622 
 
 •02704 
 
 •01912 
 
 623 
 
 •02702 
 
 •01911 
 
 624 
 
 •02700 
 
 ■01909 
 
 625 
 
 •02698 
 
 •01908 
 
 626 
 
 •02696 
 
 •01906 
 
 627 
 
 •02694 
 
 •01905 
 
 628 
 
 •02692 
 
 •01903 
 
 620 
 
 •02689 
 
 •01902 
 
 630 
 
 •02687 
 
 •01900 
 
 6S1 
 
 ■02685 
 
 •01899 
 
 632 
 
 •02683 
 
 •01897 
 
 633 
 
 •02681 
 
 •01896 
 
 634 
 
 •02679 
 
 •01894 
 
 635 
 
 •02677 
 
 •01893 
 
 636 
 
 •02675 
 
 •01891 
 
 637 
 
 •02672 
 
 •01890 
 
 638 
 
 •02670 
 
 •01888 
 
 639 
 
 •02668 
 
 •01887 
 
 640 
 
 •02666 
 
 •01885 
 
 641 
 
 •02664 
 
 •01884 
 
 642 
 
 •02662 
 
 •01822 
 
 643 
 
 •02660 
 
 •01881 
 
 644 
 
 •02658 
 
 •01879 
 
 645 
 
 •02656 
 
 •01878 
 
 646 
 
 •02654 
 
 •01876 
 
 647 
 
 •02652 
 
 ■01875 
 
 648 
 
 •02650 
 
 •01874 
 
 640 
 
 •02648 
 
 •01872 
 
 650 
 
 •02646 
 
 •01871 
 
 n 
 
 X, 
 
 X, 
 
 651 
 
 •02644 
 
 •01869 
 
 652 
 
 •02642 
 
 •01868 
 
 653 
 
 •02639 
 
 •01866 
 
 654 
 
 •02637 
 
 •01865 
 
 655 
 
 •02635 
 
 •01864 
 
 656 
 
 •02633 
 
 ■01862 
 
 657 
 
 •02631 
 
 •01861 
 
 658 
 
 •02629 
 
 •01859 
 
 659 
 
 •02627 
 
 •01858 
 
 660 
 
 •02625 
 
 •01856 
 
 661 
 
 •02623 
 
 •01855 
 
 662 
 
 •02621 
 
 •01854 
 
 663 
 
 •02620 
 
 •01852 
 
 664 
 
 •02618 
 
 •01851 
 
 665 
 
 •02616 
 
 •01849 
 
 666 
 
 •02614 
 
 •01848 
 
 667 
 
 •02612 
 
 •01847 
 
 668 
 
 •02610 
 
 •01845 
 
 669 
 
 •02608 
 
 •01844 
 
 670 
 
 •02606 
 
 •01843 
 
 671 
 
 •02604 
 
 •01841 
 
 672 
 
 •02602 
 
 •01840 
 
 673 
 
 ■02600 
 
 ■01838 
 
 674 
 
 •02598 
 
 •01837 
 
 675 
 
 •02596 
 
 •01836 
 
 676 
 
 •02594 
 
 •01834 
 
 677 
 
 •02592 
 
 ■01833 
 
 678 
 
 •02590 
 
 •01832 
 
 679 
 
 •02588 
 
 •01830 
 
 680 
 
 •02587 
 
 •01829 
 
 681 
 
 •02585 
 
 •01828 
 
 682 
 
 •02583 
 
 •01826 
 
 683 
 
 ■02581 
 
 •01825 
 
 684 
 
 •02579 
 
 •01824 
 
 685 
 
 •02577 
 
 •01822 
 
 686 
 
 •02575 
 
 •01821 
 
 687 
 
 •02573 
 
 •01820 
 
 688 
 
 •02571 
 
 •01818 
 
 689 
 
 •02570 
 
 •01817 
 
 690 
 
 •02568 
 
 •01816 
 
 691 
 
 •02566 
 
 •01814 
 
 692 
 
 •02564 
 
 •01813 
 
 693 
 
 •02562 
 
 •01812 
 
 694 
 
 •02560 
 
 •01810 
 
 695 
 
 •02558 
 
 •01809 
 
 696 
 
 •02557 
 
 •01808 
 
 697 
 
 •02555 
 
 •01807 
 
 698 
 
 •02553 
 
 •01805 
 
 699 
 
 •02551 
 
 •01804 
 
 700 
 
 •02549 
 
 •01803 
 
 n 
 
 Xt 
 
 x 2 
 
 701 
 
 •02548 
 
 •01801 
 
 702 
 
 •02546 
 
 •01800 
 
 703 
 
 •02544 
 
 •01799 
 
 704 
 
 •02542 
 
 •01798 
 
 705 
 
 •02540 
 
 •01796 
 
 706 
 
 •02538 
 
 •01795 
 
 707 
 
 •02537 
 
 •01794 
 
 708 
 
 •02535 
 
 •01792 
 
 709 
 
 •02533 
 
 ■01791 
 
 710 
 
 •02531 
 
 •01790 
 
 711 
 
 •02530 
 
 •01789 
 
 712 
 
 •02528 
 
 •01787 
 
 713 
 
 •02526 
 
 •01786 
 
 714 
 
 •02524 
 
 •01785 
 
 715 
 
 •02522 
 
 •01784 
 
 716 
 
 •02521 
 
 •01782 
 
 717 
 
 •02519 
 
 •01781 
 
 718 
 
 •02517 
 
 ■01780 
 
 719 
 
 •02515 
 
 •01779 
 
 720 
 
 •02514 
 
 •01777 
 
 721 
 
 •02512 
 
 ■01776 
 
 722 
 
 •02510 
 
 •01775 
 
 723 
 
 •02508 
 
 •01774 
 
 724 
 
 •02507 
 
 •01773 
 
 725 
 
 •02505 
 
 •01771 
 
 726 
 
 •02503 
 
 ■01770 
 
 727 
 
 •02502 
 
 ■01769 
 
 728 
 
 •02500 
 
 •01768 
 
 729 
 
 •02498 
 
 •01766 
 
 730 
 
 •02496 
 
 •01765 
 
 731 
 
 •02495 
 
 •01764 
 
 732 
 
 •02493 
 
 •01763 
 
 733 
 
 •02491 
 
 •01762 
 
 784 
 
 •02490 
 
 •01760 
 
 785 
 
 •02488 
 
 •01759 
 
 786 
 
 •02486 
 
 •01758 
 
 737 
 
 •02485 
 
 •01757 
 
 788 
 
 •02483 
 
 •01756 
 
 739 
 
 •02481 
 
 •01754 
 
 740 
 
 •02479 
 
 •01753 
 
 741 
 
 •02478 
 
 •01752 
 
 742 
 
 •02476 
 
 •01751 
 
 743 
 
 •02474 
 
 •01750 
 
 744 
 
 •02473 
 
 •01749 
 
 745 
 
 •02471 
 
 •01747 
 
 746 
 
 •02469 
 
 •01746 
 
 747 
 
 •02468 
 
 •01745 
 
 74s 
 
 ■02466 
 
 •01744 
 
 749 
 
 •02465 
 
 •01743 
 
 750 
 
 •02463 
 
 •01742 
 
Tables for Facilitating the Computation of Probable Errors 17 
 
 TABLE V. Probable Errors of Means and Standard Deviations. 
 
 n 
 
 Xi 
 
 X, 
 
 751 
 
 •02461 
 
 ■01740 
 
 751 
 
 •02460 
 
 •01739 
 
 7.-,.: 
 
 •02458 
 
 •01738 
 
 754 
 
 •02456 
 
 •01737 
 
 755 
 
 •02455 
 
 •01736 
 
 756 
 
 •02453 
 
 •01735 
 
 757 
 
 •02451 
 
 •01733 
 
 75S 
 
 •02450 
 
 •01732 
 
 750 
 
 •02448 
 
 •01731 
 
 760 
 
 •02447 
 
 •01730 
 
 761 
 
 •02445 
 
 •01729 
 
 76 ! 
 
 •02443 
 
 •01728 
 
 763 
 
 •02442 
 
 •01727 
 
 764 
 
 •02440 
 
 •01725 
 
 765 
 
 •0243!) 
 
 •01724 
 
 766 
 
 •02437 
 
 •01723 
 
 767 
 
 •02435 
 
 •01722 
 
 768 
 
 •02434 
 
 .•01721 
 
 769 
 
 •02432 
 
 •01720 
 
 770 
 
 ■02431 
 
 •01719 
 
 771 
 
 •02429 
 
 •01718 
 
 772 
 
 •02428 
 
 •01717 
 
 778 
 
 •02420 
 
 •01715 
 
 77.4 
 
 •02424 
 
 •01714 
 
 775 
 
 •02423 
 
 •01713 
 
 776 
 
 •02421 
 
 •01712 
 
 777 
 
 •02420 
 
 •01711 
 
 778 
 
 •02418 
 
 •01710 
 
 779 
 
 •02417 
 
 •01709 
 
 780 
 
 •02415 
 
 •01708 
 
 781 
 
 •02414 
 
 •01707 
 
 78! 
 
 •02412 
 
 •01706 
 
 783 
 
 •02410 
 
 •01704 
 
 784 
 
 •02409 
 
 •01703 
 
 785 
 
 •02407 
 
 •01702 
 
 786 
 
 •02406 
 
 •01701 
 
 787 
 
 •02404 
 
 •01700 
 
 78S 
 
 •02403 
 
 •01699 
 
 789 
 
 •02401 
 
 •01698 
 
 790 
 
 •02400 
 
 •01697 
 
 791 
 
 •02398 
 
 ■01696 
 
 792 
 
 •02397 
 
 •01695 
 
 793 
 
 •02395 
 
 •01094 
 
 794 
 
 •02394 
 
 •01693 
 
 795 
 
 •02392 
 
 ■01692 
 
 796 
 
 •02391 
 
 ■01690 
 
 797 
 
 •02389 
 
 •010.-.9 
 
 798 
 
 •02388 
 
 •01688 
 
 799 
 
 •02386 
 
 ■01687 
 
 800 
 
 •02385 
 
 •01686 
 
 n 
 
 X-i 
 
 *2 
 
 801 
 
 •02383 
 
 •01685 
 
 802 
 
 •02382 
 
 •01684 
 
 803 
 
 •02380 
 
 •01683 
 
 804 
 
 •02379 
 
 •01682 
 
 805 
 
 •02377 
 
 •01681 
 
 806 
 
 •02376 
 
 •01680 
 
 807 
 
 •02374 
 
 ■01679 
 
 808 
 
 •02373 
 
 •01678 
 
 809 
 
 •02371 
 
 ■01677 
 
 810 
 
 •02370 
 
 •01676 
 
 811 
 
 •02368 
 
 •01675 
 
 812 
 
 •02367 
 
 •01674 
 
 813 
 
 •02366 
 
 •01673 
 
 814 
 
 •02364 
 
 •01672 
 
 815 
 
 •02363 
 
 •01671 
 
 816 
 
 •02361 
 
 ■01670 
 
 817 
 
 •02360 
 
 •016G9 
 
 818 
 
 •02358 
 
 ■01668 
 
 819 
 
 ■02357 
 
 •01667 
 
 820 
 
 •02355 
 
 •01666 
 
 821 
 
 •02354 
 
 •01605 
 
 822 
 
 ■02363 
 
 •01664 
 
 883 
 
 •02351 
 
 •01662 
 
 824 
 
 •02350 
 
 •01661 
 
 8*5 
 
 •02348 
 
 •01GG0 
 
 826 
 
 •02347 
 
 •01659 
 
 827 
 
 •02345 
 
 •01658 
 
 828 
 
 •02344 
 
 •01657 
 
 829 
 
 ■02343 
 
 •01G56 
 
 830 
 
 •02341 
 
 •01655 
 
 831 
 
 •02340 
 
 •01654 
 
 832 
 
 •02338 
 
 •01653 
 
 833 
 
 •02337 
 
 •01652 
 
 834 
 
 •02336 
 
 •01651 
 
 835 
 
 •02334 
 
 •01651 
 
 836 
 
 •02333 
 
 •01650 
 
 837 
 
 •02331 
 
 •01649 
 
 838 
 
 •02330 
 
 •01648 
 
 839 
 
 •02329 
 
 •01647 
 
 840 
 
 •02327 
 
 •01646 
 
 8!,1 
 
 •02326 
 
 •01645 
 
 86 .' 
 
 •0232 1 
 
 •01644 
 
 8+S 
 
 •02323 
 
 •01643 
 
 844 
 
 •02322 
 
 •01642 
 
 845 
 
 •02320 
 
 •01641 
 
 846 
 
 •02319 
 
 ■01640 
 
 847 
 
 •02318 
 
 •01G3!) 
 
 848 
 
 •02316 
 
 ■01638 
 
 840 
 
 •02315 
 
 •01637 
 
 850 
 
 •02313 
 
 •01636 
 
 n 
 
 *i 
 
 * 2 
 
 851 
 
 •02312 
 
 •01635 
 
 852 
 
 •02311 
 
 •01634 
 
 853 
 
 •02309 
 
 •01G33 
 
 854 
 
 •02308 
 
 •01632 
 
 855 
 
 •02307 
 
 01631 
 
 856 
 
 •02305 
 
 •01630 
 
 857 
 
 •02304 
 
 •01629 
 
 858 
 
 •02303 
 
 •01628 
 
 859 
 
 •02301 
 
 •01627 
 
 860 
 
 •02300 
 
 •01626 
 
 861 
 
 •02299 
 
 •01625 
 
 862 
 
 •02297 
 
 •01624 
 
 863 
 
 •02296 
 
 •01624 
 
 864 
 
 •02295 
 
 ■01623 
 
 865 
 
 •02293 
 
 •01622 
 
 866 
 
 •02292 
 
 •01G21 
 
 867 
 
 •02291 
 
 •01G20 
 
 868 
 
 •02289 
 
 •01619 
 
 869 
 
 •02288 
 
 •01618 
 
 870 
 
 •02287 
 
 •01017 
 
 871 
 
 ■02285 
 
 •01616 
 
 872 
 
 •02284 
 
 •01615 
 
 873 
 
 •02283 
 
 •01614 
 
 874 
 
 •02281 
 
 •01613 
 
 875 
 
 ■02280 
 
 •01612 
 
 876 
 
 •02279 
 
 ■01611 
 
 877 
 
 •02278 
 
 •01610 
 
 878 
 
 ■02276 
 
 •01610 
 
 879 
 
 •02275 
 
 ■01609 
 
 8S0 
 
 •02274 
 
 •01608 
 
 8S1 
 
 •02272 
 
 •01607 
 
 882 
 
 ■02271 
 
 ■01606 
 
 883 
 
 •02270 
 
 •01605 
 
 884 
 
 •02269 
 
 •01604 
 
 885 
 
 •02267 
 
 •01603 
 
 886 
 
 •02266 
 
 •01602 
 
 887 
 
 •02265 
 
 •01001 
 
 888 
 
 •02263 
 
 •01600 
 
 889 
 
 •02262 
 
 •01600 
 
 890 
 
 ■02261 
 
 •01599 
 
 891 
 
 •02260 
 
 •01598 
 
 892 
 
 •02258 
 
 •01597 
 
 893 
 
 •02257 
 
 •01596 
 
 894 
 
 •02256 
 
 •01595 
 
 895 
 
 •02255 
 
 01594 
 
 896 
 
 •02253 
 
 •01593 
 
 897 
 
 •02252 
 
 •01592 
 
 898 
 
 •02251 
 
 •01592 
 
 899 
 
 ■O2250 
 
 •01591 
 
 900 
 
 •02248 
 
 •01590 
 
 1 
 
 B. 
 
18 Tables for Statisticians and Bionielricians 
 
 TABLE V. TABLE VI. 
 
 Probable Errors of Means and Standard Deviations. Probable Errors of Coefficient of Variation. 
 
 
 
 
 
 
 
 
 
 V 
 
 + 
 
 A 
 
 A 2 
 
 A 3 
 
 'a 
 
 X, 
 
 *2 
 
 
 n 
 
 Xi 
 
 *2 
 
 
 + 
 
 + 
 
 + 
 
 901 
 
 •02247 
 
 •01589 
 
 951 
 
 •02187 
 
 •01547 
 
 
 1 
 2 
 3 
 
 4 
 5 
 
 o-ooooo 
 
 1-00010 
 2-00080 
 3-00270 
 4-00639 
 5-01248 
 
 1 -oooio 
 
 
 60 
 120 
 180 
 239 
 299 
 
 60 
 
 90% 
 
 •02246 
 
 ■01588 
 
 
 952 
 
 •02186 
 
 •01546 
 
 
 1-00070 
 
 60 
 
 90S 
 
 •02245 
 
 •01587 
 
 
 953 
 
 •02185 
 
 •01545 
 
 
 1-00190 
 
 60 
 
 904 
 
 •02243 
 
 •01586 
 
 
 954 
 
 •02184 
 
 •01544 
 
 
 1-00370 
 
 60 
 
 905 
 
 ■02242 
 
 •01585 
 
 
 955 
 
 •02183 
 
 •01543 
 
 
 1-00609 
 
 59 
 
 906 
 
 •02241 
 
 •01585 
 
 
 956 
 
 •02181 
 
 •01543 
 
 
 6 
 7 
 8 
 9 
 10 
 
 6-02156 
 7-03422 
 8-05104 
 9-07261 
 10-09050 
 
 1 -00908 
 
 358 
 417 
 475 
 533 
 590 
 
 59 
 
 907 
 
 •02240 
 
 •01584 
 
 
 957 
 
 •02180 
 
 •01542 
 
 
 1-01266 
 
 59 
 
 908 
 
 •02238 
 
 •01583 
 
 
 968 
 
 •02179 
 
 •01541 
 
 
 1-01682 
 
 58 
 
 909 
 
 •02237 
 
 ■01582 
 
 
 959 
 
 •02178 
 
 •01540 
 
 
 1-02157 
 
 58 
 
 910 
 
 •02236 
 
 •01581 
 
 
 960 
 
 •02177 
 
 •01539 
 
 
 1-02690 
 
 57 
 
 911 
 
 •02235 
 
 •01580 
 
 
 961 
 
 •02176 
 
 •01539 
 
 
 11 
 12 
 18 
 
 14 
 15 
 
 11-13230 
 12-17157 
 
 13-21787 
 14-27176 
 15-33379 
 
 1-03280 
 
 647 
 703 
 759 
 814 
 
 868 
 
 57 
 
 912 
 
 •02233 
 
 •01579 
 
 
 966 
 
 •02175 
 
 ■01538 
 
 
 1-03927 
 
 56 
 
 91S 
 
 •02232 
 
 •01578 
 
 
 96S 
 
 •02174 
 
 •01537 
 
 
 1 -04630 
 
 56 
 
 914 
 
 •02231 
 
 •01578 
 
 
 964 
 
 •02172 
 
 ■01536 
 
 
 1-05389 
 
 55 
 
 915 
 
 •02230 
 
 •01577 
 
 
 965 
 
 •02171 
 
 •01535 
 
 
 1-06202 
 
 54 
 
 91G 
 
 •02229 
 
 •01576 
 
 
 966 
 
 •02170 
 
 •01535 
 
 
 16 
 17 
 18 
 19 
 
 20 
 
 16-40449 
 
 17-48440 
 18-57405 
 19-67395 
 20-78461 
 
 1-07070 
 
 921 
 
 974 
 
 1025 
 
 1076 
 
 1126 
 
 53 
 
 917 
 
 •02227 
 
 •01575 
 
 
 967 
 
 •02169 
 
 ■01534 
 
 
 1 -07991 
 
 53 
 
 918 
 
 •02226 
 
 •01574 
 
 
 968 
 
 •02168 
 
 •01533 
 
 
 1 -08965 
 
 52 
 
 919 
 
 •02225 
 
 •01573 
 
 
 969 
 
 •02167 
 
 •01532 
 
 
 1-09990 
 
 51 
 
 920 
 
 •02224 
 
 •01572 
 
 
 970 
 
 •02166 
 
 •01531 
 
 
 1-11066 
 
 50 
 
 921 
 
 •02223 
 
 •01572 
 
 
 971 
 
 •02165 
 
 ■01531 
 
 
 21 
 
 9$ 
 
 21-90653 
 23-04021 
 24-18612 
 25-34473 
 26-51650 
 
 1-12192 
 
 1175 
 1223 
 1270 
 1316 
 1362 
 
 49 
 
 92$ 
 
 •02221 
 
 •01571 
 
 
 972 
 
 •02163 
 
 •01530 
 
 
 1-13368 
 
 48 
 
 923 
 
 •02220 
 
 •01570 
 
 
 973 
 
 •02162 
 
 •01529 
 
 
 OQ 
 
 1-14591 
 
 47 
 
 924 
 996 
 
 •02219 
 •02218 
 
 •01569 
 •01568 
 
 
 974 
 975 
 
 •02161 
 •02160 
 
 •01 528 
 •01527 
 
 
 CO 
 
 ..'4 
 
 25 
 
 1-15861 
 1-17177 
 
 46 
 45 
 
 926 
 
 •02217 
 
 •01567 
 
 
 976 
 
 •02159 
 
 •01527 
 
 
 26 
 27 
 28 
 29 
 SO 
 
 27-70190 
 28-90135 
 30-11530 
 31-34416 
 32-58834 
 
 1-1853'.) 
 
 1 106 
 1449 
 1491 
 1533 
 1573 
 
 44 
 
 927 
 
 •02215 
 
 •01566 
 
 
 977 
 
 •02158 
 
 •01526 
 
 
 1-19945 
 
 43 
 
 928 
 
 •02214 
 
 •01566 
 
 
 97S 
 
 •02157 
 
 •01525 
 
 
 1-21395 
 
 42 
 
 929 
 
 •02213 
 
 •01565 
 
 
 979 
 
 •02156 
 
 •01524 
 
 
 1-22886 
 
 41 
 
 980 
 
 •02212 
 
 •01564 
 
 
 980 
 
 •02155 
 
 •01524 
 
 
 1-24418 
 
 40 
 
 931 
 
 •02211 
 
 •01563 
 
 
 981 
 
 •02153 
 
 •01523 
 
 
 31 
 82 
 38 
 
 33-84825 
 35-12428 
 36'41681 
 37-72621 
 39 05285 
 
 1-25991 
 
 1612 
 1650 
 1687 
 1723 
 1758 
 
 39 
 
 932 
 
 •02209 
 
 •01562 
 
 
 982 
 
 •02152 
 
 •01522 
 
 
 1-27603 
 
 38 
 
 933 
 
 •02208 
 
 •01561 
 
 
 988 
 
 •02151 
 
 •01521 
 
 
 1 -29253 
 
 37 
 
 934 
 
 •02207 
 
 •01561 
 
 
 984 
 
 •02150 
 
 •01520 
 
 
 1-30940 
 
 30 
 
 935 
 
 •02206 
 
 •01560 
 
 
 985 
 
 •02149 
 
 •01520 
 
 
 04 
 
 35 
 
 1-32664 
 
 35 
 
 936 
 
 •02205 
 
 ■01559 
 
 
 986 
 
 •02148 
 
 •01519 
 
 
 86 
 37 
 38 
 39 
 40 
 
 40-39707 
 41-75922 
 43-13962 
 44-53861 
 45-95650 
 
 1-34422 
 
 1793 
 1826 
 1858 
 1890 
 1920 
 
 34 
 
 987 
 
 •02203 
 
 •01558 
 
 
 987 
 
 ■02147 
 
 •01518 
 
 
 1-36215 
 
 33 
 
 938 
 
 •02202 
 
 •01557 
 
 
 988 
 
 •02146 
 
 •01517 
 
 
 1 -38041 
 
 32 
 
 939 
 
 •02201 
 
 •0155(i 
 
 
 989 
 
 •02145 
 
 •01517 
 
 
 1-39899 
 
 31 
 
 940 
 
 •02200 
 
 ■01556 
 
 
 990 
 
 •02144 
 
 •01516 
 
 
 1-41789 
 
 30 
 
 941 
 
 •02199 
 
 •01555 
 
 
 991 
 
 •02143 
 
 •01515 
 
 
 41 
 42 
 
 43 
 
 44 
 45 
 
 47-39359 
 48-85017 
 50-32654 
 51-82296 
 53-33971 
 
 1-43709 
 
 1950 
 1978 
 2006 
 2033 
 2059 
 
 30 
 
 942 
 
 •02198 
 
 •01554 
 
 
 992 
 
 •02142 
 
 ■01514 
 
 
 1 -45658 
 
 29 
 
 943 
 944 
 
 •02196 
 
 ■11219.") 
 
 •01553 
 •01552 
 
 
 998 
 994 
 
 •02140 
 •02139 
 
 •01514 
 •01513 
 
 
 1 -47636 
 1-49642 
 
 28 
 27 
 
 945 
 
 •02194 
 
 ■01551 
 
 
 995 
 
 •02138 
 
 •01512 
 
 
 1-51675 
 
 26 
 
 946 
 
 •02193 
 
 •01551 
 
 
 996 
 
 •02137 
 
 •01511 
 
 
 46 
 47 
 48 
 
 49 
 
 50 
 
 54-87706 
 56-43524 
 58-01451 
 59-61510 
 61 -83784 
 
 1-53734 
 
 2084 
 2109 
 2132 
 2155 
 2177 
 
 25 
 
 947 
 
 •(12192 
 
 •01550 
 
 
 997 
 
 •02136 
 
 •01510 
 
 
 1-55818 
 
 24 
 
 94S 
 
 •02191 
 
 •01549 
 
 
 998 
 
 •02 1 33 
 
 •01510 
 
 
 1-57927 
 
 24 
 
 9.'i9 
 
 •02189 
 
 •01548 
 
 
 999 
 
 •02131 
 
 •01509 
 
 
 1-60059 
 
 23 
 
 950 
 
 •02188 
 
 •015 17 
 
 
 1000 
 
 •02133 
 
 •01508 
 
 
 1-62214 
 
 22 
 
Probable Error of a Coefficient of Correlation 19 
 
 TABLE VII. Abac for Probable Errors of r. 
 
 
 
 
 
 
 
 
 
 
 
 
 Scale of Correlation 
 
 
 o 
 
 Scale t 
 
 >-Ocftcor-(Om ■* co 
 
 /" Probable Errors 
 
 cn •— O oi cp r~- 
 
 : - ±: 
 
 1 
 i 
 
 - 
 
 i 
 
 i 
 
 c 
 
 t 
 
 t 
 
 3 
 
 bP 
 
 V////// 
 
 {/////ft 
 
 
 8 
 8 
 
 CO 
 
 o 
 
 o 
 
 r- 
 O 
 
 o 
 to 
 
 
 
 
 
 
 
 
 
 
 
 ill 
 
 tVtV 
 
 //////// 
 
 / / / 1 / / 1 
 
 8 
 
 CO 
 
 
 
 
 8 
 
 to 
 
 
 
 
 
 
 
 
 
 
 
 ////// 
 
 ' 1 / / 
 
 1 1 1 1 1 1 1 
 
 ' / / / / / / 
 
 
 
 
 
 
 
 
 
 
 
 II III )i 1 . 
 
 '/III 1 1 I 
 
 / / / / / / 
 
 
 
 
 
 
 
 
 
 
 
 Villi 
 
 1 1 / ii 
 
 1 / / 1 I 1 I 
 
 / / / / / / 
 
 ill J . 
 
 
 
 
 
 
 
 
 
 
 
 1 / 1 
 
 1 1 1 1 1 1 I 
 
 /■1 / 1 / 
 
 - 
 
 
 
 
 
 
 
 
 -J 
 
 4-r 
 
 rQ4U 
 
 +UU- 
 
 L/ / / / / 
 
 o 
 
 o 
 
 TTTT +i"" 
 
 
 
 
 
 
 
 
 
 ^ 
 
 m 
 
 -Hm 
 
 hPfmm 
 
 / / / / / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 VVr/ 
 
 
 mill 
 
 / / / / / 
 
 ' / / 1 T 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 Wh 
 
 ///// 
 
 / / / / 
 
 o 
 o 
 
 o 
 o 
 
 CO 
 
 
 
 
 
 
 
 
 
 ///// 
 
 w, 
 
 / 11 
 
 Wf 
 
 H-hHh 
 
 tH—h-f ; 
 
 8 
 8 
 
 CO 
 
 
 
 
 
 
 
 
 
 j,.; 
 
 7 / /!/ /// 
 
 / / / / > 
 
 / ■/ f / 
 
 / / / 
 
 T " * 
 
 
 
 
 
 
 
 
 7:4 
 
 '/y/ //■/ 
 
 1 1 / / 1 
 
 
 
 
 
 
 
 
 
 
 
 i. - 
 
 / / / 
 
 //// 
 
 '// 
 
 1 f/i 
 
 III / 
 
 
 
 
 
 
 
 
 ^ 
 
 /// 
 
 i/ i 
 
 in 
 
 'ii/i 
 
 / / / 
 
 
 
 
 
 
 
 
 
 k 
 
 m 
 
 ^ 
 
 '-m 
 
 iJJ-U 
 
 / / / / 
 
 
 8 
 
 CN 
 
 8 
 8 
 
 O 
 CO 
 
 O 
 
 r- 
 
 8 
 
 i 
 
 
 
 
 
 
 
 m 
 
 w 
 
 Wi 
 
 W&- 
 
 T ^ l 
 
 ' / / / 
 
 —1 1 -/ 
 
 
 
 8 
 
 
 
 
 s! 
 
 
 
 
 
 
 
 
 7 // 
 
 / / 
 
 1 i 1 
 
 ill 
 
 1/1 
 
 / / / 
 
 
 
 
 
 
 
 
 
 /// 
 
 // 
 
 III 
 
 III I 
 
 /iii 
 
 / / / 
 
 / / ' 
 
 
 
 
 
 
 
 
 // 
 
 
 III 
 
 
 1 1 / 
 
 1 ' 1 
 
 / /' 
 
 
 
 
 
 
 
 ' 
 
 V ^ 
 
 
 lilt 
 
 /III 
 
 1 / / 
 
 /ii / 
 
 
 
 
 
 
 
 
 1 
 
 / A 
 
 
 III/ 
 
 /III 
 
 1 1 / 1 
 
 / 1 1 
 
 
 
 
 
 
 
 
 
 // 
 
 1 1 
 
 iiii 
 
 III, 
 
 111 
 
 ill 
 
 
 
 
 
 
 
 j 
 
 / ' 
 
 /y 
 
 i / 
 
 
 fill 
 
 / 1 1 
 
 1 / / 
 
 
 
 
 
 
 
 >' 
 
 > 
 
 // 
 
 1 ' 
 
 ("j / / 
 
 /' 1 
 
 /// 
 
 / / 1 
 
 
 
 
 
 
 
 'I 
 
 / ! 
 
 / ^ 
 
 //' 
 
 i / / 
 
 1 / , 
 
 
 
 
 
 
 
 
 /// 
 
 1 
 
 
 / 1 
 
 /// 
 
 ' ' / ' / / 
 
 / 
 
 
 / I 
 
 
 O" - 
 
 "0 
 
 
 
 
 
 // 
 
 f/J 
 
 -- 
 
 i 
 
 /// 
 
 // / 
 
 1 / 
 
 
 111 
 
 
 
 
 
 j 
 
 /■'/ 
 
 l 
 
 / 
 
 li 
 
 //a 
 
 ! / ' 
 
 / / 
 
 
 /// 
 
 
 0"" 
 '"o 
 
 
 
 
 21 
 
 •// 
 
 // 
 
 / , 
 
 // 
 
 v// 
 
 ' // 
 
 1 / 1 
 
 / / / 
 
 
 
 
 
 
 % 
 
 // 
 
 // 
 
 t 
 
 /// 
 
 W- 
 
 i 
 
 '-I-/- 
 
 // 
 
 ' / / 
 
 
 
 
 
 
 = &• 
 8 1 
 
 — «*■ 
 
 8* 
 
 O 
 O . 
 CO -Si 
 
 ~t 
 
 CO 
 
 p 
 
 r- 
 
 O 
 
 ■ 
 
 O 
 
 
 
 
 CO 
 
 to 
 
 
 
 CM 
 O 
 
 IBi::::;:: 
 
 
 
 
 
 7 
 
 7V7 
 
 1 / 1 
 
 It 
 
 m 
 
 III 
 
 / / 
 
 
 I ...... L 
 
 
 
 /- 
 
 % 
 
 t 
 
 y 
 
 f] 
 
 7Y7 
 
 I ! 
 
 / / 
 
 
 -/- — -/■ — ■ 
 
 — / / 
 
 -i + 
 
 
 
 / 
 
 J 
 
 ' I 
 
 ' 
 
 /,■ 
 
 / O 
 
 
 / / 
 
 1 1 1 
 
 / / 
 
 1 / 
 
 1 1 
 
 
 - -- —tTJ- 
 
 
 L. 
 
 i 
 
 
 ■ 
 
 I' 
 
 I / 
 
 ' / / 
 
 1 1 
 
 / / 
 
 / / 
 
 1 / 
 
 
 „-jm 
 
 J 
 
 H, 
 
 i 
 
 / 
 
 / 
 
 
 1 / 1 
 
 / / 
 
 / / 
 
 / / 
 
 
 
 
 Jul 
 
 1 
 
 U. 
 
 / 
 
 / 
 
 •• 
 
 / / 
 
 
 ft 
 
 / 1 
 
 / / 
 
 1 1 
 
 
 
 — - mt 
 
 ' / 
 
 / 
 
 
 ij 
 
 
 
 1 / 1 
 
 
 
 / 
 
 1 1 
 
 
 
 M/ 
 
 
 1 
 
 / 
 
 >h 
 
 / 
 
 7 
 
 I 1 
 
 
 
 / / 
 
 I 
 
 
 
 M5 
 
 / 
 
 . 
 
 / 
 
 r 
 
 / 
 
 
 t I 
 
 
 
 / / 
 
 1 
 
 
 
 Awl 
 
 / 
 
 , 
 
 " 
 
 
 / 
 
 
 TX 
 
 
 
 / / 
 
 1 
 
 
 
 i f-hrf-.-A 
 
 - 
 
 3 
 
 
 
 ■ - 
 
 
 i—t- 
 
 
 
 - — j— 
 
 —f- / 
 
 
 
 --------mitt 
 
 / 
 
 / 
 
 
 / 
 
 
 
 
 
 i=f. 
 
 — — 1 — 
 — t— 
 
 l — A 
 
 
 o 
 o 
 
 o 
 
 CO 
 
 
 \iizzjW/rf4 
 
 
 -A 
 
 - 
 
 4- 
 
 — 
 
 y- 
 
 
 —f- ; 
 
 / 1 
 
 -i — 
 
 L I 
 
 
 
 tn.uiT.i_tL 
 
 / 
 
 tt 
 
 
 i 
 
 i 
 
 / 
 
 
 / / 
 
 
 / 
 
 
 
 
 .. / ■ . . 
 
 
 ti- 
 
 i 
 
 . 
 
 • 
 
 i 
 
 
 / / 
 
 
 / / 
 
 / 
 
 
 
 t f 4-t 1 4-7-1- -I 
 
 < 
 
 it/ 
 
 i 
 
 ' 
 
 
 
 i i 
 
 / 
 
 
 / / 
 
 I 
 
 
 
 - Jitilftf LL 
 
 
 
 i 
 
 ' 
 
 
 
 i / 
 
 / 
 
 
 
 i 
 
 
 
 .- titihyz u 
 
 
 ■ i 
 
 
 
 
 / / 
 
 i 
 
 / 
 
 
 / 
 
 / 
 
 
 
 --uY/j/fU-ZJu 
 
 
 
 
 
 
 1 
 
 / 
 
 
 
 / 
 
 / 
 
 
 
 -1lLU4 Lj 
 
 
 t 
 
 
 l 
 
 / 
 
 
 / 
 
 i / 
 
 
 / 
 
 / 
 
 
 
 Itluitujlu. 
 
 
 'j 
 
 
 ' 
 
 / 
 
 i 
 
 / / 
 
 / 
 
 
 / 
 
 / 
 
 
 
 M4m 
 
 A 
 
 - 
 
 J 
 
 <k 
 
 / 
 
 / / 
 
 1 
 
 4= 
 
 
 -f- — 
 
 / ■ 
 
 
 o 
 
 CN 
 
 ||iiii:i::=ie 
 
 mm\¥$¥ 
 
 
 2 
 
 7 
 
 i 
 
 
 4— 
 
 4— 
 
 t ; 
 
 ) 
 
 1 / 
 
 
 
 / 
 
 nti *ij,i 
 
 r,'i 4r ■ r' 
 
 
 £ 
 
 
 i 
 
 
 1 
 
 
 
 
 
 
 
 
 lf t / / ' / 
 
 
 ' 
 
 
 
 
 
 
 
 / 
 
 
 
 / 
 
 / A ' 1 1 
 
 
 
 / 
 
 
 
 
 
 
 / 
 
 
 
 / 
 
 ■ , .'// 
 
 
 zx 
 
 i 
 
 
 
 
 
 
 / 
 
 
 
 / 
 
 V L' -tiLLLlLJI. 
 
 ' 
 
 
 > 
 
 
 / 
 
 / 
 
 
 
 .-■ 
 
 
 
 / 
 
 Jil ' ' ' 
 
 -J.J. J. . / / ; , 
 
 
 _, 
 
 
 
 
 
 
 t 
 
 / 
 
 
 
 7 
 
 
 ' /..,■/.. / 
 
 
 ~z 
 
 
 ! 
 
 j 
 
 
 
 t 
 
 / 
 
 
 
 / 
 
 ^ Z It^ilTtL 
 
 
 / 
 
 
 1 
 
 ' 
 
 
 / 
 
 
 
 
 
 / 
 
 ./-UtlttttZ, 
 
 ' 
 
 / 
 
 
 
 
 i 
 
 / 
 
 
 
 
 
 / 
 
 
 /J// 2222111 
 
 1 
 
 ./ 
 
 
 
 i 
 
 1 
 
 / 
 
 
 
 
 
 / 
 
 1 J f j !~ ~tl i 
 
 1 
 
 / 
 
 / 
 
 ' 
 
 
 
 / 
 
 
 
 
 
 / 
 
 fflfe' '---? 
 
 llVltTVtli 
 
 
 
 
 1 
 
 / 
 
 
 f 
 
 
 
 
 
 / 
 
 11M:#< ^ /- 
 
 LLLLLLj i iff 
 
 
 J. 
 
 
 1 
 
 / 
 
 / 
 
 
 
 
 
 
 / 
 
 I ( ! (1 
 
 iUulttlJ-4 
 
 1 
 
 L 
 
 
 J 
 
 ' 
 
 
 
 
 
 
 
 / 
 
 Wul-Ui 
 
 'III; Hill 
 
 / 
 
 
 
 I 
 
 
 1 
 
 
 
 
 
 
 / 
 
 o 
 
 CN 
 
 r-'cyoi m s. co *o xf co, 
 
 wffrrff/ 
 
 to ^f «^ io <o <0«»(o o 
 
 . ^ . ^> . <0<CHO U 
 
 Cn . — nvo,«n O «-•' co/ui r>- to/ « 
 r>- c« ^ « «> to <n ■« to 
 
 p* r~ r~ r* CD cK> *3C 
 
 eo cn « 
 
 / cO "o *" i— / *> 
 
 / ? M 8 / 1 <?/ 8 
 
 Abac for determiuing the Probable Errors of Correlation Coefficients. 
 
 3—2 
 
20 
 
 Tables for Statisticians and Biometricians 
 
 TABLE VIII. Values of 1 - r"- for r = '001 to -999. 
 Values of 1 — r-. 
 
 r 
 
 ■000 
 
 ■001 
 
 •002 
 
 •003 
 
 •004 
 
 ■005 
 
 •006 
 
 ■007 
 
 •008 
 
 ■009 
 
 ■000 
 
 1-000 000 
 
 ■999 999 
 
 •999 996 
 
 •999 991 
 
 •999 984 
 
 ■999 975 
 
 ■999 964 
 
 •999 951 
 
 •999 936 
 
 •999 919 
 
 ■010 
 
 •999 900 
 
 •999 879 
 
 ■999 856 
 
 ■999 831 
 
 •999 804 
 
 ■999 775 
 
 •999 744 
 
 •999 711 
 
 •999 676 
 
 •999 639 
 
 ■020 
 
 •999 600 
 
 •999 559 
 
 •999 516 
 
 •999 471 
 
 •999 424 
 
 •999 375 
 
 •999 324 
 
 •999 271 
 
 •999 216 
 
 ■999 159 
 
 ■030 
 
 •999 100 
 
 •999 039 
 
 •998 976 
 
 •998 911 
 
 •998 844 
 
 •998 775 
 
 •998 704 
 
 •998 631 
 
 •998 556 
 
 •998 479 
 
 ■040 
 
 •998 400 
 
 ■998 319 
 
 •998 236 
 
 •998 151 
 
 •998 064 
 
 •997 975 
 
 •997 884 
 
 •997 791 
 
 ■997 696 
 
 •997 599 
 
 ■050 
 
 •997 500 
 
 •997 399 
 
 •997 296 
 
 •997 191 
 
 •997 084 
 
 ■996 975 
 
 ■996 864 
 
 •996 751 
 
 ■996 636 
 
 •996 519 
 
 ■060 
 
 •996 400 
 
 ■996 279 
 
 •996 156 
 
 •996 031 
 
 •995 904 
 
 ■995 775 
 
 •995 644 
 
 •995 511 
 
 •995 376 
 
 ■995 239 
 
 ■070 
 
 •995 100 
 
 •994 959 
 
 '994 816 
 
 •994 671 
 
 ■994 524 
 
 •994 375 
 
 •994 224 
 
 •994 071 
 
 •993 916 
 
 •993 759 
 
 ■080 
 
 •993 600 
 
 ■993 439 
 
 •993 276 
 
 •993 111 
 
 •992 944 
 
 •992 775 
 
 ■992 604 
 
 •992 431 
 
 •992 256 
 
 •992 079 
 
 ■090 
 
 •991 900 
 
 ■991 719 
 
 •991 536 
 
 •991 351 -991 164 
 
 ■990 975 
 
 ■990 784 
 
 •990 591 
 
 ■990 396 
 
 ■990 199 
 
 ■100 
 
 •990 000 
 
 •989 799 
 
 ■989 596 
 
 •989 391 -989 184 
 
 •988 975 
 
 •988 764 
 
 •988 551 
 
 •988 336 
 
 •988 119 
 
 ■110 
 
 •987 900 
 
 •987 679 
 
 ■987 456 
 
 •987 231 
 
 •987 004 
 
 •986 775 
 
 ■986 544 
 
 •986 311 
 
 •986 076 
 
 •985 839 
 
 ■120 
 
 •985 600 
 
 •985 359 
 
 ■985 116 
 
 •984 871 
 
 •984 624 
 
 •984 375 
 
 ■984 124 
 
 •983 871 
 
 •983 616 
 
 ■983 359 
 
 ■ISO 
 
 •983 100 
 
 •982 839 
 
 ■982 576 
 
 •982 311 
 
 •982 044 
 
 •981 775 
 
 •981 504 
 
 •981 231 
 
 ■980 956 
 
 ■980 679 
 
 •lJfl 
 
 •980 400 
 
 ■980 119 
 
 ■979 836 
 
 •979 551 
 
 •979 264 
 
 •978 975 
 
 •978 684 
 
 •978 391 
 
 •978 096 
 
 •977 799 
 
 ■150 
 
 •977 500 
 
 ■977 199 
 
 •976 896 
 
 •976 591 
 
 •976 284 
 
 •975 975 
 
 •975 664 
 
 •975 351 
 
 •975 036 
 
 •974 719 
 
 ■160 
 
 •974 400 
 
 •974 079 
 
 •973 756 
 
 •973 431 
 
 •973 104 
 
 •972 775 
 
 •972 444 
 
 •972 111 
 
 •971 776 
 
 •971 439 
 
 •170 
 
 •971 100 
 
 •970 759 
 
 •970 416 
 
 •970 071 
 
 •969 724 
 
 •969 375 
 
 •969 024 
 
 •968 671 
 
 ■968 316 
 
 •967 959 
 
 ■180 
 
 •967 600 
 
 ■967 239 
 
 •966 876 
 
 •966 511 
 
 •966 144 
 
 •965 775 
 
 ■965 404 
 
 •965 031 
 
 ■904 656 
 
 •964 279 
 
 ■190 
 
 ■963 900 
 
 ■963 519 
 
 •963 136 
 
 •962 751 
 
 •962 364 
 
 •961 975 
 
 •961 584 
 
 •961 191 
 
 •960 796 
 
 •960 399 
 
 •200 
 
 •960 000 
 
 •959 599 
 
 •959 196 
 
 •958 791 
 
 •958 384 
 
 ■957 975 
 
 •957 564 
 
 •957 151 
 
 •956 736 
 
 •956 319 
 
 •210 
 
 •955 900 
 
 •955 479 
 
 •955 056 
 
 •954 631 
 
 •954 204 
 
 •953 775 
 
 •953 344 
 
 •952 911 
 
 •952 476 
 
 •952 039 
 
 ■220 
 
 ■951 600 
 
 ■951 159 
 
 •950 716 
 
 ■950 271 
 
 •949 824 
 
 •949 375 
 
 •948 924 
 
 •948 471 
 
 •948 016 
 
 •947 559 
 
 •2S0 
 
 •947 100 
 
 •946 639 
 
 •946 176 
 
 •945 711 
 
 •945 244 
 
 •944 775 
 
 •944 304 
 
 •943 831 
 
 •943 356 
 
 •942 879 
 
 ■240 
 
 •942 400 
 
 •941 919 
 
 •941 436 
 
 •940 951 
 
 ■940 464 
 
 ■939 975 
 
 •939 484 
 
 •938 991 
 
 •938 496 
 
 •937 999 
 
 ■250 
 
 •937 500 
 
 •936 999 
 
 •936 496 
 
 •935 991 
 
 •935 484 
 
 •934 975 
 
 •934 464 
 
 •933 951 
 
 •933 436 
 
 •932 919 
 
 •260 
 
 •932 400 
 
 •931 879 
 
 •931 356 
 
 •930 831 
 
 •930 304 
 
 ■929 775 
 
 •929 244 
 
 •928 711 
 
 •928 176 
 
 •927 639 
 
 •2-70 
 
 •927 100 
 
 •926 559 
 
 •926 016 
 
 •925 471 
 
 •924 924 
 
 •924 375 
 
 •923 824 
 
 •923 271 
 
 •922 716 
 
 •922 159 
 
 ■280 
 
 •921 600 
 
 •921 039 
 
 •920 476 
 
 •919 911 
 
 •919 344 
 
 •918 775 
 
 •918 204 
 
 •917 631 
 
 •917 056 
 
 •916 479 
 
 ■290 
 
 •915 900 
 
 •915 319 
 
 •914 736 
 
 ■914 151 
 
 •913 564 
 
 •912 975 
 
 •912 384 
 
 ■911 791 
 
 ■911 196 
 
 •910 599 
 
 ■800 
 
 •910 000 
 
 •909 399 
 
 •908 796 
 
 •908 191 
 
 •907 584 
 
 •906 975 
 
 •906 364 
 
 •905 751 
 
 ■905 136 
 
 •904 519 
 
 ■310 
 
 ■903 900 
 
 •903 279 
 
 •902 656 
 
 •902 031 
 
 •901 404 
 
 •900 775 
 
 •900 144 
 
 •899 511 
 
 •898 876 
 
 •898 239 
 
 ■320 
 
 •897 600 
 
 •896 959 
 
 •896 316 
 
 •895 671 
 
 •895 024 
 
 •894 375 
 
 •893 724 
 
 ■893 071 
 
 •892 416 
 
 •891 759 
 
 ■830 
 
 •891 100 
 
 •890 439 
 
 •889 776 
 
 •889 111 
 
 ■888 444 
 
 •887 775 
 
 •887 104 
 
 •886 431 
 
 •885 756 
 
 •885 079 
 
 ■3.1,0 
 
 •884 400 
 
 •883 719 
 
 •883 036 
 
 •882 351 
 
 ■881 664 
 
 •880 975 
 
 ■880 284 
 
 •879 591 
 
 ■878 896 
 
 ■878 199 
 
 ■350 
 
 •877 500 
 
 •876 799 
 
 •876 096 
 
 ■875 391 
 
 •874 684 
 
 •873 975 
 
 •873 264 
 
 •872 551 
 
 •871 836 
 
 •871 119 
 
 ■360 
 
 •870 400 
 
 •869 679 
 
 •868 956 
 
 •868 231 
 
 •867 504 
 
 ■866 775 
 
 •866 044 
 
 •865 311 
 
 •864 576 
 
 •863 839 
 
 ■370 
 
 •863 100 
 
 •862 359 
 
 •861 616 
 
 •860 871 
 
 •860 124 
 
 •859 375 
 
 •858 624 
 
 •857 871 
 
 •857 116 
 
 •856 359 
 
 ■380 
 
 •855 600 
 
 ■854 839 
 
 •854 076 
 
 •853 311 
 
 ■852 544 
 
 •851 775 
 
 •851 004 
 
 •850 231 
 
 •849 456 
 
 •848 679 
 
 •390 
 
 •847 900 
 
 •847 119 
 
 •846 336 
 
 •845 551 
 
 •844 764 
 
 •843 975 
 
 ■843 184 
 
 •842 391 
 
 •841 596 
 
 •840 799 
 
 ■400 
 
 •840 000 
 
 •839 199 
 
 •838 396 
 
 ■837 591 
 
 •836 784 
 
 •835 975 
 
 •835 164 
 
 ■834 351 
 
 •833 536 
 
 •832 719 
 
 ■410 
 
 •831 900 
 
 •831 079 
 
 •830 256 
 
 •829 431 
 
 •828 604 
 
 •827 775 
 
 •826 944 
 
 •826 111 
 
 •825 276 
 
 •824 439 
 
 ■420 
 
 •823 600 
 
 •822 759 
 
 •821 916 
 
 •821 071 
 
 •820 224 
 
 •819 375 
 
 •818 524 
 
 ■817 671 
 
 •816 816 
 
 •815 959 
 
 •430 
 
 •815 100 
 
 ■814 239 
 
 •813 376 
 
 •812 511 
 
 •811 644 
 
 •810 775 
 
 •809 904 
 
 •809 031 
 
 •808 156 
 
 •807 279 
 
 ■440 
 
 •806 400 
 
 •805 519 
 
 •804 636 
 
 •803 751 
 
 •802 864 
 
 •801 975 
 
 ■801 084 
 
 ■800 191 
 
 •799 296 
 
 •798 399 
 
 ■450 
 
 •797 500 
 
 •796 599 
 
 •795 696 
 
 •794 791 
 
 •793 884 
 
 •792 975 
 
 •792 064 
 
 •791 151 
 
 ■790 236 
 
 •789 319 
 
 •460 
 
 •788 400 
 
 •787 479 
 
 •786 556 
 
 •785 631 
 
 •784 704 
 
 •783 775 
 
 •782 844 
 
 •781 911 
 
 •780 976 
 
 •780 039 
 
 ■470 
 
 •779 100 
 
 •778 159 
 
 •777 216 
 
 •776 271 
 
 •775 324 
 
 •774 375 
 
 •773 424 
 
 •772 471 
 
 •771 516 
 
 •770 559 
 
 ■480 
 
 •769 600 
 
 •768 639 
 
 •767 676 
 
 •766 711 
 
 •765 744 
 
 •764 775 
 
 •763 804 
 
 •762 831 
 
 •761 856 
 
 •760 879 
 
 ■490 
 
 •759 900 
 
 •758 919 
 
 •757 936 
 
 •756 951 
 
 ■755 964 
 
 •754 975 
 
 ■753 984 
 
 •752 991 
 
 •751 996 
 
 •750 999 
 
Probable Error of a Coefficient of Correlation 
 
 21 
 
 TABLE VIII. Values of l-r> for r=001 to '999. 
 Values of 1— r 3 . 
 
 ■ooo 
 
 ■500 
 ■510 
 ■520 
 ■530 
 ■540 
 ■550 
 ■560 
 •570 
 ■5S0 
 ■590 
 
 ■600 
 •610 
 ■620 
 ■630 
 ■640 
 ■650 
 ■660 
 ■670 
 •680 
 ■690 
 
 ■700 
 ■710 
 ■720 
 •730 
 ■740 
 ■750 
 ■760 
 ■770 
 ■780 
 ■790 
 
 •800 
 ■810 
 ■820 
 ■830 
 ■840 
 ■850 
 ■860 
 ■870 
 ■880 
 ■890 
 
 ■900 
 •910 
 ■920 
 ■930 
 ■940 
 ■950 
 ■960 
 ■970 
 ■980 
 ■990 
 
 •750 000 
 ■739 900 
 •729 600 
 •719 100 
 •708 400 
 •697 500 
 •686 400 
 •675 100 
 •663 600 
 •651 900 
 
 ■640 000 
 •627 900 
 •615 600 
 •603 100 
 ■590 400 
 •577 500 
 •564 400 
 •551 100 
 •537 600 
 •523 900 
 
 •510 000 
 •495 900 
 •481 600 
 •467 100 
 •452 400 
 •437 500 
 ■422 400 
 •407 100 
 ■391 600 
 •375 900 
 
 •360 000 
 •343 900 
 •327 600 
 •311 100 
 •294 400 
 •277 500 
 •260 400 
 •243 100 
 •225 600 
 ■207 900 
 
 •190 000 
 •171 900 
 •153 600 
 •135 100 
 •116 400 
 •097 500 
 •078 400 
 •059 100 
 •039 600 
 019 900 
 
 ■001 
 
 748 999 
 738 879 
 728 559 
 718 039 
 707 319 
 696 399 
 685 279 
 673 959 
 662 439 
 650 719 
 
 638 799 
 626 679 
 614 359 
 601 839 
 589 119 
 576 199 
 563 079 
 549 759 
 536 239 
 522 519 
 
 508 599 
 194 479 
 480 159 
 465 639 
 450 919 
 435 999 
 420 879 
 405 559 
 390 039 
 374 319 
 
 •358 399 
 ■342 279 
 •325 959 
 •309 439 
 292 719 
 '275 799 
 •258 679 
 •241 359 
 ■223 839 
 •206 119 
 
 188 199 
 170 079 
 151 759 
 133 239 
 114 519 
 095 599 
 076 479 
 057 159 
 037 039 
 017 919 
 
 •003 
 
 •747 996 
 
 •737 856 
 
 •727 516 
 
 •716 976 
 
 •706 236 
 
 •695 296 
 
 •684 156 
 
 •672 816 
 
 •661 276 
 
 •649 536 
 
 ■637 596 
 
 •625 456 
 
 •613 116 
 
 •600 576 
 
 •587 836 
 
 •574 896 
 
 •561 756 
 
 •548 416 
 
 •534 876 
 
 •521 136 
 
 •507 196 
 
 •493 056 
 
 ■478 716 
 
 •464 176 
 
 •449 436 
 
 •434 496 
 
 •419 356 
 
 •404 016 
 
 ■388 476 
 
 •372 736 
 
 •356 796 
 
 •340 656 
 
 •324 316 
 
 •307 776 
 
 •291 036 
 
 •274 096 
 
 •256 956 
 
 •239 616 
 
 •222 076 
 
 •204 336 
 
 •186 396 
 
 •168 256 
 
 •149 916 
 
 •131 376 
 
 •112 636 
 
 •093 696 
 
 •074 556 
 
 •055 216 
 
 •035 676 
 
 •015 936 
 
 ■003 
 
 746 991 
 736 831 
 726 471 
 715 911 
 705 151 
 694 191 
 683 031 
 671 671 
 660 111 
 648 351 
 
 636 391 
 624 231 
 611 871 
 599 311 
 586 551 
 573 591 
 560 431 
 547 071 
 533 511 
 519 751 
 
 505 791 
 491 631 
 477 271 
 462 711 
 447 951 
 432 991 
 417 831 
 402 471 
 386 911 
 371 151 
 
 355 191 
 339 031 
 322 671 
 306 111 
 289 351 
 272 391 
 255 231 
 237 871 
 220 311 
 202 551 
 
 184 591 
 166 431 
 148 071 
 129 511 
 110 751 
 091 791 
 072 631 
 053 271 
 033 711 
 013 951 
 
 •004 
 
 ■005 
 
 •006 
 
 •007 
 
 •745 984 
 
 •744 975 
 
 •743 964 
 
 ■742 951 
 
 •735 804 
 
 •734 775 
 
 •733 744 
 
 •732 711 
 
 •725 424 
 
 •724 375 
 
 •723 324 
 
 •722 271 
 
 ■714 844 
 
 •713 775 
 
 •712 704 
 
 •711 631 
 
 •704 064 
 
 •702 975 
 
 •701 884 
 
 •700 791 
 
 •693 084 
 
 •691 975 
 
 •690 864 
 
 •689 751 
 
 •681 904 
 
 •680 775 
 
 ■679 644 
 
 •678 511 
 
 •670 524 
 
 •669 375 
 
 ■668 224 
 
 •667 071 
 
 •658 944 
 
 •657 775 
 
 •656 604 
 
 •655 431 
 
 •647 164 
 
 •645 975 
 
 •644 784 
 
 •643 591 
 
 •635 184 
 
 •633 975 
 
 •632 764 
 
 •631 551 
 
 •623 004 
 
 •621-775 
 
 •620 544 
 
 ■619 311 
 
 •610 624 
 
 •609 375 
 
 •608 124 
 
 •606 871 
 
 •598 044 
 
 •596 775 
 
 •595 504 
 
 •594 231 
 
 •585 264 
 
 •583 975 
 
 •582 684 
 
 •581 391 
 
 •572 284 
 
 ■570 975 
 
 •569 664 
 
 •568 351 
 
 ■559 104 
 
 •557 775 
 
 •556 444 
 
 •555 111 
 
 •545 724 
 
 •544 375 
 
 •543 024 
 
 •541 671 
 
 •532 144 
 
 •530 775 
 
 •529 404 
 
 •528 031 
 
 •518 364 
 
 •516 975 
 
 •515 584 
 
 •514 191 
 
 •504 384 
 
 •502 975 
 
 •501 564 
 
 ■500 151 
 
 •490 204 
 
 •488 775 
 
 •487 344 
 
 •485 911 
 
 ■475 824 
 
 •474 375 
 
 •472 924 
 
 ■471 471 
 
 •461 244 
 
 •459 775 
 
 •458 304 
 
 ■406 831 
 
 •446 464 
 
 •444 975 
 
 ■443 484 
 
 •441 991 
 
 ■431 484 
 
 •429 975 
 
 •428 464 
 
 •426 951 
 
 •416 304 
 
 •414 775 
 
 ■413 244 
 
 •411 711 
 
 •400 924 
 
 •399 375 
 
 •397 824 
 
 •396 271 
 
 •385 344 
 
 •383 775 
 
 •382 204 
 
 ■380 631 
 
 ■369 564 
 
 ■367 975 
 
 •366 384 
 
 •364 791 
 
 •353 584 
 
 •351 975 
 
 •350 364 
 
 •348 751 
 
 ■337 404 
 
 •335 775 
 
 •334 144 
 
 •332 511 
 
 •321 024 
 
 •319 375 
 
 •317 724 
 
 ■316 071 
 
 •304 444 
 
 •302 775 
 
 •301 104 
 
 •299 431 
 
 •287 664 
 
 •285 975 
 
 •284 284 
 
 •282 591 
 
 •270 684 
 
 •268 975 
 
 •267 264 
 
 •265 551 
 
 •253 504 
 
 •251 775 
 
 •250 044 
 
 ■248 311 
 
 •236 124 
 
 •234 375 
 
 •232 624 
 
 •230 871 
 
 •218 544 
 
 •216 775 
 
 •215 004 
 
 •213 231 
 
 •200 764 
 
 •198 975 
 
 •197 184 
 
 •195 391 
 
 •182 784 
 
 ■180 975 
 
 •179 164 
 
 •177 351 
 
 •164 604 
 
 •162 775 
 
 •160 944 
 
 •159 111 
 
 •146 224 
 
 •144 375 
 
 ■142 524 
 
 •140 671 
 
 •127 644 
 
 •125 775 
 
 •123 904 
 
 •122 031 
 
 •108 864 
 
 ■106 975 
 
 •105 084 
 
 •103 191 
 
 •089 884 
 
 •087 975 
 
 •086 064 
 
 •084 151 
 
 •070 704 
 
 •068 775 
 
 •066 844 
 
 •064 911 
 
 •051 324 
 
 •049 375 
 
 •047 424 
 
 •045 471 
 
 •031 744 
 
 •029 775 
 
 •027 804 
 
 •025 831 
 
 •Oil 964 
 
 009 975 
 
 •007 984 
 
 ■005 991 
 
 ■008 
 
 •741 936 
 •731 676 
 •721 216 
 ■710 556 
 •699 696 
 •688 636 
 •677 376 
 •665 916 
 •654 256 
 •642 396 
 
 ■630 336 
 •618 076 
 •605 616 
 •592 956 
 •580 096 
 •567 036 
 •553 776 
 •540 316 
 •526 656 
 ■512 796 
 
 •498 736 
 •484 476 
 •470 016 
 •455 356 
 •440 496 
 •425 436 
 ■410 176 
 •394 716 
 •379 056 
 ■363 196 
 
 •347 136 
 •330 876 
 •314 416 
 •297 756 
 •280 896 
 •263 836 
 •246 576 
 ■229 116 
 •211 456 
 •193 596 
 
 •175 536 
 •157 276 
 •138 816 
 •120 156 
 •101 296 
 082 236 
 •062 976 
 •043 516 
 •023 856 
 •003 996 
 
 •009 
 
 •740 919 
 •730 639 
 •720 159 
 •709 479 
 •698 599 
 •687 519 
 •676 239 
 •664 759 
 •653 079 
 •641 199 
 
 •629 119 
 •616 839 
 ■604 359 
 •591 679 
 •578 799 
 ■565 719 
 •552 439 
 •538 959 
 ■525 279 
 •511 399 
 
 •497 319 
 •483 039 
 •468 559 
 ■453 879 
 •438 999 
 •423 919 
 •408 639 
 •393 159 
 ■377 479 
 •361 599 
 
 •345 519 
 •329 239 
 •312 759 
 •296 079 
 •279 199 
 •262 119 
 •244 839 
 •227 359 
 •209 679 
 191 799 
 
 •173 719 
 •155 439 
 •136 959 
 •118 279 
 •099 399 
 •080 319 
 •061 039 
 •041 559 
 ■021 879 
 •001 999 
 
22 Tables for Statisticians and Biometricians 
 
 TABLE IX. Values of the Incomplete Normal Moment Function /j, n (x). 
 A. Odd Moments m, (x) = /t„ (#)/{(» - 1) (n - 3) (« - 5) ... 2{. 
 
 X 
 
 mi(x) 
 
 »"3 (*) 
 
 m, (x) 
 
 J«7 (x) 
 
 "'9 (x) 
 
 o-o 
 
 •0000000 
 
 •0000000 
 
 •0000000 
 
 •ooooooo 
 
 ■ooooooo 
 
 0-1 
 
 •0019897 
 
 •0000050 
 
 •ooooooo 
 
 •ooooooo 
 
 •ooooooo 
 
 0-2 
 
 •0078996 
 
 •0000787 
 
 •0000005 
 
 •ooooooo 
 
 ■ooooooo 
 
 OS 
 
 •0175545 
 
 •0003920 
 
 •0000059 
 
 •0000001 
 
 •ooooooo 
 
 o-4 
 
 •0306721 
 
 ■0012105 
 
 •0000321 
 
 •0000006 
 
 •ooooooo 
 
 0-5 
 
 •0468770 
 
 ■0028688 
 
 ■0001183 
 
 ■0000037 
 
 •0000001 
 
 0-6 
 
 •0657177 
 
 •0057372 
 
 •0003390 
 
 •0000151 
 
 •0000005 
 
 0-7 
 
 •0866883 
 
 ■0101861 
 
 •0008140 
 
 •0000493 
 
 •0000024 
 
 0-8 
 
 •1092507 
 
 ■0166494 
 
 •0017172 
 
 •0001350 
 
 •0000086 
 
 0-9 
 
 •1328570 
 
 •0^.")0925 
 
 •0032702 
 
 •0003242 
 
 •0000259 
 
 1-0 
 
 •1569716 
 
 •0359862 
 
 •0057399 
 
 •0006988 
 
 •0000687 
 
 VI 
 
 •1810901 
 
 •0492895 
 
 •0094199 
 
 •0013795 
 
 ■0001634 
 
 V2 
 
 •2047562 
 
 •0649423 
 
 •0146092 
 
 •0025293 
 
 •0003549 
 
 V3 
 
 •2275737 
 
 ' -0827672 
 
 •0215865 
 
 •0043539 
 
 •0007135 
 
 1'4 
 
 •2492148 
 
 •1024819 
 
 •0305828 
 
 •0070957 
 
 •0013414 
 
 1-5 
 
 •2694247 
 
 •1237174 
 
 •0417570 
 
 •0110219 
 
 •0023776 
 
 1-6 
 
 •2880214 
 
 •1460428 
 
 •0551764 
 
 •0164068 
 
 •0040005 
 
 V7 
 
 •3048932 
 
 •1689923 
 
 •0708039 
 
 •0235098 
 
 •0064248 
 
 1-8 
 
 •3199921 
 
 •1920929 
 
 •0884945 
 
 •0325513 
 
 ■0098944 
 
 V9 
 
 •3333265 
 
 •2148899 
 
 •1080009 
 
 •0436894 
 
 ■0146688 
 
 2-0 
 
 •3449513 
 
 •23690!) [ 
 
 •1289874 
 
 •0569995 
 
 •0210055 
 
 2-1 
 
 •3549587 
 
 •2579749 
 
 •1510502 
 
 •0724606 
 
 •0291380 
 
 2-2 
 
 •3634677 
 
 ■2776192 
 
 •1737425 
 
 •0899486 
 
 •0392533 
 
 2-8 
 
 •3706152 
 
 •2956902 
 
 •1966019 
 
 •1092390 
 
 •0514703 
 
 2-4 
 
 •3765478 
 
 •3120515 
 
 •2191769 
 
 •1300173 
 
 •0658224 
 
 2-5 
 
 •3814140 
 
 •3266380 
 
 •2410506 
 
 •1518971 
 
 •0822459 
 
 2-6 
 
 •3853593 
 
 •3394489 
 
 •2618602 
 
 •1744437 
 
 •1005767 
 
 2-7 
 
 •3885213 
 
 •3505370 
 
 ■2813106 
 
 •1972006 
 
 •1205553 
 
 2-8 
 
 •3910268 
 
 •3599983 
 
 •2991823 
 
 •2197160 
 
 •1418391 
 
 2-9 
 
 •3929897 
 
 •3679593 
 
 •3153329 
 
 •2415682 
 
 •1640231 
 
 8-0 
 
 •3945104 
 
 •3745671 
 
 •3296946 
 
 •2623860 
 
 •1866637 
 
 3-1 
 
 •3956755 
 
 •3799784 
 
 •3422662 
 
 •2818638 
 
 •2093055 
 
 ■ S-2 
 
 •3965582 
 
 •3843517 
 
 •3531029 
 
 •2997718 
 
 •2315079 
 
 8-3 
 
 •3972197 
 
 •3878403 
 
 ■3623049 
 
 •3159582 
 
 •2528687 
 
 3-4 
 
 •3977101 
 
 •3905878 
 
 •3700046 
 
 •3303476 
 
 •2730432 
 
 8-5 
 
 •3980696 
 
 •3927244 
 
 •3763548 
 
 •3429335 
 
 •2917571 
 
 s-n 
 
 •3983304 
 
 •3943653 
 
 •3815183 
 
 •3537687 
 
 •3088145 
 
 3-7 
 
 •3985175 
 
 •3956099 
 
 •3856585 
 
 •3629529 
 
 •3240979 
 
 8-8 
 
 •3986503 
 
 •3965425 
 
 •3889331 
 
 •3706199 
 
 •3375646 
 
 8-9 
 
 •3987436 
 
 •3972329 
 
 •3914881 
 
 •3769253 
 
 •3492376 
 
 40 
 
 •3988085 
 
 •3977378 
 
 •3934552 
 
 •3820351 
 
 •3591947 
 
 4-1 
 
 •3988530 
 
 •3981028 
 
 •3949499 
 
 •3861165 
 
 •3675554 
 
 4-2 
 
 •3988833 
 
 •3983635 
 
 •3960708 
 
 •3893304 
 
 •3744677 
 
 4-8 
 
 •3989037 
 
 •3985475 
 
 •3969007 
 
 •3918258 
 
 •3800964 
 
 4-4 
 
 •3989173 
 
 ■3986759 
 
 •3975073 
 
 •3937367 
 
 •3846117 
 
 4-rj 
 
 •3989263 
 
 •3987645 
 
 •3979452 
 
 •3951801 
 
 •3881809 
 
 4-6 
 
 •3989321 
 
 •3988248 
 
 •3982573 
 
 •3962557 
 
 •3909614 
 
 4-7 
 
 •3989359 
 
 ■3988656 
 
 •3984770 
 
 •3970466 
 
 •3930967 
 
 4-8 
 
 •3989383 
 
 •3988927 
 
 •3986298 
 
 •3976205 
 
 •3947135 
 
 4-9 
 
 •3989398 
 
 •3989106 
 
 •3987348 
 
 •3980315 
 
 •3959207 
 
 5-0 
 
 •3989408 
 
 •3989222 
 
 •3988061 
 
 •3983221 
 
 •3968097 
 
 00 
 
 •3989423 
 
 •3989423 
 
 •3989423 
 
 •3989423 
 
 •3989423 
 
Incomplete Normal Moment Functions 
 TABLE IX. Values of the Incomplete Normal Moment Function. 
 
 23 
 
 B. Even Moments m n (x) = /x n (x)/{(n — !)(«• — 3) (n — 5) ...1 
 
 X 
 
 n*2 (x) 
 
 m 4 {x) 
 
 m (x) 
 
 m 8 (x) 
 
 ot 10 (.t) 
 
 o-o 
 
 ■ooooooo 
 
 •ooooooo 
 
 ooooooo 
 
 ooooooo 
 
 OOOOOOO 
 
 o-i 
 
 •0001325 
 
 •0000002 
 
 ooooooo 
 
 00(30000 
 
 OOOOOOO 
 
 OS 
 
 •0010512 
 
 •0000084 
 
 ooooooo 
 
 ooooooo 
 
 OOOOOOO 
 
 OS 
 
 •0034951 
 
 •0000626 
 
 0000008 
 
 ooooooo 
 
 ooooooo 
 
 0-4 
 
 •0081136 
 
 ■0002572 
 
 0000058 
 
 0000001 
 
 ooooooo 
 
 OS 
 
 •0154298 
 
 •0007604 
 
 0000270 
 
 0000008 
 
 0000001 
 
 0-6 
 
 •0258121 
 
 •0018200 
 
 0000925 
 
 0000037 
 
 0000001 
 
 0-7 
 
 •0394585 
 
 •0037575 
 
 0002588 
 
 0000139 
 
 0000006 
 
 OS 
 
 •0563914 
 
 •0069507 
 
 0006223 
 
 0000437 
 
 0000025 
 
 0-9 
 
 •0764632 
 
 •0118045 
 
 0013297 
 
 0001177 
 
 0000086 
 
 1-0 
 
 •0993740 
 
 •0187171 
 
 0025857 
 
 0002812 
 
 0000251 
 
 1-1 
 
 •1246965 
 
 •0280428 
 
 0046525 
 
 0006094 
 
 0000658 
 
 12 
 
 •1519070 
 
 •0400559 
 
 0078427 
 
 0012160 
 
 0001558 
 
 1-3 
 
 •1804203 
 
 •0549214 
 
 0125028 
 
 0022017 
 
 0003386 
 
 t'k 
 
 •2096248 
 
 •0726741 
 
 0189894 
 
 0039577 
 
 0006842 
 
 1-5 
 
 •2389164 
 
 •0932091 
 
 0276408 
 
 0065653 
 
 0012964 
 
 1-6 
 
 •2677274 
 
 •1162835 
 
 0387442 
 
 0103869 
 
 0023209 
 
 V7 
 
 •2955511 
 
 •1415300 
 
 0525059 
 
 0157516 
 
 0039494 
 
 1-8 
 
 •3219594 
 
 •1684803 
 
 0090258 
 
 0229926 
 
 0064207 
 
 VO 
 
 •3466134 
 
 •1905937 
 
 0882796 
 
 0324204 
 
 0100147 
 
 2-0 
 
 •3692680 
 
 •2252921 
 
 1101113 
 
 0442938 
 
 0150415 
 
 2-1 
 
 •3897700 
 
 •2539927 
 
 1342371 
 
 0587910 
 
 0218224 
 
 2-2 
 
 •4080525 
 
 •2821413 
 
 1602593 
 
 0759806 
 
 0306667 
 
 2-3 
 
 •4241237 
 
 •3092387 
 
 1876903 
 
 0958345 
 
 0418437 
 
 2-4 
 
 •4380556 
 
 ■3348616 
 
 2159821 
 
 1181613 
 
 0555560 
 
 2-5 
 
 •4499695 
 
 •3586763 
 
 2445598 
 
 1426700 
 
 0719132 
 
 2-6 
 
 •4600231 
 
 •3804450 
 
 2728554 
 
 1689546 
 
 0909136 
 
 2-7 
 
 •4683905 
 
 ■4000247 
 
 3003387 
 
 1965228 
 
 1124320 
 
 2-8 
 
 •4752816 
 
 •4173616 
 
 3265431 
 
 2248263 
 
 1362197 
 
 2-9 
 
 •4808719 
 
 •4324798 
 
 3510842 
 
 2532933 
 
 1619132 
 
 SO 
 
 •4853546 
 
 •4454079 
 
 3736720 
 
 2813629 
 
 1890538 
 
 3-1 
 
 •4889053 
 
 •4564647 
 
 3941138 
 
 3085150 
 
 2171145 
 
 8-2 
 
 •491683S 
 
 •4656432 
 
 4123121 
 
 3342962 
 
 2455315 
 
 3-3 
 
 •4938321 
 
 •4731975 
 
 4282552 
 
 3583379 
 
 2737379 
 
 s-4 
 
 •4954736 
 
 •4793298 
 
 4420056 
 
 3803672 
 
 3011962 
 
 3-5 
 
 •4967130 
 
 •4842409 
 
 4536843 
 
 4002102 
 
 3274261 
 
 8-6 
 
 •4976381 
 
 •4881218 
 
 4634555 
 
 4177877 
 
 3520261 
 
 S-7 
 
 •4983205 
 
 •4911484 
 
 4715111 
 
 4331061 
 
 3746880 
 
 S-8 
 
 •4988183 
 
 •4934784 
 
 4780568 
 
 4462441 
 
 3952025 
 
 S-9 
 
 •4991771 
 
 •4952491 
 
 4833001 
 
 4573300 
 
 4134583 
 
 fo 
 
 ■4994330 
 
 •4965779 
 
 4874418 
 
 4665592 
 
 4294345 
 
 4-1 
 
 •4996133 
 
 •4975627 
 
 4906683 
 
 4741120 
 
 4431886 
 
 4-9 
 
 •4997391 
 
 •4982835 
 
 4931479 
 
 4802003 
 
 4548407 
 
 4-3 
 
 •4998258 
 
 •4988045 
 
 4950279 
 
 4850521 
 
 4645574 
 
 4-4 
 
 •4998849 
 
 •4991766 
 
 4964343 
 
 4888500 
 
 4725352 
 
 4-5 
 
 •4999247 
 
 •4994392 
 
 4974729 
 
 4917840 
 
 4789861 
 
 4-6 
 
 •4999512 
 
 •4996222 
 
 4982298 
 
 4940207 
 
 4841246 
 
 4-~ 
 
 •4999688 
 
 •4997483 
 
 4987744 
 
 4957010 
 
 4881574 
 
 4-8 
 
 •4999802 
 
 •4998342 
 
 4991613 
 
 4969464 
 
 4912765 
 
 4-9 
 
 •4999876 
 
 •4998919 
 
 4994326 
 
 4978572 
 
 4936544 
 
 5-0 
 
 •4999923 
 
 •4999303 
 
 4996206 
 
 4985144 
 
 4954417 
 
 00 
 
 •5000000 
 
 •5000000 
 
 5000000 
 
 5000000 
 
 5000000 
 
24 
 
 Tables for Statisticians and Biometrieians 
 
 TABLE X. Diagram of Generalised ' Probable Error.' 
 Table of Generalised ' Probable Errors.' 
 
 Number of 
 Variables 
 
 Probable Error 
 
 1 
 
 o 
 
 3 
 4 
 
 0-674,4898 
 1-177,4062 
 1-538,1667 
 1-832,1239 
 
 5 
 6 
 
 2-086,0146 
 2-312,5982 
 
 7 
 
 S 
 
 9 
 
 10 
 
 2-519,0869 
 2-710,0022 
 2-888,3962 
 3-056,4366 
 
 11 
 
 3-215,7402 
 
 
 
 
 
 
 Diagram for Value of Probable Error for 
 
 ii Variables. 
 
 
 
 
 
 40 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3-6 
 3-4 
 3-2 
 30 
 2-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 | 2-4 
 
 J! 2-2, 
 
 2 
 
 S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 | 1-6 
 
 1 1-4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 * in 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 •4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -2. 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '. 
 
 1 C 
 
 4 
 
 1 
 
 > ! 
 
 » 
 
 ' f 
 
 i ~~? 
 
 1 
 
 1 
 
 1 1 
 
 2 1 
 
 3 1 
 
 * IS 
 
 Number of Variables 
 
Determination of Normal Curve from Tail 
 
 25 
 
 TABLE XI. Constants of Normal Curve from Moments of Tail 
 
 about Stump. 
 
 Values of the Functions -v^, and ^, required to determine the Constants of a 
 Normal Frequency Distribution from the Moments of its Truncated Tail. 
 
 h' 
 
 ft 
 
 fi 
 
 ^3 
 
 h' 
 
 tl 
 
 H 
 
 <h 
 
 o-oo 
 
 •571 
 
 1-253 
 
 2-000 
 
 1-1 
 
 •734 
 
 1-977 
 
 7-371 
 
 o-oi 
 
 •573 
 
 1-259 
 
 2-016 
 
 V2 
 
 •746' 
 
 2-051 
 
 8-690 
 
 0-02 
 
 •574 
 
 1-265 
 
 2-032 
 
 IS 
 
 •757 
 
 2-126 
 
 10-331 
 
 0-03 
 
 •576 
 
 1-271 
 
 2-049 
 
 1-k 
 
 •767 
 
 2-202 
 
 12-383 
 
 0-04 
 
 •578 
 
 1-276 
 
 2-066 
 
 1-5 
 
 •777 
 
 2-280 
 
 14-968 
 
 0-05 
 
 •580 
 
 1-282 
 
 2-083 
 
 1-6 
 
 •787 
 
 2-358 
 
 18-248 
 
 0-06 
 
 •581 
 
 1-288 
 
 2-100 
 
 V7 
 
 •796 
 
 2-437 
 
 22-439 
 
 0-07 
 
 •583 
 
 1-294 
 
 2-118 
 
 IS 
 
 •804 
 
 2-517 
 
 27-832 
 
 0-08 
 
 ■585 
 
 1-300 
 
 2136 
 
 1-9 
 
 •813 
 
 2-598 
 
 34-823 
 
 0-09 
 
 •587 
 
 1-305 
 
 2-155 
 
 2-0 
 
 •820 
 
 2-679 
 
 43-956 
 
 o-i 
 
 •588 
 
 1-311 
 
 2-173 
 
 2-1 
 
 •828 
 
 2-762 
 
 55-977 
 
 0-2 
 
 •605 
 
 1-371 
 
 2-377 
 
 2-2 
 
 •835 
 
 2-845 
 
 71-925 
 
 OS 
 
 •622 
 
 1-432 
 
 2-617 
 
 2S 
 
 •842 
 
 2-929 
 
 93-248 
 
 0-4 
 
 •638 
 
 1-495 
 
 2-902 
 
 2-4 
 
 •848 
 
 3-013 
 
 121-988 
 
 OS 
 
 •653 
 
 1-560 
 
 3-241 
 
 2-5 
 
 •854 
 
 3-098 
 
 161 038 
 
 0-6 
 
 •668 
 
 1-626 
 
 3 646 
 
 2S 
 
 •860 
 
 3-184 
 
 214-537 
 
 0-7 
 
 •682 
 
 1-693 
 
 4133 
 
 2-7 
 
 •866 
 
 3-270 
 
 288-434 
 
 0-8 
 
 •696 
 
 1-762 
 
 4-720 
 
 2S 
 
 •871 
 
 3-357 
 
 391-374 
 
 0-9 
 
 •709 
 
 1-833 
 
 5-433 
 
 2-9 
 
 •876 
 
 3-445 
 
 535-963 
 
 1-0 
 
 •722 
 
 1-904 
 
 6-303 
 
 s-o 
 
 •880 
 
 3-532 
 
 740-796 
 
 1-1 
 
 •734 
 
 1-977 
 
 7-371 
 
 S-B 
 
 •888 
 
 3-969 
 
 [4299-226] 
 
 Let d equal distance of centroid of tail from stump, 2 = standard deviation of 
 the tail about its mean, and n = its area. 
 
 (i) Find ^ from ^ = S'/d*. Hence from table determine h'. 
 
 (ii) From this value of h' find >/r 2 , then o-=dxi|r 2 gives the standard 
 deviation of the uncurtailed normal curve. 
 
 (iii) h = h'x(T gives the origin of the uncurtailed normal curve. 
 
 (iv) Knowing h', Table II gives | (1 + a) and therefore the ratio £(1 — a) 
 of tail to total area of curve JV, or JV = nj% (1 - a). For many purposes it is 
 sufficient to use N — nx yjr,. 
 
26 
 
 Tables for Statisticians and Bio metricians 
 
 TABLE XII. Test for Goodness of Fit. Values of P. 
 
 X 2 
 
 n' = 3 
 
 n'=4 
 
 n'=5 
 
 n' = 6 
 
 n' = 7 
 
 n' = 8 
 
 n' = 9 
 
 n'=10 
 
 ri = 11 
 
 1 
 
 •606531 
 
 •801253 
 
 •909796 
 
 •962566 
 
 •985612 
 
 •994829 
 
 •998249 
 
 •999438 
 
 ■999828 
 
 2 
 
 •367870 
 
 •572407 
 
 •735759 
 
 •849146 
 
 •919699 
 
 •959840 
 
 •981012 
 
 •991468 
 
 •996340 
 
 S 
 
 •223130 
 
 •391625 
 
 •557825 
 
 •699986 
 
 •808847 
 
 •885002 
 
 •934357 
 
 •964295 
 
 •981424 
 
 4 
 
 •135335 
 
 •261464 
 
 •406006 
 
 •549416 
 
 •676676 
 
 •779778 
 
 •857123 
 
 •911413 
 
 947347 
 
 5 
 
 •082085 
 
 •171797 
 
 •287298 
 
 •415880 
 
 •543813 
 
 •659963 
 
 •757576 
 
 •834308 
 
 •891178 
 
 6 
 
 ■049787 
 
 •111610 
 
 •199148 
 
 •306219 
 
 •423190 
 
 •539750 
 
 ■647232 
 
 •739919 
 
 •815263 
 
 7 
 
 •030197 
 
 ■071897 
 
 •135888 
 
 •220640 
 
 •320847 
 
 •428880 
 
 ■536632 
 
 •637119 
 
 •725444 
 
 8 
 
 •018316 
 
 •046012 
 
 •091578 
 
 •156236 
 
 •238103 
 
 •332594 
 
 •433470 
 
 •534146 
 
 ■628837 
 
 
 
 •011109 
 
 •029291 
 
 ■061099 
 
 •109064 
 
 •173578 
 
 •252656 
 
 •342296 
 
 •437274 
 
 •532104 
 
 10 
 
 •006738 
 
 •018566 
 
 •040428 
 
 •075235 
 
 •124652 
 
 •188573 
 
 •265026 
 
 •350485 
 
 •440493 
 
 11 
 
 •004087 
 
 •011726 
 
 •026564 
 
 •051380 
 
 •088376 
 
 •138619 
 
 •201699 
 
 •275709 
 
 •357518 
 
 12 
 
 •002479 
 
 •007383 
 
 •017351 
 
 •034787 
 
 •061969 
 
 •100558 
 
 •151204 
 
 •213308 
 
 •285057 
 
 13 
 
 •001503 
 
 •004637 
 
 •011276 
 
 •023379 
 
 •043036 
 
 •072109 
 
 •111850 
 
 •162607 
 
 ■223672 
 
 U 
 
 •000912 
 
 ■002905 
 
 •007295 
 
 •015609 
 
 •029636 
 
 051181 
 
 ■081765 
 
 •122325 
 
 •172992 
 
 IS 
 
 •000553 
 
 •001817 
 
 •004701 
 
 •010363 
 
 •020256 
 
 •036000 
 
 •059145 
 
 090937 
 
 ■132061 
 
 16 
 
 •000335 
 
 •001134 
 
 •003019 
 
 •006844 
 
 •013754 
 
 •025116 
 
 042380 
 
 066881 
 
 •099632 
 
 17 
 
 •000203 
 
 •000707 
 
 •001933 
 
 •004500 
 
 •009283 
 
 •017396 
 
 •030109 
 
 048716 
 
 •074364 
 
 18 
 
 •000123 
 
 •000440 
 
 •001234 
 
 •002947 
 
 •006232 
 
 •011970 
 
 •021226 
 
 035174 
 
 054964 
 
 19 
 
 •000075 
 
 •000273 
 
 •000786 
 
 •001922 
 
 ■004164 
 
 •008187 
 
 •014860 
 
 025193 
 
 040263 
 
 20 
 
 •000045 
 
 •000170 
 
 •000499 
 
 •001250 
 
 •002769 
 
 •005570 
 
 •010336 
 
 •017913 
 
 029253 
 
 21 
 
 •000028 
 
 •000105 
 
 •000317 
 
 •000810 
 
 •001835 
 
 003770 
 
 •007147 
 
 •012650 
 
 021093 
 
 22 
 
 •000017 
 
 •000065 
 
 •000200 
 
 •000524 
 
 •001211 
 
 •002541 
 
 •004916 
 
 008880 
 
 015105 
 
 23 
 
 •000010 
 
 •000040 
 
 •000127 
 
 •000338 
 
 •000796 
 
 •001705 
 
 •003364 
 
 •006197 
 
 •010747 
 
 24 
 
 •000006 
 
 •000025 
 
 •000080 
 
 •000217 
 
 •000522 
 
 •001139 
 
 •002292 
 
 004301 
 
 •007600 
 
 25 
 
 ■000004 
 
 ■000016 
 
 •000050 
 
 •000139 
 
 •000341 
 
 000759 
 
 •001554 
 
 002971 
 
 005345 
 
 26 
 
 •000002 
 
 ■000010 
 
 •000032 
 
 •000090 
 
 •000223 
 
 •000504 
 
 •001050 
 
 002043 
 
 003740 
 
 27 
 
 •000001 
 
 •000006 
 
 •000020 
 
 •000057 
 
 •000145 
 
 •000333 
 
 •000707 
 
 •001399 
 
 002604 
 
 28 
 
 •000001 
 
 •000004 
 
 •000012 
 
 ■000037 
 
 •000094 
 
 •000220 
 
 •000474 
 
 •000954 
 
 001805 
 
 29 
 
 •000001 
 
 •000002 
 
 •000008 
 
 •000023 
 
 •000061 
 
 •000145 
 
 000317 
 
 000648 
 
 001246 
 
 SO 
 
 •oooooo 
 
 •000001 
 
 •000005 
 
 ■000015 
 
 •000039 
 
 •000095 
 
 ■000211 
 
 000439 
 
 000857 
 
 40 
 
 oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •000001 
 
 •000001 
 
 ■000003 
 
 ■000008 
 
 •000017 
 
 50 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 OOOOOO 
 
 ■oooooo 
 
 •oooooo 
 
 60 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •OOOOOO 
 
 •oooooo 
 
 •oooooo 
 
 70 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •oooooo 
 
 •OOOOOO 
 
 OOOOOO 
 
 •oooooo 
 
Tables for Testing Goodness of Fit 
 
 27 
 
 TABLE HTl— {continued). 
 
 x" 
 
 n' = 12 
 
 n'=13 
 
 h' = 14 
 
 )i'=15 
 
 n'=16 
 
 «'=17 
 
 n' = 18 
 
 n' = 19 
 
 n' = 20 
 
 1 
 
 •999950 
 
 •999986 
 
 •999997 
 
 ■999999 
 
 I- 
 
 1" 
 
 1- 
 
 1' 
 
 1- 
 
 2 
 
 •998496 
 
 •999406 
 
 •999774 
 
 •999917 
 
 •999970 
 
 •999990 
 
 •999997 
 
 •999999 
 
 1- 
 
 S 
 
 •990726 
 
 •995544 
 
 •997934 
 
 ■999074 
 
 •999598 
 
 ■999830 
 
 •999931 
 
 •999972 
 
 •999989 
 
 4 
 
 •969917 
 
 •983436 
 
 •991191 
 
 •995466 
 
 •997737 
 
 •998903 
 
 •999483 
 
 •999763 
 
 •999894 
 
 5 
 
 •931167 
 
 •957979 
 
 •975193 
 
 •985813 
 
 •992127 
 
 •995754 
 
 •997771 
 
 •998860 
 
 •999431 
 
 6 
 
 •873365 
 
 ■916082 
 
 •946153 
 
 •966491 
 
 •979749 
 
 •988095 
 
 •993187 
 
 •996197 
 
 •997929 
 
 7 
 
 •799073 
 
 •857613 
 
 •902151 
 
 •934711 
 
 •957650 
 
 •973260 
 
 •983549 
 
 •990125 
 
 •994213 
 
 8 
 
 •713304 
 
 •785131 
 
 •843601 
 
 •889327 
 
 •923783 
 
 •948867 
 
 ■966547 
 
 •978637 
 
 •986671 
 
 9 
 
 •621892 
 
 •702931 
 
 •772943 
 
 •831051 
 
 •877517 
 
 •913414 
 
 •940261 
 
 •959743 
 
 •973479 
 
 10 
 
 •530387 
 
 •615960 
 
 •693934 
 
 •762183 
 
 •819739 
 
 •866628 
 
 •903610 
 
 •931906 
 
 •952946 
 
 11 
 
 •443263 
 
 ■528919 
 
 •610817 
 
 •686036 
 
 •752594 
 
 ■809485 
 
 •856564 
 
 •894357 
 
 •923839 
 
 IS 
 
 •362642 
 
 •445680 
 
 •527643 
 
 •606303 
 
 •679028 
 
 •743980 
 
 •800136 
 
 •847237 
 
 •885624 
 
 13 
 
 •293326 
 
 •369041 
 
 •447812 
 
 •526524 
 
 •602298 
 
 •672758 
 
 •736186 
 
 •791573 
 
 •838571 
 
 U 
 
 •232993 
 
 •300708 
 
 •373844 
 
 •449711 
 
 •525529 
 
 •598714 
 
 ■607102 
 
 •729091 
 
 •783691 
 
 IS 
 
 •182498 
 
 •241436 
 
 •307354 
 
 •378154 
 
 •451418 
 
 •524638 
 
 •595482 
 
 •661967 
 
 •722598 
 
 16 
 
 •141130 
 
 •191 230 
 
 •249129 
 
 •313374 
 
 •382051 
 
 •452961 
 
 •523834 
 
 •592547 
 
 •657277 
 
 17 
 
 •107876 
 
 •149597 
 
 •199304 
 
 •256178 
 
 •318864 
 
 •385597 
 
 •454366 
 
 •523105 
 
 •589868 
 
 18 
 
 •081581 
 
 •115691 
 
 •157520 
 
 ■206781 
 
 •262666 
 
 •323897 
 
 •388841 
 
 •455653 
 
 •522438 
 
 19 
 
 •061094 
 
 •088529 
 
 •123104 
 
 164949 
 
 •213734 
 
 •268663 
 
 •328532 
 
 •391823 
 
 •456836 
 
 SO 
 
 •045341 
 
 •067086 
 
 ■095210 
 
 •130141 
 
 •171932 
 
 •220220 
 
 •274229 
 
 •332819 
 
 •394578 
 
 SI 
 
 •033371 
 
 •050380 
 
 •072929 
 
 •101632 
 
 ■136830 
 
 •178510 
 
 •226291 
 
 •279413 
 
 •336801 
 
 ss 
 
 •024374 
 
 •037520 
 
 •055362 
 
 •078614 
 
 •107804 
 
 •143191 
 
 •184719 
 
 •231985 
 
 •284256 
 
 S3 
 
 ■017676 
 
 •027726 
 
 •041677 
 
 060270 
 
 O84140 
 
 •113735 
 
 •149251 
 
 •190590 
 
 ■237342 
 
 *k 
 
 •012733 
 
 •020341 
 
 •031130 
 
 ■045822 
 
 065093 
 
 •089504 
 
 •119435 
 
 •155028 
 
 •196152 
 
 25 
 
 •009117 
 
 •014822 
 
 •023084 
 
 •034566 
 
 •049943 
 
 ■069824 
 
 •094710 
 
 •124915 
 
 •160542 
 
 26 
 
 O06490 
 
 ■010734 
 
 •017001 
 
 •025887 
 
 038023 
 
 ■054028 
 
 074461 
 
 ■099758 
 
 ■130189 
 
 27 
 
 •004595 
 
 •007727 
 
 •012441 
 
 •019254 
 
 •028736 
 
 041483 
 
 •058068 
 
 •078995 
 
 •104653 
 
 28 
 
 •003238 
 
 005532 
 
 •009050 
 
 •014228 
 
 •021569 
 
 •031620 
 
 •044938 
 
 ■062055 
 
 083428 
 
 29 
 
 •002270 
 
 ■003940 
 
 006546 
 
 •010450 
 
 •016085 
 
 023936 
 
 •034526 
 
 •048379 
 
 •065985 
 
 SO 
 
 •001585 
 
 ■002792 
 
 004710 
 
 •007632 
 
 011921 
 
 •018002 
 
 ■026345 
 
 037446 
 
 •051798 
 
 40 
 
 •000036 
 
 •000072 
 
 •000138 
 
 ■000255 
 
 000453 
 
 •000778 
 
 •001294 
 
 •002087 
 
 003272 
 
 50 
 
 •000001 
 
 •000001 
 
 ■000003 
 
 •000006 
 
 •000012 
 
 •000023 
 
 ■000042 
 
 •000075 
 
 000131 
 
 60 
 
 ■000000 
 
 •oooooo 
 
 ■OOOOOO 
 
 •oooooo 
 
 ■oooooo 
 
 •000001 
 
 •000001 
 
 •000002 
 
 •000004 
 
 70 
 
 •oooooo 
 
 •oooooo 
 
 •OOOOOO 
 
 •oooooo 
 
 ■oooooo 
 
 ■oooooo 
 
 ■OOOOOO 
 
 •oooooo 
 
 •OOOOOO 
 
 4—2 
 
28 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XII. Test for Goodness of Fit. Values of P. 
 
 x a 
 
 n' = 21 
 
 n' = 22 
 
 n' = 23 
 
 n' = 24 
 
 n'=25 
 
 n' = 26 
 
 n' = 27 
 
 n' = 28 
 
 n' = 29 
 
 n' = 30 
 
 1 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 
 2 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 V 
 
 !• 
 
 3 
 
 ■999996 
 
 •999998 
 
 •999999 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 I- 
 
 
 It 
 
 •999954 
 
 •999980 
 
 ■999992 
 
 •999997 
 
 ■999999 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 1- 
 
 5 
 
 •999722 
 
 •999808 
 
 •999939 
 
 •999972 
 
 •999987 
 
 •999994 
 
 •999998 
 
 •999999 
 
 1- 
 
 
 6 
 
 •998898 
 
 ■999427 
 
 •999708 
 
 •999855 
 
 •999929 
 
 •999966 
 
 •999984 
 
 •999993 
 
 •999997 
 
 •999999 
 
 7 
 
 •996085 
 
 •998142 
 
 •998980 
 
 •999452 
 
 •999711 
 
 •999851 
 
 •999924 
 
 •999962 
 
 •999981 
 
 •999991 
 
 8 
 
 •991868 
 
 •995143 
 
 •997160 
 
 •998371 
 
 •999085 
 
 ■999494 
 
 ■999726 
 
 •999853 
 
 •999924 
 
 •999960 
 
 9 
 
 •982907 
 
 •989214 
 
 •993331 
 
 ■995957 
 
 •997595 
 
 ■998596 
 
 •999194 
 
 •999546 
 
 •999748 
 
 •999863 
 
 10 
 
 •968171 
 
 ■978912 
 
 •986304 
 
 ■991277 
 
 •994547 
 
 ■996653 
 
 •997981 
 
 •998803 
 
 •999302 
 
 ■999599 
 
 11 
 
 •946223 
 
 •962787 
 
 •974749 
 
 •983189 
 
 •989012 
 
 •992946 
 
 •995549 
 
 •997239 
 
 •998315 
 
 •998988 
 
 12 
 
 •91C076 
 
 •939617 
 
 ■957379 
 
 •970470 
 
 •979908 
 
 •986567 
 
 •991173 
 
 •994294 
 
 •996372 
 
 •997728 
 
 13 
 
 •877384 
 
 •908624 
 
 •933161 
 
 •951990 
 
 •966121 
 
 •976501 
 
 •983974 
 
 •989247 
 
 ■992900 
 
 •995384 
 
 U 
 
 •830496 
 
 •869599 
 
 •901479 
 
 •926871 
 
 •940050 
 
 •961732 
 
 •973000 
 
 •981254 
 
 •987189 
 
 •991377 
 
 15 
 
 •776408 
 
 ■822952 
 
 862238 
 
 •894634 
 
 •920759 
 
 •941383 
 
 •957334 
 
 •969432 
 
 •978436 
 
 •985015 
 
 16 
 
 •716624 
 
 •769050 
 
 •815886 
 
 •855268 
 
 •888076 
 
 •914828 
 
 •936203 
 
 •952947 
 
 •965819 
 
 •975536 
 
 17 
 
 •652974 
 
 •711106 
 
 •763362 
 
 •809251 
 
 •848662 
 
 •881793 
 
 ■909083 
 
 •931122 
 
 •948589 
 
 •962181 
 
 18 
 
 •587408 
 
 ■649004 
 
 •705988 
 
 •757489 
 
 •803008 
 
 •842390 
 
 •875773 
 
 •903519 
 
 ■926149 
 
 •944272 
 
 19 
 
 •521826 
 
 •585140 
 
 •645328 
 
 •701224 
 
 •751990 
 
 ■797120 
 
 •836430 
 
 •870001 
 
 •898136 
 
 •921288 
 
 20 
 
 ■457930 
 
 •521261 
 
 •583040 
 
 •641912 
 
 •696776 
 
 •746825 
 
 ■791556 
 
 •830756 
 
 •864464 
 
 •892927 
 
 21 
 
 •397132 
 
 •458944 
 
 520738 
 
 •581087 
 
 •638725 
 
 •692009 
 
 ■741904 
 
 •786288 
 
 •825349 
 
 •859149 
 
 22 
 
 •340511 
 
 •399510 
 
 •459889 
 
 •520252 
 
 ■579267 
 
 •635744 
 
 •688697 
 
 •737377 
 
 •781291 
 
 •820189 
 
 23 
 
 •288795 
 
 •343979 
 
 •401730 
 
 •460771 
 
 •519798 
 
 •577564 
 
 •632947 
 
 •685013 
 
 •733041 
 
 ■776543 
 
 H 
 
 •242392 
 
 •293058 
 
 •347229 
 
 •403808 
 
 •401597 
 
 •519373 
 
 •575965 
 
 ■630316 
 
 •681535 
 
 •728932 
 
 25 
 
 •201431 
 
 •247164 
 
 •297075 
 
 ■350285 
 
 405700 
 
 •462373 
 
 •518975 
 
 •574462 
 
 ■627835 
 
 ■678248 
 
 26 
 
 •165812 
 
 •206449 
 
 •251682 
 
 •300866 
 
 •353105 
 
 •407598 
 
 •463105 
 
 •518600 
 
 •573045 
 
 ■625491 
 
 27 
 
 •135264 
 
 •170853 
 
 •211226 
 
 •255967 
 
 •304453 
 
 •355884 
 
 •409333 
 
 •463794 
 
 •518247 
 
 •571705 
 
 28 
 
 •109399 
 
 •140151 
 
 •175681 
 
 •215781 
 
 •200040 
 
 •307853 
 
 •358458 
 
 •410973 
 
 •464447 
 
 •517913 
 
 29 
 
 •087759 
 
 •114002 
 
 •144861 
 
 •180310 
 
 •220131 
 
 •263916 
 
 •311082 
 
 •360899 
 
 •412528 
 
 •465066 
 
 SO 
 
 •069854 
 
 •091988 
 
 •118464 
 
 •149402 
 
 •184752 
 
 •224289 
 
 •267611 
 
 •314154 
 
 •363218 
 
 •414004 
 
 40 
 
 •004995 
 
 •007437 
 
 ■010812 
 
 •015369 
 
 •021387 
 
 •029164 
 
 •039012 
 
 •051237 
 
 •066128 
 
 •083937 
 
 50 
 
 •000221 
 
 •000365 
 
 •000586 
 
 •000921 
 
 •001416 
 
 •002131 
 
 •003144 
 
 •004551 
 
 •006467 
 
 •009032 
 
 60 
 
 •000007 
 
 •000013 
 
 •000022 
 
 ■000038 
 
 •000064 
 
 •000104 
 
 •000168 
 
 •000264 
 
 •000407 
 
 •000618 
 
 70 
 
 •000000 
 
 •000000 
 
 •000001 
 
 •000001 
 
 •000002 
 
 •000004 
 
 •000007 
 
 •000011 
 
 •000019 
 
 •000030 
 
Tables for Testing Goodness of Fit 
 
 29 
 
 TABLE XITI. Auxiliary Table A. 
 
 X 2 
 
 ioe {x \fi e ~ h "} 
 
 log e ix* 
 
 X* 
 
 log {x N /L"**j 
 
 loge-*** 
 
 1 
 
 1-68479282 
 
 T-78285276 
 
 51 
 
 TT-68121586 
 
 T2-92549071 
 
 2 
 
 1 -61816058 
 
 1-56570552 
 
 52 
 
 11-46828520 
 
 12-70834347 
 
 s 
 
 1-48905897 
 
 1-34855828 
 
 5S 
 
 1 1-25527422 
 
 12-49119623 
 
 4 
 
 1-33438109 
 
 1-13141104 
 
 64 
 
 11-04218593 
 
 1227404899 
 
 r, 
 
 1-16568886 
 
 2-91426380 
 
 55 
 
 1282902315 
 
 1205690175 
 
 6 
 
 2-98813224 
 
 2-69711655 
 
 56 
 
 1261578858 
 
 13-83975450 
 
 7 
 
 2-80445839 
 
 2-47996931 
 
 57 
 
 12-40248475 
 
 13-62260726 
 
 S 
 
 2-61630713 
 
 2-26282207 
 
 58 
 
 12-18911408 
 
 13-40546002 
 
 9 
 
 2-42473615 
 
 2-04567483 
 
 59 
 
 13-97567885 
 
 13-18831278 
 
 10 
 
 2-23046765 
 
 3-82852759 
 
 60 
 
 13-76218123 
 
 14-97116554 
 
 11 
 
 203401675 
 
 3-61138035 
 
 61 
 
 13-54862328 
 
 14-75401830 
 
 12 
 
 3-83576379 
 
 3-39423311 
 
 62 
 
 13-33500696 
 
 14-53687106 
 
 IS 
 
 3-63599760 
 
 3-17708587 
 
 63 
 
 1312133415 
 
 1431972382 
 
 U 
 
 3-43494271 
 
 4-95993863 
 
 64 
 
 14-90760662 
 
 14-10257658 
 
 15 
 
 3-23277708 
 
 4-74279139 
 
 65 
 
 14-69382607 
 
 15-88542934 
 
 1G 
 
 3-02964420 
 
 452564414 
 
 66 
 
 14-48099412 
 
 15-66828209 
 
 n 
 
 4-82566143 
 
 4-30849690 
 
 67 
 
 14-26611232 
 
 15-45113485 
 
 18 
 
 4-62092598 
 
 4-09134966 
 
 68 
 
 1405218213 
 
 15-23398761 
 
 19 
 
 4-41551928 
 
 5-87420242 
 
 69 
 
 15-83820498 
 
 15-01684037 
 
 20 
 
 4-20951024 
 
 5-65705518 
 
 70 
 
 15-62418221 
 
 16-79969313 
 
 21 
 
 4-00295765 
 
 5-43990794 
 
 71 
 
 1541011512 
 
 16-58254589 
 
 22 
 
 5-79591210 
 
 5-22276070 
 
 72 
 
 15-19600496 
 
 1636539865 
 
 23 
 
 5-58841744 
 
 5-00561346 
 
 73 
 
 16-98185290 
 
 16-14825141 
 
 n 
 
 5-38051190 
 
 6-78846622 
 
 74 
 
 16-76766009 
 
 17-93110417 
 
 25 
 
 5-17222904 
 
 6-57131898 
 
 75 
 
 16-55342762 
 
 17-71395693 
 
 26 
 
 6-96359847 
 
 635417173 
 
 76 
 
 1633915654 
 
 17-49680968 
 
 27 
 
 6-75464644 
 
 6-13702449 
 
 77 
 
 16-12484787 
 
 17-27966244 
 
 28 
 
 6-54539633 
 
 7-91987725 
 
 78 
 
 17-91050256 
 
 17-06251520 
 
 29 
 
 6-33586907 
 
 7-70273001 
 
 79 
 
 17-69612157 
 
 18-84536796 
 
 30 
 
 6-12608346 
 
 7-48558277 
 
 80 
 
 17-48170578 
 
 13-62822072 
 
 31 
 
 7-91605644 
 
 7-26843553 
 
 81 
 
 17-26725605 
 
 18-41107348 
 
 S2 
 
 7-70580334 
 
 7-05128829 
 
 82 
 
 17-05277323 
 
 18-19392624 
 
 S3 
 
 7-49533808 
 
 8-83414105 
 
 83 
 
 1883825810 
 
 19-97677900 
 
 S4 
 
 7-28467333 
 
 8-61699381 
 
 84 
 
 18-62371146 
 
 19-75963176 
 
 S5 
 
 7-07382065 
 
 8-39984657 
 
 85 
 
 18-40913404 
 
 19-54248452 
 
 86 
 
 8-86279064 
 
 8-18269932 
 
 86 
 
 18-19452656 
 
 19-32533727 
 
 37 
 
 8-65159301 
 
 996555208 
 
 87 
 
 19-97988972 
 
 19-10819003 
 
 S8 
 
 844023670 
 
 9-74840484 
 
 88 
 
 19-76522419 
 
 20-89104279 
 
 39 
 
 8-22872997 
 
 9-53125760 
 
 89 
 
 1955053062 
 
 20-67389555 
 
 40 
 
 8-01708042 
 
 931411036 
 
 90 
 
 19-33580963 
 
 20-45674831 
 
 U 
 
 9-80529511 
 
 909696312 
 
 91 
 
 19-12106183 
 
 20-23960107 
 
 42 
 
 9-59338058 
 
 10-87981588 
 
 92 
 
 20-90628780 
 
 2002245383 
 
 43 
 
 9-38134293 
 
 10-66266864 
 
 93 
 
 20-69148812 
 
 21-80530659 
 
 u 
 
 916918780 
 
 10-44552140 
 
 94 
 
 20-47666333 
 
 21-58815935 
 
 45 
 
 10-95692047 
 
 10-22837416 
 
 95 
 
 20-26181397 
 
 21-37101211 
 
 46 
 
 10-74454589 
 
 10-01122691 
 
 96 
 
 20-04694054 
 
 21-15386486 
 
 47 
 
 10-53206866 
 
 11-79407967 
 
 97 
 
 21-83204355 
 
 22-93671762 
 
 48 
 
 10-31949311 
 
 11-57693243 
 
 98 
 
 21-61712348 
 
 22-71957038 
 
 49 
 
 10-10682329 
 
 11-35978519 
 
 99 
 
 21-40218080 
 
 22-50242314 
 
 50 
 
 11-89406301 
 
 11-14263795 
 
 100 
 
 21-18721596 
 
 22-28527590 
 
30 
 
 Tables for Statisticians and Biometricians 
 
 TABLES XIV— XVI. Auxiliary Ta 
 TABLE XIV (B). 
 Table of colog [n] :— [n] = n (n - 2) (n - 4") 
 
 odd nos. 
 
 colog [n] 
 
 n 
 even nos. 
 
 colog [n] 
 
 1 
 
 •00000000 
 
 2 
 
 T -69897000 
 
 s 
 
 1-52287875 
 
 4 
 
 1-09691001 
 
 5 
 
 282390874 
 
 6 
 
 2-31875876 
 
 7 
 
 3-97881070 
 
 8 
 
 3-41566878 
 
 9 
 
 302456819 
 
 10 
 
 4-41566878 
 
 11 
 
 5 98317551 
 
 12 
 
 5-33648753 
 
 13 
 
 6-86923215 
 
 14 
 
 619035949 
 
 15 
 
 7-69314089 
 
 16 
 
 898623951 
 
 17 
 
 8-46269197 
 
 18 
 
 9-73096701 
 
 19 
 
 9-18393837 
 
 20 
 
 10-42993701 
 
 SI 
 
 11-86171908 
 
 22 
 
 11-08751433 
 
 23 
 
 12-49999124 
 
 24 
 
 13-70730309 
 
 25 
 
 13-10205123 
 
 26 
 
 14-29232974 
 
 27 
 
 15-67068747 
 
 28 
 
 1684517171 
 
 29 
 
 16-20828947 
 
 80 
 
 17-36805045 
 
 SI 
 
 18-71692778 
 
 32 
 
 1986290048 
 
 33 
 
 19-19841384 
 
 84 
 
 20-33142156 
 
 35 
 
 21-65434579 
 
 36 
 
 22-77511906 
 
 37 
 
 22-08614407 
 
 38 
 
 23-19533546 
 
 39 
 
 24-49507946 
 
 40 
 
 25-59327547 
 
 41 
 
 26-88229561 
 
 4^ 
 
 27-97002618 
 
 43 
 
 27-24882715 
 
 44 
 
 28-32657350 
 
 45 
 
 29-59561464 
 
 46 
 
 3066381567 
 
 47 
 
 31-92351678 
 
 48 
 
 32-98257443 
 
 49 
 
 3223332070 
 
 50 
 
 3328360443 
 
 51 
 
 34-52575052 
 
 52 
 
 35-56760109 
 
 53 
 
 3680147465 
 
 54 
 
 37-83520733 
 
 55 
 
 3706111196 
 
 56 
 
 3808701930 
 
 57 
 
 39-30523711 
 
 58 
 
 4032359131 
 
 59 
 
 4153438510 
 
 60 
 
 42-54544006 
 
 61 
 
 4374905526 
 
 62 
 
 44-75304837 
 
 63 
 
 45-94971471 
 
 64 
 
 4694686839 
 
 65 
 
 4613680135 
 
 66 
 
 47-12732446 
 
 67 
 
 48-31072655 
 
 68 
 
 4929481554 
 
 69 
 
 50-47187746 
 
 70 
 
 51-44971750 
 
 71 
 
 52-62061911 
 
 72 
 
 53-59238501 
 
 73 
 
 54-75729625 
 
 74 
 
 55-72315329 
 
 75 
 
 56-88223499 
 
 76 
 
 57-84233970 
 
 77 
 
 58-99574426 
 
 78 
 
 59-95024509 
 
 79 
 
 59-09811717 
 
 80 
 
 6004715511 
 
 81 
 
 61-18963215 
 
 82 
 
 62-13334125 
 
 83 
 
 63-27055406 
 
 84 
 
 64-20906197 
 
 85 
 
 65-34113514 
 
 86 
 
 88-27466368 
 
 87 
 
 67-40161588 
 
 88 
 
 68-33008084 
 
 89 
 
 69-45222588 
 
 90 
 
 70-37583833 
 
 91 
 
 7l -49318448 
 
 92 
 
 72-41205051 
 
 93 
 
 73-52470154 
 
 94 
 
 74-43892265 
 
 95 
 
 75-54697793 
 
 96 
 
 76-45665142 
 
 97 
 
 77-56020620 
 
 98 
 
 78-46542534 
 
 99 
 
 79-56457100 
 
 100 
 
 80-46542534 
 
 B, C and D. 
 
 TABLE XV (C). 
 
 
 
 
 /2 f 00 
 
 X 2 
 
 W- j e-ix-dx 
 
 
 v * Jx 
 
 1 
 
 •3173106 
 
 2 
 
 •1572992 
 
 3 
 
 ■0832646 
 
 4 
 
 •0455003 
 
 5 
 
 •0253474 
 
 6 
 
 ■0143060 
 
 7 
 
 •0081506 
 
 8 
 
 •0046776 
 
 9 
 
 •0026998 
 
 10 
 
 •0015654 
 
 11 
 
 •0009112 
 
 12 
 
 •0005321 
 
 13 
 
 •0003115 
 
 14 
 
 •0001828 
 
 15 
 
 •0001076 
 
 16 
 
 •0000634 
 
 17 
 
 •0000374 
 
 18 
 
 •0000221 
 
 19 
 
 •0000132 
 
 20 
 
 •0000078 
 
 21 
 
 •0000046 
 
 22 
 
 •0000027 
 
 28 
 
 •0000016 
 
 24 
 
 •0000011 
 
 25 
 
 •0000007 
 
 26 
 
 •0000004 
 
 27 
 
 •0000003 
 
 28 
 
 •0000002 
 
 29 
 
 •0000001 
 
 so 
 
 •0000000 
 
 TABLE XVI (D). 
 
 Function 
 
 Log. Function 
 
 e-i 
 
 1-7828527590 
 1-9019400615 
 
Probability of Association on Correlation-Scale 
 
 31 
 
 TABLE XVII. 
 
 Values of (— log P) corresponding to given values of % 3 in a fourfold table. 
 (Extension of Table XII for n = 4.) 
 
 X 5 
 
 -logP 
 
 X 2 
 
 -logP 
 
 x' 
 
 -logP 
 
 X a 
 
 -logP 
 
 X a 
 
 -logP 
 
 X 1 
 
 -logP 
 
 / 
 
 0096 
 
 26 
 
 5-021 
 
 50 
 
 10097 
 
 1100 
 
 237-439 
 
 2600 
 
 562-973 
 
 13500 
 
 2929521 
 
 fl 
 
 0-242 
 
 27 
 
 5-230 
 
 60 
 
 12-231 
 
 1150 
 
 248-287 
 
 2700 
 
 584-680 
 
 14000 
 
 3038-086 
 
 3 
 
 0407 
 
 28 
 
 5-440 
 
 70 
 
 14-370 
 
 1200 
 
 259-135 
 
 2800 
 
 606-387 
 
 14500 
 
 3146-652 
 
 4 
 
 0583 
 
 29 
 
 5-650 
 
 80 
 
 16513 
 
 1250 
 
 269-983 
 
 2900 
 
 628 094 
 
 15000 
 
 3255-219 
 
 5 
 
 0765 
 
 SO 
 
 5-860 
 
 90 
 
 18-659 
 
 1300 
 
 280-832 
 
 8000 
 
 649-801 
 
 15500 
 
 3363785 
 
 6 
 
 0952 
 
 SI 
 
 6-071 
 
 100 
 
 20-809 
 
 1350 
 
 291-681 
 
 S500 
 
 758-341 
 
 16000 
 
 3472-352 
 
 7 
 
 1143 
 
 32 
 
 6-281 
 
 150 
 
 31579 
 
 1400 
 
 302-531 
 
 4000 
 
 866-886 16500 
 
 3580-919 
 
 X 
 
 1-337 
 
 88 
 
 6-492 
 
 200 
 
 42-375 
 
 1450 
 
 313381 
 
 4500 
 
 975-434 \ 17000 
 
 3689-486 
 
 9 
 
 1-533 
 
 34 
 
 6-703 
 
 250 
 
 53-184 
 
 1500 
 
 324-231 
 
 5000 
 
 1083-995 1 17500 
 
 3798053 
 
 10 
 
 1-731 
 
 35 
 
 6914 
 
 800 
 
 64O02 
 
 1550 
 
 335 081 
 
 5500 
 
 1192-538 
 
 18000 
 
 3906-621 
 
 11 
 
 1-931 
 
 86 
 
 7-126 
 
 850 
 
 74-826 
 
 1600 
 
 345-931 
 
 6000 
 
 1301-092 
 
 18500 
 
 4015-188 
 
 12 
 
 2-132 
 
 37 
 
 7337 
 
 400 
 
 85-655 
 
 1650 
 
 356-782 
 
 6500 
 
 1409-649 
 
 19000 
 
 4123-756 
 
 IS 
 
 2-334 
 
 ,18 
 
 7-549 
 
 450 
 
 96-487 
 
 1700 
 
 367-633 
 
 7000 
 
 1518-206 
 
 19500 
 
 4232-324 
 
 tt 
 
 2-537 
 
 39 
 
 7-761 
 
 500 
 
 107-321 
 
 1750 
 
 378-484 
 
 7500 
 
 1626-765 
 
 20000 
 
 4340-892 
 
 15 
 
 2-741 
 
 40 
 
 7-972 
 
 550 
 
 118-158 
 
 1800 
 
 389335 
 
 8000 
 
 1735-324 
 
 20500 
 
 4449-461 
 
 16 
 
 2-945 
 
 41 
 
 8-184 
 
 600 
 
 128-997 
 
 1850 
 
 400-187 
 
 8500 
 
 1843-885 
 
 21000 
 
 4558-029 
 
 n 
 
 3-151 
 
 >& 
 
 8-397 
 
 650 
 
 139-837 
 
 1900 
 
 411038 
 
 9000 
 
 1952-446 
 
 21500 
 
 4666-597 
 
 18 
 
 3-357 
 
 4-1 
 
 8-609 
 
 700 
 
 150678 
 
 1950 
 
 421-890 
 
 9500 
 
 2061-008 
 
 22000 
 
 4775-166 
 
 19 
 
 3-564 
 
 4h 
 
 8-821 
 
 750 
 
 161-520 
 
 2000 
 
 432-742 
 
 10000 
 
 2169-570 
 
 22500 
 
 4883735 
 
 no 
 
 3-770 
 
 45 
 
 9034 
 
 800 
 
 172-364 
 
 2050 
 
 443-594 
 
 10500 
 
 2278-133 
 
 23000 
 
 4992-304 
 
 9,1 
 
 3-978 
 
 46 
 
 9-246 
 
 850 
 
 183-208 
 
 2100 
 
 454-446 
 
 11000 
 
 2386-697 
 
 23500 
 
 5100-873 
 
 22 
 
 4-186 
 
 If 
 
 9-459 
 
 900 
 
 194 053 
 
 2200 
 
 476151 
 
 11500 
 
 2495-261 
 
 24000 
 
 5209-442 
 
 2.1 
 
 4-394 
 
 48 
 
 9672 
 
 950 
 
 204-899 
 
 2800 
 
 497-856 
 
 12000 
 
 2603-825 
 
 24500 
 
 5318011 
 
 21, 
 
 4-602 
 
 49 
 
 9-885 
 
 1000 
 
 215-745 
 
 2400 
 
 519561 
 
 12500 
 
 2712-390 
 
 25000 
 
 5426-580 
 
 25 
 
 4811 
 
 50 
 
 10097 
 
 1050 
 
 226-592 
 
 2500 
 
 541267 
 
 13000 
 
 2820955 
 
 
 
 26 
 
 5-021 
 
 
 
 1100 
 
 237-439 
 
 $600 
 
 562973 
 
 18500 
 
 2929-521 
 
 
 
 TABLE XVIII. 
 
 Values of (— log P), entering with r and „<t t . 
 Values of „a r . 
 
 
 
 ■01 
 
 ■OS 
 
 ■OS 
 
 ■04 
 
 ■05 
 
 ■06 
 
 •07 
 
 ■08 
 
 
 0-05 
 
 6-248 
 
 1-907 
 
 1020 
 
 0675 
 
 0-498 
 
 0-392 
 
 0-322 
 
 0-273 
 
 
 0075 
 
 13-228 
 
 3-760 
 
 1-908 
 
 1-217 
 
 0-874 
 
 0674 
 
 0545 
 
 0-456 
 
 4- 
 
 0-1 
 
 22-924 
 
 6-267 
 
 3-076 
 
 1-910 
 
 1-343 
 
 1-019 
 
 0-814 
 
 0675 
 
 O 
 
 0-15 
 
 50687 
 
 13-329 
 
 6-298 
 
 3-784 
 
 2-586 
 
 1-916 
 
 1-498 
 
 1-218 
 
 
 0-2 
 
 90035 
 
 23254 
 
 10-771 
 
 6-343 
 
 4-259 
 
 3-100 
 
 2-384 
 
 1-924 
 
 c 
 
 OS 
 
 206-348 
 
 52-453 
 
 23-836 
 
 13-758 
 
 9057 
 
 6-478 
 
 4-906 
 
 3903 
 
 
 0-4 
 
 380266 
 
 96013 
 
 43-254 
 
 24-726 
 
 16112 
 
 11-407 
 
 8-552 
 
 6686 
 
 !> 
 
 0-5 
 
 626428 
 
 157607 
 
 70-669 
 
 40177 
 
 26025 
 
 18-312 
 
 13-642 
 
 10-597 
 
 
 06 
 
 970-879 
 
 243-753 
 
 108-980 
 
 61-747 
 
 39-845 
 
 27-922 
 
 20-713 
 
 16-020 
 
 
 0-7 
 
 1463-946 
 
 367033 
 
 163-781 
 
 92-579 
 
 59-584 
 
 41-634 
 
 30-792 
 
 23-740 
 
 
 08 
 
 2220-267 
 
 556100 
 
 247-801 
 
 139-832 
 
 89-819 
 
 62-625 
 
 46-209 
 
 35-539 
 
 
 0-9 
 
 3607-924 
 
 902-949 
 
 401-907 
 
 226-479 
 
 145-241 
 
 101085 
 
 74-442 
 
 57-134 
 
 
 095 
 
 5056547 
 
 1265 013 
 
 562-757 
 
 316904 
 
 203069 
 
 141-207 
 
 103-886 
 
 79-671 
 
32 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XIX. 
 
 Values of %' corresponding to the values of (— log P) in Table X VIII. 
 
 Values of „oy. 
 
 
 
 ■01 
 
 ■02 
 
 ■03 
 
 ■04 
 
 ■05 
 
 ■06 
 
 ■07 
 
 ■08 
 
 
 0-05 
 
 31-84 
 
 10-88 
 
 6-36 
 
 451 
 
 3-52 
 
 2-91 
 
 2-48 
 
 219 
 
 
 0-075 
 
 64-66 
 
 1995 
 
 10-89 
 
 7-38 
 
 5-58 
 
 4-51 
 
 3-78 
 
 3-28 
 
 *. 
 
 o-i 
 
 109-82 
 
 31-93 
 
 1664 
 
 10-90 
 
 8 03 
 
 6-35 
 
 5-26 
 
 4-51 
 
 <** 
 
 0-15 
 
 238-45 
 
 65-13 
 
 3208 
 
 20-07 
 
 14-24 
 
 10-93 
 
 8-82 
 
 7-39 
 
 
 0-2 
 
 422-29 
 
 111-35 
 
 53-16 
 
 32-29 
 
 22 35 
 
 16-75 
 
 13-25 
 
 1097 
 
 a) 
 
 0-8 
 
 95668 
 
 246-62 
 
 114-05 
 
 67-14 
 
 4511 
 
 32-93 
 
 25-45 
 
 20-64 
 
 £ 
 
 o-4 
 
 1758-21 
 
 447-81 
 
 204-07 
 
 11818 
 
 78-13 
 
 5614 
 
 42-73 
 
 33-92 
 
 0-5 
 
 2892-33 
 
 731-95 
 
 33080 
 
 189-82 
 
 124-22 
 
 88-38 
 
 66-60 
 
 52-34 
 
 
 0-6 
 
 447902 
 
 1129-10 
 
 507-65 
 
 289-58 
 
 188-28 
 
 13302 
 
 99-55 
 
 77-70 
 
 
 0-7 
 
 6750-09 
 
 1697-24 
 
 760-43 
 
 431-96 
 
 279-58 
 
 19657 
 
 146-35 
 
 113-61 
 
 
 08 
 
 10233-49 
 
 2568-34 
 
 1147-76 
 
 649-98 
 
 41922 
 
 29364 
 
 217-74 
 
 lfi.S-34 
 
 
 0-9 
 
 16624-37 
 
 4166-12 
 
 1857-93 
 
 1049-48 
 
 674-92 
 
 471-22 
 
 348-23 
 
 26826 
 
 
 0-95 
 
 23295-86 
 
 583382 
 
 2599-00 
 
 1466-24 
 
 941-56 
 
 656-32 
 
 484-15 
 
 372-37 
 
 4 
 
 > 
 
 TABLE XX. 
 
 Vahies of log %* corresponding to values of r and „<r r in Tables 
 XVII and XV III. 
 
 Values of O oy. 
 
 
 ■01 
 
 ■02 
 
 ■03 
 
 ■04 
 
 ■05 
 
 •06 
 
 ■07 
 
 ■08 
 
 0-05 
 
 1-5030 
 
 10366 
 
 08035 
 
 0-6542 
 
 0-5465 
 
 0-4639 
 
 0-3945 
 
 0-3404 
 
 0-075 
 
 1-8106 
 
 1-2999 
 
 1-0370 
 
 0-8681 
 
 0-7466 
 
 06542 
 
 0-5775 
 
 0-5159 
 
 0-1 
 
 2-0407 
 
 1-5042 
 
 1-2212 
 
 1-0374 
 
 0-9047 
 
 0-8028 
 
 0-7210 
 
 0-6522 
 
 015 
 
 2-3774 
 
 1-8138 
 
 1-5062 
 
 1-3025 
 
 1-1535 
 
 1-0386 
 
 0-9455 
 
 0-8686 
 
 0-2 
 
 2-6256 
 
 2-0467 
 
 1-7256 
 
 1-5091 
 
 1-3493 
 
 1-2240 
 
 1-1222 
 
 1-0400 
 
 OS 
 
 2-9808 
 
 2-3920 
 
 2 0571 
 
 1-8270 
 
 1-6543 
 
 1-5176 
 
 1-4057 
 
 1-3148 
 
 0-4 
 
 3-2451 
 
 2-6511 
 
 2-3098 
 
 2-0725 
 
 1-8928 
 
 1-7493 
 
 1-6307 
 
 1-5305 
 
 0-5 
 
 3-4612 
 
 2-8645 
 
 2-5196 
 
 2-2783 
 
 2 0942 
 
 1-9464 
 
 1-8235 
 
 1-7188 
 
 0-6 
 
 3-6512 
 
 30527 
 
 2-7056 
 
 2-4618 
 
 2-2748 
 
 21239 
 
 1-9980 
 
 1-8904 
 
 0-7 
 
 3-8293 
 
 3-2297 
 
 2-8811 
 
 2-6354 
 
 2-4465 
 
 2-2935 
 
 2-1654 
 
 2-0554 
 
 0-8 
 
 4-0100 
 
 3-4097 
 
 30598 
 
 2-8129 
 
 2 6224 
 
 2-4678 
 
 2-3379 
 
 2-2262 
 
 0-9 
 
 4-2207 
 
 3-6197 
 
 32690 
 
 30210 
 
 2-8293 
 
 2-6732 
 
 2-5419 
 
 2-4286 
 
 0-95 
 
 4-3673 
 
 3-7660 
 
 34148 
 
 3-1662 
 
 2-9738 
 
 2-8171 
 
 2-6850 
 
 2-5710 
 
Abacs for Determining the Equiprobable Tetrachoric r. 33 
 
 XXI. Abac to determine cr r . 
 
 •60-70 -80.84 -88 '90 -92-84 95 -96 97 «80 985 990 -992 994-995 996 997 998« 
 
 Scale of 4(1 + 00 and \(\ + a$. 
 B. 5 
 
 •999 
 
34 
 
 Tables for Statisticiam and Biometricians 
 
 XXII. Abac to determine r r 
 
 «\| I 1 1 1 1 1 1 1 I 1 1 1 1 I 1 1 1 1 1 1 I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 1 1 1 1 1 1 I 
 
 
 III ' 
 
 M iiiib 
 
 *° iiiiiis 
 
 ^iiiiii^^ 
 
 !8 fililli^ll^ 
 
 7 «iiiili^§^>k 
 
 3C , 8|||li|lil§^^4v 
 
 5 Rliliiiiii^^^^s^^ 
 
 14 Sll^lilili^^^^^^b^ 
 
 1 it tl ll|ili iilllillpiife ^^^^4- 
 
 '' ^llll|l|ll^ft^^P^^^^^^^5t-L 
 
 ^Illllllll^^^^S^^^^^^^^S^S^?^:--. 
 
 ° wllll|llll^P^^^fei^^^^^§^^^^S^p; ;: gr--~. 
 
 11 ^l^l^lllll^p^^pR^^fe^^^l^p^^s^^^^jS?^-^^ 
 
 ^l^lmill^ll^l^ft^^^^^l^iS^l^^^^p^^sSsS^i^^----.^ 
 
 7 ^^^^ll^lll^^^^l^^^^^^^^^§§^^^^^^^§§§^-35S=S^^=;~:r~--^L 
 
 6 ^^^^^^^^^^^^^^^^^^^P^^^^fc^^^K^^fi^^-S**-^^-^^*^^^^^^^ 
 
 ^^^^^$^^^l^^ll|l||^^^lll§illllllSSllll§S*--li3ll = §l3 = S5 = 5|55 = iS 
 
 o 3 ^^^^^^^^^^^Illl^lll^l^llllllSlllllllllllllllll^gslalsissSlS 
 
 oo ^^^S^^^x^^^^^^^ll^llllllllllisJ^lllllli^llllsllllllllsl'lils^ 
 
 1 \\\^^^$\^^>^p|^$$5$$5^§llllll^l|g^|§l||||§§||lsgslss|l§sg£|ls=l5 
 
 ^^^^^VX\\^;n^$5$^5$3$333$3^^ 
 
 \\\\^s\^^^^^^55$^^$55i|:3~ S5355§§*=|| l^5|^Sg| = §|l§|l5|l si lls|l|ll 
 
 \\V\\\ s \^^v^:^^^^$^:?::5£ 5 5=^533333 3355££3-5i= 5 UsS 3 5S3g £5555 5|S££ 
 
 7 \ \\^ V v^ N \ x ^^^^^^^^5^5$ --35 5 = = = 333333 = 333^3 s£^ 
 
 6 ^ \ \V^ N 0^\^^^3^3^5^i5^;;-:;- = === = = = =3?3=.== = 3-333=3 
 
 \ \\X s On n n n ^v v ^v^^^>s55 = ::~H 5 £ = 5=5 = = | = 55 = = ^ 
 
 \ N Os x ^C n ^ v n^T5>>3s^;: 3;>-3 ^-^^i^S^ ========== =============3=3==== 
 
 \ V ^^^^^T^^-;^^^-:;-^^-^^ 
 
 1 I ^ s v ^^^^^^^^C;^53-33-33-i;;-=£-="£--r^ 
 
 v v ^^^^^^ ^^^^^^^^^-5" ^-3^ 33-~33~~^^~:r~-r;~;~~==~~ = = = = = = = = = = = 
 
 >v ^ ^^ ^1 ^^^^^^^^~~;^--~^--;~~~;~ = ~;~~-:~~-^~~-=~--== :::= == ::: 
 
 v -^ ^-^^ c: ^-^^"~-^^^---^"~--->.;~---^^3~-r~;~:;~~~rrr~ = -=z:r ::::: -- 
 
 08 "*""*» ""*"■■"« ^"-- "**--~."*~---' ~^---- — ----- — --2E-- 
 
 07 "**"">» """- — — ^ * ~ ™* -■■;----»■! 
 
 or ~~~~^-- *"" --I~ ---~~-----~-~""" 
 
 ■- — _ ------ - — ■ 
 
 
 
 
 
 XT J_ XII 
 
 & 
 
 •95 . 
 k 
 
 • 90 J 
 
 •85 I 
 
 ■80 o 
 
 •75 •£. 
 
 "70 2 
 
 -65-= 
 
 •60 
 
 ■55 
 
 •50 
 
 ■45 
 
 •40 
 
 ■35 
 •30 
 -25 
 -20 
 
 10 
 
 05 
 
 01 
 
 02 
 
 03 
 
 •04 05 
 
 Value of „<r r . 
 
 ■06 
 
 •07 
 
 08 
 
Approximate Values of Probable Error of r 
 
 35 
 
 Approximate values of Probable Error of r from a four-fold Correlation 
 
 Table (to be used with xi of Table V). 
 
 TABLE XXIII. Valves of % r for Values of r. 
 
 r 
 
 Xr 
 
 r 
 
 Xr 
 
 r 
 
 Xr 
 
 r 
 
 Xr 
 
 r 
 
 Xr 
 
 ■00 
 
 1-0000 
 
 •go 
 
 •9717 
 
 •40 
 
 ■8845 
 
 ■GO 
 
 •7298 
 
 ■80 
 
 •4843 
 
 ■01 
 
 '9999 
 
 ■21 
 
 -9688 
 
 ■41 
 
 •8785 
 
 •61 
 
 •7200 
 
 ■81 
 
 •4687 
 
 ■OS 
 
 •9997 
 
 ■22 
 
 •9657 
 
 ■1,2 
 
 •8723 
 
 ■62 
 
 •7099 
 
 ■82 
 
 ■4528 
 
 ■OS 
 
 •9994 
 
 ■2S 
 
 •9625 
 
 •1,3 
 
 •8659 
 
 ■63 
 
 •6997 
 
 •83 
 
 •4362 
 
 ■oh 
 
 •9989 
 
 ■24 
 
 •9591 
 
 ■u 
 
 •8594 
 
 ■64 
 
 ■6892 
 
 ■84 
 
 •4192 
 
 ■05 
 
 •998-2 
 
 ■25 
 
 •9556 
 
 ■45 
 
 •8527 
 
 ■65 
 
 •6785 
 
 ■85 
 
 •4018 
 
 ■06 
 
 •9975 
 
 ■26 
 
 •9520 
 
 •46 
 
 •8458 
 
 ■66 
 
 •6675 
 
 ■86 
 
 •8838 
 
 •07 
 
 •9966 
 
 ■27 
 
 •9482 
 
 ■47 
 
 ■8388 
 
 ■67 
 
 •6563 
 
 ■87 
 
 •3652 
 
 ■08 
 
 •9955 
 
 ■28 
 
 •9442 
 
 ■48 
 
 •8315 
 
 ■68 
 
 •6448 
 
 ■88 
 
 •3461 
 
 ■09 
 
 ■9943 
 
 ■29 
 
 •9401 
 
 ■49 
 
 •8241 
 
 ■69 
 
 •6331 
 
 ■89 
 
 •3262 
 
 ■10 
 
 •9930 
 
 ■so 
 
 ■9358 
 
 ■50 
 
 •8165 
 
 ■70 
 
 •6211 
 
 ■90 
 
 ■3057 
 
 ■11 
 
 •9915 
 
 ■si 
 
 •9314 
 
 ■51 
 
 •8087 
 
 ■71 
 
 •6088 
 
 ■91 
 
 •2843 
 
 ■12 
 
 •9899 
 
 •S2 
 
 •9268 
 
 ■52 
 
 •8007 
 
 ■72 
 
 •5962 
 
 ■92 
 
 ■2620 
 
 ■is 
 
 ■9881 
 
 ■S3 
 
 •9221 
 
 ■53 
 
 •7926 
 
 ■73 
 
 •5834 
 
 ■93 
 
 •2387 
 
 •n 
 
 •9862 
 
 Sir 
 
 •9172 
 
 ■54 
 
 •7842 
 
 ■74 
 
 •5702 
 
 ■94 
 
 •2142 
 
 ■15 
 
 •9841 
 
 ■35 
 
 •9122 
 
 ■55 
 
 •7756 
 
 ■75 
 
 •5568 
 
 ■95 
 
 •1882 
 
 ■1G 
 
 ■9819 
 
 ■36 
 
 •9070 
 
 ■56 
 
 •7669 
 
 ■76 
 
 •5430 
 
 ■9G 
 
 •1605 
 
 •17 
 
 •9796 
 
 '37 
 
 •9016 
 
 ■57 
 
 •7579 
 
 ■77 
 
 ■5288 
 
 ■97 
 
 •1305 
 
 ■18 
 
 •9771 
 
 •38 
 
 •8961 
 
 ■58 
 
 •7488 
 
 ■78 
 
 •5144 
 
 ■98 
 
 •0972 
 
 ■19 
 
 •9745 
 
 ■39 
 
 •8904 
 
 ■59 
 
 •7394 
 
 ■79 
 
 •4995 
 
 ■99 
 
 •0585 
 
 
 
 
 
 
 
 
 
 1-00 
 
 •0000 
 
 TABLE XXIV. 
 
 Values of x* for Values of \ (1 + a). 
 
 *(! + •) 
 
 Xa 
 
 i(l + a) 
 
 Xa 
 
 4(1 + ") 
 
 Xa. 
 
 *(!+•) 
 
 X« 
 
 •50 
 
 1-2533 
 
 ■65 
 
 1-2877 
 
 ■80 
 
 1-4288 
 
 ■95 
 
 2-1132 
 
 ■51 
 
 1-2535 
 
 ■66 
 
 1-2928 
 
 ■81 
 
 1-4457 
 
 ■96 
 
 2-2740 
 
 •52 
 
 1-2539 
 
 •67 
 
 1-2984 
 
 •82 
 
 1-4641 
 
 ■97 
 
 2-5071 
 
 ■53 
 
 1-2546 
 
 •68 
 
 1-3044 
 
 ■83 
 
 1-4844 
 
 ■98 
 
 2-8915 
 
 ■54 
 
 1-2556 
 
 ■69 
 
 1-3109 
 
 ■84 
 
 1-5067 
 
 •985 
 
 3-2097 
 
 ■55 
 
 1-2569 
 
 •70 
 
 1-3180 
 
 ■85 
 
 1-5315 
 
 ■990 
 
 3-7333 
 
 ■56 
 
 1-2585 
 
 ■71 
 
 1-3256 
 
 ■86 
 
 1-5590 
 
 ■991 
 
 3-8854 
 
 ■57 
 
 1 -2604 
 
 '72 
 
 1-3338 
 
 ■S7 
 
 1-5897 
 
 •992 
 
 4-0639 
 
 •58 
 
 1-2626 
 
 ■73 
 
 1-3427 
 
 ■88 
 
 1-6245 
 
 ■993 
 
 4-2784 
 
 ■59 
 
 1-2052 
 
 ■74 
 
 1-3523 
 
 •89 
 
 1-6640 
 
 •994 
 
 4-5419 
 
 ■60 
 
 1-2680 
 
 •75 
 
 1-3626 
 
 ■90 
 
 1-7094 
 
 •995 
 
 4-8779 
 
 ■Gl 
 
 1-2712 
 
 •76 
 
 1-3738 
 
 •91 
 
 1-7623 
 
 ■996 
 
 5-3278 
 
 ■62 
 
 1 -2748 
 
 ■77 
 
 1-3859 
 
 ■92 
 
 1-8249 
 
 ■997 
 
 5-9776 
 
 •63 
 
 1-2787 
 
 ■78 
 
 1-3990 
 
 •93 
 
 1-9003 
 
 •998 
 
 7-0465 
 
 ■64 
 
 1-2830 
 
 ■79 
 
 1-4133 
 
 ■94 
 
 1-9937 
 
 •999 
 
 9-3870 
 
 6—2 
 
36 
 
 Tables for Statisticians and Bio metricians 
 
 TABLE XXV. 
 
 Values of y = 
 
 n — 2 to — 4 
 to — 3 ' n — 5 
 
 l- 
 
 4 2.,, ^ 
 
 - . — it n be even , M 
 
 3 v v I [« 
 
 '„ . -x if n be odd 
 
 (1 + z-) *.dz. 
 
 Or, the probability that the mean of a sample of n, draiun at random from a 
 normal population, will not exceed (in algebraic sense) the mean of the popu- 
 lation by more than z times the standard deviation of the sample. 
 
 z 
 
 71 = 4 
 
 n=5 
 
 n=6 
 
 n=7 
 
 n=8 
 
 n=9 
 
 n = 10 
 
 ■1 
 
 •5633 
 
 •5745 
 
 •5841 
 
 5928 
 
 •6006 
 
 •60787 
 
 •61462 
 
 •2 
 
 •6241 
 
 •6458 
 
 •6634 
 
 6798 
 
 •6936 
 
 •70705 
 
 •71846 
 
 ■3 
 
 •0804 
 
 •7096 
 
 •7340 
 
 7549 
 
 •7733 
 
 •78961 
 
 •80423 
 
 ■4 
 
 •730!) 
 
 •7657 
 
 •7939 
 
 8175 
 
 •8376 
 
 •85465 
 
 •86970 
 
 ■5 
 
 •7749 
 
 •8131 
 
 •8428 
 
 8C67 
 
 •8863 
 
 •90251 
 
 •91609 
 
 ■6 
 
 •8125 
 
 •8518 
 
 •8813 
 
 9040 
 
 •9218 
 
 •93600 
 
 •94732 
 
 ■7 
 
 •8440 
 
 •8830 
 
 •9109 
 
 9314 
 
 •9468 
 
 •95851 
 
 •96747 
 
 ■8 
 
 •8701 
 
 •9076 
 
 ■9332 
 
 9512 
 
 •9640 
 
 •97328 
 
 •98007 
 
 ■9 
 
 •8915 
 
 •9269 
 
 •9498 
 
 9652 
 
 •9756 
 
 •98279 
 
 ■98780 
 
 1-0 
 
 ■9092 
 
 •9419 
 
 •9622 
 
 9751 
 
 •9834 
 
 ■98890 
 
 •99252 
 
 VI 
 
 •9236 
 
 •9537 
 
 •9714 
 
 9821 
 
 •9887 
 
 •99280 
 
 •99539 
 
 V2 
 
 •9354 
 
 •9628 
 
 •9782 
 
 9870 
 
 •9922 
 
 •99528 
 
 ■99713 
 
 1-3 
 
 •9451 
 
 •9700 
 
 •9832 
 
 9905 
 
 •9946 
 
 •99688 
 
 •99819 
 
 1-4 
 
 •9531 
 
 •9756 
 
 •9870 
 
 9930 
 
 •9962 
 
 •99791 
 
 •99885 
 
 IS 
 
 •9598 
 
 •9800 
 
 •9899 
 
 9948 
 
 •9973 
 
 •99859 
 
 •99926 
 
 1-6 
 
 ■9653 
 
 ■9836 
 
 •9920 
 
 9961 
 
 •9981 
 
 •99903 
 
 •99951 
 
 1-7 
 
 •9699 
 
 •9864 
 
 •9937 
 
 9970 
 
 •9986 
 
 •99933 
 
 •99968 
 
 1-8 
 
 •9737 
 
 •9886 
 
 •9950 
 
 9977 
 
 •9990 
 
 •99953 
 
 •99978 
 
 1-9 
 
 •9770 
 
 •9904 
 
 •9959 
 
 9983 
 
 ■9992 
 
 •99967 
 
 •99985 
 
 2-0 
 
 •9797 
 
 •9919 
 
 •9967 
 
 9986 
 
 •9994 
 
 •99976 
 
 •99990 
 
 2-1 
 
 •9821 
 
 •9931 
 
 •9973 
 
 9989 
 
 •9996 
 
 •99983 
 
 •99993 
 
 2-2 
 
 •9841 
 
 •9941 
 
 •9978 
 
 9992 
 
 •9997 
 
 •99987 
 
 •99995 
 
 2-3 
 
 •9858 
 
 •9950 
 
 •9982 
 
 9993 
 
 •9998 
 
 •99991 
 
 •99996 
 
 $-4 
 
 •9873 
 
 •9957 
 
 •9985 
 
 9995 
 
 •9998 
 
 •99993 
 
 •99997 
 
 2-5 
 
 •9886 
 
 •9963 
 
 •9987 
 
 9996 
 
 •9998 
 
 •99995 
 
 •99998 
 
 2-6 
 
 •9898 
 
 •9967 
 
 •9989 
 
 9996 
 
 •9999 
 
 •99996 
 
 •99999 
 
 2-7 
 
 •9908 
 
 •9972 
 
 •9991 
 
 9997 
 
 •9999 
 
 •99997 
 
 •99999 
 
 2-8 
 
 •9916 
 
 •9975 
 
 •9992 
 
 9998 
 
 •9999 
 
 •99998 
 
 •99999 
 
 2-9 
 
 •9924 
 
 •9978 
 
 •9993 
 
 9998 
 
 •9999 
 
 •99998 
 
 •99999 
 
 3-0 
 
 •9931 
 
 •9981 
 
 •9994 
 
 9998 
 
 — 
 
 •99999 
 
 — 
 
Table for Type III Curves 
 
 37 
 
 TABLE XXVI. 
 
 Table for use in plotting Type III Curves, i.e. 
 
 -P- (-. , x\p 
 y = y e •(! + -) . 
 
 X 
 
 logio(l + .X)-Xlog 1 oe 
 
 X 
 
 log 10 (l + X)-Xlog, e 
 
 -•95 
 
 •888 450 
 
 + -80 
 
 092 163 
 
 -•90 
 
 •609 135 
 
 + -85 
 
 •101 979 
 
 -•85 
 
 •454 759 
 
 + -90 
 
 •112 111 
 
 -•80 
 
 •351 534 
 
 + -95 
 
 •122 545 
 
 -•75 
 
 •276 339 
 
 + 1-00 
 
 •133 265 
 
 -•70 
 
 •218 873 
 
 + 1-05 
 
 •144 255 
 
 -■65 
 
 •173 641 
 
 + 1-10 
 
 •155 505 
 
 -•60 
 
 •137 363 
 
 + 1-15 
 
 •167 000 
 
 -•55 
 
 ■107 926 
 
 + 1-20 
 
 •178 731 
 
 -•50 
 
 •083 883 
 
 + 1-25 
 
 •190 685 
 
 -•45 
 
 •064 205 
 
 + 1-30 
 
 •202 855 
 
 -■40 
 
 •048 131 
 
 + 1-35 
 
 •215 230 
 
 -•35 
 
 035 084 
 
 + 1-40 
 
 •227 801 
 
 -•30 
 
 •024 614 
 
 + 1-45 
 
 ■240 561 
 
 -•25 
 
 016 365 
 
 + 1-50 
 
 •253 502 
 
 -■20 
 
 010 051 
 
 + 1-55 
 
 ■266 616 
 
 -•15 
 
 •005 437 
 
 + 1-60 
 
 •279 898 
 
 -•10 
 
 •002 328 
 
 + 1-65 
 
 ■293 340 
 
 -•05 
 
 •000 562 
 
 + 1-70 
 
 •306 937 
 
 •00 
 
 •000 000 
 
 + 1-75 
 
 •320 683 
 
 + •05 
 
 •000 525 
 
 + 1-80 
 
 •334 572 
 
 + •10 
 
 •002 037 
 
 + 1-85 
 
 •348 600 
 
 + 15 
 
 •004 446 
 
 + 1-90 
 
 •362 761 
 
 + •20 
 
 •007 678 
 
 + 1-95 
 
 •377 052 
 
 + •25 
 
 •Oil 664 
 
 +200 
 
 •391 468 
 
 + •30 
 
 •016 345 
 
 + 2-05 
 
 ■406 004 
 
 + •35 
 
 •021 669 
 
 + 2-10 
 
 •420 657 
 
 + •40 
 
 •027 590 
 
 + 2-15 
 
 •435 422 
 
 + -15 
 
 •034 065 
 
 + 2'20 
 
 •450 298 
 
 + •50 
 
 •041 056 
 
 +2-25 
 
 •465 279 
 
 + •55 
 
 •048 530 
 
 +2-30 
 
 •480 363 
 
 + •60 
 
 •056 457 
 
 + 2-35 
 
 •495 547 
 
 + •05 
 
 ■064 808 
 
 + 2-40 
 
 •510 828 
 
 + •70 
 
 •073 557 
 
 + 2-45 
 
 •526 202 
 
 + •75 
 
 •082 683 
 
 + 2-50 
 
 •541 668 
 
38 
 
 Tables for Statisticians and Biometriciaus 
 
 TABLE XXV II. 
 
 Powers of Natural Numbers. 
 
 n 
 
 n* 
 
 n 3 
 
 n* 
 
 ?t 5 
 
 to 6 
 
 a 
 
 n 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 i 
 
 1 
 
 2 
 
 4 
 
 8 
 
 16 
 
 32 
 
 64 
 
 128 
 
 2 
 
 8 
 
 9 
 
 27 
 
 81 
 
 243 
 
 729 
 
 2187 
 
 3 
 
 4 
 
 16 
 
 64 
 
 256 
 
 1024 
 
 4096 
 
 16384 
 
 4 
 
 5 
 
 25 
 
 125 
 
 625 
 
 3125 
 
 15625 
 
 78125 
 
 5 
 
 G 
 
 3G 
 
 216 
 
 1296 
 
 7770 
 
 46656 
 
 279936 
 
 6 
 
 7 
 
 49 
 
 343 
 
 2401 
 
 16807 
 
 117649 
 
 823543 
 
 7 
 
 8 
 
 84 
 
 512 
 
 4096 
 
 32768 
 
 262144 
 
 2097152 
 
 8 
 
 9 
 
 81 
 
 729 
 
 6561 
 
 59049 
 
 531441 
 
 4782969 
 
 9 
 
 10 
 
 100 
 
 1000 
 
 10000 
 
 100000 
 
 1000000 
 
 10000000 
 
 10 
 
 11 
 
 121 
 
 1331 
 
 14641 
 
 161051 
 
 1771561 
 
 19487171 
 
 11 
 
 U 
 
 144 
 
 1728 
 
 20736 
 
 248832 
 
 2985984 
 
 35831808 
 
 12 
 
 IS 
 
 1G9 
 
 2197 
 
 28561 
 
 371293 
 
 4826809 
 
 62748517 
 
 IS 
 
 14 
 
 196 
 
 2744 
 
 38416 
 
 537824 
 
 7529536 
 
 105413504 
 
 H 
 
 15 
 
 225 
 
 3375 
 
 50025 
 
 759375 
 
 11390625 
 
 170859375 
 
 10 
 
 1G 
 
 250 
 
 4090 
 
 65536 
 
 1018576 
 
 16777216 
 
 268435456 
 
 10 
 
 77 
 
 289 
 
 4913 
 
 83521 
 
 1419857 
 
 24137569 
 
 410338673 
 
 17 
 
 18 
 
 324 
 
 5832 
 
 104976 
 
 1889568 
 
 34012221 
 
 612220032 
 
 18 
 
 19 
 
 361 
 
 6859 
 
 130321 
 
 2476099 
 
 47045881 
 
 893871739 
 
 19 
 
 20 
 
 400 
 
 8000 
 
 160000 
 
 3200000 
 
 64000000 
 
 1280000000 
 
 20 
 
 21 
 
 441 
 
 9261 
 
 194481 
 
 4084101 
 
 85766121 
 
 1801088541 
 
 21 
 
 ..'.' 
 
 484 
 
 10648 
 
 234256 
 
 5153632 
 
 113379904 
 
 2494357888 
 
 22 
 
 28 
 
 529 
 
 12167 
 
 279841 
 
 6436343 
 
 148035889 
 
 3404825447 
 
 23 
 
 24 
 
 576 
 
 13824 
 
 331776 
 
 7962624 
 
 191102976 
 
 4586471424 
 
 24 
 
 25 
 
 625 
 
 15625 
 
 390625 
 
 9765625 
 
 244140025 
 
 6103515625 
 
 25 
 
 2G 
 
 676 
 
 17576 
 
 456976 
 
 11881376 
 
 308915776 
 
 8031810176 
 
 26 
 
 27 
 
 729 
 
 19683 
 
 531441 
 
 14318907 
 
 387420489 
 
 10460353203 
 
 27 
 
 28 
 
 784 
 
 21952 
 
 614656 
 
 17210368 
 
 481890304 
 
 13492928512 
 
 28 
 
 29 
 
 841 
 
 24389 
 
 707281 
 
 20511149 
 
 694823321 
 
 17249876309 
 
 29 
 
 SO 
 
 900 
 
 27000 
 
 810000 
 
 24300000 
 
 729000000 
 
 21870000000 
 
 SO 
 
 81 
 
 961 
 
 29791 
 
 923521 
 
 28629151 
 
 887503681 
 
 27512614111 
 
 81 
 
 82 
 
 1024 
 
 32768 
 
 1048576 
 
 33554432 
 
 1073741824 
 
 34359738368 
 
 82 
 
 S3 
 
 1089 
 
 35937 
 
 1185921 
 
 39135393 
 
 1291467969 
 
 42618442977 
 
 oo 
 
 Sit 
 
 1156 
 
 39304 
 
 1336336 
 
 45435424 
 
 1544804416 
 
 52523350144 
 
 34 
 
 S5 
 
 1225 
 
 42875 
 
 1500625 
 
 52521875 
 
 1838265625 
 
 64339296875 
 
 So 
 
 SG 
 
 1296 
 
 46656 
 
 1679616 
 
 60466176 
 
 2176782336 
 
 78364164096 
 
 36 
 
 87 
 
 1369 
 
 50653 
 
 1874161 
 
 69343957 
 
 2565726409 
 
 94931877133 
 
 87 
 
 88 
 
 1444 
 
 54872 
 
 2085136 
 
 79235168 
 
 3010936384 
 
 114415582592 
 
 38 
 
 89 
 
 1521 
 
 59319 
 
 2313441 
 
 90224199 
 
 3518743761 
 
 137231006679 
 
 89 
 
 40 
 
 1600 
 
 64000 
 
 2560000 
 
 102400000 
 
 4096000000 
 
 163840000000 
 
 40 
 
 41 
 
 1681 
 
 68921 
 
 2825761 
 
 115856201 
 
 4750104241 
 
 194754273881 
 
 41 
 
 42 
 
 1764 
 
 74088 
 
 3111696 
 
 130091232 
 
 5489031744 
 
 230539333248 
 
 42 
 
 43 
 
 1849 
 
 79507 
 
 3418801 
 
 147008443 
 
 6321363049 
 
 271818611107 
 
 43 
 
 44 
 
 1936 
 
 85184 
 
 3748096 
 
 164916224 
 
 7256313856 
 
 319277809664 
 
 44 
 
 45 
 
 2025 
 
 91125 
 
 4100625 
 
 184528125 
 
 8303765625 
 
 373009453125 
 
 45 
 
 4G 
 
 2116 
 
 97336 
 
 4477456 
 
 205902976 
 
 9474296896 
 
 435817657216 
 
 .',<; 
 
 47 
 
 2209 
 
 103823 
 
 4879681 
 
 229345007 
 
 10779215329 
 
 506623120403 
 
 4< 
 
 48 
 
 2304 
 
 110592 
 
 5308416 
 
 2548031)68 
 
 12230590464 
 
 587008342272 
 
 48 
 
 49 
 
 2401 
 
 117649 
 
 5764801 
 
 282475249 
 
 13841287201 
 
 678223072849 
 
 49 
 
 50 
 
 2500 
 
 125000 
 
 6250000 
 
 312500000 
 
 15625000000 
 
 781250000000 
 
 00 
 
Tables of Potvers and Sums of Powers 
 
 39 
 
 TABLE XXVII.— {continued). 
 Powers of Natural Numbers. 
 
 n 
 
 n 2 
 
 n 3 
 
 n* 
 
 n* 
 
 n e 
 
 n? 
 
 n 
 
 51 
 
 2C01 
 
 132651 
 
 6765201 
 
 345025251 
 
 17590287801 
 
 897410677851 
 
 51 
 
 52 
 
 2704 
 
 140608 
 
 7311616 
 
 380204032 
 
 19770009064 
 
 1028071702528 
 
 52 
 
 58 
 
 2809 
 
 148877 
 
 7890481 
 
 418195493 
 
 22164361129 
 
 1174711139837 
 
 53 
 
 «* 
 
 2916 
 
 157464 
 
 8503056 
 
 459165024 
 
 24794911296 
 
 1338925209984 
 
 54 
 
 55 
 
 3025 
 
 166375 
 
 9150625 
 
 503284375 
 
 • 27680640625 
 
 1522435234375 
 
 55 
 
 56 
 
 3136 
 
 175616 
 
 9834496 
 
 550731776 
 
 30840979456 
 
 1727094849536 
 
 56 
 
 57 
 
 3249 
 
 185193 
 
 10556001 
 
 601692057 
 
 34296447249 
 
 1954897493193 
 
 57 
 
 5S 
 
 3364 
 
 195112 
 
 11316496 
 
 650356768 
 
 38068692544 
 
 2207984167552 
 
 58 
 
 59 
 
 3481 
 
 205379 
 
 12117361 
 
 714924299 
 
 42180533641 
 
 2488651484819 
 
 59 
 
 60 
 
 3600 
 
 216000 
 
 12960000 
 
 777600000 
 
 46656000000 
 
 2799360000000 
 
 60 
 
 61 
 
 3721 
 
 226981 
 
 13845841 
 
 814596301 
 
 51520374361 
 
 3142742836021 
 
 61 
 
 62 
 
 3844 
 
 23S328 
 
 14776336 
 
 916132832 
 
 56800235584 
 
 3521614606208 
 
 62 
 
 63 
 
 3969 
 
 250047 
 
 15752961 
 
 992436543 
 
 62523502209 
 
 3938980639167 
 
 63 
 
 6J t 
 
 4096 
 
 262144 
 
 16777216 
 
 1073741824 
 
 68719476736 
 
 4398046511104 
 
 64 
 
 65 
 
 4225 
 
 274625 
 
 17850625 
 
 1160290625 
 
 75418890625 
 
 4902227890625 
 
 65 
 
 66 
 
 4356 
 
 287496 
 
 18974736 
 
 1252332576 
 
 82653950016 
 
 5455160701056 
 
 66 
 
 67 
 
 4489 
 
 300763 
 
 20151121 
 
 1350125107 
 
 90458382169 
 
 6060711605323 
 
 67 
 
 6S 
 
 4624 
 
 314432 
 
 21381376 
 
 1453933568 
 
 98867482624 
 
 6722988818432 
 
 68 
 
 69 
 
 4761 
 
 32S509 
 
 22667121 
 
 1564031349 
 
 107918163081 
 
 7446353252589 
 
 69 
 
 70 
 
 4900 
 
 343000 
 
 24010000 
 
 1680700000 
 
 117649000000 
 
 8235430000000 
 
 70 
 
 71 
 
 5041 
 
 357911 
 
 25411681 
 
 1804229351 
 
 128100283921 
 
 9095120158391 
 
 71 
 
 72 
 
 5184 
 
 373248 
 
 26873856 
 
 19349176:52 
 
 139314069504 
 
 10030613004288 
 
 72 
 
 73 
 
 5329 
 
 389017 
 
 28398241 
 
 2073071593 
 
 151334226289 
 
 11047398519097 
 
 73 
 
 74 
 
 5476 
 
 405224 
 
 29986576 
 
 2219006024 
 
 164206490176 
 
 12151280273024 
 
 74 
 
 75 
 
 5625 
 
 421875 
 
 31640625 
 
 2373046875 
 
 177978515625 
 
 13348388071875 
 
 75 
 
 76 
 
 5776 
 
 438976 
 
 33362176 
 
 2535525376 
 
 192699928576 
 
 14645194571776 
 
 76 
 
 77 
 
 5929 
 
 456533 
 
 35153041 
 
 2706784157 
 
 208422380089 
 
 16048523266853 
 
 77 
 
 78 
 
 6084 
 
 474552 
 
 37015056 
 
 2887174368 
 
 225199600704 
 
 17565568854912 
 
 78 
 
 79 
 
 6241 
 
 493039 
 
 38950081 
 
 3077056399 
 
 243087455521 
 
 19203908986159 
 
 79 
 
 80 
 
 6400 
 
 612000 
 
 40960000 
 
 3276800000 
 
 262144000000 
 
 20971520000000 
 
 80 
 
 81 
 
 6561 
 
 531441 
 
 43046721 
 
 3486784401 
 
 282429536481 
 
 22876792454961 
 
 81 
 
 82 
 
 6724 
 
 551368 
 
 45212176 
 
 3707398432 
 
 304006671424 
 
 24928547056768 
 
 82 
 
 83 
 
 6889 
 
 571787 
 
 47458321 
 
 3939040643 
 
 326940373369 
 
 27136050989627 
 
 83 
 
 84 
 
 7056 
 
 592704 
 
 49787136 
 
 4182119424 
 
 351298031616 
 
 29509034655744 
 
 84 
 
 85 
 
 7225 
 
 614125 
 
 52200625 
 
 4437053125 
 
 377149515625 
 
 32057708828125 
 
 85 
 
 86 
 
 7396 
 
 636056 
 
 54700816 
 
 4704270176 
 
 404567235136 
 
 34792782221696 
 
 86 
 
 87 
 
 7569 
 
 658503 
 
 57289761 
 
 4984209207 
 
 433626201009 
 
 37725479487783 
 
 87 
 
 88 
 
 7744 
 
 681472 
 
 59969536 
 
 5277319168 
 
 464404086784 
 
 40867559636992 
 
 88 
 
 89 
 
 7921 
 
 704969 
 
 62742241 
 
 5584059449 
 
 496981290961 
 
 44231334895529 
 
 89 
 
 90 
 
 8100 
 
 729000 
 
 65610000 
 
 5904900000 
 
 531441000000 
 
 47829690000000 
 
 90 
 
 91 
 
 8281 
 
 753571 
 
 68574961 
 
 6240321451 
 
 567869252041 
 
 51676101935731 
 
 91 
 
 92 
 
 8464 
 
 778688 
 
 71639296 
 
 6590815232 
 
 606355001344 
 
 65784660123648 
 
 92 
 
 93 
 
 8649 
 
 804357 
 
 74805201 
 
 6956883093 
 
 646990183449 
 
 60170087060757 
 
 93 
 
 n 
 
 8836 
 
 830584 
 
 78074896 
 
 7339040224 
 
 689809781056 
 
 64847759419264 
 
 94 
 
 95 
 
 9025 
 
 857375 
 
 81450625 
 
 7737809375 
 
 735091890625 
 
 69833729609375 
 
 95 
 
 96 
 
 9216 
 
 884736 
 
 84934656 
 
 8153726976 
 
 782757789696 
 
 75144747810816 
 
 96 
 
 97 
 
 9409 
 
 912673 
 
 88529281 
 
 8587340257 
 
 832972004929 
 
 80798284478113 
 
 97 
 
 98 
 
 9604 
 
 941192 
 
 92236816 
 
 9039207968 
 
 885842380864 
 
 86812553324672 
 
 98 
 
 99 
 
 9801 
 
 970299 
 
 96059601 
 
 9509900499 
 
 941480149401 
 
 93206534790099 
 
 99 
 
 100 
 
 10000 
 
 1000000 
 
 100000000 
 
 10000000000 
 
 1000000000000 
 
 100000000000000 
 
 100 
 
40 
 
 Table for Statisticians and Biometricians 
 
 TABLE XXVIII. 
 
 Sums of Poivers of Natural Numbers. 
 
 n 
 
 S(n) 
 
 S(k 2 ) 
 
 S(n 3 ) 
 
 »(«*) 
 
 <S(n 6 ) 
 
 <S(m 6 ) 
 
 S(tf) 
 
 n 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 2 
 
 3 
 
 5 
 
 9 
 
 17 
 
 33 
 
 65 
 
 129 
 
 2 
 
 S 
 
 6 
 
 14 
 
 36 
 
 98 
 
 276 
 
 794 
 
 2316 
 
 3 
 
 4 
 
 10 
 
 30 
 
 100 
 
 354 
 
 1300 
 
 4890 
 
 18700 
 
 4 
 
 5 
 
 15 
 
 55 
 
 225 
 
 979 
 
 4425 
 
 20515 
 
 96825 
 
 5 
 
 6 
 
 21 
 
 91 
 
 441 
 
 2275 
 
 12201 
 
 67171 
 
 376701 
 
 6 
 
 7 
 
 28 
 
 140 
 
 784 
 
 4676 
 
 2H008 
 
 184820 
 
 1200304 
 
 4 
 
 8 
 
 36 
 
 204 
 
 1296 
 
 8772 
 
 61776 
 
 446964 
 
 3297456 
 
 8 
 
 9 
 
 45 
 
 285 
 
 2025 
 
 15333 
 
 120825 
 
 978405 
 
 8080425 
 
 9 
 
 10 
 
 55 
 
 385 
 
 3025 
 
 25333 
 
 220825 
 
 1978405 
 
 18080425 
 
 10 
 
 11 
 
 66 
 
 506 
 
 4356 
 
 39974 
 
 381876 
 
 3749966 
 
 37567596 
 
 11 
 
 12 
 
 78 
 
 650 
 
 6084 
 
 60710 
 
 630708 
 
 6735950 
 
 73399404 
 
 12 
 
 IS 
 
 91 
 
 819 
 
 8281 
 
 89271 
 
 1002001 
 
 11562759 
 
 136147921 
 
 IS 
 
 U 
 
 105 
 
 1015 
 
 11025 
 
 127687 
 
 1539825 
 
 19092295 
 
 241561425 
 
 u 
 
 15 
 
 120 
 
 1240 
 
 14400 
 
 178312 
 
 2299200 
 
 30482920 
 
 412420800 
 
 15 
 
 JO 
 
 136 
 
 1496 
 
 18196 
 
 243848 
 
 3347776 
 
 47260136 
 
 680856256 
 
 16 
 
 n 
 
 153 
 
 1785 
 
 23 109 
 
 327309 
 
 4767633 
 
 71397705 
 
 1091194929 
 
 17 
 
 18 
 
 171 
 
 2109 
 
 29241 
 
 432345 
 
 6657201 
 
 105409929 
 
 1703414961 
 
 IS 
 
 19 
 
 190 
 
 2470 
 
 36100 
 
 502666 
 
 9133300 
 
 152155810 
 
 2597286700 
 
 19 
 
 20 
 
 210 
 
 2870 
 
 44100 
 
 722666 
 
 12333300 
 
 216455810 
 
 3877286700 
 
 20 
 
 21 
 
 231 
 
 3311 
 
 53361 
 
 917147 
 
 16417401 
 
 302221931 
 
 5678375241 
 
 21 
 
 22 
 
 253 
 
 3795 
 
 64009 
 
 1151403 
 
 21571033 
 
 415601835 
 
 8172733129 
 
 22 
 
 23 
 
 276 
 
 4324 
 
 76176 
 
 1431244 
 
 28007370 
 
 563637724 
 
 11577558576 
 
 23 
 
 n 
 
 300 
 
 4900 
 
 90000 
 
 1763020 
 
 35970000 
 
 754740700 
 
 16164030000 
 
 a* 
 
 25 
 
 325 
 
 5525 
 
 105625 
 
 2153645 
 
 45735025 
 
 998881325 
 
 22267545625 
 
 25 
 
 20 
 
 351 
 
 6201 
 
 123201 
 
 2610621 
 
 57617001 
 
 1307797101 
 
 30299355801 
 
 26 
 
 27 
 
 378 
 
 6930 
 
 142884 
 
 3142062 
 
 71965908 
 
 1695217590 
 
 40759709004 
 
 27 
 
 28 
 
 406 
 
 7714 
 
 164836 
 
 3756718 
 
 89176276 
 
 2177107894 
 
 54252637516 
 
 28 
 
 29 
 
 435 
 
 8555 
 
 189225 
 
 4463999 
 
 109687425 
 
 2771931215 
 
 71502513825 
 
 29 
 
 SO 
 
 465 
 
 9155 
 
 216225 
 
 5273999 
 
 133987425 
 
 3500931215 
 
 93372513825 
 
 30 
 
 SI 
 
 496 
 
 10416 
 
 246016 
 
 6197520 
 
 162616576 
 
 4388434896 
 
 120885127936 
 
 SI 
 
 82 
 
 528 
 
 11440 
 
 278784 
 
 7246090 
 
 196171008 
 
 5462176720 
 
 155244866304 
 
 S2 
 
 S3 
 
 561 
 
 12529 
 
 314721 
 
 8432017 
 
 135306401 
 
 6753644689 
 
 197863309281 
 
 S3 
 
 34 
 
 595 
 
 13685 
 
 354025 
 
 9768353 
 
 180741825 
 
 8298449105 
 
 250386659425 
 
 34 
 
 35 
 
 630 
 
 14910 
 
 396900 
 
 11268978 
 
 333263700 
 
 10136714730 
 
 314725956300 
 
 35 
 
 S6 
 
 666 
 
 16206 
 
 443556 
 
 12948594 
 
 393729876 
 
 12313497066 
 
 393090120396 
 
 36 
 
 37 
 
 703 
 
 17575 
 
 494209 
 
 14822755 
 
 463073833 
 
 14879223475 
 
 488021997529 
 
 37 
 
 38 
 
 741 
 
 19019 
 
 5 19081 
 
 16907891 
 
 542309001 
 
 17890159859 
 
 602437580121 
 
 38 
 
 39 
 
 780 
 
 20540 
 
 608400 
 
 19221332 
 
 632533200 
 
 21408903620 
 
 739668586800 
 
 39 
 
 40 
 
 820 
 
 22140 
 
 672400 
 
 21781332 
 
 734933200 
 
 25504903620 
 
 903508586800 
 
 40 
 
 41 
 
 861 
 
 23821 
 
 741321 
 
 24607093 
 
 850789401 
 
 30255007861 
 
 1098262860681 
 
 41 
 
 42 
 
 903 
 
 25585 
 
 815409 
 
 27718789 
 
 981480633 
 
 35744039605 
 
 1328802193929 
 
 42 
 
 1,3 
 
 946 
 
 27434 
 
 894916 
 
 31137590 
 
 1128489076 
 
 42065402654 
 
 1600620805036 
 
 43 
 
 44 
 
 990 
 
 29370 
 
 980100 
 
 34885686 
 
 1293405300 
 
 49321716510 
 
 1919898614700 
 
 44 
 
 45 
 
 1035 
 
 31395 
 
 1071225 
 
 38986311 
 
 1477933425 
 
 57625482135 
 
 2293568067825 
 
 45 
 
 46 
 
 1081 
 
 33511 
 
 1168561 
 
 43463767 
 
 1683896401 
 
 67099779031 
 
 2729385725041 
 
 46 
 
 47 
 
 1128 
 
 35720 
 
 1272384 
 
 48343448 
 
 1913241408 
 
 77878994360 
 
 3236008845504 
 
 47 
 
 48 
 
 1176 
 
 38024 
 
 1382976 
 
 53651864 
 
 2168045376 
 
 90109584824 
 
 3823077187776 
 
 48 
 
 49 
 
 1225 
 
 40428 
 
 1500625 
 
 59416665 
 
 2450520625 
 
 103950872025 
 
 4501300260625 
 
 49 
 
 50 
 
 , — 
 
 1275 
 
 42925 
 
 1625625 
 
 65666665 
 
 2763020625 
 
 119575872025 
 
 5282550260625 
 
 50 
 
Tables of Powers and Sums of Powers 
 
 41 
 
 TABLE XXVIII.— (continued). 
 Hums of Powers of Natural Numbers. 
 
 n 
 
 S(n) 
 
 £(m 2 ) 
 
 S(n 3 ) 
 
 «(»«) 
 
 S (?» 5 ) 
 
 S(n") 
 
 S(» 7 ) 
 
 n 
 
 51 
 
 1326 
 
 45520 
 
 1758276 
 
 72431866 
 
 3108045876 
 
 137172159826 
 
 6179960938476 
 
 51 
 
 52 
 
 1378 
 
 48230 
 
 1898884 
 
 79743482 
 
 3488249908 
 
 156942769490 
 
 7208032641004 
 
 52 
 
 58 
 
 1431 
 
 51039 
 
 2047761 
 
 87633963 
 
 3906445401 
 
 179107130619 
 
 8382743780841 
 
 53 
 
 54 
 
 1485 
 
 53955 
 
 2205225 
 
 96137019 
 
 4365610425 
 
 203902041915 
 
 9721668990825 
 
 54 
 
 55 
 
 1540 
 
 56980 
 
 2371600 
 
 105287644 
 
 4868894800 
 
 231582682540 
 
 11244104225200 
 
 55 
 
 56 
 
 1596 
 
 60116 
 
 2547216 
 
 115122140 
 
 5419626576 
 
 262423661996 
 
 12971199074736 
 
 56 
 
 57 
 
 1653 
 
 63365 
 
 2732409 
 
 125678141 
 
 6021318633 
 
 296720109245 
 
 14926096567929 
 
 57 
 
 58 
 
 1711 
 
 66729 
 
 2927521 
 
 136994637 
 
 6677675401 
 
 334788801789 
 
 17134080735481 
 
 58 
 
 59 
 
 1770 
 
 70210 
 
 3132900 
 
 149111998 
 
 7392599700 
 
 376969335430 
 
 19622732220300 
 
 59 
 
 60 
 
 1830 
 
 73810 
 
 3348900 
 
 162071998 
 
 8170199700 
 
 423625335430 
 
 22422092220300 
 
 60 
 
 61 
 
 1891 
 
 77531 
 
 3575881 
 
 175917839 
 
 9014796001 
 
 475145709791 
 
 25564835056321 
 
 61 
 
 62 
 
 1953 
 
 81375 
 
 3814209 
 
 190694175 
 
 9930928833 
 
 531945945375 
 
 29086449662529 
 
 62 
 
 6S 
 
 2016 
 
 85344 
 
 4064256 
 
 206447136 
 
 10923365376 
 
 594469447584 
 
 33025430301696 
 
 63 
 
 64 
 
 2080 
 
 89440 
 
 4326400 
 
 223224352 
 
 11997107200 
 
 663188924320 
 
 37423476812800 
 
 64 
 
 65 
 
 2145 
 
 93665 
 
 4001025 
 
 241074977 
 
 13157397825 
 
 738607814945 
 
 42325704703425 
 
 65 
 
 66 
 
 2211 
 
 98021 
 
 4888521 
 
 260049713 
 
 14409730401 
 
 821261764961 
 
 47780865404481 
 
 66 
 
 67 
 
 2278 
 
 102510 
 
 5189284 
 
 280200834 
 
 15759855508 
 
 911720147130 
 
 53841577009804 
 
 67 
 
 68 
 
 2346 
 
 107134 
 
 5503716 
 
 301582210 
 
 17213789076 
 
 1010587629754 
 
 60564565828236 
 
 68 
 
 69 
 
 2415 
 
 111895 
 
 5832225 
 
 324249331 
 
 18777820425 
 
 1118505792835 
 
 68010919080825 
 
 69 
 
 70 
 
 2485 
 
 116795 
 
 6175225 
 
 348259331 
 
 20458520425 
 
 1236154792835 
 
 76246349080825 
 
 70 
 
 71 
 
 2556 
 
 121836 
 
 6533136 
 
 373671012 
 
 22262749776 
 
 1364255076756 
 
 85341469239216 
 
 71 
 
 72 
 
 2628 
 
 127020 
 
 6906384 
 
 400544868 
 
 24197667408 
 
 1503569146260 
 
 95372082243504 
 
 72 
 
 73 
 
 2701 
 
 132349 
 
 7295401 
 
 428943109 
 
 26270739001 
 
 1654903372549 
 
 106419480762601 
 
 73 
 
 74 
 
 2775 
 
 137825 
 
 7700625 
 
 458929685 
 
 28489745625 
 
 1819109862725 
 
 118570761035625 
 
 74 
 
 75 
 
 2850 
 
 143450 
 
 8122500 
 
 490570310 
 
 30862792500 
 
 1997088378350 
 
 131919149707500 
 
 75 
 
 76 
 
 2926 
 
 149226 
 
 8561476 
 
 523932486 
 
 33398317876 
 
 2189788306926 
 
 146564344279276 
 
 76 
 
 77 
 
 3003 
 
 155155 
 
 9018009 
 
 559085527 
 
 36105102033 
 
 2398210687015 
 
 162612867546129 
 
 77 
 
 78 
 
 3081 
 
 161239 
 
 9492561 
 
 596100583 
 
 38992276401 
 
 2623410287719 
 
 180178436401041 
 
 78 
 
 79 
 
 3160 
 
 167480 
 
 9985600 
 
 635050664 
 
 42069332800 
 
 2866497743240 
 
 199382345387200 
 
 79 
 
 80 
 
 3240 
 
 173880 
 
 10497600 
 
 676010664 
 
 45346132800 
 
 3128641743240 
 
 220353865387200 
 
 80 
 
 81 
 
 3321 
 
 180441 
 
 11029041 
 
 719057385 
 
 48832917201 
 
 3411071279721 
 
 243230657842161 
 
 81 
 
 82 
 
 3403 
 
 187165 
 
 11580409 
 
 764269561 
 
 52540315633 
 
 3715077951145 
 
 268159204898929 
 
 82 
 
 83 
 
 3486 
 
 194054 
 
 12152196 
 
 811727882 
 
 56479356276 
 
 4042018324514 
 
 295295255888556 
 
 83 
 
 H 
 
 3570 
 
 201110 
 
 12744900 
 
 861515018 
 
 60661475700 
 
 4393316356130 
 
 324804290544300 
 
 84 
 
 85 
 
 3655 
 
 208335 
 
 13359025 
 
 913715643 
 
 65098528825 
 
 4770465871755 
 
 356861999372425 
 
 85 
 
 86 
 
 3741 
 
 215731 
 
 13995081 
 
 968416459 
 
 69802799001 
 
 5175033106891 
 
 391654781594121 
 
 86 
 
 87 
 
 3828 
 
 2233U0 
 
 14653584 
 
 1025706220 
 
 74787008208 
 
 5608659307900 
 
 429380261081904 
 
 87 
 
 88 
 
 3916 
 
 231044 
 
 15335056 
 
 1085675756 
 
 80064327376 
 
 6073063394684 
 
 470247820718896 
 
 88 
 
 89 
 
 4005 
 
 238965 
 
 16040025 
 
 1148417997 
 
 85648386825 
 
 6570044685645 
 
 514479155614425 
 
 89 
 
 90 
 
 4095 
 
 247065 
 
 16769025 
 
 1214027997 
 
 91553286825 
 
 7101485685645 
 
 562308845614425 
 
 90 
 
 91 
 
 4186 
 
 255346 
 
 17522596 
 
 1282602958 
 
 97793608276 
 
 7669354937686 
 
 613984947550156 
 
 91 
 
 92 
 
 4278 
 
 263810 
 
 18301284 
 
 1354242254 
 
 104384423508 
 
 8275709939030 
 
 669769607673804 
 
 92 
 
 93 
 
 4371 
 
 272459 
 
 19105641 
 
 1429047455 
 
 111341307201 
 
 8922700122479 
 
 729939694734561 
 
 93 
 
 94 
 
 4465 
 
 281295 
 
 19936225 
 
 1507122351 
 
 118680347425 
 
 9612569903535 
 
 794787454153825 
 
 94 
 
 95 
 
 4560 
 
 290320 
 
 20793600 
 
 1588572976 
 
 126418156800 
 
 10347661794160 
 
 864621183763200 
 
 95 
 
 96 
 
 4656 
 
 299536 
 
 21678336 
 
 1673507632 
 
 134571883776 
 
 11130419583856 
 
 939765931574016 
 
 96 
 
 97 
 
 4753 
 
 308945 
 
 22591009 
 
 1762036913 
 
 143159224033 
 
 11963391588785 
 
 1020564216052129 
 
 97 
 
 98 
 
 4851 
 
 318549 
 
 23532201 
 
 1854273729 
 
 152198432001 
 
 12849233969649 
 
 1107376769376801 
 
 98 
 
 99 
 
 4950 
 
 328350 
 
 24502500 
 
 1950333330 
 
 161708332500 
 
 13790714119050 
 
 1200583304167500 
 
 99 
 
 100 
 
 5050 
 
 338350 
 
 25502500 
 
 2050333330 
 
 171708332500 
 
 14790714119050 
 
 1300583304167500 
 
 100 
 
42 Tables for Statisticians and Biomctricians 
 
 TABLE XXIX. Tetrachoric Functions for Fourfold Correlation Tables. 
 
 £(i-«) 
 
 n 
 
 T t 
 
 ^3 
 
 U 
 
 n 
 
 T6 
 
 k 
 
 •001 
 
 •00337 
 
 •00736 
 
 + •01175 
 
 + -01391 
 
 + •01 134 
 
 + -00415 
 
 3-09023 
 
 ■002 
 
 •00634 
 
 •01290 
 
 + -01885 
 
 + -01968 
 
 + -01269 
 
 + -00053 
 
 2-87816 
 
 ■006 
 
 •00915 
 
 •01778 
 
 + -02446 
 
 + -02335 
 
 + -01228 
 
 - -00328 
 
 2-74778 
 
 ■001, 
 
 •01185 
 
 •02222 
 
 + -02918 
 
 + -02587 
 
 + 01111 
 
 - -00687 
 
 2-65207 
 
 •005 
 
 •01446 
 
 •02634 
 
 + -03326 
 
 + -02764 
 
 + -00952 
 
 - -01017 
 
 2-57583 
 
 ■006 
 
 •01700 
 
 •03020 
 
 + -03686 
 
 + -02887 
 
 + -00770 
 
 - 01318 
 
 2-51214 
 
 •007 
 
 •01949 
 
 •03386 
 
 + -04008 
 
 + -02970 
 
 + -00575 
 
 - -01592 
 
 2-45726 
 
 ■008 
 
 •02192 
 
 •03734 
 
 + -04298 
 
 + -03021 
 
 + -00371 
 
 - -01841 
 
 2-40892 
 
 ■009 
 
 •02431 
 
 •04066 
 
 + -04561 
 
 + -03047 
 
 + -00164 
 
 - -02067 
 
 2-30562 
 
 ■010 
 
 ■02665 
 
 •04384 
 
 + -04800 
 
 + -03053 
 
 - -00044 
 
 - -02271 
 
 2 32635 
 
 ■on 
 
 ■02896 
 
 •04690 
 
 + -05020 
 
 + -03041 
 
 - -00253 
 
 - -02457 
 
 2-29037 
 
 •012 
 
 •03123 
 
 •04985 
 
 + -05221 
 
 + -03014 
 
 - -00460 
 
 - -02625 
 
 2-25713 
 
 •013 
 
 •03348 
 
 •05270 
 
 + -05406 
 
 + -02975 
 
 - -00664 
 
 - -02777 
 
 2-22021 
 
 ■014 
 
 •03069 
 
 •05545 
 
 + -05577 
 
 + -02926 
 
 - -00866 
 
 - -02914 
 
 2-19729 
 
 ■015 
 
 •03787 
 
 •05811 
 
 + -05735 
 
 + -02867 
 
 - -01064 
 
 - -03037 
 
 2-17009 
 
 ■016 
 
 •04003 
 
 •06069 
 
 + -05880 
 
 + -02801 
 
 - -01259 
 
 - -03147 
 
 2-14441 
 
 ■017 
 
 •04216 
 
 •06320 
 
 + -06015 
 
 + -02727 
 
 - -01449 
 
 - -03246 
 
 2-12007 
 
 ■018 
 
 •04427 
 
 •06564 
 
 + -06139 
 
 + -02647 
 
 - -01636 
 
 - -03334 
 
 2 09693 
 
 ■019 
 
 •04635 
 
 •06801 
 
 + -06254 
 
 + -02562 
 
 - -01818 
 
 - -03411 
 
 2-07485 
 
 ■020 
 
 •04842 
 
 •07031 
 
 + •06361 
 
 + -02472 
 
 - -01996 
 
 - -03479 
 
 2 05375 
 
 ■021 
 
 •05046 
 
 •07256 
 
 + -06459 
 
 + -02378 
 
 - -02170 
 
 - -03538 
 
 2-03352 
 
 •022 
 
 •05249 
 
 •07475 
 
 + -06549 
 
 + -02280 
 
 - -02340 
 
 - -03589 
 
 2-01409 
 
 ■023 
 
 •05449 
 
 •07688 
 
 + "06633 
 
 + -02179 
 
 - -02505 
 
 - -03632 
 
 1-99539 
 
 ■024 
 
 •05648 
 
 •07897 
 
 + -06709 
 
 + -02074 
 
 - -02666 
 
 - -03667 
 
 1-97737 
 
 ■025 
 
 •05845 
 
 •08100 
 
 + -06780 
 
 + -01968 
 
 - -02823 
 
 - "03696 
 
 1-95996 
 
 ■026 
 
 •06040 
 
 •08299 
 
 + -06844 
 
 + -01858 
 
 - -02976 
 
 - -03718 
 
 1-94313 
 
 •027 
 
 •06233 
 
 •08493 
 
 + -06903 
 
 + •01747 
 
 - -03125 
 
 - -03734 
 
 1-92684 
 
 •028 
 
 •06425 
 
 •08682 
 
 + -06956 
 
 + -01634 
 
 - -03270 
 
 - -03744 
 
 1-91104 
 
 •029 
 
 •06615 
 
 •08868 
 
 + -07005 
 
 + -01520 
 
 - -03411 
 
 - -03749 
 
 1-89570 
 
 ■030 
 
 •06804 
 
 •09049 
 
 + -07048 
 
 + -01404 
 
 - -03547 
 
 - -03749 
 
 1-88079 
 
 ■osi 
 
 •06992 
 
 •09227 
 
 + -07087 
 
 + -01287 
 
 - -03680 
 
 - -03744 
 
 1-86630 
 
 ■032 
 
 •07177 
 
 •09400 
 
 + -07122 
 
 + -01168 
 
 - -03810 
 
 - -03734 
 
 1-85218 
 
 ■OSS 
 
 •07362 
 
 •09570 
 
 + -07153 
 
 + -01049 
 
 - 03935 
 
 - -03720 
 
 1-83842 
 
 ■034 
 
 •07545 
 
 •09737 
 
 + -07179 
 
 + -00929 
 
 - -01057 
 
 - -03702 
 
 1-82501 
 
 ■035 
 
 •07727 
 
 •09900 
 
 + -07202 
 
 + -00809 
 
 - -04176 
 
 - -03680 
 
 1-81191 
 
 ■036 
 
 •07908 
 
 •10060 
 
 + -07221 
 
 + -00688 
 
 - -04291 
 
 - -03654 
 
 1-79912 
 
 ■037 
 
 •08087 
 
 •10216 
 
 + -07237 
 
 + -00566 
 
 - -04402 
 
 - -03624 
 
 1-78661 
 
 ■038 
 
 •08265 
 
 •10370 
 
 + -07249 
 
 + -00444 
 
 - -04510 
 
 - -03592 
 
 1-77438 
 
 ■039 
 
 •08442 
 
 •10520 
 
 + -07258 
 
 + -00322 
 
 - -04615 
 
 - -03556 
 
 1-76241 
 
 ■040 
 
 •08617 
 
 •10668 
 
 + -07264 
 
 + -00200 
 
 - -04717 
 
 - -03517 
 
 1-75069 
 
 ■041 
 
 ■08792 
 
 •10812 
 
 + -07268 
 
 + -00077 
 
 - -04815 
 
 - -03475 
 
 1-73920 
 
 ■042 
 
 •08965 
 
 •10954 
 
 + -07268 
 
 - -00045 
 
 - -04910 
 
 - -03431 
 
 1-72793 
 
 ■04S 
 
 •09137 
 
 •11093 
 
 + -07266 
 
 - -00167 
 
 - -05003 
 
 - -03384 
 
 1-71689 
 
 ■044 
 
 •09309 
 
 •11229 
 
 + -07261 
 
 - -00290 
 
 - -05092 
 
 - -03335 
 
 1-70604 
 
 ■045 
 
 •09479 
 
 ■11363 
 
 + -07253 
 
 - -00412 
 
 - -05178 
 
 - -03283 
 
 1-69540 
 
 ■046 
 
 •09648 
 
 •11495 
 
 + -07243 
 
 - -00534 
 
 - -05261 
 
 - -03229 
 
 1-68494 
 
 •047 
 
 •09816 
 
 •11623 
 
 + -07231 
 
 - -00656 
 
 - -05342 
 
 - -03173 
 
 1-67466 
 
 ■048 
 
 •09983 
 
 •11750 
 
 + •07217 
 
 - -00778 
 
 - -05420 
 
 -•03115 
 
 1-66456 
 
 ■049 
 
 •10149 
 
 •11874 
 
 + -07200 
 
 - -00899 
 
 - -05495 
 
 - -03055 
 
 1-65463 
 
 ■050 
 
 ■10314 
 
 •11996 
 
 + -07181 
 
 - -01020 
 
 - -05567 
 
 - -02994 
 
 1 -64485 
 
 ti, ra and h are essentially positive. 
 
Tables of the Tetrachoric Functions 
 TABLE XXIX.— (continued). 
 
 43 
 
 i(l-a) 
 
 «1 
 
 Ti 
 
 T3 
 
 U 
 
 r» 
 
 re 
 
 h 
 
 •051 
 
 •10478 
 
 •12115 
 
 + •07160 
 
 - -01140 
 
 - -05637 
 
 - -02931 
 
 1-63523 
 
 ■052 
 
 •10641 
 
 •12232 
 
 + •07138 
 
 - -01260 
 
 - -05704 
 
 - -02866 
 
 1-62576 
 
 ■05S 
 
 •10803 
 
 •12347 
 
 + •07113 
 
 - -01380 
 
 - -05769 
 
 - -02799 
 
 1-61644 
 
 ■054 
 
 •10964 
 
 •12460 
 
 + -07087 
 
 - -01499 
 
 - -05831 
 
 - -02732 
 
 1-60725 
 
 ■055 
 
 •11124 
 
 •12571 
 
 + -07058 
 
 - -01618 
 
 - -05891 
 
 - -02662 
 
 1-59819 
 
 ■056 
 
 •11284 
 
 •12680 
 
 + -07028 
 
 - -01736 
 
 - -05949 
 
 - -02592 
 
 1-58927 
 
 ■057 
 
 •11442 
 
 •12787 
 
 + -06997 
 
 - -01854 
 
 - -06004 
 
 - -02520 
 
 1-58047 
 
 ■058 
 
 •11600 
 
 •12892 
 
 + -06964 
 
 - -01971 
 
 - -06057 
 
 - -02447 
 
 1-57179 
 
 ■059 
 
 11756 
 
 •12995 
 
 + -06929 
 
 - -02087 
 
 - -06107 
 
 - -02373 
 
 1-56322 
 
 ■060 
 
 •11912 
 
 •13096 
 
 + -06893 
 
 - -02203 
 
 - 06155 
 
 - -02298 
 
 1-55477 
 
 ■061 
 
 •12067 
 
 •13196 
 
 + -06855 
 
 - 02318 
 
 - 06202 
 
 - -02222 
 
 1 -54643 
 
 ■062 
 
 •12222 
 
 •13293 
 
 + -06816 
 
 - -02433 
 
 - -06246 
 
 - -02145 
 
 1 53820 
 
 06S 
 
 12375 
 
 •13389 
 
 + -06775 
 
 - -02547 
 
 - -06288 
 
 - -02068 
 
 1-53007 
 
 ■064 
 
 •12528 
 
 13483 
 
 + 06734 
 
 - -02660 
 
 - -06328 
 
 - -01989 
 
 1-52204 
 
 ■065 
 
 •12679 
 
 •13575 
 
 + '06690 
 
 - -02773 
 
 - -06365 
 
 - -01910 
 
 1-51410 
 
 ■066 
 
 •12830 
 
 •13666 
 
 + -06646 
 
 - -02884 
 
 - -06401 
 
 - 01830 
 
 1-50626 
 
 ■067 
 
 •12981 
 
 ■13754 
 
 + -06601 
 
 - -02996 
 
 - 06435 
 
 - -01749 
 
 1-49851 
 
 ■068 
 
 ■13130 
 
 ■13842 
 
 + -06554 
 
 - 03106 
 
 - -06467 
 
 - -01668 
 
 1-49085 
 
 ■069 
 
 •13279 
 
 ■13927 
 
 + -06506 
 
 - -03216 
 
 - -06498 
 
 - -01586 
 
 1 -48328 
 
 ■070 
 
 •13427 
 
 14011 
 
 + -06457 
 
 - 03325 
 
 - -06526 
 
 - -01504 
 
 1-47579 
 
 •071 
 
 •13574 
 
 •14094 
 
 + -06407 
 
 - 03433 
 
 - -06552 
 
 - -01421 
 
 1-46838 
 
 ■072 
 
 •13720 
 
 •14175 
 
 + -06356 
 
 - 03541 
 
 - -06577 
 
 - -01337 
 
 1 -46106 
 
 ■078 
 
 •13866 
 
 •14254 
 
 + -06304 
 
 - 03648 
 
 - -06600 
 
 - -01253 
 
 1-45381 
 
 ■074 
 
 •14011 
 
 14332 
 
 + '06251 
 
 - -03754 
 
 - -06621 
 
 - -01169 
 
 1-44663 
 
 •075 
 
 •14156 
 
 •14409 
 
 + ■06197 
 
 - -03859 
 
 - 06641 
 
 - -01085 
 
 1-43953 
 
 ■076 
 
 ■14299 
 
 •14484 
 
 + -06142 
 
 - -03963 
 
 - -06659 
 
 - -oiooo 
 
 1-43250 
 
 •077 
 
 ■14442 
 
 •14558 
 
 + -06086 
 
 - -04067 
 
 - -06675 
 
 - -00915 
 
 1-42554 
 
 •078 
 
 •14584 
 
 •14630 
 
 + 06029 
 
 - -04170 
 
 - -06690 
 
 - -00829 
 
 1-41865 
 
 ■079 
 
 ■14726 
 
 •14701 
 
 + '05971 
 
 - -04272 
 
 - -06703 
 
 - -00743 
 
 1-41183 
 
 ■080 
 
 •14867 
 
 •14771 
 
 + -05913 
 
 - 04374 
 
 - 06715 
 
 - -00658 
 
 1-40507 
 
 ■081 
 
 •15007 
 
 •14839 
 
 + -05854 
 
 - 04474 
 
 - -06725 
 
 - -00572 
 
 1-39838 
 
 ■082 
 
 •15146 
 
 •14906 
 
 + -05794 
 
 - -04574 
 
 - -06733 
 
 - -00485 
 
 1-39174 
 
 ■083 
 
 •15285 
 
 •14971 
 
 + 05733 
 
 - -04673 
 
 - -06741 
 
 - -00399 
 
 1-38517 
 
 ■084 
 
 •15423 
 
 •16036 
 
 + -05671 
 
 - -04771 
 
 - -06746 
 
 - -00312 
 
 1-37866 
 
 ■085 
 
 •15561 
 
 15099 
 
 + -05609 
 
 - -04869 
 
 - 06751 
 
 - -00226 
 
 1-37220 
 
 ■086 
 
 •15698 
 
 •15160 
 
 + -05546 
 
 - -04965 
 
 - -06753 
 
 - -00139 
 
 1-36581 
 
 ■087 
 
 15834 
 
 •15221 
 
 + -05483 
 
 - -05061 
 
 - -06755 
 
 - -00053 
 
 1-35946 
 
 ■088 
 
 •15970 
 
 •15280 
 
 + 05418 
 
 - -05156 
 
 - -06755 
 
 + -00034 
 
 1-35317 
 
 089 
 
 ■16105 
 
 •15339 
 
 + -05353 
 
 - -05250 
 
 - 06754 
 
 + -00120 
 
 1-34694 
 
 ■090 
 
 •16239 
 
 •15396 
 
 + -05288 
 
 - 05344 
 
 - -06751 
 
 + -00207 
 
 1-34076 
 
 ■091 
 
 16373 
 
 15451 
 
 + -05222 
 
 - -05436 
 
 - -06748 
 
 + 00294 
 
 1-33462 
 
 ■092 
 
 •16506 
 
 •15506 
 
 + -05155 
 
 - -05528 
 
 - -06743 
 
 + -00380 
 
 1-32854 
 
 •093 
 
 •16639 
 
 •15560 
 
 + -05088 
 
 - -05619 
 
 - -06736 
 
 + -00467 
 
 1-32251 
 
 •094 
 
 •16770 
 
 •15612 
 
 + -05020 
 
 - -05709 
 
 - -06729 
 
 + -00553 
 
 1-31652 
 
 ■095 
 
 •16902 
 
 •15663 
 
 + -04952 
 
 - -05798 
 
 - -06720 
 
 + -00639 
 
 1-31058 
 
 ■096 
 
 •17033 
 
 •15713 
 
 + -04883 
 
 - -05887 
 
 - 06710 
 
 + -00725 
 
 1-30469 
 
 ■097 
 
 •17163 
 
 •15763 
 
 + 04813 
 
 - -05975 
 
 - -06699 
 
 + -00811 
 
 1-29884 
 
 ■098 
 
 •17292 
 
 •15811 
 
 + -04744 
 
 - -06061 
 
 - -06687 
 
 + -00897 
 
 1-29303 
 
 ■099 
 
 •17421 
 
 •15858 
 
 + -04673 
 
 - -06148 
 
 - 06674 
 
 + -00982 
 
 1-28727 
 
 ■100 
 
 •17550 
 
 •15904 
 
 + '04602 
 
 - 06233 
 
 - -06660 
 
 + -01068 
 
 1-28156 
 
 C— 2 
 
44 Tables for Statisticians and Biometricians 
 
 TABLE XXIX. Tetrackorio Functions for Fourfold Correlation Tables. 
 
 *(!-«) 
 
 Tl 
 
 T2 
 
 T"3 
 
 u 
 
 n 
 
 Tt 
 
 h 
 
 ■101 
 
 •17678 
 
 •15948 
 
 + •04531 
 
 - -06317 
 
 - -06644 
 
 + 01153 
 
 1-27587 
 
 •102 
 
 •17805 
 
 •15992 
 
 + -04459 
 
 - -06401 
 
 - -06628 
 
 + -01238 
 
 1-27024 
 
 ■10S 
 
 •17932 
 
 •16035 
 
 + -04387 
 
 - -06484 
 
 - -06610 
 
 + -01322 
 
 1-26464 
 
 ■104 
 
 •18058 
 
 •16077 
 
 + -04315 
 
 - -06566 
 
 - -06592 
 
 + -01407 
 
 1 -25908 
 
 ■105 
 
 •18184 
 
 •16118 
 
 + •04242 
 
 - -06647 
 
 - -06572 
 
 + -01491 
 
 1-25357 
 
 ■100 
 
 •18309 
 
 •16158 
 
 + "04169 
 
 - -06727 
 
 - 06551 
 
 + -01575 
 
 1-24808 
 
 ■107 
 
 •18433 
 
 •16197 
 
 + -04095 
 
 - -06807 
 
 - -06530 
 
 + -01659 
 
 1-24264 
 
 ■108 
 
 •18557 
 
 •16235 
 
 + -04021 
 
 - -06886 
 
 - -06507 
 
 + -01742 
 
 1-23723 
 
 ■109 
 
 •18681 
 
 •16272 
 
 + -03947 
 
 - -0696 1 
 
 - -06484 
 
 + -01825 
 
 1-23186 
 
 ■110 
 
 •18804 
 
 •16308 
 
 + -03872 
 
 - -07041 
 
 - -06459 
 
 + -01908 
 
 1 -22653 
 
 ■111 
 
 •18926 
 
 •16343 
 
 + 03797 
 
 - 07117 
 
 - -06434 
 
 + -01990 
 
 1-22123 
 
 •112 
 
 •19048 
 
 •16378 
 
 + -03721 
 
 - -07193 
 
 - -06408 
 
 + -02072 
 
 1-21596 
 
 ■118 
 
 •19169 
 
 •16411 
 
 + -03646 
 
 - -07268 
 
 - -06381 
 
 + -02154 
 
 1-21073 
 
 ■114 
 
 •19290 
 
 •16443 
 
 + -03570 
 
 - -07342 
 
 - -06353 
 
 + -02235 
 
 1 -20553 
 
 ■115 
 
 •19410 
 
 •16476 
 
 + 03493 
 
 - -07415 
 
 - -06324 
 
 + -02316 
 
 1-20036 
 
 ■116 
 
 •19530 
 
 •16506 
 
 + -03417 
 
 - -07488 
 
 - -06294 
 
 + -02397 
 
 1-19522 
 
 ■in 
 
 •19649 
 
 •16536 
 
 + 03340 
 
 - -07559 
 
 - -06264 
 
 + -02477 
 
 1-19012 
 
 ■118 
 
 •19768 
 
 •16565 
 
 + -03263 
 
 - -07630 
 
 - -06233 
 
 + -02557 
 
 1-18504 
 
 ■119 
 
 ■19886 
 
 •16593 
 
 + -03186 
 
 - -07700 
 
 - -06201 
 
 + -02636 
 
 1-18000 
 
 •120 
 
 •20004 
 
 •16620 
 
 + -03108 
 
 - -07770 
 
 - -06168 
 
 + -02716 
 
 1-17499 
 
 •121 
 
 ■20121 
 
 •16647 
 
 + -03030 
 
 - -07838 
 
 - -06134 
 
 + -02794 
 
 1-17000 
 
 •122 
 
 •20238 
 
 •16672 
 
 + -02952 
 
 - -07906 
 
 - 06100 
 
 + -02873 
 
 1-16505 
 
 ■12S 
 
 •20354 
 
 •16697 
 
 + -02874 
 
 - -07973 
 
 - -06065 
 
 + -02950 
 
 1-16012 
 
 •124 
 
 •20470 
 
 •16721 
 
 + -02796 
 
 - -08039 
 
 - -06029 
 
 + -03028 
 
 1-15522 
 
 ■125 
 
 •20585 
 
 •16745 
 
 + -02717 
 
 - -08105 
 
 - -05992 
 
 + -03105 
 
 1-15035 
 
 •126 
 
 •20700 
 
 •16767 
 
 + -02638 
 
 - -08169 
 
 - -05955 
 
 + -03181 
 
 1-14551 
 
 ■127 
 
 •20814 
 
 •16789 
 
 + -02559 
 
 - -08233 
 
 - -05917 
 
 + -03257 
 
 1-14069 
 
 •128 
 
 •20928 
 
 •16810 
 
 + -02480 
 
 - -08297 
 
 - -05878 
 
 + -03333 
 
 1-13590 
 
 ■129 
 
 •21042 
 
 •16830 
 
 + -02401 
 
 - -08359 
 
 - -05839 
 
 + 03408 
 
 1-13113 
 
 •180 
 
 •21155 
 
 •16849 
 
 + •02321 
 
 - -08421 
 
 - -05799 
 
 + -03483 
 
 1-12639 
 
 ■131 
 
 •21267 
 
 •16868 
 
 + •02241 
 
 - -08482 
 
 - -05758 
 
 + -03557 
 
 1-12168 
 
 ■182 
 
 •21379 
 
 •16886 
 
 + -02162 
 
 - -08542 
 
 - -05717 
 
 + 03631 
 
 1-11699 
 
 •1SS 
 
 •21490 
 
 •16903 
 
 + -02082 
 
 - 08601 
 
 - 05675 
 
 + -03704 
 
 1-11232 
 
 •184 
 
 •21601 
 
 •16919 
 
 + -02001 
 
 - -08660 
 
 - -05632 
 
 + 03777 
 
 1-10768 
 
 ■185 
 
 •21712 
 
 •16935 
 
 + -01921 
 
 - -08718 
 
 - 05589 
 
 + -03850 
 
 1-10306 
 
 ■186 
 
 •21822 
 
 •16950 
 
 + -01841 
 
 - -08775 
 
 - -05546 
 
 + 03921 
 
 1-09847 
 
 ■187 
 
 •21932 
 
 •16964 
 
 + -01760 
 
 - -08831 
 
 - -05501 
 
 + -03993 
 
 1-09390 
 
 ■188 
 
 •22041 
 
 •16978 
 
 + -01680 
 
 - -08887 
 
 - -05456 
 
 + -04064 
 
 1-08935 
 
 ■139 
 
 •22149 
 
 •16990 
 
 + -01599 
 
 - -08942 
 
 - -05411 
 
 + -04134 
 
 1-08482 
 
 140 
 
 •22258 
 
 •17003 
 
 + -01518 
 
 - -08996 
 
 - -05365 
 
 + -04204 
 
 1-08032 
 
 ■141 
 
 •22365 
 
 17014 
 
 + -01437 
 
 - -09050 
 
 - -05318 
 
 + -04273 
 
 1-07584 
 
 ■142 
 
 •22473 
 
 •17025 
 
 + 01350 
 
 - 09103 
 
 - -05271 
 
 + -04342 
 
 1-07138 
 
 ■143 
 
 •22580 
 
 •17035 
 
 + -01275 
 
 - -09155 
 
 - -05224 
 
 + -04410 
 
 1 06694 
 
 ■144 
 
 •22686 
 
 •17044 
 
 + •01194 
 
 - -09206 
 
 - 05176 
 
 + -04478 
 
 1 -06252 
 
 •145 
 
 •22792 
 
 •17053 
 
 +01113 
 
 - -09257 
 
 - -05127 
 
 + -04545 
 
 1 05812 
 
 ■146 
 
 •22898 
 
 •17061 
 
 + -01032 
 
 - -09307 
 
 - -05078 
 
 + •046 12 
 
 1-05374 
 
 ■147 
 
 •23003 
 
 •17069 
 
 + -00950 
 
 - -09356 
 
 - -05028 
 
 + -04678 
 
 1-04939 
 
 ■148 
 
 •23108 
 
 •17076 
 
 + -00869 
 
 - -09405 
 
 - -04978 
 
 + -04744 
 
 1-04505 
 
 ■149 
 
 •23212 
 
 •17082 
 
 + ■00788 
 
 - -09452 
 
 - -04928 
 
 + -04809 
 
 1 -04073 
 
 •150 
 
 •23316 
 
 •17087 
 
 + -00706 
 
 - 09499 
 
 - -04877 
 
 + -04874 
 
 1 03643 
 
Tables of the Tetrachoric Functions 
 TABLE XXIX— (continued). 
 
 45 
 
 1(1-.) 
 
 ft 
 
 T 2 
 
 • 
 
 U 
 
 U 
 
 tb 
 
 h 
 
 ■151 
 
 •23419 
 
 •17092 
 
 + -00025 
 
 - -09546 
 
 - -04825 
 
 + -04938 
 
 1-03215 
 
 •152 
 
 •23522 
 
 •17097 
 
 + -00543 
 
 - 09592 
 
 - -04774 
 
 + -05002 
 
 1-02789 
 
 ■153 
 
 •23625 
 
 •17100 
 
 + -00462 
 
 - -09637 
 
 - -04721 
 
 + -05065 
 
 1 -02365 
 
 '154 
 
 •23727 
 
 •17103 
 
 + -00380 
 
 - -09081 
 
 - -04669 
 
 + -05127 
 
 1-01943 
 
 •155 
 
 •23829 
 
 •17106 
 
 + -00298 
 
 - -09725 
 
 - -04615 
 
 + -05189 
 
 1-01522 
 
 ■156 
 
 •23930 
 
 •17108 
 
 + -00217 
 
 - -09768 
 
 - -04562 
 
 + -05250 
 
 1-01103 
 
 ■157 
 
 •24031 
 
 •17109 
 
 + •00135 
 
 - -09810 
 
 - -04508 
 
 + -05311 
 
 1-00686 
 
 ■158 
 
 •24131 
 
 •17110 
 
 + •00053 
 
 - -09852 
 
 - -04454 
 
 + -05371 
 
 1-00271 
 
 •159 
 
 •24232 
 
 •17110 
 
 -■00028 
 
 - -09892 
 
 - -04399 
 
 + -05431 
 
 •99858 
 
 •160 
 
 •24331 
 
 •17109 
 
 - 00110 
 
 - -09933 
 
 - -04344 
 
 + -05490 
 
 •99446 
 
 ■161 
 
 •24430 
 
 •17108 
 
 - -00191 
 
 - -09972 
 
 - -04288 
 
 + -05549 
 
 •99036 
 
 ■162 
 
 •24529 
 
 •17107 
 
 - -00273 
 
 - -10011 
 
 - -04232 
 
 + -05607 
 
 •98627 
 
 ■16S 
 
 •24628 
 
 •17104 
 
 -■00355 
 
 - -10049 
 
 - -04176 
 
 + -05664 
 
 •98220 
 
 ■164 
 
 •24726 
 
 •17102 
 
 - -00436 
 
 - -10087 
 
 - -04120 
 
 + -05721 
 
 ■97815 
 
 •165 
 
 •24823 
 
 •17098 
 
 - -00518 
 
 - -10124 
 
 - -04063 
 
 + -05778 
 
 ■97411 
 
 ■166 
 
 •24921 
 
 •17094 
 
 - -00599 
 
 - -10160 
 
 - -04006 
 
 + -05834 
 
 •97009 
 
 •167 
 
 •25017 
 
 •17090 
 
 - -00681 
 
 - -10196 
 
 - -03948 
 
 + -05889 
 
 •96609 
 
 ■168 
 
 •25114 
 
 •17085 
 
 - -00762 
 
 - -10231 
 
 - -03890 
 
 + -05943 
 
 •96210 
 
 •169 
 
 •25210 
 
 •17080 
 
 - -00844 
 
 - -10265 
 
 - -03832 
 
 + -05998 
 
 ■95812 
 
 •170 
 
 •25305 
 
 •17073 
 
 - -00925 
 
 - -10299 
 
 - -03774 
 
 + -06051 
 
 ■95417 
 
 ■171 
 
 •25401 
 
 •17067 
 
 - -01007 
 
 - -10332 
 
 - -03715 
 
 + -06104 
 
 •95022 
 
 •172 
 
 •25495 
 
 •17060 
 
 - -01088 
 
 - -10364 
 
 - -03656 
 
 + 06156 
 
 •94629 
 
 •173 
 
 •25590 
 
 •17052 
 
 - -01169 
 
 - -10396 
 
 - -03597 
 
 + -06208 
 
 •94238 
 
 ■174 
 
 •25684 
 
 •17044 
 
 - 01251 
 
 - -10427 
 
 - -03537 
 
 + -06260 
 
 •93848 
 
 ■175 
 
 •25778 
 
 •17035 
 
 - 01332 
 
 - -10458 
 
 - -03478 
 
 + -06310 
 
 •93459 
 
 •176 
 
 ■25871 
 
 •17026 
 
 - -01413 
 
 - -10487 
 
 - -03417 
 
 + 06360 
 
 •93072 
 
 ■177 
 
 •25964 
 
 •17016 
 
 - -01494 
 
 - -10517 
 
 - -03357 
 
 + -06410 
 
 •92686 
 
 ■178 
 
 •26056 
 
 •17006 
 
 - -01575 
 
 - -10545 
 
 - -03296 
 
 + -06459 
 
 •92301 
 
 ■179 
 
 •26148 
 
 •16995 
 
 - -01656 
 
 - -10573 
 
 - -03236 
 
 + -06507 
 
 •91918 
 
 ■180 
 
 •26240 
 
 •16984 
 
 - -01737 
 
 - -10601 
 
 - 03175 
 
 + •06555 
 
 •91537 
 
 •181 
 
 •26331 
 
 •16972 
 
 - -01817 
 
 - -10627 
 
 - 03113 
 
 + -06603 
 
 •91156 
 
 •182 
 
 •26422 
 
 ■16960 
 
 - -01898 
 
 - -10653 
 
 - -03052 
 
 + -06649 
 
 •90777 
 
 ■183 
 
 •26513 
 
 •16948 
 
 - -01979 
 
 - -10679 
 
 - -02990 
 
 + -06695 
 
 •90399 
 
 •184 
 
 •26603 
 
 •16934 
 
 - -02059 
 
 - -10704 
 
 - -02928 
 
 + -06741 
 
 ■90023 
 
 •185 
 
 •26693 
 
 •16921 
 
 - -02140 
 
 - -10728 
 
 - -02866 
 
 + -06786 
 
 •89647 
 
 ■18G 
 
 •26782 
 
 •16907 
 
 - -02220 
 
 - -10752 
 
 - -02803 
 
 + -06830 
 
 •89273 
 
 ■187 
 
 •26871 
 
 •16892 
 
 - -02300 
 
 - -10775 
 
 - -02741 
 
 + -06874 
 
 •88901 
 
 ■188 
 
 •26960 
 
 •16877 
 
 - -02380 
 
 - -10798 
 
 - -02678 
 
 + -06917 
 
 •88529 
 
 ■189 
 
 •27049 
 
 •16861 
 
 - -02460 
 
 - -10819 
 
 - -02615 
 
 + -06960 
 
 •88159 
 
 ■190 
 
 •27137 
 
 •16845 
 
 - 02540 
 
 - -10841 
 
 - -02552 
 
 + -07002 
 
 •87790 
 
 •191 
 
 •27224 
 
 •16829 
 
 - -02620 
 
 - -10861 
 
 - -02489 
 
 + -07044 
 
 •87422 
 
 •192 
 
 •27311 
 
 •16812 
 
 - -02700 
 
 - -10882 
 
 - -02425 
 
 + -07085 
 
 •87055 
 
 ■193 
 
 •27398 
 
 •16795 
 
 - -02779 
 
 - -10901 
 
 - 02362 
 
 +07125 
 
 •86689 
 
 ■194 
 
 •27485 
 
 •16777 
 
 - -02859 
 
 -•10920 
 
 - -02298 
 
 + -07165 
 
 •86325 
 
 ■195 
 
 •27571 
 
 •16759 
 
 - -02938 
 
 - -10939 
 
 - 02234 
 
 + -07204 
 
 •85962 
 
 •196 
 
 •27657 
 
 •16740 
 
 - 03018 
 
 - -10956 
 
 - -02170 
 
 + -07243 
 
 •85600 
 
 •197 
 
 •27742 
 
 •16721 
 
 - -03097 
 
 - -10974 
 
 - -02106 
 
 + -07281 
 
 •85239 
 
 •198 
 
 ■27827 
 
 •16701 
 
 - -03176 
 
 - -10990 
 
 - -02041 
 
 + 07319 
 
 •84879 
 
 •199 
 
 •27912 
 
 •16681 
 
 - -03255 
 
 - -11007 
 
 - -01977 
 
 + -07356 
 
 •84520 
 
 ■200 
 
 •27996 
 
 •16661 
 
 - 03334 
 
 - -11022 
 
 - 01912 
 
 + -07392 
 
 •84162 
 
46 Tables for Statisticians and Biometricians 
 
 TABLE XXIX. Tetrachoric Functions for Fourfold Correlation Tables. 
 
 i(l-a) 
 
 n 
 
 r-2 
 
 ^3 
 
 U 
 
 *l 
 
 m 
 
 h 
 
 ■201 
 
 •28080 
 
 •16640 
 
 - 03412 
 
 - -11037 
 
 - -01848 
 
 + -07428 
 
 •83805 
 
 ■202 
 
 •28164 
 
 •16619 
 
 - -03491 
 
 - -11051 
 
 - -01783 
 
 + -07464 
 
 •83450 
 
 ■203 
 
 •28247 
 
 •16597 
 
 - -03569 
 
 - 11065 
 
 - -01718 
 
 + -07498 
 
 •83095 
 
 ■20k 
 
 •28330 
 
 •16575 
 
 - -03648 
 
 - -11079 
 
 - -01653 
 
 + -07532 
 
 •82742 
 
 'SOB 
 
 •28413 
 
 •16553 
 
 - -03726 
 
 - -11091 
 
 - -01587 
 
 + -07566 
 
 ■82389 
 
 ■MM 
 
 •28495 
 
 •16530 
 
 - -03804 
 
 -11104 
 
 - -01522 
 
 + -07599 
 
 •82038 
 
 ■207 
 
 •28577 
 
 •16506 
 
 - -03882 
 
 - -11115 
 
 - -01457 
 
 + -07632 
 
 •81687 
 
 ■208 
 
 ■28658 
 
 16483 
 
 - -03959 
 
 - -11126 
 
 - 01391 
 
 + -07664 
 
 •81338 
 
 ■209 
 
 ■28739 
 
 •16459 
 
 - -04037 
 
 - -11137 
 
 - -01326 
 
 + -07695 
 
 ■80990 
 
 ■210 
 
 •28820 
 
 •16434 
 
 - -04114 
 
 — •11147 
 
 - -01260 
 
 + -07726 
 
 •80642 
 
 ■211 
 
 •28901 
 
 •16409 
 
 - -04192 
 
 - -11157 
 
 -•01194 
 
 + -07756 
 
 •80296 
 
 •9.12 
 
 •28981 
 
 •16384 
 
 - -04269 
 
 - -11166 
 
 - -01129 
 
 + "07786 
 
 •79950 
 
 ■21S 
 
 •29060 
 
 •16358 
 
 - -04346 
 
 - -11174 
 
 - -01063 
 
 + -07815 
 
 •79606 
 
 ■214 
 
 •29140 
 
 •16332 
 
 - -04423 
 
 - -11182 
 
 - -00997 
 
 + -07844 
 
 •79262 
 
 ■215 
 
 •29219 
 
 •16305 
 
 - -04499 
 
 - -11189 
 
 - -00931 
 
 + -07872 
 
 •78919 
 
 ■216 
 
 •29298 
 
 •16279 
 
 - -04576 
 
 -11196 
 
 - -00865 
 
 + -07899 
 
 •78577 
 
 ■217 
 
 •29376 
 
 •16251 
 
 - -04652 
 
 - -11203 
 
 - -00799 
 
 + -07926 
 
 ■78237 
 
 •218 
 
 •29454 
 
 •16224 
 
 - -04728 
 
 - -11208 
 
 - 00733 
 
 + -07952 
 
 •77897 
 
 ■219 
 
 •29532 
 
 •16196 
 
 - -04804 
 
 -•11214 
 
 - -00667 
 
 + -07978 
 
 •77557 
 
 ■220 
 
 •29609 
 
 16167 
 
 - -04880 
 
 -•11218 
 
 - -00600 
 
 + -08004 
 
 •77219 
 
 ■221 
 
 •29686 
 
 •16139 
 
 - -04956 
 
 - -11223 
 
 - -00534 
 
 + -08028 
 
 •76882 
 
 •222 
 
 •29763 
 
 •16110 
 
 - 05031 
 
 - -11226 
 
 - -00468 
 
 + -08052 
 
 •76546 
 
 •223 
 
 •29840 
 
 •16080 
 
 - -05107 
 
 - -11230 
 
 - -00402 
 
 + -08076 
 
 •76210 
 
 ■224 
 
 •29916 
 
 •16050 
 
 - -05182 
 
 - -11233 
 
 - 00335 
 
 + -08099 
 
 •75875 
 
 ■225 
 
 •29991 
 
 •16020 
 
 - -05257 
 
 - -11235 
 
 - -00269 
 
 + -08122 
 
 •75541 
 
 ■226 
 
 •30067 
 
 ■15990 
 
 - 05332 
 
 -11237 
 
 - -00203 
 
 + -08144 
 
 •75208 
 
 ■227 
 
 ■30142 
 
 •15959 
 
 - 05406 
 
 - -11238 
 
 - 00136 
 
 + -08165 
 
 •74876 
 
 •228 
 
 •30216 
 
 •15927 
 
 - -05481 
 
 - -11239 
 
 - -00070 
 
 + -08186 
 
 •74545 
 
 ■229 
 
 •30291 
 
 •15896 
 
 - -05555 
 
 - -11239 
 
 -•00004 
 
 + -08207 
 
 •74214 
 
 ■230 
 
 •30365 
 
 •15864 
 
 - -05629 
 
 -11239 
 
 + -00063 
 
 + -08226 
 
 •73885 
 
 •231 
 
 •30439 
 
 •15832 
 
 - -05703 
 
 - -11238 
 
 + -00129 
 
 + -08246 
 
 •73556 
 
 •232 
 
 ■30512 
 
 •15799 
 
 - -05777 
 
 -11237 
 
 + •00195 
 
 + -08265 
 
 •73228 
 
 ■233 
 
 •30585 
 
 ■15766 
 
 - -05851 
 
 - -11235 
 
 + -00262 
 
 + -08283 
 
 •72900 
 
 ■234. 
 
 •30658 
 
 •15733 
 
 - 05924 
 
 - -11233 
 
 + -00328 
 
 + -08301 
 
 •72574 
 
 ■235 
 
 •30730 
 
 •15699 
 
 - -05997 
 
 - -11230 
 
 + -00394 
 
 + -08318 
 
 •72248 
 
 •236 
 
 •30802 
 
 15665 
 
 - -06070 
 
 - -11227 
 
 + -00461 
 
 + -08334 
 
 •71923 
 
 ■237 
 
 •30874 
 
 •15631 
 
 - -06143 
 
 - -11224 
 
 + -00527 
 
 + -08351 
 
 •71599 
 
 ■238 
 
 •30945 
 
 ■15596 
 
 - -06215 
 
 - -11220 
 
 + -00593 
 
 + -08366 
 
 •71275 
 
 ■239 
 
 ■31017 
 
 •15561 
 
 - -06288 
 
 - -11215 
 
 + -00659 
 
 + -08381 
 
 •70952 
 
 ■240 
 
 •31087 
 
 •15526 
 
 - 06360 
 
 - -11210 
 
 + -00726 
 
 + -08396 
 
 •70630 
 
 ■241 
 
 •31158 
 
 •15490 
 
 - -06432 
 
 - -11205 
 
 + -00792 
 
 + -08410 
 
 •70309 
 
 ■242 
 
 •31228 
 
 •15454 
 
 - -06504 
 
 - -11199 
 
 + -00858 
 
 + -08423 
 
 •69988 
 
 ■243 
 
 •31298 
 
 •15418 
 
 - -06576 
 
 -•11192 
 
 + -00924 
 
 + -08436 
 
 •69668 
 
 •244 
 
 •31367 
 
 •15382 
 
 - -06647 
 
 -•11185 
 
 + -00990 
 
 + 08449 
 
 •69349 
 
 ■245 
 
 •31436 
 
 ■15345 
 
 - -06718 
 
 -•11178 
 
 + -01056 
 
 + -08461 
 
 •69031 
 
 ■246 
 
 •31505 
 
 •15308 
 
 - -06789 
 
 -•11170 
 
 + -01122 
 
 + -08472 
 
 •68713 
 
 ■247 
 
 •31574 
 
 •15270 
 
 - O6860 
 
 -•11162 
 
 + •01188 
 
 + -08483 
 
 ■68396 
 
 ■243 
 
 •31G42 
 
 •15232 
 
 - 06931 
 
 -•11154 
 
 + •01253 
 
 + -08494 
 
 •68080 
 
 •249 
 
 •31710 
 
 •15194 
 
 - -07001 
 
 -•11145 
 
 + •01319 
 
 + -08504 
 
 •67764 
 
 ■250 
 
 •31778 
 
 •15156 
 
 - -07071 
 
 -11135 
 
 + -01385 
 
 + -08513 
 
 ■67449 
 
Tables of the Tetraclwric Functions 
 TABLE XXIX.— {continued). 
 
 47 
 
 i(i-«) 
 
 T\ 
 
 T2 
 
 »| 
 
 U 
 
 ti 
 
 T o 
 
 h 
 
 ■251 
 
 •31845 
 
 •15117 
 
 - 07141 
 
 - -11125 
 
 + -01450 
 
 + -08522 
 
 •67135 
 
 ■252 
 
 •31912 
 
 •15078 
 
 -•07211 
 
 - -11115 
 
 + -01516 
 
 + -08530 
 
 •66821 
 
 •253 
 
 •31979 
 
 •15039 
 
 - -07280 
 
 - -11104 
 
 + -01581 
 
 + -08538 
 
 •66508 
 
 ■254 
 
 •32045 
 
 •14999 
 
 - -07350 
 
 - -11093 
 
 + -01647 
 
 + -08546 
 
 •66196 
 
 ■255 
 
 •32111 
 
 •14959 
 
 - 07419 
 
 - -11081 
 
 + -01712 
 
 + -08553 
 
 •65884 
 
 •256 
 
 ■32177 
 
 •14919 
 
 - -07488 
 
 - -11069 
 
 + -01777 
 
 + -08559 
 
 •65573 
 
 ■257 
 
 •32242 
 
 ■14879 
 
 - -07557 
 
 - -11056 
 
 + -01842 
 
 + -08565 
 
 •65262 
 
 ■258 
 
 ■32307 
 
 •14838 
 
 - -07625 
 
 - -11043 
 
 + -01907 
 
 + -08571 
 
 •64952 
 
 ■259 
 
 •32372 
 
 •14797 
 
 - -07693 
 
 - -11030 
 
 + -01972 
 
 + -08575 
 
 •64643 
 
 ■260 
 
 •32437 
 
 •14756 
 
 - -07761 
 
 - -11016 
 
 + -02037 
 
 + -08580 
 
 •64335 
 
 ■261 
 
 ■32501 
 
 •14714 
 
 - -07829 
 
 -■11002 
 
 + -02102 
 
 + -08584 
 
 •64027 
 
 ■262 
 
 •32565 
 
 •14672 
 
 - -07897 
 
 - -10987 
 
 + -02166 
 
 + -08587 
 
 •63719 
 
 ■263 
 
 ■32628 
 
 •14630 
 
 - -07964 
 
 - -10972 
 
 + -02231 
 
 + -08590 
 
 •63412 
 
 ■26k 
 
 •32691 
 
 •14588 
 
 - -08031 
 
 - -10956 
 
 + -02295 
 
 + -08593 
 
 •63106 
 
 ■265 
 
 •32754 
 
 •14545 
 
 - -08098 
 
 - -10940 
 
 + -02360 
 
 + -08595 
 
 •62801 
 
 ■266 
 
 •32817 
 
 •14502 
 
 - -08165 
 
 - -10924 
 
 + -02424 
 
 + -08596 
 
 •62496 
 
 ■267 
 
 •32879 
 
 •14459 
 
 - -08231 
 
 - -10907 
 
 + -02488 
 
 + -08597 
 
 •62191 
 
 ■268 
 
 •32941 
 
 •14415 
 
 - -08298 
 
 - -10890 
 
 + -02552 
 
 + -08598 
 
 •61887 
 
 ■269 
 
 •33003 
 
 •14372 
 
 - -08364 
 
 - -10873 
 
 + -02616 
 
 + -08598 
 
 •61684 
 
 ■270 
 
 •33065 
 
 •14328 
 
 - -08429 
 
 - -10855 
 
 + -02680 
 
 + -08598 
 
 •61281 
 
 ■271 
 
 •33126 
 
 •14283 
 
 - -08495 
 
 -•10837 
 
 + -02743 
 
 + -08597 
 
 •60979 
 
 ■272 
 
 •33187 
 
 •14239 
 
 - -08560 
 
 - -10818 
 
 + '02807 
 
 + -08596 
 
 •60678 
 
 ■373 
 
 •33247 
 
 •14194 
 
 - -08625 
 
 - -10799 
 
 + -02870 
 
 + -08594 
 
 •60376 
 
 ■274 
 
 •33307 
 
 •1414!) 
 
 - -08690 
 
 - -10779 
 
 + -02933 
 
 + -08591 
 
 •60076 
 
 ■275 
 
 •33367 
 
 •14104 
 
 - -08755 
 
 - -10759 
 
 + ■02997 
 
 + -08589 
 
 •59776 
 
 ■276 
 
 •33427 
 
 •14058 
 
 - -08819 
 
 - -10739 
 
 + -03060 
 
 + -08586 
 
 •59477 
 
 ■277 
 
 •33486 
 
 •14012 
 
 - -08883 
 
 -•10718 
 
 + -03122 
 
 4- -08582 
 
 •59178 
 
 ■278 
 
 •33545 
 
 •13966 
 
 - -08947 
 
 -•10697 
 
 + -03185 
 
 + -08578 
 
 •58879 
 
 ■279 
 
 •33604 
 
 •13920 
 
 - -09011 
 
 -■10676 
 
 + -03248 
 
 + -08573 
 
 •58581 
 
 ■280 
 
 •33662 
 
 •13873 
 
 - -09074 
 
 - -10654 
 
 + -03310 
 
 + -08568 
 
 •68284 
 
 ■281 
 
 •33720 
 
 •13826 
 
 - -09137 
 
 - -10632 
 
 + 03372 
 
 + -08563 
 
 •57987 
 
 ■282 
 
 •33778 
 
 •13779 
 
 - -09200 
 
 - -10609 
 
 + -03434 
 
 + -08557 
 
 •57691 
 
 ■283 
 
 •33836 
 
 •13732 
 
 - -09263 
 
 - -10587 
 
 + -03496 
 
 + -08551 
 
 •57395 
 
 ■284 
 
 •33893 
 
 •13685 
 
 - -09325 
 
 - -10563 
 
 + -03558 
 
 + -08544 
 
 •57100 
 
 ■285 
 
 •33950 
 
 •13637 
 
 - -09388 
 
 - -10540 
 
 + -03620 
 
 + -08536 
 
 •56805 
 
 ■286 
 
 •34007 
 
 •13589 
 
 - -09450 
 
 - -10516 
 
 4- -03681 
 
 + -08529 
 
 •56511 
 
 ■287 
 
 •34063 
 
 •13541 
 
 - -09511 
 
 - -10491 
 
 + -03743 
 
 + -08521 
 
 •56217 
 
 ■288 
 
 •34119 
 
 •13492 
 
 - -09573 
 
 - -10466 
 
 + -03804 
 
 + -08512 
 
 •55924 
 
 ■389 
 
 •34175 
 
 •13443 
 
 - -09634 
 
 - -10441 
 
 + -03865 
 
 + -08503 
 
 •55631 
 
 ■890 
 
 •34230 
 
 •13394 
 
 - -09695 
 
 - -10416 
 
 + -03926 
 
 4- -08494 
 
 •55338 
 
 ■291 
 
 ■34286 
 
 •13345 
 
 - -09756 
 
 - -10390 
 
 + -03987 
 
 + -08484 
 
 •55047 
 
 ■292 
 
 •34341 
 
 •13296 
 
 - -09816 
 
 - -10364 
 
 + -04047 
 
 + -08473 
 
 •64755 
 
 ■298 
 
 •34395 
 
 •13240 
 
 - -09876 
 
 - -10337 
 
 + -04107 
 
 + -08463 
 
 •54464 
 
 ■294 
 
 •34449 
 
 •13196 
 
 - -09936 
 
 - -10310 
 
 + -04168 
 
 + -08451 
 
 •64174 
 
 ■295 
 
 ■34503 
 
 •13146 
 
 - -09996 
 
 - -10283 
 
 + -04228 
 
 + -08440 
 
 •53884 
 
 •296 
 
 •34557 
 
 ■13096 
 
 - -10056 
 
 - -10256 
 
 + -04287 
 
 + -08428 
 
 •53594 
 
 ■297 
 
 •34611 
 
 •13046 
 
 -•10115 
 
 - -10228 
 
 + -04347 
 
 + -08415 
 
 •53305 
 
 ■298 
 
 ■34664 
 
 •12995 
 
 - -10174 
 
 - -10199 
 
 + -04407 
 
 + -08402 
 
 •53016 
 
 •299 
 
 •34717 
 
 •12944 
 
 - -10233 
 
 - -10171 
 
 + -04466 
 
 + -08389 
 
 •52728 
 
 ■300 
 
 •34769 
 
 •12893 
 
 - -10291 
 
 - -10142 
 
 + -04525 
 
 + -08375 
 
 ■52440 
 
48 Tables for Statisticians and Bio metricians 
 
 TABLE XXIX. Tetraohoric Functions for Fourfold Correlation Tables. 
 
 i(i-«) 
 
 1 
 
 r i 
 
 n 
 
 U 
 
 n 
 
 T e 
 
 h 
 
 ■301 
 
 •34822 
 
 •12841 
 
 - -10349 
 
 -10113 
 
 + -04584 
 
 + 08361 
 
 •52153 
 
 ■302 
 
 •34874 
 
 •12790 
 
 - -10407 
 
 - -10083 
 
 + -04643 
 
 + -08347 
 
 •51806 
 
 ■303 
 
 •34925 
 
 •12738 
 
 - '10465 
 
 - -10053 
 
 + -04701 
 
 + -08332 
 
 •51579 
 
 ■304 
 
 •34977 
 
 •12686 
 
 - -10522 
 
 - -10023 
 
 + -04759 
 
 + -08316 
 
 •51293 
 
 ■305 
 
 •35028 
 
 •12634 
 
 - -10580 
 
 - -09992 
 
 + •04818 
 
 + -08301 
 
 •51007 
 
 ■306 
 
 •35079 
 
 •12581 
 
 - 10636 
 
 - 09961 
 
 + -04876 
 
 + -08284 
 
 •50722 
 
 •307 
 
 •35129 
 
 ■12529 
 
 - -10693 
 
 - -09930 
 
 + 04933 
 
 + -08268 
 
 •50437 
 
 ■308 
 
 •35180 
 
 •12476 
 
 - -10750 
 
 - -09899 
 
 + -04991 
 
 + -08251 
 
 •50153 
 
 ■309 
 
 •35230 
 
 ■12423 
 
 - -10806 
 
 - -09867 
 
 + -05048 
 
 + -08233 
 
 •49869 
 
 ■310 
 
 •35279 
 
 •12370 
 
 - -10862 
 
 - -09834 
 
 + -05105 
 
 + -08216 
 
 •49585 
 
 ■311 
 
 •35329 
 
 •12316 
 
 - -10917 
 
 - -09802 
 
 + •05162 
 
 + -08197 
 
 •49302 
 
 ■312 
 
 •35378 
 
 •12263 
 
 - -10973 
 
 - -09769 
 
 + -05219 
 
 + -08179 
 
 •49019 
 
 ■313 
 
 •35427 
 
 •12209 
 
 - -11028 
 
 - -09736 
 
 + -05276 
 
 + -08100 
 
 •48736 
 
 ■311, 
 
 •35475 
 
 •12155 
 
 - -11082 
 
 - -09703 
 
 + -05332 
 
 + -08140 
 
 •48454 
 
 •315 
 
 •35524 
 
 12101 
 
 -11137 
 
 -•09669 
 
 + -05388 
 
 + 08121 
 
 •48173 
 
 ■316 
 
 •35572 
 
 •12046 
 
 -11191 
 
 - -09635 
 
 + -05444 
 
 + -08101 
 
 •47891 
 
 ■317 
 
 •35620 
 
 •11992 
 
 - -11245 
 
 - -09600 
 
 + -05500 
 
 + -08080 
 
 •47610 
 
 ■318 
 
 •35667 
 
 •11937 
 
 - -11299 
 
 - -09566 
 
 + -05555 
 
 + -08059 
 
 •47330 
 
 ■319 
 
 •35714 
 
 •11882 
 
 - 11353 
 
 - -09531 
 
 + -05610 
 
 + -08038 
 
 •47050 
 
 •320 
 
 •35761 
 
 •11827 
 
 - -11406 
 
 - -09495 
 
 + -05665 
 
 + -08016 
 
 •46770 
 
 ■321 
 
 ■35808 
 
 •11771 
 
 - -11459 
 
 - -09460 
 
 + -05720 
 
 + -07994 
 
 •46490 
 
 ■322 
 
 •35854 
 
 •11716 
 
 - -11512 
 
 - -09424 
 
 + -05775 
 
 + -07972 
 
 •46211 
 
 •323 
 
 •35900 
 
 •11660 
 
 -•11564 
 
 - -09388 
 
 + -05829 
 
 + -07949 
 
 •45933 
 
 ■321,. 
 
 •35946 
 
 •11604 
 
 -•11616 
 
 - -09351 
 
 + -05883 
 
 + -07920 
 
 •45654 
 
 ■325 
 
 •35991 
 
 •11548 
 
 - -11668 
 
 - -09315 
 
 + -05937 
 
 + -07902 
 
 •45376 
 
 ■326 
 
 •36037 
 
 •11492 
 
 - -11720 
 
 - -09278 
 
 + -05991 
 
 + -07878 
 
 •45099 
 
 ■327 
 
 •36082 
 
 •11436 
 
 -•11771 
 
 - -09240 
 
 + -06044 
 
 + -07854 
 
 •44821 
 
 ■328 
 
 •36126 
 
 •11379 
 
 - -11822 
 
 - -09203 
 
 + -06097 
 
 + -07829 
 
 •44544 
 
 •329 
 
 •36171 
 
 •11322 
 
 -•11873 
 
 - -09165 
 
 + -06150 
 
 + -07804 
 
 ■44268 
 
 ■330 
 
 •36215 
 
 •11265 
 
 -•11923 
 
 - -09127 
 
 + -06203 
 
 + -07779 
 
 •43991 
 
 ■331 
 
 •36259 
 
 •11208 
 
 -•11974 
 
 - -09088 
 
 + -06255 
 
 + -07753 
 
 •43715 
 
 •332 
 
 •36302 
 
 •11151 
 
 -12024 
 
 - -09049 
 
 + -06308 
 
 + -07727 
 
 •43440 
 
 ■333 
 
 •36346 
 
 •11093 
 
 - -12073 
 
 - -09010 
 
 + -06360 
 
 + -07701 
 
 •43164 
 
 ■334 
 
 •36389 
 
 •11036 
 
 -12123 
 
 - -08971 
 
 + -06412 
 
 + -07674 
 
 ■42889 
 
 ■335 
 
 •36431 
 
 •10978 
 
 - -12172 
 
 - -08932 
 
 + 06463 
 
 + -07647 
 
 •42615 
 
 •336 
 
 •36474 
 
 •10920 
 
 - -12221 
 
 - -08892 
 
 + -06514 
 
 + -07620 
 
 •42340 
 
 ■337 
 
 •36516 
 
 •10862 
 
 - -12270 
 
 - -08852 
 
 + -06565 
 
 + -07592 
 
 •42066 
 
 ■338 
 
 •36558 
 
 •10804 
 
 -■12318 
 
 -•08811 
 
 + -06616 
 
 + •07564 
 
 •41793 
 
 •339 
 
 •36600 
 
 •10745 
 
 - -12366 
 
 - -08771 
 
 + -06667 
 
 + -07535 
 
 •41519 
 
 •31,0 
 
 •36641 
 
 •10687 
 
 - -12414 
 
 - -08730 
 
 + -06717 
 
 + -07507 
 
 •41246 
 
 ■31,1 
 
 •36682 
 
 •10628 
 
 - -12461 
 
 - -08689 
 
 + -06767 
 
 + -07477 
 
 •40974 
 
 ■31$ 
 
 •36723 
 
 ■10569 
 
 - -12509 
 
 - -08647 
 
 + -06817 
 
 + -07448 
 
 ■40701 
 
 ■343 
 
 •36764 
 
 •10510 
 
 -•12555 
 
 - -08606 
 
 + -06867 
 
 + -07418 
 
 •40429 
 
 8U 
 
 •36804 
 
 •10451 
 
 - -12602 
 
 - -08564 
 
 + -06916 
 
 + -07388 
 
 •40157 
 
 ■345 
 
 •36844 
 
 •10391 
 
 - -12649 
 
 - -08522 
 
 + -06965 
 
 + -07358 
 
 •39886 
 
 ■346 
 
 •36884 
 
 •10332 
 
 - -12695 
 
 - -08479 
 
 + -07014 
 
 + •07327 
 
 •39614 
 
 ■34.7 
 
 •36923 
 
 •10272 
 
 - -12741 
 
 - -08437 
 
 + -07062 
 
 + -07296 
 
 •39343 
 
 ■348 
 
 •36962 
 
 •10212 
 
 - -12786 
 
 - -08394 
 
 + •07110 
 
 + -07264 
 
 •39073 
 
 •S49 
 
 •37001 
 
 •10152 
 
 - -12831 
 
 - 08351 
 
 + •07158 
 
 + -07232 
 
 •38802 
 
 ■350 
 
 •37010 
 
 •10092 
 
 - -12876 
 
 - -08307 
 
 + -07206 
 
 + -07200 
 
 •38532 
 
Tables of the Tetrachorie Functions 
 TABLE XXIX.— (continued). 
 
 40 
 
 *<!-•) 
 
 n 
 
 T2 
 
 T3 
 
 U 
 
 ^6 
 
 U 
 
 h 
 
 ■351 
 
 •37078 
 
 •10032 
 
 - -12921 
 
 - -08264 
 
 + -07254 
 
 + -07168 
 
 •38262 
 
 ■352 
 
 •37116 
 
 •09971 
 
 - -12966 
 
 - -08220 
 
 + -07301 
 
 + -07135 
 
 •37993 
 
 ■353 
 
 •37154 
 
 •09911 
 
 -•13010 
 
 - -08176 
 
 + -07348 
 
 + -07102 
 
 •37723 
 
 ■354 
 
 •37192 
 
 •09850 
 
 - -13054 
 
 - -08131 
 
 + -07395 
 
 + -07069 
 
 •37454 
 
 ■355 
 
 •37229 
 
 •09789 
 
 - -13097 
 
 - -08087 
 
 + -07441 
 
 + -07035 
 
 •37186 
 
 ■356 
 
 •37266 
 
 •09728 
 
 - -13140 
 
 - 08042 
 
 + -07487 
 
 + -07002 
 
 •36917 
 
 ■357 
 
 •37303 
 
 •09667 
 
 - -13183 
 
 - -07997 
 
 + -07533 
 
 + -06967 
 
 •36649 
 
 ■358 
 
 •37340 
 
 •09606 
 
 - -13226 
 
 - -07952 
 
 + -07579 
 
 + -06933 
 
 •36381 
 
 ■359 
 
 •37376 
 
 •09544 
 
 - -13269 
 
 - -07906 
 
 + -07624 
 
 + -06898 
 
 •36113 
 
 ■360 
 
 ■37412 
 
 •09483 
 
 - -13311 
 
 - -07861 
 
 + -07669 
 
 + -06863 
 
 •35846 
 
 ■361 
 
 ■37447 
 
 •09421 
 
 -13353 
 
 - 07815 
 
 + -07714 
 
 + -06827 
 
 •35579 
 
 ■362 
 
 •37483 
 
 •09359 
 
 - -13394 
 
 - -07768 
 
 + -07758 
 
 + -06792 
 
 •35312 
 
 ■363 
 
 •37518 
 
 •09297 
 
 - -13436 
 
 - -07722 
 
 + -07803 
 
 + 06756 
 
 •35045 
 
 ■364 
 
 •37553 
 
 •09235 
 
 - -13477 
 
 - -07675 
 
 + -07847 
 
 + -06719 
 
 •34779 
 
 •365 
 
 •37588 
 
 •09173 
 
 - -13517 
 
 - -07629 
 
 + -07890 
 
 + -06683 
 
 •34513 
 
 •366 
 
 •37622 
 
 •09111 
 
 - -13558 
 
 - -07582 
 
 + -07934 
 
 + -06646 
 
 ■34247 
 
 ■367 
 
 •37656 
 
 •09048 
 
 - -13598 
 
 - -07534 
 
 + -07977 
 
 + -06609 
 
 •33981 
 
 ■368 
 
 •37690 
 
 •08985 
 
 - -13638 
 
 - -07487 
 
 + -08020 
 
 + -06571 
 
 •33715 
 
 ■369 
 
 •37724 
 
 •08923 
 
 - -13677 
 
 - -07439 
 
 + -08062 
 
 + -06534 
 
 •33450 
 
 ■370 
 
 •37757 
 
 •08860 
 
 - -13717 
 
 - -07391 
 
 + -08105 
 
 + -06496 
 
 •33185 
 
 ■371 
 
 •37790 
 
 •08797 
 
 - -13756 
 
 - -07343 
 
 + -08147 
 
 + -06458 
 
 •32921 
 
 ■S72 
 
 •37823 
 
 •08734 
 
 - -13794 
 
 - -07295 
 
 + -08188 
 
 + -06419 
 
 •32656 
 
 ■373 
 
 •37855 
 
 •08671 
 
 - -13833 
 
 - -07246 
 
 + -08230 
 
 + -06380 
 
 •32392 
 
 ■374 
 
 •37888 
 
 •08607 
 
 - -13871 
 
 - -07198 
 
 + -08271 
 
 + -06341 
 
 •32128 
 
 ■375 
 
 •37920 
 
 •08544 
 
 - -13909 
 
 - -07149 
 
 + 08312 
 
 + -06302 
 
 •31864 
 
 ■376 
 
 ■37951 
 
 •08480 
 
 - 13946 
 
 - -07100 
 
 + -08352 
 
 + -06262 
 
 •31600 
 
 ■377 
 
 •37983 
 
 ■08416 
 
 - -13984 
 
 - -07050 
 
 + 08392 
 
 + -06222 
 
 •31337 
 
 ■378 
 
 •38014 
 
 •08353 
 
 - -14021 
 
 - -07001 
 
 + -08432 
 
 + -06182 
 
 •31074 
 
 ■379 
 
 •38045 
 
 •08289 
 
 - -14057 
 
 - -06951 
 
 + 08472 
 
 + -06142 
 
 •30811 
 
 ■380 
 
 •38076 
 
 •08225 
 
 - -14094 
 
 - 06901 
 
 + -08512 
 
 + 06101 
 
 •30548 
 
 ■381 
 
 •38106 
 
 •08160 
 
 - -14130 
 
 - -06851 
 
 + -08551 
 
 + -06061 
 
 •30286 
 
 ■382 
 
 •38136 
 
 •08096 
 
 - -14166 
 
 - -06801 
 
 + -08589 
 
 + -06019 
 
 •30023 
 
 ■383 
 
 •38166 
 
 •08032 
 
 - -14201 
 
 - -06750 
 
 + -08628 
 
 + -05978 
 
 •29761 
 
 ■384 
 
 •38196 
 
 •07967 
 
 - -14236 
 
 - -06700 
 
 + -08666 
 
 + -05936 
 
 •29499 
 
 ■385 
 
 •38225 
 
 •07903 
 
 - -14271 
 
 - -06649 
 
 + -08704 
 
 + -05895 
 
 •29237 
 
 ■386 
 
 •38254 
 
 •07838 
 
 - -14306 
 
 - -06598 
 
 + -08742 
 
 + -05853 
 
 •28976 
 
 ■387 
 
 •38283 
 
 •07773 
 
 - -14340 
 
 - -06547 
 
 + -08779 
 
 + -05810 
 
 •28715 
 
 ■388 
 
 •38312 
 
 •07708 
 
 - -14374 
 
 - -06495 
 
 + -08816 
 
 + 05766 
 
 •28454 
 
 ■389 
 
 •38340 
 
 •07643 
 
 - -14408 
 
 - -06444 
 
 + -08853 
 
 + -05725 
 
 •28193 
 
 ■390 
 
 •38368 
 
 •07578 
 
 - -14442 
 
 - -06392 
 
 + -08889 
 
 + -05682 
 
 •27932 
 
 •391 
 
 •38396 
 
 •07513 
 
 - -14475 
 
 - -06340 
 
 + -08925 
 
 + 05638 
 
 •27671 
 
 ■392 
 
 •38423 
 
 •07447 
 
 - -14508 
 
 - -06288 
 
 + -08961 
 
 + -05595 
 
 •27411 
 
 •S9S 
 
 •38451 
 
 •07382 
 
 - -14540 
 
 - -06236 
 
 + -08997 
 
 + -05551 
 
 •27151 
 
 ■394 
 
 •38478 
 
 •07316 
 
 - -14573 
 
 - -06183 
 
 + -09032 
 
 + -05507 
 
 •26891 
 
 ■395 
 
 •38504 
 
 •07251 
 
 - -14604 
 
 - -06131 
 
 + -09067 
 
 + -05463 
 
 •26631 
 
 ■396 
 
 •38531 
 
 •07185 
 
 - -14636 
 
 - -06078 
 
 + -09101 
 
 + -05419 
 
 •26371 
 
 ■397 
 
 •38557 
 
 ■07119 
 
 - -14668 
 
 - -06025 
 
 + -09136 
 
 + -05374 
 
 •26112 
 
 ■398 
 
 •38583 
 
 •07053 
 
 - -14699 
 
 - -05972 
 
 + -09170 
 
 + -05329 
 
 ■25853 
 
 ■399 
 
 •38609 
 
 •06987 
 
 - -14730 
 
 - -05919 
 
 + -09203 
 
 + -05284 
 
 •25694 
 
 ■400 
 
 •38634 
 
 •06921 
 
 - -14760 
 
 - -05866 
 
 + -09237 
 
 + -05239 
 
 •25335 
 
50 Tables for Statisticians and Biometricians 
 
 TABLE XXIX. Tetrachoric Functions for Fourfold Correlation Tables. 
 
 i(l-a) 
 
 n 
 
 Tt 
 
 ^3 
 
 U 
 
 n 
 
 r» 
 
 h 
 
 ■4OI 
 
 ■402 
 
 •38659 
 
 •06855 
 
 - -14790 
 
 - -05812 
 
 + -09270 
 
 + -05193 
 
 •25076 
 
 •38684 
 
 •06789 
 
 -•14820 
 
 - -05758 
 
 + -09303 
 
 + -05148 
 
 •24817 
 
 ■403 
 
 ■38709 
 
 •06722 
 
 - -14850 
 
 - -05705 
 
 + -09335 
 
 + -05102 
 
 •24559 
 
 ■404 
 
 •38734 
 
 •06656 
 
 -•14879 
 
 - -05651 
 
 + -09367 
 
 + -05056 
 
 •24301 
 
 ■405 
 
 •38758 
 
 •06589 
 
 - -14908 
 
 - -05596 
 
 + -09399 
 
 + -05010 
 
 •24043 
 
 ■406 
 
 •38782 
 
 ■06522 
 
 - -14937 
 
 - -05542 
 
 + -09430 
 
 + -04963 
 
 •23785 
 
 ■407 
 
 •38805 
 
 •06456 
 
 - -14965 
 
 - -05488 
 
 + -09462 
 
 + -04916 
 
 •23527 
 
 ■40S 
 
 •38829 
 
 •06389 
 
 - -14993 
 
 - -05433 
 
 + -09493 
 
 + -04869 
 
 •23269 
 
 ■409 
 
 •38852 
 
 •06322 
 
 - -15021 
 
 - -05378 
 
 + -09523 
 
 + -04822 
 
 •23012 
 
 ■410 
 
 •38875 
 
 •06255 
 
 - -15049 
 
 - -05323 
 
 + -09553 
 
 + -04775 
 
 •22754 
 
 ■411 
 
 •38897 
 
 •06188 
 
 - -15076 
 
 - -05268 
 
 + -09583 
 
 + -04728 
 
 ■22497 
 
 •412 
 
 •38920 
 
 •06121 
 
 - -15103 
 
 - -05213 
 
 + •09613 
 
 + -04680 
 
 •22240 
 
 ■413 
 
 •38942 
 
 •06053 
 
 -•15130 
 
 - -05158 
 
 + -09642 
 
 + -04632 
 
 •21983 
 
 ■414 
 
 •38964 
 
 •05986 
 
 -■15156 
 
 - -05102 
 
 + -09671 
 
 + -04584 
 
 •21727 
 
 ■415 
 
 •38985 
 
 05919 
 
 -•15182 
 
 - -05047 
 
 + -09700 
 
 + -04536 
 
 •21470 
 
 ■4if> 
 
 •39007 
 
 •05851 
 
 - -15208 
 
 - -04991 
 
 + -09728 
 
 + -04488 
 
 •21214 
 
 ■417 
 
 •39028 
 
 •05784 
 
 - -15233 
 
 - -04935 
 
 + -09756 
 
 + -04439 
 
 •20957 
 
 ■418 
 
 •39049 
 
 •05716 
 
 - -15258 
 
 - -04879 
 
 + -09784 
 
 + -04390 
 
 •20701 
 
 ■419 
 
 •39069 
 
 •05648 
 
 - -15283 
 
 - -04823 
 
 + -09811 
 
 + -04341 
 
 ■20445 
 
 •420 
 
 •39089 
 
 •05580 
 
 - -15308 
 
 - -04767 
 
 + -09838 
 
 + -04292 
 
 •20189 
 
 •421 
 
 •39109 
 
 •05513 
 
 - -15332 
 
 - -04711 
 
 + -09865 
 
 + -04243 
 
 •19934 
 
 •422 
 
 •39129 
 
 •05445 
 
 - -15356 
 
 - -04654 
 
 + -09891 
 
 + -04194 
 
 •19678 
 
 ■423 
 
 •39149 
 
 •05377 
 
 - -15380 
 
 - -04598 
 
 + -09918 
 
 + -04144 
 
 •19422 
 
 ■424 
 
 •39168 
 
 •05309 
 
 - -15403 
 
 - -04541 
 
 + -09943 
 
 + -04094 
 
 •19167 
 
 •425 
 
 •39187 
 
 •05240 
 
 - -15426 
 
 - -04484 
 
 + -09969 
 
 + -04044 
 
 •18912 
 
 •426 
 
 •39206 
 
 •05172 
 
 - -15449 
 
 - -04427 
 
 + -09994 
 
 + -03994 
 
 •18657 
 
 •427 
 
 •39224 
 
 •05104 
 
 - -15471 
 
 - -04370 
 
 + -10019 
 
 + 03944 
 
 •18402 
 
 ■428 
 
 •39243 
 
 •05036 
 
 - -15493 
 
 - -04313 
 
 + -10043 
 
 + -03894 
 
 •18147 
 
 •429 
 
 ■39261 
 
 •04967 
 
 -•15515 
 
 - -04256 
 
 + -10067 
 
 + -03843 
 
 •17892 
 
 ■430 
 
 •39279 
 
 •04899 
 
 - -15537 
 
 - -04198 
 
 + -10091 
 
 + -03793 
 
 •17637 
 
 •431 
 
 •39296 
 
 •04830 
 
 - -15558 
 
 - -04141 
 
 + -10115 
 
 + -03742 
 
 •17383 
 
 ■432 
 
 •39313 
 
 •04761 
 
 - -15579 
 
 - -04083 
 
 + -10138 
 
 + -03691 
 
 •17128 
 
 ■433 
 
 •39330 
 
 •04693 
 
 - -15599 
 
 - -04026 
 
 + -10161 
 
 + -03640 
 
 •16874 
 
 •434 
 
 •39347 
 
 •04624 
 
 -•15620 
 
 - -03968 
 
 + -10183 
 
 + -03589 
 
 •16620 
 
 ■435 
 
 •39364 
 
 •045. r ,f> 
 
 -■15640 
 
 - -03910 
 
 + -10205 
 
 + -03537 
 
 •16366 
 
 ■436 
 
 •39380 
 
 •04486 
 
 - -15659 
 
 - -03852 
 
 + -10227 
 
 + -03486 
 
 •16112 
 
 ■437 
 
 •39396 
 
 ■04418 
 
 - 15679 
 
 - -03794 
 
 + -10249 
 
 + -03434 
 
 •15858 
 
 ■438 
 
 •39411 
 
 •04349 
 
 - -15698 
 
 - -03735 
 
 + -10270 
 
 + -03382 
 
 •15604 
 
 •439 
 
 •39427 
 
 •04280 
 
 - -15717 
 
 - '03677 
 
 + •10291 
 
 + -03330 
 
 •15351 
 
 ■440 
 
 •39442 
 
 •04211 
 
 - -15735 
 
 - -03619 
 
 + •10311 
 
 + -03278 
 
 •15097 
 
 ■441 
 
 •39457 
 
 •04141 
 
 - -15753 
 
 - '03560 
 
 + -10331 
 
 + -03226 
 
 •14843 
 
 •442 
 
 •39472 
 
 •04072 
 
 - -15771 
 
 - -03502 
 
 + -10351 
 
 + -03174 
 
 •14590 
 
 ■44s 
 
 •39486 
 
 •04003 
 
 - -15789 
 
 - -03443 
 
 + -10371 
 
 + •03121 
 
 •14337 
 
 ■444 
 
 •39601 
 
 03934 
 
 - -15806 
 
 - -03384 
 
 + -10390 
 
 + -03069 
 
 •14084 
 
 ■446 
 
 •39514 
 
 ■03864 
 
 - -15823 
 
 - -03325 
 
 + -10409 
 
 + -03016 
 
 •13830 
 
 ■446 
 
 •39528 
 
 •03795 
 
 - -15840 
 
 - -03266 
 
 + -10427 
 
 + -02963 
 
 •13577 
 
 ■447 
 
 •39542 
 
 •03726 
 
 - -15856 
 
 - -03207 
 
 + -10446 
 
 + -02910 
 
 •13324 
 
 ■448 
 
 •39555 
 
 03656 
 
 - -15872 
 
 - -03148 
 
 + -10463 
 
 + -02858 
 
 •13072 
 
 ■449 
 
 •39568 
 
 •03587 
 
 - -15888 
 
 - -03089 
 
 + -10481 
 
 + -02804 
 
 12819 
 
 ■450 
 
 •39580 
 
 •03517 
 
 - -15904 
 
 - -03030 
 
 + -10498 
 
 + -02751 
 
 •12566 
 
Tables of the Tetrachoric Functions 
 TABLE XXIX.— {continued). 
 
 51 
 
 HI -«) 
 
 U 
 
 Ti 
 
 n 
 
 1* 
 
 H 
 
 n 
 
 h 
 
 ■451 
 
 •39593 
 
 •03447 
 
 - -15919 
 
 - -02970 
 
 + •10515 
 
 + -02698 
 
 •12314 
 
 ■1,5% 
 
 •39605 
 
 •03378 
 
 - -15934 
 
 -•02911 
 
 + -10532 
 
 + -02644 
 
 •12061 
 
 ■45S 
 
 •39617 
 
 •03308 
 
 - -15948 
 
 - -02851 
 
 + -10548 
 
 + -02591 
 
 •11809 
 
 ■454 
 
 •39629 
 
 •03238 
 
 - -15962 
 
 - -02792 
 
 + -10564 
 
 + 02537 
 
 •11556 
 
 •455 
 
 •39640 
 
 •03168 
 
 - -15976 
 
 - -02732 
 
 + -10579 
 
 + -02484 
 
 •11304 
 
 ■456 
 
 •39651 
 
 •03099 
 
 - -15990 
 
 - -02673 
 
 +10594 
 
 + -02430 
 
 •11052 
 
 ■457 
 
 •39662 
 
 ■03029 
 
 - -16003 
 
 - -02613 
 
 + -10609 
 
 + -02376 
 
 •10799 
 
 ■458 
 
 •39673 
 
 02959 
 
 - -16016 
 
 - -02553 
 
 + -10624 
 
 + -02322 
 
 •10547 
 
 ■459 
 
 •39683 
 
 02889 
 
 - -16029 
 
 - -02493 
 
 + -10638 
 
 + -02268 
 
 •10295 
 
 ■46O 
 
 •39694 
 
 02819 
 
 - -16041 
 
 - 02433 
 
 + -10652 
 
 + -02214 
 
 •10043 
 
 ■461 
 
 •39703 
 
 •02749 
 
 - -16053 
 
 - 02373 
 
 + -10665 
 
 + -02159 
 
 •09791 
 
 '462 
 
 ■39713 
 
 02679 
 
 - -16065 
 
 - -02313 
 
 + -10678 
 
 + -02105 
 
 •09540 
 
 ■46S 
 
 •39723 
 
 02609 
 
 - -16077 
 
 - -02253 
 
 + •10691 
 
 + '02051 
 
 •09288 
 
 ■404 
 
 •39732 
 
 02539 
 
 - -16088 
 
 - -02193 
 
 + •10704 
 
 + -01996 
 
 •09036 
 
 465 
 
 •39741 
 
 •02469 
 
 - -16099 
 
 - -02132 
 
 + •10716 
 
 + -01941 
 
 •08784 
 
 ■466 
 
 •39749 
 
 02398 
 
 - -16109 
 
 - -02072 
 
 + •10727 
 
 + -01887 
 
 08533 
 
 ■467 
 
 •39758 
 
 02328 
 
 - -16120 
 
 - -02012 
 
 + •10739 
 
 + -01832 
 
 •08281 
 
 ■468 
 
 •39766 
 
 •02258 
 
 -16130 
 
 - -01951 
 
 + -10750 
 
 + -01777 
 
 •08030 
 
 ■469 
 
 •39774 
 
 ■02188 
 
 -16139 
 
 - -01891 
 
 + -10761 
 
 + 01722 
 
 •07778 
 
 ■470 
 
 ■39781 
 
 02117 
 
 -•16149 
 
 - -01830 
 
 + •10771 
 
 + -01668 
 
 •07527 
 
 ■471 
 
 •39789 
 
 •02047 
 
 -•16158 
 
 - -01770 
 
 + ■10781 
 
 + 01613 
 
 •07276 
 
 ■472 
 
 •39796 
 
 •01977 
 
 - -16166 
 
 - -01709 
 
 + •10791 
 
 + -01558 
 
 •07024 
 
 ■473 
 
 •39803 
 
 •01906 
 
 - -16175 
 
 - -01648 
 
 + -10801 
 
 + -01502 
 
 •06773 
 
 •&4 
 
 ■39809 
 
 ■01836 
 
 - 16183 
 
 - -01588 
 
 + -10810 
 
 + 01447 
 
 •06522 
 
 ■475 
 
 •39816 
 
 •01765 
 
 - -16191 
 
 - -01527 
 
 + ■10818 
 
 + 01392 
 
 •06271 
 
 ■476 
 
 •39822 
 
 •01695 
 
 - -16198 
 
 - -01460 
 
 + -10827 
 
 + -01337 
 
 •06020 
 
 •^77 
 
 •39828 
 
 •01625 
 
 - -16206 
 
 - -01405 
 
 + -10835 
 
 + -01281 
 
 •05768 
 
 ■478 
 
 •39834 
 
 •01554 
 
 - -16212 
 
 - 01344 
 
 + -10842 
 
 + -01226 
 
 •05517 
 
 ■479 
 
 •39839 
 
 •01484 
 
 - -16219 
 
 - -01284 
 
 + -10850 
 
 +01171 
 
 •05266 
 
 ■48O 
 
 •39844 
 
 •01413 
 
 - -16225 
 
 - -01223 
 
 + -10857 
 
 + -01115 
 
 •05015 
 
 ■481 
 
 •39849 
 
 01342 
 
 - -16231 
 
 - -01162 
 
 + -10864 
 
 + -01060 
 
 •04764 
 
 ■482 
 
 •39854 
 
 •01272 
 
 - -16237 
 
 - 01101 
 
 + -10870 
 
 + -01004 
 
 04513 
 
 ■48S 
 
 •39858 
 
 01201 
 
 - -16242 
 
 - -01040 
 
 + -10876 
 
 + -00949 
 
 •04263 
 
 ■m 
 
 ■39862 
 
 01131 
 
 - -16247 
 
 - -00979 
 
 + -10882 
 
 + -00893 
 
 •04012 
 
 •485 
 
 •39866 
 
 •01060 
 
 - -16252 
 
 - -00918 
 
 + -10887 
 
 + -00837 
 
 •03761 
 
 ■486 
 
 •39870 
 
 •00990 
 
 - -16257 
 
 - -00857 
 
 + -10892 
 
 + -00782 
 
 •03510 
 
 ■487 
 
 •39873 
 
 •00919 
 
 - -16261 
 
 - 00796 
 
 + -10896 
 
 + -00726 
 
 •03259 
 
 ■488 
 
 •39876 
 
 •00848 
 
 - -16265 
 
 - -00734 
 
 + 10901 
 
 + -00670 
 
 •03008 
 
 ■489 
 
 •39879 
 
 •00778 
 
 - -16268 
 
 - 00673 
 
 + -10905 
 
 + -00614 
 
 •02758 
 
 ■490 
 
 ■39882 
 
 •00707 
 
 - -16271 
 
 - -00612 
 
 + 10908 
 
 + -00559 
 
 •02507 
 
 ■491 
 
 ■39884 
 
 ■00636 
 
 - -16274 
 
 - -00551 
 
 + -10912 
 
 + -00503 
 
 •02256 
 
 ■492 
 
 •39886 
 
 ■00566 
 
 - -16277 
 
 - -00490 
 
 + -10914 
 
 + -00447 
 
 •02005 
 
 ■493 
 
 •39888 
 
 •00495 
 
 - -16279 
 
 - -00429 
 
 + -10917 
 
 + -00391 
 
 •01755 
 
 ■494 
 
 •39890 
 
 •00424 
 
 - -16281 
 
 - -00367 
 
 + 10919 
 
 + -00335 
 
 •01504 
 
 ■495 
 
 •39891 
 
 00354 
 
 - -16283 
 
 - -00306 
 
 + -10921 
 
 + -00279 
 
 01253 
 
 ■496 
 
 •39892 
 
 •00283 
 
 - -16284 
 
 - -00245 
 
 + -10923 
 
 + -00224 
 
 •01003 
 
 ■497 
 
 •39893 
 
 •00212 
 
 - -16285 
 
 - -00184 
 
 + •10924 
 
 + -00168 
 
 •00752 
 
 ■498 
 
 •39894 
 
 •00141 
 
 - -16286 
 
 - 00122 
 
 + •10925 
 
 + -00112 
 
 •00501 
 
 ■499 
 
 ■39894 
 
 •00071 
 
 - -16287 
 
 - -00061 
 
 + -10925 
 
 + -00056 
 
 •00251 
 
 ■500 
 
 •39894 
 
 •00000 
 
 - -16287 
 
 •00000 
 
 + -10925 
 
 •00000 
 
 •00000 
 
 7—2 
 
52 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XXX. Supplementary Tables for determining High 
 r = -80. 
 
 A= 
 
 
 
 •1 
 
 '2 
 
 •3 
 
 ■4 
 
 ■5 
 
 •6 
 
 •7 
 
 ■8 
 
 ■9 
 
 1-0 
 
 1-1 
 
 1-2 
 
 k = 0-0 
 
 •3976 
 
 •3766 
 
 •3538 
 
 •3294 
 
 •3039 
 
 •2778 
 
 •2515 
 
 •2254 
 
 ■2001 
 
 •1759 
 
 •1531 
 
 •1320 
 
 •1127 
 
 o-i 
 
 •3766 
 
 •3583 
 
 •3380 
 
 •3162 
 
 •2930 
 
 •2689 
 
 •2445 
 
 •2200 
 
 •1960 
 
 •1728 
 
 •1509 
 
 •1304 
 
 •1116 
 
 0-2 
 
 •3538 
 
 •3380 
 
 •3204 
 
 •3011 
 
 •2804 
 
 •2586 
 
 ■2361 
 
 •2134 
 
 •1909 
 
 •1690 
 
 •1481 
 
 •1284 
 
 •1102 
 
 OS 
 
 •3294 
 
 •3162 
 
 •3011 
 
 •2843 
 
 •2661 
 
 ■2466 
 
 •2263 
 
 •2056 
 
 •1848 
 
 •1643 
 
 •1446 
 
 •1258 
 
 •1083 
 
 0-4 
 
 ■3039 
 
 •2930 
 
 •2804 
 
 •2661 
 
 •2503 
 
 •2332 
 
 •2152 
 
 •1965 
 
 •1775 
 
 •1587 
 
 •1402 
 
 •1226 
 
 •1060 
 
 0-5 
 
 •2778 
 
 •2689 
 
 •2586 
 
 •2466 
 
 •2332 
 
 •2186 
 
 •2028 
 
 •1862 
 
 •1692 
 
 •1520 
 
 •1351 
 
 •1187 
 
 •1031 
 
 0-6 
 
 •2515 
 
 •2445 
 
 •2361 
 
 •2263 
 
 •2152 
 
 •2028 
 
 •1893 
 
 •1748 
 
 •1598 
 
 •1444 
 
 •1291 
 
 •1140 
 
 •0995 
 
 0-7 
 
 •2254 
 
 •2200 
 
 •2134 
 
 •2056 
 
 •1965 
 
 •1862 
 
 •1748 
 
 •1625 
 
 •1494 
 
 •1359 
 
 •1222 
 
 •1086 
 
 •0954 
 
 0-8 
 
 •2001 
 
 •I960 
 
 •1909 
 
 •1848 
 
 ■1775 
 
 •1692 
 
 •1598 
 
 •1494 
 
 •1383 
 
 •1266 
 
 •1146 
 
 •1025 
 
 •0906 
 
 09 
 
 •1759 
 
 •1728 
 
 •1690 
 
 •1643 
 
 •1587 
 
 •1520 
 
 •1444 
 
 •1359 
 
 •1266 
 
 •1167 
 
 •1064 
 
 •0958 
 
 •0852 
 
 1-0 
 
 •1531 
 
 •1509 
 
 •1481 
 
 •1446 
 
 •1402 
 
 •1351 
 
 •1291 
 
 •1222 
 
 •1146 
 
 •1064 
 
 •0976 
 
 •0886 
 
 •0794 
 
 1-1 
 
 •1320 
 
 •1304 
 
 •1284 
 
 •1258 
 
 •1226 
 
 •1187 
 
 •1140 
 
 •1086 
 
 •1025 
 
 •0958 
 
 •0886 
 
 •0809 
 
 •0731 
 
 IS 
 
 •1127 
 
 •1116 
 
 •1102 
 
 •1083 
 
 •1060 
 
 •1031 
 
 •0995 
 
 •0954 
 
 •0906 
 
 •0852 
 
 •0794 
 
 •0731 
 
 ■0665 
 
 IS 
 
 •0953 
 
 •0946 
 
 •0936 
 
 •0923 
 
 •0906 
 
 •0885 
 
 •0859 
 
 •0828 
 
 •0791 
 
 ■0749 
 
 •0702 
 
 0652 
 
 ■0597 
 
 1-4 
 
 •0798 
 
 •0793 
 
 •0787 
 
 •0778 
 
 •0766 
 
 ■0751 
 
 •0733 
 
 •0710 
 
 •0682 
 
 ■0650 
 
 •0614 
 
 •0574 
 
 •0530 
 
 1-5 
 
 •0662 
 
 •0659 
 
 •0655 
 
 •0649 
 
 •0641 
 
 •0631 
 
 •0618 
 
 •0601 
 
 ■0581 
 
 •0557 
 
 ■0529 
 
 •0498 
 
 •0464 
 
 V6 
 
 •0545 
 
 •0543 
 
 •0540 
 
 •0536 
 
 •0531 
 
 •0524 
 
 •0515 
 
 •0503 
 
 •0489 
 
 •0471 
 
 •0451 
 
 •0427 
 
 •0401 
 
 1-7 
 
 •0444 
 
 ■0443 
 
 •0441 
 
 •0438 
 
 •0435 
 
 •0430 
 
 ■0424 
 
 ■0416 
 
 •0406 
 
 •0394 
 
 •0379 
 
 •0362 
 
 •0342 
 
 1-8 
 
 •0358 
 
 ■0357 
 
 •0357 
 
 •0355 
 
 •0353 
 
 •0350 
 
 •0346 
 
 •0341 
 
 •0334 
 
 •0325 
 
 •0315 
 
 •0302 
 
 •0287 
 
 1-9 
 
 •0287 
 
 •0286 
 
 •0286 
 
 •0285 
 
 ■0283 
 
 •0281 
 
 •0279 
 
 •0275 
 
 •0271 
 
 •0265 
 
 •0258 
 
 •0249 
 
 •0238 
 
 2-0 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0226 
 
 •0225 
 
 •0224 
 
 •0223 
 
 •0220 
 
 •0217 
 
 •0213 
 
 •0209 
 
 •0202 
 
 •0195 
 
 2-1 
 
 •0178 
 
 •0178 
 
 •0178 
 
 •0178 
 
 •0177 
 
 •0177 
 
 •0176 
 
 •0174 
 
 •0172 
 
 •0170 
 
 •0167 
 
 •0163 
 
 •0158 
 
 2-2 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0138 
 
 •0138 
 
 •0137 
 
 •0137 
 
 •0135 
 
 •0134 
 
 •0132 
 
 •0129 
 
 0126 
 
 2-8 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0106 
 
 •0106 
 
 •0105 
 
 •0104 
 
 •0103 
 
 •0101 
 
 •0099 
 
 2-4 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0081 
 
 •0081 
 
 •0080 
 
 •0079 
 
 •0078 
 
 •0077 
 
 2S 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0061 
 
 •0061 
 
 •0061 
 
 •0060 
 
 •0059 
 
 2-6 
 
 ■0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0046 
 
 •0046 
 
 •0046 
 
 •0046 
 
 •0046 
 
 •0045 
 
 •0045 
 
 
 
 
 
 
 
 r = ■ 
 
 85. 
 
 
 
 
 
 
 
 h = 
 
 
 
 •1 
 
 •2 
 
 ■3 
 
 ■4 
 
 ■5 
 
 •6 
 
 •7 
 
 ■8 
 
 ■9 
 
 1-0 
 
 1-1 
 
 1'8 
 
 l=o-o 
 
 •4117 
 
 •3905 
 
 •3670 
 
 •3417 
 
 •3149 
 
 •2873 
 
 •2595 
 
 •2319 
 
 •2052 
 
 •1798 
 
 •1560 
 
 •1341 
 
 •1141 
 
 o-i 
 
 •3905 
 
 •3723 
 
 ■3518 
 
 •3292 
 
 •3050 
 
 •2796 
 
 •2537 
 
 ■2277 
 
 •2022 
 
 ■1777 
 
 •1546 
 
 •1332 
 
 •1136 
 
 0-2 
 
 •3670 
 
 •3518 
 
 •3342 
 
 •3145 
 
 •2930 
 
 •2702 
 
 •2464 
 
 •2222 
 
 •1983 
 
 •1749 
 
 •1527 
 
 •1319 
 
 •1127 
 
 OS 
 
 •3417 
 
 •3292 
 
 •3145 
 
 •2978 
 
 •2791 
 
 •2588 
 
 •2374 
 
 •2154 
 
 •1931 
 
 •1712 
 
 •1501 
 
 •1301 
 
 •1116 
 
 0-4 
 
 •3149 
 
 •3050 
 
 •2930 
 
 •2791 
 
 •2632 
 
 •2467 
 
 •2268 
 
 •2070 
 
 •1867 
 
 •1665 
 
 •1467 
 
 •1277 
 
 •1099 
 
 0-5 
 
 •2873 
 
 •2796 
 
 •2702 
 
 •2588 
 
 •2457 
 
 •2309 
 
 •2146 
 
 •1972 
 
 •1790 
 
 •1606 
 
 •1423 
 
 •1246 
 
 ■1078 
 
 0-6 
 
 •2595 
 
 •2537 
 
 •2464 
 
 •2374 
 
 •2268 
 
 •2146 
 
 •2008 
 
 •1859 
 
 •1700 
 
 •1535 
 
 •1370 
 
 •1206 
 
 •1049 
 
 0-7 
 
 •2319 
 
 •2277 
 
 •2222 
 
 •2154 
 
 •2070 
 
 •1972 
 
 •1859 
 
 •1733 
 
 •1597 
 
 •1453 
 
 •1306 
 
 •1158 
 
 •1014 
 
 0-8 
 
 •2052 
 
 •2022 
 
 •1983 
 
 •1931 
 
 •1867 
 
 •1790 
 
 •1700 
 
 •1597 
 
 •1483 
 
 •1360 
 
 •1232 
 
 •1101 
 
 •0971 
 
 0-9 
 
 •1798 
 
 •1777 
 
 •1749 
 
 •1712 
 
 •1665 
 
 •1606 
 
 •1535 
 
 •1453 
 
 •1360 
 
 •1258 
 
 •1149 
 
 •1035 
 
 •0920 
 
 1-0 
 
 •1560 
 
 •1546 
 
 •1527 
 
 •1501 
 
 •1467 
 
 •1423 
 
 •1370 
 
 •1306 
 
 •1232 
 
 •1149 
 
 •1058 
 
 •0962 
 
 •0862 
 
 1-1 
 
 •1341 
 
 •1332 
 
 •1319 
 
 •1301 
 
 •1277 
 
 •1246 
 
 •1206 
 
 •1158 
 
 •1101 
 
 •1035 
 
 •0962 
 
 •0882 
 
 •0798 
 
 1-2 
 
 •1141 
 
 •1136 
 
 •1127 
 
 •1116 
 
 •1099 
 
 •1078 
 
 •1049 
 
 •1014 
 
 •0971 
 
 •0920 
 
 •0862 
 
 •0798 
 
 •0729 
 
 l'S 
 
 •0963 
 
 •0959 
 
 •0954 
 
 •0947 
 
 •0936 
 
 •0921 
 
 •0901 
 
 •0876 
 
 •0845 
 
 •0807 
 
 •0763 
 
 •0712 
 
 •0656 
 
 1-4 
 
 •0805 
 
 •0803 
 
 •0800 
 
 ■0795 
 
 •0788 
 
 •0778 
 
 •0765 
 
 ■0748 
 
 •0725 
 
 •0698 
 
 •0665 
 
 •0626 
 
 •0583 
 
 1-5 
 
 •0666 
 
 •0665 
 
 •0664 
 
 •0661 
 
 •0656 
 
 •0650 
 
 •0642 
 
 •0630 
 
 •0615 
 
 •0595 
 
 •0571 
 
 •0543 
 
 •0510 
 
 1-6 
 
 •0547 
 
 •0547 
 
 •0546 
 
 •0544 
 
 •0541 
 
 •0538 
 
 •0532 
 
 •0525 
 
 •0514 
 
 •0501 
 
 •0484 
 
 •0464 
 
 •0439 
 
 1-7 
 
 •0445 
 
 •0445 
 
 •0444 
 
 •0443 
 
 •0442 
 
 •0440 
 
 •0436 
 
 •0432 
 
 •0425 
 
 •0416 
 
 ■0405 
 
 •0390 
 
 •0373 
 
 1-8 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0358 
 
 •0357 
 
 •0356 
 
 •0354 
 
 •0351 
 
 •0347 
 
 •0341 
 
 •0334 
 
 •0324 
 
 •0312 
 
 1-9 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0286 
 
 •0285 
 
 •0284 
 
 •0283 
 
 •0280 
 
 •0276 
 
 •0272 
 
 •0265 
 
 •0257 
 
 2-0 
 
 ■0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0226 
 
 •0225 
 
 •0224 
 
 •0221 
 
 •0218 
 
 •0214 
 
 •0209 
 
 2-1 
 
 •0179 
 
 •0179 
 
 ■0179 
 
 •0178 
 
 ■0178 
 
 •0178 
 
 •0178 
 
 •0177 
 
 •0176 
 
 •0175 
 
 •0173 
 
 •0171 
 
 •0167 
 
 2-2 
 
 •0139 
 
 •0139 
 
 ■0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0138 
 
 •0138 
 
 •0137 
 
 •0136 
 
 •0135 
 
 •0133 
 
 2-3 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0106 
 
 •0106 
 
 •0105 
 
 •0104 
 
 2-4 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 ■0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0081 
 
 •0081 
 
 •0081 
 
 •0080 
 
 2-6 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0061 
 
 ■0061 
 
 2-6 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 ■0046 
 
 •0046 
 
 •0046 
 
 •0046 
 
Tables for High Fourfold Correlation 
 
 53 
 
 Correlations from Tetrachoric Groupings. 
 
 r=-80. 
 
 h= 
 
 IS 
 
 M 
 
 1-6 
 
 1-6 
 
 1-7 
 
 1-8 
 
 1-9 
 
 2-0 
 
 2-1 
 
 2-2 
 
 2-3 
 
 2-4 
 
 2S 
 
 2-6 
 
 k=00 
 
 ■0953 
 
 •0798 
 
 •0662 
 
 •0545 
 
 •0444 
 
 •0358 
 
 •0287 
 
 •0227 
 
 •0178 
 
 •0139 
 
 •0107 
 
 ■0082 
 
 •0062 
 
 •0047 
 
 o-i 
 
 •0946 
 
 •0793 
 
 •0659 
 
 •0543 
 
 •0443 
 
 •0357 
 
 •0286 
 
 •0227 
 
 •0178 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 ■0047 
 
 OS 
 
 •0936 
 
 •0787 
 
 •0655 
 
 •0540 
 
 •0441 
 
 •0357 
 
 •0288 
 
 •0227 
 
 •0178 
 
 •0139 
 
 ■0107 
 
 •0082 
 
 •0062 
 
 ■0047 
 
 OS 
 
 •0923 
 
 •0778 
 
 •0649 
 
 •0536 
 
 •0438 
 
 •0355 
 
 ■0285 
 
 •0226 
 
 •0178 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 0-b 
 
 •0906 
 
 •0766 
 
 •0641 
 
 •0531 
 
 ■0435 
 
 •0353 
 
 •0283 
 
 •0225 
 
 •0177 
 
 •0138 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 OS 
 
 •0885 
 
 •0751 
 
 •0631 
 
 •0524 
 
 •0430 
 
 •0350 
 
 •0281 
 
 •0224 
 
 •0177 
 
 •0138 
 
 •0107 
 
 ■0082 
 
 •0062 
 
 •0047 
 
 0-6 
 
 •0859 
 
 ■0733 
 
 ■0618 
 
 •0515 
 
 •0424 
 
 ■0346 
 
 ■0279 
 
 •0223 
 
 •0176 
 
 •0137 
 
 •0106 
 
 •0082 
 
 •0062 
 
 •0046 
 
 0-7 
 
 •0828 
 
 •0710 
 
 •0601 
 
 •0503 
 
 •0416 
 
 •0341 
 
 •0275 
 
 •0220 
 
 •0174 
 
 •0137 
 
 0106 
 
 •0081 
 
 •0062 
 
 •0046 
 
 0-8 
 
 •0791 
 
 •0682 
 
 •0581 
 
 •0489 
 
 •0406 
 
 •0334 
 
 •0271 
 
 •0217 
 
 •0172 
 
 •0135 
 
 •0105 
 
 •0081 
 
 •0061 
 
 •0046 
 
 0-9 
 
 •0749 
 
 •0650 
 
 •0557 
 
 •0471 
 
 •0394 
 
 •0325 
 
 •0265 
 
 •0213 
 
 •0170 
 
 •0134 
 
 •0104 
 
 •0080 
 
 •0061 
 
 •0046 
 
 1-0 
 
 •0702 
 
 •0614 
 
 •0529 
 
 •0451 
 
 •0379 
 
 •0315 
 
 ■0258 
 
 •0209 
 
 •0167 
 
 •0132 
 
 •0103 
 
 •0079 
 
 •0061 
 
 •0046 
 
 1-1 
 
 •0652 
 
 •0574 
 
 •0498 
 
 •0427 
 
 •0362 
 
 •0302 
 
 •0249 
 
 •0202 
 
 •0163 
 
 •0129 
 
 •0101 
 
 •0078 
 
 •0060 
 
 •0045 
 
 IS 
 
 •0597 
 
 •0530 
 
 •0464 
 
 •0401 
 
 •0342 
 
 •0287 
 
 •0238 
 
 ■0195 
 
 •0158 
 
 •0126 
 
 •0099 
 
 •0077 
 
 •0059 
 
 •0045 
 
 IS 
 
 •0541 
 
 •0484 
 
 •0427 
 
 •0372 
 
 •0319 
 
 •0271 
 
 •0226 
 
 •0186 
 
 •0151 
 
 ■0121 
 
 •0096 
 
 •0075 
 
 •0058 
 
 •0044 
 
 I'Jt 
 
 •0484 
 
 •0436 
 
 ■0388 
 
 •0341 
 
 •0295 
 
 •0252 
 
 •0212 
 
 •0176 
 
 •0144 
 
 •0117 
 
 •0093 
 
 •0073 
 
 •0057 
 
 •0043 
 
 1-5 
 
 •0427 
 
 •0388 
 
 •0348 
 
 •0309 
 
 •0270 
 
 •0232 
 
 •0197 
 
 •0165 
 
 •0136 
 
 •0111 
 
 •0089 
 
 •0070 
 
 •0055 
 
 •0042 
 
 1-6 
 
 •0372 
 
 •0341 
 
 •0309 
 
 •0276 
 
 •0243 
 
 ■0211 
 
 •0181 
 
 •0153 
 
 •0127 
 
 •0104 
 
 •0084 
 
 •0067 
 
 •0053 
 
 •0041 
 
 1-7 
 
 •0319 
 
 •0295 
 
 •0270 
 
 •0243 
 
 •0216 
 
 •0190 
 
 •0164 
 
 •0140 
 
 •0117 
 
 •0097 
 
 •0079 
 
 •0063 
 
 •0050 
 
 •0039 
 
 IS 
 
 •0271 
 
 •0252 
 
 ■0232 
 
 •0211 
 
 •0190 
 
 •0168 
 
 •0146 
 
 •0126 
 
 •0107 
 
 •0089 
 
 •0073 
 
 •0059 
 
 •0047 
 
 •0037 
 
 1-9 
 
 •0226 
 
 •0212 
 
 •0197 
 
 •0181 
 
 •0164 
 
 •0146 
 
 •0129 
 
 •0112 
 
 •0096 
 
 •0081 
 
 •0067 
 
 •0055 
 
 ■0044 
 
 •0035 
 
 2-0 
 
 •0186 
 
 •0176 
 
 •0165 
 
 0153 
 
 •0140 
 
 •0126 
 
 •0112 
 
 •0098 
 
 •0085 
 
 ■0072 
 
 •0060 
 
 •0050 
 
 •0040 
 
 •0032 
 
 2-1 
 
 •0151 
 
 •0144 
 
 •0136 
 
 •0127 
 
 •0117 
 
 •0107 
 
 •0096 
 
 ■0085 
 
 •0074 
 
 •0064 
 
 •0054 
 
 •0045 
 
 •0037 
 
 •0030 
 
 2-2 
 
 •0121 
 
 •0117 
 
 •0111 
 
 •0104 
 
 •0097 
 
 •0089 
 
 •0081 
 
 •0072 
 
 ■0064 
 
 •0055 
 
 ■0047 
 
 •0040 
 
 •0033 
 
 •0027 
 
 2-3 
 
 •0096 
 
 •0093 
 
 •0089 
 
 •0084 
 
 •0079 
 
 •0073 
 
 •0067 
 
 •0060 
 
 •0054 
 
 •0047 
 
 •0041 
 
 •0035 
 
 •0029 
 
 •0024 
 
 2-4 
 
 •0075 
 
 •0073 
 
 •0070 
 
 •0067 
 
 •0063 
 
 •0059 
 
 •0055 
 
 •0050 
 
 ■0045 
 
 •0040 
 
 •0035 
 
 •0030 
 
 •0025 
 
 •0021 
 
 2-5 
 
 •0058 
 
 ■0057 
 
 •0055 
 
 •0053 
 
 •0050 
 
 ■0047 
 
 •0044 
 
 •0040 
 
 •0037 
 
 •0033 
 
 •0029 
 
 •0025 
 
 •0022 
 
 •0018 
 
 2-G 
 
 •0044 
 
 •0043 
 
 •0042 
 
 •0041 
 
 •0039 
 
 •0037 
 
 •0035 
 
 .OK 
 
 •0032 
 
 ■0030 
 
 •0027 
 
 •0024 
 
 •0021 
 
 •0018 
 
 •0016 
 
 h = 
 
 1-3 
 
 14 
 
 IS 1-6 
 
 1-7 
 
 r 
 
 1-8 
 
 = oo. 
 1-9 
 
 2-0 
 
 2-1 
 
 2-2 
 
 2-S 
 
 2-4 
 
 2-5 
 
 2-G 
 
 k=00 
 
 •0963 
 
 •0805 
 
 •0666 -0547 
 
 •0445 
 
 •0359 
 
 •0287 
 
 •0227 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 o-i 
 
 •0959 
 
 •0803 
 
 ■0665 
 
 •0547 
 
 •0445 
 
 •0359 
 
 •0287 
 
 •0227 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 0-2 
 
 •0954 
 
 •0800 
 
 •0664 
 
 •0546 
 
 •0444 
 
 •0359 
 
 ■0287 
 
 •0227 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 OS 
 
 •0947 
 
 •0795 
 
 •0661 
 
 •0544 
 
 •0443 
 
 •0358 
 
 •0287 
 
 •0227 
 
 •0178 
 
 •0139 
 
 •0107 
 
 •0082 
 
 ■0062 
 
 •0047 
 
 0-4 
 
 •0936 
 
 •0788 
 
 •0656 
 
 •0541 
 
 •0442 
 
 •0357 
 
 •0286 
 
 •0227 
 
 •0178 
 
 ■0139 
 
 •0107 
 
 •0082 
 
 ■0062 
 
 •0047 
 
 0-5 
 
 0921 
 
 •0778 
 
 •0650 
 
 •0538 
 
 •0440 
 
 •0356 
 
 •0285 
 
 •0227 
 
 ■0178 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 06 
 
 •0901 
 
 •0765 
 
 •0642 
 
 •0532 
 
 •0436 
 
 •0354 
 
 •0284 
 
 •0226 
 
 •0178 
 
 •0139 
 
 •0107 
 
 ■0082 
 
 0062 
 
 •0047 
 
 0-7 
 
 •0876 
 
 •0748 
 
 •0630 
 
 •0525 
 
 •0432 
 
 •0351 
 
 •0283 
 
 •0225 
 
 •0177 
 
 •0138 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 OS 
 
 •0845 
 
 •0725 
 
 •0615 
 
 •0514 
 
 •0425 
 
 •0347 
 
 •0280 
 
 •0224 
 
 •0176 
 
 •0138 
 
 •0107 
 
 •0082 
 
 •0062 
 
 ■0047 
 
 0-9 
 
 •0807 
 
 •0698 
 
 •0595 
 
 •0501 
 
 •0416 
 
 •0341 
 
 •0276 
 
 •0221 
 
 •0175 
 
 •0137 
 
 ■0106 
 
 •0081 
 
 •0062 
 
 •0046 
 
 1-0 
 
 •0763 
 
 •0665 
 
 •0571 
 
 •0484 
 
 •0405 
 
 •0334 
 
 •0272 
 
 •0218 
 
 •0173 
 
 •0136 
 
 •0106 
 
 •0081 
 
 •0062 
 
 ■0046 
 
 1-1 
 
 •0712 
 
 •0626 
 
 •0543 
 
 •0464 
 
 •0390 
 
 •0324 
 
 •0265 
 
 •0214 
 
 •0171 
 
 •0135 
 
 •0105 
 
 •0081 
 
 •0061 
 
 ■0046 
 
 1-2 
 
 •0656 
 
 ■0583 
 
 •0510 
 
 •0439 
 
 •0373 
 
 •0312 
 
 •0257 
 
 •0209 
 
 •0167 
 
 •0133 
 
 •0104 
 
 •0080 
 
 •0061 
 
 •0046 
 
 1-3 
 
 •0597 
 
 •0535 
 
 •0473 
 
 •0411 
 
 0352 
 
 •0297 
 
 •0247 
 
 •0202 
 
 •0163 
 
 ■0130 
 
 •0102 
 
 ■0079 
 
 •0060 
 
 •0046 
 
 1-4 
 
 0535 
 
 •0485 
 
 •0432 
 
 •0380 
 
 •0329 
 
 ■0280 
 
 •0234 
 
 •0194 
 
 •0157 
 
 •0126 
 
 •0100 
 
 •0078 
 
 •0060 
 
 •0045 
 
 1-5 
 
 •0473 
 
 •0432 
 
 ■0390 
 
 •0346 
 
 •0302 
 
 •0260 
 
 •0220 
 
 •0183 
 
 •0150 
 
 •0121 
 
 •0097 
 
 •0076 
 
 •0058 
 
 •0045 
 
 1-6 
 
 •0411 
 
 •0380 
 
 •0346 
 
 •0311 
 
 •0274 
 
 •0239 
 
 •0204 
 
 ■0172 
 
 ■0142 
 
 ■0116 
 
 •0093 
 
 •0073 
 
 •0057 
 
 •0044 
 
 1-7 
 
 ■0352 
 
 •0329 
 
 ■0302 
 
 •0274 
 
 •0245 
 
 •0216 
 
 •0186 
 
 •0159 
 
 0133 
 
 •0109 
 
 ■0088 
 
 •0070 
 
 •0055 
 
 •0043 
 
 1-8 
 
 •0297 
 
 •0280 
 
 •0260 
 
 •0239 
 
 •0216 
 
 •0192 
 
 •0168 
 
 •0144 
 
 ■0122 
 
 •0102 
 
 •0083 
 
 •0067 
 
 •0053 
 
 •0041 
 
 1-9 
 
 •0247 
 
 •0234 
 
 •0220 
 
 •0204 
 
 •0186 
 
 •0168 
 
 •0149 
 
 •0129 
 
 0111 
 
 •0093 
 
 •0077 
 
 •0063 
 
 •0050 
 
 ■0039 
 
 2-0 
 
 •0202 
 
 •0194 
 
 •0183 
 
 •0172 
 
 •0159 
 
 •0144 
 
 •0129 
 
 0114 
 
 •0099 
 
 •0084 
 
 ■0070 
 
 •0058 
 
 •0047 
 
 ■0037 
 
 2-1 
 
 •0163 
 
 •0157 
 
 •0150 
 
 •0142 
 
 •0133 
 
 •0122 
 
 •0111 
 
 •0099 
 
 •0087 
 
 •0075 
 
 •0063 
 
 ■0053 
 
 0043 
 
 •0034 
 
 2-2 
 
 •0130 
 
 •0126 
 
 •0121 
 
 •0116 
 
 •0109 
 
 •0102 
 
 •0093 
 
 •0084 
 
 •0075 
 
 •0065 
 
 •0056 
 
 ■0047 
 
 •0039 
 
 •0032 
 
 23 
 
 •0102 
 
 •0100 
 
 ■0097 
 
 •0093 
 
 •0088 
 
 •0083 
 
 •0077 
 
 •0070 
 
 •0063 
 
 •0056 
 
 •0049 
 
 •0042 
 
 •0035 
 
 •0029 
 
 2-4 
 
 •0079 
 
 •0078 
 
 •0076 
 
 •0073 
 
 •0070 
 
 •0067 
 
 •0063 
 
 •0058 
 
 •0053 
 
 •0047 
 
 •0042 
 
 •0036 
 
 •0031 
 
 •0025 
 
 2-5 
 
 •0060 
 
 •0060 
 
 •0058 
 
 •0057 
 
 •0055 
 
 •0053 
 
 •0050 
 
 •0047 
 
 •0043 
 
 •0039 
 
 •0035 
 
 •0031 
 
 •0026 
 
 •0022 
 
 2-6 
 
 •0046 
 
 •0045 
 
 •0045 
 
 •0044 
 
 •0043 
 
 •0041 
 
 •0039 
 
 •0037 
 
 •0034 
 
 •0032 
 
 •0029 
 
 •0025 
 
 •0022 
 
 •0019 
 
54 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XXX. Supplementary Tables for determining High 
 r = 90. 
 
 A= 
 
 
 
 ■1 
 
 •2 
 
 •s 
 
 ■4 
 
 •5 
 
 ■6 
 
 •7 
 
 ■8 
 
 ■9 
 
 1-0 
 
 VI 
 
 1-2 
 
 h=00 
 
 •4282 
 
 •4067 
 
 •3822 
 
 •3552 
 
 •3266 
 
 •2969 
 
 •2670 
 
 •2377 
 
 •2094 
 
 •1827 
 
 •1579 
 
 •1353 
 
 •1149 
 
 o-i 
 
 •4067 
 
 •3887 
 
 •3678 
 
 •3441 
 
 •3183 
 
 •2910 
 
 •2630 
 
 •2350 
 
 •2077 
 
 •1817 
 
 •1574 
 
 •1350 
 
 •1147 
 
 0-2 
 
 •3822 
 
 •3678 
 
 •3504 
 
 •3302 
 
 •3076 
 
 •2830 
 
 •2573 
 
 •2311 
 
 •2052 
 
 •1801 
 
 ■1564 
 
 •1345 
 
 •1144 
 
 OS 
 
 ■3552 
 
 •3441 
 
 •3302 
 
 •3135 
 
 •2943 
 
 •2728 
 
 •2498 
 
 ■2258 
 
 •2016 
 
 •1778 
 
 •1550 
 
 •1336 
 
 •1140 
 
 0-4 
 
 •3266 
 
 •3183 
 
 •3076 
 
 •2943 
 
 •2784 
 
 •2602 
 
 •2401 
 
 •2187 
 
 •1966 
 
 •1744 
 
 •1528 
 
 •1322 
 
 •1132 
 
 OS 
 
 •2969 
 
 •2910 
 
 •2830 
 
 ■2728 
 
 •2602 
 
 •2453 
 
 •2284 
 
 •2097 
 
 •1900 
 
 •1698 
 
 •1497 
 
 •1302 
 
 •1119 
 
 o-a 
 
 ■2670 
 
 •2630 
 
 ■2573 
 
 •2498 
 
 •2401 
 
 •2284 
 
 ■2145 
 
 •1988 
 
 •1817 
 
 •1037 
 
 •1454 
 
 •1274 
 
 •1101 
 
 0-7 
 
 •2377 
 
 •2350 
 
 •2311 
 
 •2258 
 
 ■2187 
 
 •2097 
 
 •1988 
 
 •1860 
 
 •1717 
 
 •1561 
 
 •1399 
 
 •1236 
 
 •1075 
 
 08 
 
 •2094 
 
 •2077 
 
 ■2052 
 
 •2016 
 
 •1966 
 
 •1900 
 
 •1817 
 
 •1717 
 
 •1600 
 
 •1470 
 
 •1331 
 
 •1186 
 
 •1041 
 
 0-9 
 
 •1827 
 
 •1817 
 
 ■1801 
 
 •1778 
 
 •1744 
 
 •1698 
 
 •1637 
 
 •1561 
 
 •1470 
 
 •1365 
 
 •1249 
 
 •1124 
 
 •0997 
 
 VO 
 
 •1579 
 
 •1574 
 
 •1564 
 
 •1550 
 
 •1528 
 
 ■1497 
 
 •1454 
 
 •1399 
 
 •1331 
 
 •1249 
 
 •1155 
 
 •1052 
 
 •0942 
 
 VI 
 
 •1353 
 
 •1350 
 
 •1345 
 
 •1336 
 
 •1322 
 
 •1302 
 
 •1274 
 
 •1236 
 
 •1186 
 
 •1124 
 
 ■1052 
 
 •0969 
 
 ■0878 
 
 1-2 
 
 •1149 
 
 •1147 
 
 •1144 
 
 •1140 
 
 •1132 
 
 1119 
 
 •1101 
 
 •1075 
 
 •1041 
 
 •0997 
 
 •0942 
 
 •0878 
 
 •080(5 
 
 IS 
 
 •0967 
 
 •0966 
 
 •0905 
 
 •0962 
 
 •0958 
 
 0950 
 
 •0939 
 
 •0923 
 
 •0900 
 
 •0869 
 
 •0830 
 
 •0783 
 
 •0727 
 
 1-4 
 
 •0807 
 
 •0807 
 
 •0806 
 
 •0805 
 
 •0802 
 
 •0798 
 
 •0792 
 
 •0782 
 
 •0767 
 
 •0747 
 
 •0720 
 
 •0686 
 
 •0645 
 
 1-5 
 
 0668 
 
 •0668 
 
 •0667 
 
 •0667 
 
 •0665 
 
 •0663 
 
 •0660 
 
 •0654 
 
 •0645 
 
 •0632 
 
 •0614 
 
 •0591 
 
 •0562 
 
 1-0 
 
 •0548 
 
 ■0548 
 
 •0548 
 
 •0547 
 
 •0547 
 
 •0546 
 
 •0544 
 
 •0540 
 
 •0535 
 
 •0528 
 
 •0516 
 
 •0501 
 
 •0481 
 
 1-7 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0445 
 
 •0445 
 
 •0445 
 
 •0444 
 
 •0442 
 
 •0439 
 
 •0435 
 
 •0428 
 
 •0418 
 
 •0405 
 
 IS 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0358 
 
 •0357 
 
 •0356 
 
 •0353 
 
 •0350 
 
 •0344 
 
 •0338 
 
 1-9 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0286 
 
 •0286 
 
 •0284 
 
 •0282 
 
 •0279 
 
 •0274 
 
 2-0 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0226 
 
 •0225 
 
 •0223 
 
 •0220 
 
 2-1 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0178 
 
 •0178 
 
 •0178 
 
 •0177 
 
 •0176 
 
 ■0175 
 
 2-2 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 ■0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0138 
 
 •0138 
 
 •0137 
 
 2S 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •010(5 
 
 2-4 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 2-5 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 2-0 
 
 •0047 1 
 
 ■0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 ■0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 
 
 
 
 
 
 r 
 
 = •95. 
 
 
 
 
 
 
 
 /* = 
 
 
 
 •1 
 
 * 
 
 s 
 
 •4 
 
 •5 
 
 ■6 
 
 •7 
 
 •8 
 
 •9 
 
 VO 
 
 VI 
 
 V2 
 
 k=00 
 
 •4495 
 
 •4271 
 
 •4005 
 
 •3705 
 
 •3385 
 
 •3055 
 
 •2729 
 
 •2414 
 
 •2116 
 
 •1840 
 
 •1586 
 
 •1357 
 
 •1151 
 
 01 
 
 •4271 
 
 •4099 
 
 •3880 
 
 •3622 
 
 •3333 
 
 •3026 
 
 •2713 
 
 ■2407 
 
 •2113 
 
 •1839 
 
 •1586 
 
 •1356 
 
 •1151 
 
 0-2 
 
 •4005 
 
 •3880 
 
 •3712 
 
 •3500 
 
 •3252 
 
 •2976 
 
 •2685 
 
 ■2392 
 
 •2106 
 
 •1835 
 
 •1585 
 
 •1356 
 
 •1150 
 
 OS 
 
 ■3705 
 
 •3622 
 
 •3500 
 
 •3338 
 
 •3135 
 
 •2898 
 
 •2637 
 
 •2365 
 
 •2092 
 
 •1829 
 
 •1582 
 
 •1355 
 
 •1150 
 
 0-4 
 
 •3385 
 
 •3333 
 
 •3252 
 
 •3135 
 
 •2980 
 
 •2787 
 
 •2564 
 
 •2320 
 
 •2067 
 
 •1816 
 
 •1576 
 
 •1352 
 
 •1149 
 
 OS 
 
 •3055 
 
 •3026 
 
 •2976 
 
 •2898 
 
 •2787 
 
 ■2640 
 
 •2459 
 
 •2250 
 
 •2024 
 
 •1792 
 
 •1563 
 
 •1346 
 
 •1147 
 
 0-6 
 
 •2729 
 
 •2713 
 
 •2685 
 
 •2637 
 
 •2564 
 
 •2459 
 
 •2321 
 
 •2153 
 
 •1960 
 
 •1753 
 
 •1542 
 
 •1335 
 
 •1141 
 
 0-7 
 
 •2414 
 
 •2407 
 
 •2392 
 
 •2365 
 
 •2320 
 
 •2250 
 
 •2153 
 
 •2025 
 
 •1870 
 
 •1694 
 
 •1506 
 
 •1315 
 
 •1131 
 
 0-8 
 
 •2116 
 
 •2113 
 
 •2106 
 
 ■2092 
 
 •2067 
 
 •2024 
 
 •1960 
 
 •1870 
 
 •1753 
 
 •1611 
 
 ■1452 
 
 •1283 
 
 •1113 
 
 0-9 
 
 •1840 
 
 •1839 
 
 •1835 
 
 •1829 
 
 •1816 
 
 •1792 
 
 •1753 
 
 •1694 
 
 •1611 
 
 •1505 
 
 •1377 
 
 •1234 
 
 ■1084 
 
 VO 
 
 •1586 
 
 •1586 
 
 •1585 
 
 •1582 
 
 •1576 
 
 •1563 
 
 •1542 
 
 •1506 
 
 •1452 
 
 •1377 
 
 •1281 
 
 •1167 
 
 •1041 
 
 VI 
 
 •1357 
 
 •1356 
 
 •1356 
 
 •1355 
 
 •1352 
 
 •1346 
 
 •1335 
 
 •1315 
 
 •1283 
 
 •1234 
 
 •1167 
 
 •1082 
 
 ■0981 
 
 VM 
 
 •1151 
 
 •1151 
 
 •1150 
 
 •1150 
 
 •1149 
 
 •1147 
 
 •1141 
 
 •1131 
 
 •1113 
 
 •1084 
 
 •1041 
 
 ■0981 
 
 •0906 
 
 VS 
 
 •0968 
 
 ■0968 
 
 •0968 
 
 •0968 
 
 ■0967 
 
 •0966 
 
 •0964 
 
 •0959 
 
 •0950 
 
 •0934 
 
 •0908 
 
 •0870 
 
 •0818 
 
 1-4 
 
 •0808 
 
 •0808 
 
 •0808 
 
 •0808 
 
 •0807 
 
 •0807 
 
 •0806 
 
 •0804 
 
 •0800 
 
 •0792 
 
 •0778 
 
 •0755 
 
 ■0721 
 
 V5 
 
 •0668 
 
 •0668 
 
 •0668 
 
 •0668 
 
 ■0668 
 
 •0668 
 
 •0668 
 
 •0667 
 
 •0665 
 
 •0661 
 
 •0654 
 
 •0642 
 
 •0622 
 
 1-6 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0547 
 
 ■0545 
 
 •0542 
 
 •0536 
 
 •0525 
 
 1-7 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0445 
 
 •0445 
 
 •0445 
 
 •0443 
 
 •0440 
 
 •0135 
 
 1-8 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0358 
 
 •0357 
 
 •0355 
 
 1-9 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0286 
 
 •0285 
 
 2-0 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 ■0227 
 
 •0227 
 
 •0227 
 
 ■0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 •0227 
 
 2-1 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 ■0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0178 
 
 2-2 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 ■0139 
 
 2-S 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 ■0107 
 
 2-4 
 
 •0082 
 
 •0082 
 
 ■0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 ■0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 2-5 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 ■0062 
 
 •0062 
 
 ■0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 2-6 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 ■0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 ■0047 
 
 ■0047 
 
Tables for High Fourfold Correlation 
 
 55 
 
 Correlations from Tetrachoric Groupings. 
 
 r = 90. 
 
 /*= 
 
 k = 0-0 
 0-1 
 0-2 
 OS 
 0-4 
 OS 
 OS 
 0-7 
 OS 
 OS 
 1-0 
 1-1 
 1-2 
 1-3 
 
 14 
 
 IS 
 1-6 
 1-7 
 1-8 
 1-9 
 20 
 2-1 
 2-2 
 
 2:1 
 
 2-Jf 
 2-5 
 2-6 
 
 1-3 
 
 •0967 
 •0966 
 •0965 
 •0962 
 •0958 
 •0950 
 •0939 
 •0923 
 •0900 
 •0869 
 •0830 
 •0783 
 •0727 
 •0664 
 •0596 
 •0526 
 •0456 
 •0388 
 •0325 
 •0267 
 •0216 
 •0173 
 •0136 
 •0106 
 •0081 
 ■0062 
 •0046 
 
 1-* 
 
 •0807 
 •0807 
 •0806 
 •0805 
 •0802 
 •0798 
 •0792 
 •0782 
 •0767 
 •0747 
 •0720 
 •0686 
 •0645 
 •0596 
 •0543 
 •0485 
 •0426 
 •0367 
 •0310 
 •0258 
 •0211 
 •0169 
 •0134 
 •0105 
 •0081 
 •0062 
 •0046 
 
 IS 
 
 ■0668 
 •0668 
 •0667 
 •0667 
 •0665 
 •0663 
 •0660 
 •0654 
 •0645 
 •0632 
 •0614 
 •0591 
 •0562 
 •0526 
 •0485 
 •0439 
 •0391 
 ■0341 
 •0292 
 •0246 
 •0203 
 •0164 
 •0131 
 •0103 
 •0080 
 ■0061 
 •0046 
 
 1-6 
 
 •0548 
 •0548 
 •0548 
 ■0547 
 •0547 
 •0546 
 •0544 
 •0540 
 •0535 
 •0528 
 ■0516 
 •0501 
 •0481 
 •0456 
 •0426 
 •0391 
 ■0353 
 ■0312 
 •0271 
 •0231 
 •0193 
 •0158 
 •0127 
 •0101 
 •0079 
 •0060 
 •0046 
 
 1-7 
 
 •0446 
 •0446 
 •0446 
 •0445 
 •0445 
 •0445 
 •0444 
 •0442 
 •0439 
 •0435 
 •0428 
 •0418 
 •0405 
 •0388 
 •0367 
 •0341 
 •0312 
 •0281 
 •0247 
 ■0214 
 •0181 
 •0150 
 •0122 
 •0098 
 •0077 
 •0059 
 •0045 
 
 •0359 
 •0359 
 •0359 
 •0359 
 •0359 
 0359 
 •0358 
 •0357 
 ■0356 
 •0353 
 •0350 
 •0344 
 •0338 
 •0325 
 •0310 
 •0292 
 •0271 
 •0247 
 •0221 
 •0194 
 •0167 
 •0140 
 •0116 
 •0094 
 •0074 
 •0058 
 •0044 
 
 IS 
 
 •0287 
 •0287 
 •0287 
 •0287 
 •0287 
 •0287 
 •0287 
 •0286 
 •0286 
 •0284 
 •0282 
 •0279 
 •0274 
 •0267 
 •0258 
 •0246 
 •0231 
 •0214 
 •0194 
 •0173 
 •0151 
 •0129 
 •0108 
 •0088 
 •0071 
 ■0056 
 •0043 
 
 2-0 
 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 ■0226 
 •0225 
 •0223 
 •0220 
 •0216 
 •0211 
 •0203 
 •0193 
 •0181 
 •0167 
 •0151 
 •0134 
 •0116 
 •0099 
 •0082 
 •0067 
 •0053 
 •0042 
 
 2'1 
 
 •0179 
 •0179 
 •0179 
 •0179 
 •0179 
 •0179 
 •0179 
 •0178 
 •0178 
 •0178 
 •0177 
 •0176 
 •0175 
 •0173 
 •0169 
 •0164 
 •0158 
 •0150 
 •0140 
 •0129 
 ■0116 
 •0102 
 •0088 
 •0075 
 •0062 
 •0050 
 •0040 
 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0138 
 •0138 
 •0137 
 •0136 
 •0134 
 •0131 
 •0127 
 ■0122 
 •0116 
 •0108 
 ■0099 
 •0088 
 ■0078 
 •0067 
 •0056 
 •0046 
 ■0037 
 
 2-3 
 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0106 
 •0106 
 •0105 
 •0103 
 •0101 
 •0098 
 •0094 
 •0088 
 •0082 
 •0075 
 •0067 
 •0058 
 •0050 
 •0042 
 •0034 
 
 x-4 
 
 2-5 
 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0081 
 •0081 
 ■0080 
 •0079 
 •0077 
 ■0074 
 •0071 
 •0067 
 •0062 
 •0056 
 •0050 
 •0044 
 •0037 
 •0031 
 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0061 
 •0060 
 •0059 
 •0058 
 •0056 
 •0053 
 •0050 
 •0046 
 •0042 
 ■0037 
 •0032 
 •0027 
 
 2-6 
 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0046 
 •0046 
 •0046 
 •0046 
 •0045 
 •0044 
 •0043 
 •0042 
 •0040 
 ■0037 
 •0034 
 •0031 
 •0027 
 •0024 
 
 r = !)5. 
 
 V3 
 
 1-4 
 
 IS 
 
 1-6 
 
 1-7 
 
 1-8 
 
 1-9 2-0 
 
 2-3 
 
 2-4 
 
 ■5 2-6 
 
 --OS 
 01 
 0-2 
 OS 
 0-4 
 05 
 0-6 
 0-7 
 08 
 0-9 
 VO 
 1-1 
 1-2 
 IS 
 
 1-4 
 IS 
 1-6 
 
 1-7 
 1-8 
 1-9 
 
 2-0 
 21 
 2-2 
 2-3 
 2-4 
 2-5 
 2-6 
 
 •0968 
 •0968 
 •0968 
 •09C8 
 •0967 
 •0966 
 •0964 
 •0959 
 •0950 
 •0934 
 •0908 
 •0870 
 •0818 
 •0752 
 •0676 
 •0593 
 •0508 
 •0426 
 •0350 
 •0283 
 •0226 
 •0178 
 •0139 
 •0107 
 •0082 
 •0062 
 •0047 
 
 •0808 
 •0808 
 •0808 
 •0808 
 •0807 
 •0807 
 •0806 
 •0804 
 •0800 
 •0792 
 •0778 
 •0755 
 •0721 
 •0676 
 •0619 
 •0554 
 •0483 
 •0411 
 •0342 
 •0279 
 •0224 
 •0177 
 •0139 
 •0107 
 •0082 
 •0062 
 •0047 
 
 •0668 
 •0668 
 •0668 
 •0668 
 •0668 
 •0668 
 •0668 
 •0667 
 •0665 
 •0661 
 •0654 
 •0642 
 •0622 
 •0593 
 •0554 
 •0505 
 •0450 
 •0390 
 •0330 
 •0273 
 •0221 
 •0176 
 •0138 
 •0107 
 •0082 
 •0062 
 •0047 
 
 •0548 
 •0548 
 •0548 
 •0548 
 •0548 
 •0548 
 •0548 
 •0548 
 •0547 
 •0545 
 ■0542 
 •0536 
 •0525 
 •0508 
 •0483 
 •0450 
 •0409 
 •0362 
 •0312 
 •0263 
 •0215 
 •0173 
 •0137 
 •0106 
 •0082 
 •0062 
 •0047 
 
 •0440 
 •0446 
 •0446 
 •0446 
 •0446 
 •0446 
 •0446 
 •0445 
 •0445 
 •0445 
 •0443 
 •0440 
 •0435 
 •0426 
 •0411 
 •0390 
 •0362 
 •0328 
 •0289 
 •0248 
 •0207 
 •0169 
 •0135 
 •0105 
 •0081 
 •0062 
 •0047 
 
 ■0359 
 •0359 
 •0359 
 •0359 
 •0359 
 •0359 
 •0359 
 •0359 
 •0359 
 •0359 
 •0358 
 •0357 
 •0355 
 •0350 
 •0342 
 •0330 
 •0312 
 •0289 
 •0261 
 •0229 
 •0195 
 •0162 
 •0131 
 •0104 
 •0080 
 •0062 
 •0046 
 
 •0287 
 •0287 
 ■0287 
 •0287 
 •0287 
 •0287 
 •0287 
 •0287 
 •0287 
 •0287 
 •0287 
 •0286 
 •0285 
 •0283 
 •0279 
 •0273 
 •0263 
 •0248 
 •0229 
 •0205 
 •0179 
 •0152 
 •0125 
 •0101 
 •0079 
 •0061 
 •0046 
 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0227 
 •0226 
 •0224 
 •0221 
 •0215 
 •0207 
 •0195 
 •0179 
 •0160 
 •0139 
 •0117 
 •0096 
 •0077 
 •0060 
 •0046 
 
 •0179 
 •0179 
 •0179 
 •0179 
 •0179 
 •0179 
 •0179 
 •0179 
 •0179 
 ■0179 
 •0179 
 •0179 
 •0178 
 •0178 
 •0177 
 •0176 
 •0173 
 •0169 
 •0162 
 •0152 
 ■0139 
 ■0124 
 •0107 
 •0090 
 •0073 
 •0058 
 •0045 
 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0139 
 •0138 
 •0137 
 •0135 
 •0131 
 •0125 
 •0117 
 •0107 
 •0095 
 •0082 
 •0068 
 •0055 
 •0043 
 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 ■0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0107 
 •0106 
 •0105 
 •0104 
 •0101 
 •0096 
 •0090 
 •0082 
 •0072 
 •0062 
 •0051 
 •0041 
 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 ■0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0082 
 •0081 
 •0080 
 •0079 
 •0077 
 •0073 
 •0068 
 •0062 
 •0054 
 •0046 
 •0038 
 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 ■0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0062 
 •0061 
 •0060 
 •0058 
 •0055 
 •0051 
 •0046 
 •0040 
 •0034 
 
 •0047 
 ■0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 ■0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0047 
 •0046 
 •0046 
 •0046 
 •0045 
 •0043 
 •0041 
 •0038 
 •0034 
 •0030 
 
56 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XXX. Supplementary Tables for determining High 
 r=100. 
 
 h= 
 
 
 
 ■1 
 
 ■2 
 
 S 
 
 ■4 
 
 ■5 
 
 ■6 
 
 ■7 
 
 S 
 
 ■9 
 
 VO 
 
 VI 
 
 1-2 
 
 1=00 
 
 •51)00 
 
 ■4602 
 
 •4207 
 
 •3821 
 
 •3446 
 
 •3085 
 
 ■2743 
 
 •2420 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 o-i 
 
 •4602 
 
 •4602 
 
 •4207 
 
 •3821 
 
 •3446 
 
 •3085 
 
 •2743 
 
 •2420 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 0-2 
 
 •4207 
 
 •4207 
 
 •4207 
 
 •3821 
 
 •3446 
 
 •3085 
 
 •2743 
 
 •2420 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 OS 
 
 •3821 
 
 •3821 
 
 •3821 
 
 •3821 
 
 •3446 
 
 •3085 
 
 •2743 
 
 •2420 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 0-4 
 
 •3446 
 
 •3446 
 
 •3446 
 
 •3446 
 
 •3446 
 
 •3085 
 
 •2743 
 
 •2420 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 0-5 
 
 •3085 
 
 •3085 
 
 •3085 
 
 •3085 
 
 ■3085 
 
 •3085 
 
 •2743 
 
 •2420 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 0-6 
 
 •2743 
 
 •2743 
 
 •2743 
 
 ■2743 
 
 •2743 
 
 •2743 
 
 •2743 
 
 •2420 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 0-7 
 
 •2420 
 
 •2420 
 
 •2420 
 
 •2420 
 
 •2420 
 
 ■2420 
 
 •2420 
 
 •2420 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 0-8 
 
 •2119 
 
 •2119 
 
 •2119 
 
 •2119 
 
 •2119 
 
 •2119 
 
 •2119 
 
 •2119 
 
 •2119 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 0-9 
 
 •1841 
 
 •1841 
 
 •1841 
 
 •1841 
 
 •1841 
 
 •1841 
 
 •1841 
 
 •1841 
 
 •1841 
 
 •1841 
 
 •1587 
 
 •1357 
 
 •1151 
 
 VO 
 
 •1587 
 
 •1587 
 
 •1587 
 
 •1587 
 
 ■1587 
 
 •1587 
 
 •1587 
 
 •1587 
 
 •1587 
 
 •1587 
 
 •1587 
 
 •1357 
 
 •1151 
 
 1-1 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1357 
 
 •1151 
 
 V2 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 •1151 
 
 IS 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 •0968 
 
 1-4 
 
 •0808 
 
 •0808 
 
 •0808 
 
 •0808 
 
 •0808 
 
 •0808 
 
 •0808 
 
 •0808 
 
 ■0808 
 
 •0808 
 
 •0808 
 
 •0808 
 
 ■0808 
 
 1-5 
 
 •0668 
 
 •0668 
 
 ■0668 
 
 •0668 
 
 •0668 
 
 •0668 
 
 ■0668 
 
 •0668 
 
 •0668 
 
 •0668 
 
 •0668 
 
 •0668 
 
 •0668 
 
 1-6 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 ■0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 1-7 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 ■0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 IS 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 ■0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 1-9 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 2-0 
 
 •0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 ■0228 
 
 •0228 
 
 ■0228 
 
 2-1 
 
 ■0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 ■0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 ■0179 
 
 2-2 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 2S 
 
 ■0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 ■0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 2-4 
 
 ■0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 2-5 
 
 ■0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 ■0062 
 
 •0062 
 
 ■0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 2-6 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 ■0047 
 
 ■0047 
 
 ■0047 
 
 •0047 
 
 •0047 
 
 ■0047 
 
 ■0047 
 
 •0047 
 
 •0047 
 
Tables for High Fourfold Correlation 
 
 57 
 
 Correlations from Tetrachoric Groupings. 
 
 r = 1-00. 
 
 h= 
 
 1-3 
 
 1-4 
 
 1-5 
 
 1-6 
 
 1-7 
 
 IS 
 
 V9 
 
 2-0 
 
 2-1 
 
 2-2 
 
 2-3 
 
 2-k 
 
 2-5 
 
 2-6 
 
 k-o-o 
 
 •0968 
 
 ■0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 01 
 
 •0968 
 
 ■0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 ■0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 on 
 
 ■0968 
 
 ■0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 ■0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 OS 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 0-4 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 ■0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 ■0062 
 
 •0047 
 
 0-5 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 ■0107 
 
 •0082 
 
 ■0062 
 
 •0047 
 
 06 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 ■0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 ■0062 
 
 •0047 
 
 0-7 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 0-8 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 0-9 
 
 •0968 
 
 •0808 
 
 ■0668 
 
 ■0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 10 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 ■0359 
 
 •0287 
 
 •0228 
 
 ■0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 l-l 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 ■0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 ■0062 
 
 •0047 
 
 1-2 
 
 •0968 
 
 ■0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 ■01 39 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 IS 
 
 •0968 
 
 •0808 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 ■0062 
 
 ■0047 
 
 14 
 
 •0808 
 
 •0808 
 
 ■0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 1-5 
 
 •0668 
 
 •0668 
 
 •0668 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 ■0082 
 
 •0062 
 
 •0047 
 
 1-6 
 
 ■0548 
 
 •0548 
 
 •0548 
 
 •0548 
 
 •0446 
 
 •0359 
 
 •0287 
 
 •0228 
 
 •0179 
 
 •0139 
 
 ■0107 
 
 ■0082 
 
 •0062 
 
 •0047 
 
 1-7 
 
 •0446 
 
 •0446 
 
 •0446 
 
 •0446 
 
 ■0446 
 
 ■0359 
 
 •0287 
 
 •0228 
 
 ■0179 
 
 •(1139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 1-8 
 
 •03. r )9 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 •0359 
 
 ■0287 
 
 ■0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 10 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 •0287 
 
 ■0228 
 
 •0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 ■0062 
 
 •0047 
 
 2-0 
 
 •0228 
 
 •0228 
 
 •0228 
 
 ■0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 •0228 
 
 0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 21 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 •0179 
 
 ■0179 
 
 •0139 
 
 •0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 2-2 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 •0139 
 
 ■0107 
 
 •0082 
 
 •0062 
 
 ■0047 
 
 2-3 
 
 ■0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 •0107 
 
 ■0107 
 
 •0107 
 
 •0107 
 
 ■0107 
 
 •0082 
 
 •0062 
 
 •0047 
 
 n 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 ■0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 •0082 
 
 ■0062 
 
 •0047 
 
 2-5 
 
 •0062 
 
 •0062 
 
 •0062 
 
 ■0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0062 
 
 •0047 
 
 2-r, 
 
 ■0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 ■0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 •0047 
 
 B. 
 
58 Tables for Statisticians and Biometricians 
 
 TABLE XXXI. T/te V-Function. 
 
 p 
 
 
 
 
 Loa r(j>), Negative Charaeterist 
 
 c, 1 
 
 
 
 
 
 
 I 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1-00 
 
 •999,9999 
 
 7497 
 
 5001 
 
 2512 
 
 0030 
 
 ■998,7555 
 
 5087 
 
 2627 
 
 0173 
 
 -7727 
 
 1-01 
 
 •997,5287 
 
 2855 
 
 0430 
 
 -8011 
 
 - 5600 
 
 •996,3196 
 
 0798 
 
 -8408 
 
 -6025 
 
 3648 
 
 V02 
 
 •995,1279 
 
 -8916 
 
 -6561 
 
 -4212 
 
 -1870 
 
 •993,9535 
 
 7207 
 
 4886 
 
 2572 
 
 0265 
 
 1-03 
 
 •992,7964 
 
 5671 
 
 3384 
 
 1104 
 
 - 8831 
 
 •991,6564 
 
 4305 
 
 2052 
 
 -9806 
 
 -7567 
 
 1-04 
 
 •990,5334 
 
 3108 
 
 0889 
 
 -867? 
 
 -6471 
 
 •989,4273 
 
 2080 
 
 -9895 
 
 -7716 
 
 -5544 
 
 1-05 
 
 •988,3379 
 
 1220 
 
 -9068 
 
 -6922 
 
 - 4783 
 
 ■987,2651 
 
 0525 
 
 -8406 
 
 (1291 
 
 -4188 
 
 1-06 
 
 •986,2089 
 
 -9996 
 
 -7910 
 
 - 5830 
 
 - 3757 
 
 •985,1690 
 
 -9630 
 
 -7577 
 
 - 5530 
 
 -3489 
 
 1-07 
 
 •984,1455 
 
 -9 128" 
 
 -7407 
 
 - 5392 
 
 -3384 
 
 •983,1382 
 
 -9387 
 
 -7398 
 
 -5415 
 
 -3439 
 
 1-08 
 
 •982,1469 
 
 -9506 
 
 -7535 
 
 - 5599 
 
 3655 
 
 •981,1717 
 
 -9785 
 
 -7860 
 
 - 5941 
 
 - I'i29 
 
 1-09 
 
 •980,2123 
 
 0223 
 
 -8329 
 
 -6442 
 
 -4561 
 
 ■979,2686 
 
 0818 
 
 -8956 
 
 -7100 
 
 -5250 
 
 1-10 
 
 •978,3407 
 
 1570 
 
 -9738 
 
 -7914 
 
 -6095 
 
 •977,4283 
 
 2476 
 
 0676 
 
 -8882 
 
 -7095 
 
 in 
 
 •976,5313 
 
 3538 
 
 1768 
 
 0005 
 
 -8248 
 
 •975,6497 
 
 4753 
 
 3014 
 
 1281 
 
 - 9555 
 
 vvt 
 
 ■974,7834 
 
 6120 
 
 4411 
 
 2709 
 
 1013 
 
 •973,9323 
 
 7638 
 
 5960 
 
 4288 
 
 2622 
 
 1-13 
 
 •973,0962 
 
 -9308 
 
 - 7659 
 
 -6017 
 
 -4381 
 
 •972,2751 
 
 1126 
 
 -9508 
 
 -7896 
 
 -6289 
 
 1-lk 
 
 •971,4689 
 
 3094 
 
 1505 
 
 -9922 
 
 - 831.-, 
 
 •970,6774 
 
 5209 
 
 3650 
 
 2096 
 
 05 19 
 
 1-15 
 
 •969,9007 
 
 7471 
 
 5941 
 
 4417 
 
 2898 
 
 •969,1386 
 
 - 9S79 
 
 -8378 
 
 - 6883 
 
 - 5393 
 
 1-16 
 
 •968,3910 
 
 2432 
 
 0960 
 
 -9493 
 
 - 8033 
 
 •967,6578 
 
 5129 
 
 3686 
 
 2248 
 
 0816 
 
 V17 
 
 •966,9390 
 
 7969 
 
 6554 
 
 5145 
 
 3742 
 
 •966,2344 
 
 0952 
 
 -9566 
 
 -8185 
 
 -6810 
 
 1-18 
 
 •965,5440 
 
 4076 
 
 2718 
 
 1366 
 
 0019 
 
 •964,8677 
 
 7341 
 
 6011 
 
 4687 
 
 3368 
 
 1-19 
 
 •964,2054 
 
 0746 
 
 -9444 
 
 -8147 
 
 -6856 
 
 •963,5570 
 
 4290 
 
 3016 
 
 1747 
 
 0483 
 
 1-20 
 
 •962,9225 
 
 7973 
 
 6725 
 
 5484 
 
 4248 
 
 •962,3017 
 
 1792 
 
 0573 
 
 -5358 
 
 -8150 
 
 1-21 
 
 •961,6946 
 
 5748 
 
 4556 
 
 3369 
 
 2188 
 
 •961,1011 
 
 -9841 
 
 - 86?5 
 
 -7515 
 
 -6361 
 
 1-22 
 
 •960,5212 
 
 4068 
 
 2930 
 
 1796 
 
 0669 
 
 •959,9546 
 
 8430 
 
 7318 
 
 6212 
 
 5111 
 
 1-23 
 
 •959,4015 
 
 2925 
 
 1840 
 
 0760 
 
 -9685 
 
 •958,8616 
 
 7553 
 
 6494 
 
 5441 
 
 4393 
 
 1-24 
 
 •958,3350 
 
 2313 
 
 1280 
 
 0253 
 
 - 9232 
 
 •957,8215 
 
 7204 
 
 6198 
 
 5197 
 
 4201 
 
 1-25 
 
 •957,3211 
 
 2226 
 
 1246 
 
 0271 
 
 -9301 
 
 •956,8337 
 
 7377 
 
 6423 
 
 5474 
 
 4530 
 
 1 -86 
 
 ■956,3592 
 
 2658 
 
 1730 
 
 0806 
 
 -9888 
 
 •955,8975 
 
 8067 
 
 7165 
 
 6267 
 
 5374 
 
 1-27 
 
 •955,4487 
 
 3604 
 
 2727 
 
 1855 
 
 0988 
 
 ■955,0126 
 
 - 9268 
 
 -64id 
 
 7570 
 
 0728 
 
 1-28 
 
 •954,5891 
 
 5059 
 
 4232 
 
 3410 
 
 2593 
 
 •954,1782 
 
 0975 
 
 0173 
 
 9370 
 
 -8585 
 
 1-29 
 
 •953,7798 
 
 7016 
 
 6239 
 
 5467 
 
 47(10 
 
 •953,3938 
 
 3181 
 
 2429 
 
 1682 
 
 0940 
 
 ISO 
 
 •953,0203 
 
 -9470 
 
 - 8743 
 
 -8021 
 
 -7303 
 
 •952,6590 
 
 5883 
 
 5180 
 
 4482 
 
 3789 
 
 1-31 
 
 •952,3100 
 
 2417 
 
 1739 
 
 1065 
 
 0390 
 
 •951,9732 
 
 9073 
 
 8419 
 
 7770 
 
 7125 
 
 1-S2 
 
 •951,6485 
 
 5850 
 
 5220 
 
 4595 
 
 3975 
 
 •951,3359 
 
 2748 
 
 2142 
 
 1541 
 
 0944 
 
 1-33 
 
 •951,0353 
 
 -9766 
 
 -9184 
 
 -8606 
 
 - 8034 
 
 •950,7466 
 
 6903 
 
 6344 
 
 5791 
 
 5242 
 
 1-Slt 
 
 •950,4698 
 
 4158 
 
 3624 
 
 3094 
 
 2568 
 
 •950,2048 
 
 1532 
 
 1021 
 
 0514 
 
 0012 
 
 1-35 
 
 •949,9515 
 
 9023 
 
 8535 
 
 8052 
 
 7573 
 
 ■949,7100 
 
 6630 
 
 6166 
 
 5706 
 
 5251 
 
 1-36 
 
 •949,4800 
 
 4355 
 
 3913 
 
 3477 
 
 3044 
 
 •949,2617 
 
 2194 
 
 1776 
 
 1362 
 
 0953 
 
 1-37 
 
 •949,0549 
 
 0149 
 
 - 9754 
 
 -9363 
 
 -8977 
 
 ■948,8595 
 
 8218 
 
 7846 
 
 7478 
 
 7115 
 
 1-38 
 
 •948,6756 
 
 6402 
 
 6052 
 
 5707 
 
 5366 
 
 •948,5030 
 
 4698 
 
 4371 
 
 4049 
 
 3731 
 
 1-89 
 
 •948,3417 
 
 3108 
 
 2803 
 
 2503 
 
 2208 
 
 •948,1916 
 
 1630 
 
 1348 
 
 1070 
 
 0797 
 
 l'Jfi 
 
 •948,0528 
 
 0263 
 
 0003 
 
 - 9748 
 
 -9497 
 
 •947,9250 
 
 9008 
 
 8770 
 
 8537 
 
 8308 
 
 1-41 
 
 •947,8084 
 
 7864 
 
 7648 
 
 7437 
 
 7230 
 
 •947,7027 
 
 6829 
 
 6636 
 
 6446 
 
 6261 
 
 1-42 
 
 •947,6081 
 
 5905 
 
 5733 
 
 5565 
 
 5402 
 
 ■947,5243 
 
 5089 
 
 4939 
 
 4793 
 
 4652 
 
 1-43 
 
 •947,4515 
 
 4382 
 
 4254 
 
 4130 
 
 4010 
 
 •947,3894 
 
 3783 
 
 3676 
 
 3574 
 
 3476 
 
 1-44 
 
 •947,3382 
 
 3292 
 
 3207 
 
 3125 
 
 3049 
 
 •947,2976 
 
 2908 
 
 2844 
 
 2784 
 
 2728 
 
 1-45 
 
 •947,2677 
 
 2630 
 
 2587 
 
 2549 
 
 2514 
 
 •947,2484 
 
 2459 
 
 2437 
 
 2419 
 
 2406 
 
 1-46 
 
 •947,2397 
 
 2393 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2392 
 2617 
 
 2396 
 2662 
 
 2404 
 2712 
 
 •947,2416 
 •947,2766 
 
 2432 
 
 2824 
 
 2452 
 2886 
 
 2477 
 2952 
 
 2506 
 3022 
 
 1-47 
 
 •947,2539 
 
 2576 
 
 1-48 
 
 •947,3097 
 
 3175 
 
 3258 
 
 3345 
 
 3436 
 
 •947,3531 
 
 3630 
 
 3734 
 
 3841 
 
 3953 
 
 1.49 
 
 •947,4068 
 
 4188 
 
 4312 
 
 4440 
 
 4572 
 
 947,4708 
 
 4848 
 
 4992 
 
 5141 
 
 5293 
 
 1.50 
 
 •947,5449 
 
 5610 
 
 5774 
 
 5943 
 
 6116 
 
 •947,6292 
 
 6473 
 
 6658 
 
 6847 
 
 7040 
 
 A horizontal bar means that the third figure of the mantissa has changed, a negative sign that it must be 
 lowered one unit. 
 
Tables of the Y-Function 
 
 59 
 
 Differences : — Negative down to rule 
 
 P 
 
 
 
 1 
 
 % 
 
 S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 S 
 
 9 
 
 2503 
 
 2496 
 
 2489 
 
 2482 
 
 2475 
 
 2468 
 
 2460 
 
 2454 
 
 ■ 
 
 2446 
 
 2440 
 
 V00 
 
 2432 
 
 2425 
 
 2419 
 
 2411 
 
 2404 
 
 2398 
 
 2390 
 
 2383 
 
 2377 
 
 2369 
 
 V01 
 
 2363 
 
 2355 
 
 2349 
 
 2342 
 
 2335 
 
 2328 
 
 2321 
 
 2314 
 
 2307 
 
 2301 
 
 V02 
 
 2293 
 
 2287 
 
 2280 
 
 2273 
 
 2267 
 
 2259 
 
 2253 
 
 2246 
 
 2239 
 
 2233 
 
 1-03 
 
 2226 
 
 2219 
 
 2212 
 
 2206 
 
 2198 
 
 2193 
 
 2185 
 
 2179 
 
 2172 
 
 2165 
 
 V04 
 
 2159 
 
 2152 
 
 2146 
 
 2139 
 
 2132 
 
 2126 
 
 2119 
 
 2112 
 
 2106 
 
 2099 
 
 V05 
 
 2093 
 
 2086 
 
 2080 
 
 2073 
 
 2067 
 
 2060 
 
 2053 
 
 2047 
 
 2041 
 
 2034 
 
 V06 
 
 2027 
 
 2021 
 
 2015 
 
 2008 
 
 2002 
 
 1995 
 
 1989 
 
 1983 
 
 1976 
 
 1970 
 
 1-07 
 
 1963 
 
 1957 
 
 1950 
 
 1944 
 
 1938 
 
 1932 
 
 1925 
 
 1919 
 
 1912 
 
 1906 
 
 V08 
 
 19( X) 
 
 1894 
 
 1887 
 
 1881 
 
 1875 
 
 1868 
 
 1862 
 
 1856 
 
 1850 
 
 1843 
 
 V09 
 
 1837 
 
 1832 
 
 1824 
 
 1819 
 
 1812 
 
 1807 
 
 1800 
 
 1794 
 
 1787 
 
 1782 
 
 V10 
 
 1775 
 
 1770 
 
 1763 
 
 1757 
 
 1751 
 
 1744 
 
 1739 
 
 1733 
 
 1726 
 
 1721 
 
 I'll 
 
 1714 
 
 1709 
 
 1702 
 
 1696 
 
 1690 
 
 1685 
 
 1678 
 
 1672 
 
 1666 
 
 1660 
 
 V12 
 
 1654 
 
 1649 
 
 1642 
 
 1636 
 
 1630 
 
 1625 
 
 1618 
 
 1612 
 
 1607 
 
 1600 
 
 VIS 
 
 1595 
 
 1589 
 
 1583 
 
 1577 
 
 1571 
 
 1565 
 
 1559 
 
 1554 
 
 1547 
 
 1542 
 
 1-14 
 
 1536 
 
 1530 
 
 1524 
 
 1519 
 
 1512 
 
 1507 
 
 1501 
 
 1495 
 
 1490 
 
 1483 
 
 1-15 
 
 1478 
 
 1472 
 
 1467 
 
 1460 
 
 1455 
 
 1449 
 
 1443 
 
 1438 
 
 1432 
 
 1426 
 
 via 
 
 1421 
 
 1415 
 
 1409 
 
 1403 
 
 1398 
 
 1392 
 
 1386 
 
 1381 
 
 1375 
 
 1370 
 
 V17 
 
 1364 
 
 1358 
 
 1352 
 
 1347 
 
 1342 
 
 1336 
 
 1330 
 
 1324 
 
 1319 
 
 1314 
 
 VIS 
 
 1308 
 
 1302 
 
 1297 
 
 1291 
 
 1286 
 
 1280 
 
 1274 
 
 1269 
 
 1264 
 
 1258 
 
 VIS 
 
 1252 
 
 1248 
 
 1241 
 
 1236 
 
 1231 
 
 1225 
 
 1219 
 
 1215 
 
 1208 
 
 1204 
 
 1-20 
 
 1198 
 
 1192 
 
 1187 
 
 1181 
 
 1177 
 
 1170 
 
 1166 
 
 1160 
 
 1154 
 
 1149 
 
 V21 
 
 1144 
 
 1138 
 
 1134 
 
 1127 
 
 1123 
 
 1116 
 
 1112 
 
 1106 
 
 1101 
 
 1096 
 
 V22 
 
 1090 
 
 1085 
 
 1080 
 
 1075 
 
 1069 
 
 1063 
 
 1059 
 
 1053 
 
 1048 
 
 1043 
 
 vss 
 
 1037 
 
 1033 
 
 1027 
 
 1021 
 
 1017 
 
 1011 
 
 1006 
 
 1001 
 
 996 
 
 990 
 
 v*4 
 
 985 
 
 980 
 
 975 
 
 970 
 
 964 
 
 960 
 
 954 
 
 949 
 
 944 
 
 938 
 
 VKB 
 
 934 
 
 928 
 
 924 
 
 918 
 
 913 
 
 908 
 
 902 
 
 898 
 
 893 
 
 887 
 
 ISO 
 
 883 
 
 877 
 
 872 
 
 867 
 
 862 
 
 858 
 
 852 
 
 846 
 
 842 
 
 837 
 
 V27 
 
 832 
 
 827 
 
 822 
 
 817 
 
 811 
 
 807 
 
 802 
 
 797 
 
 791 
 
 787 
 
 1-28 
 
 782 
 
 777 
 
 772 
 
 767 
 
 762 
 
 757 
 
 752 
 
 747 
 
 742 
 
 737 
 
 V29 
 
 733 
 
 727 
 
 722 
 
 718 
 
 713 
 
 707 
 
 703 
 
 698 
 
 693 
 
 689 
 
 VSO 
 
 683 
 
 678 
 
 674 
 
 669 
 
 664 
 
 659 
 
 654 
 
 649 
 
 645 
 
 640 
 
 VS1 
 
 635 
 
 630 
 
 625 
 
 620 
 
 616 
 
 611 
 
 606 
 
 601 
 
 597 
 
 591 
 
 VS2 
 
 587 
 
 582 
 
 578 
 
 572 
 
 568 
 
 563 
 
 559 
 
 553 
 
 549 
 
 544 
 
 1-33 
 
 540 
 
 534 
 
 530 
 
 526 
 
 520 
 
 516 
 
 511 
 
 507 
 
 502 
 
 497 
 
 1*4 
 
 492 
 
 488 
 
 483 
 
 479 
 
 473 
 
 470 
 
 464 
 
 460 
 
 455 
 
 451 
 
 VSS 
 
 445 
 
 442 
 
 436 
 
 433 
 
 427 
 
 423 
 
 418 
 
 414 
 
 409 
 
 404 
 
 VSO 
 
 400 
 
 395 
 
 391 
 
 386 
 
 382 
 
 377 
 
 372 
 
 368 
 
 363 
 
 359 
 
 VS7 
 
 354 
 
 350 
 
 345 
 
 341 
 
 336 
 
 332 
 
 327 
 
 322 
 
 318 
 
 314 
 
 VSS 
 
 309 
 
 305 
 
 300 
 
 295 
 
 292 
 
 286 
 
 282 
 
 278 
 
 273 
 
 269 
 
 1-S9 
 
 265 
 
 260 
 
 255 
 
 251 
 
 247 
 
 242 
 
 238 
 
 233 - 
 
 229 
 
 224 
 
 1-40 
 
 220 
 
 216 
 
 211 
 
 207 
 
 203 
 
 198 
 
 193 
 
 190 
 
 185 
 
 180 
 
 V41 
 
 176 
 
 172 
 
 168 
 
 163 
 
 159 
 
 154 
 
 150 
 
 146 
 
 141 
 
 137 
 
 V42 
 
 133 
 
 128 
 
 124 
 
 120 
 
 116 
 
 111 
 
 107 
 
 102 
 
 98 
 
 94 
 
 V4S 
 
 90 
 
 85 
 
 82 
 
 76 
 
 73 
 
 68 
 
 64 
 
 60 
 
 56 
 
 51 
 
 1-44 
 
 47 
 - 4 
 
 43 
 - 1 
 
 -38 
 
 -35 
 
 -30 
 
 -25 
 
 -22 
 
 -18 
 
 -13 
 
 - 9 
 
 1-45 
 
 + 4 
 
 + 8 
 
 + 12 
 
 + 16 
 
 + 20 
 
 + 25 
 
 + 29 
 
 + 33 
 
 Vlfi 
 
 
 
 + 37 
 
 + 41 
 
 45 
 
 50 
 
 54 
 
 58 
 
 62 
 
 66 
 
 70 
 
 75 
 
 1-47 
 
 78 
 
 83 
 
 87 
 
 91 
 
 95 
 
 99 
 
 104 
 
 107 
 
 112 
 
 115 
 
 V4S 
 
 120 
 
 124 
 
 128 
 
 132 
 
 136 
 
 140 
 
 144 
 
 149 
 
 152 
 
 156 
 
 V49 
 
 161 
 
 164 
 
 169 
 
 173 
 
 176 
 
 181 
 
 185 
 
 189 
 
 193 
 
 197 
 
 1-50 
 
 * Differences change sign at horizontal rule. 
 
 8—2 
 
60 Tables for Statisticians and Biometricians 
 
 TABLE XXXI. The T-Function. 
 
 p 
 
 
 
 
 Loo T (p), Negative Characteristic, 1 
 
 
 
 1 
 
 
 
 1 
 
 2 
 
 S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1-51 
 
 •947,7237 
 
 7437 
 
 7642 
 
 7851 
 
 8064 
 
 •947,8281 
 
 8502 
 
 8727 
 
 8956 
 
 9189 
 
 V52 
 
 •947,9426 
 
 9667 
 
 9912 
 
 + 0161 
 
 + 0414 
 
 •948,0671 
 
 0932 
 
 1196 
 
 1465 
 
 1738 
 
 1-5S 
 
 •948,2015 
 
 2295 
 
 2580 
 
 2868 
 
 3161 
 
 •948,3457 
 
 3758 
 
 4062 
 
 4370 
 
 4682 
 
 1-54 
 
 •948,4998 
 
 5318 
 
 5642 
 
 5970 
 
 6302 
 
 •948,6638 
 
 6977 
 
 7321 
 
 7668 
 
 8019 
 
 1-55 
 
 •948,8374 
 
 8733 
 
 9096 
 
 9463 
 
 9834 
 
 •949,020S 
 
 0587 
 
 0969 
 
 1355 
 
 1745 
 
 1-56 
 
 •949,2139 
 
 2537 
 
 2938 
 
 3344 
 
 3753 
 
 949,4166 
 
 4583 
 
 5004 
 
 5429 
 
 5857 
 
 1-57 
 
 •949,6289 
 
 6725 
 
 7165 
 
 7609 
 
 8056 
 
 •949,8508 
 
 8963 
 
 9422 
 
 9885 
 
 + 0351 
 
 V58 
 
 •950,0822 
 
 1296 
 
 1774 
 
 2255 
 
 2741 
 
 •950,3230 
 
 3723 
 
 4220 
 
 4720 
 
 5225 
 
 1-59 
 
 ■950,5733 
 
 6245 
 
 6760 
 
 7280 
 
 7803 
 
 •950,8330 
 
 8860 
 
 9395 
 
 9933 
 
 +0475 
 
 1-60 
 
 •951,1020 
 
 1569 
 
 2122 
 
 2679 
 
 3240 
 
 •951,3804 
 
 4372 
 
 4943 
 
 5519 
 
 6098 
 
 1-61 
 
 •951,6680 
 
 7267 
 
 7857 
 
 8451 
 
 9048 
 
 •951,9649 
 
 +0254 
 
 +0862 
 
 + 1475 
 
 + 2091 
 
 V62 
 
 ■952,2710 
 
 3333 
 
 3960 
 
 4591 
 
 5225 
 
 '952,5863 
 
 6504 
 
 7149 
 
 7798 
 
 8451 
 
 1-63 
 
 •952,9107 
 
 9766 
 
 +0430 
 
 + 1097 
 
 + 1767 
 
 953,2442 
 
 3120 
 
 3801 
 
 4486 
 
 5175 
 
 VG.i 
 
 •953,5867 
 
 6563 
 
 7263 
 
 7966 
 
 8673 
 
 •953,9383 
 
 + 0097 
 
 + 0815 
 
 + 1536 
 
 + 2260 
 
 1-65 
 
 •954,2989 
 
 3721 
 
 4456 
 
 5195 
 
 5938 
 
 •954,6684 
 
 7434 
 
 8187 
 
 8944 
 
 9704 
 
 1-66 
 
 •955,0468 
 
 1236 
 
 2007 
 
 2782 
 
 3560 
 
 •955,4342 
 
 5127 
 
 5916 
 
 6708 
 
 7504 
 
 1-67 
 
 •955,8303 
 
 9106 
 
 9913 
 
 +0723 
 
 + 1536 
 
 •956,2353 
 
 3174 
 
 3998 
 
 4825 
 
 5656 
 
 1-68 
 
 •956,6491 
 
 7329 
 
 8170 
 
 9015 
 
 9864 
 
 •957,0716 
 
 1571 
 
 2430 
 
 3293 
 
 4159 
 
 1-69 
 
 •957,5028 
 
 5901 
 
 6777 
 
 7657 
 
 8540 
 
 •957,9427 
 
 +031? 
 
 + 1211 
 
 + 2108 
 
 + 3008 
 
 V70 
 
 •958,3912 
 
 4820 
 
 5731 
 
 6645 
 
 7563 
 
 '958,8484 
 
 9409 
 
 +o:m 
 
 + 1268 
 
 + 2203 
 
 1-71 
 
 •959,3141 
 
 4083 
 
 5028 
 
 5977 
 
 6929 
 
 •959,7884 
 
 8843 
 
 9805 
 
 +0771 
 
 + 1740 
 
 1-72 
 
 •960,2712 
 
 3688 
 
 4667 
 
 5650 
 
 6636 
 
 •960,7625 
 
 8618 
 
 9614 
 
 + 0613 
 
 + 1616 
 
 1-73 
 
 •961,2622 
 
 3632 
 
 4645 
 
 5661 
 
 6681 
 
 ■961,7704 
 
 8730 
 
 9760 
 
 + 0793 
 
 + 1830 
 
 1-74 
 
 •962,2869 
 
 3912 
 
 4959 
 
 6009 
 
 7062 
 
 •962,8118 
 
 9178 
 
 +0241 
 
 + 1308 
 
 + 2378 
 
 1-75 
 
 •963,3451 
 
 4527 
 
 5607 
 
 6690 
 
 7776 
 
 •963,8866 
 
 9959 
 
 + 1055 
 
 + 2155 
 
 + 3258 
 
 1-7G 
 
 •964,4364 
 
 5473 
 
 6586 
 
 7702 
 
 8821 
 
 ■964,9944 
 
 + 1070 
 
 + 2199 
 
 +3331 
 
 + 4467 
 
 1-77 
 
 •965,5606 
 
 6749 
 
 7894 
 
 9043 
 
 +0195 
 
 •966,1350 
 
 2509 
 
 3671 
 
 4836 
 
 6004 
 
 1-78 
 
 •966,7176 
 
 8351 
 
 9529 
 
 +0710 
 
 + 1895 
 
 •967,3082 
 
 4274 
 
 5468 
 
 6665 
 
 7866 
 
 1-79 
 
 •967,9070 
 
 + 0277 
 
 + 1488 
 
 +2701 
 
 + 3918 
 
 •968,5138 
 
 6361 
 
 7588 
 
 8818 
 
 + 0051 
 
 V80 
 
 ■969,1287 
 
 2526 
 
 3768 
 
 5014 
 
 6263 
 
 •969,7515 
 
 8770 
 
 + 0029 
 
 + 1291 
 
 + 2555 
 
 181 
 
 •970,3823 
 
 5095 
 
 6369 
 
 7646 
 
 8927 
 
 •971,0211 
 
 1498 
 
 2788 
 
 4082 
 
 5378 
 
 1-82 
 
 •971,6678 
 
 7981 
 
 9287 
 
 +0596 
 
 + 1908 
 
 •972,3224 
 
 4542 
 
 . 5864 
 
 7189 
 
 8517 
 
 1-8S 
 
 •972,9848 
 
 + 1182 
 
 + 2520 
 
 + 3860 
 
 + 5204 
 
 •973,6551 
 
 7900 
 
 9254 
 
 + 0610 
 
 + 1969 
 
 V8 J, 
 
 •974,3331 
 
 4697 
 
 6065 
 
 7437 
 
 8812 
 
 •975,0190 
 
 1571 
 
 2955 
 
 4342 
 
 5733 
 
 V85 
 
 •975,7126 
 
 8522 
 
 9922 
 
 + 1325 
 
 + 2730 
 
 •976,4139 
 
 5551 
 
 6966 
 
 8384 
 
 9805 
 
 1-86 
 
 ■977,1230 
 
 2657 
 
 4087 
 
 5521 
 
 6957 
 
 •977,8397 
 
 9839 
 
 + 1285 
 
 + 2734 
 
 + 4186 
 
 1-87 
 
 •978,5640 
 
 7098 
 
 8559 
 
 + 0023 
 
 + 1490 
 
 •979,2960 
 
 4433 
 
 5909 
 
 7389 
 
 8871 
 
 1-88 
 
 •980,0356 
 
 1844 
 
 3335 
 
 4830 
 
 6327 
 
 •980,7827 
 
 9331 
 
 +0837 
 
 + 2346 
 
 + 3859 
 
 189 
 
 •981,5374 
 
 6893 
 
 8414 
 
 9939 
 
 + 1466 
 
 •982,2996 
 
 4530 
 
 6066 
 
 7606 
 
 9148 
 
 1-90 
 
 •983,0693 
 
 2242 
 
 3793 
 
 5348 
 
 6905 
 
 •983,8465 
 
 + 0028 
 
 + 1595 
 
 + 3164 
 
 +4736 
 
 1-91 
 
 ■984,6311 
 
 7890 
 
 9471 
 
 + 1055 
 
 + 2642 
 
 •985,4232 
 
 5825 
 
 7421 
 
 9020 
 
 +0621 
 
 V92 
 
 •986,2226 
 
 3834 
 
 5445 
 
 7058 
 
 8675 
 
 •987,0294 
 
 1917 
 
 3542 
 
 5170 
 
 6802 
 
 1-93 
 
 •987,8436 
 
 +0073 
 
 + 1713 
 
 +3356 
 
 + 5002 
 
 •988,6651 
 
 8302 
 
 9957 
 
 + 1614 
 
 + 3275 
 
 1M 
 
 •989,4938 
 
 6605 
 
 8274 
 
 9946 
 
 + 1621 
 
 ■990,3299 
 
 4980 
 
 6663 
 
 8350 
 
 + 0039 
 
 1-95 
 
 •991,1732 
 
 3427 
 
 5125 
 
 6826 
 
 8530 
 
 •992,0237 
 
 1947 
 
 3659 
 
 5375 
 
 7093 
 
 1-96 
 
 •992,8815 
 
 +0539 
 
 + 2266 
 
 + 3995 
 
 + 5728 
 
 ■993,7464 
 
 9202 
 
 + 0943 
 
 + 2688 
 
 + 4435 
 
 1-97 
 
 •994,6185 
 
 7937 
 
 9693 
 
 + 1451 
 
 + 3213 
 
 •995,4977 
 
 6744 
 
 8513 
 
 +»±m 
 
 + 2062 
 
 1-98 
 
 •996,3840 
 
 5621 
 
 7405 
 
 9192 
 
 + 0982 
 
 •997,2774 
 
 4569 
 
 6368 
 
 8169 
 
 9972 
 
 1-99 
 
 •998,1779 
 
 3588 
 
 5401 
 
 7216 
 
 9034 
 
 •999,0854 
 
 2678 
 
 4504 
 
 6333 
 
 8165 
 
 A horizontal bar means that the third figure of the mantissa has changed, a positive sign that it must he 
 raised one unit. 
 
Tables of the T-Function 
 
 61 
 
 
 
 
 Dirt'EKENCES : — on 
 
 this page 
 
 Positive 
 
 
 
 
 P 
 
 
 
 1 
 
 t 
 
 S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 200 
 
 205 
 
 209 
 
 213 
 
 217 
 
 221 
 
 225 
 
 229 
 
 233 
 
 237 
 
 161 
 
 241 
 
 245 
 
 249 
 
 253 
 
 257 
 
 261 
 
 264 
 
 269 
 
 273 
 
 277 
 
 V52 
 
 280 
 
 285 
 
 288 
 
 293 
 
 296 
 
 301 
 
 304 
 
 308 
 
 312 
 
 316 
 
 V5S 
 
 320 
 
 324 
 
 328 
 
 332 
 
 336 
 
 339 
 
 344 
 
 347 
 
 351 
 
 355 
 
 1-5^ 
 
 359 
 
 363 
 
 367 
 
 371 
 
 374 
 
 379 
 
 382 
 
 386 
 
 390 
 
 394 
 
 1-55 
 
 398 
 
 401 
 
 406 
 
 409 
 
 413 
 
 417 
 
 421 
 
 425 
 
 428 
 
 432 
 
 l'Se 
 
 436 
 
 440 
 
 444 
 
 447 
 
 452 
 
 455 
 
 459 
 
 463 
 
 466 
 
 471 
 
 1-57 
 
 474 
 
 478 
 
 481 
 
 486 
 
 489 
 
 493 
 
 497 
 
 500 
 
 505 
 
 508 
 
 1-58 
 
 512 
 
 515 
 
 520 
 
 523 
 
 527 
 
 530 
 
 535 
 
 538 
 
 542 
 
 545 
 
 1-59 
 
 549 
 
 553 
 
 557 
 
 561 
 
 564 
 
 568 
 
 571 
 
 576 
 
 579 
 
 582 
 
 1-60 
 
 587 
 
 590 
 
 594 
 
 597 
 
 601 
 
 605 
 
 608 
 
 613 
 
 616 
 
 619 
 
 1-61 
 
 623 
 
 627 
 
 631 
 
 634 
 
 638 
 
 641 
 
 645 
 
 649 
 
 653 
 
 656 
 
 VG2 
 
 659 
 
 664 
 
 667 
 
 670 
 
 675 
 
 678 
 
 681 
 
 685 
 
 689 
 
 692 
 
 1-63 
 
 696 
 
 700 
 
 703 
 
 707 
 
 710 
 
 714 
 
 718 
 
 721 
 
 724 
 
 729 
 
 1-OJ, 
 
 732 
 
 735 
 
 739 
 
 743 
 
 746 
 
 750 
 
 753 
 
 757 
 
 760 
 
 764 
 
 vac, 
 
 768 
 
 771 
 
 775 
 
 778 
 
 782 
 
 785 
 
 789 
 
 792 
 
 796 
 
 799 
 
 1-C1U 
 
 803 
 
 807 
 
 810 
 
 813 
 
 817 
 
 821 
 
 824 
 
 827 
 
 831 
 
 835 
 
 1-07 
 
 838 
 
 841 
 
 845 
 
 849 
 
 852 
 
 855 
 
 859 
 
 863 
 
 866 
 
 869 
 
 1-08 
 
 873 
 
 876 
 
 880 
 
 883 
 
 887 
 
 890 
 
 894 
 
 897 
 
 900 
 
 904 
 
 1-69 
 
 908 
 
 911 
 
 914 
 
 918 
 
 921 
 
 925 
 
 928 
 
 931 
 
 935 
 
 938 
 
 1-70 
 
 942 
 
 945 
 
 949 
 
 952 
 
 955 
 
 969 
 
 962 
 
 966 
 
 969 
 
 972 
 
 1-71 
 
 976 
 
 979 
 
 983 
 
 986 
 
 989 
 
 993 
 
 996 
 
 999 
 
 1003 
 
 1006 
 
 V72 
 
 1010 
 
 1013 
 
 1016 
 
 1020 
 
 1023 
 
 1026 
 
 1030 
 
 1033 
 
 1037 
 
 1039 
 
 V7S 
 
 1043 
 
 1047 
 
 1050 
 
 1053 
 
 1056 
 
 1060 
 
 1063 
 
 1067 
 
 1070 
 
 1073 
 
 1-74 
 
 1076 
 
 1080 
 
 1083 
 
 1086 
 
 1090 
 
 1093 
 
 1096 
 
 1100 
 
 1103 
 
 1106 
 
 1-75 
 
 1109 
 
 1113 
 
 1116 
 
 1119 
 
 1123 
 
 1126 
 
 1129 
 
 1132 
 
 1136 
 
 1139 
 
 1-70 
 
 1143 
 
 1145 
 
 1149 
 
 1152 
 
 1155 
 
 1159 
 
 1162 
 
 1165 
 
 1168 
 
 1172 
 
 V77 
 
 1175 
 
 1178 
 
 1181 
 
 1185 
 
 1187 
 
 1192 
 
 1194 
 
 1197 
 
 1201 
 
 1204 
 
 1-78 
 
 1207 
 
 1211 
 
 1213 
 
 1217 
 
 1220 
 
 1223 
 
 1227 
 
 1230 
 
 1233 
 
 1236 
 
 1-79 
 
 1239 
 
 1242 
 
 1246 
 
 1249 
 
 1252 
 
 1255 
 
 1259 
 
 1262 
 
 1264 
 
 1268 
 
 1-80 
 
 1272 
 
 1274 
 
 1277 
 
 1281 
 
 1284 
 
 1287 
 
 1290 
 
 1294 
 
 1296 
 
 1300 
 
 1-81 
 
 1303 
 
 1306 
 
 1309 
 
 1312 
 
 1316 
 
 1318 
 
 1322 
 
 1325 
 
 1328 
 
 1331 
 
 V82 
 
 1334 
 
 1338 
 
 1340 
 
 1344 
 
 1347 
 
 1349 
 
 1354 
 
 1356 
 
 1359 
 
 1302 
 
 1-83 
 
 1366 
 
 1368 
 
 1372 
 
 1375 
 
 1378 
 
 1381 
 
 1384 
 
 1387 
 
 1391 
 
 1393 
 
 1-8 J t 
 
 1396 
 
 1400 
 
 1403 
 
 1405 
 
 1409 
 
 1412 
 
 1415 
 
 1418 
 
 1421 
 
 1425 
 
 1-85 
 
 1427 
 
 1430 
 
 1434 
 
 1436 
 
 1440 
 
 1442 
 
 1446 
 
 1449 
 
 1452 
 
 1454 
 
 1-86 
 
 1458 
 
 1461 
 
 1464 
 
 1467 
 
 1470 
 
 1473 
 
 1476 
 
 1480 
 
 1482 
 
 1485 
 
 1-87 
 
 1488 
 
 1491 
 
 1495 
 
 1497 
 
 1500 
 
 1504 
 
 1506 
 
 1509 
 
 1513 
 
 1515 
 
 1-SS 
 
 1519 
 
 1521 
 
 1525 
 
 1527 
 
 1530 
 
 1534 
 
 1536 
 
 1540 
 
 1542 
 
 1545 
 
 1-89 
 
 1549 
 
 1551 
 
 1555 
 
 1557 
 
 1560 
 
 1563 
 
 1567 
 
 1569 
 
 1572 
 
 1575 
 
 1-90 
 
 1579 
 
 1581 
 
 1584 
 
 1587 
 
 1590 
 
 1593 
 
 1596 
 
 1599 
 
 1601 
 
 1605 
 
 1-91 
 
 1608 
 
 1611 
 
 1613 
 
 1617 
 
 1619 
 
 1623 
 
 1625 
 
 1628 
 
 1632 
 
 1634 
 
 1-92 
 
 1637 
 
 1640 
 
 1643 
 
 1646 
 
 1649 
 
 1651 
 
 1655 
 
 1657 
 
 1661 
 
 1663 
 
 1-93 
 
 1667 
 
 1669 
 
 1672 
 
 1675 
 
 1678 
 
 1681 
 
 1683 
 
 1687 
 
 1689 
 
 1693 
 
 1-94 
 
 1695 
 
 1698 
 
 1701 
 
 1704 
 
 1707 
 
 1710 
 
 1712 
 
 1716 
 
 1718 
 
 1722 
 
 1-95 
 
 1724 
 
 1727 
 
 1729 
 
 1733 
 
 1736 
 
 1738 
 
 1741 
 
 1745 
 
 1747 
 
 1750 
 
 V9G 
 
 1752 
 
 1756 
 
 1758 
 
 1762 
 
 1764 
 
 1767 
 
 1769 
 
 1773 
 
 1776 
 
 1778 
 
 1-97 
 
 1781 
 
 1784 
 
 1787 
 
 1790 
 
 1792 
 
 1795 
 
 1799 
 
 1801 
 
 1803 
 
 1807 
 
 1-98 
 
 1809 
 
 1813 
 
 1815 
 
 1818 
 
 1820 
 
 1824 
 
 1826 
 
 1829 
 
 1832 
 
 1835 
 
 1-99 
 
62 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XXXII. Subtense from Arc and Chord 
 
 Table to pass from measured index /3 = 100 (arc — chord) I chord of a curve to the index 
 and may be closely represented by a common catenary. Suggested use: to pass 
 
 Values of a for given values of /3 as argument. 
 
 (3 
 
 ■0 
 
 ■1 
 
 ■2 
 
 ■3 
 
 ■4 
 
 ■5 
 
 •6 
 
 •7 
 
 ■8 
 
 •9 
 
 13 
 
 23-1 
 
 23-2 
 
 23-2 
 
 23 3 
 
 23-4 
 
 23-5 
 
 23 6 
 
 23-7 
 
 23-8 
 
 23-9 
 
 n 
 
 24-0 
 
 24-1 
 
 24-2 
 
 24-3 
 
 24-4 
 
 24-5 
 
 24-6 
 
 24-7 
 
 24-7 
 
 24-8 
 
 15 
 
 24-9 
 
 25-0 
 
 25-1 
 
 25-2 
 
 25-3 
 
 25-4 
 
 25-5 
 
 25-6 
 
 25-6 
 
 25-7 
 
 10 
 
 25-8 
 
 25-9 
 
 20-0 
 
 20-1 
 
 26-2 
 
 26-3 
 
 26 '4 
 
 26-4 
 
 26-5 
 
 26 6 
 
 17 
 
 267 
 
 20 -8 
 
 2G-9 
 
 27-0 
 
 27-0 
 
 27-1 
 
 27-2 
 
 27-3 
 
 27-4 
 
 27-5 
 
 18 
 
 27-6 
 
 27-7 
 
 27-7 
 
 27-8 
 
 27-9 
 
 28-0 
 
 28-1 
 
 28-2 
 
 28-3 
 
 28-3 
 
 19 
 
 28-4 
 
 28-5 
 
 28-6 
 
 28-7 
 
 28-7 
 
 28-8 
 
 28-9 
 
 29-0 
 
 29-1 
 
 29-2 
 
 20 
 
 29-2 
 
 29 3 
 
 29-4 
 
 29-5 
 
 29 -G 
 
 29 6 
 
 29-7 
 
 29-8 
 
 29-9 
 
 30 
 
 21 
 
 30 '0 
 
 301 
 
 30-2 
 
 30-3 
 
 30 4 
 
 30-4 
 
 30-5 
 
 30-6 
 
 30-7 
 
 30 8 
 
 22 
 
 30 8 
 
 309 
 
 31-0 
 
 31 1 
 
 31-2 
 
 31-2 
 
 313 
 
 314 
 
 31-5 
 
 31-6 
 
 23 
 
 316 
 
 31-7 
 
 31-8 
 
 31 '9 
 
 319 
 
 32-0 
 
 32 1 
 
 32-2 
 
 32-3 
 
 32 3 
 
 24 
 
 32-4 
 
 32-5 
 
 32 -6 
 
 32 -G 
 
 32-7 
 
 32-8 
 
 32-9 
 
 32-9 
 
 33 
 
 33 1 
 
 25 
 
 33 2 
 
 33 3 
 
 33 3 
 
 33 4 
 
 33 5 
 
 33 6 
 
 33 G 
 
 33-7 
 
 33-8 
 
 339 
 
 2G 
 
 33-9 
 
 34 
 
 341 
 
 34-2 
 
 34-2 
 
 34 3 
 
 34-4 
 
 34-5 
 
 34 5 
 
 34-0 
 
 27 
 
 34-7 
 
 34-8 
 
 34-8 
 
 34-9 
 
 35-0 
 
 35-1 
 
 35-1 
 
 35-2 
 
 35 3 
 
 35 3 
 
 28 
 
 35-4 
 
 35 5 
 
 35 6 
 
 35 6 
 
 35-7 
 
 35-8 
 
 35 9 
 
 35-9 
 
 36 
 
 36-1 
 
 2'J 
 
 36-2 
 
 36-2 
 
 363 
 
 3G4 
 
 36-4 
 
 36 5 
 
 36-6 
 
 36-7 
 
 307 
 
 36-8 
 
 30 
 
 36 "9 
 
 36 9 
 
 37-0 
 
 371 
 
 37-2 
 
 37-2 
 
 37 3 
 
 37 4 
 
 37 5 
 
 37-5 
 
 31 
 
 37-6 
 
 37-7 
 
 37-7 
 
 37 8 
 
 37-9 
 
 38-0 
 
 38-0 
 
 38-1 
 
 38-2 
 
 38-2 
 
 S3 
 
 38-3 
 
 38-4 
 
 38-4 
 
 38-5 
 
 38 G 
 
 38-7 
 
 38-7 
 
 38-8 
 
 38-9 
 
 38-9 
 
 33 
 
 39-0 
 
 31) 1 
 
 39-2 
 
 39 2 
 
 39 3 
 
 39-4 
 
 39-4 
 
 39-5 
 
 396 
 
 39 6 
 
 Sit 
 
 397 
 
 39-8 
 
 39 8 
 
 39 9 
 
 40-0 
 
 40-1 
 
 40-1 
 
 40-2 
 
 40-3 
 
 403 
 
 35 
 
 40-4 
 
 40-5 
 
 40-5 
 
 40 6 
 
 40-7 
 
 40-7 
 
 40-8 
 
 40-9 
 
 41 
 
 41-0 
 
 30 
 
 41-1 
 
 41-2 
 
 41-2 
 
 41-3 
 
 41-4 
 
 414 
 
 41-5 
 
 41-6 
 
 41-6 
 
 41-7 
 
 37 
 
 418 
 
 41-8 
 
 41-9 
 
 42-0 
 
 42-0 
 
 42-1 
 
 42-2 
 
 42-2 
 
 42-3 
 
 42-4 
 
 38 
 
 42-4 
 
 42-5 
 
 42-6 
 
 42-6 
 
 42-7 
 
 42-8 
 
 42 '9 
 
 42-9 
 
 43-0 
 
 43-1 
 
 39 
 
 431 
 
 43-2 
 
 43 3 
 
 43 3 
 
 43-4 
 
 43-5 
 
 43-5 
 
 43 
 
 43-7 
 
 43-7 
 
 40 
 
 43-8 
 
 43-9 
 
 43 9 
 
 44-0 
 
 44-1 
 
 44-1 
 
 44-2 
 
 44-3 
 
 44 3 
 
 44-4 
 
 41 
 
 44-5 
 
 44-5 
 
 44-6 
 
 44-6 
 
 44-7 
 
 44-8 
 
 44-8 
 
 44-9 
 
 45-0 
 
 45-0 
 
 42 
 
 45-1 
 
 45-2 
 
 45 2 
 
 45-3 
 
 45-4 
 
 45-4 
 
 45-5 
 
 45-6 
 
 45-6 
 
 45-7 
 
 43 
 
 45-8 
 
 45-8 
 
 45-9 
 
 40 -0 
 
 40 
 
 46-1 
 
 40-2 
 
 46-2 
 
 463 
 
 46-4 
 
 44 
 
 46-4 
 
 46-5 
 
 4G-5 
 
 46 6 
 
 46-7 
 
 46-7 
 
 46-8 
 
 40-9 
 
 46-9 
 
 47-0 
 
 45 
 
 47-1 
 
 47-1 
 
 47-2 
 
 47 3 
 
 47-3 
 
 47-4 
 
 47-5 
 
 47-5 
 
 47-6 
 
 47-6 
 
 40 
 
 47-7 
 
 47-8 
 
 47-8 
 
 47-9 
 
 48-0 
 
 48-0 
 
 48-1 
 
 48-2 
 
 48-2 
 
 48-3 
 
 47 
 
 48-1 
 
 48-4 
 
 48-5 
 
 48-5 
 
 48-0 
 
 48-7 
 
 48-7 
 
 48-8 
 
 48-9 
 
 48-9 
 
 48 
 
 49-0 
 
 49-1 
 
 49-1 
 
 49 '2 
 
 49 '2 
 
 49-3 
 
 49-4 
 
 49-4 
 
 49-5 
 
 49-6 
 
 49 
 
 49 -G 
 
 49-7 
 
 49-8 
 
 49-8 
 
 49-9 
 
 49 9 
 
 50-0 
 
 50-1 
 
 50-1 
 
 50-2 
 
 50 
 
 50 3 
 
 50-3 
 
 50-4 
 
 50-5 
 
 50-5 
 
 50-6 
 
 50-6 
 
 50-7 
 
 50-8 
 
 50-8 
 
 51 
 
 50-9 
 
 51 
 
 51-0 
 
 51-1 
 
 51-1 
 
 51-2 
 
 513 
 
 51-3 
 
 51-4 
 
 51-5 
 
 52 
 
 515 
 
 51-6 
 
 51 G 
 
 51-7 
 
 51-8 
 
 51-8 
 
 51-9 
 
 52-0 
 
 52-0 
 
 52-1 
 
 53 
 
 52-1 
 
 52-2 
 
 52 3 
 
 52-3 
 
 52-4 
 
 52-5 
 
 62-5 
 
 520 
 
 52-6 
 
 52-7 
 
 54 
 
 52-8 
 
 52-8 
 
 52-9 
 
 53-0 
 
 53 
 
 53-1 
 
 53-1 
 
 53 2 
 
 53-3 
 
 53 3 
 
 65 
 
 534 
 
 53 4 
 
 53-5 
 
 53-6 
 
 53-6 
 
 53-7 
 
 53-8 
 
 53-8 
 
 53 9 
 
 53-9 
 
 56 
 
 54 
 
 54-1 
 
 54-1 
 
 54-2 
 
 54-3 
 
 54 '3 
 
 54-4 
 
 54-4 
 
 54-5 
 
 54-0 
 
 57 
 
 54-6 
 
 54-7 
 
 54-7 
 
 54-8 
 
 54 9 
 
 54-9 
 
 55-0 
 
 55-0 
 
 551 
 
 55-2 
 
 58 
 
 55-2 
 
 55-3 
 
 55-4 
 
 55-4 
 
 55-5 
 
 55-5 
 
 55-6 
 
 55-7 
 
 55-7 
 
 55-8 
 
 59 
 
 55-8 
 
 559 
 
 56-0 
 
 56-0 
 
 561 
 
 56-1 
 
 56-2 
 
 50 3 
 
 56 3 
 
 56-4 
 
 60 
 
 56-5 
 
 565 
 
 56-6 
 
 56-6 
 
 56-7 
 
 56 8 
 
 56-8 
 
 56-9 
 
 56-9 
 
 57-0 
 
 61 
 
 57-1 
 
 57-1 
 
 57 2 
 
 57-2 
 
 57-3 
 
 57-4 
 
 57-4 
 
 57-5 
 
 57-5 
 
 57-0 
 
 62 
 
 57-7 
 
 57-7 
 
 57-8 
 
 57 8 
 
 57-9 
 
 58-0 
 
 58 
 
 58-1 
 
 58-1 
 
 58-2 
 
 63 
 
 58 3 
 
 58-3 
 
 58-4 
 
 58-4 
 
 58-5 
 
 58-6 
 
 58-6 
 
 58-7 
 
 58-7 
 
 58-8 
 
 64 
 
 58-9 
 
 58-9 
 
 59'0 
 
 59-0 
 
 59-1 
 
 59-2 
 
 59-2 
 
 59-3 
 
 59 '3 
 
 59-4 
 
Tables of Catenary Indices 
 
 63 
 
 in the case of the Common Catenary, 
 
 a =100 subtense /chord, on the assumption that the curve is symmetrical about the subtense 
 from callipers and tape measurements of the nasal bridge to the ratio of "rise" to "span." 
 
 Values of a for given values of /3 as argument. 
 
 |8 
 
 ■o 
 
 ■1 
 
 ■2 
 
 ■3 
 
 ■4 
 
 ■5 
 
 •6 
 
 •7 
 
 ■8 
 
 ■9 
 
 65 
 
 59 -5 
 
 59 -5 
 
 59-6 
 
 59-6 
 
 59-7 
 
 59-8 
 
 59-8 
 
 59 '9 
 
 59 9 
 
 60 '0 
 
 66 
 
 60-1 
 
 60-1 
 
 60-2 
 
 60-2 
 
 60-3 
 
 60-4 
 
 60-4 
 
 60-5 
 
 60 
 
 5 
 
 60-6 
 
 67 
 
 60-7 
 
 60-7 
 
 60-8 
 
 60-8 
 
 60-9 
 
 61 
 
 61-0 
 
 61-1 
 
 61 
 
 1 
 
 61-2 
 
 68 
 
 61-3 
 
 613 
 
 61-4 
 
 61-4 
 
 615 
 
 616 
 
 61-6 
 
 61-7 
 
 61 
 
 7 
 
 61-8 
 
 69 
 
 61-9 
 
 61-9 
 
 62-0 
 
 62-0 
 
 62-1 
 
 62-1 
 
 62-2 
 
 62-3 
 
 62 
 
 3 
 
 62-4 
 
 70 
 
 62-4 
 
 62-5 
 
 62-6 
 
 62-6 
 
 62-7 
 
 62-7 
 
 62-8 
 
 62-9 
 
 62 
 
 9 
 
 63 
 
 71 
 
 630 
 
 63-1 
 
 631 
 
 63-2 
 
 63 3 
 
 63 3 
 
 63-4 
 
 634 
 
 63 
 
 5 
 
 63 6 
 
 72 
 
 636 
 
 63-7 
 
 637 
 
 63-8 
 
 63 9 
 
 63-9 
 
 64-0 
 
 64 
 
 64 
 
 1 
 
 64-1 
 
 73 
 
 64 2 
 
 64-3 
 
 64 3 
 
 64-4 
 
 64-4 
 
 64 5 
 
 64-6 
 
 64-6 
 
 64 
 
 7 
 
 64-7 
 
 74 
 
 64-8 
 
 64 9 
 
 64-9 
 
 65-0 
 
 65-0 
 
 65 1 
 
 65-1 
 
 65-2 
 
 65 
 
 3 
 
 65 3 
 
 75 
 
 65-4 
 
 65-4 
 
 65-5 
 
 65-6 
 
 65 -6 
 
 65-7 
 
 65-7 
 
 65 -8 
 
 65 
 
 8 
 
 65-9 
 
 76 
 
 66 '0 
 
 66-0 
 
 66-1 
 
 66 2 
 
 662 
 
 66-3 
 
 66 3 
 
 66 4 
 
 66 
 
 4 
 
 66-5 
 
 77 
 
 66-5 
 
 66-6 
 
 66-7 
 
 66-7 
 
 66-8 
 
 66-8 
 
 66 9 
 
 66 9 
 
 67 
 
 
 
 67-1 
 
 78 
 
 67-1 
 
 67-2 
 
 67-2 
 
 67 3 
 
 67-4 
 
 67-4 
 
 67-5 
 
 67-5 
 
 67 
 
 6 
 
 67-6 
 
 79 
 
 67-7 
 
 67-8 
 
 67-8 
 
 679 
 
 67 9 
 
 68-0 
 
 68-0 
 
 681 
 
 68 
 
 2 
 
 68-2 
 
 80 
 
 68-3 
 
 68 3 
 
 68-4 
 
 68-5 
 
 68-5 
 
 68 6 
 
 68-6 
 
 68-7 
 
 68 
 
 7 
 
 68-8 
 
 81 
 
 68-9 
 
 68-9 
 
 69-0 
 
 69-0 
 
 69-1 
 
 69-1 
 
 69-2 
 
 69 3 
 
 69 
 
 3 
 
 69 4 
 
 82 
 
 69-4 
 
 69-5 
 
 69 5 
 
 69-6 
 
 69-7 
 
 69-7 
 
 69-8 
 
 69-8 
 
 69 
 
 9 
 
 70 
 
 83 
 
 70-0 
 
 70-1 
 
 70-1 
 
 70-2 
 
 70-2 
 
 70 3 
 
 70-4 
 
 70-4 
 
 70 
 
 5 
 
 70 5 
 
 84 
 
 70-6 
 
 70-6 
 
 70-7 
 
 70-8 
 
 70-8 
 
 70-9 
 
 70-9 
 
 71-0 
 
 71 
 
 
 
 711 
 
 85 
 
 71-2 
 
 71-2 
 
 71-3 
 
 71-3 
 
 71-4 
 
 71-4 
 
 71-5 
 
 71-6 
 
 71 
 
 6 
 
 71-7 
 
 86 
 
 717 
 
 71-8 
 
 71-8 
 
 71-9 
 
 72-0 
 
 72-0 
 
 72-1 
 
 72-1 
 
 72 
 
 2 
 
 72-2 
 
 87 
 
 72-3 
 
 72-4 
 
 72-4 
 
 72-5 
 
 72-5 
 
 72-6 
 
 72-6 
 
 72-7 
 
 72 
 
 8 
 
 72-8 
 
 88 
 
 729 
 
 72-9 
 
 73 
 
 73 
 
 73-1 
 
 73-2 
 
 73-2 
 
 73 3 
 
 73 
 
 3 
 
 73-4 
 
 89 
 
 734 
 
 73 '5 
 
 73-6 
 
 73-6 
 
 73-7 
 
 73-7 
 
 73-8 
 
 73-8 
 
 73 
 
 9 
 
 73-9 
 
 90 
 
 74 
 
 74-1 
 
 74-1 
 
 74-2 
 
 74-2 
 
 74 3 
 
 74-3 
 
 74 '4 
 
 74 
 
 5 
 
 74-5 
 
 91 
 
 74-6 
 
 74-6 
 
 74-7 
 
 74-7 
 
 74-8 
 
 749 
 
 74 9 
 
 75-0 
 
 75 
 
 
 
 751 
 
 92 
 
 75-1 
 
 75-2 
 
 75-3 
 
 75-3 
 
 75-4 
 
 75-4 
 
 75-5 
 
 75-5 
 
 75 
 
 6 
 
 75-6 
 
 93 
 
 75-7 
 
 75-8 
 
 75-8 
 
 75-9 
 
 75-9 
 
 76-0 
 
 76-0 
 
 76-1 
 
 76 
 
 2 
 
 76-2 
 
 94 
 
 76-3 
 
 76-3 
 
 76-4 
 
 76-4 
 
 76-5 
 
 76-6 
 
 76-6 
 
 76-7 
 
 76 
 
 7 
 
 76-8 
 
 95 
 
 76-8 
 
 76-9 
 
 769 
 
 77-0 
 
 77-1 
 
 77-1 
 
 77-2 
 
 77-2 
 
 77 
 
 3 
 
 77-3 
 
 96 
 
 77-4 
 
 77-5 
 
 77-5 
 
 77-6 
 
 77-6 
 
 77-7 
 
 77-7 
 
 77-8 
 
 77 
 
 8 
 
 77-9 
 
 97 
 
 78-0 
 
 78-0 
 
 78-1 
 
 78-1 
 
 78-2 
 
 78-2 
 
 78-3 
 
 78-3 
 
 78 
 
 4 
 
 78-5 
 
 98 
 
 78-5 
 
 78-6 
 
 78-6 
 
 78-7 
 
 78-7 
 
 78-8 
 
 78-9 
 
 78-9 
 
 79 
 
 
 
 79-0 
 
 99 
 
 79-1 
 
 79-1 
 
 792 
 
 792 
 
 79-3 
 
 79-4 
 
 79-4 
 
 79 5 
 
 79 
 
 5 
 
 79 6 
 
 100 
 
 79-6 
 
 797 
 
 79-8 
 
 79 8 
 
 79 9 
 
 799 
 
 80-0 
 
 80 
 
 80 
 
 1 
 
 80-1 
 
 101 
 
 80-2 
 
 80-3 
 
 80-3 
 
 80-4 
 
 80-4 
 
 80-5 
 
 80-5 
 
 80-6 
 
 80 
 
 6 
 
 80-7 
 
 102 
 
 80-8 
 
 80-8 
 
 80-9 
 
 80-9 
 
 81-0 
 
 81-0 
 
 81-1 
 
 81-1 
 
 81 
 
 2 
 
 81-3 
 
 1CS 
 
 81-3 
 
 81-4 
 
 81 '4 
 
 81-5 
 
 81-5 
 
 816 
 
 81-6 
 
 81-7 
 
 81 
 
 8 
 
 81-8 
 
 104 
 
 8T9 
 
 819 
 
 82-0 
 
 82-0 
 
 82-1 
 
 82-1 
 
 82-2 
 
 82-3 
 
 82 
 
 3 
 
 82-4 
 
 105 
 
 82-4 
 
 82-5 
 
 82-5 
 
 826 
 
 82-6 
 
 82-7 
 
 82-8 
 
 82-8 
 
 82 
 
 9 
 
 82-9 
 
 106 
 
 83-0 
 
 83-0 
 
 83-1 
 
 83-1 
 
 83 2 
 
 83 3 
 
 83-3 
 
 83 4 
 
 83 
 
 4 
 
 83-5 
 
 107 
 
 835 
 
 83-6 
 
 83 6 
 
 83-7 
 
 83-8 
 
 83-8 
 
 83-9 
 
 83-9 
 
 84 
 
 
 
 84-0 
 
 108 
 
 84-1 
 
 84-1 
 
 84-2 
 
 84-3 
 
 84 3 
 
 84-4 
 
 84-4 
 
 84-5 
 
 84 
 
 5 
 
 84-6 
 
 109 
 
 84-6 
 
 84-7 
 
 84-8 
 
 84-8 
 
 84-9 
 
 84-9 
 
 85-0 
 
 85-0 
 
 85 
 
 1 
 
 85-1 
 
 110 
 
 85-2 
 
 85-3 
 
 85-3 
 
 85-4 
 
 85-4 
 
 85-5 
 
 85-5 
 
 85 6 
 
 85 
 
 6 
 
 85-7 
 
 111 
 
 85-8 
 
 85-8 
 
 85-9 
 
 85-9 
 
 86-0 
 
 86-0 
 
 86-1 
 
 86-1 
 
 86 
 
 2 
 
 86-2 
 
 112 
 
 86-3 
 
 86-4 
 
 86-4 
 
 86-5 
 
 86-5 
 
 86-6 
 
 86-6 
 
 86-7 
 
 86 
 
 7 
 
 86-8 
 
 113 
 
 86-9 
 
 86-9 
 
 87-0 
 
 87-0 
 
 87-1 
 
 87-1 
 
 87-2 
 
 87-2 
 
 87 
 
 3 
 
 87-4 
 
 114 
 
 87-4 
 
 87-5 
 
 87-5 
 
 87-0 
 
 87-6 
 
 87-7 
 
 87-7 
 
 87-8 
 
 87 
 
 8 
 
 87-9 
 
 115 
 
 88-0 
 
 88-0 
 
 88-1 
 
 88-1 
 
 88-2 
 
 88-2 
 
 88-3 
 
 88-3 
 
 88 
 
 4 
 
 88-5 
 
 IV, 
 
 88 '5 
 
 88-6 
 
 88-6 
 
 88-7 
 
 88-7 
 
 88-8 
 
 88-8 
 
 88-9 
 
 88-9 
 
 89-0 
 
64 
 
 Tables for Statisticians and Bio metricians 
 
 TABLE XXXIII, A and B. Supplementary Tables of Subtense from 
 
 Arc and Chord. 
 
 TABLE XXXIII. 
 
 Supplementary Tables for Subtense Index a as calculated from tlie 
 arcual value B on the Catenary Hypothesis. 
 
 
 
 
 (A) 
 
 Values of a 
 
 for low B- 
 
 
 
 
 /9 
 
 ■0 
 
 •1 
 
 '2 
 
 ■8 
 
 ■4 
 
 •5 
 
 ■6 
 
 •7 
 
 ■8 
 
 ■9 
 
 6 
 
 153 
 
 15-4 
 
 15-6 
 
 15-7 
 
 15-8 
 
 16-0 
 
 161 
 
 162 
 
 163 
 
 16-5 
 
 7 
 
 16-6 
 
 16-7 
 
 168 
 
 17-0 
 
 171 
 
 17-2 
 
 17-3 
 
 174 
 
 17-6 
 
 17-7 
 
 8 
 
 17-8 
 
 17-9 
 
 18-0 
 
 18-1 
 
 18-3 
 
 18-4 
 
 18-5 
 
 18-6 
 
 18-7 
 
 18-8 
 
 9 
 
 18-9 
 
 19-1 
 
 19-2 
 
 193 
 
 19-4 
 
 19-5 
 
 19-6 
 
 19-7 
 
 19-8 
 
 19 1) 
 
 10 
 
 20-0 
 
 20-1 
 
 20-2 
 
 20-3 
 
 20-4 
 
 20-6 
 
 20-7 
 
 20-8 
 
 20-9 
 
 21-0 
 
 11 
 
 21-1 
 
 21-2 
 
 2T3 
 
 21-4 
 
 21-5 
 
 21-6 
 
 21-7 
 
 21-8 
 
 219 
 
 22-0 
 
 12 
 
 22-1 
 
 22-2 
 
 22-3 
 
 22-4 
 
 22-5 
 
 22-6 
 
 22-7 
 
 22-8 
 
 229 
 
 23-0 
 
 (B) Values of a. for high B. 
 
 a 
 
 a 
 
 
 
 a 
 
 
 
 a 
 
 f» 
 
 a 
 
 /3 
 
 a 
 
 
 
 a 
 
 101 
 
 80-2 
 
 126 
 
 94-0 
 
 151 
 
 107-5 
 
 176 
 
 120-8 
 
 201 
 
 133-9 
 
 226 
 
 147-0 
 
 102 
 
 80 
 
 8 
 
 127 
 
 94 
 
 6 
 
 152 
 
 108-0 
 
 177 
 
 121 
 
 3 
 
 202 
 
 134-5 
 
 227 
 
 147-5 
 
 10S 
 
 81 
 
 3 
 
 128 
 
 95 
 
 I 
 
 153 
 
 108-5 
 
 178 
 
 121 
 
 8 
 
 203 
 
 13.V0 
 
 228 
 
 148-0 
 
 104, 
 
 81 
 
 :) 
 
 129 
 
 95 
 
 e 
 
 154 
 
 109-1 
 
 179 
 
 122 
 
 4 
 
 204 
 
 135-5 
 
 229 
 
 148-6 
 
 105 
 
 82 
 
 i 
 
 ISO 
 
 96 
 
 2 
 
 155 
 
 109-6 
 
 180 
 
 122 
 
 it 
 
 205 
 
 136-0 
 
 230 
 
 149-1 
 
 106 
 
 83 
 
 
 
 181 
 
 96 
 
 7 
 
 156 
 
 110-2 
 
 181 
 
 123 
 
 4 
 
 206 
 
 136 6 
 
 281 
 
 149 6 
 
 107 
 
 83 
 
 5 
 
 132 
 
 97 
 
 2 
 
 157 
 
 110-7 
 
 182 
 
 124 
 
 
 
 207 
 
 137-1 
 
 232 
 
 150-1 
 
 108 
 
 84 
 
 1 
 
 133 
 
 97 
 
 8 
 
 158 
 
 111-2 
 
 183 
 
 124 
 
 5 
 
 208 
 
 1376 
 
 233 
 
 150 6 
 
 109 
 
 84 
 
 6 
 
 131, 
 
 98 
 
 a 
 
 159 
 
 111-8 
 
 184 
 
 125 
 
 a 
 
 209 
 
 138 1 
 
 284 
 
 151-2 
 
 110 
 
 85 
 
 2 
 
 185 
 
 98 
 
 8 
 
 160 
 
 112-3 
 
 185 
 
 125 
 
 5 
 
 210 
 
 138-7 
 
 235 
 
 151-7 
 
 111 
 
 85 
 
 H 
 
 136 
 
 99 
 
 1 
 
 101 
 
 112-8 
 
 186 
 
 126 
 
 1 
 
 211 
 
 139-2 
 
 236 
 
 152-2 
 
 112 
 
 86 
 
 3 
 
 137 
 
 99 
 
 9 
 
 162 
 
 113-4 
 
 187 
 
 126 
 
 6 
 
 212 
 
 139-7 
 
 237 
 
 152-7 
 
 118 
 
 86 
 
 it 
 
 188 
 
 100 
 
 6 
 
 163 
 
 113-9 
 
 188 
 
 127 
 
 1 
 
 218 
 
 140-2 
 
 238 
 
 153-2 
 
 111, 
 
 87 
 
 4 
 
 139 
 
 101 
 
 
 
 164 
 
 114-4 
 
 189 
 
 127 
 
 7 
 
 214 
 
 140-8 
 
 23'J 
 
 153 8 
 
 115 
 
 88 
 
 
 
 HO 
 
 101 
 
 6 
 
 165 
 
 114-9 
 
 190 
 
 128 
 
 2 
 
 215 
 
 141-3 
 
 240 
 
 154 3 
 
 116 
 
 88 
 
 5 
 
 11,1 
 
 102 
 
 ] 
 
 166 
 
 115-5 
 
 191 
 
 128 
 
 7 
 
 216 
 
 141-8 
 
 241 
 
 154-8 
 
 117 
 
 89 
 
 1 
 
 142 
 
 102 
 
 7 
 
 167 
 
 1160 
 
 192 
 
 129 
 
 2 
 
 217 
 
 142-3 
 
 242 
 
 155-3 
 
 118 
 
 89 
 
 (i 
 
 148 
 
 103 
 
 ■2 
 
 168 
 
 116-5 
 
 193 
 
 129 
 
 8 
 
 218 
 
 142-8 
 
 243 
 
 155-8 
 
 119 
 
 90 
 
 1 
 
 W 
 
 103 
 
 7 
 
 169 
 
 117-1 
 
 194 
 
 130 
 
 3 
 
 219 
 
 143-4 
 
 244 
 
 156-3 
 
 120 
 
 90 
 
 7 
 
 145 
 
 104 
 
 8 
 
 170 
 
 117-6 
 
 195 
 
 130 
 
 8 
 
 220 
 
 143-9 
 
 245 
 
 156-9 
 
 121 
 
 91 
 
 2 
 
 146 
 
 104 
 
 8 
 
 171 
 
 118-1 
 
 196 
 
 131 
 
 3 
 
 221 
 
 144-4 
 
 246 
 
 157-4 
 
 122 
 
 91 
 
 8 
 
 l/,7 
 
 105 
 
 3 
 
 172 
 
 118-7 
 
 197 
 
 131 
 
 8 
 
 222 
 
 144-9 
 
 247 
 
 157-9 
 
 123 
 
 92 
 
 :s 
 
 148 
 
 105 
 
 9 
 
 173 
 
 119-2 
 
 198 
 
 132 
 
 1 
 
 223 
 
 145-4 
 
 248 
 
 158-4 
 
 121, 
 
 92 
 
 8 
 
 149 
 
 106 
 
 1 
 
 174 
 
 119-7 
 
 199 
 
 132 
 
 it 
 
 224 
 
 146-0 
 
 249 
 
 159-0 
 
 125 
 
 93-4 
 
 150 
 
 106-9 
 
 175 
 
 120-2 
 
 200 
 
 133-4 
 
 225 
 
 146 5 
 
 250 
 
 159-5 
 
s 
 r2 
 
 ^ 
 
 © 
 I 
 
 to 
 
 a 
 5 
 
 fe> 
 
 I 
 
 is 
 
 g 
 
 s 
 
 s 
 
 
 
 X 
 M 
 
 X 
 
 Diagram of Mean Contingency 
 
 \ 
 
 X 
 
 v 
 
 : 
 
 i: 
 
 *> s % % $ « * 
 
 m 
 
 65 
 
06 Tables for Statisticians and Biometricians 
 
 XXXV. Diagram to determine the type of a Frequency Distribution from a. knowledge 
 of the Constants j3 t and /3 a . Customary Values of yS, and /8 2 . 
 
 •1 2 -3 -4 -5 -6 7 -8 
 
 A 
 
 •9 
 
 1-0 M 1-2 1-3 1-4 1-5 1-6 17 1-8 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (f 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 { k 
 
 r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 Uj 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 /t 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 u 
 
 ^ 
 
 I, 
 
 
 
 
 
 
 
 
 u T 
 
 
 
 
 
 
 
 
 
 
 
 j i 
 
 
 
 
 
 Jl 
 
 
 
 
 
 
 
 
 
 
 
 (( 
 
 i\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n 
 
 
 
 
 i>*N^ 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 U I 
 
 
 
 
 
 
 
 
 l 
 
 I 
 
 
 
 
 
 T 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J I 
 
 
 
 
 
 
 
 6j 
 
 TC 
 
 
 
 
 
 mV— 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 r 
 
 
 
 
 
 iiJ/^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 T 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 XvX 
 
 
 
 
 
 1 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 "tiii^ 
 
 
 
 
 
 
 
 
 > 
 
 
 
 
 
 r 
 
 r 
 
 
 
 
 
 r^^ 
 
 
 
 
 
 
 
 f\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 VI 
 
 
 tnil 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^V>^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 p 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 VI 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 IV 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Heti 
 
 r0 $Ki 
 
 P/C 
 
 
 
 
 
 
 
 
 
 
 
 
 Tv 
 
 7 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 5>^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0> -^ 
 
 Q_ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Frequency Type from ft and ft 
 
 67 
 
 XXXVI. Diagram showing Distribution of Frequency Types for High Values 
 
 for ft and ft. 
 
 ft 
 
 
 
 10 20 30 
 
 40 
 
 50 80 
 
 
 vvvXX. 
 
 
 
 
 
 o Vsn. ^ 
 
 
 
 
 
 
 
 
 
 50 
 
 
 
 u, ^~~"- 
 
 
 .10 
 
 
 
 
 
 
 \ \ w 
 
 J, N^ 
 
 
 • 
 
 
 \ \ V s 
 
 
 
 
 eo 
 
 \ \ \ 
 
 V \ * 
 
 \ \ 
 
 
 J, 
 
 
 80 
 
 \ \ 
 
 \ \ 
 
 Heterotypic \ \ 
 
 
 
 
 100 
 
 i \ 
 
 
 
 
 
 
 
 
 m 
 
 y \ 
 
 HHmtyOtc 
 
 - 
 
 
 (Q. >'-'o 
 
 i Heterotypic^ 
 
 
 
 
 140 
 
 \ 
 
 
 
 
 
 \ V, 1 
 
 A-« 
 
 
 A , so 
 
 
 i 
 
 •o 
 
 
 
 100 
 
 i 
 
 ■6 
 
 
 i 
 
 
 i \ 
 
 * 
 
 
 
 
 \ 
 
 | 
 
 
 <3 
 
 
 ^ 
 
 
 * 
 
 180 
 
 \ 
 \ 
 
 
 
 1 
 
 1 
 
 200 
 
 \ 
 
 \ 
 
 
 
 
 >?.o 
 
 \ 
 i 
 \ 
 
 \ 
 
 
 
 940 
 
 
 
 
 
 9—2 
 
68 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XXXVII. To find the Probable Error of ft. 
 
 Values of ViVSp . 
 
 ft 
 
 
 o-oo 
 
 0-05 
 
 o-io 
 
 0-15 
 
 0-20 
 
 0-25 
 
 0-30 
 
 0-85 
 
 040 
 
 0-45 
 
 OSO 
 
 0-55 
 
 o-co 
 
 0-65 
 
 0-70 
 
 0-75 
 
 2-0 
 2-1 
 2-2 
 2S 
 2-4. 
 2-5 
 2-6 
 2-7 
 2-8 
 2-9 
 SO 
 3-1 
 8-2 
 SS 
 
 S'4 
 8-5 
 8-6 
 3-7 
 3-8 
 8-9 
 
 h-o 
 
 Jf-2 
 4-3 
 
 4'4 
 
 k-5 
 4'6 
 4-7 
 4-8 
 4-9 
 5-0 
 5-1 
 5-2 
 5-8 
 5-4 
 5-5 
 6-6 
 5-7 
 6-8 
 5-9 
 6-0 
 6-1 
 
 6-8 
 6-4 
 6-5 
 6-6 
 6-7 
 6-8 
 6-9 
 7-0 
 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 
 000 
 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 
 0-58 
 0-59 
 
 o-eo 
 
 0-62 
 0-64 
 0-66 
 0-69 
 0-73 
 0-77 
 0-81 
 0-87 
 0-94 
 1-02 
 1-12 
 1-24 
 1-37 
 1-50 
 1-64 
 1-78 
 1-93 
 2-10 
 
 0-93 
 0-95 
 0-97 
 0-99 
 1-02 
 1-05 
 1-10 
 1-15 
 1-22 
 1-30 
 1-40 
 1-53 
 1-67 
 1-82 
 1-99 
 2-16 
 2-33 
 2-50 
 2-67 
 2-86 
 3-07 
 3-29 
 3-53 
 3-78 
 4 05 
 4-33 
 
 1-15 
 1-12 
 1-13 
 1-15 
 1-20 
 1-26 
 1-34 
 1-42 
 1-51 
 1-61 
 1-73 
 1-86 
 2-02 
 2-20 
 2-38 
 2-57 
 2-78 
 2-99 
 3-20 
 3-43 
 3-69 
 3-87 
 4-19 
 4-52 
 4-85 
 5-18 
 
 1-37 
 
 1-30 
 1-30 
 1-32 
 1-37 
 1-45 
 1-54 
 1-64 
 1-75 
 1-87 
 2-01 
 2-17 
 2-33 
 2-50 
 2-68 
 2-89 
 311 
 3-36 
 3-62 
 3-89 
 4-17 
 4-47 
 4-79 
 5-13 
 5-49 
 5-88 
 
 1-57 
 1-50 
 1-48 
 1-49 
 1-54 
 1-61 
 1-72 
 1-83 
 1-94 
 2-06 
 2-20 
 2-35 
 2-52 
 2-71 
 2-93 
 317 
 3-43 
 3-70 
 3-97 
 4-25 
 4-55 
 4-87 
 5-21 
 5-58 
 5-98 
 6-42 
 
 1-77 
 1-70 
 1-67 
 1-67 
 1-70 
 1-76 
 1-86 
 1-96 
 2-07 
 2-20 
 2 34 
 2-51 
 2-71 
 2-92 
 3-14 
 3-39 
 3-65 
 3-93 
 4-23 
 4-54 
 4-87 
 5-21 
 5-58 
 5-97 
 6-40 
 6-87 
 7-37 
 7-90 
 8-46 
 9 05 
 9-66 
 
 1-97 
 
 1-90 
 1-86 
 1-81 
 1-86 
 1-91 
 2-00 
 2-09 
 2-20 
 2-33 
 2-47 
 2-64 
 2-84 
 3-06 
 3-30 
 3-56 
 3-84 
 4-14 
 4-46 
 4-79 
 5-13 
 5-49 
 5-88 
 6-29 
 6-74 
 7-23 
 7-76 
 8-31 
 8-88 
 9-47 
 10-08 
 
 2-17 
 2-10 
 2-05 
 2-02 
 2-02 
 2-05 
 2-13 
 2-22 
 2-32 
 2-44 
 2-57 
 2-75 
 2-95 
 3-18 
 3-43 
 3-69 
 3-99 
 4-31 
 4-64 
 4-97 
 5-32 
 5-69 
 6-10 
 6-54 
 7-01 
 7-52 
 8-07 
 8-64 
 9-24 
 9-86 
 10-50 
 
 2-38 
 2-29 
 2-22 
 2-19 
 2-18 
 2-19 
 2-25 
 2-32 
 2-41 
 2-53 
 2-67 
 2-85 
 3-05 
 3-28 
 3-53 
 3-81 
 4-12 
 4-44 
 4-77 
 5-11 
 5-48 
 5-87 
 6-30 
 6-75 
 7-24 
 7-75 
 8-30 
 8-90 
 9-54 
 10-21 
 10-90 
 
 2-58 
 2-48 
 2-41 
 2-36 
 2-34 
 2-33 
 2-37 
 2-42 
 2-50 
 2-62 
 2-76 
 2-94 
 3-14 
 3-37 
 3 63 
 3-91 
 4-22 
 4-54 
 4-87 
 5-23 
 5-62 
 6-03 
 6-46 
 6 93 
 7-42 
 7-95 
 8-51 
 9-11 
 9-75 
 10-42 
 11-19 
 
 2-80 
 2-69 
 2-61 
 2-55 
 2-51 
 2-49 
 2-50 
 2-53 
 2-60 
 2-70 
 2-84 
 3-00 
 3-22 
 3-44 
 3-70 
 3-98 
 4-29 
 4-61 
 4-95 
 5-32 
 5-72 
 6-15 
 6-60 
 7-07 
 7-57 
 8-10 
 8-66 
 9-25 
 9-89 
 10-58 
 11-33 
 
 3-02 
 2-91 
 2-81 
 2-74 
 2-68 
 2-65 
 2-64 
 2-65 
 2-70 
 2-79 
 2-92 
 3-08 
 3-27 
 3-50 
 3 '75 
 4-02 
 4-33 
 4-66 
 5-00 
 5-38 
 5-79 
 6-23 
 6-69 
 7-18 
 7-68 
 8-21 
 8-76 
 9-35 
 9-99 
 10-69 
 11-44 
 12-26 
 13-10 
 13-98 
 14-91 
 15-90 
 
 3-24 
 3-12 
 
 3-0! 
 8-93 
 
 2-85 
 
 2-81 
 
 2-78 
 
 2-77 
 
 2-80 
 
 2-89 
 
 3-00 
 
 3-15 
 
 333 
 
 3 - 55 
 
 3-79 
 
 4-06 
 
 4-37 
 
 4-70 
 
 6-05 
 
 5-43 
 
 5-84 
 
 6-28 
 
 6'75 
 
 7-25 
 
 7-76 
 
 8-29 
 
 8-85 
 
 9-44 
 
 10-08 
 
 10-78 
 
 11-53 
 
 12-36 
 
 13-26 
 
 14-15 
 
 15-05 
 
 15-98 
 
 3-46 
 
 3-34 
 
 3-22 
 
 3-12 
 
 3 03 
 
 2-97 
 
 2-92 
 
 2-90 
 
 2-93 
 
 2-99 
 
 3-09 
 
 3-23 
 
 3-40 
 
 3-60 
 
 3-83 
 
 4-10 
 
 4-40 
 
 4-72 
 
 5-07 
 
 5-46 
 
 5-88 
 
 6-32 
 
 6-80 
 
 7-29 
 
 7-80 
 
 8-34 
 
 8-91 
 
 9-50 
 
 10-14 
 
 10-84 
 
 11-60 
 
 12-42 
 
 13-29 
 
 14-18 
 
 15-10 
 
 16 05 
 
 3-71 
 
 3 57 
 3-44 
 3-32 
 3-22 
 3-14 
 3-08 
 3-05 
 3-05 
 3-09 
 3-18 
 3-30 
 3-46 
 3-65 
 3-87 
 4-12 
 
 4 41 
 4-74 
 5-09 
 5-48 
 5-89 
 33 
 681 
 7-31 
 7-83 
 8-37 
 8-95 
 9-54 
 
 10-18 
 10-80 
 11-64 
 12-43 
 13-29 
 14-18 
 15-11 
 16-07 
 
Probable Errors of Frequency Constants 
 
 69 
 
 TABLE XXXVII.— (continued). 
 
 Values of ViV%,. 
 
 ft 
 
 0-80 
 
 0-85 
 
 0-90 
 
 0-95 
 
 1-00 
 
 1-05 
 
 1-10 
 
 1-15 
 
 V20 
 
 V25 
 
 ISO 
 
 1-86 
 
 'l'40 
 
 1-45 
 
 1-5U 
 
 
 3-96 
 
 4-21 
 
 4-47 
 
 4-73 
 
 5-00 
 
 5-27 
 
 5-55 
 
 5-83 
 
 6-12 
 
 6-41 
 
 6-71 
 
 7-01 
 
 7-31 
 
 7-62 
 
 7-94 
 
 2-0 
 
 3-80 
 
 4-03 
 
 4 
 
 87 
 
 4-53 
 
 4-80 
 
 5-07 
 
 5-34 
 
 5-62 
 
 5-90 
 
 6-18 
 
 6-48 
 
 6-77 
 
 7-07 
 
 7-37 
 
 7-69 
 
 2-1 
 
 3-66 
 
 3-88 
 
 4 
 
 11 
 
 4 36 
 
 4-63 
 
 4-88 
 
 5-15 
 
 5-42 
 
 5-69 
 
 5-96 
 
 6-25 
 
 6-54 
 
 6-84 
 
 7-14 
 
 7-45 
 
 2-2 
 
 3-52 
 
 3-74 
 
 3 
 
 !)i; 
 
 4-20 
 
 4-46 
 
 4-71 
 
 4-96 
 
 5-22 
 
 5-48 
 
 5-75 
 
 6-02 
 
 6-31 
 
 6-61 
 
 6-91 
 
 7-21 
 
 2S 
 
 3-41 
 
 3-62 
 
 3 
 
 83 
 
 4-05 
 
 4-29 
 
 4-54 
 
 4-78 
 
 5-03 
 
 5-28 
 
 5-55 
 
 5-82 
 
 6-10 
 
 6-38 
 
 6-68 
 
 6-97 
 
 2-4 
 
 3-32 
 
 3-51 
 
 3 
 
 71 
 
 3 92 
 
 4-15 
 
 4-38 
 
 4-61 
 
 4-85 
 
 5-10 
 
 5-36 
 
 5-62 
 
 5-89 
 
 6-16 
 
 6-45 
 
 6-74 
 
 2-6 
 
 325 
 
 3-42 
 
 3 
 
 SO 
 
 3-80 
 
 4-01 
 
 4-23 
 
 4-45 
 
 4-68 
 
 4-92 
 
 5-17 
 
 5-43 
 
 5-68 
 
 5-94 
 
 6-22 
 
 6-51 
 
 2-6 
 
 3-20 
 
 3-35 
 
 3 
 
 51 
 
 3-69 
 
 3-89 
 
 4-10 
 
 4-32 
 
 4-54 
 
 4-76 
 
 5-00 
 
 5-24 
 
 5-48 
 
 5-73 
 
 6-00 
 
 6-28 
 
 2-7 
 
 3-18 
 
 3-32 
 
 3 
 
 -17 
 
 3-63 
 
 3-80 
 
 4-00 
 
 4-21 
 
 4-41 
 
 4-62 
 
 4-84 
 
 5-07 
 
 5 30 
 
 553 
 
 5-79 
 
 6-06 
 
 2-8 
 
 3-19 
 
 3-32 
 
 3 
 
 •15 
 
 3-60 
 
 3-75 
 
 3-92 
 
 4-11 
 
 4-30 
 
 4-49 
 
 4-70 
 
 4-91 
 
 5-12 
 
 5-34 
 
 5-59 
 
 5-85 
 
 2-9 
 
 3-27 
 
 3-38 
 
 3 
 
 49 
 
 3-61 
 
 3-74 
 
 3-87 
 
 4 03 
 
 4-21 
 
 4-39 
 
 4-58 
 
 4-78 
 
 4-98 
 
 5-19 
 
 5-42 
 
 5-68 
 
 s-o 
 
 3-38 
 
 3-47 
 
 3 
 
 57 
 
 3-67 
 
 3-77 
 
 3-89 
 
 4-02 
 
 4-16 
 
 4-31 
 
 4-48 
 
 4-66 
 
 4-85 
 
 5-05 
 
 6-28 
 
 5-53 
 
 s-i 
 
 3-53 
 
 3-60 
 
 3 
 
 68 
 
 3-76 
 
 3-84 
 
 3-93 
 
 4-03 
 
 4-15 
 
 4-28 
 
 4-43 
 
 4-59 
 
 4-75 
 
 4-92 
 
 5-15 
 
 5-40 
 
 3-2 
 
 3-70 
 
 3-75 
 
 3 
 
 81 
 
 3-88 
 
 3-95 
 
 4-02 
 
 4-10 
 
 4-19 
 
 4-28 
 
 4-42 
 
 4-56 
 
 4-69 
 
 4-84 
 
 5-04 
 
 5-28 
 
 8-3 
 
 3-90 
 
 3-93 
 
 3 
 
 97 
 
 4-03 
 
 4-08 
 
 4-14 
 
 4-20 
 
 4-26 
 
 434 
 
 4-45 
 
 4-56 
 
 4-66 
 
 4-78 
 
 4-97 
 
 5-18 
 
 8-4 
 
 4-14 
 
 4-17 
 
 4 
 
 18 
 
 4-22 
 
 4-26 
 
 4-30 
 
 4-34 
 
 4-39 
 
 4-45 
 
 4-52 
 
 4-60 
 
 4-68 
 
 4-79 
 
 4-93 
 
 5-12 
 
 8-5 
 
 4-42 
 
 4-44 
 
 4 
 
 45 
 
 4-47 
 
 4-49 
 
 4-51 
 
 4-54 
 
 4-57 
 
 4-62 
 
 4-67 
 
 4-74 
 
 4-78 
 
 4-81 
 
 4-95 
 
 5-09 
 
 8-6 
 
 4-74 
 
 4-75 
 
 4 
 
 76 
 
 4-76 
 
 4-77 
 
 4-78 
 
 4-79 
 
 4-81 
 
 4-84 
 
 4-87 
 
 4-90 
 
 4 92 
 
 4-95 
 
 5 04 
 
 5-13 
 
 8-7 
 
 5-10 
 
 5-10 
 
 5 
 
 08 
 
 6-08 
 
 5-08 
 
 6-07 
 
 5-07 
 
 5-06 
 
 5-06 
 
 5-08 
 
 5-09 
 
 5-11 
 
 5-14 
 
 5-18 
 
 5-22 
 
 8-8 
 
 5-49 
 
 5-48 
 
 5 
 
 46 
 
 5-44 
 
 5-42 
 
 5-40 
 
 537 
 
 5-35 
 
 5-33 
 
 5-32 
 
 5-32 
 
 5-33 
 
 5-34 
 
 5-35 
 
 5*37 
 
 8-9 
 
 5-89 
 
 5-88 
 
 5 
 
 86 
 
 5-83 
 
 5-80 
 
 5-76 
 
 5-72 
 
 5-69 
 
 5-65 
 
 5-62 
 
 5-60 
 
 5-58 
 
 5-57 
 
 5-57 
 
 5-59 
 
 4-0 
 
 6 33 
 
 6-32 
 
 6 
 
 30 
 
 6-26 
 
 6-21 
 
 6-16 
 
 6-11 
 
 6-06 
 
 6-02 
 
 5-98 
 
 5-94 
 
 5-91 
 
 5-88 
 
 5-86 
 
 5-86 
 
 4-1 
 
 6-80 
 
 6-79 
 
 6 
 
 76 
 
 6-71 
 
 6-65 
 
 6-60 
 
 6-54 
 
 6-48 
 
 6-42 
 
 6-36 
 
 631 
 
 6-27 
 
 6-24 
 
 6-21 
 
 6-18 
 
 4-2 
 
 7-30 
 
 7-28 
 
 7 
 
 25 
 
 7-19 
 
 7-13 
 
 7-07 
 
 7-01 
 
 6-93 
 
 6-87 
 
 6-80 
 
 6-74 
 
 6-67 
 
 6-62 
 
 6-57 
 
 6 53 
 
 4-8 
 
 7-83 
 
 7-80 
 
 7 
 
 70 
 
 7-71 
 
 7-65 
 
 7-58 
 
 7-51 
 
 7-44 
 
 7-37 
 
 7-28 
 
 7-20 
 
 7-12 
 
 7-05 
 
 6-98 
 
 6-92 
 
 4-4 
 
 8-38 
 
 8-36 
 
 8 
 
 32 
 
 8-28 
 
 8-21 
 
 8-14 
 
 8-07 
 
 7-99 
 
 7-90 
 
 7-81 
 
 7-71 
 
 7-61 
 
 7-51 
 
 7-42 
 
 7-34 
 
 4-5 
 
 8-96 
 
 8-95 
 
 8 
 
 91 
 
 8-86 
 
 8-79 
 
 8-72 
 
 8-64 
 
 8-55 
 
 8-45 
 
 8-35 
 
 8-24 
 
 8-13 
 
 8-00 
 
 7-90 
 
 7-80 
 
 4-6 
 
 9-57 
 
 9-57 
 
 9 
 
 53 
 
 9-47 
 
 9-40 
 
 9-33 
 
 9-24 
 
 9-14 
 
 9-04 
 
 8-93 
 
 8-82 
 
 8-69 
 
 8-55 
 
 8-42 
 
 8-31 
 
 4-7 
 
 10-20 
 
 10-23 
 
 10 
 
 10 
 
 10-10 
 
 10-05 
 
 '9-97 
 
 9-88 
 
 9-77 
 
 9-67 
 
 9-55 
 
 9-42 
 
 9-28 
 
 9-14 
 
 9-00 
 
 8-87 
 
 4-8 
 
 10-91 
 
 10-92 
 
 10 
 
 87 
 
 10-80 
 
 10-74 
 
 10-66 
 
 10-57 
 
 10-44 
 
 10-32 
 
 10-18 
 
 10-04 
 
 9-90 
 
 9-76 
 
 9-63 
 
 9-50 
 
 4-9 
 
 11-66 
 
 11-65 
 
 11 
 
 Gl 
 
 11-55 
 
 11-48 
 
 11-39 
 
 11-29 
 
 11-17 
 
 11-04 
 
 10-90 
 
 10-77 
 
 10-62 
 
 10-46 
 
 1030 
 
 10-14 
 
 5-0 
 
 12-45 
 
 12-43 
 
 12 
 
 38 
 
 12-32 
 
 12-24 
 
 12-14 
 
 12-03 
 
 11-91 
 
 11-78 
 
 11-64 
 
 11-50 
 
 11-34 
 
 11-15 
 
 10-96 
 
 10-82 
 
 5-1 
 
 13-28 
 
 13-25 
 
 13 
 
 SO 
 
 1313 
 
 13-04 
 
 12-92 
 
 12-80 
 
 12-67 
 
 12-54 
 
 12-40 
 
 12-24 
 
 12-07 
 
 11-88 
 
 11-72 
 
 11-54 
 
 5-2 
 
 14-16 
 
 14-12 
 
 14 
 
 07 
 
 13-98 
 
 13-87 
 
 13-76 
 
 13-63 
 
 13-47 
 
 13-35 
 
 13-20 
 
 13-02 
 
 12-84 
 
 12-66 
 
 12-48 
 
 12-30 
 
 5-8 
 
 15 09 
 
 15 06 
 
 15 
 
 00 
 
 14-90 
 
 14-78 
 
 14-65 
 
 14-51 
 
 14-36 
 
 14-22 
 
 14-05 
 
 13-81 
 
 13-67 
 
 13-48 
 
 13-29 
 
 13-11 
 
 5-4 
 
 16-06 
 
 16-02 
 
 15 
 
 96 
 
 15-87 
 
 15-76 
 
 15-63 
 
 15-49 
 
 15-33 
 
 15-17 
 
 15-00 
 
 14-81 
 
 14-61 
 
 14-40 
 
 14-18 
 
 13 97 
 
 5-5 
 
 
 
 — 
 
 17 
 
 02 
 
 16-91 
 
 1679 
 
 16-67 
 
 16-51 
 
 16-34 
 
 16-18 
 
 15-95 
 
 15-70 
 
 15-50 
 
 15-30 
 
 15-07 
 
 14-84 
 
 5-6 
 
 
 
 
 
 18 
 
 14 
 
 17-99 
 
 17-88 
 
 17-75 
 
 17-58 
 
 17-40 
 
 17-23 
 
 16-94 
 
 16-70 
 
 16-47 
 
 16-26 
 
 16-04 
 
 15-77 
 
 5-7 
 
 
 
 
 
 19 
 
 34 
 
 19-13 
 
 19-02 
 
 18-87 
 
 18-69 
 
 18-48 
 
 18-26 
 
 17-98 
 
 17-74 
 
 17-50 
 
 17-26 
 
 17-01 
 
 16-76 
 
 5-8 
 
 
 
 
 
 20 
 
 57 
 
 20-36 
 
 20-20 
 
 20-03 
 
 19-84 
 
 19-62 
 
 19-39 
 
 19-11 
 
 18-84 
 
 18-59 
 
 18-32 
 
 18-05 
 
 17-78 
 
 5-9 
 
 
 
 
 
 21 
 
 86 
 
 21-65 
 
 21-45 
 
 21-25 
 
 21-03 
 
 20-79 
 
 20-54 
 
 20-29 
 
 20-02 
 
 19-76 
 
 19-47 
 
 19-18 
 
 18-90 
 
 6-0 
 
 
 
 
 
 
 
 
 
 
 
 _ 
 
 22-36 
 
 22-18 
 
 21-92 
 
 21-61 
 
 21-31 
 
 20-97 
 
 20-61 
 
 20-30 
 
 20-13 
 
 6-1 
 
 
 
 
 
 
 
 23-77 
 
 23-61 
 
 23-32 
 
 23-00 
 
 22 63 
 
 22-22 
 
 21-82 
 
 21-50 
 
 21-29 
 
 6-2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2533 
 
 25-09 
 
 24-74 
 
 24-38 
 
 24-00 
 
 23-55 
 
 23-13 
 
 22-78 
 
 22-50 
 
 0-3 
 
 
 
 
 
 . 
 
 
 
 
 
 
 
 26-95 
 
 26-64 
 
 26-27 
 
 25-86 
 
 25-43 
 
 26-00 
 
 24-52 
 
 24-12 
 
 23-82 
 
 6-4 
 
 
 
 
 
 
 
 28-61 
 
 28-18 
 
 27-73 
 
 27-30 
 
 26-89 
 
 26-46 
 
 26-06 
 27-67 
 29-40 
 31-15 
 3302 
 34-89 
 
 25-65 
 27-21 
 28-90 
 3061 
 32-41 
 34-16 
 
 25'24 
 26-75 
 28-35 
 29-94 
 31-72 
 33-59 
 
 6' -5 
 0-6 
 6-7 
 6-8 
 6-9 
 7-0 
 
70 
 
 Tables for Statisticians and Biornetricians 
 
 TABLE XXXVIII. To find Probable Error of fi» 
 
 Values of JNZp,. 
 
 ft 
 
 
 o-oo 
 
 0-05 
 
 o-io 
 
 0-15 
 
 0-20 
 
 0-25 
 
 oso 
 
 0S5 
 
 0-40 
 
 0-45 
 
 OSO 
 
 OSS 
 
 0-60 
 
 0-65 
 
 0-70 
 
 ' 0-75 
 
 i 
 
 2-0 
 
 1-41 
 
 1-60 
 
 1-74 
 
 1-93 
 
 2-11 
 
 2-28 
 
 2-44 
 
 2-60 
 
 2-77 
 
 2-94 
 
 3-12 
 
 3-29 
 
 3-46 
 
 3-65 
 
 3-86 
 
 4-05 
 
 s-i 
 
 1-57 
 
 1-70 
 
 1-89 
 
 2-00 
 
 2-10 
 
 2-20 
 
 2-35 
 
 2-51 
 
 2-68 
 
 2-86 
 
 305 
 
 3-24 
 
 3-44 
 
 364 
 
 3-84 
 
 I 404 
 
 ** 
 
 1-75 
 
 1-94 
 
 2-07 
 
 216 
 
 2-20 
 
 2-28 
 
 2-40 
 
 2-53 
 
 2-68 
 
 2-85 
 
 304 
 
 1 3-23 
 
 3-43 
 
 3-63 
 
 3-83 
 
 4-03 
 
 as 
 
 1-95 
 
 2-16 
 
 2-28 
 
 2-35 
 
 2-42 
 
 2-49 
 
 2-58 
 
 2-67 
 
 2-78 
 
 2-92 
 
 3 09 
 
 | 3-27 
 
 3-45 
 
 3-64 
 
 3-84 
 
 4-03 
 
 m-4 
 
 2-18 
 
 2-39 
 
 2-53 
 
 2-60 
 
 2-72 
 
 2-82 
 
 2-90 
 
 2-96 
 
 3 02 
 
 3-10 
 
 3-22 
 
 335 
 
 3-50 
 
 36S 
 
 3-86 
 
 4 03 
 
 2-5 
 
 2-46 
 
 2-68 
 
 2-83 
 
 2-97 
 
 309 
 
 3-19 
 
 327 
 
 3-31 
 
 3-32 
 
 3-36 
 
 3-47 
 
 3-53 
 
 3-63 
 
 3-75 
 
 3-88 
 
 4-03 
 
 2-6 
 
 2-78 
 
 303 
 
 3-24 
 
 3-38 
 
 3-52 
 
 3-60 
 
 3-66 
 
 3-69 
 
 3-69 
 
 3-70 
 
 3-75 
 
 3-78 
 
 3-83 
 
 3-87 
 
 3-95 
 
 4-07 
 
 2-7 
 
 3-17 
 
 3-48 
 
 3-71 
 
 3-87 
 
 3-98 
 
 4-03 
 
 4-08 
 
 4-12 
 
 4-11 
 
 4-09 
 
 4-07 
 
 4-06 
 
 4-06 
 
 4-06 
 
 4-07 
 
 4-15 
 
 2-8 
 
 3-64 
 
 4-02 
 
 4-26 
 
 4-42 
 
 4-52 
 
 4-58 
 
 4-60 
 
 4-59 
 
 4-57 
 
 4-52 
 
 4-44 
 
 4-39 
 
 4-34 
 
 432 
 
 4-31 
 
 4-34 
 
 2-9 
 
 4-22 
 
 4-65 
 
 4-94 
 
 5-11 
 
 5-20 
 
 5-22 
 
 5-18 
 
 5-13 
 
 5-07 
 
 4-99 
 
 490 
 
 4-80 
 
 4-70 
 
 463 
 
 4-60 
 
 4-61 
 
 3-0 
 
 4-90 
 
 5-48 
 
 5-76 
 
 5-89 
 
 5-95 
 
 5-93 
 
 5-86 
 
 5-76 
 
 5 65 
 
 5-53 
 
 5-41 
 
 5-30 
 
 5-20 
 
 5-12 
 
 5-05 
 
 5-00 
 
 8-1 
 
 5-75 
 
 6-41 
 
 6-72 
 
 6-88 
 
 6-90 
 
 6-82 
 
 6-70 
 
 6-54 
 
 6-38 
 
 6-22 
 
 6-07 
 
 5-92 
 
 5-79 
 
 5-69 
 
 560 
 
 5-53 
 
 3-2 
 
 6-77 
 
 7-55 
 
 7-90 
 
 8-00 
 
 7-97 
 
 7-83 
 
 7-63 
 
 7-42 
 
 7-21 
 
 7-03 
 
 6-86 
 
 6-70 
 
 6-53 
 
 6-39 
 
 626 
 
 6-14 
 
 3S 
 
 8-00 
 
 8-83 
 
 9-22 
 
 9-30 
 
 9-22 
 
 9-02 
 
 8-80 
 
 8-53 
 
 8-29 
 
 8-05 
 
 7-83 
 
 7-60 
 
 7-38 
 
 7-20 
 
 7-02 
 
 6-84 
 
 3-4 
 
 9-37 
 
 10-28 
 
 10-68 
 
 10-76 
 
 10-67 
 
 10-46 
 
 10-20 
 
 9-91 
 
 9-62 
 
 9-31 
 
 901 
 
 8-73 
 
 8-44 
 
 8-18 
 
 7-92 
 
 7-66 
 
 3-5 
 
 10-85 
 
 11-75 
 
 12-31 
 
 1252 
 
 12-46 
 
 12-25 
 
 11-95 
 
 11-60 
 
 11-24 
 
 10-86 
 
 10-45 
 
 10-03 
 
 9-63 
 
 9-26 
 
 8-90 
 
 8-54 
 
 S-6 
 
 12-67 
 
 13-74 
 
 14-40 
 
 14-78 
 
 14-53 
 
 14-21 
 
 13-80 
 
 13-38 
 
 12-95 
 
 12-55 
 
 1210 
 
 11-60 
 
 1106 
 
 1054 
 
 1002 
 
 955 
 
 3-7 
 
 14-78 
 
 15-98 
 
 16-78 
 
 17-09 
 
 16-93 
 
 16-53 
 
 1605 
 
 15-58 
 
 15 09 
 
 14-61 
 
 14-08 
 
 13-49 
 
 12-74 
 
 12-02 
 
 11-36 
 
 10-80 
 
 3-8 
 
 17-50 
 
 18-83 
 
 19-83 
 
 20-03 
 
 19-78 
 
 19-36 
 
 18-76 
 
 18-22 
 
 17-64 
 
 16-98 
 
 16-25 
 
 15-30 
 
 14-42 
 
 13-60 
 
 12-88 
 
 12-27 
 
 3-9 
 
 20-80 
 
 22-50 
 
 23-68 
 
 23-81 
 
 23-34 
 
 22-67 
 
 21-98 
 
 21-14 
 
 20-29 
 
 19-45 
 
 18-58 
 
 17-54 
 
 16-50 
 
 1554 
 
 14-77 
 
 14-06 
 
 4-o 
 
 24-74 
 
 26-83 
 
 28-47 
 
 28-05 
 
 27-24 
 
 26-29 
 
 25-25 
 
 24-18 
 
 23-03 
 
 22-02 
 
 2101 
 
 20-01 
 
 19-04 
 
 18-12 
 
 17-23 
 
 1636 
 
 4-1 
 
 — 
 
 — 
 
 35-00 
 
 34-17 
 
 32-88 
 
 31-36 
 
 29-77 
 
 28-13 
 
 26-60 
 
 25-12 
 
 23-82 
 
 22-64 
 
 21-54 
 
 2053 
 
 19-56 
 
 18-62 
 
 4-2 
 
 — 
 
 — 
 
 43-3 
 
 41-4 
 
 39-2 
 
 37-2 
 
 35-2 
 
 33-2 
 
 31-2 
 
 29-2 
 
 27-4 
 
 260 
 
 24-7 
 
 23-4 
 
 22-3 
 
 213 
 
 ^•3 
 
 — 
 
 — 
 
 55-3 
 
 51-6 
 
 48-0 
 
 44-6 
 
 41-2 
 
 38-6 
 
 36-2 
 
 33-8 
 
 31-8 
 
 3lM 
 
 28-5 
 
 2(>-9 
 
 25-6 
 
 24-3 
 
 4-4 
 
 — 
 
 — 
 
 72-7 
 
 66-0 
 
 59- 
 
 54-1 
 
 49-5 
 
 45-7 
 
 42-1 
 
 39-2 
 
 36-8 
 
 34-8 
 
 32-9 
 
 310 
 
 292 
 
 27-7 
 
 4-5 
 
 — 
 
 — 
 
 96-5 
 
 82-7 
 
 72-7 
 
 65-3 
 
 598 
 
 54-7 
 
 50-8 
 
 47-2 
 
 44-0 
 
 41-0 
 
 38-5 
 
 36-2 
 
 341 
 
 320 
 
 4-6 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 75-0 
 
 68-0 
 
 62-2 
 
 56-9 
 
 52-2 
 
 48-2 
 
 45-1 
 
 42 2 
 
 39 6 
 
 37-2 
 
 4-7 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 101-3 
 
 87-2 
 
 76-8 
 
 68-3 
 
 620 
 
 56-9 
 
 52-7 
 
 49-1 
 
 45-9 
 
 42-8 
 
 4-8 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 140-0 
 
 115-2 
 
 96-2 
 
 826 
 
 72-7 
 
 66-1 
 
 60-9 
 
 56-7 
 
 52-9 
 
 49-3 
 
 4-9 
 
 
 
 
 
 
 
 204-5 
 
 150-8 
 
 122-3 
 
 102-5 
 
 89-1 
 
 80 2 
 
 72-4 
 
 66-6 
 
 61-4 
 
 56-7 
 
 5-0 
 
 — 
 
 — 
 
 
 
 — 
 
 
 
 
 
 325-7 
 
 200-0 
 
 1542 
 
 1268 
 
 110-1 
 
 96-9 
 
 86-6 
 
 78-1 
 
 71-2 
 
 65-6 
 
 5-1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1036 
 
 94-4 
 
 85 9 
 
 780 
 
 5-2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 130-4 
 
 116-4 
 
 1040 
 
 91-0 
 
 5S 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 175-2 
 
 144-8 
 
 124-4 
 
 109-6 
 
 5-4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 224-4 
 
 1780 
 
 1510 
 
 132-6 
 
 5-5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 340-8 
 
 246 
 
 195-3 
 
 1632 
 
 5-6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 BS 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-1 
 
 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r>s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 G-4 
 6-5 
 6-6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Probable Errors of Frequency Constants 
 
 71 
 
 TABLE XXXVm.— (continued). 
 
 Values of ViV £„,. 
 ft. 
 
 0-80 
 
 0-85 
 
 0-90 
 
 0-05 
 
 VOO 
 
 1-05 
 
 1-10 
 
 1-15 
 
 1-20 
 
 V25 
 
 ISO 
 
 1-35 
 
 1-40 
 
 1-45 
 
 ISO 
 
 
 4-24 
 
 4-43 
 
 4-62 
 
 4-81 
 
 5-00 
 
 5-19 
 
 5-38 
 
 5-56 
 
 5-75 
 
 5-94 
 
 612 
 
 6-30 
 
 6-49 
 
 6-67 
 
 6-84 
 
 2-0 
 
 4-23 
 
 441 
 
 4-59 
 
 4-77 
 
 496 
 
 5-15 
 
 5-34 
 
 5-53 
 
 5-72 
 
 5-90 
 
 6-08 
 
 6-27 
 
 6-47 
 
 665 
 
 6-83 
 
 2-1 
 
 4-22 
 
 4-39 
 
 4-56 
 
 4-74 
 
 4-93 
 
 5-12 
 
 5-31 
 
 5-50 
 
 5-69 
 
 5-87 
 
 605 
 
 6-24 
 
 6-44 
 
 6-63 
 
 6-82 
 
 2-2 
 
 4-20 
 
 4-36 
 
 4-53 
 
 4-71 
 
 490 
 
 508 
 
 5-27 
 
 5-46 
 
 5-65 
 
 5-84 
 
 602 
 
 6-21 
 
 641 
 
 6-61 
 
 6-80 
 
 2-8 
 
 419 
 
 4-35 
 
 4-51 
 
 4-69 
 
 4-87 
 
 5-05 
 
 5-23 
 
 5-42 
 
 561 
 
 5-80 
 
 5-99 
 
 618 
 
 6-38 
 
 6-58 
 
 6-78 
 
 2-4 
 
 4-18 
 
 434 
 
 4-50 
 
 4-67 
 
 4-85 
 
 5-03 
 
 5-21 
 
 5-39 
 
 5-58 
 
 5-77 
 
 5-96 
 
 6-15 
 
 635 
 
 6-54 
 
 6-74 
 
 2-5 
 
 420 
 
 4-35 
 
 4-50 
 
 4-67 
 
 4-84 
 
 5-01 
 
 5-20 
 
 5-36 
 
 554 
 
 5-72 
 
 5-91 
 
 611 
 
 630 
 
 6-49 
 
 6-68 
 
 2-6 
 
 4-26 
 
 4-38 
 
 4-52 
 
 468 
 
 4-84 
 
 5-01 
 
 5-18 
 
 5-34 
 
 5-51 
 
 5-67 
 
 5-85 
 
 6-05 
 
 6-25 
 
 6-44 
 
 6-62 
 
 2-7 
 
 4-40 
 
 4-50 
 
 460 
 
 4-72 
 
 4-86 
 
 5-03 
 
 5-19 
 
 5-34 
 
 5-49 
 
 5-65 
 
 5-83 
 
 602 
 
 621 
 
 6-39 
 
 6-58 
 
 2-8 
 
 4-63 
 
 4-67 
 
 4-73 
 
 4-82 
 
 4-93 
 
 5-05 
 
 5-20 
 
 535 
 
 5-50 
 
 5-66 
 
 5-82 
 
 6-00 
 
 6-18 
 
 6-36 
 
 654 
 
 2-9 
 
 4-98 
 
 4-97 
 
 4-99 
 
 5-03 
 
 5-10 
 
 5-18 
 
 5-28 
 
 5-39 
 
 5-52 
 
 5-66 
 
 5-82 
 
 5-98 
 
 6-15 
 
 633 
 
 6-51 
 
 8-0 
 
 5-47 
 
 5-42 
 
 5-38 
 
 5-34 
 
 5-36 
 
 5-37 
 
 5-41 
 
 5-48 
 
 5-58 
 
 5-70 
 
 5-83 
 
 5-97 
 
 614 
 
 6-32 
 
 6-52 
 
 3-1 
 
 603 
 
 592 
 
 5-83 
 
 5*75 
 
 5-67 
 
 5-62 
 
 5-60 
 
 5-62 
 
 5-68 
 
 5-78 
 
 5-90 
 
 603 
 
 618 
 
 634 
 
 6-52 
 
 3-2 
 
 6-67 
 
 6-51 
 
 6-35 
 
 6-22 
 
 609 
 
 6-00 
 
 5-90 
 
 5-94 
 
 5-92 
 
 5-95 
 
 6-01 
 
 6-12 
 
 6-25 
 
 6-37 
 
 653 
 
 8-3 
 
 7-41 
 
 7-17 
 
 6-95 
 
 6-77 
 
 6-61 
 
 6-48 
 
 6-29 
 
 6-26 
 
 6-24 
 
 6-22 
 
 6-22 
 
 6-26 
 
 6-34 
 
 6-47 
 
 6-61 
 
 8-4 
 
 8-22 
 
 7-92 
 
 7-64 
 
 7 38 
 
 7-17 
 
 699 
 
 6-84 
 
 6-72 
 
 663 
 
 6-57 
 
 6-54 
 
 653 
 
 6-56 
 
 661 
 
 6-71 
 
 3-5 
 
 9-14 
 
 8-80 
 
 8-51 
 
 8-23 
 
 7-98 
 
 7-70 
 
 7-53 
 
 7-40 
 
 7-22 
 
 7 -09 
 
 6-99 
 
 6-98 
 
 6-95 
 
 6-92 
 
 6-93 
 
 3-6 
 
 10-34 
 
 9-94 
 
 9-58 
 
 9-25 
 
 8-96 
 
 8-66 
 
 8-36 
 
 8-14 
 
 7-90 
 
 7-75 
 
 7-61 
 
 7-51 
 
 7-42 
 
 7-34 
 
 7-23 
 
 3-7 
 
 11-77 
 
 11-29 
 
 10-82 
 
 10-37 
 
 9-98 
 
 9-62 
 
 9-31 
 
 903 
 
 8-73 
 
 8-51 
 
 8-29 
 
 8-11 
 
 7-94 
 
 7-78 
 
 7-60 
 
 8-8 
 
 13-42 
 
 12-85 
 
 12-31 
 
 11-79 
 
 11-30 
 
 10-86 
 
 10-41 
 
 10-02 
 
 9-64 
 
 9-34 
 
 9-03 
 
 8-77 
 
 8-52 
 
 8.30 
 
 8-10 
 
 8-9 
 
 15-58 
 
 14-84 
 
 14-10 
 
 13-42 
 
 12-79 
 
 12-20 
 
 11-64 
 
 1113 
 
 10-65 
 
 10-21 
 
 9-83 
 
 9-51 
 
 9-20 
 
 8-92 
 
 8-67 
 
 4-0 
 
 17-72 
 
 16-85 
 
 16-01 
 
 15-21 
 
 14-44 
 
 13-70 
 
 1300 
 
 12 34 
 
 11-73 
 
 1117 
 
 10-67 
 
 10-24 
 
 9-87 
 
 9-58 
 
 9-32 
 
 4-1 
 
 20-2 
 
 19-2 
 
 18-3 
 
 17-3 
 
 16-4 
 
 15-5 
 
 14-7 
 
 14-0 
 
 133 
 
 12-6 
 
 12-0 
 
 11-5 
 
 11-0 
 
 10-5 
 
 10-3 
 
 4-2 
 
 23-1 
 
 22-0 
 
 20-9 
 
 19-8 
 
 18-7 
 
 17-6 
 
 16-7 
 
 15-8 
 
 15-0 
 
 14-2 
 
 13-5 
 
 12-8 
 
 12-3 
 
 11-8 
 
 11-3 
 
 4-8 
 
 26-3 
 
 25-0 
 
 23-8 
 
 22-5 
 
 21-3 
 
 20-1 
 
 19-0 
 
 180 
 
 17-1 
 
 161 
 
 15-3 
 
 146 
 
 13 9 
 
 132 
 
 126 
 
 4-4 
 
 30-1 
 
 28-4 
 
 26-8 
 
 25-3 
 
 23-9 
 
 22-6 
 
 21-4 
 
 20-3 
 
 19-3 
 
 18-3 
 
 17-3 
 
 164 
 
 15-6 
 
 14-8 
 
 14-1 
 
 4-5 
 
 34-7 
 
 32-5 
 
 30-5 
 
 28-8 
 
 27-3 
 
 25-6 
 
 24-2 
 
 22-9 
 
 21-7 
 
 20-6 
 
 19-5 
 
 18-4 
 
 17-5 
 
 16-7 
 
 161 
 
 4-6 
 
 40-0 
 
 37 4 
 
 35-0 
 
 32-8 
 
 30-9 
 
 29-2 
 
 27-6 
 
 26-1 
 
 24-7 
 
 23-3 
 
 22-0 
 
 20-9 
 
 19-8 
 
 18-8 
 
 18-1 
 
 4-7 
 
 461 
 
 43-1 
 
 40-3 
 
 37-7 
 
 353 
 
 33-2 
 
 31-4 
 
 29-7 
 
 28-0 
 
 26-4 
 
 25-0 
 
 236 
 
 22-3 
 
 21-2 
 
 20-3 
 
 4-8 
 
 52-4 
 
 48-8 
 
 46-8 
 
 431 
 
 40-2 
 
 37-8 
 
 35-6 
 
 336 
 
 31-6 
 
 29-8 
 
 28-1 
 
 266 
 
 25-1 
 
 238 
 
 22-7 
 
 4-9 
 
 60-6 
 
 561 
 
 52-3 
 
 48-8 
 
 45-5 
 
 42-6 
 
 40-0 
 
 37 6 
 
 35-4 
 
 33-4 
 
 31-5 
 
 29-8 
 
 28-1 
 
 266 
 
 25-2 
 
 5-0 
 
 712 
 
 651 
 
 606 
 
 56-5 
 
 •52-6 
 
 49-1 
 
 45'8 
 
 431 
 
 40-5 
 
 38-0 
 
 35-6 
 
 33-6 
 
 31-7 
 
 30-0 
 
 28-4 
 
 6-1 
 
 83-0 
 
 76-4 
 
 70-5 
 
 65-4 
 
 60-6 
 
 563 
 
 52-5 
 
 193 
 
 46-3 
 
 43-4 
 
 40-4 
 
 38-0 
 
 35-6 
 
 336 
 
 31-7 
 
 5-2 
 
 98-8 
 
 89-6 
 
 81-9 
 
 75-6 
 
 70-2 
 
 650 
 
 60-2 
 
 56-2 
 
 52-5 
 
 48-9 
 
 45-5 
 
 42-6 
 
 39-9 
 
 37-4 
 
 352 
 
 5-3 
 
 118-4 
 
 105-2 
 
 960 
 
 876 
 
 80-4 
 
 74 
 
 68 3 
 
 634 
 
 58-8 
 
 54-7 
 
 51-0 
 
 47-7 
 
 44-5 
 
 41-5 
 
 389 
 
 5-4 
 
 141-4 
 
 124-0 
 
 111-2 
 
 99-6 
 
 91-2 
 
 84-0 
 
 77-4 
 
 71-2 
 
 65-7 
 
 612 
 
 56-9 
 
 52-9 
 
 49-4 
 
 46-2 
 
 43-5 
 
 5-5 
 
 — 
 
 — 
 
 131-2 
 
 117-4 
 
 105-2 
 
 96-0 
 
 87-3 
 
 793 
 
 72-8 
 
 67-8 
 
 63-2 
 
 58-6 
 
 54-8 
 
 51-4 
 
 48-8 
 
 5-6 
 
 
 
 — 
 
 I60-O 
 
 142-4 
 
 126-4 
 
 113-4 
 
 102-2 
 
 930 
 
 84-4 
 
 773 
 
 71-1 
 
 65-6 
 
 60-8 
 
 57-2 
 
 54-7 
 
 5-7 
 
 — 
 
 — 
 
 IU9-2 
 
 175-8 
 
 154-8 
 
 134-2 
 
 119-6 
 
 107-0 
 
 97-2 
 
 88-4 
 
 80-6 
 
 74-4 
 
 69-4 
 
 64-9 
 
 61-5 
 
 5-8 
 
 — 
 
 
 266-0 
 
 221-6 
 
 192-8 
 
 163-6 
 
 142-8 
 
 1280 
 
 114-6 
 
 104-0 
 
 94-6 
 
 86-0 
 
 796 
 
 74-4 
 
 70-2 
 
 5-9 
 
 — 
 
 — 
 
 :;78-l 
 
 2840 
 
 2315 
 
 198-2 
 
 171-6 
 
 151-5 
 
 136-2 
 
 123-8 
 
 112-8 
 
 103-4 
 
 948 
 
 87-5 
 
 81-4 
 
 6-0 
 
 
 
 — 
 
 
 
 — 
 
 — 
 
 
 
 206-3 
 
 186-3 
 
 167-5 
 
 1500 
 
 134-2 
 
 121-5 
 
 111-0 
 
 101-8 
 
 92-8 
 
 6-1 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 264 
 
 232 
 
 205 
 
 280 
 
 160 
 
 141 
 
 128 
 
 116 
 
 107 
 
 6-2 
 
 — 
 
 — 
 
 
 
 
 
 — 
 
 — 
 
 350 
 
 297 
 
 251 
 
 216 
 
 188 
 
 164 
 
 148 
 
 132 
 
 122 
 
 6-3 
 
 
 
 
 
 
 
 510 
 
 376 
 
 308 
 
 263 
 
 225 
 
 196 
 
 172 
 
 152 
 
 138 
 
 6-4 
 
 
 
 
 
 
 
 889 
 
 524 
 
 387 
 
 313 
 
 264 
 
 229 
 
 200 
 237 
 286 
 
 177 
 204 
 249 
 
 161 
 
 184 
 220 
 
 6-5 
 6-6 
 6-7 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 363 
 
 305 
 
 268 
 
 6-8 
 
 
 
 _. 
 
 
 
 
 
 
 
 
 
 
 
 485 
 
 392 
 
 333 
 
 6-9 
 
 — • 
 1 
 
 — 
 
 — - 
 
 
 — 
 
 — 
 
 • — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 747 
 
 510 
 
 416 
 
 7-0 
 
<u 
 
 Tables for Statisticians and Biometricians 
 
 fr 
 
 TABLE XXXIX. To find the correlation in errors of /3, and /3 2 . 
 
 Values of .ftp,^. 
 
 ft 
 
 
 o-oo 
 
 0-05 
 
 o-io 
 
 0-15 
 
 0-20 
 
 0-25 
 
 0-30 
 
 0-35 
 
 0-40 
 
 0-45 
 
 0-60 
 
 0-55 
 
 0-60 
 
 0-65 
 
 0-70 
 
 0-75 
 
 2-0 
 2-1 
 2-2 
 H'S 
 2-4 
 2-5 
 2-6 
 2-7 
 2-8 
 2-9 
 3-0 
 3-1 
 ■ ;:.> 
 8-8 
 8-Jf 
 3-5 
 3-6 
 3-7 
 8'8 
 3-9 
 
 4-0 
 4-1 
 4-2 
 4-s 
 
 4-4 
 4-5 
 4-6 
 4-7 
 4-8 
 4-9 
 5-0 
 5-1 
 5-2 
 5-3 
 5'4 
 5-5 
 5-6 
 5-7 
 5-8 
 5-9 
 6-0 
 6-1 
 6-2 
 6-3 
 6-4 
 6-5 
 6-6 
 6-7 
 6-8 
 6-9 
 7-0 
 
 o-oo 
 o-oo 
 o-ou 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 
 O'OO 
 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 o-oo 
 
 000 
 
 o-oo 
 
 •570 
 •557 
 •551 
 •550 
 •551 
 •554 
 •557 
 •557 
 ■556 
 •550 
 •542 
 •534 
 •524 
 •512 
 ■501 
 •490 
 •477 
 •462 
 •450 
 •438 
 •422 
 
 •706 
 •685 
 •672 
 •663 
 •660 
 •659 
 •662 
 •668 
 •674 
 •680 
 •684 
 ■687 
 •688 
 •688 
 •6S6 
 •681 
 •676 
 •670 
 •662 
 •654 
 •645 
 •630 
 •608 
 •580 
 •540 
 •481 
 
 •770 
 •755 
 •728 
 •719 
 •712 
 •706 
 •706 
 •710 
 •716 
 •724 
 •738 
 •744 
 •746 
 •747 
 •748 
 •747 
 •745 
 •741 
 •736 
 •720 
 •713 
 •702 
 •682 
 •658 
 •628 
 •590 
 
 •823 
 •798 
 •771 
 •765 
 •752 
 •745 
 •742 
 •744 
 ■750 
 ■760 
 •774 
 •781 
 •786 
 •788 
 •790 
 •790 
 •788 
 ■784 
 •779 
 •770 
 •760 
 •748 
 •733 
 •712 
 •688 
 •657 
 
 •863 
 •838 
 •814 
 •799 
 •787 
 •776 
 •773 
 •773 
 •779 
 •787 
 •796 
 •808 
 •811 
 •814 
 •816 
 •815 
 •813 
 ■810 
 •803 
 •796 
 •788 
 •780 
 •770 
 •753 
 •732 
 •709 
 
 •894 
 •870 
 •847 
 ■829 
 •814 
 •805 
 •799 
 •800 
 •803 
 •810 
 •816 
 •825 
 •830 
 •832 
 •833 
 •833 
 •832 
 •831 
 •828 
 •822 
 •816 
 •807 
 ■793 
 •784 
 •770 
 •749 
 •716 
 •674 
 •615 
 •532 
 •362 
 
 •917 
 
 •895 
 •874 
 •859 
 •843 
 •834 
 •825 
 •825 
 •826 
 •830 
 •835 
 •840 
 ■842 
 •845 
 •848 
 •849 
 ■850 
 •848 
 •845 
 •841 
 •837 
 •830 
 ■822 
 •811 
 •796 
 •780 
 •754 
 •723 
 •681 
 •620 
 •534 
 
 •935 
 •914 
 •896 
 •880 
 •867 
 ■858 
 •851 
 •846 
 •842 
 •844 
 •847 
 •850 
 •852 
 •855 
 •858 
 •860 
 •860 
 •859 
 •858 
 ■856 
 •853 
 •849 
 •842 
 •832 
 •819 
 •804 
 •784 
 •759 
 •727 
 •680 
 •628 
 
 •949 
 •936 
 •919 
 •900 
 •886 
 •878 
 •871 
 •863 
 •858 
 •857 
 •857 
 •858 
 •860 
 •863 
 •865 
 •867 
 •867 
 •867 
 •866 
 •866 
 •865 
 •862 
 •857 
 •848 
 •837 
 •824 
 •808 
 •788 
 •761 
 •728 
 •687 
 
 •960 
 •948 
 •934 
 ■919 
 •905 
 •893 
 •883 
 •876 
 •871 
 •868 
 •867 
 •867 
 •868 
 •870 
 ■872 
 •873 
 •874 
 •874 
 ■874 
 •875 
 •873 
 •871 
 •867 
 •860 
 •851 
 •841 
 •828 
 ■812 
 •791 
 ■766 
 •731 
 
 •968 
 •959 
 •948 
 •935 
 •920 
 •908 
 •898 
 ■889 
 •883 
 •878 
 •875 
 •874 
 ■875 
 •876 
 •878 
 ■879 
 •880 
 •881 
 •882 
 •882 
 •881 
 •880 
 •877 
 •871 
 •863 
 •853 
 •842 
 •830 
 •815 
 •795 
 •767 
 
 •976 
 •969 
 •960 
 
 •948 
 •936 
 •924 
 •913 
 
 ■902 
 •893 
 •887 
 •883 
 •882 
 •882 
 ■882 
 •883 
 •884 
 •886 
 •887 
 •888 
 •889 
 ■888 
 •887 
 •884 
 •878 
 •872 
 •865 
 •856 
 •846 
 •834 
 •818 
 •798 
 •768 
 •730 
 •679 
 •608 
 •496 
 
 •983 
 •977 
 •968 
 •957 
 •945 
 •933 
 •921 
 •911 
 •901 
 •895 
 •890 
 •889 
 •889 
 •888 
 •889 
 •890 
 •891 
 •892 
 •893 
 •894 
 •894 
 •892 
 •890 
 •885 
 •880 
 •874 
 •868 
 •860 
 •849 
 •835 
 •822 
 ■799 
 •76S 
 •729 
 •686 
 •601 
 
 •989 
 •983 
 •975 
 ■965 
 •954 
 •941 
 •930 
 •920 
 •910 
 •903 
 •898 
 •897 
 •896 
 •895 
 •895 
 •895 
 •896 
 •897 
 •898 
 •899 
 •899 
 •897 
 •894 
 •890 
 •887 
 •882 
 •877 
 •870 
 •861 
 •850 
 •837 
 •820 
 •799 
 •769 
 •736 
 ■674 
 
 •992 
 •986 
 •980 
 •971 
 •962 
 •949 
 ■938 
 •928 
 •919 
 •912 
 •906 
 •903 
 •902 
 •901 
 •900 
 •900 
 •900 
 •901 
 ■901 
 •903 
 •903 
 •901 
 •899 
 •897 
 •894 
 •890 
 •886 
 •879 
 •872 
 •862 
 •851 
 •837 
 •820 
 •799 
 •774 
 •724 
 
Probable Errors of Frequency Constants 
 
 TABLE XXXIX.— (continued). 
 Values of £*,£• 
 
 A 
 
 73 
 
 0-80 
 
 0-85 
 
 o-oo 
 
 0-95 
 
 1-00 
 
 1-05 
 
 1-10 
 
 1-15 
 
 V20 
 
 1-25 
 
 ISO 
 
 1-85 
 
 1-Jfi 
 
 V45 
 
 ISO 
 
 
 •993 
 
 ■995 
 
 •997 
 
 ■999 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 l'OOO 
 
 1-000 
 
 1-000 
 
 2-0 
 
 •989 
 
 •991 
 
 •994 
 
 •996 
 
 •998 
 
 •998 
 
 •999 
 
 •999 
 
 •999 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 2-1 
 
 •983 
 
 •986 
 
 •989 
 
 •992 
 
 •995 
 
 ■996 
 
 •997 
 
 •998 
 
 •998 
 
 •999 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 1-000 
 
 2-2 
 
 •976 
 
 •980 
 
 •984 
 
 •988 
 
 •992 
 
 •993 
 
 •994 
 
 •995 
 
 ■997 
 
 •998 
 
 '999 
 
 •999 
 
 •999 
 
 1-000 
 
 1-000 
 
 2-3 
 
 •968 
 
 •973 
 
 •978 
 
 •983 
 
 •987 
 
 •989 
 
 •991 
 
 •993 
 
 •995 
 
 •996 
 
 •998 
 
 •998 
 
 •999 
 
 •999 
 
 1-000 
 
 2- k 
 
 •958 
 
 •965 
 
 ■972 
 
 •977 
 
 •982 
 
 •985 
 
 •988 
 
 •990 
 
 •992 
 
 •994 
 
 •996 
 
 •997 
 
 •998 
 
 •999 
 
 1-000 
 
 2S 
 
 •947 
 
 •956 
 
 ■964 
 
 •970 
 
 ■976 
 
 •980 
 
 ■984 
 
 •986 
 
 •988 
 
 •991 
 
 •993 
 
 •995 
 
 ■997 
 
 •998 
 
 •999 
 
 8-6 
 
 •937 
 
 •947 
 
 •957 
 
 •963 
 
 •968 
 
 •973 
 
 •977 
 
 •980 
 
 •983 
 
 •986 
 
 •989 
 
 •992 
 
 •995 
 
 •997 
 
 •998 
 
 2-7 
 
 •928 
 
 ■939 
 
 •949 
 
 •955 
 
 •960 
 
 •965 
 
 •970 
 
 •974 
 
 •978 
 
 •981 
 
 •985 
 
 •989 
 
 •992 
 
 •994 
 
 •996 
 
 2-8 
 
 •921 
 
 •932 
 
 ■942 
 
 •947 
 
 ■952 
 
 •957 
 
 •963 
 
 ■968 
 
 •972 
 
 •976 
 
 •980 
 
 •984 
 
 •988 
 
 •990 
 
 •992 
 
 2-0 
 
 •915 
 
 ■923 
 
 •931 
 
 •937 
 
 •943 
 
 •948 
 
 •954 
 
 •960 
 
 ■966 
 
 •971 
 
 •975 
 
 •979 
 
 •983 
 
 •986 
 
 •988 
 
 3-0 
 
 •909 
 
 •915 
 
 •922 
 
 •929 
 
 •936 
 
 •942 
 
 •947 
 
 •953 
 
 •959 
 
 •965 
 
 •970 
 
 •974 
 
 •978 
 
 ■981 
 
 •984 
 
 3-1 
 
 •907 
 
 •912 
 
 •918 
 
 •924 
 
 •930 
 
 •936 
 
 •941 
 
 •946 
 
 •952 
 
 •958 
 
 •963 
 
 •968 
 
 •973 
 
 •977 
 
 •980 
 
 3-2 
 
 •906 
 
 •909 
 
 •914 
 
 •919 
 
 •925 
 
 ■930 
 
 •935 
 
 •940 
 
 '946 
 
 •951 
 
 •956 
 
 •961 
 
 •966 
 
 •971 
 
 •975 
 
 3-3 
 
 •905 
 
 •908 
 
 •912 
 
 •916 
 
 •920 
 
 ■925 
 
 •930 
 
 •935 
 
 •940 
 
 •945 
 
 •950 
 
 •954 
 
 •958 
 
 ■964 
 
 •974 
 
 3-4 
 
 •904 
 
 •907 
 
 •910 
 
 •914 
 
 •918 
 
 •922 
 
 •926 
 
 •931 
 
 •936 
 
 •940 
 
 •944 
 
 •948 
 
 •952 
 
 •958 
 
 •965 
 
 3-5 
 
 •904 
 
 •907 
 
 •910 
 
 •914 
 
 •918 
 
 •921 
 
 •924 
 
 •928 
 
 •932 
 
 •935 
 
 •938 
 
 •942 
 
 •946 
 
 •952 
 
 •959 
 
 3-6 
 
 •905 
 
 •907 
 
 •910 
 
 ■914 
 
 •917 
 
 •920 
 
 •923 
 
 •927 
 
 •930 
 
 •933 
 
 •935 
 
 •937 
 
 •940 
 
 •946 
 
 •953 
 
 3-7 
 
 •905 
 
 •908 
 
 •911 
 
 ■914 
 
 •917 
 
 •920 
 
 •922 
 
 •925 
 
 •928 
 
 •930 
 
 •932 
 
 •934 
 
 •938 
 
 •941 
 
 •948 
 
 3-8 
 
 •906 
 
 ■909 
 
 •911 
 
 •914 
 
 •917 
 
 •919 
 
 •921 
 
 •924 
 
 ■927 
 
 •929 
 
 •931 
 
 •933 
 
 •935 
 
 •939 
 
 •944 
 
 3-9 
 
 •906 
 
 •909 
 
 ■912 
 
 •914 
 
 •917 
 
 •919 
 
 •921 
 
 •923 
 
 •926 
 
 •928 
 
 •930 
 
 •932 
 
 ■934 
 
 •936 
 
 •940 
 
 4-0 
 
 •905 
 
 •908 
 
 ■911 
 
 ■914 
 
 •917 
 
 •919 
 
 •921 
 
 •923 
 
 •925 
 
 •927 
 
 •930 
 
 •931 
 
 •932 
 
 •933 
 
 •934 
 
 4-1 
 
 •905 
 
 •907 
 
 •910 
 
 •913 
 
 •916 
 
 •919 
 
 •921 
 
 •923 
 
 •924 
 
 •926 
 
 •929 
 
 •929 
 
 •930 
 
 •930 
 
 •929 
 
 4-2 
 
 •903 
 
 ■906 
 
 •910 
 
 •913 
 
 •916 
 
 •918 
 
 •920 
 
 •922 
 
 ■924 
 
 •926 
 
 •929 
 
 •928 
 
 ■928 
 
 •927 
 
 •924 
 
 4-S 
 
 •900 
 
 ■904 
 
 •908 
 
 •912 
 
 •916 
 
 •918 
 
 ■920 
 
 •922 
 
 •923 
 
 •926 
 
 •928 
 
 •927 
 
 •927 
 
 •925 
 
 •922 
 
 4-4 
 
 •897 
 
 •902 
 
 •906 
 
 •910 
 
 •915 
 
 •918 
 
 •920 
 
 •922 
 
 •923 
 
 •926 
 
 •928 
 
 •927 
 
 •926 
 
 •923 
 
 •920 
 
 4-5 
 
 •893 
 
 •898 
 
 •903 
 
 ■908 
 
 •913 
 
 •916 
 
 •919 
 
 •920 
 
 •922 
 
 •925 
 
 •927 
 
 •926 
 
 •925 
 
 •923 
 
 •920 
 
 4-6 
 
 ■887 
 
 •894 
 
 •900 
 
 •905 
 
 •910 
 
 •913 
 
 •917 
 
 •919 
 
 •921 
 
 •924 
 
 •926 
 
 •925 
 
 ■925 
 
 •923 
 
 •922 
 
 4-7 
 
 •881 
 
 •890 
 
 ■896 
 
 •901 
 
 •906 
 
 •910 
 
 •914 
 
 •917 
 
 •920 
 
 •923 
 
 •925 
 
 •926 
 
 •926 
 
 •925 
 
 •925 
 
 4-8 
 
 •874 
 
 •884 
 
 •890 
 
 ■895 
 
 •901 
 
 •907 
 
 •911 
 
 •915 
 
 •919 
 
 •922 
 
 •925 
 
 •926 
 
 •927 
 
 927 
 
 •928 
 
 4-9 
 
 •863 
 
 •875 
 
 •883 
 
 ■889 
 
 •896 
 
 •903 
 
 •908 
 
 •913 
 
 •918 
 
 •922 
 
 •925 
 
 ■927 
 
 •928 
 
 ■930 
 
 •932 
 
 5-0 
 
 •851 
 
 •864 
 
 •875 
 
 •882 
 
 •890 
 
 •898 
 
 •905 
 
 •911 
 
 •917 
 
 •922 
 
 •925 
 
 •928 
 
 •931 
 
 •933 
 
 •936 
 
 5-1 
 
 •837 
 
 •852 
 
 •866 
 
 ■875 
 
 ■884 
 
 ■892 
 
 •901 
 
 •909 
 
 •916 
 
 •921 
 
 •924 
 
 •928 
 
 •933 
 
 •937 
 
 •941 
 
 5-2 
 
 •820 
 
 •839 
 
 •853 
 
 •865 
 
 •876 
 
 •885 
 
 •895 
 
 •904 
 
 ■913 
 
 •918 
 
 •923 
 
 929 
 
 •935 
 
 •940 
 
 •945 
 
 5-3 
 
 •798 
 
 •818 
 
 •837 
 
 •853 
 
 •867 
 
 •877 
 
 ■888 
 
 •898 
 
 •908 
 
 '915 
 
 •92] 
 
 •928 
 
 •935 
 
 ■941 
 
 •947 
 
 5-4 
 
 •764 
 
 ■792 
 
 •817 
 
 •837 
 
 •854 
 
 •867 
 
 ■880 
 
 •890 
 
 •900 
 
 •910 
 
 •918 
 
 •925 
 
 •933 
 
 •940 
 
 •947 
 
 5-5 
 
 — 
 
 — 
 
 •789 
 
 •815 
 
 •835 
 
 •852 
 
 •868 
 
 •880 
 
 •890 
 
 •904 
 
 •911 
 
 917 
 
 •926 
 
 ■935 
 
 •944 
 
 5-6 
 
 — 
 
 — 
 
 •750 
 
 •786 
 
 •811 
 
 •835 
 
 •854 
 
 •869 
 
 •880 
 
 •892 
 
 •901 
 
 •909 
 
 •917 
 
 •927 
 
 •938 
 
 5-7 
 
 — 
 
 — 
 
 •701 
 
 •748 
 
 •783 
 
 •811 
 
 •835 
 
 •852 
 
 •866 
 
 •879 
 
 •890 
 
 •897 
 
 •905 
 
 •915 
 
 •928 
 
 5-8 
 
 — 
 
 — 
 
 •640 
 
 •700 
 
 •748 
 
 •781 
 
 ■810 
 
 •828 
 
 •846 
 
 •861 
 
 •875 
 
 •883 
 
 •892 
 
 ■901 
 
 •913 
 
 5-9 
 
 — 
 
 — 
 
 •544 
 
 •639 
 
 •703 
 
 •746 
 
 •778 
 
 •802 
 
 •825 
 
 •842 
 
 •857 
 
 •867 
 
 ■879 
 
 •886 
 
 •893 
 
 6-0 
 
 _ 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •741 
 
 •769 
 
 •796 
 
 •820 
 
 •837 
 
 •852 
 
 •866 
 
 •872 
 
 •873 
 
 6-1 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •691 
 
 •727 
 
 •762 
 
 •792 
 
 •815 
 
 •836 
 
 •852 
 
 •858 
 
 •856 
 
 6-3 
 
 — 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 •628 
 
 ■678 
 
 •724 
 
 •761 
 
 •790 
 
 •818 
 
 •838 
 
 ■845 
 
 •842 
 
 6-8 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 ■526 
 
 •606 
 
 •675 
 
 •724 
 
 •763 
 
 •793 
 
 •818 
 
 •831 
 
 •834 
 
 6-4 
 
 . — 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 ■354 
 
 •526 
 
 •619 
 
 •680 
 
 •726 
 
 •761 
 
 •791 
 
 •814 
 
 ■831 
 
 6S 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■761 
 •721 
 •670 
 ■600 
 
 •790 
 •760 
 •727 
 •683 
 
 •832 
 •837 
 ■845 
 •857 
 
 6-6 
 6-7 
 6-8 
 6-9 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 —■" 
 
 
 
 •468 
 
 •602 
 
 •876 
 
 7-0 
 
 ft 
 
 B. 
 
 10 
 
74 
 
 Tables for Statisticians and Biometriciaus 
 
 TABLE XL. To find the Probable Error of the distance from Mean to Mode. 
 
 Values of ~ d . 
 
 J a 
 
 ft. 
 
 
 o-oo 
 
 0-05 
 
 0-10 
 
 0-15 
 
 0-20 
 
 0-25 
 
 0-30 
 
 0-35 
 
 0-40 
 
 0-45 
 
 0-50 
 
 0-55 
 
 0-60 
 
 0-65 
 
 0-70 
 
 0-75 
 
 0-80 
 
 2-0 
 
 3-54 
 
 
 
 _ 
 
 ._ 
 
 
 
 
 
 
 3-03 
 
 2-44 
 
 2-10 
 
 1-80 
 
 1-58 
 
 1-42 
 
 1-30 
 
 2-1 
 
 2-15 
 
 4-36 
 
 — 
 
 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-10 
 
 2-53 
 
 2-16 
 
 1-88 
 
 2-2 
 
 1-87 
 
 2-75 
 
 9-65 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-91 
 
 3-17 
 
 2-60 
 
 M 
 
 1-64 
 
 1-86 
 
 3-00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3-78 
 
 2-4 
 
 1-46 
 
 1-58 
 
 2-07 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2-5 
 
 1-35 
 
 1-46 
 
 1-67 
 
 2-08 
 
 2-87 
 
 4-04 
 
 5-21 
 
 
 
 
 
 
 
 
 
 
 
 2-6 
 
 1-28 
 
 1-37 
 
 1-58 
 
 1-98 
 
 2-60 
 
 3-42 
 
 4-43 
 
 6-72 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 2-7 
 
 1-25 
 
 1-30 
 
 1-50 
 
 1-83 
 
 2-34 
 
 2-98 
 
 3-75 
 
 5-06 
 
 7-48 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 2-8 
 
 1-23 
 
 1-28 
 
 1-43 
 
 1-71 
 
 2-11 
 
 2-60 
 
 3-17 
 
 4-12 
 
 5-28 7-45 
 
 
 
 
 
 
 
 
 29 
 
 1-22 
 
 1-27 
 
 1-38 
 
 1-60 
 
 T90 
 
 2-27 
 
 2-69 
 
 3-20 
 
 3-84 
 
 4-78 
 
 6-65 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 8-0 
 
 1-23 
 
 1-26 
 
 1-34 
 
 1-51 
 
 1-73 
 
 1-98 
 
 2-29 
 
 2-63 
 
 3-06 
 
 3-58 
 
 4-28 
 
 5-18 
 
 6-43 
 
 8-24 
 
 10-89 
 
 — 
 
 — 
 
 8-1 
 
 1-25 
 
 1-27 
 
 1-32 
 
 1-44 
 
 1-58 
 
 1-76 
 
 2-00 
 
 2-23 
 
 2-54 
 
 2-94 
 
 3-42 
 
 3-93 
 
 4-52 
 
 5-50 
 
 6-76 
 
 8-66 
 
 — 
 
 8-2 
 
 1-27 
 
 1-28 
 
 1-30 
 
 1-38 
 
 1-48 
 
 1-60 
 
 1-75 
 
 1-92 
 
 2-13 
 
 2-37 
 
 2-72 
 
 312 
 
 3-04 
 
 4-21 
 
 6-07 
 
 6-22 
 
 8-00 
 
 8.3 
 
 1-29 
 
 1-29 
 
 1-28 
 
 1-32 
 
 1-39 
 
 1-47 
 
 1-55 
 
 1-68 
 
 1-83 
 
 2-03 
 
 2-27 
 
 2-57 
 
 2-90 
 
 3-39 
 
 4-04 
 
 4-66 
 
 5-53 
 
 84 
 
 1-30 
 
 1-29 
 
 1-28 
 
 1-29 
 
 1-31 
 
 1-37 
 
 1-45 
 
 1-54 
 
 1-63 
 
 1-79 
 
 2-00 
 
 2-24 
 
 2-51 
 
 2-88 
 
 3-36 
 
 3-86 
 
 4-50 
 
 8* 
 
 1-31 
 
 1-30 
 
 1-29 
 
 1-27 
 
 1-25 
 
 1-30 
 
 1-37 
 
 1-45 
 
 1-54 
 
 1-66 
 
 1-83 
 
 2-03 
 
 2-26 
 
 2-55 
 
 2-89 
 
 3-18 
 
 3-61 
 
 3-6 
 
 1-32 
 
 1-31 
 
 1-30 
 
 1-26 
 
 1-22 
 
 1-26 
 
 1-32 
 
 1-40 
 
 1-50 
 
 1-61 
 
 1-74 
 
 1-89 
 
 2-08 
 
 2-31 
 
 2-56 
 
 2-86 
 
 3-24 
 
 3-7 
 
 1-31 
 
 1-31 
 
 1-31 
 
 1-26 
 
 1-22 
 
 1-25 
 
 1-30 
 
 1-37 
 
 1-46 
 
 1-57 
 
 1-69 
 
 1-82 
 
 1-97 
 
 2-14 
 
 2-34 
 
 2-62 
 
 2-95 
 
 3-8 
 
 1-30 
 
 1-31 
 
 1-32 
 
 1-28 
 
 1-25 
 
 1-27 
 
 1-32 
 
 1-38 
 
 1-46 
 
 1-55 
 
 1-65 
 
 1-76 
 
 1-88 
 
 2-03 
 
 2-20 
 
 2-43 
 
 2-69 
 
 3-9 
 
 1-29 
 
 1-33 
 
 1-35 
 
 1-33 
 
 1-30 
 
 1-32 
 
 1-36 
 
 1-41 
 
 1-48 
 
 1-56 
 
 1-64 
 
 1-73 
 
 1-84 
 
 1-96 
 
 2-11 
 
 2-27 
 
 2-49 
 
 4-0 
 
 1-27 
 
 1-37 
 
 1-40 
 
 1-39 
 
 1-39 
 
 1-40 
 
 1-42 
 
 1-46 
 
 I'M 
 
 1-58 
 
 1-65 
 
 1-73 
 
 1-83 
 
 1-94 
 
 2-Ofi 
 
 2-19 
 
 2-36 
 
 4-1 
 
 — 
 
 — 
 
 1-47 
 
 1-48 
 
 1-50 
 
 1-51 
 
 1-53 
 
 1-55 
 
 1-57 
 
 1-61 
 
 1-66 
 
 1-75 
 
 1-85 
 
 1-94 
 
 2-03 
 
 2-13 
 
 2-26 
 
 4-2 
 
 — 
 
 — 
 
 1-58 
 
 1-62 
 
 1-64 
 
 1-65 
 
 1-65 
 
 1-65 
 
 1-65 
 
 1-66 
 
 1-70 
 
 1-77 
 
 1-85 
 
 1-94 
 
 2-03 
 
 2-12 
 
 2-22 
 
 4-3 
 
 — 
 
 — 
 
 1-75 
 
 1-77 
 
 1-78 
 
 1-79 
 
 1-78 
 
 1-76 
 
 1-75 
 
 1-75 
 
 1-78 
 
 1-83 
 
 1-90 
 
 1-97 
 
 2-05 
 
 2-13 
 
 2-23 
 
 4'4 
 
 — 
 
 — 
 
 1-98 
 
 1-97 
 
 1-95 
 
 1-94 
 
 1-93 
 
 1-90 
 
 1-88 
 
 1-89 
 
 1-92 
 
 1-96 
 
 2-01 
 
 2-07 
 
 2-13 
 
 2-20 
 
 2-29 
 
 4-5 
 
 — 
 
 — 
 
 2-27 
 
 2-20 
 
 2-15 
 
 2-11 
 
 2-10 
 
 2-09 
 
 2-09 
 
 2-10 
 
 2-12 
 
 2-15 
 
 2-18 
 
 2-22 
 
 2-26 
 
 2-32 
 
 2-38 
 
 4-6 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 2-44 
 
 2-40 
 
 2-38 
 
 2-36 
 
 234 
 
 2-36 
 
 2-38 
 
 2-40 
 
 2-42 
 
 2-45 
 
 2-48 
 
 4-7 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 2-93 
 
 2-85 
 
 2-78 
 
 2-71 
 
 2-67 
 
 2-65 
 
 2-64 
 
 2-63 
 
 2-62 
 
 2-61 
 
 2-61 
 
 4-8 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-74 
 
 3-52 
 
 3-33 
 
 3-16 
 
 3-07 
 
 3-00 
 
 2-95 
 
 2-90 
 
 2-85 
 
 2-81 
 
 2-80 
 
 4-9 
 
 ■ — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 5-44 
 
 4-64 
 
 4-16 
 
 3-87 
 
 3-63 
 
 3-45 
 
 332 
 
 3-24 
 
 3-11 
 
 3-05 
 
 3-03 
 
 50 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 10-66 
 
 6-83 
 
 5-53 
 
 4-84 
 
 4-37 
 
 4-04 
 
 3-79 
 
 3-62 
 
 3-47 
 
 3-37 
 
 3-31 
 
 51 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 4-46 
 
 4-21 
 
 3-99 
 
 3-85 
 
 3-74 
 
 5-2 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 5-38 
 
 5-05 
 
 4-73 
 
 4-47 
 
 4-24 
 
 5-3 
 
 — 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — . 
 
 6-84 
 
 6-19 
 
 5-66 
 
 5-27 
 
 4-92 
 
 5-4 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 9-24 
 
 7-96 
 
 7-00 
 
 6-24 
 
 5-74 
 
 5-5 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 14-81 
 
 10 89 
 
 8-87 
 
 7-64 
 
 6-81 
 
 5-6 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-2 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-3 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 65 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-6 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-7 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Probable Errors of Frequency Constants 
 
 75 
 
 TABLE XL— (continued). 
 
 
 
 
 
 
 
 
 Jn 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Values of - 
 
 T 2 - 
 
 
 
 
 
 
 0-85 
 
 0-90 
 
 0-95 
 
 1-00 
 
 1-05 
 
 1-10 
 
 1-15 
 
 ISO 
 
 V25 
 
 ISO 
 
 1S5 
 
 1-40 
 
 1-46 
 
 ISO 
 
 
 1-20 
 
 1-13 
 
 1-07 
 
 1-02 
 
 •97 
 
 •92 
 
 ■87 
 
 •83 
 
 •80 
 
 •76 
 
 •72 
 
 •68 
 
 •64 
 
 •60 
 
 2-0 
 
 1-64 
 
 1-49 
 
 1-38 
 
 1-29 
 
 1-21 
 
 1-14 
 
 1-08 
 
 103 
 
 •99 
 
 •94 
 
 •90 
 
 •86 
 
 •82 
 
 •78 
 
 2-1 
 
 2-22 
 
 1-97 
 
 1-80 
 
 1-66 
 
 1-54 
 
 1-45 
 
 1-38 
 
 1-32 
 
 1-26 
 
 1-20 
 
 1-14 
 
 1-08 
 
 1-02 
 
 •96 
 
 2-2 
 
 3-10 
 
 2-63 
 
 2-46 
 
 2-15 
 
 1-98 
 
 1-85 
 
 1-75 
 
 1-67 
 
 1-58 
 
 1-50 
 
 1-41 
 
 1-32 
 
 1-24 
 
 1-16 
 
 2S 
 
 — 
 
 3-72 
 
 3-20 
 
 2-81 
 
 2-56 
 
 2-36 
 
 2-22 
 
 2-09 
 
 1-96 
 
 1-84 
 
 1-73 
 
 1-62 
 
 1-51 
 
 1-39 
 
 2-4 
 
 — 
 
 — 
 
 — 
 
 3-94 
 
 3-40 
 
 3-08 
 
 2-87 
 
 2-66 
 
 2-47 
 
 2-29 
 
 2-12 
 
 1-96 
 
 1-80 
 
 1-64 
 
 2-5 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 4-32 
 
 3-82 
 
 3-48 
 
 314 
 
 2-86 
 
 2-60 
 
 2-34 
 
 2-10 
 
 1-91 
 
 2-0 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 4-82 
 
 4-20 
 
 3-66 
 
 3-18 
 
 2-78 
 
 2-46 
 
 2-19 
 
 2*7 
 
 
 
 
 
 
 
 
 
 5-73 
 
 4-82 
 6-63 
 
 3-88 
 4-81 
 6-15 
 
 3-28 
 3-94 
 
 4-80 
 
 2-87 
 3-31 
 3-79 
 
 4-29 
 
 2-49 
 2-80 
 3-12 
 
 3-48 
 
 2-8 
 2-9 
 3-0 
 3-1 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 3-88 
 
 3-2 
 
 6-80 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3-3 
 
 5-38 
 
 6-55 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s-4 
 
 4-21 
 
 4-95 
 
 6-00 
 
 7-33 
 
 9-30 
 
 12-16 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 3S 
 
 3 74 
 
 4-34 
 
 5'12 
 
 6-03 
 
 7-17 
 
 8-80 
 
 11-52 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 3-6 
 
 3-34 
 
 3-80 
 
 4-32 
 
 5-01 
 
 5-78 
 
 6-65 
 
 8-28 
 
 11-12 
 
 
 
 
 
 — 
 
 
 
 — 
 
 — 
 
 3-7 
 
 3-00 
 
 3-35 
 
 3*74 
 
 4*23 
 
 4-72 
 
 5-41 
 
 6-36 
 
 8-00 
 
 10-22 
 
 
 
 
 
 
 3-8 
 
 2-74 
 
 3-00 
 
 3-32 
 
 3-6G 
 
 4-04 
 
 4-50 
 
 5-18 
 
 616 
 
 7-43 
 
 9-32 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-9 
 
 2-55 
 
 2-77 
 
 3-02 
 
 3-29 
 
 3-60 
 
 3-98 
 
 4-50 
 
 5-12 
 
 5-92 
 
 691 
 
 8-00 
 
 9-23 
 
 11-08 
 
 — ■ 
 
 4-0 
 
 2-42 
 
 2-60 
 
 2-79 
 
 3-00 
 
 3-27 
 
 3-61 
 
 4-06 
 
 4-61 
 
 5-20 
 
 5-90 
 
 6-80 
 
 7-86 
 
 9-48 
 
 11-58 
 
 4-1 
 
 2-35 
 
 2-50 
 
 2-67 
 
 2-86 
 
 3-09 
 
 3-38 
 
 3-71 
 
 413 
 
 4-60 
 
 5-15 
 
 5-86 
 
 6-72 
 
 7-76 
 
 9-10 
 
 4-2 
 
 2-35 
 
 2-48 
 
 2-G2 
 
 2-77 
 
 2-96 
 
 3-21 
 
 3-49 
 
 3-82 
 
 4-18 
 
 4-59 
 
 5-14 
 
 5-82 
 
 6 66 
 
 7-59 
 
 4-s 
 
 2-39 
 
 2-50 
 
 2-61 
 
 2-73 
 
 2-89 
 
 3-10 
 
 3-32 
 
 3-55 
 
 3-83 
 
 4-16 
 
 4-60 
 
 5-16 
 
 5-82 
 
 6-60 
 
 4'4 
 
 2 '45 
 
 2-53 
 
 2-63 
 
 2-74 
 
 2-87 
 
 3-02 
 
 3-20 
 
 3-39 
 
 3-61 
 
 3-87 
 
 4-21 
 
 4-66 
 
 5-18 
 
 5-86 
 
 4S 
 
 2-52 
 
 2-60 
 
 2-69 
 
 2-79 
 
 2-89 
 
 3-01 
 
 3-15 
 
 331 
 
 3-48 
 
 3-67 
 
 3-95 
 
 4-31 
 
 4-77 
 
 5-36 
 
 4-0 
 
 2-64 
 
 2-69 
 
 2-77 
 
 2-85 
 
 2-93 
 
 3-02 
 
 313 
 
 3-25 
 
 3 39 
 
 3-55 
 
 3 78 
 
 4-10 
 
 4-50 
 
 4-99 
 
 4-7 
 
 2-81 
 
 2-82 
 
 2-86 
 
 2-93 
 
 3-00 
 
 3-08 
 
 3-17 
 
 3-27 
 
 3-38 
 
 3-50 
 
 3-68 
 
 3-94 
 
 4-26 
 
 4-66 
 
 4'8 
 
 3-02 
 
 3 03 
 
 3-04 
 
 3-08 
 
 3'12 
 
 3-16 
 
 3-22 
 
 3-30 
 
 3-39 
 
 3-50 
 
 3-63 
 
 3-81 
 
 4-07 
 
 4-35 
 
 4'9 
 
 3-28 
 
 3-2G 
 
 3-25 
 
 3-26 
 
 3-28 
 
 3-31 
 
 3 34 
 
 3-39 
 
 3-44 
 
 3-51 
 
 3-61 
 
 3-73 
 
 3-90 
 
 4-09 
 
 5'0 
 
 3-64 
 
 3-55 
 
 3-51 
 
 3-49 
 
 3-47 
 
 3-47 
 
 3-48 
 
 3-49 
 
 3-52 
 
 3-56 
 
 3-61 
 
 3-68 
 
 3-77 
 
 3-90 
 
 5-1 
 
 4-04 
 
 3-90 
 
 3-81 
 
 3-74 
 
 3-68 
 
 3-65 
 
 3-63 
 
 3-62 
 
 361 
 
 3-62 
 
 3-63 
 
 3-65 
 
 3-69 
 
 3-76 
 
 5-2 
 
 4-60 
 
 4-35 
 
 4-18 
 
 4-05 
 
 3-95 
 
 3-88 
 
 3-81 
 
 3-75 
 
 3-72 
 
 3-70 
 
 3-69 
 
 3-68 
 
 3-69 
 
 3-70 
 
 5-3 
 
 5-33 
 
 4-98 
 
 4-71 
 
 4'52 
 
 4-37 
 
 4-24 
 
 4-12 
 
 4-01 
 
 3-92 
 
 3-85 
 
 3-79 
 
 3-75 
 
 3-73 
 
 3-72 
 
 5'4 
 
 6-21 
 
 5-74 
 
 5-36 
 
 5-08 
 
 4-86 
 
 4-66 
 
 4-48 
 
 4-32 
 
 4-20 
 
 4-09 
 
 3-99 
 
 3-92 
 
 3-87 
 
 3-82 
 
 5-6 
 
 — 
 
 6-69 
 
 6-27 
 
 5-83 
 
 5-49 
 
 5-19 
 
 4-94 
 
 4-73 
 
 4-56 
 
 4-42 
 
 4-30 
 
 4-18 
 
 4-11 
 
 4-04 
 
 5-6 
 
 — 
 
 8-11 
 
 7-48 
 
 6-82 
 
 6-32 
 
 5-90 
 
 5-55 
 
 5-27 
 
 5-03 
 
 4-84 
 
 4-68 
 
 4-54 
 
 4-43 
 
 4-35 
 
 5'7 
 
 — 
 
 10-18 
 
 9-11 
 
 8-12 
 
 7-45 
 
 6-85 
 
 6-35 
 
 5-95 
 
 5-62 
 
 5-37 
 
 5-16 
 
 5-00 
 
 4-87 
 
 4-76 
 
 5S 
 
 — 
 
 13-53 
 
 11-44 
 
 9-84 
 
 8-71 
 
 7-94 
 
 7-32 
 
 6-82 
 
 6-45 
 
 613 
 
 5-85 
 
 5-61 
 
 5-43 
 
 5-29 
 
 5-9 
 
 — 
 
 19-95 
 
 14*26 
 
 11-92 
 
 10-48 
 
 9-38 
 
 8-55 
 
 7-89 
 
 7-38 
 
 6-95 
 
 6-62 
 
 6 33 
 
 6-10 
 
 5-90 
 
 6-0 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 11-64 
 
 10-26 
 
 9-31 
 
 8-62 
 
 8-03 
 
 7-53 
 
 7-15 
 
 6-87 
 
 6-60 
 
 0-1 
 
 
 
 
 
 
 14-83 
 
 12-55 
 
 11-19 
 
 10-24 
 
 9-40 
 
 8-64 
 
 8-08 
 
 7-68 
 
 7-36 
 
 6-2 
 
 
 
 
 
 
 19-65 
 
 15-85 
 
 13-69 
 
 1221 
 
 11-01 
 
 10-02 
 
 9-19 
 
 8-65 
 
 8-22 
 
 6S 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 28-03 
 
 20-85 
 
 17-09 
 
 14-56 
 
 12-84 
 
 11-45 
 
 10-45 
 
 9-69 
 
 9-11 
 
 o-4 
 
 
 
 
 
 
 47-99 
 
 28-04 
 
 21-30 
 
 17-44 
 
 15-07 
 
 13-20 
 
 11-90 
 
 14-2 
 
 17-2 
 
 21-8 
 
 29-4 
 
 43 
 
 10-83 
 
 12-9 
 
 15-6 
 
 19-6 
 
 26-0 
 
 37-8 
 
 10-07 
 
 12-4 
 
 14-7 
 
 18-3 
 
 24-1 
 
 34-9 
 
 OS 
 
 6-7 
 6S 
 6-9 
 7-0 
 
 ft 
 
 10—2 
 
76 
 
 Tables for Statisticians and Biometriciam 
 
 ft 
 
 TABLE XLI. To find the Probable Error of the Bkeivness sk. 
 Values of ViV2 S j6. 
 
 
 o-oo 
 
 0-05 
 
 o-io 
 
 0-15 
 
 0-20 
 
 0-25 
 
 0-80 
 
 0-85 
 
 0-40 
 
 0-46 
 
 0-50 
 
 6-55 
 
 0-60 
 
 0-65 
 
 0-70 
 
 0-75 
 
 2-0 
 
 3-54 
 
 
 _ 
 
 
 
 
 
 _ 
 
 
 
 
 _ 
 
 3-41 
 
 2-80 
 
 2-43 
 
 2-12 
 
 1-89 
 
 1-72 
 
 2-1 
 
 2-16 
 
 4-20 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 3-67 
 
 3-02 
 
 2-58 
 
 2-2 
 
 1-87 
 
 2-63 
 
 9-50 
 
 
 
 
 
 
 
 
 
 
 
 
 5-57 
 
 4-10 
 
 2-8 
 
 1-64 
 
 1-78 
 
 2-88 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 2 '4 
 
 1-46 
 
 1-49 
 
 2-02 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 , — 
 
 
 
 — 
 
 2-5 
 
 1-35 
 
 1-41 
 
 1-62 
 
 2-02 
 
 2-80 
 
 4-06 
 
 5-08 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 2-6 
 
 1-28 
 
 1-30 
 
 1-43 
 
 1-75 
 
 2-18 
 
 2-82 
 
 3-65 
 
 4-90 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 , | 
 
 — 
 
 2-7 
 
 1-25 
 
 1-25 
 
 1-31 
 
 1-52 
 
 1-84 
 
 2-29 
 
 2-85 
 
 3-08 
 
 4-84 
 
 
 
 
 
 
 
 
 2-8 
 
 1-23 
 
 1-22 
 
 1-24 
 
 1-36 
 
 1-59 
 
 1-89 
 
 2-21 
 
 2-70 
 
 3 -30 
 
 4-36 
 
 
 
 
 
 
 
 2-9 
 
 1-22 
 
 1-20 
 
 1-20 
 
 1-28 
 
 1-43 
 
 1G3 
 
 1-86 
 
 2-19 
 
 2-60 
 
 3-33 
 
 4-30 
 
 — 
 
 
 
 
 
 — 
 
 — 
 
 s-o 
 
 1-23 
 
 1-21 
 
 1-20 
 
 1-25 
 
 1-34 
 
 1-48 
 
 1-62 
 
 1-84 
 
 2-12 
 
 2-49 
 
 3-00 
 
 3-7G 
 
 4-62 
 
 6-02 
 
 8-12 
 
 — 
 
 8-1 
 
 1-25 
 
 1-22 
 
 1-21 
 
 1-23 
 
 1-27 
 
 1-36 
 
 1-50 
 
 1-67 
 
 1-88 
 
 2-16 
 
 2-50 
 
 3-06 
 
 376 
 
 4-72 
 
 6-10 
 
 8-08 
 
 8-2 
 
 1-27 
 
 1-23 
 
 1-22 
 
 1-22 
 
 1-23 
 
 1-29 
 
 1-40 
 
 1-53 
 
 1-70 
 
 1-90 
 
 2-17 
 
 2-58 
 
 3-17 
 
 3-97 
 
 4-87 
 
 5-83 
 
 8-3 
 
 1-29 
 
 1-25 
 
 1-23 
 
 1-21 
 
 1-21 
 
 1-24 
 
 1-33 
 
 1-44 
 
 1-58 
 
 1-74 
 
 1-94 
 
 2'27 
 
 2-73 
 
 3-28 
 
 3-88 
 
 4-53 
 
 8-4 
 
 1-30 
 
 1-27 
 
 1-24 
 
 1-21 
 
 1-20 
 
 1-23 
 
 1-29 
 
 1-38 
 
 1-49 
 
 1-61 
 
 1-78 
 
 2-04 
 
 2-36 
 
 2-75 
 
 3-18 
 
 3-63 
 
 8-5 
 
 1-31 
 
 1-29 
 
 1-25 
 
 1-21 
 
 1-20 
 
 1-22 
 
 1-28 
 
 1-35 
 
 1-43 
 
 1-54 
 
 1-68 
 
 1-87 
 
 2-09 
 
 2-37 
 
 2-68 
 
 2-98 
 
 S-6 
 
 1-32 
 
 1-30 
 
 1-26 
 
 1-22 
 
 1-20 
 
 1-22 
 
 1-26 
 
 1-32 
 
 1-40 
 
 1-50 
 
 1-61 
 
 1-75 
 
 1-91 
 
 2-13 
 
 2-39 
 
 2-65 
 
 8-7 
 
 1-31 
 
 1-31 
 
 1-28 
 
 1-24 
 
 1-20 
 
 1-23 
 
 1-27 
 
 1-32 
 
 1-39 
 
 1-47 
 
 1-56 
 
 1-67 
 
 1-80 
 
 1-98 
 
 2-19 
 
 2-40 
 
 8-8 
 
 1-30 
 
 1-32 
 
 1-30 
 
 1-27 
 
 1-23 
 
 1-25 
 
 1-28 
 
 1-33 
 
 1-38 
 
 1-46 
 
 1-54 
 
 1-63 
 
 1-75 
 
 1-88 
 
 2-04 
 
 2-22 
 
 3-9 
 
 1-28 
 
 1-34 
 
 1-33 
 
 1-30 
 
 1-28 
 
 1-28 
 
 1-30 
 
 1-34 
 
 1-39 
 
 1-45 
 
 1-53 
 
 1-61 
 
 1-71 
 
 1-82 
 
 1-96 
 
 2-12 
 
 4-0 
 
 1-27 
 
 1-36 
 
 1-38 
 
 1-36 
 
 1-35 
 
 1-35 
 
 1-36 
 
 1-39 
 
 1-43 
 
 1-48 
 
 1-56 
 
 1-63 
 
 1-72 
 
 1-81 
 
 1-92 
 
 2-04 
 
 4-1 
 
 — 
 
 — 
 
 1-46 
 
 1-45 
 
 1-45 
 
 1-44 
 
 1-44 
 
 1-46 
 
 1-50 
 
 1-54 
 
 1-60 
 
 1-67 
 
 1-74 
 
 1-82 
 
 1-91 
 
 2-03 
 
 4-2 
 
 — 
 
 — 
 
 1-58 
 
 1-57 
 
 1-57 
 
 1-56 
 
 1-55 
 
 1-56 
 
 1-59 
 
 1-62 
 
 1-67 
 
 1-72 
 
 1-78 
 
 1-85 
 
 1-93 
 
 2-02 
 
 4-8 
 
 — 
 
 — 
 
 1-75 
 
 1-74 
 
 1-73 
 
 1-71 
 
 1-70 
 
 1-69 
 
 1-70 
 
 1-72 
 
 1-76 
 
 1-81 
 
 1-86 
 
 1-92 
 
 1-98 
 
 2-06 
 
 4'4 
 
 — 
 
 — 
 
 1-95 
 
 1-94 
 
 1-92 
 
 1-90 
 
 1-88 
 
 1-85 
 
 1-86 
 
 1-87 
 
 1-90 
 
 1-93 
 
 1-97 
 
 2-01 
 
 2-06 
 
 2-12 
 
 4-5 
 
 — 
 
 — 
 
 2-26 
 
 2-19 
 
 2-13 
 
 2-09 
 
 2-07 
 
 2-06 
 
 2-05 
 
 2-06 
 
 2-07 
 
 2-08 
 
 2-11 
 
 2-15 
 
 2-19 
 
 2-23 
 
 4-o 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 2-48 
 
 2-42 
 
 2-37 
 
 2-35 
 
 2-33 
 
 2-31 
 
 2-31 
 
 2-32 
 
 2-33 
 
 2-35 
 
 4-7 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-11 
 
 2-93 
 
 2-82 
 
 2-73 
 
 2-65 
 
 2-60 
 
 2-57 
 
 2-54 
 
 2-52 
 
 2-50 
 
 4-8 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-78 
 
 3-53 
 
 3-35 
 
 3-21 
 
 3-08 
 
 2-97 
 
 2-89 
 
 2-82 
 
 2-7G 
 
 2-71 
 
 4-9 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 5-48 
 
 4-66 
 
 4-17 
 
 3-87 
 
 3G3 
 
 3-44 
 
 3-30 
 
 3-17 
 
 3-07 
 
 3 01 
 
 5-0 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 11-12 
 
 6-96 
 
 5-52 
 
 4-82 
 
 4-36 
 
 4-02 
 
 3 77 
 
 3-58 
 
 3-45 
 
 3-37 
 
 5-1 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 4-45 
 
 4-16 
 
 4-00 
 
 3-87 
 
 5-2 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 6-38 
 
 5-02 
 
 4-72 
 
 4-49 
 
 5-3 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 6-84 
 
 6-18 
 
 5-G6 
 
 5-26 
 
 5-4 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 9-24 
 
 7-76 
 
 6-80 
 
 6-22 
 
 5-5 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 14-80 
 
 10-67 
 
 8-87 
 
 7-71 
 
 5-6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 G-G 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Probable Errors of Frequency Constants 
 
 77 
 
 TABLE XLI— {continued). 
 Values of "JN2 sk . 
 
 ft 
 
 0-80 
 
 0-85 
 
 0-90 
 
 0-95 
 
 1-00 
 
 1-05 
 
 1-10 
 
 1-15 
 
 1-20 
 
 V25 
 
 ISO 
 
 1S5 
 
 VJfi 
 
 1-46 
 
 VBO 
 
 
 1-59 
 
 1-48 
 
 1-39 
 
 1-30 
 
 1-24 
 
 1-19 
 
 1-14 
 
 1-10 
 
 1-06 
 
 1-02 
 
 •99 
 
 •95 
 
 •91 
 
 •87 
 
 •83 
 
 2-0 
 
 2-20 
 
 1-95 
 
 1-80 
 
 1-68 
 
 1-58 
 
 1-52 
 
 1-47 
 
 1-42 
 
 1-37 
 
 1-32 
 
 1-26 
 
 1-20 
 
 1-15 
 
 1-10 
 
 1-05 
 
 2-1 
 
 3-22 
 
 2-65 
 
 2-29 
 
 2-08 
 
 1-98 
 
 1-91 
 
 1-84 
 
 1-78 
 
 1-72 
 
 1-66 
 
 1-59 
 
 1-52 
 
 1-45 
 
 1-38 
 
 1-31 
 
 2-2 
 
 5-23 
 
 3-80 
 
 3-04 
 
 2-75 
 
 2-53 
 
 2-40 
 
 2-30 
 
 2-21 
 
 2-12 
 
 2-03 
 
 1-94 
 
 1-85 
 
 1-76 
 
 1-67 
 
 1-58 
 
 2-3 
 
 — 
 
 — 
 
 4-29 
 
 3-64 
 
 331 
 
 3-12 
 
 2-94 
 
 2-78 
 
 2-63 
 
 2-49 
 
 2-36 
 
 2-23 
 
 2-10 
 
 1-98 
 
 1-86 
 
 2-4 
 
 — 
 
 — 
 
 — 
 
 — 
 
 4-77 
 
 4-27 
 
 3-84 
 
 3-55 
 
 3-29 
 
 3-06 
 
 2-85 
 
 2-66 
 
 2-49 
 
 2-32 
 
 2-15 
 
 2-5 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 5-72 
 
 5-00 
 
 4-39 
 
 3-94 
 
 3-54 
 
 3-20 
 
 2-93 
 
 2-67 
 
 2-44 
 
 2-0 
 
 
 
 
 
 
 
 
 
 6-25 
 
 5-20 
 7-05 
 
 4-46 
 5-68 
 7-45 
 
 3-88 
 4-78 
 6-00 
 7-80 
 
 3-42 
 4-03 
 4-85 
 6-05 
 
 3-08 
 3-54 
 4-11 
 4-77 
 5-57 
 
 2-74 
 3-05 
 3-37 
 3-70 
 4-10 
 
 2-7 
 2-8 
 2-9 
 3'U 
 8-1 
 
 7-00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4-58 
 
 8-2 
 
 5-30 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8-3 
 
 4-16 
 
 4-71 
 
 5-60 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3-4 
 
 3-38 
 
 3-89 
 
 4-59 
 
 5-58 
 
 6-85 
 
 8-38 
 
 11-48 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 8-5 
 
 2-95 
 
 3-38 
 
 3-99 
 
 4-G6 
 
 5-50 
 
 6-48 
 
 7-55 
 
 9-92 
 
 — 
 
 
 
 
 
 
 
 8-6 
 
 2-67 
 
 3 05 
 
 3-49 
 
 3-97 
 
 4-52 
 
 5-22 
 
 6-00 
 
 7-36 
 
 9-42 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-7 
 
 2-44 
 
 2-75 
 
 3 08 
 
 3 43 
 
 3-80 
 
 4 25 
 
 4-78 
 
 6-64 
 
 7-08 
 
 9-02 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-8 
 
 2-29 
 
 2-53 
 
 2-79 
 
 3-07 
 
 3-38 
 
 3-70 
 
 4-15 
 
 4-77 
 
 5-62 
 
 6-89 
 
 8-76 
 
 — 
 
 — 
 
 — 
 
 — 
 
 3-J 
 
 2-20 
 
 2-38 
 
 2-58 
 
 2-82 
 
 3-0G 
 
 3-33 
 
 3-69 
 
 4-14 
 
 4-77 
 
 5-50 
 
 6-42 
 
 7-40 
 
 8-57 
 
 10-12 
 
 — 
 
 4-0 
 
 2-16 
 
 2-31 
 
 2-47 
 
 2-65 
 
 2-85 
 
 3 09 
 
 3-34 
 
 3-72 
 
 4-15 
 
 4-61 
 
 5-14 
 
 5-84 
 
 6-80 
 
 8-44 
 
 11-00 
 
 4-1 
 
 2-14 
 
 2-26 
 
 2-40 
 
 2-55 
 
 2-71 
 
 2-88 
 
 3-10 
 
 3-37 
 
 3-67 
 
 4-03 
 
 4-44 
 
 5-04 
 
 5-94 
 
 7-12 
 
 8-67 
 
 4-2 
 
 2-16 
 
 2-27 
 
 2-38 
 
 2-50 
 
 2-65 
 
 2-80 
 
 2-96 
 
 3-14 
 
 3-37 
 
 3-65 
 
 4-01 
 
 4-55 
 
 5-28 
 
 6-18 
 
 7-21 
 
 4'8 
 
 2-20 
 
 2-30 
 
 2-41 
 
 2-52 
 
 2-64 
 
 2-76 
 
 2-89 
 
 3-04 
 
 3-23 
 
 3-48 
 
 3-78 
 
 4-22 
 
 4-78 
 
 5-44 
 
 6-26 
 
 4'4 
 
 2-29 
 
 2-35 
 
 2-44 
 
 2-53 
 
 2-63 
 
 2-75 
 
 2-88 
 
 3-02 
 
 3-20 
 
 3-40 
 
 3 65 
 
 3 97 
 
 4-40 
 
 4-91 
 
 5-56 
 
 4-5 
 
 2-39 
 
 2-43 
 
 2-48 
 
 2-56 
 
 2-66 
 
 2-77 
 
 2-89 
 
 3 01 
 
 316 
 
 3-34 
 
 3-55 
 
 3-79 
 
 4-16 
 
 4-55 
 
 5-10 
 
 4-0 
 
 2-52 
 
 2-54 
 
 2-58 
 
 2 63 
 
 2-71 
 
 2-80 
 
 2-90 
 
 3-01 
 
 3-14 
 
 3-29 
 
 3-47 
 
 3-67 
 
 3-96 
 
 4-28 
 
 4-72 
 
 4-7 
 
 2-70 
 
 2-72 
 
 2-74 
 
 2-78 
 
 2-83 
 
 2-89 
 
 2-97 
 
 3-06 
 
 3-16 
 
 3-28 
 
 3-41 
 
 3-58 
 
 3-80 
 
 4-06 
 
 4-42 
 
 4-8 
 
 2-98 
 
 2-97 
 
 2-96 
 
 2-97 
 
 3-00 
 
 3-04 
 
 3-08 
 
 314 
 
 3-21 
 
 3-30 
 
 3-40 
 
 3-53 
 
 3-68 
 
 3-88 
 
 4-16 
 
 4-9 
 
 3-31 
 
 3-25 
 
 3-21 
 
 3-20 
 
 3-21 
 
 3'22 
 
 3-24 
 
 3-26 
 
 3-31 
 
 3-36 
 
 3-43 
 
 3-51 
 
 3-60 
 
 3-73 
 
 3-96 
 
 5-0 
 
 3-75 
 
 3-64 
 
 3-55 
 
 3-49 
 
 3 45 
 
 3-42 
 
 341 
 
 3-41 
 
 3-43 
 
 3-44 
 
 3-46 
 
 351 
 
 3-57 
 
 3-65 
 
 3-78 
 
 5-1 
 
 4-28 
 
 4-10 
 
 3-96 
 
 3-85 
 
 3-76 
 
 3-68 
 
 3-63 
 
 3-60 
 
 3-57 
 
 ' 355 
 
 3-53 
 
 3-53 
 
 3 55 
 
 3-60 
 
 3-67 
 
 5-2 
 
 4-93 
 
 4-69 
 
 4-48 
 
 4-29 
 
 4-13 
 
 4-02 
 
 3-92 
 
 3-84 
 
 3-76 
 
 3-08 
 
 3-63 
 
 3-60 
 
 3-58 
 
 3-59 
 
 3-62 
 
 5-3 
 
 5-78 
 
 5-42 
 
 5-09 
 
 4-80 
 
 4-56 
 
 4*40 
 
 4-26 
 
 4-13 
 
 4-00 
 
 3-89 
 
 3-80 
 
 373 
 
 3-68 
 
 3-65 
 
 3-63 
 
 5-4 
 
 6-94 
 
 6-32 
 
 5-82 
 
 5-40 
 
 5-07 
 
 4-84 
 
 4-64 
 
 4-46 
 
 4-30 
 
 4-17 
 
 4-06 
 
 3-95 
 
 3-87 
 
 3-80 
 
 3-75 
 
 5-5 
 
 — 
 
 — 
 
 6-75 
 
 6-22 
 
 5-79 
 
 5-46 
 
 5-19 
 
 4-97 
 
 4-77 
 
 4-60 
 
 4-44 
 
 4-30 
 
 4-19 
 
 4-10 
 
 4-01 
 
 5-6 
 
 — 
 
 — 
 
 8-15 
 
 7-30 
 
 6-73 
 
 6-26 
 
 5-91 
 
 5-61 
 
 5-34 
 
 5-10 
 
 4-90 
 
 4-73 
 
 4-59 
 
 4-47 
 
 4-38 
 
 5-7 
 
 — 
 
 — 
 
 10-20 
 
 8-76 
 
 7-98 
 
 7-26 
 
 6-76 
 
 6-34 
 
 5-99 
 
 5-68 
 
 5-44 
 
 5-24 
 
 5-06 
 
 4-91 
 
 4-78 
 
 5-8 
 
 — 
 
 — 
 
 13-53 
 
 10-83 
 
 9-66 
 
 8-71 
 
 7-90 
 
 7-28 
 
 6-78 
 
 6-40 
 
 6-10 
 
 5-84 
 
 5-62 
 
 6-44 
 
 5-28 
 
 5-9 
 
 — 
 
 — 
 
 19 96 
 
 14-30 
 
 12-02 
 
 10-51 
 
 9-39 
 
 8-56 
 
 7-90 
 
 7-39 
 
 6-96 
 
 6-61 
 
 6-33 
 
 610 
 
 5-89 
 
 6-0 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 
 — 
 
 11-64 
 
 10-26 
 
 9-31 
 
 8-56 
 
 8-03 
 
 7-53 
 
 7-15 
 
 6-82 
 
 6-54 
 
 6-1 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 14-83 
 
 12-55 
 
 11-19 
 
 1013 
 
 9 30 
 
 8-64 
 
 8-08 
 
 7-63 
 
 7-24 
 
 G-2 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 19-65 
 
 15-85 
 
 13-69 
 
 12-21 
 
 11-01 
 
 10-02 
 
 9-19 
 
 8-55 
 
 8-01 
 
 6-3 
 
 
 
 
 
 
 
 28-03 
 
 20-85 
 
 17-09 
 
 14-56 
 
 12-84 
 
 11-45 
 
 10-45 
 
 9-60 
 
 8-92 
 
 6-4 
 
 
 
 
 
 
 
 47-99 
 
 28-04 
 
 21-30 
 
 17-44 
 
 15-07 
 
 13-20 
 
 11-90 
 
 14-2 
 
 17-2 
 
 21-8 
 
 29-4 
 
 43-0 
 
 10-83 
 
 12-9 
 
 15-6 
 
 19-6 
 
 26-0 
 
 37-8 
 
 10-07 
 
 12-4 
 
 14-7 
 
 18-3 
 
 24-1 
 
 34-9 
 
 0-5 
 6-0 
 6-7 
 0-8 
 6-9 
 7-0 
 
 & 
 
78 
 
 Tables for Statisticians and Biometricians 
 
 A 
 
 TABLE XLII. To give values of /3 3 , /3 4 , /3 6 and /3, in terms 
 
 TABLE XLII (a). 
 
 Values of /3 3 . 
 
 A 
 
 
 2-0 
 
 2S 
 
 8-0 
 
 SS 
 
 4-0 
 
 J,S 
 
 6-0 
 
 BS 
 
 6-0 
 
 6S 
 
 7-0 
 
 o-o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0-1 
 
 0-48493 
 
 0-68971 
 
 0-94286 
 
 1-25688 
 
 1-64906 
 
 2-14375 
 
 
 
 
 
 
 0-2 
 
 0-91585 
 
 1 -32958 
 
 1-78182 
 
 2-37049 
 
 3-10000 
 
 4-01176 
 
 
 
 
 
 
 OS 
 
 1-29873 
 
 1-85270 
 
 2-53043 
 
 3-36094 
 
 4-38305 
 
 5-65000 
 
 7-2368 
 
 
 
 
 
 O-Jf 
 
 1-63902 
 
 2-34286 
 
 3-20000 
 
 4-24478 
 
 5-52277 
 
 7-09474 
 
 9-0462 
 
 
 
 
 
 OS 
 
 1-94118 
 
 2-78126 
 
 3-80000 
 
 5-03582 
 
 6-53846 
 
 8-37500 
 
 10-6364 
 
 
 
 
 
 OS 
 
 2-20909 
 
 3-17350 
 
 4-33846 
 
 5-80178 
 
 7-44706 
 
 9-51429 
 
 12-0414 
 
 15-1585 
 
 
 
 
 0-7 
 
 2-44615 
 
 3-52441 
 
 4-82222 
 
 6-38287 
 
 8-26202 
 
 10-53182 
 
 13-2885 
 
 16-6624 
 
 
 
 
 0-8 
 
 2-65532 
 
 3-83820 
 
 5-25714 
 
 6-95698 
 
 8-99462 
 
 11-44375 
 
 14-4000 
 
 17-9932 
 
 
 
 
 0-9 
 
 2-83917 
 
 4-12064 
 
 5-64828 
 
 7-47438 
 
 9-65454 
 
 12-26250 
 
 15-3940 
 
 19-1758 
 
 23-7791 
 
 
 
 VO 
 
 3-00000 
 
 4-36842 
 
 6-00000 
 
 7-94121 
 
 10-24999 
 
 13-00000 
 
 16-2857 
 
 20-2308 
 
 25-0000 
 
 
 
 1-1 
 
 3-13980 
 
 4-59081 
 
 6-31613 
 
 8-36246 
 
 10-78796 
 
 13-66538 
 
 17-0877 
 
 21-1750 
 
 26-0857 
 
 32-0328 
 
 
 V2 
 
 3-26038 
 
 4-78812 
 
 6-60000 
 
 8-74286 
 
 11-27443 
 
 14-26667 
 
 17-8105 
 
 22-0225 
 
 27-0546 
 
 33-1082 
 
 
 IS 
 
 3-36330 
 
 4-96250 
 
 6-85454 
 
 9-08619 
 
 11-71461 
 
 14-81071 
 
 18-4633 
 
 22-7851 
 
 27-9217 
 
 34-0635 
 
 
 1-4 
 
 3-45000 
 
 5-11589 
 
 7-08235 
 
 9-39582 
 
 12-11304 
 
 15-30345 
 
 190536 
 
 23-4727 
 
 28-7000 
 
 34-9164 
 
 42-3013 
 
 IS 
 
 3-52174 
 
 5-25000 
 
 7-28571 
 
 9-67501 
 
 12-47368 
 
 15-75000 
 
 19-5864 
 
 24-0937 
 
 29-4000 
 
 35-6786 
 
 43-1538 
 
 A 
 
 TABLE XLII (b). 
 
 Values of /3 4 . 
 ft 
 
 
 2S 
 
 2S 
 
 8-0 
 
 SS 
 
 4-0 
 
 4S 
 
 6-0 
 
 5S 
 
 G-0 
 
 6S 
 
 7-0 
 
 o-o 
 
 5-00000 
 
 8-92856 
 
 15-0000 
 
 23-7288 
 
 31-0000 
 
 
 
 
 
 
 
 0-1 
 
 5-27356 
 
 9-41054 
 
 15-7973 
 
 25-7430 
 
 41-7660 
 
 69-3682 
 
 
 
 
 
 
 0-2 
 
 5-44361 
 
 9-75086 
 
 16-2648 
 
 26-4018 
 
 42-5000 
 
 69-4796 
 
 
 
 
 
 
 OS 
 
 5-53293 
 
 9-86224 
 
 16-4907 
 
 26-6520 
 
 42-5613 
 
 68-5776 
 
 114-4732 
 
 
 
 
 
 0-4 
 
 5-55998 
 
 9-91072 
 
 16-5385 
 
 26-6144 
 
 42-1807 
 
 67-0888 
 
 109-4534 
 
 
 
 
 
 OS 
 
 5-53802 
 
 9-87751 
 
 16-4545 
 
 26-3742 
 
 41-5076 
 
 65-2679 
 
 104-4652 
 
 
 
 
 
 OS 
 
 5-47791 
 
 9-81734 
 
 16-2732 
 
 26-1077 
 
 40-6453 
 
 63-2707 
 
 99-6442 
 
 162-125 
 
 
 
 
 0-7 
 
 5-38824 
 
 9-63895 
 
 16-0200 
 
 25-5026 
 
 39-6623 
 
 61-1940 
 
 95-0525 
 
 151-253 
 
 
 
 
 OS 
 
 5-27513 
 
 9-45991 
 
 15-7143 
 
 24-9478 
 
 38-6061 
 
 59-1016 
 
 90-7143 
 
 141 -707 
 
 
 
 
 OS 
 
 5-14437 
 
 9-25645 
 
 15-3706 
 
 24-3462 
 
 87*6090 
 
 57-0279 
 
 86-6331 
 
 133-240 
 
 210-995 
 
 
 
 1-0 
 
 5-00000 
 
 9-02746 
 
 15-0000 
 
 23-7495 
 
 36-3971 
 
 55-0000 
 
 82-8022 
 
 125-664 
 
 195-000 
 
 
 
 VI 
 
 4-84537 
 
 8-78075 
 
 14-6111 i 23-0744 
 
 35-2835 
 
 53-0316 
 
 79-2091 
 
 118-839 
 
 181-299 
 
 286-374 
 
 
 1-2 
 
 4-68319 
 
 8-53522 
 
 14-2105 22-4107 
 
 34-1811 
 
 51-1309 
 
 75-8392 
 
 112-653 
 
 169-394 
 
 261-436 
 
 
 1-8 
 
 4-51562 
 
 8-27700 
 
 13-8032 
 
 21-7535 
 
 33-0971 
 
 49-3447 
 
 72-6772 
 
 107-016 
 
 158-930 
 
 240-845 
 
 
 I'A 
 
 4-34440 
 
 8-01454 
 
 13-3931 
 
 21-1002 
 
 32-0367 
 
 47-5471 
 
 69-7076 
 
 101-850 
 
 149-643 
 
 223-304 
 
 343-147 
 
 IS 
 
 4-17097 
 
 7-75000 
 
 12-9832 
 
 20-4546 
 
 31-0037 
 
 45-9038 
 
 66-9117 
 
 97112 
 
 141333 
 
 208-129 
 
 313-704 
 
A 
 
 Probable Errors of Frequency Constants 79 
 
 of fj t and /3 3 on the assumption that the Frequency falls into 
 one or other of Pearson's Types. 
 
 TABLE XLII(c). 
 
 Values of /3 5 . 
 ft 
 
 1 
 
 2-0 
 
 2-5 
 
 3-0 
 
 3-5 
 
 4-0 
 
 4S 
 
 6-0 
 
 6-5 
 
 6-0 
 
 6S 
 
 7-0 
 
 0-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o-i 
 
 1-99086 
 
 4-39480 
 
 9-3207 
 
 19-9714 
 
 45-9387 
 
 128-529 
 
 
 
 
 
 
 0-2 
 
 3-59438 
 
 8-03374 
 
 16-5960 
 
 34-7825 
 
 76-6000 
 
 193-361 
 
 
 
 
 
 
 OS 
 
 4-86677 
 
 10-68765 
 
 22-2196 
 
 45-7142 
 
 97-2263 
 
 228-104 
 
 668-284 
 
 
 
 
 
 0-4 
 
 5-85929 
 
 12-85477 
 
 26-5187 
 
 53-7090 
 
 111-0237 
 
 246-506 
 
 650 398 
 
 
 
 
 
 OS 
 
 6-51704 
 
 14-51540 
 
 29-7545 
 
 59-4655 
 
 120-0543 
 
 255-295 
 
 614633 
 
 
 
 
 
 0-6 
 
 7-17383 
 
 15-7892 
 
 32-1362 
 
 63-9266 
 
 125-6629 
 
 258-147 
 
 581 -205 
 
 1618-635 
 
 
 
 
 0-7 
 
 7-56616 
 
 16-6546 
 
 33-8306 
 
 66-2045 
 
 128-8283 
 
 257-225 
 
 550-107 
 
 1368-373 
 
 
 
 
 0-8 
 
 7-81963 
 
 17-2668 
 
 349714 
 
 67-8804 
 
 130-2010 
 
 253-872 
 
 521-257 
 
 1196-612 
 
 
 
 
 0-9 
 
 7-95777 
 
 17-6667 
 
 35-6658 
 
 68-7533 
 
 130-2587 
 
 248-937 
 
 495-375 
 
 1068-877 
 
 2769-42 
 
 
 
 1-0 
 
 8-00000 
 
 17-8291 
 
 36-0000 
 
 69-0644 
 
 129-3434 
 
 243-000 
 
 469-637 
 
 968-318 
 
 2280-00 
 
 
 
 VI 
 
 7-96281 
 
 17-8472 
 
 36 0437 
 
 68-7730 
 
 127-7158 
 
 236-441 
 
 446-547 
 
 886-541 
 
 1945-69 
 
 5313-80 
 
 
 V2 
 
 7-86015 
 
 17-7503 
 
 35-8535 
 
 68-1357 
 
 125-5684 
 
 229-524 
 
 425-062 
 
 818-040 
 
 1700-98 
 
 4135-56 
 
 
 IS 
 
 7-70375 
 
 17-5396 
 
 35-4754 
 
 67-2181 
 
 123-0362 
 
 222-562 
 
 405-663 
 
 759-486 
 
 1512-94 
 
 3388-18 
 
 
 1-4 
 
 7-50358 
 
 17-2423 
 
 34-9467 
 
 66 0678 
 
 120-5142 
 
 214-828 
 
 386-347 
 
 708-620 
 
 1363-20 
 
 2870-08 
 
 7265-31 
 
 1-5 
 
 7-26808 
 
 16-8768 
 
 34-2983 
 
 64-7210 
 
 117-2460 
 
 207-227 
 
 368-843 
 
 663-926 
 
 1240-65 
 
 2488-62 
 
 5719-68 
 
 ft 
 
 TABLE XLII (d). 
 
 Values of /3 6 . 
 
 ft 
 
 
 2-0 
 
 2-5 
 
 3-0 
 
 3-5 
 
 4-0 
 
 4-5 
 
 6-0 
 
 6 5 
 
 6-0 
 
 6-5 
 
 7-0 
 
 o-o 
 
 14-0000 
 
 39 0649 
 
 105-000 
 
 290-678 
 
 868-015 
 
 
 
 
 
 
 
 o-i 
 
 16-4616 
 
 45-7741 
 
 124-835 
 
 355-508 
 
 1243-832 
 
 10228-33 
 
 
 
 
 
 
 0-2 
 
 17-7296 
 
 50-2472 
 
 132-998 
 
 369-894 
 
 1190-700 
 
 6204-69 
 
 
 
 
 
 
 OS 
 
 18-1764 
 
 51-0927 
 
 134-215 
 
 361-909 
 
 1089-739 
 
 4485-38 
 
 107697-95 
 
 
 
 
 
 0-4 
 
 18-0667 
 
 50-2458 
 
 131-337 
 
 344-886 
 
 977-506 
 
 3471-87 
 
 25413-18 
 
 
 
 
 
 0-5 
 
 17-5474 
 
 48-7896 
 
 126-107 
 
 323-447 
 
 877-884 
 
 2792-19 
 
 13737-63 
 
 
 
 
 
 0-6 
 
 16-8560 
 
 46-8558 
 
 119-601 
 
 303-252 
 
 784-431 
 
 2303-07 
 
 9048-43 
 
 119230-33 
 
 
 
 
 0-7 
 
 15-9787 
 
 44-3106 
 
 112-492 
 
 277-658 
 
 701-500 
 
 1934-79 
 
 6534-78 
 
 40994-77 
 
 
 
 
 OS 
 
 15-0148 
 
 41-7081 
 
 105-200 
 
 255-716 
 
 628-450 
 
 1648-52 
 
 5045-80 
 
 22660-09 
 
 
 
 
 0-9 
 
 14-0113 
 
 39-0906 
 
 97-984 
 
 235-072 
 
 564-277 
 
 1420-51 
 
 4024-45 
 
 14836-90 
 
 137288-7 
 
 
 
 VO 
 
 13-0000 
 
 36-4119 
 
 91-000 
 
 216-137 
 
 507-894 
 
 1235-50 
 
 3286-65 
 
 10612-25 
 
 57584-9 
 
 
 
 VI 
 
 12-0030 
 
 33-7916 
 
 84-339 
 
 198-263 
 
 456-575 
 
 1083-04 
 
 2741-39 
 
 8135-91 
 
 33078-5 
 
 797653-2 
 
 
 V2 
 
 11-0354 
 
 31-3418 
 
 78-047 
 
 181-987 
 
 414-455 
 
 955-78 
 
 2322-13 
 
 6314-06 
 
 21891-8 
 
 155693-9 
 
 
 IS 
 
 10-1070 
 
 28-9775 
 
 72-146 
 
 167-142 
 
 375-834 
 
 848-97 
 
 1994-05 
 
 5108-55 
 
 15690-2 
 
 75009-7 
 
 
 1-4 
 
 9-2240 
 
 26-7355 
 
 66-637 
 
 153-582 
 
 342-057 
 
 765-79 
 
 1726-18 
 
 4219-50 
 
 11846-9 
 
 44891-9 
 
 565740 
 
 V5 
 
 8-3899 
 
 246268 
 
 61-512 
 
 141-477 
 
 310-976 
 
 676-32 
 
 1508-92 
 
 354482 
 
 9281-2 
 
 30280-3 
 
 180793 
 
80 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XLIII. 
 
 Probable Error of Criterion k„. Values of ViVS,, for valves of ft,, ft, 
 
 
 ■00 
 
 ■05 
 
 •10 
 
 •15 
 
 ■20 
 
 •25 
 
 ■30 
 
 ■35 
 
 ■40 
 
 ■45 
 
 ■50 
 
 ■55 
 
 ■60 
 
 ■65 
 
 ■70 
 
 2-0 
 
 •000 
 
 •242 
 
 •332 
 
 •399 
 
 •454 
 
 •498 
 
 •545 
 
 •582 
 
 ■631 
 
 •671 
 
 ■716 
 
 •758 
 
 •806 
 
 •854 
 
 ■899 
 
 2-1 
 
 •000 
 
 •271 
 
 •367 
 
 •430 
 
 •483 
 
 •521 
 
 •557 
 
 •600 
 
 •639 
 
 •678 
 
 •717 
 
 •760 
 
 •800 
 
 •843 
 
 •890 
 
 2-2 
 
 •000 
 
 •310 
 
 •415 
 
 ■480 
 
 •527 
 
 •565 
 
 •597 
 
 •626 
 
 •656 
 
 •691 
 
 •728 
 
 •767 
 
 •809 
 
 •845 
 
 •890 
 
 2S 
 
 •000 
 
 •355 
 
 •477 
 
 •550 
 
 •596 
 
 ■635 
 
 •660 
 
 •678 
 
 •700 
 
 •725 
 
 •753 
 
 •790 
 
 •826 
 
 •858 
 
 •895 
 
 2- 4 
 
 ■000 
 
 •417 
 
 ■560 
 
 •642 
 
 •691 
 
 •722 
 
 •748 
 
 •759 
 
 •770 
 
 •787 
 
 •806 
 
 •830 
 
 •857 
 
 •884 
 
 •914 
 
 2S 
 
 •000 
 
 •500 
 
 •697 
 
 •771 
 
 •816 
 
 ■841 
 
 •855 
 
 •860 
 
 •867 
 
 ■873 
 
 •881 
 
 •892 
 
 •909 
 
 •928 
 
 •947 
 
 2-6 
 
 •000 
 
 •660 
 
 •840 
 
 •946 
 
 1-01 
 
 103 
 
 104 
 
 1-02 
 
 101 
 
 1-00 
 
 100 
 
 1-00 
 
 1-00 
 
 1-00 
 
 1-00 
 
 2-7 
 
 •000 
 
 1-04 
 
 1-30 
 
 1-35 
 
 1-34 
 
 1-33 
 
 1-30 
 
 1-27 
 
 1-24 
 
 119 
 
 1-15 
 
 1-12 
 
 1-10 
 
 109 
 
 1-08 
 
 2-8 
 
 •000 
 
 1-83 
 
 1-97 
 
 1-98 
 
 1-93 
 
 1-84 
 
 1-74 
 
 1-64 
 
 1-55 
 
 1-47 
 
 1-40 
 
 1-33 
 
 1-27 
 
 1-23 
 
 1-20 
 
 2-9 
 
 •000 
 
 351 
 
 3-71 
 
 3-42 
 
 3-00 
 
 2-66 
 
 2-42 
 
 2-22 
 
 2-05 
 
 1-89 
 
 1-73 
 
 1-61 
 
 1-51 
 
 1-45 
 
 1-39 
 
 3-0 
 
 — 
 
 18-8 
 
 9-89 
 
 694 
 
 5 30 
 
 4-30 
 
 3-62 
 
 3-10 
 
 273 
 
 243 
 
 2-18 
 
 2O0 
 
 1-84 
 
 1-71 
 
 1-61 
 
 3-1 
 
 •ooo 
 
 
 
 62-0 
 
 20-5 
 
 8-47 
 
 6-36 
 
 5-22 
 
 4-42 
 
 378 
 
 334 
 
 296 
 
 2-66 
 
 2-41 
 
 2-20 
 
 201 
 
 3-2 
 
 •ooo 
 
 7-82 
 
 91-7 
 
 
 
 49-7 
 
 20-2 
 
 12-2 
 
 8-48 
 
 644 
 
 5 09 
 
 4-28 
 
 3-71 
 
 321 
 
 2-86 
 
 2-58 
 
 3-3 
 
 •ooo 
 
 2-99 
 
 11-7 
 
 70-2 
 
 
 
 142 
 
 32 4 
 
 15-4 
 
 108 
 
 8-56 
 
 6-88 
 
 562 
 
 466 
 
 3-98 
 
 346 
 
 3-4 
 3-5 
 
 •ooo 
 
 1-82 
 
 4-80 
 
 13-8 
 
 55-5 
 
 — 
 
 344 
 
 182 
 
 28-9 
 
 16-8 
 
 11-8 
 
 8-69 
 
 6-88 
 
 5 60 
 
 476 
 
 •ooo 
 
 1-43 
 
 2 '89 
 
 6-43 
 
 15-8 
 
 50-6 
 
 380 
 
 — 
 
 127 
 
 465 
 
 25-1 
 
 16-1 
 
 11-5 
 
 855 
 
 691 
 
 3-6 
 
 •ooo 
 
 117 
 
 2-18 
 
 4-08 
 
 8-00 
 
 17-2 
 
 46-5 
 
 215 
 
 — 
 
 277 
 
 76-4 
 
 354 
 
 224 
 
 158 
 
 11-5 
 
 3-7 
 
 •ooo 
 
 1-04 
 
 1-79 
 
 3-08 
 
 5-18 
 
 9 36 
 
 24 '6 
 
 44-3 
 
 155 
 
 — 
 
 767 
 
 133 
 
 546 
 
 28-5 
 
 19-2 
 
 3-8 
 
 •ooo 
 
 •979 
 
 1-54 
 
 2-54 
 
 4-09 
 
 6 33 
 
 14-3 
 
 26-2 
 
 44-0 
 
 126 
 
 855 
 
 — 
 
 230 
 
 80-0 
 
 40-8 
 
 3-9 
 
 •ooo 
 
 •920 
 
 1-41 
 
 2-20 
 
 332 
 
 4-91 
 
 8-84 
 
 131 
 
 20-5 
 
 382 
 
 105 
 
 448 
 
 — 
 
 — 
 
 125 
 
 4-0 
 4-1 
 
 iy2 
 
 •ooo 
 
 •869 
 
 1-35 
 
 2-00 
 
 2-83 
 
 4-OS 
 
 5-98 
 
 9-08 
 
 137 
 
 223 
 
 41-4 
 
 98-0 
 
 296 
 
 — 
 
 — 
 
 
 
 
 1-30 
 
 1-94 
 
 2-60 
 
 3-61 
 
 5-08 
 
 7-25 
 
 10-4 
 
 15-8 
 
 24-4 
 
 40-5 
 
 78-7 
 
 216 
 
 — 
 
 
 
 
 
 1-33 
 
 1-94 
 
 2-58 
 
 3 43 
 
 4-54 
 
 6-09 
 
 8-49 
 
 11-8 
 
 168 
 
 253 
 
 394 
 
 67-2 
 
 169 
 
 4-3 
 
 *'4 
 
 4-5 
 4-6 
 4-7 
 4* 
 4-9 
 5-0 
 5-1 
 6-2 
 5-3 
 5-4 
 5-6 
 
 
 
 
 
 1-44 
 
 2-01 
 
 2-63 
 
 3 37 
 
 4-20 
 
 5-41 
 
 7-19 
 
 9-52 
 
 128 
 
 178 
 
 262 
 
 45-2 
 
 72-2 
 
 
 
 
 1-58 
 
 2-15 
 
 2-74 
 
 3-39 
 
 4-20 
 
 5-18 
 
 6-58 
 
 856 
 
 109 
 
 146 
 
 20-2 
 
 290 
 
 40-4 
 
 
 
 1-81 
 
 2 32 
 
 2-94 
 
 3-59 
 
 4 36 
 
 5-29 
 
 6-45 
 
 776 
 
 9-85 
 
 12-4 
 
 161 
 
 211 
 
 29-9 
 
 
 
 
 
 
 
 4 90 
 
 5-63 
 
 660 
 
 7-84 
 
 920 
 
 111 
 
 136 
 
 17-2 
 
 242 
 
 
 
 
 
 
 
 
 
 5-87 
 
 646 
 
 7-11 
 
 7-97 
 
 8-96 
 
 10-2 
 
 12-2 
 
 153 
 
 200 
 
 
 
 
 
 
 
 
 7-43 
 
 7-61 
 
 7-94 
 
 8-41 
 
 916 
 
 101 
 
 12-0 
 
 14-3 
 
 175 
 
 
 
 
 
 
 
 10-1 
 
 9-45 
 
 908 
 
 936 
 
 980 
 
 106 
 
 120 
 
 137 
 
 159 
 
 
 
 
 
 
 
 IV1 
 
 11-3 
 
 10-4 
 
 10-4 
 
 108 
 
 11-5 
 
 12-4 
 
 136 
 
 15-4 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 135 
 153 
 183 
 
 22-8 
 
 140 
 155 
 
 17-8 
 20-6 
 
 153 
 160 
 177 
 200 
 
 
 
 _ 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 306 
 
 260 
 
 233 
 
 5-6 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 
 
 — 
 
 5-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-8 
 5-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o-o 
 
 — 
 
 — 
 
 — 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 - — 
 
 — 
 
 
 
 
 
 
 6-1 
 6-2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7-0 
 
 — 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
Probable Errors of Frequency Constants 81 
 
 TABLE XLIII— (continued). 
 
 Probable Error of Criterion k 2 . Values of ViV2„, for values of #,, /?,. 
 
 A 
 
 •75 
 
 ■so 
 
 ■8S 
 
 ■00 
 
 ■95 
 
 1-00 
 
 1-05 
 
 1-10 
 
 1-15 
 
 1-20 
 
 1-25 
 
 ISO 
 
 1SS 
 
 VJfi 
 
 VJfi 
 
 ISO 
 
 
 •949 
 
 1-00 
 
 1-06 
 
 1-12 
 
 1-18 
 
 1-25 
 
 1-31 
 
 1-39 
 
 1-46 
 
 1-56 
 
 1-64 
 
 1-75 
 
 1-86 
 
 1-97 
 
 211 
 
 2'25 
 
 $-0 
 
 •937 
 
 •992 
 
 1-05 
 
 1-10 
 
 1-16 
 
 1-22 
 
 1-28 
 
 1-35 
 
 1-42 
 
 1-49 
 
 1-58 
 
 1-67 
 
 1-77 
 
 1-88 
 
 2-00 
 
 2-12 
 
 2-1 
 
 ■936 
 
 ■987 
 
 1-04 
 
 1-09 
 
 114 
 
 1-19 
 
 1-25 
 
 1-31 
 
 1-37 
 
 1-43 
 
 1-51 
 
 1-59 
 
 1-69 
 
 1-79 
 
 1-89 
 
 1-99 
 
 2-2 
 
 •939 
 
 •982 
 
 1-04 
 
 1-08 
 
 112 
 
 1-17 
 
 1-22 
 
 1-27 
 
 1-32 
 
 1-38 
 
 1-45 
 
 1-53 
 
 1-62 
 
 1-70 
 
 1-79 
 
 1-87 
 
 2-3 
 
 •950 
 
 •990 
 
 1-03 
 
 1-07 
 
 I'll 
 
 1-15 
 
 119 
 
 1-24 
 
 1-29 
 
 1-34 
 
 1-40 
 
 1-47 
 
 1-55 
 
 1-62 
 
 1-69 
 
 1-77 
 
 2-4 
 
 •972 
 
 •998 
 
 1-03 
 
 107 
 
 1-10 
 
 1-14 
 
 1-18 
 
 1-22 
 
 1-27 
 
 1-32 
 
 1-37 
 
 1-43 
 
 1-49 
 
 1-55 
 
 1-61 
 
 1-68 
 
 2-5 
 
 1-01 
 
 103 
 
 1-06 
 
 1-09 
 
 1-12 
 
 1-15 
 
 1-18 
 
 1-21 
 
 1-26 
 
 1-31 
 
 1-36 
 
 1-41 
 
 1-46 
 
 1-51 
 
 1-57 
 
 1-63 
 
 2-6 
 
 1-07 
 
 1-08 
 
 1 -09 
 
 111 
 
 1-14 
 
 1-17 
 
 1-19 
 
 1-21 
 
 1-25 
 
 1-30 
 
 1-35 
 
 1-39 
 
 1-43 
 
 1-48 
 
 1-53 
 
 1-59 
 
 2-7 
 
 117 
 
 1-15 
 
 1-16 
 
 1-17 
 
 1-18 
 
 1-19 
 
 1-21 
 
 1-23 
 
 1-27 
 
 1-31 
 
 1-35 
 
 1-39 
 
 1-43 
 
 1-47 
 
 1-52 
 
 1-58 
 
 2-8 
 
 1-33 
 
 1-28 
 
 1-26 
 
 1-26 
 
 1-25 
 
 1-25 
 
 1-25 
 
 1-27 
 
 1-29 
 
 1-32 
 
 1-35 
 
 1-39 
 
 1-43 
 
 1-47 
 
 1-52 
 
 1-57 
 
 2-9 
 
 1-53 
 
 1-47 
 
 1-42 
 
 1\38 
 
 1-36 
 
 1-34 
 
 1-34 
 
 1-34 
 
 1-35 
 
 1-36 
 
 1-38 
 
 1-40 
 
 1-43 
 
 1-46 
 
 1-51 
 
 1-56 
 
 s-o 
 
 1-85 
 
 1-72 
 
 1-62 
 
 1-56 
 
 1-52 
 
 1-49 
 
 1-46 
 
 1-44 
 
 1-43 
 
 1-43 
 
 1-43 
 
 1-44 
 
 1-45 
 
 1-47 
 
 1-51 
 
 1-57 
 
 3-1 
 
 2-34 
 
 213 
 
 1-96 
 
 1-83 
 
 1-75 
 
 1'68 
 
 1-62 
 
 1-57 
 
 1-54 
 
 1-52 
 
 1-51 
 
 1-50 
 
 1-50 
 
 1-51 
 
 1-54 
 
 1-58 
 
 S-2 
 
 303 
 
 2 69 
 
 2-44 
 
 2-23 
 
 2-06 
 
 1-94 
 
 1-85 
 
 1-77 
 
 1-72 
 
 1-67 
 
 1-63 
 
 1-60 
 
 1-59 
 
 1-58 
 
 1-59 
 
 1-61 
 
 3-3 
 
 4-08 
 
 350 
 
 3-06 
 
 2-74 
 
 2-49 
 
 2-29 
 
 2-14 
 
 2-02 
 
 1-92 
 
 1-84 
 
 1-78 
 
 1-73 
 
 1-70 
 
 1-68 
 
 1-68 
 
 1-69 
 
 S-Jf 
 
 572 
 
 475 
 
 4-12 
 
 3-56 
 
 3-15 
 
 2-83 
 
 2-59 
 
 2-38 
 
 2-22 
 
 2-09 
 
 1-99 
 
 1-91 
 
 1-85 
 
 1-81 
 
 1-79 
 
 1-79 
 
 3-5 
 
 8-85 
 
 6-90 
 
 5-71 
 
 4-79 
 
 4-15 
 
 3-68 
 
 3-22 
 
 2-90 
 
 2-67 
 
 2-49 
 
 2-31 
 
 2-20 
 
 2-09 
 
 2-00 
 
 1-96 
 
 1-92 
 
 3-6 
 
 14-2 
 
 104 
 
 8'22 
 
 6 69 
 
 5-70 
 
 4-81 
 
 4-17 
 
 3-67 
 
 3-31 
 
 3 05 
 
 2-81 
 
 2-62 
 
 2-45 
 
 2-29 
 
 2-19 
 
 2-10 
 
 3-7 
 
 249 
 
 181 
 
 138 
 
 10-1 
 
 8-16 
 
 6-71 
 
 5-76 
 
 4-92 
 
 4-28 
 
 3-84 
 
 3-48 
 
 3-16 
 
 291 
 
 2-68 
 
 2-50 
 
 2 36 
 
 8-8 
 
 05 5 
 
 365 
 
 21-7 
 
 15-8 
 
 12-2 
 
 10-0 
 
 8-17 
 
 675 
 
 5-75 
 
 5-00 
 
 4 43 
 
 3 93 
 
 352 
 
 3-18 
 
 2-91 
 
 2-72 
 
 3-9 
 
 242 
 
 8G-6 
 
 48-5 
 
 30 
 
 20-9 
 
 15-4 
 
 12-0 
 
 961 
 
 7-86 
 
 660 
 
 5-67 
 
 4-90 
 
 4-34 
 
 3-88 
 
 3-47 
 
 319 
 
 4-0 
 
 — 
 
 374 
 
 127 
 
 62-9 
 
 38 3 
 
 26-0 
 
 18-3 
 
 14-4 
 
 11-5 
 
 9-15 
 
 7-53 
 
 6-45 
 
 5 66 
 
 4-98 
 
 4-38 
 
 3-80 
 
 4-1 
 
 — 
 
 — 
 
 — 
 
 200 
 
 91-0 
 
 516 
 
 327 
 
 23 4 
 
 17-4 
 
 13-4 
 
 10-6 
 
 8-70 
 
 7-50 
 
 6-48 
 
 5-52 
 
 4-61 
 
 4-2 
 
 144 
 
 478 
 
 — 
 
 — 
 
 314 
 
 112 
 
 70 
 
 42-9 
 
 29-1 
 
 21-3 
 
 16-1 
 
 12-7 
 
 10-3 
 
 8-55 
 
 7-24 
 
 6-25 
 
 4-3 
 
 62-8 
 
 135 
 
 ._ 
 
 — 
 
 — 
 
 580 
 
 192 
 
 93-9 
 
 55-8 
 
 37 1 
 
 26-5 
 
 19-8 
 
 15-4 
 
 121 
 
 9-74 
 
 8-26 
 
 4-4 
 
 42-8 
 
 68-3 
 
 119 
 
 280 
 
 — 
 
 — 
 
 — 
 
 286 
 
 126 
 
 72-8 
 
 46-4 
 
 322 
 
 23-9 
 
 18-3 
 
 146 
 
 11-8 
 
 4-5 
 
 32 3 
 
 44-7 
 
 62-7 
 
 99-6 
 
 240 
 
 742 
 
 — 
 
 — 
 
 — 
 
 181 
 
 91-0 
 
 58-9 
 
 41-0 
 
 30-0 
 
 22 3 
 
 16'9 
 
 4'6 
 
 26-2 
 
 33 8 
 
 46-1 
 
 68-0 
 
 105 
 
 240 
 
 632 
 
 — 
 
 — 
 
 — 
 
 260 
 
 128 
 
 76-7 
 
 50-4 
 
 30-0 
 
 26-6 
 
 4-7 
 
 21-6 
 
 27-1 
 
 35-0 
 
 47-3 
 
 66 8 
 
 104 
 
 182 
 
 413 
 
 — 
 
 — 
 
 — 
 
 403 
 
 172 
 
 99-3 
 
 63-0 
 
 44-8 
 
 4-8 
 
 18-9 
 
 226 
 
 269 
 
 33-7 
 
 44-0 
 
 61-5 
 
 84-2 
 
 115 
 
 337 
 
 — 
 
 — 
 
 — 
 
 — 
 
 249 
 
 140 
 
 80-8 
 
 4-9 
 
 17-8 
 
 20-7 
 
 24-6 
 
 30-1 
 
 37-8 
 
 48-9 
 
 66 
 
 97-0 
 
 157 
 
 286 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 172 
 
 5-0 
 
 17-2 
 
 20O 
 
 23-0 
 
 27-1 
 
 32-5 
 
 40-6 
 
 51-5 
 
 69-2 
 
 99-8 
 
 147 
 
 253 
 
 559 
 
 — 
 
 — 
 
 — 
 
 — 
 
 5-1 
 
 173 
 
 193 
 
 21-7 
 
 24-8 
 
 29-5 
 
 35-4 
 
 431 
 
 54-7 
 
 70-6 
 
 94 6 
 
 138 
 
 216 
 
 — 
 
 — 
 
 — 
 
 — 
 
 5-2 
 
 18-0 
 
 19-7 
 
 21-4 
 
 239 
 
 27-1 
 
 31-5 
 
 37-0 
 
 44-2 
 
 54-5 
 
 69 3 
 
 93 2 
 
 132 
 
 205 
 
 380 
 
 — 
 
 — 
 
 5-3 
 
 19-6 
 
 20-5 
 
 21-6 
 
 23-3 
 
 25-6 
 
 28-6 
 
 32-8 
 
 38 
 
 44-8 
 
 54-0 
 
 68 6 
 
 94-0 
 
 130 
 
 185 
 
 — 
 
 — 
 
 5-4 
 
 22 '3 
 
 220 
 
 22-5 
 
 23 6 
 
 25-2 
 
 27-5 
 
 30-5 
 
 34 3 
 
 394 
 
 .45-4 
 
 54-7 
 
 67 3 
 
 86-5 
 
 116 
 
 169 
 
 275 
 
 5-5 
 
 
 
 — 
 
 
 
 24-5 
 
 25 '8 
 
 27-6 
 
 29-8 
 
 32-2 
 
 35-6 
 
 39 6 
 
 44-8 
 
 51-4 
 
 63-2 
 
 83-0 
 
 118 
 
 168 
 
 5-6 
 
 — 
 
 — 
 
 — 
 
 26-8 
 
 27-8 
 
 28-8 
 
 30-0 
 
 31-5 
 
 33-8 
 
 36-5 
 
 39-8 
 
 44-4 
 
 53-1 
 
 67-0 
 
 85-2 
 
 116 
 
 5-7 
 
 
 
 
 
 
 
 30-7 
 
 30-8 
 
 31 
 
 313 
 
 32 
 
 33-4 
 
 35-3 
 
 38 '2 
 
 42-2 
 
 48-4 
 
 56 6 
 
 69 !) 
 
 87-9 
 
 5-8 
 
 — 
 
 — 
 
 
 
 38-4 
 
 36-0 
 
 34-4 
 
 33 5 
 
 33 1 
 
 34-1 
 
 36-0 
 
 38-7 
 
 42-2 
 
 47-0 
 
 53-1 
 
 61-9 
 
 74-8 
 
 5-9 
 
 
 
 
 
 
 
 50-4 
 
 42-5 
 
 38-3 
 
 36 8 
 
 30 -4 
 
 36-5 
 
 37 9 
 
 40-0 
 
 43 
 
 46-6 
 
 513 
 
 578 
 
 66-2 
 
 6-0 
 
 
 
 — 
 
 
 
 — 
 
 
 
 
 
 
 
 41-0 
 
 41 
 
 4V6 
 
 42-7 
 
 44-6 
 
 47-2 
 
 50 3 
 
 55-0 
 
 61-5 
 
 6-1 
 
 
 
 
 
 
 
 
 
 
 48-2 
 
 47-0 
 
 46-1 
 
 46-0 
 
 46-7 
 
 48-0 
 
 50-0 
 
 53-5 
 
 58-5 
 
 6-2 
 
 
 
 
 
 
 
 
 573 
 
 54-0 
 
 51-6 
 
 49 8 
 
 49-1 
 
 49-4 
 
 50-0 
 
 52-5 
 
 56-8 
 
 6-3 
 
 
 
 
 
 
 
 
 79-4 
 
 60-6 
 
 58-6 
 
 54-2 
 
 51-8 
 
 51-2 
 
 50-8 
 
 52-5 
 
 55-5 
 
 6-4 
 
 
 
 
 
 
 
 
 128 
 
 84-0 
 
 66-9 
 
 599 
 
 55-9 
 
 53-9 
 
 53-1 
 57-4 
 63 5 
 73-6 
 97-2 
 
 53 6 
 57-0 
 61-5 
 678 
 
 77-7 
 
 55-4 
 56-7 
 59-2 
 63 5 
 70 5 
 
 6-5 
 6-6 
 6-7 
 6-8 
 6-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 134 
 
 96-0 
 
 82-0 
 
 7-0 
 
 11 
 
82 
 
 (8, 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XLIV. To find probable Frequency Type. 
 
 Values of l - 77 *JN'l l for given values of ft u y3 a {Semi-Minor Axis 
 of Probability Ellipse). 
 
 ft 
 
 
 
 
 ■05 
 
 ■1 
 
 ■15 
 
 ■2 
 
 ■25 
 
 ■8 
 
 ■35 
 
 '4 
 
 ■45 
 
 ■5 
 
 ■55 
 
 ■6 
 
 ■65 
 
 ■7 
 
 ■75 
 
 2-0 
 2-1 
 2-2 
 2S 
 2-4 
 2-5 
 2-6 
 2-7 
 2-8 
 2-9 
 8-0 
 3-1 
 8-2 
 8-8 
 3-4 
 8-5 
 3-6 
 3-7 
 3-8 
 3-9 
 4-0 
 4-1 
 4-2 
 4-s 
 4-4 
 4-5 
 4-6 
 4-7 
 4-S 
 4-9 
 5-0 
 5-1 
 5-2 
 5-3 
 
 5-4 
 5-5 
 5-6 
 5-7 
 5-8 
 5-9 
 6-0 
 6-1 
 6-2 
 6-3 
 6-4 
 6-5 
 6-6 
 6-7 
 6-8 
 6-9 
 7-0 
 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 o-o 
 
 •00 
 •00 
 •00 
 •00 
 •00 
 •00 
 •00 
 •00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 1 
 1 
 1 
 
 1 
 1 
 1 
 
 2 
 2 
 
 5 
 5 
 5 
 5 
 
 e 
 
 6 
 6 
 
 7 
 7 
 8 
 8 
 9 
 
 1 
 2 
 4 
 5 
 6 
 S 
 
 2 
 
 0-7 
 0-7 
 0-8 
 0-8 
 0-9 
 0-9 
 1-0 
 1-0 
 1-1 
 1-2 
 1-2 
 1-3 
 1-4 
 1-6 
 1-7 
 
 1'cS 
 
 2-0 
 2-1 
 2 3 
 2-5 
 2-8 
 3-0 
 33 
 3-6 
 4-0 
 4-4 
 
 
 
 
 
 
 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 
 2 
 2 
 2 
 2 
 2 
 3 
 3 
 3 
 4 
 4 
 4 
 
 8 
 8 
 9 
 9 
 
 
 1 
 2 
 3 
 3 
 4 
 5 
 6 
 8 
 9 
 
 1 
 3 
 5 
 7 
 
 3 
 6 
 
 4 
 9 
 
 0-8 
 0-9 
 0-9 
 1-0 
 1-0 
 1-1 
 1-1 
 1-2 
 1-3 
 1-3 
 1-4 
 -1-5 
 1-6 
 1-8 
 1-9 
 2-1 
 2-2 
 2 -4 
 2-6 
 2-9 
 3-2 
 3-5 
 3-8 
 4-2 
 4-7 
 5-2 
 
 
 
 
 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 
 2 
 2 
 2 
 2 
 2 
 3 
 3 
 3 
 3 
 4 
 4 
 5 
 
 8 
 9 
 9 
 
 
 1 
 2 
 3 
 3 
 4 
 5 
 6 
 7 
 9 
 
 ] 
 3 
 5 
 7 
 
 3 
 6 
 9 
 3 
 8 
 3 
 
 0-8 
 0-8 
 0-9 
 1-0 
 1-0 
 1-1 
 1-2 
 1-3 
 1-3 
 1-4 
 1-5 
 1-6 
 1-7 
 1-9 
 2'0 
 2-1 
 2-3 
 2-5 
 2-7 
 3-0 
 3-3 
 3-6 
 4-0 
 4-4 
 4-8 
 5-3 
 5 9 
 6-7 
 7-6 
 8 '5 
 11 
 
 
 
 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 2 
 2 
 2 
 2 
 2 
 3 
 3 
 3 
 4 
 4 
 4 
 5 
 5 
 6 
 7 
 8 
 1 
 
 8 
 8 
 9 
 
 
 1 
 2 
 3 
 3 
 4 
 5 
 6 
 7 
 9 
 
 1 
 3 
 .") 
 7 
 
 3 
 (. 
 
 4 
 8 
 3 
 9 
 7 
 6 
 B 
 
 
 0-7 
 0-8 
 0-9 
 1-0 
 10 
 1-1 
 1-1 
 1-2 
 1-3 
 1-4 
 1-5 
 1-6 
 1-7 
 1-8 
 2-0 
 21 
 2-3 
 2-5 
 2-7 
 3-0 
 3 3 
 3G 
 3-9 
 4-3 
 4-7 
 5-2 
 5-8 
 6-6 
 7-4 
 8-4 
 9-7 
 
 0-7 
 0-8 
 0-8 
 0'9 
 1-0 
 1-1 
 1-1 
 1-2 
 1-3 
 1-4 
 1-5 
 1-6 
 1-7 
 1-8 
 2-0 
 2-1 
 2-3 
 2-5 
 2-7 
 2-9 
 3-2 
 3-5 
 3 8 
 4-2 
 4-6 
 5-1 
 5-7 
 6-4 
 7-2 
 8-1 
 9-3 
 
 0-7 
 0-7 
 0-8 
 0-9 
 1-0 
 1-0 
 1-1 
 1-2 
 1-3 
 1-4 
 1-5 
 1-6 
 1-7 
 1-8 
 1-9 
 2-1 
 2-3 
 2-4 
 2-6 
 2-9 
 3-1 
 3-4 
 3-8 
 4-1 
 4-5 
 5-0 
 5-6 
 6-2 
 6-9 
 7-8 
 8-9 
 
 0-6 
 0-7 
 0-8 
 0-9 
 0-9 
 1-0 
 1-] 
 1-2 
 1-3 
 1-4 
 1-5 
 1-6 
 1-7 
 1-8 
 1-9 
 2-1 
 2-2 
 2-4 
 2-6 
 2-8 
 3-1 
 3 4 
 3-7 
 4-0 
 4-4 
 4-9 
 5-4 
 6-0 
 6-6 
 7-5 
 8-5 
 
 
 
 
 
 
 
 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 
 2 
 2 
 2 
 2 
 2 
 3 
 3 
 3 
 3 
 4 
 4 
 5 
 5 
 6 
 7 
 8 
 9 
 1 
 1 
 1 
 1 
 
 6 
 
 7 
 7 
 8 
 9 
 
 1 
 1 
 2 
 3 
 4 
 6 
 7 
 8 
 9 
 1 
 
 2 
 1 
 
 (i 
 6 
 
 
 :? 
 6 
 9 
 3 
 7 
 
 2 
 8 
 ■1 
 2 
 1 
 
 3 
 1 
 2 
 
 i 
 6 
 
 0-5 
 0-6 
 0-7 
 0-8 
 0-9 
 1-0 
 1-0 
 l-l 
 1-2 
 1-3 
 1-4 
 IT) 
 1-6 
 1-7 
 1-9 
 2-0 
 2-1 
 23 
 2-5 
 27 
 2'9 
 3-2 
 3-5 
 3-8 
 4-2 
 4-6 
 5-1 
 5-7 
 63 
 7-0 
 7-7 
 8-6 
 9-5 
 11 
 13 
 15 
 
 
 
 
 
 
 1 
 1 
 1 
 1 
 1 
 1 
 I 
 1 
 1 
 1 
 2 
 2 
 2 
 2 
 2 
 2 
 3 
 3 
 3 
 4 
 4 
 5 
 5 
 6 
 6 
 7 
 8 
 8 
 1 
 1 
 1 
 
 5 
 6 
 
 7 
 8 
 9 
 
 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 9 
 
 1 
 3 
 5 
 7 
 i) 
 2 
 5 
 8 
 1 
 5 
 
 6 
 2 
 8 
 4 
 1 
 9 
 
 2 
 4 
 
 0-4 
 0-5 
 0-6 
 0-7 
 0-8 
 9 
 1-0 
 1-1 
 1-2 
 1-3 
 1-4 
 1-5 
 1-6 
 1-7 
 1-8 
 1-9 
 2-1 
 2-3 
 2-4 
 2-6 
 2-8 
 3-1 
 8-4 
 3-7 
 4-0 
 4-4 
 4-8 
 63 
 5-9 
 6-5 
 7-1 
 7-7 
 8-6 
 9(> 
 11 
 13 
 
Probable Errors of Frequency Types 
 
 TABLE XLIV— {continued). 
 
 Values of 177 \ r N 2, for given values of &, /3 2 (Semi-AImor Axis 
 of Probability Ellipse). 
 
 A 
 
 83 
 
 ■8 
 
 ■85 
 
 •9 
 
 ■05 
 
 1-0 
 
 1-05 
 
 VI 
 
 1-15 
 
 V2 
 
 1-36 
 
 IS 
 
 1-SB 
 
 1-4 
 
 1-45 
 
 1 -5 
 
 
 0-4 
 
 3 
 
 03 
 
 0-2 
 
 O'l 
 
 o-i 
 
 
 
 o-o 
 
 00 
 
 o-o 
 
 o-o 
 
 o-o 
 
 o-o 
 
 o-o 
 
 o-o 
 
 2-0 
 
 0-5 
 
 0-5 
 
 0-4 
 
 0-3 
 
 0-2 
 
 0-2 
 
 
 
 2 
 
 O'l 
 
 
 
 1 
 
 0-1 
 
 00 
 
 o-o 
 
 
 
 
 
 o-o 
 
 o-o 
 
 2-1 
 
 0-6 
 
 0-6 
 
 O-o 
 
 0-5 
 
 0-4 
 
 0-4 
 
 
 
 :■> 
 
 03 
 
 
 
 3 
 
 0-2 
 
 o-i 
 
 o-o 
 
 
 
 
 
 o-o 
 
 o-o 
 
 2-2 
 
 0-7 
 
 0-7 
 
 0-6 
 
 0-6 
 
 0-5 
 
 0-5 
 
 
 
 I 
 
 0-4 
 
 
 
 4 
 
 0-3 
 
 0-2 
 
 0-1 
 
 
 
 1 
 
 
 
 o-o 
 
 2-3 
 
 0-8 
 
 0-8 
 
 0-7 
 
 0-7 
 
 0-G 
 
 0-6 
 
 
 
 5 
 
 0'5 
 
 
 
 4 
 
 0-4 
 
 3 
 
 0-2 
 
 
 
 2 
 
 0-1 
 
 o-o 
 
 2-Jf 
 
 0-9 
 
 0-8 
 
 0-8 
 
 0-7 
 
 0-7 | 0-7 
 
 
 
 (i 
 
 06 
 
 
 
 5 
 
 0-5 
 
 0-4 
 
 0-3 
 
 
 
 3 
 
 0-2 
 
 o-i 
 
 2-5 
 
 1-0 
 
 0-9 
 
 9 
 
 0-8 
 
 0-8 
 
 0-8 
 
 
 
 7 
 
 0-7 
 
 
 
 7 
 
 0-6 
 
 0-6 
 
 5 
 
 
 
 4 
 
 0-3 
 
 0-2 
 
 2-6 
 
 11 
 
 1-0 
 
 1-0 
 
 0-9 
 
 0-9 
 
 0-9 
 
 
 
 H 
 
 0-8 
 
 
 
 8 
 
 0-7 
 
 07 
 
 0-6 
 
 
 
 5 
 
 0-4 
 
 03 
 
 
 1-1 
 
 11 
 
 1-1 
 
 1-0 
 
 1-0 
 
 1-0 
 
 
 
 !) 
 
 0-9 
 
 
 
 9 
 
 0-8 
 
 0-8 
 
 0-7 
 
 
 
 6 
 
 0-5 
 
 0-5 
 
 2-8 
 
 1-2 
 
 1-2 
 
 1-2 
 
 l'l 
 
 1-1 
 
 1-0 
 
 1 
 
 
 
 1-0 
 
 1 
 
 
 
 1-0 
 
 0-9 
 
 0-8 
 
 
 
 7 
 
 0-7 
 
 0-7 
 
 2-9 
 
 1-3 
 
 1-3 
 
 1-3 
 
 1-2 
 
 1-2 
 
 1-2 
 
 1 
 
 1 
 
 11 
 
 1 
 
 1 
 
 1-0 
 
 l'O 
 
 0-9 
 
 
 
 9 
 
 0-8 
 
 0-8 
 
 s-o 
 
 1-4 
 
 1-4 
 
 1-4 
 
 1-3 
 
 1-3 
 
 1-3 
 
 1 
 
 2 
 
 1-2 
 
 1 
 
 2 
 
 1-1 
 
 1-1 
 
 1-0 
 
 1 
 
 
 
 0-9 
 
 0-9 
 
 8-1 
 
 1-5 
 
 1-5 
 
 1-5 
 
 1-4 
 
 1-4 
 
 1-4 
 
 1 
 
 :; 
 
 1-3 
 
 1 
 
 2 
 
 1-2 
 
 1-2 
 
 11 
 
 1 
 
 1 
 
 10 
 
 10 
 
 S-2 
 
 1-7 
 
 1-6 
 
 1-6 
 
 1-5 
 
 1-4 
 
 1-4 
 
 1 
 
 :; 
 
 1-3 
 
 1 
 
 :; 
 
 1-2 
 
 1-2 
 
 1-2 
 
 1 
 
 1 
 
 1-1 
 
 1-0 
 
 3-8 
 
 1-8 
 
 1-7 
 
 1-7 
 
 1-6 
 
 1-6 1-6 
 
 1 
 
 5 
 
 1-5 
 
 1 
 
 i 
 
 1-4 
 
 13 
 
 1-3 
 
 1 
 
 2 
 
 1-2 
 
 1-1 
 
 s-j f 
 
 1-9 
 
 1-9 
 
 1-8 
 
 1-8 
 
 1-7 
 
 1-7 
 
 1 
 
 6 
 
 1-6 
 
 1 
 
 5 
 
 1-6 
 
 1-4 
 
 1-4 
 
 1 
 
 3 
 
 1-3 
 
 1-2 
 
 3-5 
 
 2-0 
 
 2-0 
 
 2-0 
 
 1-9 
 
 1-8 
 
 1-8 
 
 1 
 
 7 
 
 1-7 
 
 1 
 
 6 
 
 1-6 
 
 1-5 
 
 1-5 
 
 1 
 
 4 
 
 1-4 
 
 1-4 
 
 3-6 
 
 2-2 
 
 2-2 
 
 2-1 
 
 2-1 
 
 2-0 
 
 2 
 
 1 
 
 9 
 
 1-8 
 
 1 
 
 7 
 
 1-7 
 
 1-6 
 
 1-6 
 
 1 
 
 (i 
 
 1-5 
 
 1-6 
 
 87 
 
 2-4 
 
 23 
 
 2-2 
 
 2-2 
 
 2-1 
 
 2-1 
 
 2 
 
 
 
 2-0 
 
 1 
 
 9 
 
 1-9 
 
 1-8 
 
 1-7 
 
 1 
 
 7 
 
 1-6 
 
 1-6 
 
 3-8 
 
 2-6 
 
 2-5 
 
 2 4 
 
 2-4 
 
 23 
 
 2-3 
 
 2 
 
 2 
 
 2-1 
 
 2 
 
 
 
 2-0 
 
 19 
 
 19 
 
 1 
 
 8 
 
 1-7 
 
 1-7 
 
 39 
 
 2-8 
 
 2-7 
 
 2-6 
 
 2-6 
 
 2-6 
 
 25 
 
 2 
 
 4 
 
 2-3 
 
 2 
 
 2 
 
 22 
 
 2-1 
 
 2-1 
 
 2 
 
 
 
 2-0 
 
 1-9 
 
 4-0 
 
 3-0 
 
 2-9 
 
 2-8 
 
 2-8 
 
 2-7 
 
 2-7 
 
 2 
 
 6 
 
 2-5 
 
 2 
 
 4 
 
 2-4 
 
 2-3 
 
 2-3 
 
 2 
 
 2 
 
 2-2 
 
 2-1 
 
 4'1 
 
 3-2 
 
 3 2 
 
 31 
 
 3 
 
 2-9 
 
 2-9 
 
 ■2. 
 
 8 
 
 2-7 
 
 2 
 
 6 
 
 2 6 
 
 2-5 
 
 2-4 
 
 2 
 
 3 
 
 2-3 
 
 2-2 
 
 4-2 
 
 3-5 
 
 3 5 
 
 3-4 
 
 3-3 
 
 3-2 
 
 31 
 
 3 
 
 
 
 2-9 
 
 2 
 
 8 
 
 2-8 
 
 2-7 
 
 2-6 
 
 2 
 
 5 
 
 2-4 
 
 2-4 
 
 4-3 
 
 3!) 
 
 3-8 
 
 3-7 
 
 3-6 
 
 3 5 
 
 3 4 
 
 3 
 
 3 
 
 31 
 
 3 
 
 
 
 3-0 
 
 2-9 
 
 2-8 
 
 2 
 
 7 
 
 2-7 
 
 2-6 
 
 4' 4 
 
 4-2 
 
 4-1 
 
 4-0 
 
 3-9 
 
 3-7 
 
 3 6 
 
 3 
 
 5 
 
 3 4 
 
 3 
 
 3 3-2 
 
 3-1 
 
 31 
 
 3 
 
 
 
 3-0 
 
 2'9 
 
 4-5 
 
 4-6 
 
 4-5 
 
 4-4 
 
 4-2 
 
 4-1 
 
 39 
 
 3 
 
 8 
 
 3 7 
 
 3 
 
 5 
 
 34 
 
 3 3 
 
 33 
 
 3 
 
 ■2 
 
 3-2 
 
 3 1 
 
 4-6 
 
 5-1 
 
 4-9 
 
 4-8 
 
 4-6 
 
 4-4 
 
 4 3 
 
 4 
 
 2 
 
 4-0 
 
 3 
 
 8 
 
 37 
 
 3 6 
 
 3-5 
 
 3 
 
 4 
 
 3-4 
 
 33 
 
 4-7 
 
 5-6 
 
 6-4 
 
 5-2 
 
 5-0 
 
 4-8 
 
 4-7 
 
 4 
 
 6 
 
 4-4 
 
 4 
 
 2 
 
 4-0 
 
 3 9 
 
 3-8 
 
 3 
 
 7 
 
 3-6 
 
 35 
 
 4-8 
 
 6 ■% 
 
 6-0 
 
 5-7 
 
 5-5 
 
 6-3 
 
 5-1 
 
 5 
 
 
 
 4-8 
 
 4 
 
 6 
 
 4-4 
 
 4-2 
 
 4-1 
 
 4 
 
 (i 
 
 3-9 
 
 3-8 
 
 4-9 
 
 6-8 
 
 (i-6 
 
 6 3 
 
 61 
 
 5-8 
 
 5-6 
 
 5 
 
 4 
 
 5-2 
 
 5 
 
 
 
 4-8 
 
 4-6 
 
 4-4 
 
 4 
 
 3 
 
 4-2 
 
 4-1 
 
 5-0 
 
 7-4 
 
 7-2 
 
 7-0 
 
 6-7 
 
 6-4 
 
 6-2 
 
 5 
 
 9 
 
 5-7 
 
 5 
 
 4 
 
 5-1 
 
 4-9 
 
 4-7 
 
 4 
 
 (i 
 
 4-4 
 
 4 3 
 
 5-1 
 
 8 3 
 
 8-0 
 
 7 '7 
 
 7-4 
 
 7-1 
 
 6-8 
 
 6 
 
 5 
 
 6-2 
 
 5 
 
 8 
 
 5-5 
 
 5-2 
 
 5-0 
 
 4 
 
 9 
 
 4-7 
 
 4-5 
 
 S-2 
 
 92 
 
 8 9 
 
 8-6 
 
 8-2 
 
 7-8 
 
 7'5 
 
 7 
 
 1 
 
 6-7 
 
 6 
 
 3 
 
 5-9 
 
 5-6 
 
 5-4 
 
 5 
 
 ■2 
 
 5-0 
 
 4-7 
 
 5-3 
 
 10 
 
 10 
 
 9-6 
 
 91 
 
 8-6 
 
 8'2 
 
 7 
 
 8 
 
 7 3 
 
 6 
 
 9 
 
 6-5 
 
 6-1 
 
 5-8 
 
 5 
 
 6 
 
 5-3 
 
 5-0 
 
 0-4 
 
 12 
 
 11 
 
 11 
 
 10 
 
 9-5 
 
 9-0 
 
 8 
 
 5 
 
 8-0 
 
 7 
 
 5 
 
 7-1 
 
 6-7 
 
 6-4 
 
 8 
 
 1 
 
 5-8 
 
 5-5 
 
 5-5 
 
 — 
 
 — 
 
 12 
 
 11 
 
 10 
 
 10 
 
 9-5 
 
 8-9 
 
 8 
 
 4 
 
 7-9 
 
 7-4 
 
 7-0 
 
 6 
 
 6 
 
 6 3 
 
 6-0 
 
 5-0 
 
 — 
 
 — 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 10 
 
 9-4 
 
 8-8 
 
 8-3 
 
 7-8 
 
 7 
 
 ■1 
 
 7-0 
 
 6-7 
 
 5-7 
 
 — 
 
 — 
 
 16 
 
 14 
 
 13 
 
 12 
 
 12 
 
 11 
 
 10 
 
 10 
 
 9-4 
 
 8-8 
 
 8 
 
 3 
 
 7-8 
 
 7-4 
 
 5-8 
 
 — 
 
 — 
 
 18 
 
 16 
 
 15 
 
 14 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 10 
 
 9 
 
 5 
 
 8-9 
 
 8-4 
 
 5-9 
 
 — 
 
 — 
 
 22 
 
 20 
 
 18 
 
 10 
 
 15 
 
 14 
 
 13 
 
 12 
 
 12 
 
 11 
 
 11 
 
 10 
 
 9-5 
 
 G-0 
 
 
 
 
 
 
 
 18 
 
 16 
 
 15 
 
 14 
 
 13 
 
 13 
 
 12 
 
 12 
 
 11 
 
 G-l 
 
 
 
 
 
 
 
 21 
 
 19 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 13 
 
 12 
 
 0-2 
 
 
 
 
 
 
 
 24 
 
 21 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 0-3 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 ■21 
 
 24 
 
 22 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 6-4 
 
 
 
 
 
 
 
 
 28 
 
 25 
 
 23 
 
 21 
 
 20 
 
 19 
 
 18 
 
 16 
 
 6-5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-6 
 6-7 
 0-8 
 6-9 
 7-0 
 
 A 
 
 11-2 
 
84 Tobies for Statisticians and Biometricians 
 
 TABLE XLV. To find 'probable Frequency Type. 
 
 Values of l - 77 ViVSj for values of /3 lt /3. 2 (Semi-Major Axis 
 of Probability Ellipse). 
 
 A 
 
 
 
 
 ■05 -1 
 
 •IB 
 
 •2 
 
 ■25 
 
 
 ■35 
 
 ■4 
 
 ■45 
 
 ■5 
 
 •55 
 
 ■6 
 
 •65 
 
 •7 
 
 ■75 
 
 2-0 
 2-1 
 
 2-2 
 S2S 
 2-4 
 2-5 
 2-6 
 2-7 
 2-8 
 2 9 
 
 s-o 
 
 81 
 
 8-2 
 38 
 
 s-j, 
 
 3-5 
 36 
 3-7 
 3-8 
 8-9 
 4-0 
 
 4-1 
 4-2 
 4-3 
 
 4'4 
 4-5 
 4-6 
 4-7 
 4-8 
 4-9 
 5-0 
 5-1 
 5-2 
 5-3 
 
 5-4 
 5-5 
 56 
 5-7 
 5-8 
 5-9 
 60 
 6-1 
 6-2 
 6-3 
 6-4 
 6-5 
 6-6 
 6-7 
 6-8 
 G-9 
 70 
 
 1-6 
 1-7 
 1-9 
 2-1 
 2-4 
 2-8 
 33 
 38 
 4-4 
 51 
 5-8 
 6-7 
 7-8 
 9-2 
 11 
 13 
 15 
 18 
 21 
 25 
 29 
 
 1-9 
 2-1 
 
 2 3 
 2-5 
 2-8 
 
 3 3 
 3-8 
 4-4 
 
 5 
 5-7 
 
 6 5 
 7-5 
 8-7 
 10 
 11 
 13 
 16 
 19 
 23 
 27 
 32 
 
 2-2 
 
 2-4 
 
 2-6 
 
 2-8 
 
 31 
 
 3 5 
 
 4-0 
 
 4-6 
 
 53 
 
 6-0 
 
 69 
 
 8-0 
 
 9-2 
 
 11 
 
 12 
 
 14 
 
 17 
 
 20 
 
 24 
 
 28 
 
 34 
 
 43 
 
 55 
 
 69 
 
 87 
 
 113 
 
 2-5 
 2-7 
 2-9 
 31 
 3-3 
 3-7 
 4-2 
 4-7 
 5-4 
 6-2 
 7-1 
 8 3 
 9-6 
 11 
 12 
 14 
 17 
 20 
 24 
 28 
 33 
 41 
 51 
 62 
 76 
 98 
 
 2-8 
 2-9 
 31 
 3-3 
 3 5 
 3-8 
 4-3 
 49 
 5-6 
 63 
 7-2 
 8-4 
 9-8 
 11 
 13 
 15 
 17 
 20 
 24 
 28 
 32 
 39 
 48 
 57 
 69 
 86 
 
 3-1 
 
 32 
 34 
 3-5 
 3-7 
 4-0 
 4-5 
 5-0 
 5-7 
 6-4 
 7 3 
 8-4 
 9-8 
 11 
 13 
 15 
 17 
 20 
 23 
 27 
 31 
 37 
 45 
 53 
 64 
 78 
 
 34 
 
 3o 
 
 3'7 
 
 3-8 
 
 3 9 
 
 4-1 
 
 4-6 
 
 5-1 
 
 5-8 
 
 6-5 
 
 7-3 
 
 8-4 
 
 9-7 
 
 11 
 
 13 
 
 14 
 
 16 
 
 19 
 
 22 
 
 26 
 
 30 
 
 36 
 
 42 
 
 50 
 
 60 
 
 71 
 
 90 
 
 110 
 
 140 
 
 200 
 
 384 
 
 3-7 
 
 3-8 
 
 3-9 
 
 4-0 
 
 4-1 
 
 43 
 
 4-7 
 
 5-2 
 
 5-9 
 
 6-6 
 
 7 3 
 
 8-3 
 
 9-5 
 
 11 
 
 12 
 
 14 
 
 16 
 
 19 
 
 22 
 
 25 
 
 29 
 
 34 
 
 40 
 
 48 
 
 56 
 
 66 
 
 79 
 
 99 
 
 124 
 
 160 
 
 234 
 
 4-1 
 
 4-1 
 
 4-1 
 
 4-2 
 
 43 
 
 4-5 
 
 4-8 
 
 5 3 
 
 5-9 
 
 6-5 
 
 7-2 
 
 8-2 
 
 9-4 
 
 11 
 
 12 
 
 14 
 
 16 
 
 18 
 
 21 
 
 24 
 
 28 
 
 32 
 
 38 
 
 45 
 
 53 
 
 62 
 
 73 
 
 90 
 
 110 
 
 135 
 
 182 
 
 4-4 
 
 4-4 
 
 4-4 
 
 4-4 
 
 4-5 
 
 4-7 
 
 4-9 
 
 5-4 
 
 5-9 
 
 65 
 
 7-1 
 
 8-0 
 
 9-2 
 
 11 
 
 12 
 
 14 
 
 16 
 
 18 
 
 21 
 
 24 
 
 27 
 
 31 
 
 36 
 
 43 
 
 50 
 
 57 
 
 67 
 
 80 
 
 97 
 
 119 
 
 153 
 
 4-7 
 
 4-6 
 
 4-6 
 
 4-6 
 
 4-7 
 
 4-8 
 
 5 
 
 5-4 
 
 5-8 
 
 6-4 
 
 7-0 
 
 7-9 
 
 9-0 
 
 10 
 
 11 
 
 13 
 
 15 
 
 17 
 
 20 
 
 23 
 
 26 
 
 30 
 
 34 
 
 40 
 
 46 
 
 53 
 
 61 
 
 72 
 
 86 
 
 105 
 
 130 
 
 5-0 
 4 '9 
 4-9 
 4-9 
 4-9 
 
 5 
 5-1 
 6-4 
 5-8 
 
 6 3 
 6-9 
 7-8 
 8-8 
 
 10 
 11 
 12 
 14 
 17 
 19 
 22 
 24 
 28 
 32 
 37 
 43 
 49 
 57 
 67 
 79 
 95 
 115 
 
 5-4 
 5 3 
 
 5-2 
 
 5-2 
 
 5-1 
 
 5-2 
 
 5 3 
 
 5-5 
 
 5-8 
 
 6-3 
 
 6-8 
 
 7-7 
 
 8-6 
 
 9-6 
 
 11 
 
 12 
 
 14 
 
 16 
 
 18 
 
 21 
 
 23 
 
 26 
 
 30 
 
 35 
 
 40 
 
 46 
 
 53 
 
 62 
 
 73 
 
 86 
 
 102 
 
 124 
 
 148 
 
 188 
 
 245 
 
 405 
 
 5-7 
 
 5-6 
 
 5-5 
 
 5-5 
 
 5-4 
 
 5-4 
 
 5 '5 
 
 56 
 
 5-9 
 
 6-3 
 
 6-8 
 
 7'6 
 
 8-4 
 
 9-4 
 
 11 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 25 
 
 29 
 
 33 
 
 38 
 
 43 
 
 50 
 
 58 
 
 68 
 
 79 
 
 92 
 
 110 
 
 134 
 
 166 
 
 203 
 
 297 
 
 61 
 
 6-0 
 
 5-9 
 
 5-8 
 
 5-7 
 
 5-7 
 
 5-7 
 
 5-8 
 
 6 
 
 6-4 
 
 6-8 
 
 7-5 
 
 8-3 
 
 92 
 
 10 
 
 11 
 
 13 
 
 15 
 
 17 
 
 19 
 
 21 
 
 24 
 
 27 
 
 31 
 
 36 
 
 41 
 
 48 
 
 55 
 
 64 
 
 74 
 
 85 
 
 101 
 
 120 
 
 146 
 
 177 
 
 230 
 
 6-4 
 
 6-3 
 
 62 
 
 61 
 
 60 
 
 5-9 
 
 6-0 
 
 61 
 
 6-2 
 
 6 5 
 
 6 8 
 
 7-4 
 
 8-1 
 
 8-9 
 
 10 
 
 11 
 
 13 
 
 14 
 
 16 
 
 18 
 
 20 
 
 23 
 
 26 
 
 29 
 
 34 
 
 39 
 
 45 
 
 52 
 
 60 
 
 69 
 
 79 
 
 93 
 
 109 
 
 129 
 
 154 
 
 194 
 
 A 
 
Probable Errors of Frequency Types 
 TABLE XLV— {continued). 
 
 Values of VII -J N "2* for values of @ u j3 2 (Semi-Major Axis 
 of Probability Ellipse). 
 
 fit 
 
 85 
 
 ■s 
 
 '86 
 
 ■9 
 
 ■95 
 
 1-0 
 
 1-05 
 
 1-1 
 
 1-15 
 
 1-2 
 9-9 
 
 125 
 10 
 
 1-8 
 
 11 
 
 1S5 
 11 
 
 1-4 
 11 
 
 1-1,5 
 12 
 
 1-5 
 12 
 
 2-0 
 
 6-8 
 
 7-2 
 
 7 
 
 6 
 
 7-9 
 
 8-3 
 
 8-7 
 
 9-1 
 
 9-5 
 
 6-7 
 
 7'0 
 
 7 
 
 4 
 
 7-8 
 
 8-1 
 
 8-5 
 
 8-9 
 
 92 
 
 9-6 
 
 10 
 
 11 
 
 11 
 
 11 
 
 11 
 
 12 
 
 2-1 
 
 65 
 
 6 8 
 
 7 
 
 1 
 
 7-6 
 
 8-0 
 
 8-3 
 
 8-6 
 
 9-0 
 
 9-4 
 
 9-8 
 
 10 
 
 10 
 
 11 
 
 11 
 
 12 
 
 2-2 
 
 6-4 
 
 6-7 
 
 7 
 
 1 
 
 7-4 
 
 7-8 
 
 8-1 
 
 8-5 
 
 8-8 
 
 9-2 
 
 9 6 
 
 9 9 
 
 10 
 
 11 
 
 11 
 
 12 
 
 2-8 
 
 6-3 
 
 6-6 
 
 6 
 
 B 
 
 7-3 
 
 76 
 
 8-0 
 
 8-3 
 
 8-7 
 
 9-0 
 
 9-4 
 
 9-8 
 
 10 
 
 10 
 
 11 
 
 11 
 
 2-4 
 
 6-2 
 
 65 
 
 6 
 
 8 
 
 7-1 
 
 7 5 
 
 7-8 
 
 8-2 
 
 8-5 
 
 8-9 
 
 9-2 
 
 9-6 
 
 10 
 
 10 
 
 11 
 
 11 
 
 2-5 
 
 62 
 
 6-5 
 
 G 
 
 8 
 
 7-1 
 
 7-4 
 
 7-7 
 
 8-0 
 
 8-3 
 
 8-7 
 
 9-0 
 
 9-4 
 
 9-9 
 
 10 
 
 11 
 
 11 
 
 26 
 
 6-3 
 
 6 5 
 
 6 
 
 8 
 
 7-0 
 
 7 3 
 
 7-6 
 
 7 9 
 
 8-2 
 
 8-5 
 
 8-8 
 
 9-2 
 
 9-7 
 
 10 
 
 10 
 
 11 
 
 2'7 
 
 64 
 
 6-6 
 
 6 
 
 8 
 
 7-0 
 
 7-2 
 
 7'5 
 
 7-7 
 
 80 
 
 8-3 
 
 8-7 
 
 91 
 
 9-5 
 
 9-8 
 
 10 
 
 10 
 
 2-8 
 
 6-6 
 
 6-7 
 
 6 
 
 8 
 
 7-0 
 
 7-2 
 
 7-4 
 
 7 6 
 
 7-9 
 
 8-2 
 
 8-0 
 
 8-9 
 
 9-3 
 
 9-6 
 
 10 
 
 10 
 
 2-9 
 
 6-9 
 
 7-0 
 
 7 
 
 1 
 
 7-2 
 
 7 3 
 
 7-5 
 
 7-7 
 
 7-9 
 
 8-2 
 
 8-5 
 
 8-8 
 
 91 
 
 9-4 
 
 9-7 
 
 10 
 
 3-0 
 
 7 3 
 
 73 
 
 7 
 
 ■■', 
 
 7-4 
 
 75 
 
 7 6 
 
 7-8 
 
 8-0 
 
 8-2 
 
 8-5 
 
 8-7 
 
 9 
 
 9-3 
 
 96 
 
 9-9 
 
 3-1 
 
 7-9 
 
 7-9 
 
 7 
 
 8 
 
 7-8 
 
 7-8 
 
 79 
 
 8-1 
 
 8-2 
 
 8-3 
 
 8-5 
 
 8-7 
 
 8-9 
 
 9-2 
 
 9-5 
 
 9-8 
 
 3-2 
 
 8-7 
 
 8-5 
 
 8 
 
 4 
 
 8-3 
 
 8 3 
 
 8-3 
 
 8-3 
 
 8-4 
 
 8-5 
 
 8-7 
 
 89 
 
 91 
 
 9-3 
 
 9-5 
 
 97 
 
 38 
 
 9-5 
 
 9-3 
 
 91 
 
 9 
 
 8-8 
 
 8-8 
 
 8-8 
 
 8-8 
 
 8-9 
 
 9 
 
 91 
 
 9-2 
 
 9-4 
 
 9-5 
 
 9 7 
 
 Slf 
 
 11 
 
 10 
 
 10 
 
 9-8 
 
 9-6 
 
 95 
 
 9-4 
 
 9 3 
 
 9-3 
 
 9-3 
 
 93 
 
 9-4 
 
 9-5 
 
 9 6 
 
 9-8 
 
 3-5 
 
 12 
 
 11 
 
 11 
 
 11 
 
 10 
 
 10 
 
 10 
 
 10 
 
 99 
 
 9-9 
 
 9-8 
 
 9-7 
 
 9-7 
 
 9-8 
 
 9-9 
 
 3-G 
 
 13 
 
 12 
 
 12 
 
 12 
 
 11 
 
 11 
 
 11 
 
 11 
 
 11 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 3-7 
 
 15 
 
 14 
 
 14 
 
 13 
 
 13 
 
 13 
 
 12 
 
 12 
 
 12 
 
 11 
 
 11 
 
 10 
 
 10 
 
 10 
 
 11 
 
 S-8 
 
 17 
 
 16 
 
 16 
 
 15 
 
 14 
 
 14 
 
 14 
 
 13 
 
 13 
 
 12 
 
 12 
 
 11 
 
 11 
 
 11 
 
 11 
 
 39 
 
 19 
 
 18 
 
 18 
 
 17 
 
 16 
 
 16 
 
 15 
 
 14 
 
 14 
 
 13 
 
 13 
 
 12 
 
 12 
 
 12 
 
 12 
 
 4-0 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 17 
 
 16 
 
 15 
 
 15 
 
 14 
 
 14 
 
 13 
 
 13 
 
 13 
 
 4-1 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 18 
 
 17 
 
 16 
 
 15 
 
 15 
 
 14 
 
 14 
 
 14 
 
 4-2 
 
 28 
 
 26 
 
 25 
 
 23 
 
 22 
 
 21 
 
 20 
 
 20 
 
 19 
 
 18 
 
 17 
 
 16 
 
 15 
 
 15 
 
 15 
 
 1,-S 
 
 32 
 
 30 
 
 29 
 
 27 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 17 
 
 17 
 
 17 
 
 4-4 
 
 37 
 
 35 
 
 33 
 
 31 
 
 29 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 19 
 
 18 
 
 4-5 
 
 43 
 
 40 
 
 37 
 
 35 
 
 33 
 
 31 
 
 29 
 
 28 
 
 27 
 
 25 
 
 24 
 
 23 
 
 22 
 
 21 
 
 20 
 
 40 
 
 49 
 
 45 
 
 42 
 
 39 
 
 37 
 
 35 
 
 33 
 
 32 
 
 •30 
 
 28 
 
 27 
 
 26 
 
 24 
 
 23 
 
 22 
 
 47 
 
 56 
 
 52 
 
 48 
 
 45 
 
 42 
 
 40 
 
 38 
 
 36 
 
 34 
 
 32 
 
 31 
 
 29 
 
 27 
 
 26 
 
 25 
 
 1,-S 
 
 64 
 
 59 
 
 55 
 
 51 
 
 48 
 
 46 
 
 42 
 
 40 
 
 38 
 
 36 
 
 35 
 
 33 
 
 31 
 
 29 
 
 28 
 
 1,9 
 
 73 
 
 68 
 
 63 
 
 59 
 
 55 
 
 52 
 
 49 
 
 46 
 
 43 
 
 41 
 
 39 
 
 37 
 
 35 
 
 33 
 
 32 
 
 5-0 
 
 85 
 
 78 
 
 72 
 
 67 
 
 63 
 
 59 
 
 55 
 
 52 
 
 49 
 
 46 
 
 43 
 
 41 
 
 39 
 
 37 
 
 36 
 
 5-1 
 
 99 
 
 90 
 
 82 
 
 77 
 
 72 
 
 67 
 
 63 
 
 58 
 
 55 
 
 52 
 
 49 
 
 46 
 
 43 
 
 41 
 
 40 
 
 5-2 
 
 114 
 
 104 
 
 95 
 
 88 
 
 82 
 
 76 
 
 71 
 
 66 
 
 62 
 
 58 
 
 55 
 
 51 
 
 48 
 
 46 
 
 44 
 
 5-3 
 
 136 
 
 123 
 
 112 
 
 102 
 
 95 
 
 87 
 
 80 
 
 75 
 
 70 
 
 65 
 
 61 
 
 57 
 
 54 
 
 51 
 
 49 
 
 5-4 
 
 167 
 
 147 
 
 132 
 
 119 
 
 108 
 
 100 
 
 92 
 
 85 
 
 79 
 
 74 
 
 69 
 
 64 
 
 60 
 
 57 
 
 54 
 
 5-5 
 
 — 
 
 — 
 
 160 
 
 141 
 
 126 
 
 115 
 
 105 
 
 96 
 
 89 
 
 83 
 
 78 
 
 73 
 
 68 
 
 64 
 
 60 
 
 5-6 
 
 — 
 
 — 
 
 206 
 
 169 
 
 148 
 
 132 
 
 120 
 
 no 
 
 102 
 
 95 
 
 88 
 
 82 
 
 76 
 
 72 
 
 67 
 
 57 
 
 — 
 
 — 
 
 258 
 
 206 
 
 175 
 
 150 
 
 136 
 
 126 
 
 116 
 
 108 
 
 100 
 
 93 
 
 87 
 
 81 
 
 75 
 
 58 
 
 — 
 
 — 
 
 318 
 
 255 
 
 215 
 
 190 
 
 168 
 
 150 
 
 136 
 
 125 
 
 115 
 
 107 
 
 99 
 
 92 
 
 85 
 
 59 
 
 — 
 
 — 
 
 446 
 
 332 
 
 273 
 
 228 
 
 200 
 
 178 
 
 161 
 
 147 
 
 134 
 
 123 
 
 113 
 
 104 
 
 98 
 
 GO 
 
 
 
 
 
 
 
 264 
 
 215 
 
 190 
 
 171 
 
 157 
 
 144 
 
 130 
 
 120 
 
 112 
 
 G-l 
 
 
 
 
 
 
 
 345 
 
 268 
 
 230 
 
 207 
 
 184 
 
 167 
 
 150 
 
 138 
 
 127 
 
 G-2 
 
 
 
 
 
 
 
 480 
 
 364 
 
 294 
 
 250 
 
 215 
 
 194 
 
 174 
 
 159 
 
 144 
 
 63 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 680 
 
 477 
 
 370 
 
 299 
 
 252 
 
 224 
 
 201 
 
 181 
 
 165 
 
 G-l, 
 
 
 
 
 
 
 
 1047 
 
 680 
 
 456 
 
 3G8 
 
 312 
 
 268 
 
 237 
 
 212 
 
 191 
 
 6-6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 280 
 338 
 412 
 525 
 
 248 
 297 
 362 
 446 
 
 223 
 266 
 320 
 390 
 
 GG 
 G-7 
 G-8 
 G-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 809 
 
 584 
 
 491 
 
 7-0 
 
 A 
 
8G 
 
 Tables for Statisticians and Biometricians 
 
 A 
 
 TABLE XLVI. To find probable Frequency Type. 
 
 Angle between Major-Axis and Axis of j3- 2 {Probability Ellipse) 
 measured in degrees. 
 
 
 
 
 ■05 
 
 •1 
 
 ■15 
 
 ■2 
 
 •25 
 
 ■3 
 
 •35 
 
 ■4 
 
 •46 
 
 ■5 
 
 ■55 
 
 ■6 
 
 ■65 
 
 ■7 
 
 ■75 
 
 2-0 
 
 
 
 12 
 
 23 
 
 28 
 
 31 
 
 33 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 2-1 
 
 
 
 11 
 
 21 
 
 20 
 
 28 
 
 30 
 
 32 
 
 34 
 
 35 
 
 37 
 
 38 
 
 39 
 
 40 
 
 40 
 
 41 
 
 41 
 
 2-2 
 
 
 
 10 
 
 19 
 
 23 
 
 26 
 
 28 
 
 30 
 
 32 
 
 33 
 
 35 
 
 36 
 
 38 
 
 39 
 
 39 
 
 40 
 
 40 
 
 2-s 
 
 
 
 10 
 
 18 
 
 22 
 
 25 
 
 27 
 
 28 
 
 30 
 
 32 
 
 34 
 
 35 
 
 37 
 
 38 
 
 38 
 
 39 
 
 39 
 
 2-4 
 
 
 
 9 
 
 17 
 
 20 
 
 23 
 
 25 
 
 26 
 
 29 
 
 31 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 38 
 
 2-5 
 
 
 
 8 
 
 15 
 
 18 
 
 21 
 
 23 
 
 25 
 
 27 
 
 29 
 
 31 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 37 
 
 2-6 
 
 
 
 7 
 
 14 
 
 17 
 
 20 
 
 22 
 
 24 
 
 26 
 
 28 
 
 30 
 
 31 
 
 33 
 
 34 
 
 35 
 
 35 
 
 36 
 
 2-7 
 
 
 
 7 
 
 13 
 
 16 
 
 19 
 
 21 
 
 23 
 
 25 
 
 26 
 
 28 
 
 29 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 2-8 
 
 
 
 6 
 
 12 
 
 15 
 
 17 
 
 19 
 
 21 
 
 23 
 
 25 
 
 27 
 
 28 
 
 30 
 
 31 
 
 32 
 
 33 
 
 33 
 
 2-9 
 
 
 
 6 
 
 11 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 23 
 
 25 
 
 26 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 3-0 
 
 
 
 5 
 
 10 
 
 13 
 
 15 
 
 17 
 
 19 
 
 21 
 
 22 
 
 24 
 
 25 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 S-l 
 
 
 
 5 
 
 9 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 21 
 
 23 
 
 24 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 3-2 
 
 
 
 5 
 
 9 
 
 12 
 
 14 
 
 16 
 
 17 
 
 19 
 
 20 
 
 21 
 
 22 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 3-3 
 
 
 
 4 
 
 8 
 
 11 
 
 13 
 
 15 
 
 16 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 24 
 
 25 
 
 26 
 
 27 
 
 3-4 
 
 
 
 4 
 
 8 
 
 10 
 
 12 
 
 14 
 
 15 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 SO 
 
 3-5 
 
 
 
 3 
 
 7 
 
 9 
 
 11 
 
 13 
 
 14 
 
 15 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 S-G 
 
 
 
 3 
 
 6 
 
 8 
 
 10 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 3-7 
 
 
 
 3 
 
 5 
 
 7 
 
 9 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 3-8 
 
 
 
 3 
 
 5 
 
 7 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 3-9 
 
 
 
 2 
 
 4 
 
 6 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 4-0 
 
 
 
 2 
 
 4 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 *-l 
 
 — 
 
 — 
 
 3 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 w 
 
 — 
 
 — 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 4-3 
 
 — 
 
 — 
 
 2 
 
 4 
 
 5 
 
 5 
 
 6 
 
 7 
 
 9 
 
 10 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 ■M 
 
 — 
 
 — 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 4-5 
 
 
 
 — 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 4-0 
 
 
 
 
 
 
 
 4 
 
 5 
 
 6 
 
 7 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 4-7 
 
 
 
 
 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 4-8 
 
 
 
 
 
 
 
 '3 
 
 4 
 
 4 
 
 5 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 4-9 
 
 
 
 
 
 
 
 2 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 5-0 
 
 
 
 
 
 
 
 1 
 
 2 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 6 
 
 7 
 
 8 
 
 5-1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 5 
 
 C 
 
 7 
 
 5-2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 5 
 
 6 
 
 7 
 
 5-3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 4 
 
 5 
 
 5 
 
 5-4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 3 
 
 4 
 
 5 
 
 5-5 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 5-9 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-2 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 0-3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 is- 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-G 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 G-8 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6-9 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7-0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Probable Errors of Frequency Types 
 
 87 
 
 TABLE XLVI— (continued). 
 
 Angle between Major-Axis and Axis of y8 a (Probability Ellipse) 
 measured in degrees. 
 
 ■8 
 
 ■85 
 
 ■9 
 
 ■95 
 
 1-0 
 
 105 
 
 VI 
 
 1-15 
 
 V2 
 
 V25 
 
 1-3 
 
 1-35 
 
 1-b 
 
 1-45 
 
 1-5 
 
 
 43 
 
 43 
 
 44 
 
 44 
 
 45 
 
 45 
 
 46 
 
 46 
 
 46 
 
 46 
 
 47 
 
 47 
 
 48 
 
 48 
 
 49 
 
 2-0 
 
 42 
 
 42 
 
 43 
 
 44 
 
 44 
 
 44 
 
 45 
 
 45 
 
 45 
 
 46 
 
 46 
 
 47 
 
 47 
 
 48 
 
 48 
 
 2-1 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 44 
 
 45 
 
 45 
 
 46 
 
 46 
 
 46 
 
 47 
 
 47 
 
 2-2 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 44 
 
 45 
 
 45 
 
 46 
 
 46 
 
 47 
 
 2-3 
 
 39 
 
 39 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 44 
 
 45 
 
 45 
 
 46 
 
 2-4 
 
 38 
 
 38 
 
 39 
 
 39 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 44 
 
 45 
 
 2-5 
 
 37 
 
 37 
 
 38 
 
 39 
 
 39 
 
 39 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 2-6 
 
 36 
 
 36 
 
 37 
 
 38 
 
 38 
 
 39 
 
 39 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 43 
 
 44 
 
 2'7 
 
 34 
 
 35 
 
 36 
 
 36 
 
 37 
 
 38 
 
 38 
 
 39 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 42 
 
 43 
 
 2-8 
 
 33 
 
 34 
 
 35 
 
 35 
 
 36 
 
 37 
 
 38 
 
 38 
 
 39 
 
 39 
 
 40 
 
 40 
 
 41 
 
 41 
 
 42 
 
 2-9 
 
 32 
 
 33 
 
 34 
 
 34 
 
 35 
 
 36 
 
 37 
 
 37 
 
 38 
 
 38 
 
 39 
 
 39 
 
 40 
 
 40 
 
 41 
 
 3-0 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 35 
 
 36 
 
 37 
 
 38 
 
 38 
 
 39 
 
 39 
 
 40 
 
 40 
 
 3-1 
 
 20 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 38 
 
 39 
 
 39 
 
 39 
 
 3-2 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 37 
 
 38 
 
 38 
 
 38 
 
 3'3 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 36 
 
 37 
 
 38 
 
 38 
 
 3-4 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 35 
 
 36 
 
 36 
 
 37 
 
 3-5 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 34 
 
 35 
 
 35 
 
 36 
 
 3-6 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 31 
 
 32 
 
 33 
 
 33 
 
 34 
 
 34 
 
 35 
 
 3-7 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 33 
 
 34 
 
 3-8 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 32 
 
 33 
 
 3-9 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 31 
 
 32 
 
 4'0 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 30 
 
 31 
 
 4-1 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 30 
 
 4-2 
 
 16 
 
 17 
 
 18 
 
 19 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 4-3 
 
 15 
 
 16 
 
 17 
 
 18 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 4-4 
 
 14 
 
 15 
 
 16 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 25 
 
 26 
 
 4-5 
 
 13 
 
 14 
 
 15 
 
 15 
 
 18 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 25 
 
 4-6 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 24 
 
 4-7 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 23 
 
 4-s 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 4'9 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 20 
 
 21 
 
 5-0 
 
 .8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 13 
 
 15 
 
 16 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 20 
 
 5-1 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 19 
 
 5-2 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 18 
 
 5-3 
 
 6 
 
 7 
 
 8 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 17 
 
 5-4 
 
 5 
 
 6 
 
 7 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 5-5 
 
 — 
 
 — 
 
 6 
 
 7 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 13 
 
 14 
 
 15 
 
 16 
 
 5-6 
 
 — 
 
 — 
 
 5 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 12 
 
 13 
 
 14 
 
 15 
 
 5-7 
 
 — 
 
 — 
 
 4 
 
 5 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 11 
 
 12 
 
 13 
 
 14 
 
 5-S 
 
 — 
 
 — 
 
 3 
 
 4 
 
 5 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 10 
 
 11 
 
 12 
 
 13 
 
 5-9 
 
 — 
 
 — 
 
 2 
 
 3 
 
 4 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 9 
 
 10 
 
 11 
 
 12 
 
 6-0 
 
 — 
 
 — 
 
 
 
 
 
 
 
 6 
 
 7 
 
 8 
 
 9 
 
 9 
 
 10 
 
 11 
 
 6-1 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 5 
 
 6 
 
 7 
 
 8 
 
 8 
 
 9 
 
 10 
 
 6-8 
 
 
 
 
 
 
 
 
 
 4 
 
 5 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 6-3 
 
 
 
 
 
 
 
 
 
 3 
 
 4 
 
 5 
 
 6 
 
 6 
 
 7 
 
 8 
 
 6-4 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 5 
 
 6 
 6 
 
 7 
 7 
 
 6-5 
 6-6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 3 
 2 
 
 1 
 
 5 
 4 
 3 
 3 
 
 6 
 5 
 4 
 4 
 
 6-7 
 6-8 
 
 6-9 
 7-0 
 
 A 
 
88 
 
 Tables for Statisticians and Biontetr, 
 
 icuins 
 
 Diagram XLVII determining the probability of a given Type of Frequency. 
 
Probable Occurrences in Second Small Samples 
 
 89 
 
 TABLE XL VIII. Percentage Frequency of Successes in a Second Sample 
 " m " after drawing " p " Successes in a First Sample " n ". 
 
 Successes 
 
 p = 
 
 i>=l 
 
 i> = 2 
 
 p = 3 
 
 p = i 
 
 n- 
 
 = 6) 
 = 5] 1 
 
 58-3333 
 
 31-8182 
 
 15-9091 
 
 7-0707 
 
 
 in 
 
 26-5151 
 
 31-8182 
 
 26-5151 
 
 17-6768 
 
 
 
 2 
 
 10-6060 
 
 21-2121 
 
 26-5151 
 
 25-2525 
 
 
 
 3 
 
 3 5354 
 
 10-6060 
 
 18-9394 
 
 25-2525 
 
 
 
 4 
 
 •8838 
 
 3-7879 
 
 9-4697 
 
 17-6768 
 
 
 
 5 
 
 •1263 
 
 •7576 
 
 2-6515 
 
 7-0707 
 
 
 n- 
 
 = 6| 
 
 53-8162 
 
 26-9231 
 
 12-2378 
 
 4-8951 
 
 
 m 
 
 = 6( 1 
 
 26-9231 
 
 29-3706 
 
 22-0280 
 
 13-0536 
 
 
 
 2 
 
 12*2378 
 
 22-0280 
 
 24-4755 
 
 20-3963 
 
 
 
 3 
 
 4-8951 
 
 13-0536 
 
 20-3962 
 
 23-3100 
 
 
 
 4 
 
 1-6317 
 
 6-1189 
 
 13-1119 
 
 20-3963 
 
 
 
 5 
 
 •4079 
 
 2-0979 
 
 6-1189 
 
 13-0536 
 
 
 
 6 
 
 •0582 
 
 •4079 
 
 1-6317 
 
 4-8951 
 
 
 n = 
 
 = 7| 
 
 61-5385 
 
 35-8974 
 
 19-5804 
 
 9-7902 
 
 
 m= 
 
 = 5f 1 
 
 25-6410 
 
 32-6340 
 
 29-3706 
 
 21-7560 
 
 
 
 2 
 
 9-3240 
 
 19-5804 
 
 26-1072 
 
 27-1950 
 
 
 
 3 
 
 2-7972 
 
 8-7024 
 
 16-3170 
 
 23-3100 
 
 
 
 4 
 
 •6216 
 
 2-7195 
 
 6-9930 
 
 13-5975 
 
 
 
 6 
 
 •0777 
 
 •4662 
 
 1-6317 
 
 4-3512 
 
 
 n- 
 
 = 7| 
 
 57-1429 
 
 30-7692 
 
 15-3846 
 
 6-9930 
 
 
 m- 
 
 -ef i 
 
 26-3736 
 
 30-7692 
 
 25-1748 
 
 16-7832 
 
 
 
 2 
 
 10-9890 
 
 20-9790 
 
 25-1748 
 
 23-3100 
 
 
 
 3 
 
 3-9960 
 
 11-1888 
 
 18-6480 
 
 23-3100 
 
 
 
 4 
 
 1-1988 
 
 4-6620 
 
 10-4895 
 
 17-4825 
 
 
 
 5 
 
 •2664 
 
 1-3986 
 
 4-1958 
 
 9-3240 
 
 
 
 6 
 
 •0333 
 
 •2331 
 
 •9324 
 
 2-7972 
 
 
 n= 
 
 = 71 
 
 53-3333 
 
 26-6667 
 
 12-3077 
 
 5-1282 
 
 
 m*=7 
 
 26-6667 
 
 28-7179 
 
 21-5385 
 
 13-0536 
 
 
 
 2 
 
 12-3077 
 
 21 -5385 
 
 23-4965 
 
 19-5804 
 
 
 
 3 
 
 5-1282 
 
 13-0536 
 
 19-5804 
 
 21-7560 
 
 
 
 4 
 
 1-8648 
 
 6-5268 
 
 13-0536 
 
 19-0365 
 
 
 
 5 
 
 •5594 
 
 2-6107 
 
 6-8531 
 
 13-0536 
 
 
 
 6 
 
 •1243 
 
 •7615 
 
 2-6107 
 
 6-5268 
 
 
 
 7 
 
 •0155 
 
 •1243 
 
 •5594 
 
 1-8648 
 
 
 n = 
 
 = 8\ 
 
 64-2857 
 
 39-5604 
 
 23-0769 
 
 12-5874 
 
 6-2937 
 
 m = 
 
 -b{ 1 
 
 24-7253 
 
 32-9670 
 
 31-4685 
 
 25-1748 
 
 17-4825 
 
 
 2 
 
 8-2418 
 
 17-9820 
 
 25-1748 
 
 27-9720 
 
 26 2238 
 
 
 3 
 
 2-2478 
 
 7-1928 
 
 13-9860 
 
 20-9790 
 
 26-2238 
 
 
 4 
 
 •4495 
 
 1-9980 
 
 5-2448 
 
 10-4895 
 
 17-4825 
 
 
 5 
 
 •0499 
 
 •2997 
 
 1-0489 
 
 2-7972 
 
 6-2937 
 
 n = 
 
 = 81 
 
 60-0000 
 
 34-2857 
 
 18-4615 
 
 9-2308 
 
 4-1958 
 
 m = 
 
 ,6f 1 
 
 25-7143 
 
 31 -6484 
 
 27-6923 
 
 20-1398 
 
 12-5874 
 
 
 2 
 
 9-8901 
 
 19-7802 
 
 25-1748 
 
 25-1748 
 
 20-9790 
 
 
 3 
 
 3-2967 
 
 9-5904 
 
 16-7832 
 
 22-3776 
 
 24-4755 
 
 
 4 
 
 •8991 
 
 3-5964 
 
 8-3916 
 
 14-6853 
 
 20-9790 
 
 
 5 
 
 •1798 
 
 •9590 
 
 2-9370 
 
 6-7133 
 
 12-5874 
 
 
 6 
 
 •0200 
 
 •1399 
 
 •5594 
 
 1-6783 
 
 4-1958 
 
 n = 
 
 ■S\ 
 
 56 -2500 
 
 30-0000 
 
 15-0000 
 
 6 9231 
 
 2-8846 
 
 m = 
 
 .7? l 
 
 26-2500 
 
 30-0000 
 
 24-2308 
 
 16-1538 
 
 9-1783 
 
 
 2 
 
 11-2500 
 
 20-7692 
 
 24-2308 
 
 22-0280 
 
 16-5210 
 
 
 3 
 
 4-3269 
 
 11-5385 
 
 18-3566 
 
 22-0280 
 
 21-4161 
 
 
 4 
 
 1-4423 
 
 5-2448 
 
 11-0140 
 
 17-1329 
 
 21-4161 
 
 
 5 
 
 •3934 
 
 1-8881 
 
 5-1399 
 
 10-2797 
 
 16-5210 
 
 
 6 
 
 •0787 
 
 •4895 
 
 1-7132 
 
 4-4056 
 
 9-1783 
 
 
 7 
 
 •0087 
 
 •0609 
 
 •3147 
 
 1-0489 
 
 2-8846 
 
 l>=5 
 
 B. 
 
 12 
 
90 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XLVIII — {continued). Percentage Frequency of Successes in a Second 
 Sample " m " after drawing "p " Successes in a First Sample " n ". 
 
 p = 5 
 
 Successes 
 
 p = 
 
 p = \ 
 
 t»-a 
 
 ,,-3 
 
 p = i 
 
 n- 
 
 = 81 
 
 
 
 52-9412 
 
 26-4706 
 
 12-3529 
 
 5-2941 
 
 2-0362 
 
 m- 
 
 = 8f 
 
 1 
 
 26-4706 
 
 28-2353 
 
 21-1765 
 
 13-0317 
 
 6-7873 
 
 
 
 2 
 
 12-3529 
 
 21-1765 
 
 22-8054 
 
 19-0045 
 
 12-9576 
 
 
 
 8 
 
 5-2941 
 
 13-0317 
 
 19-0045 
 
 20-7322 
 
 18-1407 
 
 
 
 4 
 
 2-0362 
 
 6-7873 
 
 12-9576 
 
 18-1407 
 
 20-1563 
 
 
 
 6 
 
 •6787 
 
 2-9617 
 
 7-2563 
 
 12-9000 
 
 18-1407 
 
 
 
 6 
 
 •1851 
 
 1-0366 
 
 3-2250 
 
 7-2563 
 
 12-9576 
 
 
 
 7 
 
 •0370 
 
 •2633 
 
 1-0366 
 
 2-9617 
 
 6-7873 
 
 
 
 8 
 
 •0041 
 
 •0370 
 
 •1851 
 
 •6787 
 
 2-0362 
 
 n<= 
 
 :?! 
 
 
 
 66-6667 
 
 42-8571 
 
 26-3736 
 
 15-3846 
 
 8-3916 
 
 iti- 
 
 1 
 
 23-8095 
 
 32-9670 
 
 32-9670 
 
 27-9720 
 
 20-9790 
 
 
 
 2 
 
 7-3260 
 
 16-4835 
 
 23-9760 
 
 27-9720 
 
 27-9720 
 
 
 
 S 
 
 1-8315 
 
 5-9940 
 
 11-9880 
 
 18-6480 
 
 24-4755 
 
 
 
 4 
 
 •3330 
 
 1-4985 
 
 3-9960 
 
 8-1585 
 
 13-9860 
 
 
 
 5 
 
 •0333 
 
 •1998 
 
 •6993 
 
 1-8648 
 
 4-1958 
 
 n- 
 
 = 91 
 
 
 
 62-5000 
 
 37-5000 
 
 21-4286 
 
 11-5385 
 
 5-7692 
 
 m = 
 
 = 6f 
 
 1 
 
 25-0000 
 
 32-1429 
 
 29-6703 
 
 23-0769 
 
 15-7343 
 
 
 
 2 
 
 8-9286 
 
 18-5439 
 
 24-7253 
 
 26-2238 
 
 23-6014 
 
 
 
 8 
 
 2-7472 
 
 8-2418 
 
 14-9850 
 
 20-9790 
 
 24-4755 
 
 
 
 k 
 
 ■6868 
 
 2-8097 
 
 6-7433 
 
 12-2378 
 
 18-3566 
 
 
 
 5 
 
 •1249 
 
 •6743 
 
 2-0979 
 
 4-8951 
 
 9-4405 
 
 
 
 6 
 
 •0125 
 
 •0874 
 
 •3496 
 
 1-0489 
 
 2-6224 
 
 n» 
 
 = 9 l 
 
 
 
 58-8235 
 
 33-0882 
 
 17-6471 
 
 8-8235 
 
 4-0724 
 
 »■ 
 
 = 71 
 
 1 
 
 25-7353 
 
 30-8824 
 
 26-4706 
 
 19-0045 
 
 11-8778 
 
 
 
 2 
 
 10-2941 
 
 19-8529 
 
 24-4344 
 
 23-7557 
 
 19-4364 
 
 
 
 8 
 
 3-6765 
 
 10-1810 
 
 16-9683 
 
 21-5961 
 
 22-6759 
 
 
 
 4 
 
 1-1312 
 
 4-2421 
 
 9-2554 
 
 15-1172 
 
 20-1563 
 
 
 
 5 
 
 •2828 
 
 1-3883 
 
 3-8873 
 
 8-0625 
 
 13-6055 
 
 
 
 6 
 
 •0514 
 
 •3239 
 
 1-1518 
 
 3-0234 
 
 6-4788 
 
 
 
 7 
 
 •0051 
 
 •0411 
 
 •1851 
 
 •6170 
 
 1-6968 
 
 B = 
 
 = 91 
 
 
 
 55-5555 
 
 29-4118 
 
 14-7059 
 
 6-8627 
 
 2-9412 
 
 1H = 
 
 = 8f 
 
 1 
 
 26-1438 
 
 29-4118 
 
 23-5294 
 
 15-6863 
 
 9-0498 
 
 
 
 2 
 
 11-4379 
 
 20-5882 
 
 23-5294 
 
 21-1161 
 
 15-8371 
 
 
 
 S 
 
 4-5752 
 
 11-7647 
 
 18-0995 
 
 21-1161 
 
 20-1563 
 
 
 
 It 
 
 1-6340 
 
 5-6561 
 
 11-3122 
 
 16-7969 
 
 20-1563 
 
 
 
 5 
 
 •5027 
 
 2-2624 
 
 5-7589 
 
 10-7500 
 
 16-1250 
 
 
 
 6 
 
 •1257 
 
 •7199 
 
 2-3036 
 
 5-3750 
 
 10-0782 
 
 
 
 7 
 
 •0229 
 
 •1645 
 
 •6582 
 
 1-9197 
 
 4-5249 
 
 
 
 8 
 
 •0023 
 
 •0206 
 
 •1028 
 
 •3771 
 
 1-1312 
 
 n= 
 
 =9| 
 
 
 
 52-6316 
 
 26-3158 
 
 12-3839 
 
 5-4180 
 
 2-1672 
 
 m= 
 
 -9( 
 
 1 
 
 26-3158 
 
 27-8638 
 
 20-8978 
 
 13-0031 
 
 6-9659 
 
 
 
 2 
 
 12-3839 
 
 20-8978 
 
 22-2910 
 
 18-5759 
 
 12-8602 
 
 
 
 8 
 
 5-4180 
 
 13-0031 
 
 18-5759 
 
 20-0047 
 
 17-5042 
 
 
 
 U 
 
 2-1672 
 
 6-9659 
 
 12-8602 
 
 17-5042 
 
 19-0955 
 
 
 
 5 
 
 •7740 
 
 3-2151 
 
 7-5018 
 
 12-7303 
 
 17-1859 
 
 
 
 6 
 
 •2381 
 
 1-2503 
 
 3-6372 
 
 7-6382 
 
 12-7303 
 
 
 
 7 
 
 •0595 
 
 •3897 
 
 1-4029 
 
 3-6372 
 
 7-5018 
 
 
 
 8 
 
 •0108 
 
 •0877 
 
 •3897 
 
 1-2503 
 
 3-2150 
 
 
 
 9 
 
 •0011 
 
 •0108 
 
 •0595 
 
 •2381 
 
 •7740 
 
 n= 
 
 = 5 l 
 
 
 
 54-5454 
 
 27-2727 
 
 12-1212 
 
 4-5454 
 
 
 m - 
 
 -b) 
 
 1 
 
 27-2727 
 
 30-3030 
 
 22-7273 
 
 12-9870 
 
 
 
 
 2 
 
 12-1212 
 
 22-7273 
 
 25-9740 
 
 21-6450 
 
 
 
 
 8 
 
 4-5454 
 
 12-9870 
 
 21-6450 
 
 25-9740 
 
 
 
 
 4 
 
 1-2987 
 
 5-4112 
 
 12-9870 
 
 22-7273 
 
 
 
 
 6 
 
 •2165 
 
 1-2987 
 
 4-6454 
 
 12-1212 
 
 
Probable Occurrences in Second Small Samples 91 
 
 TABLE XLVIII— {continued). 
 Successes p = p = l i> = 2 p = 3 y = 4 p=S p=d p = l 
 
 n=10l 
 m= 5j 
 
 68-7500 45-8333 29-4643 18-1318 10-5769 5-7692 
 
 1 22-9167 32-7381 33-9972 30-2198 24-0385 17-3077 
 
 2 6-5476 15-1099 22-6648 27-4725 28-8461 26-9230 
 
 5 1-5110 5-0366 10-3022 164835 22-4359 269230 
 4 -2518 1-1447 3-0907 6-4103 11-2179 17-3077 
 
 6 -0229 -1373 -4807 1-2820 2-8846 5-7692 
 
 n = 10l 
 to = 10| 1 
 
 52-3809 26-1905 12-4060 5-5138 2-2704 -8514 
 
 26-1905 27-5689 20-6767 12-9736 7-0949 3-4056 
 
 12-4060 206767 21-8930 18-2441 12-7709 7-6G25 
 
 5 5-5138 12-9736 18'2441 19-4604 17-0278 12-5744 
 4 2-2704 7-0949 12-7709 17-0278 18-3377 16-5039 
 
 6 -8514 3-4056 7-6625 12-5744 16-5039 18-0043 
 
 6 -2838 1-4190 3-9295 7-8590 12-5030 16-5039 
 
 7 -0811 -4990 1-6841 4-0826 7-8590 12-5744 
 
 8 -0187 -1403 -5741 1-6841 3-9295 7-6625 
 
 9 -0031 -0284 -1403 "4990 1-4190 3-4056 
 10 -0003 -0031 -0187 -0811 -2838 -8514 
 
 w = 15| 
 m= 5f 1 
 
 76-1905 57-1429 42-1053 30-4094 21-4654 14-7575 9-8383 6-3246 
 
 190476 30-0752 35-0877 35-7757 33-5397 29-5149 24-5958 19-4604 
 
 2 4-0100 10-0251 16-5119 22-3598 26-8318 29-5149 30-2717 29-1906 
 
 5 -6683 2-3588 6-1599 8-9439 13-4159 18-1631 22-7038 26-5369 
 4 -0786 -3685 1-0320 2-2360 4-1280 6-8111 10-3199 14-5953 
 
 6 -0049 -0295 -1032 -2752 -6192 1-2384 2-2704 38921 
 
 n = 15l 
 «i = 10( 1 
 
 61-5384 36-9231 21-5385 12-1739 6-6403 3-4783 1-7391 -8238 
 
 24-6154 30-7692 280936 22-1344 15-8103 10-4348 6-4073 3-6613 
 
 9-2308 18-0602 22-9857 23-7154 21-3439 17-2997 12-8146 8-7226 
 
 $ 3-2107 8-7565 14-5941 18-9723 209694 20-5034 18-0913 14-5376 
 
 4 1-0216 3-6485 7 '6619 12-2322 16-3095 18-9958 19-7873 18-6566 
 
 6 -2919 1-3135 3-3874 6-5238 10-3613 14-2469 17-4128 19-1897 
 
 6 -0730 -4032 1-2546 2-8781 5-3965 8-7064 12-4377 16-9914 
 
 7 -0153 -1024 -3795 1-0279 2-2614 4-2644 7-1073 10-6609 
 
 8 -0025 -0203 "0889 -2827 -7269 1-5991 3-1094 5-4516 
 
 9 -0003 -0028 -0145 -0538 -1615 -4146 -9423 1-9383 
 10 -0000 -0002 -0012 -0054 -0189 -0565 -1508 -3661 
 
 w = 16l 
 m=15( 1 
 
 » = 15| 51-6129 25-8066 12-4583 5-7842 2-5707 1-0876 -4351 -1631 
 
 25-8065 26-6963 200222 128538 7-4156 3-9155 1-9033 -8512 
 
 2 12-4583 20-0222 20-7638 17-3032 12-4583 7-9941 4-6342 2-4375 
 
 5 5-7842 12-8538 17'3032 17'9953 15-7459 12-0490 8-2152 6-0297 
 It 2-5707 7-4156 12-4583 16-7459 16-4305 14-7874 11-7361 8-2991 
 
 6 1-0876 3-9155 7-9941 12-0490 14-7874 16-4916 14-2006 11-5313 
 
 6 -4351 1-9033 46342 8-2152 11-7361 14-2006 H'9480 13-8803 
 
 7 -1631 -8512 2-4375 5-0297 8-2991 11-5313 13-8803 14-6968 
 
 8 -0567 -3482 1-1607 2-7663 5-2415 8-3282 11-4309 13-7783 
 
 9 -0181 -1289 -4965 1-3589 29443 5-3344 8-3350 11-4309 
 
 10 -0052 -0426 -1882 -5889 1-4548 30006 5-3344 8-3282 
 
 11 -0013 -0122 -0618 -2204 -6200 1-4548 2-9443 5-2415 
 
 12 -0003 -0029 -0169 -0689 -2204 -5889 1-3589 2-7664 
 IS -0001 -0006 -0037 -0170 -0618 -1882 -4965 1-1607 
 
 14 -0000 -0001 -0006 -0029 -0122 -0426 -1290 -3482 
 
 15 -0000 -0000 -0000 -0003 -0013 -0052 -0181 -0567 
 
 12—2 
 
92 Tables for Statisticians and Biometriciam 
 
 TABLE XLVIII — (continued). Percentage Frequency of Successes in a Second 
 
 Sample "m" after drawing "p" Successes in a First Sample "n". 
 
 Successes p = p = l p = 2 p = 3 p — 4 p = 5 
 
 m=20| 80-7692 04-6154 51-1538 40-0334 30-9349 23-5695 
 
 M= 5 i 16-1538 26-9231 33-3612 36-3940 36-8273 35-3542 
 
 ft 2-6923 7-0234 12-1313 17-3305 22-0964 26-0505 
 
 S -3512 1-2770 2'8884 5-1992 8-1408 11-5780 
 
 4 -0319 -1520 -4333 -9577 1-8090 3-0647 
 
 5 -0015 -0091 -0319 -0851 -1915 -3831 
 
 w=20l 67-7419 45-1613 29-5884 19-0211 11-9763 7-3700 
 
 «i=10[ 1 22-5806 31-1457 31-7019 28-1795 23-0313 17-6880 
 
 ft 7-0078 15-0167 21-1346 24-3861 24-8738 23-2155 
 
 S 2-0022 5-9325 10-8382 15-6071 19-3463 21-5333 
 
 4 -5191 1-9965 4-5521 7-9661 H'7760 15-4158 
 
 5 -1198 -5750 1-5932 3-3250 5'7809 8-8091 
 
 6 -0240 -1398 -4618 1-1335 2-2940 4-0375 
 
 7 -0040 -0278 -1079 -3084 -7210 1-4571 
 
 8 -0005 -0043 -0193 -0636 -1707 -3946 
 
 9 -0000 -0004 -0024 -0089 -0274 -0722 
 10 -0000 -0000 -0002 -0006 -0023 -0068 
 
 w = 20( 58-3333 33-3333 18-6275 10-1604 5'3977 2-7859 
 
 to = 15| 1 25-0000 29-4118 25-4011 19-0508 13-0590 8-3578 
 
 ft 10-2941 18-7166 22-2259 21-5090 18-2826 14-1218 
 
 S 4-0553 10-1381 155343 18-6411 19-1232 17-4841 
 
 4 1-5207 4-9056 9-3206 13-4987 16-3913 17-4841 
 
 5 -5396 2-1584 4'9495 8'4849 12-0203 147942 
 
 6 -1799 -8683 2-3569 4-7138 7-7053 10-8491 
 
 7 -0558 -3190 1-0101 2-3310 4-3590 6'9744 
 
 8 -0159 -1063 -3885 1-0256 21795 3-9421 
 
 9 -0041 -0318 -1330 -3989 -9581 1-9511 
 
 10 -0010 -0084 -0399 '1353 -3658 -8362 
 
 11 -0002 -0019 -0102 -0391 '1188 -3041 
 
 12 -0000 -0004 -0022 -0093 -0317 -0907 
 
 13 -0000 -0001 -0004 -0017 -0065 -0209 
 
 14 -0000 -0000 -0000 -0002 -0009 -0033 
 
 15 -0000 -0000 -0000 -0000 -0001 -0003 
 
 
 = 201 
 = 20f 1 
 
 51-2195 25-6098 12-4765 5-9099 2-7154 1-2068 
 
 25-6098 26-2664 19-6998 12-7783 7-5427 4-1377 
 
 12-4765 19-6998 20-2323 16-8602 12-2839 8-0929 
 
 S 5-9099 12-7783 16-8602 17-3419 151742 11-7715 
 
 4 2-7154 7-5427 12-2839 15-1742 15-6340 14-0706 
 
 5 1-2068 4-1377 ' 8-0929 11-7715 140706 145245 
 
 6 -5172 2-1297 4-9048 8-2768 11-3473 133141 
 
 7 -0839 1-0326 2"7589 5-3399 8-3213 11-0186 
 
 8 -0315 -4719 1-4462 3-1817 5-5954 8-3131 
 
 9 -0112 -2030 -7070 . 1-7554 3-4638 5-7473 
 
 10 -0037 -0819 -3218 -8965 1-9757 3 6474 
 
 11 -0012 -0308 -1358 -4226 1'0362 2-1221 
 
 12 -0003 -0107 -0528 -1829 -4974 1-1274 
 IS -0001 -0034 -0188 -0720 -2168 -5429 
 
 14 -0000 -0010 -0060 -0255 -0848 -2345 
 
 15 — -0003 -0017 -0080 -0293 -0893 
 10 — -0000 -0004 -0022 -0087 "0293 
 
 17 — — -0001 -0005 -0022 -0080 
 
 18 — — -0000 -0001 -0004 -0017 
 
 19 — — — -0000 -0001 -0003 
 
 so — — — — -oooo -oooo 
 
Probable Occurrences in Second Small Samples 
 
 93 
 
 TABLE XLVIII— (continued). 
 
 Successes 
 
 p = 
 
 p=l 
 
 p=2 
 
 p = 3 
 
 p = 4 
 
 p — 5 
 
 re = 25| 
 m= 5| 1 
 
 83-8710 
 
 69-8925 
 
 57-8421 
 
 475131 
 
 38-7144 
 
 312693 
 
 13-9785 
 
 24-1008 
 
 30-9868 
 
 35-1949 
 
 37-2254 
 
 37-5232 
 
 2 
 
 1-9281 
 
 5-1645 
 
 9-1813 
 
 13-5365 
 
 17-8682 
 
 21 -8885 
 
 S 
 
 •2065 
 
 •7651 
 
 1-7656 
 
 3-2488 
 
 5-2115 
 
 7-6134 
 
 4 
 
 •0153 
 
 •0736 
 
 •2119 
 
 •4738 
 
 ■9064 
 
 1-5573 
 
 5 
 
 •000(5 
 
 •0035 
 
 •0123 
 
 •0329 
 
 •0741 
 
 •1483 
 
 n = 25l 
 m=\o\ 1 
 
 72-2222 
 
 51-5873 
 
 36-4146 
 
 25-3798 
 
 17-4486 
 
 11-8200 
 
 20-6349 
 
 303455 
 
 33-1041 
 
 31-7248 
 
 28-1430 
 
 23-6401 
 
 2 
 
 5-4622 
 
 12-4141 
 
 18-6211 
 
 23-0261 
 
 25-3287 
 
 25-6780 
 
 S 
 
 1-3242 
 
 4-1380 
 
 8-0091 
 
 12-2806 
 
 16-3035 
 
 19-5642 
 
 4 
 
 •2897 
 
 1-1680 
 
 2-8032 
 
 5-1875 
 
 8-1518 
 
 11-4125 
 
 5 
 
 •0561 
 
 •2803 
 
 •8119 
 
 1-7786 
 
 3-2607 
 
 5-2673 
 
 6 
 
 •0093 
 
 •0564 
 
 •1933 
 
 •4940 
 
 1-0451 
 
 1-9313 
 
 7 
 
 •0013 
 
 •0092 
 
 •0368 
 
 •1086 
 
 •2628 
 
 •5518 
 
 8 
 
 •0001 
 
 •0011 
 
 •0053 
 
 •0179 
 
 •0493 
 
 •1170 
 
 9 
 
 •0000 
 
 •0001 
 
 ■0005 
 
 •0020 
 
 •0062 
 
 •0165 
 
 10 
 
 •0000 
 
 •0000 
 
 •0000 
 
 •0001 
 
 •0004 
 
 •0012 
 
 n'=25| 
 m=15( 1 
 
 63-4146 
 
 39-6341 
 
 24-3902 
 
 14-7625 
 
 8-7777 
 
 6-1203 
 
 23-7805 
 
 30-4878 
 
 28-8832 
 
 23-9392 
 
 18-2869 
 
 13-1666 
 
 2 
 
 8-5366 
 
 16-8485 
 
 21-8575 
 
 23-2742 
 
 21 -9443 
 
 18-9753 
 
 3 
 
 2-9204 
 
 7-8930 
 
 13-1550 
 
 17-2894 
 
 19-5777 
 
 19-9337 
 
 4 
 
 •9472 
 
 3-2888 
 
 6-7654 
 
 10-6788 
 
 14-2383 
 
 16-8191 
 
 5 
 
 •2894 
 
 1-2403 
 
 3-0643 
 
 5-6953 
 
 8-8100 
 
 11-9361 
 
 6 
 
 •0827 
 
 •4256 
 
 1-2381 
 
 2-6697 
 
 4-7366 
 
 7-2943 
 
 7 
 
 •0218 
 
 •1326 
 
 •4477 
 
 1-1072 
 
 2-2329 
 
 3-8807 
 
 8 
 
 ■0053 
 
 •0373 
 
 •1444 
 
 •4060 
 
 •9240 
 
 1-8017 
 
 9 
 
 •0012 
 
 •0094 
 
 •0412 
 
 •1307 
 
 •3337 
 
 •7266 
 
 10 
 
 •0002 
 
 •0020 
 
 ■0102 
 
 •0364 
 
 •1038 
 
 •2515 
 
 11 
 
 •0000 
 
 •0004 
 
 •0022 
 
 •0086 
 
 •0272 
 
 •0732 
 
 12 
 
 •0000 
 
 •0001 
 
 •0004 
 
 •0016 
 
 •0058 
 
 •0173 
 
 is 
 
 •0000 
 
 •0000 
 
 •0001 
 
 •0002 
 
 •0009 
 
 •0031 
 
 U 
 
 — 
 
 •0000 
 
 •0000 
 
 •0000 
 
 •0001 
 
 •0004 
 
 10 
 
 — 
 
 — 
 
 •0000 
 
 •0000 
 
 •0000 
 
 •0000 
 
 n = 25| 
 m*=20f 1 
 
 56-5217 
 
 31-4010 
 
 17-1278 
 
 9-1614 
 
 4-7988 
 
 2-4579 
 
 25-1208 
 
 28-5463 
 
 23-8993 
 
 17-4502 
 
 11-7044 
 
 7 3738 
 
 2 
 
 10-8476 
 
 18-9202 
 
 21-6231 
 
 20-2168 
 
 16-6788 
 
 12-5733 
 
 S 
 
 4-5409 
 
 10-8116 
 
 15-8218 
 
 18-1951 
 
 17-9618 
 
 15-8820 
 
 4 
 
 1-8380 
 
 5-6036 
 
 10-0864 
 
 13-8796 
 
 16-0711 
 
 16-4186 
 
 5 
 
 •7172 
 
 2-6897 
 
 5-7932 
 
 9-3504 
 
 12-5094 
 
 14-5943 
 
 6 
 
 •2690 
 
 1-2069 
 
 3-0491 
 
 5-6861 
 
 8-6871 
 
 11-4669 
 
 7 
 
 •0965 
 
 •5082 
 
 1-4833 
 
 3-1589 
 
 5-4604 
 
 8-0943 
 
 8 
 
 •0330 
 
 •2009 
 
 •6695 
 
 1-6133 
 
 31317 
 
 5-1816 
 
 9 
 
 •0107 
 
 •0744 
 
 ■2806 
 
 •7592 
 
 1-6449 
 
 3 0226 
 
 10 
 
 •0033 
 
 •0257 
 
 ■1089 
 
 •3290 
 
 •7916 
 
 1-6088 
 
 11 
 
 •0009 
 
 •0082 
 
 •0390 
 
 •1308 
 
 ■3482 
 
 •7800 
 
 12 
 
 •0002 
 
 •0024 
 
 •0128 
 
 •0475 
 
 •1393 
 
 •3429 
 
 18 
 
 •0001 
 
 •0006 
 
 •0038 
 
 •0156 
 
 •0502 
 
 •1357 
 
 14 
 
 •0000 
 
 •0002 
 
 •0010 
 
 •0046 
 
 •0162 
 
 •0477 
 
 15 
 
 — 
 
 •0000 
 
 •0002 
 
 •0012 
 
 •0045 
 
 •0147 
 
 16 
 
 — 
 
 — 
 
 •0001 
 
 •0002 
 
 •0011 
 
 •0039 
 
 17 
 
 — 
 
 — 
 
 — 
 
 •0000 
 
 •0002 
 
 •0008 
 
 18 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •0000 
 
 •0001 
 
 19 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •0000 
 
 20 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
94 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XLVIII— (continued). Percentage Ft 
 
 •equency 
 
 of Successes 
 
 in a Se 
 
 Sample " 
 
 m " after 
 
 drawing " p" Successes in a 
 
 First Sample " n ". 
 
 Successes 
 
 p = 
 
 P = i 
 
 p = 2 
 
 i> = 3 
 
 p=4 
 
 p.fi 
 
 w = 25| 
 m = 25| ; 
 
 50-9804 
 
 25-4902 
 
 12-4850 
 
 5-9824 
 
 2-8003 
 
 1-2784 
 
 25-4902 
 
 26-0104 
 
 19-5078 
 
 12-7285 
 
 7-6094 
 
 4-2613 
 
 2 
 
 12-4850 
 
 19-5078 
 
 19-9229 
 
 16-6024 
 
 12-1751 
 
 8-1352 
 
 3 
 
 5-9824 
 
 12-7285 
 
 16-6024 
 
 16-9713 
 
 14-8499 
 
 11-6037 
 
 4 
 
 2-8003 
 
 7-6094 
 
 12-1751 
 
 14-8499 
 
 15-1953 
 
 13-6757 
 
 5 
 
 1-2784 
 
 42613 
 
 8-1352 
 
 11-6037 
 
 13-6757 
 
 14-0093 
 
 6 
 
 •5682 
 
 2-2598 
 
 5-0451 
 
 8-2883 
 
 11-1185 
 
 12-8418 
 
 7 
 
 ■2453 
 
 1-1411 
 
 2-9344 
 
 5-4870 
 
 8-2990 
 
 10-7251 
 
 8 
 
 •1027 
 
 •5502 
 
 1-6103 
 
 3-3951 
 
 5-7456 
 
 8-2555 
 
 9 
 
 •0416 
 
 •2535 
 
 •8365 
 
 1-9732 
 
 3-7128 
 
 5-9003 
 
 10 
 
 •0162 
 
 •1115 
 
 •4118 
 
 1-0801 
 
 2-2477 
 
 3-9335 
 
 11 
 
 •0061 
 
 •0468 
 
 •1921 
 
 ■5573 
 
 1-2771 
 
 2-4521 
 
 n 
 
 •0022 
 
 •0187 
 
 •0848 
 
 •2709 
 
 •6811 
 
 1-4304 
 
 is 
 
 •0007 
 
 •0070 
 
 •0353 
 
 •1238 
 
 •340C 
 
 •7802 
 
 H 
 
 •0002 
 
 •0025 
 
 •0138 
 
 •0531 
 
 •1592 
 
 •3971 
 
 15 
 
 — 
 
 •0008 
 
 •0051 
 
 •0212 
 
 •0693 
 
 •1879 
 
 16 
 
 — 
 
 •0003 
 
 •0017 
 
 •0079 
 
 •0280 
 
 •0822 
 
 17 
 
 — 
 
 •0001 
 
 •0005 
 
 •0027 
 
 •0104 
 
 •0330 
 
 18 
 
 — 
 
 — 
 
 •0002 
 
 ■0008 
 
 •0035 
 
 •0120 
 
 19 
 
 — 
 
 — 
 
 •0000 
 
 •0002 
 
 •0010 
 
 •0039 
 
 20 
 
 — 
 
 — 
 
 — 
 
 •0001 
 
 •0003 
 
 •0011 
 
 SI 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •0001 
 
 •0003 
 
 ss 
 ss 
 *4 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •0001 
 
 25 
 
 «=50l 
 m= 5| 1 
 
 91-0714 
 
 82-7922 
 
 75-1263 
 
 68-0389 
 
 61-4967 
 
 55-4676 
 
 8-2792 
 
 15-3319 
 
 21-2621 
 
 26-1688 
 
 30-1454 
 
 33-2805 
 
 2 
 
 •6133 
 
 1-7357 
 
 3-2711 
 
 5-1311 
 
 7-2349 
 
 9-5087 
 
 S 
 
 •0347 
 
 •1335 
 
 •3207 
 
 •6157 
 
 1-0335 
 
 1-5848 
 
 4 
 
 •0013 
 
 •0065 
 
 •0192 
 
 •0440 
 
 •0861 
 
 •1517 
 
 5 
 
 ■0000 
 
 •0001 
 
 •0005 
 
 •0014 
 
 •0033 
 
 •0066 
 
 »=50l 
 m = lO{ 1 
 
 83-6065 
 
 69-6721 
 
 57-8633 
 
 47-8869 
 
 39-4857 
 
 32-4346 
 
 13-9344 
 
 23-6177 
 
 29-9293 
 
 33-6048 
 
 35-2551 
 
 35 3833 
 
 2 
 
 2-1256 
 
 5-4972 
 
 9-4513 
 
 13-5019 
 
 17-3070 
 
 20-6402 
 
 3 
 
 •2932 
 
 1-0287 
 
 2-2503 
 
 3-9278 
 
 5-9827 
 
 8-3080 
 
 4 
 
 •0360 
 
 •1607 
 
 •4296 
 
 •8910 
 
 1-5803 
 
 2-5164 
 
 5 
 
 •0039 
 
 •0210 
 
 •0668 
 
 •1614 
 
 •3282 
 
 •5921 
 
 6 
 
 •0003 
 
 •0023 
 
 •0084 
 
 •0233 
 
 •0536 
 
 •1085 
 
 7 
 
 •0000 
 
 •0002 
 
 •0008 
 
 •0026 
 
 •0067 
 
 •0152 
 
 8 
 
 •0000 
 
 •0000 
 
 •0001 
 
 •0002 
 
 •0006 
 
 •0015 
 
 9 
 
 •0000 
 
 ■0000 
 
 •0000 
 
 •0000 
 
 •0000 
 
 •0001 
 
 10 
 
 •0000 
 
 •0000 
 
 •0000 
 
 •0000 
 
 •0000 
 
 ■0000 
 
 »=50| 
 »i=15| 1 
 
 77-2727 
 
 59-4406 
 
 45-5092 
 
 34-6737 
 
 26-2849 
 
 19-8214 
 
 17-8322 
 
 27-8628 
 
 32-5066 
 
 33-5552 
 
 32-3175 
 
 29-7321 
 
 2 
 
 3-9008 
 
 9-2876 
 
 14-6804 
 
 19-2530 
 
 22-6222 
 
 24-6927 
 
 3 
 
 •8049 
 
 2-5965 
 
 5-2143 
 
 8-3429 
 
 11-6306 
 
 14-7589 
 
 4 
 
 •1558 
 
 •6385 
 
 1-5643 
 
 2-9695 
 
 4-8127 
 
 6-9910 
 
 6 
 
 •0281 
 
 •1405 
 
 •4083 
 
 •9011 
 
 1-6718 
 
 2-7465 
 
 6 
 
 •0047 
 
 •0278 
 
 •0939 
 
 •2371 
 
 •4976 
 
 •9155 
 
 7 
 
 •0007 
 
 •0049 
 
 •0191 
 
 •0544 
 
 •1279 
 
 •2616 
 
 8 
 
 •0001 
 
 •0008 
 
 •0034 
 
 •0109 
 
 •0284 
 
 •0642 
 
 9 
 
 
 
 •0001 
 
 •0005 
 
 •0019 
 
 •0054 
 
 •0132 
 
 10 
 
 
 
 
 
 •0001 
 
 •0003 
 
 •0009 
 
 •0024 
 
 11 
 
 
 
 — 
 
 — 
 
 — 
 
 •0001 
 
 •0003 
 
 12 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •0000 
 
 13, 14, 15 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
Probable Occurrences in Second Small Samples 95 
 
 TABLE XLVIII— (continued). 
 
 Successes 
 
 p = 
 
 p=l 
 
 p = 2 
 
 p=3 
 
 p = i 
 
 p=5 
 
 n = 50l 
 
 «j = 20| 1 
 
 71-8310 
 
 51-3078 
 
 36-4360 
 
 25-7195 
 
 18-0421 
 
 12-5748 
 
 20-5231 
 
 29-7437 
 
 32-1494 
 
 30-7099 
 
 27-3365 
 
 23-2150 
 
 S 
 
 5-6513 
 
 12-4661 
 
 18-2340 
 
 22-1018 
 
 23-9720 
 
 24-1218 
 
 S 
 
 1-4959 
 
 4-4G55 
 
 8-2882 
 
 12-2410 
 
 15-7316 
 
 18-3785 
 
 4 
 
 •3796 
 
 1-4377 
 
 3-2515 
 
 5-6902 
 
 8-4901 
 
 11-3384 
 
 6 
 
 •0920 
 
 •4247 
 
 1-1380 
 
 2-3122 
 
 3-9438 
 
 5-9480 
 
 6 
 
 •0212 
 
 •1161 
 
 •3613 
 
 ■8391 
 
 1-6163 
 
 2-7262 
 
 7 
 
 •0046 
 
 •0295 
 
 •1049 
 
 •2751 
 
 •5926 
 
 1-1089 
 
 8 
 
 ■0010 
 
 •0070 
 
 •0279 
 
 •0820 
 
 •1959 
 
 •4039 
 
 9 
 
 •0002 
 
 •0015 
 
 •0068 
 
 •0222 
 
 •0585 
 
 •1323 
 
 10 
 
 •0000 
 
 •0003 
 
 •0015 
 
 •0055 
 
 •0158 
 
 •0390 
 
 11 
 
 — 
 
 •0001 
 
 •0003 
 
 •0012 
 
 •0039 
 
 •0103 
 
 IS 
 
 — 
 
 — 
 
 •0001 
 
 •0002 
 
 •0008 
 
 •0024 
 
 IS 
 
 — 
 
 — 
 
 — 
 
 •oooo 
 
 •0002 
 
 •0005 
 
 14 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •oooo 
 
 •0001 
 
 15— SO 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •oooo 
 
 w=50| 
 fn=25( 1 
 
 67-1053 
 
 44-7368 
 
 29-6230 
 
 19-4782 
 
 12-7149 
 
 8-2378 
 
 22-3684 
 
 30-2276 
 
 30 4346 
 
 27-0530 
 
 22-3854 
 
 17-6525 
 
 2 
 
 7-2546 
 
 14-9068 
 
 20-2898 
 
 22-8617 
 
 23 0250 
 
 21-4900 
 
 3 
 
 2-2857 
 
 6-3492 
 
 10-9546 
 
 15-0234 
 
 17-9083 
 
 19-3831 
 
 4 
 
 •6984 
 
 2-4592 
 
 5-1643 
 
 8-3826 
 
 11-5877 
 
 14-3204 
 
 6 
 
 •2066 
 
 •8853 
 
 2-2004 
 
 4-1420 
 
 6-5376 
 
 9-1130 
 
 6 
 
 •0590 
 
 •2994 
 
 •8629 
 
 1-8546 
 
 3-3018 
 
 5-1406 
 
 7 
 
 •0163 
 
 •0956 
 
 ■3146 
 
 •7627 
 
 1-5167 
 
 2-6162 
 
 8 
 
 •0043 
 
 •0289 
 
 •1073 
 
 •2904 
 
 •6398 
 
 1-2147 
 
 9 
 
 •0011 
 
 •0083 
 
 ■0343 
 
 •1029 
 
 •2494 
 
 •5181 
 
 10 
 
 •0003 
 
 •0022 
 
 •0103 
 
 •0340 
 
 •0901 
 
 •2038 
 
 11 
 
 •0001 
 
 •0006 
 
 •0029 
 
 - -0105 
 
 •0302 
 
 •0741 
 
 IS 
 
 •0000 
 
 ■0001 
 
 •0008 
 
 ■0030 
 
 •0094 
 
 •0249 
 
 IS 
 
 •oooo 
 
 •0000 
 
 •0002 
 
 ■0008 
 
 •0027 
 
 •0077 
 
 u 
 
 — 
 
 •oooo 
 
 •oooo 
 
 •0002 
 
 •0007 
 
 •0022 
 
 15 
 
 — 
 
 — 
 
 ■oooo 
 
 •oooo 
 
 •0002 
 
 •0006 
 
 16 
 
 — 
 
 — 
 
 — 
 
 •oooo 
 
 •oooo 
 
 •0001 
 
 17 
 
 — 
 
 — 
 
 
 
 — 
 
 •oooo 
 
 •OOOO 
 
 18—25 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •OOOO 
 
 n = 50| 
 
 50-4950 
 
 25-2475 
 
 12-4963 
 
 6-1206 
 
 2-9657 
 
 1-4210 
 
 m = 50f 1 
 
 25-2475 
 
 25-5026 
 
 19-1269 
 
 12-6198 
 
 7-7231 
 
 4-4875 
 
 S 
 
 12-4963 
 
 19-1269 
 
 19-3241 
 
 16-1034 
 
 11-9504 
 
 8-1873 
 
 s 
 
 6-1206 
 
 12-6198 
 
 16-1034 
 
 16-2729 
 
 14-2388 
 
 11-2686 
 
 4 
 
 2-9657 
 
 7-7231 
 
 11-9504 
 
 14-2388 
 
 14-3919 
 
 12-9527 
 
 5 
 
 1-4210 
 
 4-4875 
 
 8-1873 
 
 11-2686 
 
 12-9527 
 
 130951 
 
 6 
 
 •6731 
 
 2-5063 
 
 5-2821 
 
 8-2677 
 
 106753 
 
 12-0038 
 
 7 
 
 •3151 
 
 1-3552 
 
 3-2480 
 
 5-7108 
 
 8-2014 
 
 10-1734 
 
 8 
 
 •1457 
 
 •7126 
 
 1-9185 
 
 3-7517 
 
 5-9437 
 
 8-0780 
 
 9 
 
 •0665 
 
 •3654 
 
 1-0942 
 
 2-3606 
 
 4-0975 
 
 6-0662 
 
 10 
 
 •0300 
 
 •1831 
 
 •6049 
 
 1-4298 
 
 2-7034 
 
 4 3381 
 
 11 
 
 0133 
 
 •0898 
 
 •3250 
 
 •8367 
 
 1-7147 
 
 2 9694 
 
 IS 
 
 •0058 
 
 •0431 
 
 •1699 
 
 ■4743 
 
 1-0490 
 
 1-9531 
 
 IS 
 
 •0025 
 
 •0203 
 
 •0866 
 
 •2610 
 
 •6205 
 
 1-2381 
 
 14 
 
 •0011 
 
 •0093 
 
 •0431 
 
 •1396 
 
 •3557 
 
 •7582 
 
 15 
 
 •0004 
 
 •0042 
 
 •0209 
 
 •0726 
 
 •1978 
 
 •4493 
 
 16 
 
 •0002 
 
 •0019 
 
 •0099 
 
 •0368 
 
 ■1068 
 
 •2580 
 
 17 
 
 •0001 
 
 •0008 
 
 •0046 
 
 •0182 
 
 •0561 
 
 •1437 
 
 18 
 
 — 
 
 •0003 
 
 •0021 
 
 •0088 
 
 •0286 
 
 •0777 
 
 19 
 
 — 
 
 •0001 
 
 •0009 
 
 •0041 
 
 •0142 
 
 •0408 
 
 SO 
 
 — 
 
 — 
 
 •0004 
 
 •0019 
 
 ■0069 
 
 ■0208 
 
 SI 
 
 — 
 
 — 
 
 •0002 
 
 •0008 
 
 •0032 
 
 •0103 
 
 S3 
 
 — 
 
 — 
 
 •0001 
 
 •0004 
 
 ■0015 
 
 •0050 
 
 S3 
 
 — 
 
 — 
 
 — 
 
 •0002 
 
 •0007 
 
 •0023 
 
 S4 
 
 — 
 
 — 
 
 — 
 
 •0001 
 
 •0003 
 
 •0010 
 
 S5 
 
 — 
 
 — 
 
 — 
 
 •oooo 
 
 ■0001 
 
 •0005 
 
 S6 
 
 — 
 
 — 
 
 — 
 
 — 
 
 ■oooo 
 
 •0002 
 
 S7 
 
 — 
 
 — 
 
 
 
 
 
 •oooo 
 
 •0001 
 
 28—50 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •oooo 
 
96 
 
 Tables for Statisticians and Biometricians 
 
 Successes 
 
 ?t=100l 
 m= 10 1 1 
 
 5 
 6 
 7 
 8 
 9 
 10 
 
 TABLE XLVIII — {continued). Percentage Frequency of Successes in a Second 
 Sample "m" after drawing " p" Successes in a First Sample " n". 
 
 p=0 
 
 90-9910 
 8-2719 
 •6830 
 •0506 
 •0033 
 •0002 
 •0000 
 
 p = l 
 
 82-7191 
 15-1778 
 1 -8972 
 •1891 
 •0156 
 •0011 
 •0001 
 
 •oooo 
 
 j, = 2 
 
 75-1302 
 20-8695 
 3-5107 
 •4416 
 •0442 
 •0036 
 •0002 
 ■0000 
 
 2>=3 
 
 68-1737 
 25-4855 
 5-4097 
 ■8243 
 •0971 
 •0090 
 •0007 
 •0000 
 
 p=i 
 
 61-8023 
 
 29-1520 
 
 7-4962 
 
 1-3455 
 
 ■1829 
 
 •0194 
 
 •0016 
 
 •0001 
 
 •0000 
 
 p = 5 
 
 55-9719 
 
 31-9839 
 
 9-6874 
 
 2-0065 
 
 ■3098 
 
 •0368 
 
 •0034 
 
 •0002 
 
 •0000 
 
 P 
 50 
 34 
 11 
 
 
 
 ■6412 
 •0855 
 •9134 
 •8031 
 ■4857 
 •0641 
 •0065 
 0005 
 •0300 
 
 P 
 45 
 35 
 14 
 
 3 
 
 = 7 
 
 •7719 
 •5510 
 •1158 
 ■7269 
 •7174 
 •1044 
 •0115 
 •0010 
 •0001 
 •0000 
 
 41-3280 
 
 36 4659 
 
 162472 
 
 4-7658 
 
 1-0109 
 
 •1609 
 
 •0194 
 
 •0017 
 
 •0001 
 
 •0000 
 
 p-^ 
 
 37-2763 
 
 36-9072 
 
 18-2690 
 
 5-9052 
 
 1-3708 
 
 •2374 
 
 •0309 
 
 •0030 
 
 •0002 
 
 •0000 
 
 2) = 10 
 
 
 1 
 
 2 
 8 
 
 J> 
 5 
 6 
 7 
 8 
 9 
 10 
 
 •5855 
 •9441 
 •1513 
 •1284 
 •8005 
 •3376 
 •0474 
 ■0049 
 •0003 
 ■0000 
 
 p = 15 
 
 19-6056 
 
 33-0200 
 
 26-8727 
 
 13-8698 
 
 5-0127 
 
 1-3220 
 
 •2571 
 
 •0363 
 
 •0036 
 
 •0002 
 
 •0000 
 
 p = 20 
 
 11-0992 
 
 25-8982 
 
 28-8081 
 
 20-0784 
 
 9-6930 
 
 3-3813 
 
 •8619 
 
 •1583 
 
 •0200 
 
 ■0015 
 
 •0001 
 
 p = 25 
 
 6'0712 
 
 18-5708 
 
 26-8613 
 
 24-1644 
 
 14-9554 
 
 6-6468 
 
 2-1464 
 
 •4968 
 
 •0788 
 
 •0077 
 
 •0004 
 
 p-30 p = S5 
 
 3-1945 
 
 12-3788 
 
 22-5639 
 
 25-4567 
 
 19-6711 
 
 10-8709 
 
 4-3483 
 
 1-2424 
 
 •2425 
 
 •0292 
 
 •0017 
 
 1 
 7 
 17 
 24 
 28 
 1", 
 7 
 
 •6083 
 •7198 
 •3696 
 •1112 
 •8554 
 ■4516 
 ■5418 
 ■6232 
 •6221 
 •0908 
 •0062 
 
 p = 40 
 
 •7697 
 
 4-5082 
 
 12-3485 
 
 20-8229 
 
 23-9308 
 
 19-5797 
 
 11-5470 
 
 4-8456 
 
 1-3845 
 
 •2432 
 
 •0199 
 
 p = 45 
 
 •3473 
 
 2-4580 
 
 8-1229 
 
 16-5036 
 
 22-8255 
 
 22-4513 
 
 15-9030 
 
 8-0093 
 
 2-7446 
 
 •5778 
 
 •0567 
 
 p = o0 
 
 •1463 
 
 1-2434 
 
 4-9313 
 
 12-0167 
 
 19-9224 
 
 23-4799 
 
 19-9224 
 
 12-0167 
 
 4-9313 
 
 1-2434 
 
 •1463 
 
 Successes 
 
 m=100I 
 »i= 5 1 1 
 2 
 S 
 4 
 5 
 
 n = 100| 
 m= 15 ( 
 
 8 
 
 ■4 
 5 
 6 
 7 
 8 
 9 
 10 
 11—15 
 
 p = 
 
 95-2830 
 4-5373 
 •1745 
 •0051 
 •0001 
 •0000 
 
 87-0690 
 11-3568 
 1-3947 
 •1604 
 •0172 
 •0017 
 •0002 
 •0000 
 
 p=l 
 
 90-7457 
 8-7256 
 •5083 
 •0199 
 •0005 
 •0000 
 
 p = 1 
 
 •7121 
 •9243 
 •7027 
 •5730 
 •0774 
 •0093 
 •0010 
 •0001 
 
 •oooo 
 
 3830 
 5800 
 9867 
 0488 
 0015 
 0000 
 
 7500 
 1836 
 5459 
 2777 
 2091 
 0295 
 0037 
 0004 
 0000 
 
 y = 3 
 
 82-1896 
 
 16-1156 
 
 1-5956 
 
 •0957 
 
 •0034 
 
 •0001 
 
 •0221 
 •5476 
 •6321 
 •2767 
 •4386 
 •0715 
 •0100 
 •0012 
 •0001 
 
 •oooo 
 
 p = 4 
 
 TO-1607 
 
 19-3467 
 
 2-3216 
 
 •1642 
 
 •0067 
 
 •0001 
 
 49-3852 
 
 33-3684 
 
 12-7407 
 
 3-5456 
 
 •7879 
 
 •1458 
 
 •0229 
 
 •0031 
 
 •0004 
 
 •0000 
 
 2> = 5 
 
 74-2914 
 
 22-2874 
 
 3-1517 
 
 •2573 
 
 •0119 
 
 •0003 
 
 •7116 
 •9458 
 •7096 
 ■0426 
 •2724 
 •2641 
 •0461 
 •0068 
 ■0009 
 •0001 
 •0000 
 
 p = 6 
 
 70-5768 
 
 24-9514 
 
 4-0737 
 
 •3780 
 
 •0197 
 
 •0004 
 
 36-8873 
 
 35-5336 
 
 18-4248 
 
 6-7156 
 
 1-9006 
 
 •4381 
 
 •0842 
 
 •0137 
 
 ■0019 
 
 ■0002 
 
 •0000 
 
 j) = 7 
 
 67-0123 
 
 27-3520 
 
 5-0756 
 
 •5287 
 
 •0306 
 
 •0008 
 
 31-8110 
 
 35-3456 
 
 20-8110 
 
 8-5076 
 
 2 6738 
 
 •6787 
 
 •1428 
 
 •0252 
 
 •0037 
 
 •0005 
 
 •0001 
 
 •0000 
 
 p = 8 
 
 63-5933 
 
 29-5020 
 
 61463 
 
 •7117 
 
 •0454 
 
 •0013 
 
 27-3928 
 
 34-5611 
 
 22-8233 
 
 10-3611 
 
 3-5865 
 
 •9959 
 
 •2278 
 
 •0435 
 
 •0070 
 
 •0009 
 
 •0001 
 
 •0000 
 
 p = 9 
 
 60-3153 
 
 31-4142 
 
 7-2749 
 
 •9287 
 
 0649 
 
 •0020 
 
 23-5527 
 
 333293 
 
 24-4415 
 
 12-2207 
 
 4-6273 
 
 1-3973 
 
 •3459 
 
 •0711 
 
 •0122 
 
 •0017 
 
 •0002 
 
 •0000 
 
 p = 10 
 
 57-1739 
 
 33-1007 
 
 8-4512 
 
 1-1813 
 
 •0899 
 
 •0030 
 
 20-2198 
 
 31-7739 
 
 25-6636 
 
 14-0361 
 
 5-7796 
 
 1-8884 
 
 •5036 
 
 ■1112 
 
 •0204 
 
 •0031 
 
 •0004 
 
 ■OOOO 
 
 = 1001 
 » 20 j 
 
 8 
 
 1 1 
 
 0) 
 
 8 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 11 
 
 n 
 
 13— SO 
 
 •4711 
 •9119 
 •2212 
 •3388 
 •0492 
 •0068 
 •0009 
 •0001 
 ■OOOO 
 
 5592 
 
 57-8686 
 
 3813 
 
 29-4247 
 
 6472 
 
 9-5567 
 
 1584 
 
 2-4716 
 
 2122 
 
 •5480 
 
 0354 
 
 •1077 
 
 0054 
 
 •0191 
 
 0008 
 
 •0031 
 
 0001 
 
 •0004 
 
 OOOO 
 
 •0001 
 
 — 
 
 •oooo 
 
 46 
 
 32 
 
 13 
 
 4 
 
 1 
 
 •0604 
 
 39- 
 
 •8618 
 
 34- 
 
 •4563 
 
 17- 
 
 •2124 
 
 6- 
 
 •0993 
 
 1- 
 
 •2490 
 
 
 •0500 
 
 
 •0090 
 
 1 
 
 •0015 
 
 • 
 
 •0002 
 
 • 
 
 •oooo 
 
 • 
 • 
 
 •8449 
 •3491 
 •0252 
 ■2724 
 ■8873 
 •4853 
 •1093 
 •0219 
 ■0039 
 •0006 
 •0001 
 •OOOO 
 
 32-9751 
 
 34-4088 
 
 20-0718 
 
 8-5261 
 
 2-9118 
 
 •8394 
 
 •2099 
 
 •0462 
 
 •0090 
 
 •0016 
 
 •0002 
 
 •OOOO 
 
 27-2403 
 
 33-4530 
 
 22-4994 
 
 10-8479 
 
 4-1535 
 
 1-3291 
 
 •3658 
 
 •0881 
 
 ■0187 
 
 •0035 
 
 •0006 
 
 •0001 
 
 •OOOO 
 
 22-4613 
 
 31-8036 
 
 24-2787 
 
 13-1236 
 
 5-5775 
 
 1-9649 
 
 •5913 
 
 •1547 
 
 •0356 
 
 •0072 
 
 •0013 
 
 •0002 
 
 •OOOO 
 
 18-4859 
 
 29-7094 
 
 25-4270 
 
 15-2562 
 
 7-1382 
 
 2-7495 
 
 •8994 
 
 •2545 
 
 •0630 
 
 •0137 
 
 •0026 
 
 •0005 
 
 •0001 
 
 •OOOO 
 
 15-1848 
 
 27-3600 
 
 25-9920 
 
 17-1690 
 
 8-7832 
 
 3-6775 
 
 1-3010 
 
 •3965 
 
 •1053 
 
 •0245 
 
 •0050 
 
 •0009 
 
 •0001 
 
 •OOOO 
 
 12-4488 
 
 24-8976 
 
 26-0397 
 
 18-8065 
 
 10-4578 
 
 4-7356 
 
 1-8040 
 
 •5898 
 
 •1675 
 
 •0416 
 
 •0091 
 
 •0017 
 
 •0003 
 
 ■OOOO 
 
Probable Occurrences in Second Small Samples 97 
 
 TABLE XLVIII— (continued). 
 
 Successes 
 
 »=100( 
 
 m= 25 j 
 
 so 
 le 
 
 3 
 
 4 
 
 5 
 6 
 7 
 £ 
 .9 
 
 10 
 
 11 
 
 12 
 IS 
 
 u 
 
 15—25 
 
 p=0 
 
 ■1587 
 0317 
 •1029 
 •5802 
 •1046 
 •0182 
 •0030 
 •0005 
 •0001 
 •0000 
 
 7=1 
 
 p = 3 
 
 P = i 
 
 p = 5 
 
 j> = 6 
 
 •1270 
 
 51-1981 
 
 40-7920 
 
 32-4330 
 
 25-7320 
 
 20-3711 
 
 •8576 
 
 31-2184 
 
 33-4361 
 
 33-5052 
 
 32-1649 
 
 29-9575 
 
 •5681 
 
 12-2826 
 
 16-5799 
 
 20-1031 
 
 22-7047 
 
 24-3723 
 
 •9024 
 
 3-8912 
 
 6-3556 
 
 9-0661 
 
 11-8013 
 
 14-3734 
 
 •4323 
 
 1-0701 
 
 2-0562 
 
 3-3806 
 
 4-9929 
 
 6-8150 
 
 •0908 
 
 •2644 
 
 •5855 
 
 1-0922 
 
 1-8077 
 
 2-7378 
 
 •0178 
 
 •0597 
 
 •1501 
 
 •3138 
 
 •5764 
 
 •9606 
 
 •0033 
 
 •0125 
 
 •0351 
 
 •0815 
 
 •1647 
 
 •3000 
 
 •0005 
 
 •0024 
 
 •0075 
 
 •0193 
 
 •0426 
 
 •0844 
 
 0001 
 
 •0004 
 
 •0015 
 
 •0042 
 
 •0100 
 
 •0215 
 
 0000 
 
 •0001 
 
 •0003 
 
 •0008 
 
 •0022 
 
 •0050 
 
 , — 
 
 •0000 
 
 •0001 
 
 ■0001 
 
 ■0004 
 
 •0011 
 
 — 
 
 _ 
 
 •0000 
 
 •0000 
 
 •0001 
 
 •0002 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •oooo 
 
 •oooo 
 
 p = 7 
 
 if; 
 -27 
 SB 
 
 Ifi 
 8 
 
 a 
 
 l 
 
 p=i 
 
 p = 9 
 
 •0915 
 
 12-6823 
 
 9-9724 
 
 7- 
 
 •2737 
 
 24-3890 
 
 21-4922 
 
 18- 
 
 •1757 
 
 25-2300 
 
 24-6693 
 
 23- 
 
 •6391 
 
 18-5020 
 
 19-9086 
 
 20- 
 
 •7536 
 
 10-7117 
 
 12-5970 
 
 14- 
 
 •8700 
 
 5-1757 
 
 6-6134 
 
 8- 
 
 •4841 
 
 2-1565 
 
 2-9790 
 
 3- 
 
 •5035 
 
 •7910 
 
 1-1761 
 
 I- 
 
 •1531 
 
 •2589 
 
 •4127 
 
 • 
 
 •0421 
 
 •0763 
 
 •1299 
 
 • 
 
 •0105 
 
 •0203 
 
 •0369 
 
 •( 
 
 •0024 
 
 •0049 
 
 •0095 
 
 •( 
 
 •0005 
 
 •0011 
 
 •0022 
 
 1 
 
 •0001 
 
 •0002 
 
 ■0005 
 
 •( 
 
 •oooo 
 
 •oooo 
 
 ■0001 
 
 • 
 
 — 
 
 — 
 
 •oooo 
 
 •1 
 
 p = 10 
 
 •8232 
 ■7076 
 •6306 
 •8423 
 •3291 
 •1327 
 •9431 
 •6693 
 •6260 
 •2099 
 •0634 
 •0173 
 •0043 
 •0009 
 •0002 
 •OOOO 
 
 « = 100l 
 m= 50) 
 
 
 1 
 
 2 
 S 
 
 k 
 6 
 
 6 
 
 7 
 
 S 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 H 
 15 
 16 
 17 
 18 
 19 
 
 Slf—50 
 
 66-8874 
 
 44 ■ 
 
 22-2958 
 
 29- 
 
 7 3322 
 
 14- 
 
 2-3780 
 
 6- 
 
 •7603 
 
 2- 
 
 •2396 
 
 • 
 
 •0743 
 
 
 •0227 
 
 
 •0068 
 
 •( 
 
 •0020 
 
 •( 
 
 •0006 
 
 •1 
 
 •0002 
 
 •( 
 
 — 
 
 ■i 
 •i 
 
 ■5916 
 •9273 
 •8625 
 •4708 
 •6038 
 •9912 
 •3614 
 •1271 
 •0433 
 •0143 
 •0046 
 •0014 
 •0004 
 •0001 
 
 29-6280 
 
 30-0284 
 
 20-0189 
 
 10-9693 
 
 5-3333 
 
 2-3852 
 
 1-0008 
 
 •3987 
 
 •1520 
 
 •0557 
 
 •0197 
 
 •0068 
 
 •0022 
 
 •0007 
 
 •0002 
 
 •0001 
 
 19-6185 
 
 12 
 
 9456 
 
 8 
 
 5121 
 
 5'5769 
 
 3- 
 
 26-6919 
 
 22 
 
 1671 
 
 17 
 
 6113 
 
 13-5550 
 
 10- 
 
 22 3956 
 
 22 
 
 4728 
 
 20 
 
 9746 
 
 18-5789 
 
 15' 
 
 14-8274 
 
 17 
 
 4789 
 
 18 
 
 7745 
 
 18-8406 
 
 17- 
 
 8-4691 
 
 11 
 
 4896 
 
 13 
 
 9817 
 
 15-7005 
 
 16- 
 
 4-3589 
 
 6 
 
 6996 
 
 9 
 
 1228 
 
 11-3492 
 
 13- 
 
 2-0720 
 
 3 
 
 5636 
 
 5 
 
 3759 
 
 7-3484 
 
 9- 
 
 •9237 
 
 1 
 
 7600 
 
 2 
 
 9173 
 
 4-3512 
 
 5- 
 
 •3901 
 
 
 8167 
 
 1 
 
 4771 
 
 2-3900 
 
 3- 
 
 •1572 
 
 
 3590 
 
 
 7044 
 
 1-2301 
 
 1- 
 
 •0607 
 
 
 1504 
 
 
 3185 
 
 •5978 
 
 1- 
 
 •0226 
 
 
 0603 
 
 
 1373 
 
 •2758 
 
 > 
 
 ■0081 
 
 
 0232 
 
 
 0566 
 
 •1213 
 
 • 
 
 •0028 
 
 
 0086 
 
 
 0224 
 
 •0510 
 
 
 •0009 
 
 
 0031 
 
 
 0085 
 
 •0206 
 
 •( 
 
 •0003 
 
 
 0011 
 
 
 0031 
 
 •0080 
 
 •( 
 
 •0001 
 
 
 0004 
 
 
 0011 
 
 •0030 
 
 •( 
 
 — 
 
 
 0001 
 
 •0004 
 
 •0011 
 
 •1 
 
 
 
 — 
 
 •0001 
 
 •0004 
 
 •< 
 
 — 
 
 
 — 
 
 
 — 
 
 •0001 
 
 
 6405 
 
 2-3676 
 
 1832 
 
 7-5029 
 
 8126 
 
 13-0370 
 
 9434 
 
 16-3894 
 
 5656 
 
 16-6252 
 
 ■1572 
 
 14-4085 
 
 2958 
 
 11-0430 
 
 9710 
 
 7-6559 
 
 5398 
 
 4-8771 
 
 9578 
 
 2-8874 
 
 0184 
 
 1-6022 
 
 5012 
 
 •8386 
 
 2345 
 
 •4161 
 
 1046 
 
 •1965 
 
 0447 
 
 •0886 
 
 0183 
 
 •0382 
 
 0072 
 
 •0158 
 
 0027 
 
 •0063 
 
 0010 
 
 •0024 
 
 0003 
 
 ■0009 
 
 0001 
 
 •0003 
 
 — 
 
 •0001 
 
 1 
 
 5 
 
 10 
 
 14 
 
 18 
 
 15 
 12 
 
 
 
 6 
 3 
 2 
 1 
 
 •5339 
 •4395 
 •4710 
 ■4635 
 •0095 
 •0512 
 •4505 
 •2753 
 •3248 
 ■9946 
 ■3574 
 •3088 
 •6871 
 •3425 
 •1627 
 •0738 
 •0320 
 •0133 
 ■0053 
 •0020 
 •0008 
 •0003 
 •0001 
 
 •9900 
 •8892 
 •2261 
 •3988 
 •8876 
 •1066 
 •4281 
 •7081 
 •7895 
 •2324 
 •2752 
 ■9239 
 •0664 
 •5601 
 •2797 
 •1332 
 •0606 
 •0264 
 •0110 
 •0044 
 •0017 
 •0006 
 •0002 
 •0001 
 
 13 
 
98 
 
 1—250 
 
 Tables for Statisticians and Biometricians 
 
 TABLE XLIX. Logarithms of Factorials. 
 
 log \n from n— 1 to n = 1000. 
 
 n 
 
 log[n 
 
 1 
 
 •000 0000 
 
 S 
 
 •301 0300 
 
 S 
 
 •778 1513 
 
 4 
 
 1-380 2112 
 
 5 
 
 2079 1812 
 
 6 
 
 2-857 3325 
 
 7 
 
 3-702 4305 
 
 8 
 
 4-605 5205 
 
 9 
 
 5-559 7630 
 
 10 
 
 6-559 7630 
 
 11 
 
 7-601 1557 
 
 IS 
 
 8-680 3370 
 
 IS 
 
 9-794 2803 
 
 H 
 
 10-940 4084 
 
 15 
 
 12-116 4996 
 
 16 
 
 13-320 6196 
 
 17 
 
 14-551 0685 
 
 18 
 
 15-806 3410 
 
 19 
 
 17-085 0946 
 
 SO 
 
 18-386 1246 
 
 SI 
 
 19-708 3439 
 
 22 
 
 21-050 7666 
 
 S3 
 
 22-412 4944 
 
 21> 
 
 23-792 7057 
 
 25 
 
 25190 6457 
 
 26 
 
 26-605 6190 
 
 27 
 
 28-036 9828 
 
 28 
 
 29-484 1408 
 
 29 
 
 30 946 5388 
 
 SO 
 
 32-423 6601 
 
 SI 
 
 33915 0218 
 
 S2 
 
 35-420 1717 
 
 S3 
 
 36-938 6857 
 
 94 
 
 38-470 1646 
 
 S5 
 
 40014 2326 
 
 36 
 
 41-570 5351 
 
 37 
 
 43-138 7369 
 
 38 
 
 44-718 5205 
 
 39 
 
 46-309 5851 
 
 40 
 
 47-911 6451 
 
 41 
 
 49-524 4289 
 
 42 
 
 51-147 6782 
 
 43 
 
 52-781 1467 
 
 44 
 
 54-424 5993 
 
 45 
 
 56077 8119 
 
 46 
 
 57-740 5697 
 
 47 
 
 59-412 6676 
 
 48 
 
 61-093 9088 
 
 49 
 
 62-784 1049 
 
 50 
 
 64183 0749 
 
 n 
 
 log|n 
 
 51 
 
 66-190 6450 
 
 5S 
 
 67-906 6484 
 
 53 
 
 69-630 9243 
 
 54 
 
 71-363 3180 
 
 55 
 
 73-103 6807 
 
 56 
 
 74-851 8687 
 
 57 
 
 76-607 7436 
 
 58 
 
 78-371 1716 
 
 59 
 
 80-142 0236 
 
 60 
 
 81-920 1748 
 
 61 
 
 83-705 5047 
 
 6S 
 
 85-497 8964 
 
 63 
 
 87-297 2369 
 
 64 
 
 89-103 4169 
 
 65 
 
 90-916 3303 
 
 66 
 
 92-735 8742 
 
 67 
 
 94-561 9490 
 
 68 
 
 96-394 4579 
 
 69 
 
 98-233 3070 
 
 70 
 
 100-078 4050 
 
 71 
 
 101-929 6634 
 
 7S 
 
 103-786 9959 
 
 73 
 
 105-650 3187 
 
 74 
 
 107-519 5505 
 
 75 
 
 109-394 6117 
 
 76 
 
 111-275 4253 
 
 77 
 
 113-161 9160 
 
 78 
 
 115-054 0106 
 
 79 
 
 116-951 6377 
 
 80 
 
 118-854 7277 
 
 81 
 
 120-763 2127 
 
 82 
 
 122-677 0266 
 
 83 
 
 124-596 1047 
 
 84 
 
 126-520 3840 
 
 85 
 
 128-449 8029 
 
 86 
 
 130-384 3013 
 
 87 
 
 132-323 8206 
 
 88 
 
 134-268 3033 
 
 89 
 
 136-217 6933 
 
 90 
 
 138171 9358 
 
 91 
 
 140-130 9772 
 
 92 
 
 142-094 7650 
 
 9S 
 
 144-063 2480 
 
 94 
 
 146-036 3758 
 
 95 
 
 148-014 0994 
 
 96 
 
 149-996 3707 
 
 97 
 
 151-983 1424 
 
 98 
 
 153-974 3685 
 
 99 
 
 155-970 0037 
 
 100 
 
 157-970 0037 
 
 n 
 
 log 1 n 
 
 101 
 
 159-974 3250 
 
 102 
 
 161-982 9252 
 
 103 
 
 163-995 7624 
 
 104 
 
 166-012 7958 
 
 105 
 
 168-033 9851 
 
 106 
 
 170-059 2909 
 
 107 
 
 172-088 6747 
 
 108 
 
 174-122 0985 
 
 109 
 
 176-159 6250 
 
 no 
 
 178-200 9176 
 
 111 
 
 180-246 2406 
 
 112 
 
 182-295 4586 
 
 US 
 
 184348 5371 
 
 114 
 
 186-405 4419 
 
 115 
 
 188-466 1398 
 
 116 
 
 190-530 5978 
 
 117 
 
 192-598 7836 
 
 118 
 
 194-670 6656 
 
 119 
 
 196-746 2126 
 
 120 
 
 198-825 3938 
 
 121 
 
 200-908 1792 
 
 122 
 
 202-994 5390 
 
 1SS 
 
 205-084 4442 
 
 1S4 
 
 207-177 8658 
 
 1S5 
 
 209-274 7759 
 
 126 
 
 211-375 1464 
 
 127 
 
 213-478 9501 
 
 128 
 
 215-586 1601 
 
 129 
 
 217-696 7498 
 
 ISO 
 
 219-810 6932 
 
 1S1 
 
 221-927 9645 
 
 1S2 
 
 224-048 5384 
 
 133 
 
 226-172 3900 
 
 1S4 
 
 228-299 4948 
 
 1S5 
 
 230-429 8286 
 
 1S6 
 
 232-563 3675 
 
 137 
 
 234-700 0881 
 
 138 
 
 236-839 9672 
 
 139 
 
 238-982 9820 
 
 140 
 
 241-129 1100 
 
 141 
 
 243-278 3291 
 
 142 
 
 245-430 6174 
 
 143 
 
 247-585 9535 
 
 144 
 
 249-744 3160 
 
 145 
 
 251-905 6840 
 
 146 
 
 254-070 0368 
 
 147 
 
 256237 3542 
 
 148 
 
 258-407 6159 
 
 149 
 
 260580 8022 
 
 150 
 
 262-756 8934 
 
 n 
 
 log|n 
 
 151 
 
 264-935 8704 
 
 152 
 
 267-117 7139 
 
 153 
 
 269-302 4054 
 
 154 
 
 271-489 9261 
 
 155 
 
 273-680 2578 
 
 156 
 
 275-873 3824 
 
 157 
 
 278-069 2820 
 
 158 
 
 280-267 9391 
 
 159 
 
 282-469 3363 
 
 160 
 
 284-673 4562 
 
 161 
 
 286-880 2821 
 
 162 
 
 289-089 7971 
 
 16S 
 
 291-301 9847 
 
 164 
 
 293-516 8286 
 
 165 
 
 295-734 3125 
 
 166 
 
 297-954 4206 
 
 167 
 
 300-177 1371 
 
 168 
 
 302-402 4464 
 
 169 
 
 304-630 3331 
 
 170 
 
 306-860 7820 
 
 171 
 
 309-093 7781 
 
 17S 
 
 311-329 3066 
 
 17S 
 
 313-567 3527 
 
 174 
 
 315-807 9019 
 
 175 
 
 318-050 9400 
 
 176 
 
 320-296 4526 
 
 177 
 
 322-544 4259 
 
 178 
 
 324-794 8459 
 
 179 
 
 327-047 6989 
 
 180 
 
 329-302 9714 
 
 181 
 
 331-560 6500 
 
 18S 
 
 333820 7214 
 
 183 
 
 336-083 1725 
 
 184 
 
 338-347 9903 
 
 185 
 
 340-615 1620 
 
 186 
 
 342-884 6750 
 
 187 
 
 345-156 5166 
 
 188 
 
 347-430 6744 
 
 189 
 
 349-707 1362 
 
 190 
 
 351-985 8898 
 
 191 
 
 354-266 9232 
 
 192 
 
 356-550 2244 
 
 198 
 
 358-835 7817 
 
 194 
 
 361123 5835 
 
 195 
 
 363-413 6181 
 
 196 
 
 365-705 8742 
 
 197 
 
 368000 3404 
 
 198 
 
 370-297 0056 
 
 199 
 
 372595 8586 
 
 200 
 
 374-896 8886 
 
 S05 
 
 206 
 
 207 
 208 
 209 
 210 
 
 Sll 
 
 214 
 
 216 
 217 
 S18 
 219 
 
 224 
 225 
 
 230 
 
 235 
 
 236 
 237 
 SS8 
 SS9 
 
 24S 
 
 S44 
 S45 
 
 S47 
 
 log|n 
 
 377-200 0847 
 379-505 4361 
 381-812 9321 
 384122 5623 
 386-434 3161 
 
 388-748 1834 
 391-064 1537 
 393382 2170 
 395-702 3633 
 398-024 5826 
 
 400-348 8651 
 402-675 2009 
 405-003 5805 
 407-333 9943 
 409-666 4328 
 
 412-000 8865 
 414-337 3463 
 416675 8027 
 419-016 2469 
 421-358 6695 
 
 423-703 0618 
 426-049 4148 
 428-397 7197 
 430-747 9677 
 433-100 1502 
 
 435-454 2586 
 437-810 2845 
 440-168 2193 
 442-528 0548 
 444-889 7827 
 
 447-253 3946 
 449-618 8826 
 451-986 2385 
 454-355 4544 
 456-726 5223 
 
 459099 4343 
 461-474 1826 
 463-850 7596 
 466-229 1575 
 468-609 3687 
 
 470-991 3857 
 473-375 2011 
 475-760 8074 
 478-148 1972 
 480-537 3633 
 
 482-928 2984 
 485-320 9954 
 487-715 4470 
 490-111 6464 
 492-509 5864 
 
Logarithms of Factorials 
 TABLE XLIX— {continued). 
 
 99 
 
 251—500 
 
 logl" 
 
 251 
 
 252 
 25S 
 254 
 255 
 
 256 
 257 
 258 
 259 
 260 
 
 261 
 
 262 
 26S 
 264 
 265 
 
 266 
 267 
 268 
 269 
 270 
 
 271 
 
 272 
 273 
 274 
 275 
 
 276 
 
 277 
 278 
 279 
 
 281 
 282 
 283 
 284 
 285 
 
 287 
 
 291 
 
 296 
 297 
 298 
 299 
 300 
 
 494-909 2601 
 497-310 (5607 
 499-713 7812 
 502-118 6149 
 504-525 1551 
 
 506933 3950 
 509-343 3282 
 511-754 9479 
 514-168 2476 
 516-583 2210 
 
 518-999 8615 
 521-418 1628 
 523-838 1185 
 526-259 7225 
 528-682 9683 
 
 531-107 8500 
 533-534 3612 
 635-962 4960 
 538-392 2483 
 540-823 6121 
 
 543-256 5814 
 545-691 1503 
 548-127 3129 
 550-565 0635 
 553004 3962 
 
 555-445 3052 
 557-887 7850 
 560331 8298 
 562-777 4340 
 565-224 5920 
 
 567-673 2984 
 570-123 5475 
 572-575 3339 
 575028 6523 
 577-483 4971 
 
 579-939 8631 
 582-397 7450 
 584-857 1375 
 587-318 0354 
 589-780 4334 
 
 592-244 3264 
 594-709 7092 
 597-176 5768 
 599-644 9242 
 602-114 7462 
 
 604-586 0379 
 607-058 7943 
 609-533 0106 
 612-008 6818 
 614-485 8030 
 
 n 
 
 log [« 
 
 301 
 
 616-964 3695 
 
 302 
 
 619-444 3765 
 
 SOS 
 
 621-925 8191 
 
 304 
 
 624-408 6927 
 
 305 
 
 626-892 9925 
 
 306 
 
 629-378 7140 
 
 307 
 
 631-865 8523 
 
 308 
 
 634-354 4031 
 
 309 
 
 636-844 3615 
 
 310 
 
 639335 7232 
 
 311 
 
 641-828 4836 
 
 312 
 
 644-322 6382 
 
 313 
 
 646-818 1825 
 
 314 
 
 649-315 1122 
 
 315 
 
 651813 4227 
 
 316 
 
 654-313 1098 
 
 317 
 
 656-814 1691 
 
 318 
 
 659-316 5962 
 
 319 
 
 661820 3869 
 
 820 
 
 664-325 5369 
 
 321 
 
 666-832 0419 
 
 S22 
 
 669-339 8978 
 
 S2S 
 
 671-849 1003 
 
 324 
 
 674-359 6453 
 
 325 
 
 676-871 5287 
 
 326 
 
 679-384 7463 
 
 327 
 
 681-899 2940 
 
 328 
 
 684-415 1679 
 
 329 
 
 686-932 3638 
 
 330 
 
 689-450 8777 
 
 331 
 
 691-970 7057 
 
 332 
 
 694-491 8438 
 
 338 
 
 697-014 2880 
 
 334 
 
 699-538 0345 
 
 335 
 
 702063 0793 
 
 836 
 
 704-589 4186 
 
 337 
 
 707-117 0485 
 
 S38 
 
 709-645 9652 
 
 839 
 
 712-176 1649 
 
 340 
 
 714-707 6438 
 
 841 
 
 717-240 3982 
 
 842 
 
 719-774 4243 
 
 843 
 
 722-309 7184 
 
 344 
 
 724-846 2768 
 
 845 
 
 727-384 0959 
 
 846 
 
 729-923 1720 
 
 847 
 
 732-463 5015 
 
 S48 
 
 735-005 0807 
 
 349 
 
 737-547 9062 
 
 350 
 
 740091 9742 
 
 n 
 
 log [n 
 
 851 
 
 742-637 2813 
 
 352 
 
 745-183 8240 
 
 853 
 
 747-731 5987 
 
 354 
 
 750-280 6020 
 
 855 
 
 752-830 8303 
 
 856 
 
 755-382 2803 
 
 357 
 
 757-934 9485 
 
 858 
 
 760-488 8316 
 
 359 
 
 763043 9260 
 
 360 
 
 765-600 2285 
 
 861 
 
 768-157 7357 
 
 S62 
 
 770-716 4443 
 
 363 
 
 773-276 3509 
 
 S64 
 
 775-837 4523 
 
 865 
 
 778-399 7452 
 
 866 
 
 780-963 2262 
 
 867 
 
 783-527 8923 
 
 368 
 
 786-093 7401 
 
 369 
 
 788-660 7665 
 
 870 
 
 791-228 9682 
 
 S71 
 
 793-798 3421 
 
 872 
 
 796-368 8851 
 
 878 
 
 798-940 5939 
 
 874 
 
 801-513 4655 
 
 875 
 
 804-087 4968 
 
 376 
 
 806-662 6846 
 
 377 
 
 809-239 0260 
 
 378 
 
 811-816 5178 
 
 879 
 
 814-395 1570 
 
 380 
 
 816-974 9406 
 
 881 
 
 819-555 8655 
 
 882 
 
 822-137 9289 
 
 383 
 
 824-721 1277 
 
 884 
 
 827-305 4589 
 
 385 
 
 829-890 919G 
 
 886 
 387 
 S88 
 889 
 390 
 
 391 
 392 
 S9S 
 394 
 395 
 
 396 
 397 
 898 
 399 
 400 
 
 832-477 5069 
 835065 2179 
 837-654 0496 
 840-243 9992 
 842-835 0638 
 
 845-427 2406 
 848-020 5267 
 850-614 9192 
 853-210 4154 
 855-807 0125 
 
 858-404 7077 
 861-003 4982 
 863-603 3813 
 866-204 3542 
 868-806 4142 
 
 n 
 
 log|n 
 
 401 
 
 871-409 5586 
 
 402 
 
 874-013 7846 
 
 403 
 
 876-619 0896 
 
 404 
 
 879-225 4710 
 
 405 
 
 881-832 9260 
 
 406 
 
 884-441 4521 
 
 407 
 
 887051 0465 
 
 408 
 
 889-661 7066 
 
 409 
 
 892-273 4300 
 
 410 
 
 894-886 2138 
 
 411 
 
 897-500 0556 
 
 412 
 
 900-114 9528 
 
 413 
 
 902-730 9029 
 
 414 
 
 905-347 9032 
 
 415 
 
 907-965 9513 
 
 416 
 
 910-585 0447 
 
 417 
 
 913-205 1807 
 
 418 
 
 915826 3570 
 
 419 
 
 918-448 5710 
 
 420 
 
 921-071 8203 
 
 421 
 
 923-696 1024 
 
 422 
 
 926-321 4149 
 
 42s 
 
 928-947 7552 
 
 m 
 
 931-575 1211 
 
 425 
 
 934-203 5100 
 
 426 
 
 936-832 9196 
 
 427 
 
 939463 3475 
 
 428 
 
 942-094 7913 
 
 429 
 
 944-727 2486 
 
 430 
 
 947-360 7170 
 
 431 
 
 949-995 1943 
 
 432 
 
 952-630 6780 
 
 433 
 
 955-267 1659 
 
 434 
 
 957-904 6557 
 
 435 
 
 960-543 1449 
 
 436 
 
 963-182 6314 
 
 437 
 
 965-823 1128 
 
 438 
 
 968-464 5869 
 
 439 
 
 971-107 0515 
 
 440 
 
 973-750 5041 
 
 441 
 
 976-394 9427 
 
 443 
 
 979040 3650 
 
 443 
 
 981-686 7687 
 
 444 
 
 984-334 1517 
 
 445 
 
 986-982 5117 
 
 446 
 
 989-631 8466 
 
 447 
 
 992-282 1541 
 
 448 
 
 994-933 4321 
 
 449 
 
 997585 6784 
 
 450 
 
 1000-238 8910 
 
 451 
 452 
 453 
 454 
 455 
 
 456 
 457 
 458 
 459 
 460 
 
 461 
 462 
 463 
 464 
 
 465 
 
 466 
 
 467 
 468 
 469 
 470 
 
 471 
 
 472 
 
 473 
 474 
 475 
 
 476 
 
 477 
 478 
 479 
 480 
 
 481 
 
 482 
 483 
 
 m 
 
 486 
 487 
 488 
 489 
 490 
 
 491 
 
 492 
 493 
 494 
 495 
 
 496 
 497 
 498 
 499 
 500 
 
 13—2 
 
 log |n 
 
 1002-893 0675 
 1005-548 2059 
 1008-204 3041 
 1010-861 3600 
 1013-519 3714 
 
 1016-178 3362 
 1018-838 2524 
 1021-499 1179 
 1024-160 9306 
 1026-823 6884 
 
 1029-487 3893 
 1032-152 0313 
 1034-817 6123 
 1037484 1303 
 1040-151 5832 
 
 1042-819 9692 
 1045-489 2860 
 1048159 5319 
 1050-830 7047 
 1053-502 8026 
 
 1056-175 8235 
 1058-849 7655 
 1061-524 6266 
 1064-200 4050 
 1066-877 0986 
 
 1069-554 7056 
 1072-233 2239 
 1074-912 6518 
 1077-592 9873 
 1080-274 2286 
 
 1082-956 3737 
 1085-639 4207 
 1088323 3678 
 1091-008 2132 
 1093-693 9549 
 
 1096-380 5912 
 1099068 1202 
 1101-756 5400 
 1104-445 8488 
 1107-136 0449 
 
 1109-827 1264 
 1112-519 0915 
 1115-211 9384 
 1117-905 6654 
 1120-600 2706 
 
 1123-295 7523 
 1125-992 1086 
 1128-689 3380 
 1131-387 4385 
 1134-086 4085 
 
100 
 
 501—750 
 
 Tables for Statisticians and Biometricians 
 
 Table of log \n from n = \ to n = 1000. 
 
 n 
 
 501 
 
 log|n_ 
 
 1136-786 2463 
 
 502 
 
 1139-486 9500 
 
 503 
 
 1142-188 5180 
 
 504 
 
 1144-890 9485 
 
 505 
 
 1147-594 2399 
 
 506 
 
 1150-298 3904 
 
 507 
 
 1153-003 3984 
 
 508 
 
 1155-709 2621 
 
 509 
 
 1158-415 9798 
 
 510 
 
 1161-123 5500 
 
 511 
 
 1163-831 9709 
 
 512 
 
 1166-541 2409 
 
 513 
 
 1169-251 3583 
 
 514 
 
 1171-962 3214 
 
 515 
 
 1174-674 1286 
 
 516 
 
 1177-386 7783 
 
 517 
 
 1180-100 2C88 
 
 518 
 
 1182-814 5986 
 
 519 
 
 1185-529 7660 
 
 520 
 
 1188-245 7693 
 
 521 
 
 1190-962 6070 
 
 522 
 
 1193-680 2775 
 
 523 
 
 1196-398 7792 
 
 524 
 
 1199-118 1105 
 
 525 
 
 1201-838 2698 
 
 526 
 
 1204-559 2556 
 
 527 
 
 1207-281 0862 
 
 528 
 
 1210-003 7001 
 
 529 
 
 1212-727 1558 
 
 530 
 
 1215-451 4316 
 
 531 
 
 1218-176 5262 
 
 532 
 
 1220-902 4378 
 
 5SS 
 
 1223-629 1650 
 
 534 
 
 1226-356 7063 
 
 535 
 
 1229-085 0600 
 
 536 
 
 1231-814 2248 
 
 537 
 
 1234-544 1991 
 
 538 
 
 1237-274 9814 
 
 539 
 
 1240-006 5702 
 
 540 
 
 1242-738 9639 
 
 541 
 
 1245-472 1612 
 
 542 
 
 1248-206 1605 
 
 543 
 
 1250-940 9603 
 
 544 
 
 1253-676 5592 
 
 545 
 
 1256-412 9557 
 
 546 
 
 1259-150 1483 
 
 547 
 
 1261-888 1357 
 
 548 
 
 1264-626 9162 
 
 549 
 
 1267-366 4886 
 
 550 
 1 
 
 1270-106 8513 
 
 n 
 
 log[re_ 
 
 551 
 
 1272-848 0029 
 
 552 
 
 1275-589 9419 
 
 558 
 
 1278-332 6671 
 
 554 
 
 1281-076 1768 
 
 555 
 
 1283-820 4698 
 
 556 
 
 1286-565 5446 
 
 557 
 
 1289-311 3998 
 
 558 
 
 1292-058 0340 
 
 559 
 
 1294-805 4458 
 
 560 
 
 1297-553 6338 
 
 561 
 
 1300-302 5967 
 
 562 
 
 1303-052 3330 
 
 563 
 
 1305-802 8414 
 
 564 
 
 1308-554 1205 
 
 565 
 
 1311-306 1690 
 
 566 
 
 1314-058 9854 
 
 567 
 
 1316-812 5684 
 
 568 
 
 1319-566 9168 
 
 569 
 
 1322-322 0290 
 
 570 
 
 1325-077 9039 
 
 571 
 
 1327-834 5400 
 
 572 
 
 1330-591 9360 
 
 578 
 
 1333-350 0907 
 
 574 
 
 1336-109 0026 
 
 575 
 
 1338-868 6704 
 
 576 
 
 1341-629 0929 
 
 577 
 
 1344-390 2687 
 
 578 
 
 1347-152 1965 
 
 579 
 
 1349-914 8751 
 
 580 
 
 1352-678 3031 
 
 581 
 
 1355-442 4792 
 
 582 
 
 1358-207 4022 
 
 588 
 
 1360-973 0708 
 
 584 
 
 1363-739 4836 
 
 585 
 
 1366-506 6395 
 
 586 
 
 1369-274 5371 
 
 587 
 
 1372-043 1752 
 
 588 
 
 1374-812 5525 
 
 589 
 
 1377-582 6678 
 
 590 
 
 1380-353 5198 
 
 591 
 
 1383-125 1073 
 
 592 
 
 1385-897 4290 
 
 593 
 
 1388-670 4837 
 
 594 
 
 1391-444 2702 
 
 595 
 
 1394-218 7871 
 
 596 
 
 1396-994 0334 
 
 597 
 
 1399-770 0077 
 
 598 
 
 1402-546 7089 
 
 599 
 
 1405-324 1357 
 
 600 
 
 1408-102 2870 
 
 n 
 601 
 
 log[n 
 
 1410-881 1614 
 
 602 
 
 1413-660 7579 
 
 60S 
 
 1416-441 0752 
 
 604 
 
 1419-222 1122 
 
 605 
 
 1422-003 8676 
 
 606 
 
 1424-786 3402 
 
 607 
 
 1427-569 5289 
 
 608 
 
 1430-353 4324 
 
 609 
 
 1433-138 0497 
 
 610 
 
 1435-923 3796 
 
 611 
 
 1438-709 4208 
 
 en 
 
 1441-496 1722 
 
 618 
 
 1444-283 6327 
 
 614 
 
 1447-071 8011 
 
 615 
 
 1449-860 6762 
 
 616 
 
 1452-650 2569 
 
 617 
 
 1455-440 5420 
 
 618 
 
 1458-231 5305 
 
 619 
 
 1461-023 2212 
 
 620 
 
 1463-815 6129 
 
 621 
 
 1466-608 7045 
 
 622 
 
 1469-402 4948 
 
 623 
 
 1472-19G 9829 
 
 624 
 
 1474-992 1675 
 
 625 
 
 1477-788 0475 
 
 626 
 
 1480-584 6218 
 
 627 
 
 1483-381 8894 
 
 628 
 
 1486-179 8490 
 
 629 
 
 1488-978 4997 
 
 630 
 
 1491-777 8402 
 
 631 
 
 1494-577 8696 
 
 632 
 
 1497-378 5866 
 
 633 
 
 1500-179 9904 
 
 634 
 
 1502-982 0796 
 
 635 
 
 1505-784 8533 
 
 636 
 
 1508-588 3105 
 
 687 
 
 1511-392 4499 
 
 688 
 
 1514-197 2706 
 
 639 
 
 1517-002 7714 
 
 640 
 
 1519-808 9514 
 
 641 
 
 1522-615 8094 
 
 042 
 
 1525-423 3445 
 
 64S 
 
 1528-231 5554 
 
 644 
 
 1531-040 4413 
 
 645 
 
 1533-850 0010 
 
 646 
 
 1536-660 2335 
 
 ft+7 
 
 1539-471 1378 
 
 648 
 
 1542-282 7128 
 
 649 
 
 1545-094 9575 
 
 050 
 
 1547-907 8709 
 
 n 
 
 log [n 
 
 651 
 
 1550-721 4519 
 
 652 
 
 1553-535 6995 
 
 653 
 
 1556-350 6126 
 
 654 
 
 1559-166 1904 
 
 655 
 
 1561-982 4317 
 
 656 
 
 1564-799 3355 
 
 657 
 
 1567-616 9009 
 
 658 
 
 1570-435 1268 
 
 659 
 
 1573-254 0122 
 
 660 
 
 1576-073 5561 
 
 661 
 
 1578-893 7576 
 
 662 
 
 1581-714 6156 
 
 663 
 
 1584-536 1291 
 
 664 
 
 1587-358 2972 
 
 665 
 
 1590-181 1188 
 
 666 
 
 1593-004 5931 
 
 667 
 
 1595-828 7189 
 
 668 
 
 1598-653 4954 
 
 669 
 
 1601-478 9215 
 
 670 
 
 1604-304 9963 
 
 671 
 
 1607-131 7188 
 
 672 
 
 1609-959 0881 
 
 673 
 
 1612-787 1031 
 
 674 
 
 1615-615 7630 
 
 675 
 
 1618-445 0668 
 
 676 
 
 1621-275 0135 
 
 677 
 
 1624-105 6022 
 
 078 
 
 1626-936 8319 
 
 679 
 
 1629-768 7016 
 
 680 
 
 1632-601 2106 
 
 681 
 
 1635-434 3577 
 
 682 
 
 1638-268 1420 
 
 683 
 
 1641-102 5627 
 
 684 
 
 1643-937 6189 
 
 685 
 
 1646-773 3094 
 
 686 
 
 1649-609 6335 
 
 687 
 
 1652-446 5903 
 
 688 
 
 1655-284 1787 
 
 689 
 
 1658-122 3979 
 
 690 
 
 1660-961 2470 
 
 691 
 
 1663-800 7251 
 
 69% 
 
 1666-640 8312 
 
 699 
 
 1669-481 5644 
 
 694 
 
 1672-322 9239 
 
 695 
 
 1675-164 9087 
 
 696 
 
 1678-007 5179 
 
 697 
 
 1680-850 7507 
 
 698 
 
 1683-694 6061 
 
 699 
 
 1686 539 0833 
 
 700 
 
 1689-384 1813 
 
 n 
 
 log \n 
 
 701 
 
 1692-229 8994 
 
 702 
 
 1695-070 2365 
 
 708 
 
 1697-923 1918 
 
 704 
 
 1700-770 7644 
 
 705 
 
 1703-618 9536 
 
 706 
 
 1706-467 7583 
 
 707 
 
 1709-317 1777 
 
 708 
 
 1712-167 2109 
 
 709 
 
 1715-017 8572 
 
 710 
 
 1717-869 1155 
 
 711 
 
 1720-720 9851 
 
 712 
 
 1723-573 4651 
 
 713 
 
 1726-426 5546 
 
 714 
 
 1729-280 2529 
 
 715 
 
 1732-134 5589 
 
 716 
 
 1734-989 4719 
 
 717 
 
 1737-844 9911 
 
 718 
 
 1740-701 1155 
 
 719 
 
 1743-557 8444 
 
 720 
 
 1746-415 1769 
 
 721 
 
 1749-273 1122 
 
 722 
 
 1752-131 6494 
 
 723 
 
 1754-990 7877 
 
 724 
 
 1757-850 5262 
 
 725 
 
 1760-710 8642 
 
 726 
 
 1763-571 8009 
 
 7*7 
 
 1766-433 3353 
 
 728 
 
 1769-295 4667 
 
 729 
 
 1772-158 1942 
 
 730 
 
 1775-021 5170 
 
 731 
 
 1777-885 4344 
 
 732 
 
 1780-749 9455 
 
 733 
 
 1783-615 0495 
 
 734 
 
 1786-480 7455 
 
 735 
 
 1789-347 0329 
 
 736 
 
 1792-213 9107 
 
 737 
 
 1795-081 3782 
 
 738 
 
 1797 949 4345 
 
 739 
 
 1800-818 0790 
 
 740 
 
 1803687 3107 
 
 741 
 
 1806-557 1289 
 
 74'2 
 
 1809-427 5328 
 
 743 
 
 1812-298 5216 
 
 744 
 
 1815-170 0946 
 
 745 
 
 1818-042 2508 
 
 746 
 
 1820-914 9897 
 
 lift 
 
 1823-788 3103 
 
 748 
 
 1826-662 2119 
 
 749 
 
 1829-536 6937 
 
 750 
 
 1832-411 7549 
 
Logarithms of Factorials 
 TABLE XLIX— (continued). 
 
 101 
 
 751—1000 
 
 751 
 
 752 
 75S 
 75\ 
 755 
 
 75G 
 757 
 758 
 759 
 760 
 
 761 
 762 
 763 
 764 
 765 
 
 766 
 767 
 768 
 769 
 770 
 
 771 
 772 
 773 
 774, 
 775 
 
 776 
 777 
 778 
 779 
 780 
 
 781 
 
 782 
 783 
 784 
 785 
 
 786 
 
 787 
 788 
 789 
 790 
 
 791 
 
 792 
 793 
 794 
 795 
 
 796 
 797 
 798 
 799 
 800 
 
 \!L 
 
 1835-287 3949 
 1838-163 6127 
 1841-040 4077 
 1843-917 7790 
 1846-795 7260 
 
 1849-674 2478 
 1852-553 3437 
 1855-433 0129 
 1858-313 2546 
 1861-194 0682 
 
 1864-075 4529 
 1866-957 4079 
 1869-839 9324 
 1872-723 0258 
 1875-606 6872 
 
 1878-490 9160 
 1881-375 7113 
 1884-261 0726 
 1887-146 9989 
 1890-033 4896 
 
 1892-920 5440 
 1895-808 1613 
 1898-696 3408 
 1901-585 0817 
 1904-474 3835 
 
 1907-364 2452 
 1910-254 6662 
 1913-145 6458 
 1916-037 1832 
 1918-929 2778 
 
 1921-821 9289 
 1924-715 1356 
 1927-608 8974 
 1930-503 2135 
 1933-398 0831 
 
 1936-293 5057 
 1939-189 4804 
 1942-086 0066 
 1944-983 0836 
 1947-880 7107 
 
 1950-778 8872 
 1953-677 6124 
 1956-576 8856 
 1959-476 7061 
 1962-377 0732 
 
 1965-277 9863 
 1968-179 4440 
 1971-081 4475 
 1973-983 9943 
 1970-887 0842 
 
 801 
 802 
 803 
 804 
 805 
 
 806 
 807 
 808 
 809 
 810 
 
 811 
 812 
 813 
 814 
 815 
 
 816 
 817 
 818 
 819 
 
 820 
 
 822 
 823 
 824 
 825 
 
 826 
 827 
 828 
 829 
 830 
 
 831 
 832 
 833 
 834 
 835 
 
 836 
 837 
 838 
 8S9 
 840 
 
 841 
 842 
 843 
 844 
 
 846 
 847 
 848 
 849 
 850 
 
 log I" 
 
 1979-790 7168 
 1982-694 8911 
 1985-599 6067 
 1988-504 8627 
 1991-410 6586 
 
 1994-316 9936 
 1997-223 8672 
 2000-131 2785 
 2003-039 2271 
 2005-947 7121 
 
 2008-856 7329 
 2011-766 2890 
 2014-676 3795 
 2017-587 0039 
 2020-498 1615 
 
 2023-409 8517 
 2026-322 0737 
 2029-234 8270 
 2032-148 1109 
 2035-061 9248 
 
 2037-976 2679 
 2040-891 1398 
 2043-806 5396 
 2046-722 4008 
 2049 638 9208 
 
 2052-555 9008 
 2055-473 4063 
 2058-391 4367 
 2061-309 9912 
 2064-229 0693 
 
 2067-148 6703 
 2070-068 7936 
 2072-989 4386 
 2075-910 6047 
 2078-832 2912 
 
 2081-754 4974 
 2084-677 2229 
 2087-600 4669 
 2090-524 2289 
 2093-448 5082 
 
 2096-373 3042 
 2099-298 6162 
 2102-224 4438 
 2105-150 7863 
 2108-077 6430 
 
 2111-005 0133 
 2113-932 8967 
 2116-861 2926 
 2119-790 2003 
 2122-719 6192 
 
 n 
 
 log [»_ 
 
 851 
 
 2125-649 5488 
 
 852 
 
 2128-579 9884 
 
 853 
 
 2131-510 9374 
 
 854 
 
 2134-442 3953 
 
 855 
 
 2137-374 3614 
 
 856 
 
 2140-306 8352 
 
 857 
 
 2143-239 8160 
 
 858 
 
 2146-173 3033 
 
 859 
 
 2149-107 2964 
 
 860 
 
 2152-041 7949 
 
 861 
 
 2154-976 7980 
 
 862 
 
 2157-912 3053 
 
 863 
 
 2160-848 3161 
 
 864 
 
 2103-784 8298 
 
 865 
 
 2166-721 8459 
 
 866 
 
 2169-659 3638 
 
 867 
 
 2172-597 3829 
 
 868 
 
 2175-535 9027 
 
 869 
 
 2178-474 9224 
 
 870 
 
 2181-414 4417 
 
 871 
 
 2184-354 4598 
 
 872 
 
 2187-294 9763 
 
 873 
 
 2190-235 9906 
 
 874 
 
 2193-177 5020 
 
 875 
 
 2196-119 5101 
 
 876 
 
 2199-062 0142 
 
 877 
 
 2202-005 0138 
 
 878 
 
 2204-948 5083 
 
 879 
 
 2207-892 4971 
 
 880 
 
 2210-836 9798 
 
 881 
 
 2213-781 9557 
 
 882 
 
 2216-727 4243 
 
 883 
 
 2219-673 3850 
 
 884 
 
 2222-619 8373 
 
 885 
 
 2225-566 7805 
 
 886 
 
 2228-514 2143 
 
 887 
 
 2231-462 1379 
 
 888 
 
 2234-410 5509 
 
 889 
 
 2237-359 4526 
 
 890 
 
 2240-308 8426 
 
 891 
 
 2243-258 7203 
 
 892 
 
 2246-209 0852 
 
 893 
 
 2249-159 9366 
 
 894 
 
 2252-111 2742 
 
 895 
 
 2255-063 0972 
 
 896 
 
 2258-015 4052 
 
 897 
 
 2260-968 1976 
 
 898 
 
 2263-921 4740 
 
 899 
 
 2266-875 2337 
 
 900 
 
 2269-829 4762 
 
 n 
 
 log \n 
 
 901 
 
 2272-784 2010 
 
 902 
 
 2275-739 4075 
 
 903 
 
 2278-695 0953 
 
 904 
 
 2281-651 2637 
 
 905 
 
 2284-607 9123 
 
 906 
 
 2287-565 0405 
 
 907 
 
 2290-522 6478 
 
 908 
 
 2293-480 7336 
 
 909 
 
 2296-439 2975 
 
 910 
 
 2299-398 3389 
 
 911 
 
 2302-357 8573 
 
 912 
 
 2305-317 8521 
 
 913 
 
 2308-278 3229 
 
 914 
 
 2311-239 2691 
 
 915 
 
 2314-200 6902 
 
 916 
 
 2317-162 5856 
 
 917 
 
 2320-124 9550 
 
 918 
 
 2323-087 7977 
 
 919 
 
 2326-051 1132 
 
 920 
 
 2329-014 9010 
 
 921 
 
 2331-979 1606 
 
 922 
 
 2334-943 8915 
 
 923 
 
 2337-909 0932 
 
 924 
 
 2340-874 7652 
 
 925 
 
 2343-840 9069 
 
 926 
 
 2346-807 5179 
 
 927 
 
 2349-774 5977 
 
 928 
 
 2352-742 1456 
 
 929 
 
 2355-710 1614 
 
 930 
 
 2358-678 6443 
 
 931 
 
 2361-647 5940 
 
 932 
 
 2364-617 0099 
 
 983 
 
 2367-586 8915 
 
 934 
 
 2370-557 2384 
 
 935 
 
 2373-528 0500 
 
 936 
 
 2376-499 3259 
 
 937 
 
 2379-471 0655 
 
 938 
 
 2382-443 2683 
 
 989 
 
 2385-415 9339 
 
 940 
 
 2388-389 0618 
 
 941 
 
 2391-302 6514 
 
 942 
 
 2394-336 7023 
 
 943 
 
 2397-311 2140 
 
 944 
 
 2400-286 1860 
 
 945 
 
 2403-261 6178 
 
 946 
 
 2406-237 5089 
 
 947 
 
 2409-213 8589 
 
 948 
 
 2412-190 6672 
 
 949 
 
 2415-167 9334 
 
 950 
 
 2418-145 6570 
 
 n 
 
 log[n 
 
 951 
 
 2421-123 8376 
 
 952 
 
 2424-102 4745 
 
 953 
 
 2427-081 5674 
 
 954 
 
 2430-061 1158 
 
 955 
 
 2433-041 1192 
 
 956 
 
 2436-021 5771 
 
 957 
 
 2439-002 4890 
 
 958 
 
 2441-983 8545 
 
 959 
 
 2444965 6731 
 
 960 
 
 2447-947 9443 
 
 961 
 
 2450-930 6677 
 
 962 
 
 2453-913 8428 
 
 963 
 
 2456-897 4691 
 
 964 
 
 2459-881 5461 
 
 965 
 
 2462-866 0734 
 
 966 
 
 2465-851 0500 
 
 967 
 
 2408-836 4770 
 
 968 
 
 2471-822 3524 
 
 969 
 
 2474-808 6762 
 
 970 
 
 2477-795 4479 
 
 971 
 
 2480-782 6671 
 
 972 
 
 2483770 3334 
 
 973 
 
 2486-758 4462 
 
 974 
 
 2489-747 0052 
 
 975 
 
 2492-736 0098 
 
 976 
 
 2495-725 4596 
 
 977 
 
 2498-715 3542 
 
 978 
 
 2501-705 6930 
 
 979 
 
 2504-696 4757 
 
 980 
 
 2507-687 7018 
 
 981 
 
 2510-679 3708 
 
 982 
 
 2513-071 4823 
 
 983 
 
 2510-004 0358 
 
 984 
 
 2519-657 0309 
 
 985 
 
 2522-650 4672 
 
 986 
 
 2525-644 3441 
 
 987 
 
 2528-638 6612 
 
 988 
 
 2531-633 4182 
 
 989 
 
 2534-628 6145 
 
 990 
 
 2537-624 2497 
 
 991 
 
 2540-620 3233 
 
 992 
 
 2543-616 8350 
 
 993 
 
 2546-613 7842 
 
 994 
 
 2549-611 1706 
 
 995 
 
 2552-608 9937 
 
 996 
 
 2555-607 2530 
 
 997 
 
 2558-605 9482 
 
 998 
 
 2561-605 0787 
 
 999 
 
 2564-604 6442 
 
 1000 
 
 2567-004 6442 
 
102 
 
 Tables for Statisticians and Blotnetrlclans 
 
 TABLE L. Table of Fourth-Moments of Subgroup-Frequencies. 
 Ordinate 2 — 11. Frequency 1 — 50. 
 
 n 
 
 «a] 
 
 3 = 3 
 
 X = i 
 
 x = 5 
 
 s = 6 
 
 x = 7 
 
 x = 8 
 
 x=9 
 
 a=ll 
 
 n 
 
 1 
 
 16 
 
 81 
 
 256 
 
 625 
 
 1296 
 
 2401 
 
 4096 
 
 6561 
 
 14641 
 
 1 
 
 2 
 
 32 
 
 162 
 
 512 
 
 1250 
 
 2592 
 
 4802 
 
 8192 
 
 13122 
 
 29282 
 
 2 
 
 S 
 
 48 
 
 243 
 
 768 
 
 1875 
 
 3888 
 
 7203 
 
 12288 
 
 19683 
 
 43923 
 
 S 
 
 4 
 
 64 
 
 324 
 
 1024 
 
 2500 
 
 6184 
 
 9604 
 
 16384 
 
 26244 
 
 58564 
 
 4 
 
 5 
 
 80 
 
 405 
 
 1280 
 
 3125 
 
 6480 
 
 12005 
 
 20480 
 
 32805 
 
 73205 
 
 5 
 
 6 
 
 96 
 
 486 
 
 1536 
 
 3750 
 
 7776 
 
 14406 
 
 24576 
 
 39366 
 
 87846 
 
 6 
 
 7 
 
 112 
 
 567 
 
 1792 
 
 4375 
 
 9072 
 
 16807 
 
 28672 
 
 45927 
 
 102487 
 
 7 
 
 8 
 
 128 
 
 648 
 
 2048 
 
 5000 
 
 103G8 
 
 19208 
 
 32768 
 
 52488 
 
 117128 
 
 8 
 
 9 
 
 144 
 
 729 
 
 2304 
 
 5625 
 
 11664 
 
 21609 
 
 36864 
 
 59049 
 
 131769 
 
 9 
 
 10 
 
 160 
 
 810 
 
 2560 
 
 6250 
 
 12960 
 
 24010 
 
 40960 
 
 65610 
 
 146410 
 
 10 
 
 11 
 
 176 
 
 891 
 
 2816 
 
 6875 
 
 14256 
 
 26411 
 
 45056 
 
 72171 
 
 161051 
 
 11 
 
 12 
 
 192 
 
 972 
 
 3072 
 
 7500 
 
 15552 
 
 28812 
 
 49152 
 
 78732 
 
 175692 
 
 12 
 
 IS 
 
 208 
 
 1053 
 
 3328 
 
 8125 
 
 16848 
 
 31213 
 
 53248 
 
 85293 
 
 190333 
 
 IS 
 
 u 
 
 224 
 
 1134 
 
 3584 
 
 8750 
 
 18144 
 
 33614 
 
 57344 
 
 91854 
 
 204974 
 
 14 
 
 15 
 
 240 
 
 1215 
 
 3840 
 
 9375 
 
 19440 
 
 36015 
 
 61440 
 
 98415 
 
 219615 
 
 15 
 
 16 
 
 256 
 
 1296 
 
 4096 
 
 10000 
 
 20736 
 
 38416 
 
 65536 
 
 104976 
 
 234256 
 
 16 
 
 n 
 
 272 
 
 1377 
 
 4352 
 
 10625 
 
 22032 
 
 40817 
 
 69632 
 
 111537 
 
 248897 
 
 17 
 
 is 
 
 288 
 
 1458 
 
 4608 
 
 11250 
 
 23328 
 
 43218 
 
 73728 
 
 118098 
 
 263538 
 
 18 
 
 19 
 
 304 
 
 1539 
 
 4864 
 
 11875 
 
 24624 
 
 45619 
 
 77824 
 
 124659 
 
 278179 
 
 19 
 
 20 
 
 320 
 
 1620 
 
 5120 
 
 12500 
 
 25920 
 
 48020 
 
 81920 
 
 131220 
 
 292820 
 
 20 
 
 21 
 
 336 
 
 1701 
 
 5376 
 
 13125 
 
 27216 
 
 50421 
 
 86016 
 
 137781 
 
 307461 
 
 21 
 
 22 
 
 352 
 
 1782 
 
 5632 
 
 13750 
 
 28512 
 
 52822 
 
 90112 
 
 144342 
 
 322102 
 
 22 
 
 23 
 
 368 
 
 1863 
 
 5888 
 
 14375 
 
 29808 
 
 55223 
 
 94208 
 
 150903 
 
 336743 
 
 2S 
 
 n 
 
 384 
 
 1944 
 
 6144 
 
 15000 
 
 31104 
 
 57624 
 
 98304 
 
 157464 
 
 351384 
 
 24 
 
 25 
 
 400 
 
 2025 
 
 6400 
 
 15625 
 
 32400 
 
 60025 
 
 102400 
 
 164025 
 
 366025 
 
 25 
 
 26 
 
 416 
 
 2106 
 
 6656 
 
 16250 
 
 33696 
 
 62426 
 
 106496 
 
 170586 
 
 380666 
 
 26 
 
 27 
 
 432 
 
 2187 
 
 6912 
 
 16875 
 
 34992 
 
 64827 
 
 110592 
 
 177147 
 
 395307 
 
 27 
 
 28 
 
 448 
 
 2268 
 
 7168 
 
 17500 
 
 36288 
 
 67228 
 
 114688 
 
 183708 
 
 409948 
 
 28 
 
 29 
 
 464 
 
 2349 
 
 7424 
 
 18125 
 
 37584 
 
 69629 
 
 118784 
 
 190269 
 
 424589 
 
 29 
 
 SO 
 
 480 
 
 2430 
 
 7680 
 
 18750 
 
 38880 
 
 72030 
 
 122880 
 
 196830 
 
 439230 
 
 SO 
 
 SI 
 
 496 
 
 2511 
 
 7936 
 
 19375 
 
 40176 
 
 74431 
 
 126976 
 
 203391 
 
 453871 
 
 81 
 
 32 
 
 512 
 
 2592 
 
 8192 
 
 20000 
 
 41472 
 
 76832 
 
 131072 
 
 209952 
 
 468512 
 
 82 
 
 SS 
 
 528 
 
 2673 
 
 8448 
 
 20625 
 
 42768 
 
 79233 
 
 135168 
 
 216513 
 
 483153 
 
 SS 
 
 Sit 
 
 544 
 
 2754 
 
 8704 
 
 21250 
 
 44064 
 
 81634 
 
 139264 
 
 223074 
 
 497794 
 
 84 
 
 S5 
 
 560 
 
 2835 
 
 8960 
 
 21875 
 
 45360 
 
 84035 
 
 143360 
 
 229635 
 
 512435 
 
 35 
 
 S6 
 
 576 
 
 2916 
 
 9216 
 
 22500 
 
 46656 
 
 86436 
 
 147456 
 
 236196 
 
 527076 
 
 36 
 
 ,17 
 
 592 
 
 2997 
 
 9472 
 
 , 23125 
 
 47952 
 
 88837 
 
 151552 
 
 242757 
 
 541717 
 
 87 
 
 S8 
 
 608 
 
 3078 
 
 9728 
 
 23750 
 
 49248 
 
 91238 
 
 155648 
 
 249318 
 
 556358 
 
 38 
 
 S9 
 
 624 
 
 3159 
 
 9984 
 
 24375 
 
 50544 
 
 93639 
 
 159744 
 
 255879 
 
 570999 
 
 89 
 
 40 
 
 640 
 
 3240 
 
 10240 
 
 25000 
 
 51840 
 
 96040 
 
 163840 
 
 262440 
 
 585640 
 
 40 
 
 tl 
 
 656 
 
 3321 
 
 10496 
 
 25625 
 
 53136 
 
 98441 
 
 167936 
 
 269001 
 
 600281 
 
 41 
 
 42 
 
 672 
 
 3402 
 
 10752 
 
 26250 
 
 54432 
 
 100842 
 
 172032 
 
 275562 
 
 614922 
 
 42 
 
 4S 
 
 688 
 
 3483 
 
 11008 
 
 26875 
 
 55728 
 
 103243 
 
 176128 
 
 282123 
 
 629563 
 
 43 
 
 44 
 
 704 
 
 3564 
 
 11264 
 
 27500 
 
 57024 
 
 105644 
 
 180224 
 
 288684 
 
 644204 
 
 U 
 
 45 
 
 720 
 
 3645 
 
 11520 
 
 28125 
 
 58320 
 
 108045 
 
 184320 
 
 295245 
 
 658845 
 
 45 
 
 46 
 
 736 
 
 3726 
 
 11776 
 
 28750 
 
 59616 
 
 110446 
 
 188416 
 
 301806 
 
 673486 
 
 40 
 
 47 
 
 752 
 
 3807 
 
 12032 
 
 29375 
 
 60912 
 
 112847 
 
 192512 
 
 308367 
 
 688127 
 
 47 
 
 48 
 
 768 
 
 3888 
 
 12288 
 
 30000 
 
 62208 
 
 115248 
 
 196608 
 
 314928 
 
 702768 
 
 48 
 
 49 
 
 784 
 
 3969 
 
 12544 
 
 30625 
 
 63504 
 
 117649 
 
 200704 
 
 321489 
 
 717409 
 
 49 
 
 50 
 
 800 
 
 4050 
 
 12800 
 
 31250 
 
 64800 
 
 120050 
 
 204800 
 
 328050 
 
 732050 
 
 50 
 
Verification of the Fourth Moment 
 
 103 
 
 TABLE L— {continued). 
 Ordinate 12 — 19. Frequency 1 — 50. 
 
 n 
 1 
 
 x=n 
 
 * = 13 
 
 x=14 
 
 i = 15 
 
 *=16 
 
 x = 17 
 
 i = 18 
 
 x = 19 
 
 n 
 
 20736 
 
 28561 
 
 38416 
 
 50625 
 
 65536 
 
 83521 
 
 104976 
 
 130321 
 
 1 
 
 2 
 
 41472 
 
 57122 
 
 76832 
 
 101250 
 
 131072 
 
 167042 
 
 209952 
 
 260642 
 
 8 
 
 3 
 
 62208 
 
 85683 
 
 115248 
 
 151875 
 
 196608 
 
 250563 
 
 314928 
 
 390963 
 
 S 
 
 4 
 
 82944 
 
 114244 
 
 153664 
 
 202500 
 
 262144 
 
 334084 
 
 419904 
 
 521284 
 
 4 
 
 5 
 
 103680 
 
 142805 
 
 192080 
 
 253125 
 
 327680 
 
 417605 
 
 624880 
 
 651605 
 
 5 
 
 6 
 
 124416 
 
 171366 
 
 230496 
 
 303750 
 
 393216 
 
 501126 
 
 629856 
 
 781926 
 
 6 
 
 7 
 
 145152 
 
 199927 
 
 268912 
 
 354375 
 
 458752 
 
 584647 
 
 734832 
 
 912247 
 
 7 
 
 8 
 
 165888 
 
 228488 
 
 307328 
 
 405000 
 
 524288 
 
 668168 
 
 839808 
 
 1042568 
 
 8 
 
 9 
 
 186624 
 
 257049 
 
 345744 
 
 455625 
 
 589824 
 
 751639 
 
 944784 
 
 1172889 
 
 9 
 
 10 
 
 207360 
 
 285610 
 
 384160 
 
 506250 
 
 655360 
 
 835210 
 
 1049760 
 
 1303210 
 
 10 
 
 11 
 
 228096 
 
 314171 
 
 422576 
 
 556875 
 
 720896 
 
 918731 
 
 1154736 
 
 1433531 
 
 11 
 
 12 
 
 248832 
 
 342732 
 
 460992 
 
 607500 
 
 786432 
 
 1002252 
 
 1259712 
 
 1563852 
 
 12 
 
 IS 
 
 269568 
 
 371293 
 
 499408 
 
 658125 
 
 851968 
 
 1085773 
 
 1364688 
 
 1694173 
 
 IS 
 
 n 
 
 290304 
 
 399854 
 
 537824 
 
 708750 
 
 917504 
 
 1169294 
 
 1469664 
 
 1824494 
 
 U 
 
 15 
 
 311040 
 
 428415 
 
 576240 
 
 759375 
 
 983040 
 
 1252815 
 
 1574640 
 
 1954815 
 
 15 
 
 16 
 
 331776 
 
 456976 
 
 614656 
 
 810000 
 
 1048576 
 
 1336336 
 
 1679616 
 
 2085136 
 
 16 
 
 17 
 
 352512 
 
 485537 
 
 653072 
 
 860625 
 
 1114112 
 
 1419857 
 
 1784592 
 
 2215457 
 
 17 
 
 18 
 
 373248 
 
 514098 
 
 691488 
 
 911250 
 
 1179648 
 
 1503378 
 
 1889568 
 
 2345778 
 
 18 
 
 19 
 
 393984 
 
 542659 
 
 729904 
 
 961875 
 
 1245184 
 
 1586899 
 
 1994544 
 
 2476099 
 
 19 
 
 SO 
 
 414720 
 
 571220 
 
 768320 
 
 1012500 
 
 1310720 
 
 1670420 
 
 2099520 
 
 2606420 
 
 20 
 
 21 
 
 435456 
 
 599781 
 
 806736 
 
 1063125 
 
 1376256 
 
 1753941 
 
 2204496 
 
 2736741 
 
 21 
 
 22 
 
 456192 
 
 628342 
 
 845152 
 
 1113750 
 
 1441792 
 
 1837462 
 
 2309472 
 
 2867062 
 
 22 
 
 23 
 
 476928 
 
 656903 
 
 883568 
 
 1164375 
 
 1507328 
 
 1920983 
 
 2414448 
 
 2997383 
 
 23 
 
 24 
 
 497664 
 
 685464 
 
 921984 
 
 1215000 
 
 1572864 
 
 2004504 
 
 2519424 
 
 3127704 
 
 24 
 
 25 
 
 518400 
 
 714025 
 
 960400 
 
 1265625 
 
 1638400 
 
 2088025 
 
 2624400 
 
 3258025 
 
 25 
 
 26 
 
 539136 
 
 742586 
 
 998816 
 
 1316250 
 
 1703936 
 
 2171546 
 
 2729376 
 
 3388346 
 
 26 
 
 27 
 
 559872 
 
 771147 
 
 1037232 
 
 1366875 
 
 1769472 
 
 2255067 
 
 2834352 
 
 3518667 
 
 27 
 
 28 
 
 580608 
 
 799708 
 
 1075648 
 
 1417500 
 
 1835008 
 
 2338588 
 
 2939328 
 
 3648988 
 
 28 
 
 29 
 
 601344 
 
 828269 
 
 1114064 
 
 1468125 
 
 1900544 
 
 2422109 
 
 3044304 
 
 3779309 
 
 29 
 
 SO 
 
 622080 
 
 856830 
 
 1152480 
 
 1518750 
 
 1966080 
 
 2505630 
 
 3149280 
 
 3909630 
 
 SO 
 
 SI 
 
 642816 
 
 885391 
 
 1190896 
 
 1569375 
 
 2031616 
 
 2589151 
 
 3254256 
 
 4039951 
 
 31 
 
 32 
 
 663552 
 
 913952 
 
 1229312 
 
 1620000 
 
 2097152 
 
 2672672 
 
 3359232 
 
 4170272 
 
 32 
 
 S3 
 
 684288 
 
 942513 
 
 1267728 
 
 1670625 
 
 2162688 
 
 2756193 
 
 3464208 
 
 4300593 
 
 S3 
 
 SJ, 
 
 705024 
 
 971074 
 
 1306144 
 
 1721250 
 
 2228224 
 
 2839714 
 
 3569184 
 
 4430914 
 
 S4 
 
 35 
 
 725760 
 
 999635 
 
 1344560 
 
 1771875 
 
 2293760 
 
 2923235 
 
 3674160 
 
 4561235 
 
 35 
 
 36 
 
 746496 
 
 1028196 
 
 1382976 
 
 1822500 
 
 2359296 
 
 3006756 
 
 3779136 
 
 4691556 
 
 36 
 
 57 
 
 767232 
 
 1056757 
 
 1421392 
 
 1873125 
 
 2424832 
 
 3090277 
 
 3884112 
 
 4821877 
 
 37 
 
 38 
 
 787968 
 
 1085318 
 
 1459808 
 
 1923750 
 
 2490368 
 
 3173798 
 
 3989088 
 
 4952198 
 
 38 
 
 39 
 
 808704 
 
 1113879 
 
 1498224 
 
 1974375 
 
 2555904 
 
 3257319 
 
 4094064 
 
 5082519 
 
 39 
 
 40 
 
 829440 
 
 1142440 
 
 1536640 
 
 2025000 
 
 2621440 
 
 3340840 
 
 4199040 
 
 5212840 
 
 40 
 
 41 
 
 850176 
 
 1171001 
 
 1575056 
 
 2076626 
 
 2686976 
 
 3424361 
 
 4304016 
 
 5343161 
 
 41 
 
 42 
 
 870912 
 
 1199562 
 
 1613472 
 
 2126250 
 
 2752512 
 
 3507882 
 
 4408992 
 
 5473482 
 
 42 
 
 43 
 
 891648 
 
 1228123 
 
 1651888 
 
 2176875 
 
 2818048 
 
 3591403 
 
 4513968 
 
 5603803 
 
 43 
 
 u 
 
 912384 
 
 1256684 
 
 1690304 
 
 2227500 
 
 2883584 
 
 3674924 
 
 4618944 
 
 5734124 
 
 44 
 
 45 
 
 933120 
 
 1285245 
 
 1728720 
 
 2278125 
 
 2949120 
 
 3758445 
 
 4723920 
 
 5864445 
 
 45 
 
 46 
 
 953856 
 
 1313806 
 
 1767136 
 
 2328750 
 
 3014656 
 
 3841966 
 
 4828896 
 
 5994766 
 
 46 
 
 47 
 
 974592 
 
 1342367 
 
 1805552 
 
 2379375 
 
 3080192 
 
 3925487 
 
 4933872 
 
 6125087 
 
 47 
 
 48 
 
 995328 
 
 1370928 
 
 1843968 
 
 2430000 
 
 3145728 
 
 4009008 
 
 5038848 
 
 6255408 
 
 48 
 
 49 
 
 1016064 
 
 1399489 
 
 1882384 
 
 2480625 
 
 3211264 
 
 4092529 
 
 5143824 
 
 6385729 
 
 49 
 
 50 
 
 1036800 
 
 1428050 
 
 1920800 
 
 2531250 
 
 3276800 
 
 4176050 
 
 5248800 
 
 6516050 
 
 50 
 
104 
 
 Tables for Statisticians and Biometricians 
 
 TABLE L— (continued). 
 Ordinate 2—11. Frequency 51 — 100. 
 
 n 
 
 x=2 
 
 x = 3 
 
 x = 4 
 
 x=5 
 
 x = 6 
 
 x = 7 
 
 a=8 
 
 x=9 
 
 x=ll 
 
 71 
 
 51 
 
 816 
 
 4131 
 
 13056 
 
 31875 
 
 66096 
 
 122451 
 
 208896 
 
 334611 
 
 746691 
 
 51 
 
 52 
 
 832 
 
 4212 
 
 13312 
 
 32500 
 
 67392 
 
 124852 
 
 212992 
 
 341172 
 
 761332 
 
 52 
 
 53 
 
 848 
 
 4293 
 
 13568 
 
 33125 
 
 68688 
 
 127253 
 
 217088 
 
 347733 
 
 775973 
 
 5S 
 
 54 
 
 864 
 
 4374 
 
 13824 
 
 33750 
 
 69984 
 
 129654 
 
 221184 
 
 354294 
 
 790614 
 
 54 
 
 55 
 
 880 
 
 4455 
 
 14080 
 
 34375 
 
 71280 
 
 132055 
 
 225280 
 
 360855 
 
 805255 
 
 55 
 
 56 
 
 896 
 
 4536 
 
 14336 
 
 35000 
 
 72576 
 
 134456 
 
 229376 
 
 367416 
 
 819896 
 
 56 
 
 57 
 
 912 
 
 4617 
 
 14592 
 
 35625 
 
 73872 
 
 136857 
 
 233472 
 
 373977 
 
 834537 
 
 57 
 
 58 
 
 928 
 
 4698 
 
 14848 
 
 36250 
 
 75168 
 
 139258 
 
 237568 
 
 380538 
 
 849178 
 
 58 
 
 50 
 
 944 
 
 4779 
 
 15104 
 
 36875 
 
 76464 
 
 141659 
 
 241664 
 
 387099 
 
 863819 
 
 59 
 
 60 
 
 960 
 
 4860 
 
 15360 
 
 37500 
 
 77760 
 
 144060 
 
 245760 
 
 393660 
 
 878460 
 
 60 
 
 61 
 
 976 
 
 4941 
 
 15616 
 
 38125 
 
 79056 
 
 146461 
 
 249856 
 
 400221 
 
 893101 
 
 61 
 
 62 
 
 992 
 
 5022 
 
 15872 
 
 38750 
 
 80352 
 
 148862 
 
 253952 
 
 406782 
 
 907742 
 
 62 
 
 63 
 
 1008 
 
 5103 
 
 16128 
 
 39375 
 
 81648 
 
 151263 
 
 258048 
 
 413343 
 
 922383 
 
 63 
 
 64 
 
 1024 
 
 5184 
 
 16384 
 
 40000 
 
 82944 
 
 153664 
 
 262144 
 
 419904 
 
 937024 
 
 64 
 
 65 
 
 1040 
 
 5265 
 
 16640 
 
 40625 
 
 84240 
 
 156065 
 
 266240 
 
 426465 
 
 951665 
 
 65 
 
 66 
 
 1056 
 
 5346 
 
 16896 
 
 41250 
 
 85536 
 
 158466 
 
 270336 
 
 433026 
 
 966306 
 
 66 
 
 67 
 
 1072 
 
 5427 
 
 17152 
 
 41875 
 
 86832 
 
 160667 
 
 274432 
 
 439587 
 
 980947 
 
 67 
 
 68 
 
 1088 
 
 5508 
 
 17408 
 
 42500 
 
 88128 
 
 163268 
 
 278528 
 
 446148 
 
 995588 
 
 68 
 
 69 
 
 1104 
 
 5589 
 
 17664 
 
 43125 
 
 89424 
 
 165669 
 
 282624 
 
 452709 
 
 1010229 
 
 69 
 
 70 
 
 1120 
 
 5670 
 
 17920 
 
 43750 
 
 90720 
 
 168070 
 
 286720 
 
 459270 
 
 1024870 
 
 70 
 
 71 
 
 1136 
 
 5751 
 
 18176 
 
 44375 
 
 92016 
 
 170471 
 
 290816 
 
 465831 
 
 1039511 
 
 71 
 
 72 
 
 1152 
 
 5832 
 
 18432 
 
 45000 
 
 93312 
 
 172872 
 
 294912 
 
 472392 
 
 1054152 
 
 72 
 
 73 
 
 1168 
 
 5913 
 
 18688 
 
 45625 
 
 94608 
 
 175273 
 
 299008 
 
 478953 
 
 1068793 
 
 73 
 
 74 
 
 1184 
 
 5994 
 
 18944 
 
 46250 
 
 95904 
 
 177674 
 
 303104 
 
 485514 
 
 1083434 
 
 74 
 
 75 
 
 1200 
 
 6075 
 
 19200 
 
 46875 
 
 97200 
 
 180075 
 
 307200 
 
 492075 
 
 1098075 
 
 75 
 
 76 
 
 1216 
 
 6156 
 
 19456 
 
 47500 
 
 98496 
 
 182476 
 
 311296 
 
 498636 
 
 1112716 
 
 76 
 
 77 
 
 1232 
 
 6237 
 
 19712 
 
 48125 
 
 99792 
 
 184877 
 
 315392 
 
 505197 
 
 1127357 
 
 77 
 
 78 
 
 1248 
 
 6318 
 
 19968 
 
 48750 
 
 101088 
 
 187278 
 
 319488 
 
 511758 
 
 1141998 
 
 78 
 
 79 
 
 1264 
 
 6399 
 
 20224 
 
 49375 
 
 102384 
 
 189679 
 
 323584 
 
 518319 
 
 1156639 
 
 79 
 
 80 
 
 1280 
 
 6480 
 
 20480 
 
 50000 
 
 103680 
 
 192080 
 
 327680 
 
 524880 
 
 1171280 
 
 80 
 
 81 
 
 1296 
 
 6561 
 
 20736 
 
 50625 
 
 104976 
 
 194481 
 
 331776 
 
 531441 
 
 1185921 
 
 81 
 
 82 
 
 1312 
 
 6642 
 
 20992 
 
 51250 
 
 106272 
 
 196882 
 
 335872 
 
 538002 
 
 1200562 
 
 82 
 
 83 
 
 1328 
 
 6723 
 
 21248 
 
 51875 
 
 107568 
 
 199283 
 
 339968 
 
 544563 
 
 1215203 
 
 83 
 
 u 
 
 1344 
 
 6804 
 
 21504 
 
 52500 
 
 108864 
 
 201684 
 
 344064 
 
 551124 
 
 1229844 
 
 84 
 
 85 
 
 1360 
 
 6885 
 
 21760 
 
 53125 
 
 110160 
 
 204085 
 
 348160 
 
 557685 
 
 1244485 
 
 85 
 
 86 
 
 1376 
 
 6966 
 
 22016 
 
 53750 
 
 111456 
 
 206486 
 
 352256 
 
 564246 
 
 1259126 
 
 86 
 
 87 
 
 1392 
 
 7047 
 
 22272 
 
 54375 
 
 112752 
 
 208887 
 
 356352 
 
 570807 
 
 1273767 
 
 87 
 
 88 
 
 1408 
 
 7128 
 
 22528 
 
 55000 
 
 114048 
 
 211288 
 
 360448 
 
 577368 
 
 1288408 
 
 88 
 
 89 
 
 1424 
 
 7209 
 
 22784 
 
 55625 
 
 115344 
 
 213689 
 
 364544 
 
 583929 
 
 1303049 
 
 89 
 
 90 
 
 1440 
 
 7290 
 
 23040 
 
 56250 
 
 116640 
 
 216090 
 
 368640 
 
 590490 
 
 1317690 
 
 90 
 
 91 
 
 1456 
 
 7371 
 
 23296 
 
 56875 
 
 117936 
 
 218491 
 
 372736 
 
 597051 
 
 1332331 
 
 91 
 
 92 
 
 1472 
 
 7452 
 
 23552 
 
 57500 
 
 119232 
 
 220892 
 
 376832 
 
 603612 
 
 1346972 
 
 92 
 
 93 
 
 1488 
 
 7533 
 
 23808 
 
 58125 
 
 120528 
 
 223293 
 
 380928 
 
 610173 
 
 1361613 
 
 93 
 
 94 
 
 1504 
 
 7614 
 
 24064 
 
 58750 
 
 121824 
 
 225694 
 
 385024 
 
 616734 
 
 1376254 
 
 94 
 
 95 
 
 1520 
 
 7695 
 
 24320 
 
 59375 
 
 123120 
 
 228095 
 
 389120 
 
 623295 
 
 1390895 
 
 95 
 
 96 
 
 1536 
 
 7776 
 
 24576 
 
 60000 
 
 124416 
 
 230496 
 
 393216 
 
 629856 
 
 1405536 
 
 96 
 
 97 
 
 1552 
 
 7857 
 
 24832 
 
 60625 
 
 125712 
 
 232897 
 
 397312 
 
 636417 
 
 1420177 
 
 97 
 
 98 
 
 1568 
 
 7938 
 
 25088 
 
 61250 
 
 127008 
 
 235298 
 
 401408 
 
 642978 
 
 1434818 
 
 98 
 
 99 
 
 1584 
 
 8019 
 
 25344 
 
 61875 
 
 128304 
 
 237699 
 
 405504 
 
 649539 
 
 1449459 
 
 99 
 
 100 
 
 1600 
 
 8100 
 
 25600 
 
 62500 
 
 129600 
 
 240100 
 
 409600 
 
 656100 
 
 1464100 
 
 100 
 
Verification of the Fourth Moment 
 
 105 
 
 TABLE L— (continued). 
 Ordinate 12 — 19. Frequency 51- 
 
 -100. 
 
 re 
 
 x = 12 
 
 x = 13 
 
 a = 14 
 
 a; =15 
 
 S=16 
 
 *=17 
 
 x=18 
 
 x=19 
 
 re 
 
 51 
 
 1057536 
 
 1456611 
 
 1959216 
 
 2581875 
 
 3342336 
 
 4259571 
 
 5353776 
 
 6646371 
 
 51 
 
 52 
 
 1078272 
 
 1485172 
 
 1997632 
 
 2632500 
 
 3407872 
 
 4343092 
 
 5458752 
 
 6776692 
 
 52 
 
 53 
 
 1099008 
 
 1513733 
 
 2036048 
 
 2683125 
 
 3473408 
 
 4426613 
 
 5563728 
 
 6907013 
 
 53 
 
 54 
 
 1119744 
 
 1542294 
 
 2074464 
 
 2733750 
 
 3538944 
 
 4510134 
 
 5668704 
 
 7037334 
 
 54 
 
 55 
 
 1140480 
 
 1570855 
 
 2112880 
 
 2784375 
 
 3604480 
 
 4593655 
 
 5773680 
 
 7167655 
 
 55 
 
 56 
 
 1161216 
 
 1599416 
 
 2151296 
 
 2835000 
 
 3670016 
 
 4677176 
 
 5878656 
 
 7297976 
 
 56 
 
 57 
 
 1181952 
 
 1627977 
 
 2189712 
 
 2885625 
 
 3735552 
 
 4760697 
 
 5983632 
 
 7428297 
 
 57 
 
 58 
 
 1202688 
 
 1656538 
 
 2228128 
 
 2936250 
 
 3801088 
 
 4844218 
 
 6088608 
 
 7558618 
 
 58 
 
 59 
 
 1223424 
 
 1685099 
 
 2266544 
 
 2986875 
 
 3866624 
 
 4927739 
 
 6193584 
 
 7688939 
 
 59 
 
 60 
 
 1244160 
 
 1713660 
 
 2304960 
 
 3037500 
 
 3932160 
 
 5011260 
 
 6298560 
 
 7819260 
 
 60 
 
 61 
 
 1264896 
 
 1742221 
 
 2343376 
 
 3088125 
 
 3997696 
 
 5094781 
 
 6403536 
 
 7949581 
 
 61 
 
 62 
 
 1285632 
 
 1770782 
 
 2381792 
 
 31^8750 | 4063232 
 
 5178302 
 
 6508512 
 
 8079902 
 
 62 
 
 63 
 
 1306368 
 
 1799343 
 
 2420208 
 
 3189375 
 
 4128768 
 
 5261823 
 
 6613488 
 
 8210223 
 
 63 
 
 64 
 
 1327104 
 
 1827904 
 
 2458624 
 
 3240000 
 
 4194304 
 
 5345344 
 
 6718464 
 
 8340544 
 
 64 
 
 65 
 
 1347840 
 
 1856465 
 
 2497040 
 
 3290625 
 
 4259840 
 
 5428865 
 
 6823440 
 
 8470865 
 
 65 
 
 66 
 
 1368576 
 
 1885026 
 
 2535456 
 
 3341250 
 
 4325376 
 
 5512386 
 
 6928416 
 
 8601186 
 
 66 
 
 67 
 
 1389312 
 
 1913587 
 
 2573872 
 
 3391875 
 
 4390912 
 
 5595907 
 
 7033392 
 
 8731507 
 
 67 
 
 68 
 
 1410048 
 
 1942148 
 
 2612288 
 
 3442500 
 
 4456448 
 
 5679428 
 
 7138368 
 
 8861828 
 
 68 
 
 69 
 
 1430784 
 
 1970709 
 
 2650704 
 
 3493125 
 
 4521984 
 
 5762949 
 
 7243344 
 
 8992149 
 
 69 
 
 70 
 
 1451520 
 
 1999270 
 
 2689120 
 
 3543750 
 
 4587520 
 
 5846470 
 
 7348320 
 
 9122470 
 
 70 
 
 71 
 
 1472256 
 
 2027831 
 
 2727536 
 
 3594375 
 
 4653056 
 
 5929991 
 
 7453296 
 
 9252791 
 
 71 
 
 72 
 
 1492992 
 
 2056392 
 
 2765952 
 
 3645000 
 
 4718592 
 
 6013512 
 
 7558272 
 
 9383112 
 
 72 
 
 73 
 
 1513728 
 
 2084953 
 
 2804368 
 
 3695625 
 
 4784128 
 
 6097033 
 
 7663248 
 
 9513433 
 
 73 
 
 74 
 
 1534464 
 
 2113514 
 
 2842784 
 
 3746250 
 
 4849664 
 
 6180554 
 
 7768224 
 
 9643754 
 
 74 
 
 75 
 
 1555200 
 
 2142075 
 
 2881200 
 
 3796875 
 
 4915200 
 
 6264075 
 
 7873200 
 
 9774075 
 
 75 
 
 76 
 
 1575936 
 
 2170636 
 
 2919616 
 
 3847500 
 
 4980736 
 
 6347596 
 
 7978176 
 
 9904396 
 
 76 
 
 77 
 
 1596672 
 
 2199197 
 
 2958032 
 
 3898125 
 
 5046272 
 
 6431117 
 
 8083152 
 
 10034717 
 
 77 
 
 78 
 
 1617408 
 
 2227758 
 
 2996448 
 
 3948750 
 
 5111808 
 
 6514638 
 
 8188128 
 
 10165038 
 
 78 
 
 79 
 
 1638144 
 
 2256319 
 
 3034864 
 
 3999375 
 
 5177344 
 
 6598159 
 
 8293104 
 
 10295359 
 
 79 
 
 80 
 
 1658880 
 
 2284880 
 
 3073280 
 
 4050000 
 
 5242880 
 
 6681680 
 
 8398080 
 
 10425680 
 
 80 
 
 81 
 
 1679616 
 
 2313441 
 
 3111696 
 
 4100625 
 
 5308416 
 
 6765201 
 
 8503056 
 
 10556001 
 
 81 
 
 82 
 
 1700352 
 
 2342002 
 
 3150112 
 
 4151250 
 
 5373952 
 
 6848722 
 
 8608032 
 
 10686322 
 
 82 
 
 83 
 
 1721088 
 
 2370563 
 
 3188528 
 
 4201875 
 
 5439488 
 
 6932243 
 
 8713008 
 
 10816643 
 
 83 
 
 84 
 
 1741824 
 
 2399124 
 
 3226944 
 
 4252500 
 
 5505024 
 
 7015764 
 
 8817984 
 
 10946964 
 
 84 
 
 85 
 
 1762560 
 
 2427685 
 
 3265360 
 
 4303125 
 
 5570560 
 
 7099285 
 
 8922960 
 
 11077285 
 
 85 
 
 86 
 
 1783296 
 
 2456246 
 
 3303776 
 
 4353750 
 
 5636096 
 
 7182806 
 
 9027936 
 
 11207606 
 
 86 
 
 87 
 
 1804032 
 
 2484807 
 
 3342192 
 
 4404375 
 
 5701632 
 
 7266327 
 
 9132912 
 
 11337927 
 
 87 
 
 88 
 
 1824768 
 
 2513368 
 
 3380608 
 
 4455000 
 
 5767168 
 
 7349848 
 
 9237888 
 
 11468248 
 
 88 
 
 89 
 
 1845504 
 
 2541929 
 
 3419024 
 
 4505625 
 
 5832704 
 
 7433369 
 
 9342864 
 
 11598569 
 
 89 
 
 90 
 
 1866240 
 
 2570490 
 
 3457440 
 
 4556250 
 
 5898240 
 
 7516890 
 
 9447840 
 
 11728890 
 
 90 
 
 91 
 
 1886976 
 
 2599051 
 
 3495856 
 
 4606875 
 
 5963776 
 
 7600411 
 
 9552816 
 
 11859211 
 
 91 
 
 92 
 
 1907712 
 
 2627612 
 
 3534272 
 
 4657500 
 
 6029312 
 
 7683932 
 
 9657792 
 
 11989532 
 
 92 
 
 93 
 
 1928448 
 
 2656173 
 
 3572688 
 
 4708125 
 
 6094848 
 
 7767453 
 
 9762768 
 
 12119853 
 
 93 
 
 94 
 
 1949184 
 
 2684734 
 
 3611104 
 
 4758750 
 
 6160384 
 
 7850974 
 
 9867744 
 
 12250174 
 
 9i 
 
 95 
 
 1969920 
 
 2713295 
 
 3649520 
 
 4809375 
 
 6225920 
 
 7934495 
 
 9972720 
 
 12380495 
 
 95 
 
 96 
 
 1990656 
 
 2741856 
 
 3687936 
 
 4860000 
 
 6291456 
 
 8018016 
 
 10077696 
 
 12510816 
 
 96 
 
 .97 
 
 2011392 
 
 2770417 
 
 3726352 
 
 4910625 
 
 6356992 
 
 8101537 
 
 10182672 
 
 12641137 
 
 97 
 
 98 
 
 2032128 
 
 2798978 
 
 3764768 
 
 4961250 
 
 6422528 
 
 8185058 
 
 10287648 
 
 12771458 
 
 98 
 
 99 
 
 2052864 
 
 2827539 
 
 3803184 
 
 5011875 
 
 6488064 
 
 8268579 
 
 10392624 
 
 12901779 
 
 99 
 
 100 
 
 2073600 
 
 2856100 
 
 3841600 
 
 5062500 
 
 6553600 
 
 8352100 
 
 10497600 
 
 13032100 
 
 100 
 
 14 
 
106 
 
 Tables for Statisticians and Biometricians 
 
 TABLE L— {continued). 
 Ordinate 2—7. Frequency 101—150. 
 
 n 
 
 x = 2 
 
 x = 3 
 
 X = i 
 
 x=5 
 
 x = & 
 
 x = 7 
 
 n 
 
 101 
 
 1616 
 
 8181 
 
 25856 
 
 63125 
 
 130896 
 
 242501 
 
 101 
 
 102 
 
 1632 
 
 8262 
 
 26112 
 
 63750 
 
 132192 
 
 244902 
 
 102 
 
 10S 
 
 1648 
 
 8343 
 
 26368 
 
 64375 
 
 133488 
 
 247303 
 
 108 
 
 10 If 
 
 1664 
 
 8424 
 
 26624 
 
 65000 
 
 134784 
 
 249704 
 
 104 
 
 105 
 
 1680 
 
 8505 
 
 26880 
 
 65625 
 
 136080 
 
 252105 
 
 105 
 
 106 
 
 1696 
 
 8586 
 
 27136 
 
 66250 
 
 137376 
 
 254506 
 
 106 
 
 107 
 
 1712 
 
 8667 
 
 27392 
 
 66875 
 
 138672 
 
 256907 
 
 107 
 
 108 
 
 1728 
 
 8748 
 
 27648 
 
 67500 
 
 139968 
 
 259308 
 
 108 
 
 109 
 
 1744 
 
 8829 
 
 27904 
 
 68125 
 
 141264 
 
 261709 
 
 109 
 
 110 
 
 1760 
 
 8910 
 
 28160 
 
 68750 
 
 142560 
 
 264110 
 
 110 
 
 111 
 
 1776 
 
 8991 
 
 28416 
 
 69375 
 
 143856 
 
 266511 
 
 111 
 
 IIS 
 
 1792 
 
 9072 
 
 28672 
 
 70000 
 
 145152 
 
 268912 
 
 112 
 
 113 
 
 1808 
 
 9153 
 
 28928 
 
 70625 
 
 146448 
 
 271313 
 
 US 
 
 114 
 
 1824 
 
 9234 
 
 29184 
 
 71250 
 
 147744 
 
 273714 
 
 114 
 
 115 
 
 1840 
 
 9315 
 
 29440 
 
 71875 
 
 149040 
 
 276115 
 
 115 
 
 116 
 
 1856 
 
 9396 
 
 29696 
 
 72500 
 
 150336 
 
 278516 
 
 116 
 
 117 
 
 1872 
 
 9477 
 
 29952 
 
 73125 
 
 151632 
 
 280917 
 
 117 
 
 118 
 
 1888 
 
 9558 
 
 30208 
 
 73750 
 
 152928 
 
 283318 
 
 118 
 
 119 
 
 1904 
 
 9639 
 
 30464 
 
 74375 
 
 154224 
 
 285719 
 
 119 
 
 120 
 
 1920 
 
 9720 
 
 30720 
 
 75000 
 
 155520 
 
 288120 
 
 120 
 
 121 
 
 1936 
 
 9801 
 
 30976 
 
 75625 
 
 156816 
 
 290521 
 
 121 
 
 122 
 
 1952 
 
 9882 
 
 31232 
 
 76250 
 
 158112 
 
 292922 
 
 1M 
 
 123 
 
 1968 
 
 9963 
 
 31488 
 
 76875 
 
 159408 
 
 295323 
 
 123 
 
 124, 
 
 1984 
 
 10044 
 
 31744 
 
 77500 
 
 160704 
 
 297724 
 
 124 
 
 125 
 
 2000 
 
 10125 
 
 32000 
 
 78125 
 
 162000 
 
 300125 
 
 125 
 
 126 
 
 2016 
 
 10206 
 
 32256 
 
 78750 
 
 163296 
 
 302526 
 
 126 
 
 127 
 
 2032 
 
 10287 
 
 32512 
 
 79375 
 
 164592 
 
 304927 
 
 127 
 
 128 
 
 2048 
 
 10368 
 
 32768 
 
 80000 
 
 165888 
 
 307328 
 
 128 
 
 129 
 
 2064 
 
 10449 
 
 33024 
 
 80625 
 
 167184 
 
 309729 
 
 129 
 
 130 
 
 2080 
 
 10530 
 
 33280 
 
 81250 
 
 168480 
 
 312130 
 
 ISO 
 
 131 
 
 2096 
 
 10611 
 
 33536 
 
 81875 
 
 169776 
 
 314531 
 
 131 
 
 132 
 
 2112 
 
 10692 
 
 33792 
 
 82500 
 
 171072 
 
 316932 
 
 132 
 
 133 
 
 2128 
 
 10773 
 
 34048 
 
 83125 
 
 172368 
 
 319333 
 
 133 
 
 134 
 
 2144 
 
 10854 
 
 34304 
 
 83750 
 
 173364 
 
 321734 
 
 134 
 
 135 
 
 2160 
 
 10935 
 
 34560 
 
 84375 
 
 174960 
 
 324135 
 
 135 
 
 136 
 
 2176 
 
 11016 
 
 34816 
 
 85000 
 
 176256 
 
 326536 
 
 136 
 
 137 
 
 2192 
 
 11097 
 
 35072 
 
 85625 
 
 177552 
 
 328937 
 
 137 
 
 138 
 
 2208 
 
 11178 
 
 35328 
 
 86250 
 
 178848 
 
 331338 
 
 138 
 
 139 
 
 2224 
 
 11259 
 
 35584 
 
 86875 
 
 180144 
 
 333739 
 
 139 
 
 U0 
 
 2240 
 
 11340 
 
 35840 
 
 87500 
 
 181440 
 
 336140 
 
 140 
 
 141 
 
 2256 
 
 11421 
 
 36096 
 
 88125 
 
 182736 
 
 338541 
 
 141 
 
 142 
 
 2272 
 
 11502 
 
 36352 
 
 88750 
 
 184032 
 
 340942 
 
 142 
 
 143 
 
 2288 
 
 11583 ' 
 
 36608 
 
 89375 
 
 185328 
 
 343343 
 
 143 
 
 144 
 
 2304 
 
 11664 
 
 36864 
 
 90000 
 
 186624 
 
 345744 
 
 144 
 
 145 
 
 2320 
 
 11745 
 
 37120 
 
 90625 
 
 187920 
 
 348145 
 
 145 
 
 146 
 
 2336 
 
 11826 
 
 37376 
 
 91250 
 
 189216 
 
 350546 
 
 146 
 
 147 
 
 2352 
 
 11907 
 
 37632 
 
 91875 
 
 190512 
 
 352947 
 
 147 
 
 148 
 
 2368 
 
 11988 
 
 37888 
 
 92500 
 
 191808 
 
 355348 
 
 148 
 
 149 
 
 2384 
 
 12069 
 
 38144 
 
 93125 
 
 193104 
 
 357749 
 
 149 
 
 150 
 
 2400 
 
 12150 
 
 38400 
 
 93750 
 
 194400 
 
 360150 
 
 150 
 
Verification of the Fourth Moment 
 
 107 
 
 TABLE L— (continued). 
 Ordinate 8—14. Frequency 101—150. 
 
 n 
 
 x = S 
 
 x=9 
 
 s = ll 
 
 a; = 12 
 
 x=13 
 
 x = 14 
 
 n 
 
 101 
 
 413696 
 
 602661 
 
 1478741 
 
 2094336 
 
 2884661 
 
 3880016 
 
 101 
 
 102 
 
 417792 
 
 669222 
 
 1493382 
 
 2115072 
 
 2913222 
 
 3918432 
 
 102 
 
 103 
 
 421888 
 
 675783 
 
 1508023 
 
 2135808 
 
 2941783 
 
 3956848 
 
 103 
 
 104 
 
 425984 
 
 682344 
 
 1522664 
 
 2156544 
 
 2970344 
 
 3995264 
 
 104 
 
 105 
 
 430080 
 
 688905 
 
 1537305 
 
 2177280 
 
 2998905 
 
 4033680 
 
 105 
 
 10G 
 
 434176 
 
 695466 
 
 1551946 
 
 2198016 
 
 3027466 
 
 4072096 
 
 106 
 
 107 
 
 438272 
 
 702027 
 
 1566587 
 
 2218752 
 
 3056027 
 
 4110512 
 
 107 
 
 108 
 
 442368 
 
 708588 
 
 1581228 
 
 2239488 
 
 3084588 
 
 4148928 
 
 108 
 
 109 
 
 446464 
 
 715149 
 
 1595869 
 
 2260224 
 
 3113149 
 
 4187344 
 
 109 
 
 110 
 
 450560 
 
 721710 
 
 1610510 
 
 2280960 
 
 3141710 
 
 4225760 
 
 no 
 
 lit 
 
 454656 
 
 728271 
 
 1625151 
 
 2301696 
 
 3170271 
 
 4264176 
 
 HI 
 
 112 
 
 458752 
 
 734832 
 
 1639792 
 
 2322432 
 
 3198832 
 
 4302592 
 
 112 
 
 US 
 
 462848 
 
 741393 
 
 1654433 
 
 2343168 
 
 3227393 
 
 4341008 
 
 113 
 
 m 
 
 466944 
 
 747954 
 
 1669074 
 
 2363904 
 
 3255954 
 
 4379424 
 
 114 
 
 115 
 
 471040 
 
 754515 
 
 1683715 
 
 2384640 
 
 3284515 
 
 4417840 
 
 115 
 
 116 
 
 475136 
 
 761076 
 
 1698356 
 
 2405376 
 
 3313076 
 
 4456256 
 
 116 
 
 117 
 
 479232 
 
 767637 
 
 1712997 
 
 2426112 
 
 3341637 
 
 4494672 
 
 117 
 
 118 
 
 483328 
 
 774198 
 
 1727638 
 
 2446848 
 
 3370198 
 
 4533088 
 
 118 
 
 119 
 
 487424 
 
 780759 
 
 1742279 
 
 2467584 
 
 3398759 
 
 4571504 
 
 119 
 
 120 
 
 491520 
 
 787320 
 
 1756920 
 
 2488320 
 
 3427320 
 
 4609920 
 
 120 
 
 121 
 
 495616 
 
 793881 
 
 1771561 
 
 2509056 
 
 3455881 
 
 4648336 
 
 121 
 
 122 
 
 499712 
 
 800442 
 
 1786202 
 
 2529792 
 
 3484442 
 
 4686752 
 
 122 
 
 123 
 
 503808 
 
 807003 
 
 1800843 
 
 2550528 
 
 3513003 
 
 4725168 
 
 123 
 
 124 
 
 507904 
 
 813564 
 
 1815484 
 
 2571264 
 
 3541564 
 
 4763584 
 
 124 
 
 125 
 
 512000 
 
 820125 
 
 1830125 
 
 2592000 
 
 3570125 
 
 4802000 
 
 125 
 
 126 
 
 516096 
 
 826686 
 
 1844766 
 
 2612736 
 
 3598686 
 
 4840416 
 
 126 
 
 127 
 
 520192 
 
 833247 
 
 1859407 
 
 2633472 
 
 3627247 
 
 4878832 
 
 127 
 
 128 
 
 524288 
 
 839808 
 
 1874048 
 
 2654208 
 
 3655808 
 
 4917248 
 
 128 
 
 129 
 
 528384 
 
 846369 
 
 1888689 
 
 2674944 
 
 3684369 
 
 4955664 
 
 129 
 
 130 
 
 532480 
 
 852930 
 
 1903330 
 
 2695680 
 
 3712930 
 
 4994080 
 
 130 
 
 131 
 
 536576 
 
 859491 
 
 1917971 
 
 2716416 
 
 3741491 
 
 5032496 
 
 131 
 
 132 
 
 540672 
 
 866052 
 
 1932612 
 
 2737152 
 
 3770052 
 
 5070912 
 
 132 
 
 133 
 
 544768 
 
 872613 
 
 1947253 
 
 2757888 
 
 3798613 
 
 5109328 
 
 133 
 
 134 
 
 548864 
 
 879174 
 
 1961894 
 
 2778624 
 
 3827174 
 
 5147744 
 
 134 
 
 135 
 
 552960 
 
 885735 
 
 1976535 
 
 2799360 
 
 3855735 
 
 5186160 
 
 135 
 
 136 
 
 557056 
 
 892296 
 
 1991176 
 
 2820096 
 
 3884296 
 
 5224576 
 
 136 
 
 1S7 
 
 561152 
 
 898857 
 
 2005817 
 
 2840832 
 
 3912857 
 
 5262992 
 
 137 
 
 138 
 
 565248 
 
 905418 
 
 2020458 
 
 2861568 
 
 3941418 
 
 5301408 
 
 138 
 
 139 
 
 569344 
 
 911979 
 
 2035099 
 
 2882304 
 
 3969979 
 
 5339824 
 
 139 
 
 140 
 
 573440 
 
 918540 
 
 2049740 
 
 2903040 
 
 3998540 
 
 5378240 
 
 140 
 
 141 
 
 577536 
 
 925101 
 
 2064381 
 
 2923776 
 
 4027101 
 
 5416656 
 
 141 
 
 142 
 
 581632 
 
 931662 
 
 2079022 
 
 2944512 
 
 4055662 
 
 5455072 
 
 1)& 
 
 143 
 
 585728 
 
 938223 
 
 2093663 
 
 2965248 
 
 4084223 
 
 5493488 
 
 143 
 
 144 
 
 589824 
 
 944784 
 
 2108304 
 
 2985984 
 
 4112784 
 
 5531904 
 
 144 
 
 145 
 
 593920 
 
 951345 
 
 2122945 
 
 3006720 
 
 4141345 
 
 5570320 
 
 145 
 
 146 
 
 598016 
 
 957906 
 
 2137586 
 
 3027456 
 
 4169906 
 
 5608736 
 
 146 
 
 147 
 
 602112 
 
 964467 
 
 2152227 
 
 3048192 
 
 4198467 
 
 5647152 
 
 147 
 
 148 
 
 606208 
 
 971028 
 
 2166868 
 
 3068928 
 
 4227028 
 
 5685568 
 
 148 
 
 149 
 
 610304 
 
 977589 
 
 2181509 
 
 3089664 
 
 4255589 
 
 5723984 
 
 149 
 
 150 
 
 614400 
 
 984150 
 
 2196150 
 
 3110400 
 
 4284150 
 
 5762400 
 
 150 
 
 14—2 
 
108 
 
 Tables for Statisticians and Biometricians 
 
 TABLE h— (continued). 
 Ordinate 2—12. Frequency 151—200. 
 
 n 
 
 x = 2 
 
 x = 3 
 
 x = 4 
 
 x = 5 
 
 x = 6 
 
 x = 7 
 
 x = 8 
 
 x=9 
 
 x=ll 
 
 x = l'2 
 
 ii 
 
 151 
 
 2416 
 
 12231 
 
 38656 
 
 94375 
 
 195696 
 
 362551 
 
 618496 
 
 990711 
 
 2210791 
 
 3131136 
 
 151 
 
 152 
 
 2432 
 
 12312 
 
 38912 
 
 95000 
 
 196992 
 
 364952 
 
 622592 
 
 997272 
 
 2225432 
 
 3151872 
 
 152 
 
 153 
 
 2448 
 
 12393 
 
 39168 
 
 95625 
 
 198288 
 
 367353 
 
 626688 
 
 1003833 
 
 2240073 
 
 3172608 
 
 153 
 
 154 
 
 2464 
 
 12474 
 
 39424 
 
 96250 
 
 199584 
 
 369754 
 
 630784 
 
 1010394 
 
 2254714 
 
 3193344 
 
 154 
 
 155 
 
 2480 
 
 12555 
 
 39680 
 
 96875 
 
 200880 
 
 372155 
 
 634880 
 
 1016955 
 
 2269355 
 
 3214080 
 
 155 
 
 156 
 
 2496 
 
 12636 
 
 39936 
 
 97500 
 
 202176 
 
 374556 
 
 638976 
 
 1023516 
 
 2283996 
 
 3234816 
 
 156 
 
 157 
 
 2512 
 
 12717 
 
 40192 
 
 98125 
 
 203472 
 
 376957 
 
 643072 
 
 1030077 
 
 2298637 
 
 3255552 
 
 157 
 
 158 
 
 2528 
 
 12798 
 
 40448 
 
 98750 
 
 204768 
 
 379358 
 
 647168 
 
 1036638 
 
 2313278 
 
 3276288 
 
 158 
 
 159 
 
 2544 
 
 12879 
 
 40704 
 
 99375 
 
 206064 
 
 381759 
 
 651264 
 
 1013199 
 
 2327919 
 
 3297024 
 
 159 
 
 160 
 
 2560 
 
 12960 
 
 40960 
 
 100000 
 
 207360 
 
 384160 
 
 655360 
 
 1049760 
 
 2342560 
 
 3317760 
 
 160 
 
 161 
 
 2576 
 
 13041 
 
 41216 
 
 100625 
 
 208656 
 
 386561 
 
 659456 
 
 1056321 
 
 2357201 
 
 3338496 
 
 161 
 
 162 
 
 2592 
 
 13122 
 
 41472 
 
 101250 
 
 209952 
 
 388962 
 
 663552 
 
 1062882 
 
 2371842 
 
 3359232 
 
 10 J 
 
 163 
 
 2608 
 
 13203 
 
 41728 
 
 101875 
 
 211248 
 
 391363 
 
 667648 
 
 1069443 
 
 2386483 
 
 3379968 
 
 163 
 
 164 
 
 2624 
 
 13284 
 
 41984 
 
 102500 
 
 212544 
 
 393764 
 
 671744 
 
 1076004 
 
 2401124 
 
 3400704 
 
 164 
 
 165 
 
 2680 
 
 13365 
 
 42240 
 
 103125 
 
 213840 
 
 396165 
 
 675840 
 
 1082565 
 
 2415765 
 
 3421440 
 
 165 
 
 166 
 
 2656 
 
 13446 
 
 42496 
 
 103750 
 
 215136 
 
 398566 
 
 679936 
 
 1089126 
 
 2430406 
 
 3442176 
 
 lot; 
 
 167 
 
 2672 
 
 13527 
 
 42752 
 
 104375 
 
 216432 
 
 400967 
 
 684032 
 
 1095687 
 
 2445047 
 
 3462912 
 
 167 
 
 168 
 
 2688 
 
 13608 
 
 43008 
 
 105000 
 
 217728 
 
 403368 
 
 688128 
 
 1102248 
 
 2459688 
 
 3483648 
 
 168 
 
 169 
 
 2704 
 
 13689 
 
 43264 
 
 105625 
 
 219024 
 
 405769 
 
 692224 
 
 1108809 
 
 2474329 
 
 3504384 
 
 169 
 
 170 
 
 2720 
 
 13770 
 
 43520 
 
 106250 
 
 220320 
 
 408170 
 
 696320 
 
 1115370 
 
 2488970 
 
 3525120 
 
 170 
 
 171 
 
 2736 
 
 13851 
 
 43776 
 
 106875 
 
 221616 
 
 410571 
 
 700416 
 
 1121931 
 
 2503611 
 
 3545856 
 
 171 
 
 172 
 
 2752 
 
 13932 
 
 44032 
 
 107500 
 
 222912 
 
 412972 
 
 704512 
 
 1128492 
 
 2518252 
 
 3566592 
 
 172 
 
 173 
 
 2768 
 
 14013 
 
 44288 
 
 108125 
 
 224208 
 
 415373 
 
 708608 
 
 1135053 
 
 2532893 
 
 3587328 
 
 173 
 
 174 
 
 2784 
 
 14094 
 
 44544 
 
 108750 
 
 225504 
 
 417774 
 
 712704 
 
 1141614 
 
 2547534 
 
 3608064 
 
 174 
 
 175 
 
 2800 
 
 14175 
 
 44800 
 
 109375 
 
 226800 
 
 420175 
 
 716800 
 
 1148175 
 
 2562175 
 
 3628800 
 
 175 
 
 176 
 
 2816 
 
 14256 
 
 45056 
 
 110000 
 
 228096 
 
 422576 
 
 720896 
 
 1154736 
 
 2576816 
 
 3649536 
 
 176 
 
 177 
 
 2832 
 
 14337 
 
 45312 
 
 110625 
 
 229392 
 
 424977 
 
 724992 
 
 1161297 
 
 2591457 
 
 3670272 
 
 177 
 
 178 
 
 2848 
 
 14418 
 
 45568 
 
 111250 
 
 230688 
 
 427378 
 
 729088 
 
 1167858 
 
 2606098 
 
 3691008 
 
 178 
 
 179 
 
 2864 
 
 14499 
 
 45824 
 
 111875 
 
 231984 
 
 429779 
 
 733184 
 
 1174419 
 
 2620739 
 
 3711744 
 
 179 
 
 180 
 
 2880 
 
 14580 
 
 46080 
 
 112500 
 
 233280 
 
 432180 
 
 737280 
 
 1180980 
 
 2635380 
 
 3732480 
 
 180 
 
 181 
 
 2896 
 
 14661 
 
 46336 
 
 113125 
 
 234576 
 
 434581 
 
 741376 
 
 1187541 
 
 2650021 
 
 3753216 
 
 181 
 
 182 
 
 2912 
 
 14742 
 
 46592 
 
 113750 
 
 235872 
 
 436982 
 
 745472 
 
 1194102 
 
 2664662 
 
 3773952 
 
 182 
 
 183 
 
 2928 
 
 14823 
 
 46848 
 
 114375 
 
 237168 
 
 439383 
 
 749568 
 
 1200663 
 
 2679303 
 
 3794688 
 
 183 
 
 184 
 
 2944 
 
 14904 
 
 47104 
 
 115000 
 
 238464 
 
 441784 
 
 753664 
 
 1207224 
 
 2693944 
 
 3815424 
 
 184 
 
 185 
 
 2960 
 
 14985 
 
 47360 
 
 115625 
 
 239760 
 
 444185 
 
 757760 
 
 1213785 
 
 2708585 
 
 3836160 
 
 185 
 
 186 
 
 2976 
 
 15066 
 
 47616 
 
 116250 
 
 241056 
 
 446586 
 
 761856 
 
 1220346 
 
 2783828 
 
 3856896 
 
 186 
 
 187 
 
 2992 
 
 15147 
 
 47872 
 
 116875 
 
 242352 
 
 448987 
 
 765952 
 
 1226907 
 
 2737867 
 
 3877632 
 
 187 
 
 188 
 
 3008 
 
 15228 
 
 48128 
 
 117500 
 
 243648 
 
 451388 
 
 770048 
 
 1233468 
 
 2752508 
 
 3898368 
 
 188 
 
 189 
 
 3024 
 
 15309 
 
 48384 
 
 118125 
 
 244944 
 
 453789 
 
 774144 
 
 1240029 
 
 2767149 
 
 3919104 
 
 189 
 
 190 
 
 3040 
 
 15390 
 
 48640 
 
 118750 
 
 246240 
 
 456190 
 
 778240 
 
 1246590 
 
 2781790 
 
 3939840 
 
 190 
 
 191 
 
 3056 
 
 15471 
 
 48896 
 
 119375 
 
 247536 
 
 458591 
 
 782336 
 
 1253151 
 
 2796431 
 
 3960576 
 
 191 
 
 192 
 
 3072 
 
 15552 
 
 49152 
 
 120000 
 
 248832 
 
 460992 
 
 786432 
 
 1259712 
 
 2811072 
 
 3981312 
 
 192 
 
 193 
 
 3088 
 
 15633 
 
 49408 
 
 120625 
 
 250128 
 
 463393 
 
 790528 
 
 1266273 
 
 2825713 
 
 4002048 
 
 193 
 
 194 
 
 3104 
 
 15714 
 
 49664 
 
 121250 
 
 251424 
 
 465794 
 
 794624 
 
 1272834 
 
 2840354 
 
 4022784 
 
 194 
 
 195 
 
 3120 
 
 15795 
 
 49920 
 
 121875 
 
 252720 
 
 468195 
 
 798720 
 
 1279395 
 
 2854995 
 
 4043520 
 
 195 
 
 196 
 
 3136 
 
 15876 
 
 50176 
 
 122500 
 
 254016 
 
 470596 
 
 802816 
 
 1285956 
 
 2869636 
 
 4064256 
 
 196 
 
 197 
 
 3152 
 
 15957 
 
 50432 
 
 123125 
 
 255312 
 
 472997 
 
 806912 
 
 1292517 
 
 2884277 
 
 4084992 
 
 197 
 
 198 
 
 3168 
 
 16038 
 
 50688 
 
 123750 
 
 256608 
 
 475398 
 
 811508 
 
 1299078 
 
 2898918 
 
 4105728 
 
 19S 
 
 199 
 
 3184 
 
 16119 
 
 50944 
 
 128375 
 
 257904 
 
 477799 
 
 815104 
 
 1305639 
 
 2913559 
 
 4126464 
 
 199 
 
 200 
 
 3200 
 
 16200 
 
 51200 
 
 125000 
 
 259200 
 
 480200 
 
 819200 
 
 1312200 
 
 2928200 
 
 4147200 
 
 200 
 
Verification of the Fourth Moment 
 
 109 
 
 TABLE L— (continued). 
 
 Ordinate 2—11. Frequency 201—250. 
 
 n 
 
 x = 1 
 
 x = 3 
 
 i = 4 
 
 x = 5 
 
 a: = 6 
 
 x = 7 
 
 x=8 
 
 <E = 9 
 
 *=11 
 
 n 
 
 201 
 
 3216 
 
 16281 
 
 51456 
 
 125625 
 
 260496 
 
 482601 
 
 823296 
 
 1318761 
 
 2942841 
 
 201 
 
 202 
 
 3232 
 
 16362 
 
 51712 
 
 126250 
 
 261792 
 
 485002 
 
 827392 
 
 1325322 
 
 2957482 
 
 202 
 
 203 
 
 3248 
 
 16443 
 
 51968 
 
 126875 
 
 263088 
 
 487403 
 
 831488 
 
 1331883 
 
 2972123 
 
 203 
 
 204 
 
 3264 
 
 16524 
 
 52224 
 
 127500 
 
 264384 
 
 489804 
 
 835584 
 
 1338444 
 
 2986764 
 
 204 
 
 205 
 
 3280 
 
 16605 
 
 52480 
 
 128125 
 
 265680 
 
 492205 
 
 839680 
 
 1345005 
 
 3001405 
 
 205 
 
 206 
 
 3296 
 
 16686 
 
 52736 
 
 128750 
 
 266976 
 
 494606 
 
 843776 
 
 1351566 
 
 3016046 
 
 206 
 
 207 
 
 3312 
 
 16767 
 
 52992 
 
 129375 
 
 268272 
 
 497007 
 
 847872 
 
 1358127 
 
 3030687 
 
 207 
 
 208 
 
 3328 
 
 16848 
 
 53248 
 
 130000 
 
 269568 
 
 499408 
 
 851968 
 
 1364688 
 
 3045328 
 
 208 
 
 209 
 
 3344 
 
 16929 
 
 53504 
 
 130625 
 
 270864 
 
 501809 
 
 856064 
 
 1371249 
 
 3059969 
 
 209 
 
 210 
 
 3360 
 
 17010 
 
 53760 
 
 131250 
 
 272160 
 
 504210 
 
 860160 
 
 1377810 
 
 3074610 
 
 210 
 
 211 
 
 3376 
 
 17091 
 
 54016 
 
 131875 
 
 273456 
 
 506611 
 
 864256 
 
 1384371 
 
 3089251 
 
 211 
 
 212 
 
 3392 
 
 17172 
 
 54272 
 
 132500 
 
 274752 
 
 509012 
 
 868352 
 
 1390932 
 
 3103892 
 
 212 
 
 213 
 
 3408 
 
 17253 
 
 54528 
 
 133125 
 
 276048 
 
 511413 
 
 872448 
 
 1397493 
 
 3118533 
 
 213 
 
 214 
 
 3424 
 
 17334 
 
 54784 
 
 133750 
 
 277344 
 
 513814 
 
 876544 
 
 1404054 
 
 3133174 
 
 214 
 
 215 
 
 3440 
 
 17415 
 
 55040 
 
 134375 
 
 278640 
 
 516215 
 
 880640 
 
 1410615 
 
 3147815 
 
 215 
 
 216 
 
 3456 
 
 17496 
 
 55296 
 
 135000 
 
 279936 
 
 518616 
 
 884736 
 
 1417176 
 
 3162456 
 
 216 
 
 217 
 
 3472 
 
 17577 
 
 55552 
 
 135625 
 
 281232 
 
 521017 
 
 888832 
 
 1423737 
 
 3177097 
 
 217 
 
 218 
 
 3488 
 
 17658 
 
 55808 
 
 136250 
 
 282528 
 
 523418 
 
 892928 
 
 1430298 
 
 3191738 
 
 218 
 
 219 
 
 3504 
 
 17739 
 
 56064 
 
 136875 
 
 283824 
 
 525819 
 
 897024 
 
 1436859 
 
 3206379 
 
 219 
 
 220 
 
 3520 
 
 17820 
 
 56320 
 
 137500 
 
 285120 
 
 528220 
 
 901120 
 
 1443420 
 
 3221020 
 
 220 
 
 221 
 
 3536 
 
 17901 
 
 56576 
 
 138125 
 
 286416 
 
 530621 
 
 905216 
 
 1449981 
 
 3235661 
 
 221 
 
 222 
 
 3552 
 
 17982 
 
 56832 
 
 138750 
 
 287712 
 
 533022 
 
 909312 
 
 1456542 
 
 3250302 
 
 222 
 
 22S 
 
 3568 
 
 18063 
 
 57088 
 
 139375 
 
 289008 
 
 535423 
 
 913408 
 
 1463103 
 
 3264943 
 
 223 
 
 224 
 
 3584 
 
 18144 
 
 57344 
 
 140000 
 
 290304 
 
 537824 
 
 917504 
 
 1469664 
 
 3279584 
 
 224 
 
 225 
 
 3600 
 
 18225 
 
 57600 
 
 140625 
 
 291600 
 
 540225 
 
 921600 
 
 1476225 
 
 3294225 
 
 225 
 
 226 
 
 3616 
 
 18306 
 
 57856 
 
 141250 
 
 292896 
 
 542626 
 
 925696 
 
 1482786 
 
 3308866 
 
 226 
 
 227 
 
 3632 
 
 18387 
 
 58112 
 
 141875 
 
 294192 
 
 545027 
 
 929792 
 
 1489347 
 
 3323507 
 
 227 
 
 228 
 
 3648 
 
 18468 
 
 58368 
 
 142500 
 
 295488 
 
 547428 
 
 933888 
 
 1495908 
 
 3338148 
 
 228 
 
 229 
 
 3664 
 
 18549 
 
 58624 
 
 143125 
 
 296784 
 
 549829 
 
 937984 
 
 1502469 
 
 3352789 
 
 229 
 
 230 
 
 3680 
 
 18630 
 
 58880 
 
 143750 
 
 298080 
 
 552230 
 
 942080 
 
 1509030 
 
 3367430 
 
 230 
 
 231 
 
 3696 
 
 18711 
 
 59136 
 
 144375 
 
 299376 
 
 554631 
 
 946176 
 
 1515591 
 
 3382071 
 
 231 
 
 232 
 
 3712 
 
 18792 
 
 59392 
 
 145000 
 
 300672 
 
 557032 
 
 950272 
 
 1522152 
 
 3396712 
 
 232 
 
 233 
 
 3728 
 
 18873 
 
 59648 
 
 145625 
 
 301968 
 
 559433 
 
 954368 
 
 1528713 
 
 3411353 
 
 233 
 
 234 
 
 3744 
 
 18954 
 
 59904 
 
 146250 
 
 303264 
 
 561834 
 
 958464 
 
 1535274 
 
 3425994 
 
 234 
 
 235 
 
 3760 
 
 19035 
 
 60160 
 
 146875 
 
 304560 
 
 564235 
 
 962560 
 
 1541835 
 
 3440635 
 
 235 
 
 236 
 
 3776 
 
 19116 
 
 60416 
 
 147500 
 
 305856 
 
 566636 
 
 966656 
 
 1548396 
 
 3455276 
 
 236 
 
 237 
 
 3792 
 
 19147 
 
 60672 
 
 148125 
 
 307152 
 
 569037 
 
 970752 
 
 1554957 
 
 3469917 
 
 237 
 
 238 
 
 3808 
 
 19278 
 
 60928 
 
 148750 
 
 308448 
 
 571438 
 
 974848 
 
 1561518 
 
 3484558 
 
 238 
 
 239 
 
 3824 
 
 19359 
 
 61184 
 
 149375 
 
 309744 
 
 573839 
 
 978944 
 
 1568079 
 
 3499199 
 
 239 
 
 240 
 
 3840 
 
 19440 
 
 61440 
 
 150000 
 
 311040 
 
 576240 
 
 983040 
 
 1574640 
 
 3613840 
 
 240 
 
 241 
 
 3856 
 
 19521 
 
 61696 
 
 150625 
 
 312336 
 
 578641 
 
 987136 
 
 1581201 
 
 3528481 
 
 241 
 
 242 
 
 3872 
 
 19602 
 
 61952 
 
 151250 
 
 313632 
 
 581042 
 
 991232 
 
 1587762 
 
 3543122 
 
 242 
 
 243 
 
 3888 
 
 19683 
 
 62208 
 
 151875 
 
 314928 
 
 583443 
 
 995328 
 
 1594323 
 
 3557763 
 
 243 
 
 244 
 
 3904 
 
 19764 
 
 62464 
 
 152500 
 
 316224 
 
 585844 
 
 999424 
 
 1600884 
 
 3572404 
 
 244 
 
 245 
 
 3920 
 
 19845 
 
 62720 
 
 153125 
 
 317520 
 
 588245 
 
 1003520 
 
 1607445 
 
 3587045 
 
 245 
 
 246 
 
 3936 
 
 19926 
 
 62976 
 
 153750 
 
 318816 
 
 590646 
 
 1007616 
 
 1614006 
 
 3601686 
 
 246 
 
 247 
 
 3952 
 
 20007 
 
 68832 
 
 154375 
 
 320112 
 
 593047 
 
 1011712 
 
 1620567 
 
 3616327 
 
 247 
 
 248 
 
 3968 
 
 20088 
 
 63488 
 
 155000 
 
 321408 
 
 595448 
 
 1015808 
 
 1627128 
 
 3630968 
 
 248 
 
 249 
 
 3984 
 
 20169 
 
 63744 
 
 155625 
 
 322704 
 
 597849 
 
 1019904 
 
 1633689 
 
 3645609 
 
 249 
 
 250 
 
 4000 
 
 20250 
 
 64000 
 
 156250 
 
 324000 
 
 600250 
 
 1024000 
 
 1640250 
 
 3660250 
 
 250 
 
110 
 
 Tables for Statisticians and Biotnetricians 
 
 TABLE L— (continued). 
 
 Ordinate 2—9. Frequency 251—300. 
 
 n 
 
 x=2 
 
 * = 3 
 
 x = i 
 
 x = 5 
 
 x — 6 
 
 x = 7 
 
 i = 8 
 
 3 = 9 
 
 n 
 
 251 
 
 4016 
 
 20331 
 
 64256 
 
 156875 
 
 325296 
 
 602651 
 
 1028096 
 
 1646811 
 
 251 
 
 252 
 
 4032 
 
 20412 
 
 64512 
 
 157500 
 
 326592 
 
 605052 
 
 1032192 
 
 1653372 
 
 252 
 
 253 
 
 4048 
 
 20493 
 
 64768 
 
 158125 
 
 327888 
 
 607453 
 
 1036288 
 
 1659933 
 
 258 
 
 25Jf 
 
 4064 
 
 20574 
 
 65024 
 
 158750 
 
 329184 
 
 609854 
 
 1040384 
 
 1666494 
 
 254 
 
 255 
 
 40S0 
 
 20655 
 
 65280 
 
 159375 
 
 330480 
 
 612255 
 
 1044480 
 
 1673055 
 
 255 
 
 256 
 
 4096 
 
 20736 
 
 65536 
 
 160000 
 
 331776 
 
 614656 
 
 1048576 
 
 1679616 
 
 256 
 
 257 
 
 4112 
 
 20817 
 
 65792 
 
 160625 
 
 333072 
 
 617057 
 
 1052672 
 
 1686177 
 
 257 
 
 258 
 
 4128 
 
 20S98 
 
 66048 
 
 161250 
 
 334368 
 
 619458 
 
 1056768 
 
 1692738 
 
 258 
 
 259 
 
 4144 
 
 20979 
 
 66304 
 
 161875 
 
 335664 
 
 621859 
 
 1060864 
 
 1699299 
 
 969 
 
 260 
 
 4160 
 
 21060 
 
 66560 
 
 162500 
 
 336960 
 
 624260 
 
 1064960 
 
 1705860 
 
 260 
 
 261 
 
 4176 
 
 21141 
 
 66816 
 
 163125 
 
 338256 
 
 626661 
 
 1069056 
 
 1712421 
 
 261 
 
 262 
 
 4192 
 
 21222 
 
 67072 
 
 163750 
 
 339552 
 
 629062 
 
 1073152 
 
 1718982 
 
 262 
 
 263 
 
 4208 
 
 21303 
 
 67328 
 
 164375 
 
 340848 
 
 631463 
 
 1077248 
 
 1725543 
 
 263 
 
 26J f 
 
 4224 
 
 21384 
 
 67584 
 
 165000 
 
 342144 
 
 633864 
 
 1081344 
 
 1732104 
 
 264 
 
 265 
 
 4240 
 
 21465 
 
 67840 
 
 165625 
 
 343440 
 
 636265 
 
 1085440 
 
 1738665 
 
 265 
 
 266 
 
 4256 
 
 21546 
 
 68096 
 
 166250 
 
 344736 
 
 638666 
 
 1089536 
 
 1745226 
 
 266 
 
 267 
 
 4272 
 
 21627 
 
 68352 
 
 166875 
 
 346032 
 
 641067 
 
 1093632 
 
 1751787 
 
 267 
 
 268 
 
 4288 
 
 21708 
 
 68608 
 
 167500 
 
 347328 
 
 643468 
 
 1097728 
 
 1758348 
 
 268 
 
 269 
 
 4304 
 
 21789 
 
 68864 
 
 168125 
 
 348624 
 
 645869 
 
 1101824 
 
 1764909 
 
 269 
 
 270 
 
 4320 
 
 21870 
 
 69120 
 
 168750 
 
 349920 
 
 648270 
 
 1105920 
 
 1771470 
 
 270 
 
 271 
 
 4336 
 
 21951 
 
 69376 
 
 169375 
 
 351216 
 
 650671 
 
 1110016 
 
 1778031 
 
 271 
 
 272 
 
 4352 
 
 22032 
 
 69632 
 
 170000 
 
 352512 
 
 653072 
 
 1114112 
 
 1784592 
 
 272 
 
 273 
 
 4368 
 
 22113 
 
 69888 
 
 170625 
 
 353808 
 
 655473 
 
 1118208 
 
 1791153 
 
 278 
 
 274 
 
 4384 
 
 22194 
 
 70144 
 
 171250 
 
 355104 
 
 657874 
 
 1122304 
 
 1797714 
 
 27b 
 
 275 
 
 4400 
 
 22275 
 
 70400 
 
 171875 
 
 356400 
 
 660275 
 
 1126400 
 
 1804275 
 
 275 
 
 276 
 
 4416 
 
 22356 
 
 70656 
 
 172500 
 
 357696 
 
 662676 
 
 1130496 
 
 1810836 
 
 276 
 
 277 
 
 4432 
 
 22437 
 
 70912 
 
 173125 
 
 358992 
 
 665077 
 
 1134592 
 
 1817397 
 
 277 
 
 278 
 
 4448 
 
 22518 
 
 71168 
 
 173750 
 
 360288 
 
 667478 
 
 1138688 
 
 1823958 
 
 278 
 
 279 
 
 4464 
 
 22599 
 
 71424 
 
 174375 
 
 361584 
 
 669879 
 
 1142784 
 
 1830519 
 
 279 
 
 280 
 
 4480 
 
 22680 
 
 71680 
 
 175000 
 
 362880 
 
 672280 
 
 1146880 
 
 1837080 
 
 280 
 
 281 
 
 4496 
 
 22761 
 
 71936 
 
 175625 
 
 364176 
 
 674681 
 
 1150976 
 
 1843641 
 
 281 
 
 282 
 
 4512 
 
 22842 
 
 72192 
 
 176250 
 
 365472 
 
 677082 
 
 1155072 
 
 1850202 
 
 282 
 
 283 
 
 4528 
 
 22923 
 
 72448 
 
 176875 
 
 366768 
 
 679483 
 
 1159168 
 
 1856763 
 
 283 
 
 284 
 
 4544 
 
 23004 
 
 72704 
 
 177500 
 
 368064 
 
 681884 
 
 1163264 
 
 1863324 
 
 284 
 
 285 
 
 4660 
 
 23085 
 
 72960 
 
 178125 
 
 369360 
 
 684285 
 
 1167360 
 
 1869885 
 
 285 
 
 286 
 
 4576 
 
 23166 
 
 73216 
 
 178750 
 
 370656 
 
 686686 
 
 1171456 
 
 1876446 
 
 286 
 
 287 
 
 4592 
 
 23247 
 
 73472 
 
 179375 
 
 371952 
 
 689087 
 
 1175552 
 
 1883007 
 
 287 
 
 288 
 
 4608 
 
 23328 
 
 73728 
 
 180000 
 
 373248 
 
 691488 
 
 1179648 
 
 1889568 
 
 288 
 
 289 
 
 4624 
 
 23409 
 
 73984 
 
 180625 
 
 374544 
 
 693889 
 
 1183744 
 
 1896129 
 
 289 
 
 290 
 
 4640 
 
 23490 
 
 74240 
 
 181250 
 
 375840 
 
 696290 
 
 1187840 
 
 1902690 
 
 290 
 
 291 
 
 4656 
 
 23571 
 
 74496 
 
 181875 
 
 377136 
 
 698691 
 
 1191936 
 
 1909251 
 
 291 
 
 292 
 
 4672 
 
 23652 
 
 74752 
 
 182500 
 
 378432 
 
 701092 
 
 1196032 
 
 1915812 
 
 292 
 
 293 
 
 4688 
 
 23733 
 
 75008 
 
 183125 
 
 379728 
 
 703493 
 
 1200128 
 
 1922373 
 
 293 
 
 294 
 
 4704 
 
 23814 
 
 75264 
 
 183750 
 
 381024 
 
 705894 
 
 1204224 
 
 1928934 
 
 294 
 
 295 
 
 4720 
 
 23895 
 
 75520 
 
 184375 
 
 382320 
 
 708295 
 
 1208320 
 
 1935495 
 
 295 
 
 296 
 
 4736 
 
 23976 
 
 75776 
 
 185000 
 
 383616 
 
 710696 
 
 1212416 
 
 1942056 
 
 296 
 
 297 
 
 4752 
 
 24057 
 
 76032 
 
 185625 
 
 384912 
 
 713097 
 
 1216512 
 
 1948617 
 
 297 
 
 298 
 
 4768 
 
 24138 
 
 76288 
 
 186250 
 
 386208 
 
 715498 
 
 1220608 
 
 1955178 
 
 298 
 
 299 
 
 4784 
 
 24219 
 
 76544 
 
 186875 
 
 387504 
 
 717899 
 
 1224704 
 
 1961739 
 
 299 
 
 300 
 
 4800 
 
 24300 
 
 76800 
 
 187500 
 
 388800 
 
 720300 
 
 1228800 
 
 1968300 
 
 300 
 
Verification of the Fourth Moment 
 
 111 
 
 TABLE L — (continued). 
 Ordinate 2—8. Frequency 301—350. 
 
 n 
 
 x=2 
 
 x=3 
 
 • x=i 
 
 »=5 
 
 3=6 
 
 i=7 
 
 a=8 
 
 n 
 
 SOI 
 
 4816 
 
 24381 
 
 77056 
 
 188125 
 
 390096 
 
 722701 
 
 1232896 
 
 SOI 
 
 302 
 
 4832 
 
 24462 
 
 77312 
 
 188750 
 
 391392 
 
 725102 
 
 1236992 
 
 S02 
 
 SOS 
 
 4848 
 
 24543 
 
 77568 
 
 189375 
 
 392688 
 
 727503 
 
 1241088 
 
 SOS 
 
 304 
 
 4864 
 
 24624 
 
 77824 
 
 190000 
 
 393984 
 
 729904 
 
 1245184 
 
 S04 
 
 305 
 
 4880 
 
 24705 
 
 78080 
 
 190625 
 
 395280 
 
 732305 
 
 1249280 
 
 305 
 
 306 
 
 4896 
 
 24786 
 
 78336 
 
 191250 
 
 396576 
 
 734706 
 
 1253376 
 
 306 
 
 307 
 
 4912 
 
 24867 
 
 78592 
 
 191875 
 
 397872 
 
 737107 
 
 1257472 
 
 307 
 
 308 
 
 4928 
 
 24948 
 
 78848 
 
 192500 
 
 399168 
 
 739508 
 
 1261568 
 
 308 
 
 309 
 
 4944 
 
 25029 
 
 79104 
 
 193125 
 
 400464 
 
 741909 
 
 1265664 
 
 809 
 
 310 
 
 4960 
 
 25110 
 
 79360 
 
 193750 
 
 401760 
 
 744310 
 
 1269760 
 
 310 
 
 311 
 
 4976 
 
 25191 
 
 79616 
 
 194375 
 
 403056 
 
 746711 
 
 1273856 
 
 811 
 
 312 
 
 4992 
 
 25272 
 
 79872 
 
 195000 
 
 404352 
 
 749112 
 
 1277952 
 
 812 
 
 313 
 
 5008 
 
 25353 
 
 80128 
 
 195625 
 
 405648 
 
 751513 
 
 1282048 
 
 313 
 
 Silt 
 
 5024 
 
 25434 
 
 80384 
 
 196250 
 
 406944 
 
 753914 
 
 1286144 
 
 S14 
 
 315 
 
 5040 
 
 25515 
 
 80640 
 
 196875 
 
 408240 
 
 756315 
 
 1290240 
 
 815 
 
 S16 
 
 5056 
 
 25596 
 
 80896 
 
 197500 
 
 409536 
 
 758716 
 
 1294336 
 
 816 
 
 317 
 
 5072 
 
 25677 
 
 81152 
 
 198125 
 
 410832 
 
 761117 
 
 1298432 
 
 817 
 
 318 
 
 5088 
 
 25758 
 
 81408 
 
 198750 
 
 412128 
 
 763518 
 
 1302528 
 
 318 
 
 319 
 
 5104 
 
 25839 
 
 81664 
 
 199375 
 
 413424 
 
 765919 
 
 1306624 
 
 319 
 
 320 
 
 5120 
 
 25920 
 
 81920 
 
 200000 
 
 414720 
 
 768320 
 
 1310720 
 
 820 
 
 321 
 
 5136 
 
 26001 
 
 82176 
 
 200625 
 
 416016 
 
 770721 
 
 1314816 
 
 821 
 
 322 
 
 5152 
 
 26082 
 
 82432 
 
 201250 
 
 417312 
 
 773122 
 
 1318912 
 
 S22 
 
 323 
 
 5168 
 
 26163 
 
 82688 
 
 201875 
 
 418608 
 
 775523 
 
 1323008 
 
 323 
 
 324 
 
 5184 
 
 26244 
 
 82944 
 
 202500 
 
 419904 
 
 777924 
 
 1327104 
 
 824 
 
 325 
 
 5200 
 
 26325 
 
 83200 
 
 203125 
 
 421200 
 
 780325 
 
 1331200 
 
 S25 
 
 326 
 
 5216 
 
 26406 
 
 83456 
 
 203750 
 
 422496 
 
 782726 
 
 1335296 
 
 S26 
 
 327 
 
 5232 
 
 26487 
 
 83712 
 
 204375 
 
 423792 
 
 785127 
 
 1339392 
 
 327 
 
 328 
 
 5248 
 
 26568 
 
 83968 
 
 205000 
 
 425088 
 
 787528 
 
 1343488 
 
 S28 
 
 329 
 
 5264 
 
 26G49 
 
 84224 
 
 205625 
 
 426384 
 
 789929 
 
 1347584 
 
 829 
 
 330 
 
 5280 
 
 26730 
 
 84480 
 
 206250 
 
 427680 
 
 792330 
 
 1351680 
 
 830 
 
 331 
 
 5296 
 
 26811 
 
 84736 
 
 206875 
 
 428976 
 
 794731 
 
 1355776 
 
 S81 
 
 332 
 
 5312 
 
 26892 
 
 84992 
 
 207500 
 
 430272 
 
 797132 
 
 1359872 
 
 382 
 
 333 
 
 5328 
 
 26973 
 
 85248 
 
 208125 
 
 431568 
 
 799533 
 
 1363968 
 
 S83 
 
 334 
 
 5344 
 
 27054 
 
 85504 
 
 208750 
 
 432864 
 
 801934 
 
 1368064 
 
 SS4 
 
 835 
 
 5360 
 
 27135 
 
 85760 
 
 209375 
 
 434160 
 
 804335 
 
 1372160 
 
 835 
 
 336 
 
 5376 
 
 27216 
 
 86016 
 
 210000 
 
 435456 
 
 806736 
 
 1376256 
 
 836 
 
 837 
 
 5392 
 
 27297 
 
 86272 
 
 210625 
 
 436752 
 
 809137 
 
 1380352 
 
 SS7 
 
 338 
 
 5408 
 
 27378 
 
 86528 
 
 211250 
 
 438048 
 
 811538 
 
 1384448 
 
 838 
 
 339 
 
 5424 
 
 27459 
 
 86784 
 
 211875 
 
 439344 
 
 813939 
 
 1388544 
 
 3S9 
 
 840 
 
 5440 
 
 27540 
 
 87040 
 
 212500 
 
 440640 
 
 816340 
 
 1392640 
 
 340 
 
 341 
 
 5456 
 
 27621 
 
 87296 
 
 213125 
 
 441936 
 
 818741 
 
 1396736 
 
 841 
 
 842 
 
 5472 
 
 27702 
 
 87552 
 
 213750 
 
 443232 
 
 821142 
 
 1400832 
 
 842 
 
 343 
 
 5488 
 
 27783 
 
 87808 
 
 214375 
 
 444528 
 
 823543 
 
 1404928 
 
 843 
 
 844 
 
 5504 
 
 27864 
 
 88064 
 
 215000 
 
 445824 
 
 825944 
 
 1409024 
 
 844 
 
 345 
 
 5520 
 
 27945 
 
 88320 
 
 215625 
 
 447120 
 
 828345 
 
 1413120 
 
 345 
 
 846 
 
 5536 
 
 28026 
 
 88576 
 
 216250 
 
 448416 
 
 830746 
 
 1417216 
 
 846 
 
 347 
 
 5552 
 
 28107 
 
 88832 
 
 216875 
 
 449712 
 
 833147 
 
 1421312 
 
 347 
 
 848 
 
 5568 
 
 28188 
 
 89088 
 
 217500 
 
 451008 
 
 835548 
 
 1425408 
 
 848 
 
 349 
 
 5584 
 
 28269 
 
 89344 
 
 218125 
 
 452304 
 
 837949 
 
 1429504 
 
 849 
 
 350 
 
 5600 
 
 28350 
 
 89600 
 
 218750 
 
 453600 
 
 840350 
 
 1433600 
 
 S50 
 
112 
 
 Tables for Statisticians and Biometricians 
 
 TABLE L — (continued). 
 Ordinate 2—7. Frequency 351—400. 
 
 n 
 
 x = 2 
 
 x = d 
 
 x = i 
 
 x=5 
 
 x=6 
 
 x=7 
 
 1 
 n 
 
 351 
 
 5616 
 
 28431 
 
 89856 
 
 219375 
 
 454896 
 
 842751 
 
 SSI 
 
 352 
 
 5632 
 
 28512 
 
 90112 
 
 220000 
 
 456192 
 
 845152 
 
 Sot 
 
 S5S 
 
 5648 
 
 28593 
 
 90368 
 
 220625 
 
 457488 
 
 847553 
 
 858 
 
 854 
 
 5664 
 
 28674 
 
 90624 
 
 221250 
 
 458784 
 
 849954 
 
 854 
 
 S55 
 
 5680 
 
 28755 
 
 90880 
 
 221875 
 
 460080 
 
 852355 
 
 355 
 
 356 
 
 5696 
 
 28836 
 
 91136 
 
 222500 
 
 461376 
 
 854756 
 
 356 
 
 357 
 
 5712 
 
 28917 
 
 91392 
 
 223125 
 
 462672 
 
 857157 
 
 357 
 
 358 
 
 5728 
 
 28998 
 
 91648 
 
 223750 
 
 463968 
 
 859558 
 
 858 
 
 359 
 
 5744 
 
 29079 
 
 91904 
 
 224375 
 
 465264 
 
 861959 
 
 359 
 
 360 
 
 5760 
 
 29160 
 
 92160 
 
 225000 
 
 466560 
 
 864360 
 
 360 
 
 361 
 
 5776 
 
 29241 
 
 92416 
 
 225625 
 
 467856 
 
 866761 
 
 361 
 
 362 
 
 5792 
 
 29322 
 
 92672 
 
 226250 
 
 469152 
 
 869162 
 
 862 
 
 363 
 
 5808 
 
 29403 
 
 92928 
 
 226875 
 
 470448 
 
 871563 
 
 363 
 
 364 
 
 5824 
 
 29484 
 
 93184 
 
 227500 
 
 471744 
 
 873964 
 
 364 
 
 365 
 
 5840 
 
 29565 
 
 93440 
 
 228125 
 
 473040 
 
 876365 
 
 365 
 
 366 
 
 5856 
 
 29646 
 
 93696 
 
 228750 
 
 474336 
 
 878766 
 
 866 
 
 367 
 
 5872 
 
 29727 
 
 93952 
 
 229375 
 
 475632 
 
 881167 
 
 867 
 
 368 
 
 5888 
 
 29808 
 
 94208 
 
 230000 
 
 476928 
 
 883568 
 
 368 
 
 369 
 
 5904 
 
 29889 
 
 94464 
 
 230625 
 
 478224 
 
 885969 
 
 369 
 
 870 
 
 5920 
 
 29970 
 
 94720 
 
 231250 
 
 479520 
 
 888370 
 
 870 
 
 871 
 
 5936 
 
 30051 
 
 94976 
 
 231875 
 
 480816 
 
 890771 
 
 871 
 
 372 
 
 5952 
 
 30132 
 
 95232 
 
 232500 
 
 482112 
 
 893172 
 
 872 
 
 373 
 
 5968 
 
 30213 
 
 95488 
 
 233125 
 
 483408 
 
 895573 
 
 373 
 
 374 
 
 5984 
 
 30294 
 
 95744 
 
 233750 
 
 484704 
 
 897974 
 
 874 
 
 375 
 
 6000 
 
 30375 
 
 96000 
 
 234375 
 
 486000 
 
 900375 
 
 375 
 
 .176 
 
 6016 
 
 30456 
 
 96256 
 
 235000 
 
 487296 
 
 902776 
 
 376 
 
 ■ 177 
 
 6032 
 
 30537 
 
 96512 
 
 235625 
 
 488592 
 
 905177 
 
 377 
 
 378 
 
 6048 
 
 30618 
 
 96768 
 
 236250 
 
 489888 
 
 907578 
 
 378 
 
 879 
 
 6064 
 
 30699 
 
 97024 
 
 236875 
 
 491184 
 
 909979 
 
 879 
 
 380 
 
 6080 
 
 30780 
 
 97280 
 
 237500 
 
 492480 
 
 912380 
 
 380 
 
 381 
 
 6096 
 
 30861 
 
 97536 
 
 238125 
 
 493776 
 
 914781 
 
 881 
 
 382 
 
 6112 
 
 30942 
 
 97792 
 
 238750 
 
 495072 
 
 917182 
 
 882 
 
 ,:s.; 
 
 6128 
 
 31023 
 
 98048 
 
 239375 
 
 496368 
 
 919583 
 
 888 
 
 384 
 
 6144 
 
 31104 
 
 98304 
 
 240000 
 
 497664 
 
 921984 
 
 884 
 
 385 
 
 6160 
 
 31185 
 
 98560 
 
 240625 
 
 498960 
 
 924385 
 
 385 
 
 886 
 
 6176 
 
 31266 
 
 98816 
 
 241250 
 
 500256 
 
 926786 
 
 386 
 
 387 
 
 6192 
 
 31347 
 
 99072 
 
 241875 
 
 501552 
 
 929187 
 
 387 
 
 388 
 
 6208 
 
 31428 
 
 99328 
 
 242500 
 
 502848 
 
 931588 
 
 388 
 
 889 
 
 6224 
 
 31509 
 
 99584 
 
 243125 
 
 504144 
 
 933989 
 
 889 
 
 390 
 
 6240 
 
 31590 
 
 99840 
 
 243750 
 
 505440 
 
 936390 
 
 390 
 
 391 
 
 6256 
 
 31671 
 
 100096 
 
 244375 
 
 506736 
 
 938791 
 
 391 
 
 392 
 
 6272 
 
 31752 
 
 100352 
 
 245000 
 
 508032 
 
 941192 
 
 392 
 
 393 
 
 6288 
 
 31833 
 
 100608 
 
 245625 
 
 509328 
 
 943593 
 
 398 
 
 394 
 
 6304 
 
 31914 
 
 100864 
 
 246250 
 
 510624 
 
 945994 
 
 894 
 
 395 
 
 6320 
 
 31995 
 
 101120 
 
 246875 
 
 511920 
 
 948395 
 
 395 
 
 396 
 
 6336 
 
 32076 
 
 101376 
 
 247500 
 
 513216 
 
 950796 
 
 896 
 
 897 
 
 6352 
 
 32157 
 
 101632 
 
 248125 
 
 514512 
 
 953197 
 
 397 
 
 398 
 
 6368 
 
 32238 
 
 101888 
 
 248750 
 
 515808 
 
 955598 
 
 398 
 
 S99 
 
 6384 
 
 32319 
 
 102144 
 
 249375 
 
 517104 
 
 957999 
 
 399 
 
 400 
 
 6400 
 
 32400 
 
 102400 
 
 250000 
 
 518400 
 
 960400 
 
 400 
 
Poisson's Exponential Binomial Limit 
 
 113 
 
 TABLE LI. Tables of e~ m m x /x !: General Term of Poisson's Exponential 
 Expansion ("Law of Small Numbers"). 
 
 X 
 
 m 
 
 
 
 01 
 
 0-2 
 
 OS 
 
 0-4 
 
 OS 
 
 0-6 
 
 0-7 
 
 0-8 
 
 0-9 
 
 VO 
 
 
 
 1 
 
 2 
 S 
 * 
 
 s 
 
 6 
 7 
 8 
 9 
 
 •904837 
 
 ■090484 
 •004524 
 •000151 
 •000004 
 
 1 
 
 •818731 
 •163746 
 •016375 
 •001092 
 •000055 
 •000002 
 
 •740818 
 •222245 
 •033337 
 •003334 
 •000250 
 •000015 
 ■000001 
 
 •670320 
 •268128 
 •053626 
 •007150 
 •000715 
 •000057 
 •000004 
 
 •606531 
 •303265 
 •075816 
 •012636 
 •001580 
 ■000158 
 ■000013 
 •000001 
 
 •548812 
 •329287 
 •098786 
 •019757 
 •002964 
 •000356 
 ■000036 
 •000003 
 
 •496585 
 •347610 
 •121663 
 •028388 
 ■004968 
 •000696 
 •000081 
 ■000008 
 •000001 
 
 •449329 
 •359463 
 •143785 
 •038343 
 •007669 
 •001227 
 •000164 
 •000019 
 •000002 
 
 ■406570 
 •365913 
 •164661 
 •049398 
 •011115 
 •002001 
 •000300 
 ■000039 
 •000004 
 
 •367879 
 ■367879 
 ■183940 
 061313 
 •015328 
 •003066 
 •000511 
 ■000073 
 •000009 
 
 •oooooi 
 
 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 X 
 
 
 1 
 
 2 
 S 
 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 10 
 11 
 
 12 
 
 11 
 
 1-2 
 
 1-3 
 
 1-4 
 
 IS 
 
 ft 
 
 V7 
 
 1-8 
 
 1-9 
 
 2-0 
 
 X 
 
 •332871 
 •366158 
 •201387 
 •073842 
 •020307 
 •004467 
 •000819 
 •000129 
 ■000018 
 •000002 
 
 •301194 
 •361433 
 
 •216860 
 •086744 
 •026023 
 •006246 
 •001249 
 •000214 
 •000032 
 •000004 
 ■000001 
 
 •272532 
 •354291 
 •230289 
 ■099792 
 •032432 
 •008432 
 •001827 
 •000339 
 •000055 
 •000008 
 •000001 
 
 •246597 
 •345236 
 •241665 
 •112777 
 •039472 
 •011052 
 •002579 
 •000510 
 ■000090 
 •000014 
 •000002 
 
 •223130 
 •334695 
 •251021 
 •125510 
 ■047067 
 •014120 
 •003530 
 •000756 
 •000142 
 •000024 
 •000004 
 
 .201897 
 .323034 
 .258428 
 •137828 
 •055131 
 •017642 
 •004705 
 •001075 
 •000215 
 •000038 
 ■000006 
 •000001 
 
 •182684 
 •310562 
 •263978 
 •149587 
 •063575 
 •021615 
 •006124 
 •001487 
 •000316 
 •000060 
 •000010 
 •000002 
 
 •165299 
 •297538 
 •267784 
 •160671 
 •072302 
 •026029 
 •007809 
 •002008 
 •000452 
 •000090 
 •000016 
 •000003 
 
 ■149569 
 •284180 
 •269971 
 •170982 
 •081216 
 •030862 
 •009773 
 •002653 
 •000630 
 •000133 
 •000025 
 •000004 
 •000001 
 
 •135335 
 ■270671 
 •270671 
 •180447 
 ■090224 
 •036089 
 •012030 
 •003437 
 •000859 
 •000191 
 •000038 
 •000007 
 •000001 
 
 
 1 
 2 
 S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 X 
 
 
 
 1 
 2 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 n 
 
 15 
 
 2-1 
 
 2-2 
 
 2-3 
 
 2-4 
 
 2S 
 
 2-6 
 
 2-7 
 
 2-8 
 
 2-9 
 
 3-0 
 
 X 
 
 •122456 
 ■257159 
 ■270016 
 •189012 
 •099231 
 •041677 
 •014587 
 •004376 
 •001149 
 •000268 
 ■000056 
 •000011 
 •000002 
 
 •110803 
 •243767 
 •268144 
 ■196639 
 •108151 
 •047587 
 •017448 
 •005484 
 •001508 
 •000369 
 •000081 
 •000016 
 •000003 
 •000001 
 
 •100259 
 •230595 
 •265185 
 •203308 
 •116902 
 ■053775 
 •020614 
 ■006773 
 •001947 
 •000498 
 •000114 
 •000024 
 ■000005 
 •000001 
 
 •090718 
 •217723 
 •261268 
 •209014 
 •125409 
 •060196 
 •024078 
 •008255 
 •002477 
 •000660 
 •000158 
 •000035 
 •000007 
 •000001 
 
 •082085 
 •205212 
 •256516 
 •213763 
 •133602 
 •066801 
 •027834 
 •009941 
 •003106 
 •000863 
 000216 
 •000049 
 •000010 
 •000002 
 
 •074274 
 •193111 
 •251045 
 •217572 
 •141422 
 •073539 
 •031867 
 •011836 
 •003847 
 •001111 
 •000289 
 •000068 
 •000015 
 •00(1003 
 •000001 
 
 •067206 
 •181455 
 •244964 
 •220468 
 •148816 
 •080360 
 •036162 
 •013948 
 ■004708 
 •001412 
 •000381 
 •000094 
 •000021 
 •000004 
 ■000001 
 
 •060810 
 •170268 
 ■238375 
 ■222484 
 ■155739 
 •087214 
 •040700 
 •016280 
 •005698 
 •001773 
 •000496 
 •000126 
 •000029 
 •000006 
 •000001 
 
 ~ 1 
 
 •055023 
 •159567 
 •231373 
 •223660 
 •162154 
 •094049 
 ■045457 
 •018832 
 •006827 
 •002200 
 ■000638 
 •000168 
 •000041 
 •000009 
 •000002 
 
 ■049787 
 •149361 
 •224042 
 •224042 
 •168031 
 •100819 
 •050409 
 •021604 
 •008102 
 ■002701 
 ■000810 
 •000221 
 •000055 
 •000013 
 •000003 
 
 •oooooi 
 
 
 1 
 
 2 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 14 
 IS 
 
 B. 
 
 15 
 
114 Tables for Statisticians and Bio metricians 
 
 TABLE LI— (continued). 
 
 X 
 
 m 
 
 X 
 
 
 1 
 
 2 
 o 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 u 
 
 15 
 16 
 17 
 
 8-1 
 
 8-2 
 
 SS 
 
 8 -If 
 
 8-5 
 
 8-6 
 
 S-7 
 
 3-8 
 
 8-9 
 
 4-0 
 
 
 1 
 
 2 
 8 
 
 4 
 6 
 
 6 
 7 
 8 
 9 
 
 10 
 11 
 
 12 
 IS 
 
 n 
 
 15 
 16 
 17 
 
 X 
 
 •045049 
 •139653 
 •216461 
 •223677 
 •173350 
 •107477 
 •055530 
 ■024592 
 009529 
 •003282 
 •001018 
 •000287 
 •000074 
 •000018 
 ■000004 
 •000001 
 
 •040762 
 •130439 
 •208702 
 •222616 
 •178093 
 •113979 
 •060789 
 •027789 
 •011116 
 •003952 
 •001265 
 •000368 
 •000098 
 •000024 
 •000006 
 •000001 
 
 •036883 
 
 •121714 
 •200829 
 •220912 
 •182252 
 •120286 
 •066158 
 •031189 
 •012865 
 •004717 
 •001557 
 •000467 
 •000128 
 •000033 
 •000008 
 •000002 
 
 •033373 
 •113469 
 
 •192898 
 •218617 
 •185825 
 •126361 
 •071604 
 •034779 
 •014781 
 •005584 
 •001899 
 •000587 
 •000166 
 •000043 
 •000011 
 •000002 
 •000001 
 
 ■030197 
 •105691 
 •184959 
 •215785 
 •188812 
 •132169 
 •077098 
 •038549 
 •016865 
 •006559 
 •002296 
 •000730 
 •000213 
 •000057 
 •000014 
 •000003 
 •000001 
 
 •027324 
 •098365 
 ■177058 
 ■212169 
 •191222 
 •137680 
 •082608 
 •042484 
 •019118 
 •007647 
 •002753 
 •000901 
 •000270 
 •000075 
 •000019 
 •000005 
 •000001 
 
 •024724 
 •091477 
 •169233 
 •208720 
 •193066 
 •142869 
 •088102 
 •046568 
 •021538 
 •008854 
 •003276 
 •001102 
 •000340 
 •000097 
 •000026 
 •000006 
 •000001 
 
 •022371 
 •085009 
 •161517 
 •204588 
 •194359 
 •147713 
 •093551 
 •050785 
 •024123 
 •010185 
 •003870 
 •001337 
 •000423 
 •000124 
 •000034 
 •000009 
 •000002 
 
 •020242 
 •078943 
 •153940 
 ■200122 
 •195119 
 •152193 
 •098925 
 ■055115 
 •026869 
 •011643 
 •004541 
 •001610 
 •000523 
 •000157 
 ■000044 
 •000011 
 •000003 
 ■000001 
 
 •018316 
 •073263 
 •146525 
 •195367 
 •195367 
 ■156293 
 •104196 
 •059540 
 •029770 
 •013231 
 •005292 
 •001925 
 ■000642 
 •000197 
 •000056 
 ■000015 
 •000004 
 •000001 
 
 4'1 
 
 4'$ 
 
 4-s 
 
 4-4 
 
 4-5 
 
 k-6 
 
 4-7 
 
 4-8 
 
 4-9 
 
 5-0 
 
 X 
 
 
 1 
 
 2 
 8 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 H 
 15 
 16 
 17 
 18 
 19 
 
 X 
 
 •016573 
 •067948 
 •139293 
 •190368 
 •195127 
 •160004 
 •109336 
 •064040 
 •032S20 
 •014951 
 •006130 
 •002285 
 •000781 
 •000246 
 •000072 
 •000020 
 •000005 
 •000001 
 
 •014996 
 •062981 
 •132261 
 •185165 
 •194424 
 •163316 
 •114321 
 •068593 
 •036011 
 •016805 
 •007058 
 •002695 
 •000943 
 •000305 
 •000091 
 •000026 
 •000007 
 •000902 
 
 013569 
 058345 
 ■125441 
 •179799 
 •193284 
 •166224 
 •119127 
 •073178 
 •039333 
 •018793 
 ■008081 
 •003159 
 •001132 
 •000374 
 •000115 
 •000033 
 •000009 
 •000002 
 •000001 
 
 •012277 
 •054020 
 •118845 
 •174305 
 •191736 
 •168728 
 •123734 
 •077775 
 •042776 
 •020913 
 •009202 
 •003681 
 ■001350 
 •000457 
 •000144 
 •000042 
 ■000012 
 •000003 
 •000001 
 
 •011109 
 •049990 
 •112479 
 •168718 
 •189808 
 •170827 
 •128120 
 •082363 
 •046329 
 •023165 
 •010424 
 •004264 
 •001599 
 •000554 
 •000178 
 •000053 
 •000015 
 •000004 
 •000001 
 
 •010052 
 •046238 
 •106348 
 •163068 
 •187528 
 ■172525 
 •132270 
 •086920 
 •049979 
 •025545 
 •011751 
 •004914 
 •001884 
 ■000667 
 •000219 
 •000067 
 •000019 
 •000005 
 •000001 
 
 ■009095 
 •042748 
 •100457 
 •157383 
 •184925 
 •173830 
 •136167 
 •091426 
 •053713 
 •028050 
 •013184 
 •005633 
 •002206 
 •000798 
 •000268 
 •000084 
 •000025 
 •000007 
 •000002 
 
 •008230 
 •039503 
 •094807 
 •151691 
 •182029 
 •174748 
 •139798 
 •095862 
 •057517 
 •030676 
 •014724 
 •006425 
 •002570 
 •000949 
 •000325 
 •000104 
 •000031 
 •000009 
 •000002 
 •000001 
 
 •007447 
 •036488 
 •089396 
 •146014 
 •178867 
 •175290 
 •143153 
 •100207 
 •061377 
 ■033416 
 •016374 
 •007294 
 •002978 
 •001123 
 •000393 
 •000128 
 •000039 
 •000011 
 •000003 
 •000001 
 
 •006738 
 •033690 
 •084224 
 •140374 
 •175467 
 •175467 
 •146223 
 •104445 
 ■065278 
 •036266 
 •018133 
 •008242 
 •003434 
 •001321 
 •000472 
 •000157 
 •000049 
 •000014 
 •000004 
 •000001 
 
 
 1 
 2 
 8 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 18 
 
 U 
 
 15 
 16 
 
 17 
 18 
 19 
 
 X 
 
 5-1 
 
 5-2 
 
 5-S 
 
 5'4 
 
 5-5 
 
 5-6 
 
 5-7 
 
 5-8 
 
 5-9 
 
 6-0 
 
 
 1 
 2 
 S 
 
 •006097 
 •031093 
 •079288 
 •134790 
 
 •005517 
 •028686 
 •074584 
 •129279 
 
 004992 
 •026455 
 •070107 
 •123856 
 
 •004517 
 •024390 
 •065852 
 118533 
 
 •004087 
 •022477 
 •061812 
 •113323 
 
 •003698 
 •020708 
 ■057982 
 •108234 
 
 ■003346 
 •019072 
 •054355 
 •103275 
 
 •003028 
 •017560 
 •050923 
 •098452 
 
 •002739 
 016163 
 •047680 
 •093771 
 
 •002479 
 •014873 
 •044618 
 •089235 
 
 
 1 
 
 2 
 3 
 
Poisson's Exponential Binomial Limit 
 
 115 
 
 TABLE LI— (continued). 
 
 X 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 u 
 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 
 m 
 
 X 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 u 
 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 
 6-1 
 
 5-2 
 
 5-3 
 
 5-4 
 
 5-5 
 
 5-6 
 
 5-7 
 
 5-8 
 
 5-9 
 
 6-0 
 
 •171857 
 •175294 
 •149000 
 •108557 
 ■069205 
 •039216 
 •020000 
 •009273 
 •003941 
 •001546 
 •000563 
 •000191 
 •000061 
 •000018 
 •000005 
 •000001 
 
 •168063 
 •174785 
 •151480 
 •112528 
 ■073143 
 •042261 
 •021976 
 ■010388 
 •004502 
 ■001801 
 •000669 
 •000232 
 •000075 
 •000023 
 •000007 
 •000002 
 
 •164109 
 •173955 
 •153660 
 •116343 
 •077077 
 •045390 
 ■024057 
 •011591 
 •005119 
 •002087 
 •000790 
 •000279 
 •000092 
 •000029 
 •000008 
 •000002 
 •000001 
 
 •160020 
 •172821 
 •155539 
 ■119987 
 •080991 
 •048595 
 •026241 
 •012882 
 •005797 
 •002408 
 ■000929 
 •000334 
 •000113 
 ■000036 
 •000011 
 •000003 
 •000001 
 
 •155819 
 •171401 
 •157117 
 •123449 
 •084871 
 •051866 
 •028526 
 •014263 
 •006537 
 •002766 
 •001087 
 •000398 
 •000137 
 •000044 
 •000014 
 •000004 
 •000001 
 
 •151528 
 •169711 
 •158397 
 •126717 
 •088702 
 •055192 
 •030908 
 •015735 
 •007343 
 •003163 
 •001265 
 •000472 
 •000165 
 •000054 
 •000017 
 •000005 
 ■000001 
 
 •147167 
 •167770 
 •159382 
 •129782 
 •092470 
 •058564 
 •033382 
 •017298 
 •008216 
 •003603 
 •001467 
 •000557 
 •000199 
 •000067 
 •000021 
 ■000006 
 •000002 
 
 •142755 
 •165596 
 •160076 
 •132635 
 •096160 
 •061970 
 •035943 
 •018952 
 •009160 
 •004087 
 •001693 
 •000655 
 •000237 
 •000081 
 •000026 
 •000008 
 •000002 
 •000001 
 
 •138312 
 •163208 
 •160488 
 •135268 
 099760 
 •065398 
 •038585 
 ■020696 
 •010175 
 •004618 
 •001946 
 •000766 
 •000282 
 •000098 
 •000032 
 •000010 
 •000003 
 •000001 
 
 •133853 
 ■160623 
 •160623 
 137677 
 •103258 
 •068838 
 •041303 
 •022529 
 •011264 
 •005199 
 •002228 
 •000891 
 000334 
 •000118 
 •000039 
 ■000012 
 •000004 
 •000001 
 
 X 
 
 c-i 
 
 6-2 
 
 6-3 
 
 o-4 
 
 6-5 
 
 6-6 
 
 6-7 
 
 6-8 
 
 6-9 
 
 7-0 
 
 X 
 
 
 1 
 2 
 3 
 
 4 
 
 6 
 6 
 
 7 
 8 
 9 
 10 
 11 
 12 
 18 
 
 U 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 28 
 
 •002243 
 •013682 
 •041729 
 •084848 
 •129393 
 •157860 
 •160491 
 •139856 
 •106640 
 •072278 
 •044090 
 •024450 
 •012429 
 •005832 
 •002541 
 •001033 
 •000394 
 •000141 
 •000048 
 •000015 
 •000005 
 •000001 
 
 •002029 
 •012582 
 •039006 
 •080612 
 •124948 
 •154936 
 •160100 
 •141803 
 •109897 
 •075707 
 •046938 
 •026456 
 •013669 
 •006519 
 •002887 
 •001193 
 •000462 
 •000169 
 •000058 
 •000019 
 •000006 
 ■000002 
 
 •001836 
 •011569 
 •036441 
 •076527 
 •120530 
 •151868 
 •159461 
 •143515 
 •113018 
 •079113 
 •049841 
 •028545 
 •014986 
 •007263 
 •003268 
 •001373 
 •000540 
 •000200 
 •000070 
 •000023 
 •000007 
 •000002 
 •000001 
 
 •001662 
 •010634 
 •034029 
 •072595 
 •116151 
 •148674 
 •158585 
 •144992 
 •115994 
 •082484 
 •052790 
 •030714 
 •016381 
 ■008064 
 •003687 
 •001573 
 •000629 
 •000237 
 ■000084 
 •000028 
 ■000009 
 •000003 
 •000001 
 
 •001503 
 •009772 
 •031760 
 •068814 
 •111822 
 •145369 
 •157483 
 •146234 
 •118815 
 •085811 
 •055777 
 •032959 
 •017853 
 •008926 
 •004144 
 •001796 
 •000730 
 •000279 
 •000101 
 •000034 
 ■000011 
 000003 
 •000001 
 
 •001360 
 •008978 
 ■029629 
 •065183 
 •107553 
 •141969 
 •156166 
 •147243 
 •121475 
 •089082 
 •058794 
 •035276 
 •019402 
 •009850 
 •004644 
 •002043 
 •000843 
 •000327 
 •000120 
 ■000042 
 •000014 
 •000004 
 •000001 
 
 •001231 
 •008247 
 •027628 
 •061702 
 •103351 
 •138490 
 •154648 
 •148020 
 •123967 
 •092286 
 ■061832 
 •037661 
 •021028 
 •010837 
 •005186 
 •002317 
 •000970 
 •000382 
 •000142 
 ■000050 
 •000017 
 •000005 
 •000002 
 
 •001114 
 •007574 
 •025751 
 •058368 
 •099225 
 •134946 
 •152939 
 •148569 
 •126284 
 •095415 
 •064882 
 •040109 
 •022728 
 •011889 
 •005774 
 •002618 
 001113 
 •000445 
 000168 
 •000060 
 •000020 
 •000007 
 •000002 
 •000001 
 
 ■001008 
 •006954 
 •023990 
 ■055178 
 •095182 
 •131351 
 •151053 
 •148895 
 •128422 
 •098457 
 •067935 
 •042614 
 •024503 
 013005 
 ■006410 
 •002949 
 •001272 
 •000516 
 000198 
 •000072 
 •000025 
 •000008 
 •000003 
 ■000001 
 
 ■000912 
 •006383 
 •022341 
 •052129 
 •091226 
 •127717 
 •149003 
 •149003 
 •130377 
 •101405 
 •070983 
 •045171 
 •026350 
 •014188 
 •007094 
 •003311 
 •001448 
 •000596 
 •000232 
 •000085 
 •000030 
 •000010 
 •000003 
 •000001 
 
 
 1 
 
 2 
 S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 U 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 
 15—2 
 
116 Tables for Statisticians and Biometricians 
 
 TABLE LI— (continued). 
 
 x 
 
 
 
 1 
 2 
 S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 
 a* 
 
 25 
 
 X 
 
 in 
 
 X 
 
 
 1 
 
 2 
 3 
 
 k 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 n 
 
 16 
 
 16 
 17 
 18 
 19 
 
 20 
 
 21 
 
 22 
 23 
 
 25 
 
 7-1 
 
 7-2 
 
 7-3 
 
 7-lt 
 
 7-5 
 
 7-6 
 
 7-7 
 
 7-8 
 
 7-9 
 
 8-0 
 
 •000825 
 •005858 
 •020797 
 •049219 
 •087364 
 •124057 
 •146800 
 •148897 
 •132146 
 •104249 
 •074017 
 •047774 
 •028267 
 •015438 
 •007829 
 •003706 
 •001644 
 ■000687 
 •000271 
 ■000101 
 •000036 
 •000012 
 •000004 
 •000001 
 
 •000747 
 •005375 
 •019352 
 •046444 
 •083598 
 •120382 
 •144458 
 •148586 
 •133727 
 ■106982 
 •077027 
 •050418 
 •030251 
 •016754 
 •008616 
 ■004136 
 •001861 
 •000788 
 •000315 
 •000119 
 •000043 
 •000015 
 •000005 
 •000002 
 
 •000676 
 •004931 
 •018000 
 •043799 
 •079934 
 •116703 
 •141989 
 •148074 
 •135118 
 ■109596 
 •080005 
 •053094 
 •032299 
 ■018137 
 •009457 
 •004603 
 ■002100 
 •000902 
 •000366 
 ■000141 
 •000051 
 •000018 
 •000006 
 •000002 
 ■000001 
 
 •000611 
 •004523 
 •016736 
 •041282 
 •076372 
 •113031 
 •139405 
 •147371 
 •136318 
 •112084 
 •082942 
 •055797 
 •034408 
 •019586 
 •010353 
 •005107 
 •002362 
 •001028 
 •000423 
 •000165 
 •000061 
 •000021 
 •000007 
 •000002 
 •000001 
 
 ■000553 
 •004148 
 •015555 
 •038889 
 •072916 
 •109375 
 •136718 
 •146484 
 •137329 
 ■114440 
 ■085830 
 •058521 
 •036575 
 •021101 
 •011304 
 •005652 
 •002649 
 •001169 
 •000487 
 •000192 
 •000072 
 •000026 
 •000009 
 •000003 
 •000001 
 
 •000500 
 •003803 
 •014453 
 •036614 
 •069567 
 •105742 
 •133940 
 •145421 
 •138150 
 •116660 
 •088661 
 •061257 
 •038796 
 •022681 
 •012312 
 •006238 
 •002963 
 •001325 
 •000559 
 •000224 
 •000085 
 •000031 
 ■000011 
 •000004 
 ■000001 
 
 •000453 
 •003487 
 ■013424 
 •034455 
 •066326 
 •102142 
 •131082 
 •144191 
 •138783 
 •118737 
 •091427 
 •063999 
 •041066 
 •024324 
 •013378 
 •006867 
 •003305 
 •001497 
 •000640 
 •000259 
 •000100 
 •000037 
 •000013 
 •000004 
 •000001 
 
 •000410 
 •003196 
 ■012464 
 ■032407 
 •063193 
 •098581 
 •128156 
 •142802 
 •139232 
 •120668 
 •094121 
 ■066740 
 •043381 
 •026029 
 ■014502 
 •007541 
 •003676 
 •001687 
 •000731 
 •000300 
 •000117 
 •000043 
 •000015 
 •000005 
 •000002 
 •000001 
 
 •000371 
 ■002929 
 ■011569 
 •030465 
 •060169 
 •095067 
 •125171 
 •141264 
 •139499 
 •122449 
 •096735 
 •069473 
 •045736 
 •027794 
 •015684 
 •008260 
 •004078 
 •001895 
 •000832 
 •000346 
 •000137 
 •000051 
 •000018 
 •000006 
 •000002 
 •000001 
 
 •000335 
 •002684 
 •010735 
 •028626 
 •057252 
 •091604 
 •122138 
 •139587 
 •139587 
 •124077 
 •099262 
 •072190 
 •048127 
 •029616 
 •016924 
 •009026 
 •004513 
 ■002124 
 •000944 
 •000397 
 •000159 
 •000061 
 •000022 
 •000008 
 •000003 
 •000001 
 
 8-1 
 
 8-2 
 
 8-S 
 
 S-4 
 
 8-5 
 
 8-6 
 
 8-7 
 
 8-8 
 
 8-9 
 
 9-0 
 
 X 
 
 
 1 
 
 2 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 14 
 15 
 16 
 17 
 18 
 19 
 20 
 
 
 1 
 2 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 16 
 17 
 18 
 19 
 20 
 
 •000304 
 •002459 
 •009958 
 •026885 
 •054443 
 •088198 
 •119067 
 •137778 
 •139500 
 •125550 
 •101696 
 •074885 
 •050547 
 •031495 
 •018222 
 •009840 
 •004981 
 •002373 
 •001068 
 •000455 
 •000184 
 
 •000275 
 •002252 
 •009234 
 •025239 
 •051740 
 •084854 
 •115967 
 •135848 
 •139244 
 •126866 
 •104031 
 •077550 
 •052993 
 •033426 
 •019578 
 ■010703 
 •005485 
 •002646 
 •001205 
 ■000520 
 •000213 
 
 •000249 
 •002063 
 •008560 
 •023683 
 •049142 
 •081576 
 •112847 
 •133805 
 •138823 
 •128025 
 •106261 
 •080179 
 •055457 
 ■035407 
 •020991 
 •011615 
 •006025 
 ■002942 
 ■001356 
 000693 
 •000246 
 
 •000225 
 •001889 
 •007933 
 •022213 
 •046648 
 •078368 
 •109716 
 •131659 
 •138242 
 •129026 
 •108382 
 •082764 
 •057935 
 •037435 
 •022461 
 •012578 
 •006604 
 •003263 
 •001523 
 •000673 
 ■000283 
 
 000203 
 •001729 
 •007350 
 •020826 
 •044255 
 •075233 
 •106581 
 •129419 
 •137508 
 •129869 
 •110388 
 •085300 
 •060421 
 •039506 
 ■023986 
 •013592 
 •007221 
 •003610 
 •001705 
 •000763 
 •000324 
 
 •000184 
 •001583 
 ■006808 
 •019517 
 •041961 
 •072174 
 •103449 
 •127094 
 •136626 
 •130554 
 •112277 
 •087780 
 •062909 
 ■041617 
 •025565 
 •014657 
 •007878 
 •003985 
 •001904 
 •000862 
 •000371 
 
 •000167 
 •001449 
 ■006304 
 •018283 
 •039765 
 •069192 
 •100328 
 •124693 
 •135604 
 •131084 
 •114043 
 •090197 
 •065393 
 ■043763 
 •027196 
 ■015773 
 •008577 
 ■004389 
 ■002121 
 •000971 
 •000423 
 
 •000151 
 •001326 
 •005836 
 •017120 
 •037664 
 •066289 
 •097224 
 •122224 
 •134446 
 •131459 
 ■115684 
 •092547 
 •067868 
 •045941 
 •028877 
 •016941 
 ■009318 
 •004823 
 •002358 
 •001092 
 000481 
 
 •000136 
 •001214 
 •005402 
 •016025 
 •035656 
 •063467 
 •094143 
 •119696 
 •133161 
 •131682 
 •117197 
 ■094823 
 •070327 
 •048147 
 •030608 
 •018161 
 •010102 
 •005289 
 •002615 
 •001225 
 •000545 
 
 •000123 
 •001111 
 •004998 
 •014994 
 •033737 
 •060727 
 •091090 
 •117116 
 •131756 
 •131756 
 •118580 
 •097020 
 •072765 
 •05037G 
 •032384 
 •019431 
 ■010930 
 •005786 
 •002893 
 001370 
 •000617 
 
Poissoris Exponential Binomial Limit 
 TABLE LI— (continued). 
 
 117 
 
 X 
 
 m 
 
 X 
 
 8-1 
 
 f'J 
 
 8-8 
 
 *# 
 
 8-5 
 
 8-6 
 
 8-7 
 
 8-8 
 
 8-9 
 
 9-0 
 
 21 
 
 22 
 23 
 
 m 
 
 25 
 26 
 
 trr 
 
 •000071 
 •000026 
 ■000009 
 •000003 
 •000001 
 
 •000083 
 •000031 
 ■000011 
 •000004 
 ■000001 
 
 •000097 
 •000037 
 •000013 
 •000005 
 •000002 
 
 •000113 
 •000043 
 •000016 
 •000006 
 •000002 
 •000001 
 
 •000131 
 ■000051 
 •000019 
 •000007 
 •000002 
 •000001 
 
 •000152 
 •000059 
 •000022 
 •000008 
 •000003 
 •000001 
 
 •000175 
 •000069 
 •000026 
 •000009 
 •000003 
 •000001 
 
 •000201 
 •000081 
 •000031 
 •000011 
 •000004 
 •000001 
 
 
 000231 
 000093 
 000036 
 000013 
 000005 
 000002 
 000001 
 
 ■000864 
 
 •000108 
 •000042 
 •000016 
 •000006 
 •000002 
 
 •oooooi 
 
 21 
 22 
 23 
 
 n 
 
 25 
 26 
 27 
 
 X 
 
 
 1 
 % 
 3 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 18 
 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 28 
 
 25 
 26 
 27 
 28 
 29 
 
 X 
 
 
 1 
 2 
 8 
 
 X 
 
 9-1 
 
 9-2 
 
 9-3 
 
 *4 
 
 9-5 
 
 9-6' 
 
 9-7 
 
 9-8 
 
 9-9 
 
 10-0 
 
 
 1 
 
 2 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 U 
 
 is 
 
 10 
 17 
 18 
 19 
 20 
 
 91 
 
 ->-> 
 
 28 
 
 H 
 
 25 
 26 
 27 
 28 
 29 
 
 X 
 
 •000112 
 •001016 
 •004624 
 •014025 
 •031906 
 •058069 
 •088072 
 •114493 
 •130236 
 •131683 
 •119832 
 •099133 
 ■075176 
 •052623 
 •034205 
 •020751 
 •011802 
 •006318 
 ■003194 
 •001530 
 •000696 
 ■000302 
 ■000125 
 •000049 
 •000019 
 •000007 
 ■000002 
 •000001 
 
 •oooioi 
 
 •000930 
 •004276 
 •013113 
 •030160 
 •055494 
 •085091 
 •111834 
 ■128609 
 •131467 
 •120960 
 ■101158 
 •077555 
 •054885 
 •036067 
 ■022121 
 •012720 
 •006884 
 •003518 
 •001704 
 •000784 
 •000343 
 •000144 
 •000057 
 •000022 
 •000008 
 •000003 
 •000001 
 
 •000091 
 •000850 
 •003954 
 •012256 
 •028496 
 •053002 
 •082154 
 •109147 
 •126883 
 •131113 
 •121935 
 •103090 
 •079895 
 •057156 
 •037968 
 •023540 
 •013683 
 •007485 
 •003867 
 •001893 
 •000880 
 •000390 
 •000165 
 •000067 
 •000026 
 •000010 
 •000003 
 •000001 
 
 ■000083 
 •000778 
 •003655 
 •011452 
 •026911 
 •050593 
 •079262 
 •106438 
 •125065 
 •130623 
 •122786 
 •104926 
 •082192 
 •059431 
 •039904 
 •025006 
 •014691 
 •008123 
 •004242 
 •002099 
 •000986 
 •000442 
 •000189 
 •000077 
 ■000030 
 ■000011 
 •000004 
 •000001 
 
 •000075 
 •000711 
 •003378 
 •010696 
 •025403 
 •048266 
 •076421 
 •103714 
 •123160 
 •130003 
 •123502 
 •106661 
 •084440 
 •061706 
 •041872 
 •026519 
 •015746 
 •008799 
 •004644 
 •002322 
 •001103 
 •000499 
 •000215 
 •000089 
 •000035 
 •000013 
 •000005 
 •000002 
 •000001 
 
 •000068 
 •000650 
 •003121 
 ■009987 
 •023969 
 ■046020 
 ■073632 
 •100981 
 •121178 
 •129256 
 •124086 
 •108293 
 •086634 
 •063976 
 •043869 
 •028076 
 •016846 
 •009513 
 ■005074 
 •002563 
 •001230 
 •000563 
 •000245 
 •000102 
 •000041 
 ■000016 
 •000006 
 •000002 
 •000001 
 
 •000061 
 •000594 
 •002883 
 •009322 
 •022606 
 •043855 
 •070899 
 •098246 
 •119123 
 •128388 
 •124537 
 •109819 
 •088770 
 •066236 
 •045892 
 •029677 
 •017992 
 •010266 
 •005532 
 •002824 
 •001370 
 •000633 
 •000279 
 •000118 
 •000048 
 •000018 
 •000007 
 •000002 
 •000001 
 
 •000055 
 •000543 
 ■002663 
 •008698 
 •021311 
 •041770 
 ■068224 
 •095514 
 •117004 
 •127405 
 •124857 
 •111236 
 •090843 
 •068481 
 •047937 
 •031319 
 •019183 
 •011058 
 •006021 
 •003105 
 •001522 
 •000710 
 ■000316 
 •000135 
 ■000055 
 •000022 
 •000008 
 •000003 
 •000001 
 
 •000050 
 ■000497 
 •002459 
 •008114 
 •020082 
 •039763 
 •065609 
 •092790 
 •114827 
 •126310 
 •125047 
 •112542 
 ■092847 
 •070707 
 •050000 
 •033000 
 •020419 
 •011891 
 •006540 
 •003408 
 •001687 
 •000795 
 •000358 
 •000154 
 •000064 
 •000025 
 •000010 
 •000004 
 •000001 
 
 •000045 
 •000454 
 •002270 
 •007567 
 •018917 
 •037833 
 •063055 
 •090079 
 •112599 
 •125110 
 •125110 
 •113736 
 •094780 
 •072908 
 •052077 
 •034718 
 ■021699 
 •012764 
 •007091 
 •003732 
 ■001866 
 •000889 
 •000404 
 •000176 
 •000073 
 •000029 
 •000011 
 •000004 
 
 •oooooi 
 ■oooooi 
 
 10-1 
 
 10-2 
 
 io-s 
 
 10% 
 
 10-5 
 
 10-6- 
 
 10-7 
 
 10-8 
 
 10-9 
 
 u-o 
 
 
 1 
 
 2 
 3 
 
 •000041 | 
 •000415 
 •002095 
 •007054 
 
 ■000087 
 ■000879 
 
 •001934 
 
 •006574 
 
 ■000034 
 000346 
 •001784 
 ■006125 
 
 •000030 
 •000317 
 •001646 
 •005705 
 
 •000028 
 •000289 
 •001518 
 •005313 
 
 •000025 
 •000264 
 •001400 
 ■004946 
 
 •000023 
 •000241 
 •001291 
 004603 
 
 •000020 
 •000220 
 •001190 
 •004283 
 
 •000018 
 •000201 
 •001097 
 •003984 
 
 ■000017 
 •000184 
 •001010 
 •003705 
 
118 
 
 Tables for Statisticians and Biometricians 
 TABLE LI— (continued). 
 
 X 
 
 m 
 
 X 
 
 10-1 
 
 10-2 
 
 10-3 
 
 10-4 
 
 10-5 
 
 10-6 
 
 10-7 
 
 10-8 
 
 10-9 
 
 11-0 
 
 k 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 n 
 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 
 u 
 
 25 
 26 
 27 
 28 
 29 
 SO 
 
 ■017811 
 •035979 
 060565 
 •087387 
 •110326 
 •123810 
 ■125048 
 ■114817 
 •096637 
 •075080 
 •054165 
 •036471 
 •023022 
 •013678 
 •007675 
 •004080 
 •002060 
 •000991 
 •000455 
 •000200 
 •000084 
 •000034 
 •000013 
 •000005 
 •000002 
 •000001 
 
 •016764 
 •034199 
 •058139 
 •084716 
 •108013 
 •122415 
 ■124863 
 •115782 
 •098415 
 ■077218 
 •056259 
 •038256 
 •024388 
 ■014633 
 •008292 
 •004451 
 •002270 
 •001103 
 •000511 
 •000227 
 •000096 
 •000039 
 •000015 
 •000006 
 •000002 
 •000001 
 
 •015773 
 •032492 
 •055777 
 •082072 
 •105668 
 •120931 
 •124559 
 116633 
 •100110 
 •079318 
 •058355 
 •040071 
 •025795 
 •015629 
 •008943 
 •004848 
 •002497 
 •001225 
 ■000573 
 ■000257 
 •000110 
 •000045 
 ■000018 
 •000007 
 •000003 
 •000001 
 
 ■014834 
 •030855 
 ■053482 
 •079458 
 •103296 
 •119364 
 •124139 
 •117368 
 •101719 
 •081375 
 •060450 
 •041912 
 •027243 
 •016666 
 •009629 
 •005271 
 ■002741 
 ■001357 
 ■000642 
 •000290 
 •000126 
 •000052 
 •000021 
 ■000008 
 •000003 
 •000001 
 
 •013946 
 •029287 
 •051252 
 •076878 
 •100902 
 •117720 
 •123606 
 •117987 
 •103239 
 •083385 
 •062539 
 043777 
 ■028729 
 •017744 
 •010351 
 •005720 
 ■003003 
 •001502 
 •000717 
 •000327 
 •000143 
 •000060 
 •000024 
 •000009 
 •000004 
 •000001 
 
 •013107 
 •027786 
 •049089 
 •074334 
 ■098493 
 •116003 
 •122963 
 •118492 
 •104667 
 ■085344 
 ■064618 
 •045663 
 •030252 
 •018863 
 •011108 
 •006197 
 •003285 
 •001658 
 •000799 
 •000368 
 •000163 
 •000069 
 •000028 
 •000011 
 •000004 
 •000002 
 •000001 
 
 •012313 
 •026350 
 •046991 
 •071830 
 •096072 
 •114219 
 ■122215 
 •118882 
 ■106003 
 ■087248 
 •066683 
 ■047567 
 •031810 
 •020022 
 •011902 
 •006703 
 •003586 
 ■001827 
 •000889 
 •000413 
 •000184 
 ■000079 
 •000032 
 •000013 
 ■000005 
 •000002 
 •000001 
 
 •011564 
 ■024978 
 •044960 
 •069367 
 •093646 
 •112375 
 ■121365 
 ■119159 
 •107243 
 •089094 
 •068730 
 ■049485 
 •033403 
 •021220 
 •012732 
 •007237 
 •003908 
 •002010 
 ■000987 
 •000463 
 •000208 
 •000090 
 •000037 
 •000015 
 ■000006 
 •000002 
 •000001 
 
 •010856 
 •023667 
 •042995 
 •066949 
 •091218 
 •110475 
 •120418 
 •119323 
 •108386 
 •090877 
 •070754 
 •051415 
 •035026 
 •022458 
 •013600 
 •007802 
 •004252 
 •002207 
 •001093 
 •000518 
 •000235 
 ■000103 
 •000043 
 •000017 
 •000007 
 •000003 
 •000001 
 
 •010189 
 •022415 
 •041095 
 •064577 
 •088794 
 ■108526 
 •119378 
 •119378 
 •109430 
 ■092595 
 •072753 
 •053352 
 •036680 
 •023734 
 •014504 
 •008397 
 •004618 
 ■002419 
 ■001210 
 •000578 
 ■000265 
 •000117 
 •000049 
 •000020 
 •000008 
 •000003 
 •000001 
 
 A 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 U 
 15 
 16 
 17 
 
 18 
 19 
 20 
 21 
 
 22 
 23 
 
 n 
 
 25 
 26 
 27 
 28 
 29 
 SO 
 
 X 
 
 
 1 
 2 
 S 
 
 k 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 n 
 
 15 
 16 
 17 
 18 
 19 
 
 11 -1 
 
 11-2 
 
 US 
 
 11-4 
 
 11-5 
 
 11-6 
 
 11-7 
 
 1V8 
 
 11-9 
 
 12-0 
 
 X 
 
 •000015 
 •000168 
 000931 
 •003445 
 •009559 
 •021221 
 039259 
 •062253 
 •086376 
 •106531 
 •118249 
 •119324 
 •110375 
 ■094243 
 •074721 
 •055294 
 •038360 
 •025047 
 •015446 
 ■009023 
 
 •000014 
 •000153 
 •000858 
 •003202 
 •008965 
 •020082 
 •037487 
 •059979 
 •083970 
 •104496 
 •117036 
 •119164 
 •111220 
 •095820 
 •076656 
 ■057236 
 ■040065 
 •026396 
 •016424 
 •009682 
 
 ■000012 
 •000140 
 •000790 
 •002976 
 •008406 
 •018997 
 •035778 
 •057755 
 •081579 
 •102427 
 •115743 
 •118899 
 •111964 
 •097322 
 •078553 
 •059177 
 •041793 
 ■027780 
 ■017440 
 •010372 
 
 ■000011 
 •000128 
 •000727 
 •002764 
 •007879 
 •017963 
 •034130 
 •055584 
 ■079206 
 •100328 
 •114374 
 •118533 
 •112607 
 •098747 
 •080409 
 061110 
 •043541 
 ■029198 
 •018492 
 •011095 
 
 •000010 
 •000116 
 •000670 
 •002568 
 •007382 
 •016979 
 •032544 
 •053465 
 •076856 
 •098204 
 ■112935 
 ■118068 
 •113149 
 •100093 
 •082219 
 •063035 
 •045306 
 ■030648 
 •019581 
 •011852 
 
 •000009 
 •000106 
 •000617 
 •002386 
 •006915 
 •016043 
 •031017 
 •051400 
 ■074529 
 •096060 
 •111430 
 •117508 
 •113591 
 •101358 
 •083982 
 •064946 
 •047086 
 •032129 
 •020706 
 ■012641 
 
 •000008 
 •000097 
 •000568 
 •002214 
 •006476 
 •015153 
 •029549 
 •049388 
 •072231 
 •093900 
 •109863 
 •116854 
 •113933 
 •102539 
 •085694 
 ■066841 
 •048877 
 •033639 
 •021865 
 •013465 
 
 •000008 
 •000089 
 •000522 
 ■002055 
 •006062 
 •014307 
 •028137 
 •047432 
 •069962 
 •091728 
 •108239 
 •116110 
 •114175 
 •103636 
 •087350 
 •068716 
 •050678 
 •035176 
 •023060 
 •014322 
 
 •000007 
 ■000081 
 •000481 
 •001907 
 ■005674 
 •013504 
 •026782 
 •045530 
 •067725 
 •089548 
 ■106562 
 •115281 
 •114320 
 •104647 
 •088950 
 •070567 
 •052484 
 •036739 
 •024288 
 •015212 
 
 •000006 
 ■000074 
 ■000442 
 ■001770 
 •005309 
 ■012741 
 •025481 
 •043682 
 ■065523 
 •087364 
 •104837 
 •114368 
 •114363 
 •105570 
 •090489 
 •072391 
 •054293 
 •038325 
 •025550 
 •016137 
 
 
 1 
 
 2 
 S 
 
 k 
 5 
 
 6 
 
 r* 
 i 
 
 8 
 9 
 10 
 11 
 12 
 IS 
 
 H 
 15 
 16 
 17 
 18 
 19 
 
Poisson's Exponential Binomial Limit 
 TABLE LI— {continued). 
 
 119 
 
 20 
 
 25 
 26 
 
 £7 
 
 31 
 
 32 
 
 
 1 
 2 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 15 
 16 
 
 n 
 
 18 
 19 
 
 20 
 21 
 
 ..'2 
 23 
 
 S4 
 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 S3 
 
 11-1 
 
 •005008 
 •002647 
 001336 
 •000645 
 ■000298 
 •000132 
 •000057 
 •000023 
 ■000009 
 •000004 
 •000001 
 
 12-1 
 
 •00(1006 
 •000067 
 ■000407 
 •001641 
 •004966 
 •012017 
 •024233 
 •041889 
 •063358 
 •085181 
 
 103069 
 •113376 
 •114321 
 •106406 
 •091965 
 •074185 
 •056103 
 
 039932 
 •026843 
 •017095 
 •010342 
 •005959 
 •003278 
 •001724 
 •000869 
 •000421 
 •000196 
 •000088 
 •000038 
 •000016 
 •000006 
 •000002 
 •000001 
 
 11-2 
 
 •005422 
 •002892 
 •001472 
 •000717 
 •000335 
 ■000150 
 •000065 
 ■000027 
 •000011 
 •000004 
 •000002 
 •000001 
 
 12'2 
 
 •000005 
 •000061 
 •000374 
 •001522 
 •004643 
 •011330 
 •023037 
 •040151 
 •061230 
 •083000 
 •101261 
 •112308 
 •114180 
 ■107153 
 •093376 
 •075946 
 •057909 
 •04] 558 
 •02S167 
 ■018086 
 •011033 
 •006409 
 •003554 
 •001885 
 •000958 
 •000468 
 •000219 
 •000099 
 •000043 
 ■000018 
 •000007 
 •000003 
 000001 
 
 11-3 
 
 •005860 
 •003153 
 •001620 
 •000796 
 •000375 
 •000169 
 ■000074 
 •000031 
 •000012 
 •000005 
 •000002 
 •000001 
 
 u-4 
 
 •000005 
 •000056 
 •000344 
 •001412 
 •004341 
 •010679 
 ■021892 
 •038467 
 ■059142 
 •080828 
 •099418 
 •111168 
 •113947 
 ■107811 
 ■094720 
 •077670 
 •059709 
 •043201 
 
 029521 
 •019111 
 •011753 
 
 006884 
 ■003849 
 •002058 
 ■001055 
 •000519 
 •000246 
 •000112 
 •000049 
 •000021 
 •000009 
 •000003 
 •000001 
 
 •006324 
 •003433 
 •001779 
 •000882 
 •000419 
 •000191 
 •000084 
 •000035 
 •000014 
 •000006 
 •000002 
 •000001 
 
 12- If 
 
 11-3 
 
 •000004 
 •000051 
 •000317 
 •001309 
 •004057 
 •010062 
 •020794 
 •036836 
 •057095 
 •078665 
 •097544 
 •109959 
 •113624 
 •108380 
 •095994 
 •079355 
 •061500 
 ■044859 
 •030903 
 •020168 
 •012504 
 •007383 
 •004162 
 •002244 
 •001159 
 •000575 
 •000274 
 •000126 
 •000056 
 •000024 
 •000010 
 •000004 
 •000002 
 •000001 
 
 •006815 
 •003732 
 •001951 
 •000975 
 •000467 
 •000215 
 •000095 
 •000041 
 ■000017 
 ■000007 
 •000003 
 •000001 
 
 12-5 
 
 •000004 
 •000047 
 •000291 
 •001213 
 •003791 
 •009477 
 •019744 
 •035258 
 •055091 
 •076515 
 •095644 
 •108686 
 •113215 
 •108860 
 •097197 
 •080997 
 •063279 
 •046529 
 •032312 
 •021258 
 •013286 
 •007908 
 •004493 
 •002442 
 •001272 
 •000636 
 •000306 
 •000142 
 •000063 
 •000027 
 •000011 
 ■000005 
 •000002 
 •000001 
 
 1V6 
 
 •007332 
 •004050 
 •002136 
 •001077 
 •000521 
 •000242 
 •000108 
 ■000046 
 •000019 
 •000008 
 ■000003 
 •000001 
 
 12-6 
 
 •000003 
 •000042 
 ■000268 
 ■001124 
 •003541 
 ■008924 
 •018740 
 •033733 
 •053129 
 •074381 
 •093720 
 •107352 
 •112720 
 •109251 
 •098326 
 •082594 
 •065043 
 •048208 
 •033746 
 •022379 
 •014099 
 •008459 
 •004845 
 •002654 
 •001393 
 •000702 
 •000340 
 •000159 
 •000071 
 •000031 
 000013 
 •000005 
 •000002 
 •000001 
 
 11-7 
 
 11-8 
 
 •007877 
 
 •008450 
 
 •004388 
 
 •004748 
 
 •002334 
 
 •002547 
 
 •001187 
 
 •001307 
 
 •000579 
 
 •000642 
 
 •000271 
 
 •000303 
 
 •000122 
 
 •000138 
 
 •000053 
 
 •000060 
 
 •000022 
 
 •000025 
 
 •000009 
 
 •000010 
 
 •000003 
 
 •000004 
 
 •000001 
 
 •000002 
 
 — 
 
 •000001 
 
 12-7 
 
 12-8 
 
 •000003 
 
 ■000003 
 
 •000039 
 
 •000035 
 
 •000246 
 
 •000226 
 
 •001042 
 
 •000965 
 
 •003307 
 
 •003088 
 
 •008400 
 
 •007905 
 
 •017781 
 
 •016864 
 
 •032259 
 
 •030837 
 
 ■051212 
 
 •049339 
 
 •072266 
 
 •070171 
 
 •091777 
 
 •089819 
 
 •105961 
 
 •104516 
 
 •112142 
 
 •111484 
 
 •109554 
 
 •109769 
 
 ■099381 
 
 •100360 
 
 •084143 
 
 •085641 
 
 •066788 
 
 •068513 
 
 •049895 
 
 •051586 
 
 •035204 
 
 •036683 
 
 •023531 
 
 •024713 
 
 •014942 
 
 •015816 
 
 •009036 
 
 •009640 
 
 •005216 
 
 •005609 
 
 •002880 
 
 •003122 
 
 •001524 
 
 •001665 
 
 •000774 
 
 •000852 
 
 •000378 
 
 •000420 
 
 •000178 
 
 •000199 
 
 •000081 
 
 •000091 
 
 •000035 
 
 •000040 
 
 •000015 
 
 •000017 
 
 •000006 
 
 •000007 
 
 ■000002 
 
 •000003 
 
 •000001 
 
 ■000001 
 
 
 ^~ 
 
 11-9 
 
 •009051 
 •005129 
 •002774 
 ■001435 
 •000712 
 •000339 
 •000155 
 •000068 
 •000029 
 •000012 
 •000005 
 •000002 
 •000001 
 
 12-9 
 
 •000002 
 •000032 
 •000208 
 •000894 
 •002882 
 •007436 
 •015988 
 •029464 
 •047511 
 •068100 
 •087849 
 •103023 
 •110749 
 •109897 
 ■101263 
 •087086 
 •070213 
 •053279 
 •038183 
 •025925 
 •016721 
 •010272 
 •006023 
 •003378 
 •001816 
 •000937 
 •000465 
 •000222 
 •000102 
 •000046 
 •000020 
 •000008 
 •000003 
 •000001 
 
 12-0 
 
 •009682 
 •005533 
 •003018 
 •001575 
 •000787 
 •000378 
 ■000174 
 •000078 
 •000033 
 ■000014 
 •000005 
 •000002 
 •000001 
 
 13-0 
 
 •000002 
 •000029 
 •000191 
 •000828 
 •002690 
 •006994 
 •015153 
 •028141 
 •045730 
 •066054 
 •085870 
 •101483 
 •109940 
 •109940 
 •102087 
 •088475 
 •071886 
 •054972 
 •039702 
 •027164 
 •017657 
 •010930 
 •006459 
 •003651 
 •001977 
 •001028 
 •000514 
 •000248 
 •000115 
 •000052 
 •000022 
 •000009 
 •000004 
 •000002 
 •000001 
 
 20 
 21 
 22 
 23 
 
 n 
 
 25 
 26 
 27 
 28 
 29 
 SO 
 SI 
 82 
 
 
 1 
 
 2 
 8 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 H 
 15 
 16 
 17 
 18 
 19 
 
 27 
 
 SO 
 SI 
 32 
 S3 
 *4 
 
120 
 
 Tables for Statisticians and Biometricians 
 TABLE LI— (continued). 
 
 o 
 l 
 
 2 
 8 
 
 4 
 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 
 
 1 
 
 2 
 3 
 
 k 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 n 
 
 15 
 16 
 17 
 18 
 19 
 
 22 
 23 
 
 H 
 
 25 
 26 
 27 
 28 
 29 
 SO 
 SI 
 32 
 S3 
 Si, 
 
 13-1 
 
 ■000002 
 •000027 
 •000175 
 •000766 
 •002510 
 •006575 
 •014356 
 •026867 
 •043994 
 •064036 
 ■083887 
 •099901 
 •109059 
 •109898 
 •102833 
 •089807 
 •073530 
 •056661 
 •041237 
 •028432 
 •018623 
 •011617 
 •006917 
 •003940 
 •002151 
 •001127 
 •000568 
 •000275 
 •000129 
 •000058 
 ■000025 
 •000011 
 •000004 
 •000002 
 
 ■oooooi 
 
 U-l 
 
 •000001 
 •000011 
 •000075 
 •000352 
 •001239 
 •003494 
 •008212 
 •016541 
 •029153 
 •045673 
 •064399 
 •082547 
 
 13-2 
 
 •000002 
 ■000024 
 •000161 
 •000709 
 •002341 
 •006180 
 •013596 
 •025639 
 •042304 
 •062046 
 •081901 
 •098281 
 •108109 
 •109773 
 •103500 
 •091080 
 •075141 
 •058345 
 •042786 
 •029725 
 •019619 
 •012332 
 •007399 
 •004246 
 •002336 
 •001233 
 •000626 
 •000306 
 •000144 
 •000066 
 •000029 
 •000012 
 •000005 
 ■000002 
 
 •oooooi 
 
 U-2 
 
 •oooooi 
 
 •000010 
 •000069 
 •000325 
 •001153 
 •003275 
 •007752 
 •015726 
 •027913 
 •044040 
 •062537 
 ■080730 
 
 13-S 
 
 •000002 
 •000022 
 •000148 
 •000657 
 •002183 
 •005807 
 •012872 
 •024458 
 •040661 
 •060088 
 •079916 
 •096626 
 •107094 
 •109566 
 •104087 
 •092291 
 •076717 
 •060019 
 •044348 
 •031043 
 •020644 
 •013074 
 •007904 
 •004571 
 •002533 
 ■001348 
 •000689 
 •000340 
 •000161 
 •000074 
 ■000033 
 •000014 
 •000006 
 •000002 
 •000001 
 
 US 
 
 •000001 
 •000009 
 •000063 
 •000300 
 •001073 
 •003070 
 •007316 
 •014946 
 •026715 
 •042447 
 •060700 
 •078910 
 
 lS-lf 
 
 13-5 
 
 •000002 
 •000020 
 •000136 
 •000608 
 •002035 
 •005455 
 •012183 
 •023322 
 •039064 
 •058161 
 •077936 
 •094940 
 ■106017 
 •109279 
 •104595 
 •093439 
 •078255 
 •061683 
 •045920 
 •032385 
 •021698 
 •013846 
 •008433 
 •004913 
 •002743 
 •001470 
 •000758 
 •000376 
 •000180 
 •000083 
 •000037 
 •000016 
 •000007 
 •000003 
 •OOOOOI 
 
 U: 
 
 •OOOOOI 
 •000008 
 •000058 
 •000277 
 •000999 
 •002876 
 •006902 
 •014199 
 •025559 
 •040894 
 •058887 
 •077089 
 
 •OOOOOI 
 •000019 
 •000125 
 •000562 
 •001897 
 •005123 
 •011526 
 •022230 
 •037512 
 •056269 
 •075963 
 •093227 
 •104880 
 •108914 
 •105024 
 •094522 
 •079753 
 •063333 
 •047500 
 •033750 
 •022781 
 •014645 
 •008987 
 •005275 
 •002967 
 •001602 
 •000832 
 ■000416 
 •000201 
 •000093 
 •000042 
 •000018 
 •000008 
 •000003 
 •OOOOOI 
 
 If -5 
 
 •OOOOOI 
 •000007 
 •000053 
 •000256 
 •000929 
 •002694 
 •006510 
 •013486 
 •024443 
 ■039380 
 •057101 
 •075270 
 
 13-6 
 
 •OOOOOI 
 •000017 
 •000115 
 •000520 
 •001768 
 •004810 
 •010902 
 •021181 
 •036007 
 •054410 
 •073998 
 •091489 
 •103687 
 •108473 
 •105373 
 •095539 
 ■081208 
 •064966 
 •049086 
 •035135 
 •023892 
 •015473 
 •009565 
 •005656 
 •003205 
 •001744 
 •000912 
 •000459 
 •000223 
 •000105 
 •000047 
 •000021 
 •000009 
 •000004 
 •OOOOOI 
 •OOOOOI 
 
 U-6 
 
 •000007 
 •000049 
 •000237 
 •000864 
 •002523 
 •006139 
 •012804 
 •023367 
 •037907 
 •055343 
 •073456 
 
 1S-7 
 
 •OOOOOI 
 •000015 
 •000105 
 •000481 
 •001648 
 •004514 
 •010308 
 •020173 
 ■034547 
 •052588 
 •072046 
 •089730 
 •102441 
 •107957 
 •105644 
 •096488 
 •082618 
 •066580 
 •050675 
 •036539 
 •025030 
 •016329 
 •010168 
 •006057 
 •003457 
 •001895 
 •000998 
 •000507 
 •000248 
 •000117 
 •000053 
 •000024 
 •000010 
 •000004 
 •000002 
 •OOOOOI 
 
 U-7 
 
 •000006 
 •000045 
 •000219 
 •000803 
 •002362 
 •005787 
 •012162 
 022330 
 •036472 
 ■053614 
 •071648 
 
 13 -8 
 
 •OOOOOI 
 •000014 
 •000097 
 •000445 
 ■001535 
 •004236 
 •009743 
 •019207 
 •033132 
 •050802 
 •070107 
 •087953 
 •101146 
 •107370 
 ■105836 
 ■097369 
 •083981 
 •068173 
 •052266 
 •037962 
 •026193 
 •017213 
 •010797 
 •006478 
 •003725 
 •002056 
 •001091 
 •000558 
 •000275 
 •000131 
 •000060 
 •000027 
 •000012 
 •000005 
 •000002 
 •OOOOOI 
 
 13-9 
 
 llf-8 
 
 •000006 
 •000041 
 •000202 
 •000747 
 •002211 
 •005454 
 •011530 
 •021331 
 •035078 
 •051915 
 •069850 
 
 •OOOOOI 
 •000013 
 •000089 
 •000411 
 •001429 
 •003974 
 •009206 
 •018280 
 •031762 
 •049054 
 •068185 
 •086162 
 •099804 
 •106713 
 •105951 
 •098181 
 •085295 
 •069741 
 •053856 
 •039400 
 •027383 
 ■018125 
 •011452 
 •006921 
 •004008 
 •002229 
 •001191 
 •000613 
 •000305 
 •000146 
 •000068 
 •000030 
 •000013 
 •000006 
 •000002 
 •OOOOOI 
 
 14-9 
 
 •000005 
 •000038 
 •000186 
 •000694 
 •002069 
 •005138 
 •010937 
 •020370 
 •033723 
 •050247 
 •068062 
 
 U-0 
 
 •OOOOOI 
 •000012 
 •000081 
 •000380 
 •001331 
 •003727 
 •008696 
 •017392 
 ■030435 
 •047344 
 •066282 
 •084359 
 •098418 
 •105989 
 •105989 
 •098923 
 •086558 
 •071283 
 •055442 
 •040852 
 •028597 
 •019064 
 •012132 
 •007385 
 ■004308 
 •002412 
 •001299 
 •000674 
 •000337 
 •000163 
 •000076 
 •000034 
 •000015 
 •000006 
 •000003 
 •OOOOOI 
 
 15-0 
 
 I 
 
 •000005 
 •000034 
 •000172 
 •000645 
 •001936 
 •004839 
 ■010370 
 •019444 
 •032407 
 •048611 
 ■066287 
 
 
 
 1 
 
 2 
 
 3 
 
 k 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 IS 
 
 1* 
 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 
 25 
 26 
 27 
 28 
 29 
 SO 
 SI 
 32 
 S3 
 S4 
 35 
 
 5 
 6 
 7 
 8 
 9 
 10 
 11 
 
Pomoris Exponential Binomial Limit 
 TABLE LI— (continued). 
 
 121 
 
 X 
 
 12 
 13 
 
 n 
 
 15 
 16 
 17 
 18 
 19 
 20 
 21 
 22 
 23 
 24. 
 25 
 26 
 27 
 28 
 29 
 30 
 31 
 32 
 33 
 Sit 
 35 
 36 
 37 
 
 TO 
 
 X 
 
 12 
 IS 
 
 H 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20- 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 SO 
 
 SI 
 
 32 
 
 S3 
 
 34 
 
 35 
 
 36 
 
 37 
 
 w 
 
 14-2 
 
 14s 
 
 U'4 
 
 U-5 
 
 14-6 
 
 14-7 
 
 14-8 
 
 14-9 
 
 15-0 
 
 •096993 
 
 •105200 
 
 •105951 
 
 •099594 
 
 •087768 
 
 •072795 
 
 •057023 
 
 •042317 
 
 •029834 
 
 •020031 
 
 •012838 
 
 •007870 
 
 •004624 
 
 •002608 
 
 •001414 
 
 •000739 
 
 ■000372 
 
 •000181 
 
 •000085 
 
 •000039 
 
 •000017 
 
 •000007 
 
 •000003 ' 
 
 •000001 
 
 •095530 
 •104349 
 ■105839 
 •100195 
 •088923 
 •074277 
 •058596 
 •043793 
 •031093 
 •021025 
 •013570 
 •008378 
 •004957 
 •002816 
 ■001538 
 •000809 
 •000410 
 ■000201 
 ■000095 
 ■000044 
 •000019 
 •000008 
 •000003 
 ■000001 
 •000001 
 
 •094034 
 •103437 
 •105654 
 •100723 
 •090021 
 •075724 
 •060158 
 •045277 
 •032373 
 •022045 
 •014329 
 •008909 
 •005308 
 •003036 
 •001670 
 •000884 
 •000452 
 ■000223 
 •000106 
 •000049 
 •000022 
 •000009 
 •000004 
 •000002 
 •000001 
 
 
 092507 
 102469 
 105396 
 101181 
 0910G3 
 077135 
 061708 
 046768 
 033673 
 023090 
 015114 
 009462 
 005677 
 003270 
 001811 
 000966 
 000497 
 000247 
 000118 
 000055 
 000025 
 000011 
 000005 
 000002 
 000001 
 
 •090951 
 •101446 
 •105069 
 •101567 
 •092045 
 •078509 
 •063243 
 •048264 
 •034992 
 •024161 
 •015924 
 •010039 
 •006065 
 •003518 
 ■001962 
 •001054 
 •000546 
 •000273 
 •000132 
 •000062 
 •000028 
 •000012 
 •000005 
 •000002 
 •000001 
 
 •089371 
 •100371 
 •104672 
 •101881 
 •092967 
 •079842 
 •064761 
 •049763 
 •036327 
 •025256 
 •016761 
 •010640 
 •006472 
 •003780 
 •002123 
 •001148 
 •000598 
 •000301 
 •000147 
 •000069 
 •000032 
 ■000014 
 ■000006 
 •000002 
 •000001 
 
 •087769 
 •099247 
 •104209 
 •102125 
 •093827 
 •081133 
 •066259 
 ■051263 
 •037678 
 •026375 
 ■017623 
 •011264 
 •006899 
 •004057 
 •002294 
 •001249 
 ■000656 
 ■000332 
 •000163 
 •000077 
 •000035 
 •000016 
 •000007 
 •000003 
 •000001 
 
 •086148 
 •098076 
 •103681 
 •102298 
 •094626 
 •082380 
 •067735 
 •052762 
 •039044 
 •027517 
 •018511 
 ■011911 
 •007345 
 •004348 
 •002475 
 •001357 
 •000717 
 •000366 
 •000181 
 •000086 
 •000040 
 •000018 
 •000008 
 •000003 
 •000001 
 •000001 
 
 •084510 
 ■096862 
 ■103089 
 •102402 
 •095361 
 •083581 
 ■069187 
 •054257 
 •040422 
 •028680 
 •019424 
 •012584 
 •007812 
 •004656 
 •002668 
 •001473 
 •000784 
 •000403 
 ■000200 
 •000096 
 •000045 
 •000020 
 •000009 
 •000004 
 •000002 
 
 •000001 
 
 •082859 
 •095607 
 •102436 
 •102436 
 096034 
 •084736 
 •070613 
 •055747 
 •041810 
 •029865 
 •020362 
 •013280 
 ■008300 
 •004980 
 •002873 
 •001596 
 •000855 
 •000442 
 •000221 
 •000107 
 •000050 
 •000023 
 
 •000010 
 
 •000004 
 ■000002 
 •000001 
 
 16 
 
122 
 
 Tables for Statisticians and Biometricians 
 
 TABLE LII. 
 
 Table of Poisson-Exponential for Cell Frequencies 1 to 30. 
 Cell Frequencies 
 
 
 a; 
 
 1 
 
 s 
 
 S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 
 «* 
 
 
 
 
 
 
 
 
 
 
 
 u 
 
 o 
 
 W 
 
 
 
 
 
 
 
 
 
 
 
 *» 
 
 20 
 
 
 
 
 
 
 
 
 
 
 
 £ 
 
 19 
 
 
 
 
 
 
 
 
 
 
 
 SP 
 
 18 
 
 
 
 
 
 
 
 
 
 
 
 " E S3 
 
 17 
 16 
 
 
 
 
 
 
 
 
 
 
 
 
 15 
 
 
 
 
 
 
 
 
 
 
 
 9 3 
 
 U 
 
 
 
 
 
 
 
 
 
 
 
 J § 
 
 IS 
 
 
 
 
 
 
 
 
 
 
 
 ?e£ 
 
 12 
 
 
 
 
 
 
 
 
 
 
 
 O 4) 
 
 11 
 
 
 
 
 
 
 
 
 
 
 
 
 w <£. 
 
 10 
 9 
 
 
 
 
 
 
 
 
 
 
 "005 
 
 -_ U 
 1 — 
 
 
 
 
 
 
 
 
 
 •oi^ 
 
 •050 
 
 s 
 
 
 
 
 
 
 
 
 •034 
 •302 
 
 •123 
 "623 
 
 •277 
 1*038 
 
 3 £ 
 
 7 
 
 
 
 
 
 
 
 ■091 
 
 §1 
 
 6 
 
 
 
 
 
 
 •248 
 
 1 -735 
 
 6-197 
 
 15-120 
 
 •730 
 
 1-375 
 
 2-123 
 
 2-925 
 
 5 
 
 4 
 S 
 
 
 
 
 
 •674 
 
 4-043 
 
 12-465 
 
 26-503 
 
 44-049 
 
 2-964 
 
 8-177 
 
 17-299 
 
 4-238 
 
 9-963 
 
 19-124 
 
 5-496 
 11-569 
 
 20-678 
 
 6-708 
 13-014 
 
 a 
 
 
 
 
 j-832 
 
 D 
 
 
 
 4-979 
 
 9-158 
 
 22-022 
 
 9 
 
 2 
 
 
 13 534 
 40-601 
 
 19-915 
 
 23-810 
 
 28-506 
 
 30-071 
 
 31-337 
 
 32-390 
 
 33-282 
 
 CM 
 
 1 
 
 36-788 
 
 42-319 
 
 43-347 
 
 44-568 
 
 44-971 
 
 45-296 
 
 45-565 
 
 45-793 
 
 
 Actual 
 
 36-788 
 
 27-067 
 
 22-404 
 
 19-537 
 
 17-547 
 
 16-062 
 
 14-900 
 
 13-959 
 
 13-176 
 
 12-511 
 
 1 
 
 1 
 
 M 
 
 1 
 
 26-424 
 
 32-332 
 
 35-277 
 
 37-116 
 
 38-404 
 
 39-370 
 
 40'129 
 
 40-745 
 
 41-259 
 
 41-696 
 
 2 
 
 8-030 
 
 14-288 
 
 18-474 
 
 21-487 
 
 23-782 
 
 25-602 
 
 27-091 
 
 28-338 
 
 29-401 
 
 30-323 
 
 S 
 
 1-899 
 
 5-265 
 
 8-392 
 
 11-067 
 
 13337 
 
 15-276 
 
 16-950 
 
 18-411 
 
 19-699 
 
 20-845 
 
 a 
 
 4 
 
 •366 
 
 1-656 
 
 3 351 
 
 5-113 
 
 6-809 
 
 8-392 
 
 9-852 
 
 11-192 
 
 12-422 
 
 13-554 
 
 
 5 
 
 •059 
 
 •453 
 
 1-191 
 
 2-136 
 
 3-183 
 
 4-262 
 
 5-335 
 
 6-380 
 
 7-385 
 
 8-346 
 
 o 
 
 6 
 
 •008 
 
 •110 
 
 ■380 
 
 ■813 
 
 1-370 
 
 2-009 
 
 2-700 
 
 3-418 
 
 4-146 
 
 4-875 
 
 a 
 
 7 
 
 •001 
 
 •024 
 
 •110 
 
 •284 
 
 •545 
 
 •883 
 
 1-281 
 
 1-726 
 
 2-203 
 
 2-705 
 
 u 
 
 Q 
 
 8 
 
 ■000 
 
 •005 
 
 •029 
 
 •092 
 
 •202 
 
 •363 
 
 ■572 
 
 •823 
 
 1-110 
 
 1-428 
 
 5* 
 
 9 
 
 
 
 •001 
 
 •007 
 
 •027 
 
 •070 
 
 •140 
 
 •241 
 
 •372 
 
 •532 
 
 •719 
 
 fci 
 
 10 
 
 
 
 •000 
 
 •002 
 
 •008 
 
 •023 
 
 ■051 
 
 •096 
 
 ■159 
 
 •242 
 
 •346 
 
 X> 
 
 11 
 
 
 
 
 
 •000 
 
 •002 
 
 •007 
 
 •018 
 
 ■036 
 
 •065 
 
 •105 
 
 •160 
 
 bo 
 
 12 
 
 — 
 
 
 
 — 
 
 •001 
 
 •002 
 
 •006 
 
 •013 
 
 •025 
 
 •044 
 
 •071 
 
 
 IS 
 
 
 
 
 
 — 
 
 •000 
 
 •001 
 
 •002 
 
 ■005 
 
 •009 
 
 •017 
 
 030 
 
 »3 -*^> 
 
 U 
 
 
 
 
 
 — 
 
 — 
 
 •000 
 
 ■001 
 
 ■002 
 
 •003 
 
 •007 
 
 •013 
 
 t! <j 
 
 IB 
 
 
 
 
 
 
 
 
 
 
 
 •000 
 
 •001 
 
 •001 
 
 •002 
 
 •006 
 
 s a 
 
 .22 
 
 cS IS 
 
 16 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •000 
 
 •000 
 
 •001 
 
 ■002 
 
 17 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 •000 
 
 •001 
 
 >• 
 
 18 
 
 — 
 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •001 
 
 o 
 
 s 
 
 I 
 
 B 
 § 
 
 o 
 
 19 
 20 
 21 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 - 
 
 ■000 
 
 22 
 2S 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 a 
 8 
 
 25 
 26 
 
 — 
 
 z 
 
 z 
 
 Z 
 
 — 
 
 z 
 
 — 
 
 Z 
 
 — 
 
 — 
 
 cm 
 
 27 
 28 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
Table of Poissoris Exponential 
 
 123 
 
 TABLE LII— (continued). 
 Cell Frequencies 
 
 
 X 
 
 11 
 
 12 
 
 IS 
 
 U 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 o 
 
 22 
 21 
 20 
 19 
 18 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .5 *4 
 
 17 
 
 
 
 
 
 
 
 
 
 •000 
 •001 
 •004 
 •015 
 •052 
 •151 
 •387 
 
 •flAA 
 
 01 rt 
 
 16 
 
 
 
 
 
 
 
 •000 
 •001 
 •004 
 •018 
 •067 
 •206 
 
 •000 
 •002 
 •008 
 •032 
 •104 
 •289 
 
 
 002 
 007 
 026 
 078 
 209 
 500 
 
 15 
 
 
 
 
 
 ■000 
 ■001 
 •004 
 •021 
 •086 
 
 •000 
 •002 
 •009 
 •040 
 •138 
 
 
 n 
 
 IS 
 
 
 
 
 •000 
 •001 
 •009 
 
 •047 
 
 
 Jp 
 
 
 
 •000 
 •003 
 •022 
 
 
 £ C! 
 
 12 
 11 
 
 
 •001 
 •008 
 
 
 
 •002 
 
 
 
 10 
 
 •020 
 
 •052 
 
 •105 
 
 •181 
 
 •279 
 
 •401 
 
 •543 
 
 •706 
 
 •886 
 
 1 
 
 081 
 
 
 9 
 
 •121 
 
 •229 
 
 •374 
 
 •553 
 
 ■763 
 
 1-000 
 
 1-260 
 
 1-538 
 
 1-832 
 
 2 
 
 139 
 
 8 
 
 •492 
 
 •760 
 
 1-073 
 
 1-423 
 
 1-800 
 
 2-199 
 
 2-612 
 
 3037 
 
 3-467 
 
 3 
 
 901 
 
 g 2 
 
 7 
 
 1-510 
 
 2-034 
 
 2-589 
 
 3-162 
 
 3745 
 
 4-330 
 
 4-912 
 
 5-489 
 
 6 056 
 
 6 
 
 613 
 
 I i 
 
 6 
 
 3-752 
 
 4-582 
 
 5-403 
 
 6-206 
 
 6-985 
 
 7-740 
 
 8-467 
 
 9-167 
 
 9-840 
 
 10 
 
 486 
 
 ■*i 
 
 5 
 
 7-861 
 
 8-950 
 
 9-976 
 
 10-940 
 
 11-846 
 
 12-699 
 
 13-502 
 
 14-260 
 
 14-975 
 
 15 
 
 651 
 
 a 
 
 <0 
 
 4 
 
 14-319 
 
 15-503 
 
 16-581 
 
 17-568 
 
 18-475 
 
 19-312 
 
 20-087 
 
 20-808 
 
 21-479 
 
 22 
 
 107 
 
 1 
 
 s 
 
 23-198 
 
 24-239 
 
 25-168 
 
 26-004 
 
 26-761 
 
 27-451 
 
 28-084 
 
 28-665 
 
 29-203 
 
 29 
 
 703 
 
 2 
 
 34 051 
 
 34-723 
 
 35-317 
 
 35-846 
 
 36-322 
 
 36-753 
 
 37-146 
 
 37-505 
 
 37-836 
 
 38 
 
 142 
 
 
 1 
 
 45-989 
 
 46-150 
 
 46-311 
 
 46-445 
 
 46-565 
 
 46-674 
 
 46-774 
 
 46-865 
 
 46-948 
 
 47-026 
 
 
 Actual 
 
 11-938 
 
 11-437 
 
 10-994 
 
 10-599 
 
 10-244 
 
 9-922 
 
 9-629 
 
 9360 
 
 9112 
 
 8-884 
 
 1 
 
 1 
 
 42-073 
 
 42-404 
 
 42-695 
 
 42-956 
 
 43-191 
 
 43-404 
 
 43-597 
 
 43-776 
 
 43-939 
 
 44-091 
 
 2 
 
 31-130 
 
 31-846 
 
 32-486 
 
 33 064 
 
 33 588 
 
 34 066 
 
 34-503 
 
 34-909 
 
 35 
 
 283 
 
 35 
 
 630 
 
 9 
 
 S 
 
 21-871 
 
 22-798 
 
 23-639 
 
 24-408 
 
 25-114 
 
 25-765 
 
 26-367 
 
 26-928 
 
 27 
 
 451 
 
 27 
 
 939 
 
 a 
 
 4 
 
 14-596 
 
 15-559 
 
 16-450 
 
 17-280 
 
 18-053 
 
 18-776 
 
 19-451 
 
 20-088 
 
 20 
 
 686 
 
 21 
 
 251 
 
 g 
 
 o 
 
 5 
 
 9-261 
 
 10-129 
 
 10-953 
 
 11-736 
 
 12-478 
 
 13-184 
 
 13-852 
 
 14-491 
 
 15 
 
 099 
 
 15 
 
 677 
 
 6 
 
 5-593 
 
 6-297 
 
 6-983 
 
 7-650 
 
 8-297 
 
 8-923 
 
 9-526 
 
 10-111 
 
 10 
 
 675 
 
 11 
 
 219 
 
 a 
 
 7 
 
 3219 
 
 3-742 
 
 4-266 
 
 4-791 
 
 5-311 
 
 5-825 
 
 6329 
 
 6-826 
 
 7 
 
 313 
 
 7 
 
 789 
 
 
 8 
 
 1-769 
 
 2-128 
 
 2-501 
 
 2-884 
 
 3-275 
 
 3-669 
 
 4-064 
 
 4-461 
 
 4 
 
 856 
 
 5 
 
 248 
 
 
 
 9 
 
 •929 
 
 1-160 
 
 1-407 
 
 1-671 
 
 1-947 
 
 2-232 
 
 2-523 
 
 2-824 
 
 3 
 
 127 
 
 3 
 
 433 
 
 fc" 
 
 10 
 
 •467 
 
 •607 
 
 •762 
 
 •933 
 
 1-117 
 
 1-312 
 
 1-516 
 
 1-732 
 
 1 
 
 954 
 
 2 
 
 182 
 
 ^o 
 
 11 
 
 •225 
 
 •305 
 
 •396 
 
 •502 
 
 •619 
 
 •746 
 
 •882 
 
 1-030 
 
 1 
 
 185 
 
 1 
 
 348 
 
 
 12 
 
 •104 
 
 •148 
 
 •201 
 
 •261 
 
 ■331 
 
 •411 
 
 ■497 
 
 •595 
 
 
 699 
 
 
 809 
 
 fi 3 
 
 IS 
 
 •047 
 
 •069 
 
 •097 
 
 •131 
 
 •172 
 
 •219 
 
 •272 
 
 •333 
 
 
 400 
 
 
 473 
 
 1-3 
 
 u 
 
 •020 
 
 •031 
 
 •046 
 
 •063 
 
 •086 
 
 •114 
 
 •144 
 
 •182 
 
 
 223 
 
 
 269 
 
 « S 
 
 15 
 
 •008 
 
 •014 
 
 •021 
 
 •030 
 
 •042 
 
 •057 
 
 •074 
 
 •096 
 
 
 121 
 
 
 149 
 
 2 o 
 
 16 
 
 •003 
 
 •006 
 
 •009 
 
 •013 
 
 •020 
 
 •028 
 
 •036 
 
 •050 
 
 
 064 
 
 
 081 
 
 -a* 
 
 17 
 
 •001 
 
 •002 
 
 ■004 
 
 •006 
 
 •009 
 
 •014 
 
 •017 
 
 •025 
 
 
 033 
 
 
 042 
 
 > 
 
 18 
 
 ■001 
 
 •000 
 
 •002 
 
 •002 
 
 •004 
 
 ■006 
 
 •008 
 
 •012 
 
 
 017 
 
 
 022 
 
 o 
 
 19 
 
 •000 
 
 •000 
 
 •001 
 
 •001 
 
 •002 
 
 •003 
 
 •003 
 
 •006 
 
 
 008 
 
 
 011 
 
 OB 
 O 
 
 20 
 
 
 
 — 
 
 •001 
 
 •000 
 
 •001 
 
 •002 
 
 002 
 
 •003 
 
 
 004 
 
 
 005 
 
 
 21 
 
 
 
 
 
 •000 
 
 •000 
 
 •000 
 
 •001 
 
 ■001 
 
 •002 
 
 
 002 
 
 
 003 
 
 
 22 
 
 
 
 
 
 — 
 
 — 
 
 — 
 
 •000 
 
 •000 
 
 •001 
 
 
 001 
 
 
 001 
 
 B 
 
 23 
 
 25 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •000 
 
 •000 
 
 
 001 
 000 
 
 8 
 JP 
 
 26 
 
 27 
 28 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 16—2 
 
124 
 
 Tables for Statisticians and Biometricians 
 
 TABLE LII— {continued). 
 Cell Frequencies 
 
 
 # 
 
 21 
 
 22 
 
 2S 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 SO 
 
 9 
 
 22 
 
 .'7 
 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 •000 
 
 •ooo 
 
 •ooo 
 
 •000 
 001 
 
 ^ 
 
 20 
 
 — 
 
 
 
 — 
 
 — 
 
 000 
 
 •000 
 
 •001 
 
 
 001 
 
 
 001 
 
 •002 
 
 Jfr 
 
 19 
 
 — 
 
 — 
 
 
 000 
 
 •ooo 
 
 •001 
 
 •001 
 
 •002 
 
 
 003 
 
 
 004 
 
 ■008 
 
 00 
 
 18 
 
 
 000 
 
 •000 
 
 
 001 
 
 •001 
 
 •002 
 
 •004 
 
 •006 
 
 
 009 
 
 
 012 
 
 •017 
 
 Eh C3 
 
 n 
 
 
 001 
 
 ■002 
 
 
 003 
 
 •005 
 
 •008 
 
 •on 
 
 •016 
 
 
 023 
 
 
 031 
 
 •041 
 
 iS 3 
 
 16 
 
 
 003 
 
 •006 
 
 
 010 
 
 •015 
 
 •022 
 
 •031 
 
 •043 
 
 
 056 
 
 
 073 
 
 ■092 
 
 2 a 
 
 -3 o 
 
 15 
 
 
 012 
 
 •020 
 
 
 030 
 
 •043 
 
 •059 
 
 •078 
 
 •102 
 
 
 129 
 
 
 160 
 
 •195 
 
 H 
 
 
 039 
 
 •058 
 
 
 081 
 
 •109 
 
 •142 
 
 •180 
 
 •224 
 
 
 273 
 
 
 328 
 
 •387 
 
 IS 
 
 
 111 
 
 •150 
 
 
 198 
 
 •252 
 
 ■314 
 
 •384 
 
 •460 
 
 
 543 
 
 
 632 
 
 •727 
 
 i« 
 
 12 
 
 
 277 
 
 •355 
 
 
 443 
 
 •540 
 
 •647 
 
 •762 
 
 •884 
 
 1 
 
 014 
 
 1 
 
 161 
 
 1 293 
 
 I* 
 
 11 
 
 
 625 
 
 •763 
 
 
 912 
 
 1-072 
 
 1-240 
 
 1-417 
 
 1-601 
 
 1 
 
 791 
 
 1 
 
 987 
 
 2-187 
 
 10 
 
 1 
 
 290 
 
 1-512 
 
 1 
 
 743 
 
 1-983 
 
 2-229 
 
 2-482 
 
 2-739 
 
 3 
 
 000 
 
 3 
 
 263 
 
 3-528 
 
 9 
 
 2 
 
 455 
 
 2-778 
 
 3 
 
 107 
 
 3-440 
 
 3-775 
 
 4-111 
 
 4-446 
 
 4 
 
 781 
 
 5 
 
 114 
 
 5-444 
 
 g.s 
 
 8 
 
 4 
 
 336 
 
 4-769 
 
 5 
 
 200 
 
 5-626 
 
 6-048 
 
 6-463 
 
 6-872 
 
 7 
 
 274 
 
 7 
 
 669 
 
 8-057 
 
 
 7 
 
 7 
 
 157 
 
 7-689 
 
 8 
 
 208 
 
 8-713 
 
 9-204 
 
 9-682 
 
 10-147 
 
 10 
 
 599 
 
 11 
 
 038 
 
 11-465 
 
 J 1 
 
 6 
 
 11 
 
 107 
 
 11-704 
 
 12 
 
 277 
 
 12827 
 
 13-358 
 
 13-867 
 
 14-357 
 
 14 
 
 830 
 
 15 
 
 285 
 
 15-724 
 
 5 
 
 16 
 
 292 
 
 16-900 
 
 17 
 
 477 
 
 18-025 
 
 18-549 
 
 19-048 
 
 19-525 
 
 19 
 
 981 
 
 20 
 
 417 
 
 20-836 
 
 a 
 
 s 
 
 4 
 
 22 
 
 696 
 
 23-250 
 
 23 
 
 771 
 
 24-263 
 
 24-730 
 
 25-172 
 
 25-591 
 
 25 
 
 990 
 
 26 
 
 371 
 
 26-734 
 
 3 
 
 30 
 
 168 
 
 30-603 
 
 31 
 
 010 
 
 31-391 
 
 31-753 
 
 32-094 
 
 32-416 
 
 32 
 
 721 
 
 33 
 
 Oil 
 
 33-287 
 
 ■ 
 
 2 
 
 38 
 
 426 
 
 38-691 
 
 38 
 
 938 
 
 39-168 
 
 39-387 
 
 39-693 
 
 39-786 
 
 39 
 
 970 
 
 40 
 
 143 
 
 40-308 
 
 Ph 
 
 1 
 
 47-097 
 
 47-164 
 
 47-226 
 
 47-283 
 
 47-340 
 
 47-392 
 
 47-440 
 
 47-486 
 
 47 530 
 
 47-572 
 
 
 Actual 
 
 8-671 
 
 8-473 
 
 8-288 
 
 8-115 
 
 7-952 
 
 7-799 
 
 7-654 
 
 7-517 
 
 7-387 
 
 7-264 
 
 ■ 
 
 1 
 
 44-232 
 
 44-363 
 
 44-485 
 
 44-603 
 
 44-708 
 
 44-810 
 
 44-906 
 
 44-997 
 
 45-083 
 
 45-165 
 
 1 
 
 2 
 
 35 
 
 955 
 
 36-258 
 
 36-542 
 
 36 
 
 812 
 
 37-062 
 
 37-299 
 
 37-525 
 
 37 
 
 739 
 
 37-942 
 
 38-135 
 
 O 
 
 S 
 
 28 
 
 397 
 
 28-828 
 
 29-235 
 
 29 
 
 620 
 
 29-982 
 
 30-326 
 
 30-653 
 
 30 
 
 965 
 
 31-262 
 
 31-546 
 
 eg 
 
 4 
 
 21 
 
 785 
 
 22-290 
 
 22-770 
 
 23 
 
 227 
 
 23-660 
 
 24-074 
 
 24-469 
 
 24 
 
 847 
 
 25-208 
 
 25-555 
 
 
 5 
 
 16 
 
 230 
 
 16-758 
 
 17-264 
 
 17 
 
 748 
 
 18-211 
 
 18-655 
 
 19-083 
 
 19 
 
 493 
 
 19-888 
 
 20-269 
 
 o 
 
 6 
 
 11 
 
 744 
 
 12-251 
 
 12-740 
 
 13 
 
 213 
 
 13-669 
 
 14-110 
 
 14-538 
 
 14 
 
 951 
 
 15-351 
 
 15-738 
 
 a 
 
 7 
 
 8 
 
 254 
 
 8-709 
 
 9-153 
 
 9 
 
 585 
 
 10-007 
 
 10-418 
 
 10-819 
 
 11 
 
 210 
 
 11-591 
 
 11-962 
 
 fa 
 
 8 
 
 5 
 
 637 
 
 6-022 
 
 6-402 
 
 6 
 
 777 
 
 7-146 
 
 7-509 
 
 7-866 
 
 8 
 
 218 
 
 8-562 
 
 8-901 
 
 H 
 
 9 
 
 3 
 
 742 
 
 4-052 
 
 4-362 
 
 4 
 
 670 
 
 4-978 
 
 5-284 
 
 5-588 
 
 5 
 
 890 
 
 6-188 
 
 6-484 
 
 >> 
 
 10 
 
 2 
 
 415 
 
 2-654 
 
 2-895 
 
 3 
 
 138 
 
 3-385 
 
 3-632 
 
 3-880 
 
 4 
 
 129 
 
 4-377 
 
 4-625 
 
 X> 
 
 11 
 
 1 
 
 517 
 
 1-692 
 
 1-873 
 
 2 
 
 057 
 
 2-246 
 
 2-438 
 
 2-633 
 
 2 
 
 831 
 
 3-030 
 
 3-230 
 
 
 12 
 
 
 927 
 
 1-051 
 
 1-181 
 
 1 
 
 315 
 
 1-456 
 
 1-599 
 
 1-747 
 
 1 
 
 899 
 
 2-053 
 
 2-210 
 
 > 3 
 
 IS 
 
 
 552 
 
 •637 
 
 •727 
 
 
 821 
 
 •921 
 
 1-025 
 
 1-134 
 
 1 
 
 247 
 
 1-362 
 
 1-481 
 
 jg-s 
 
 U 
 
 
 320 
 
 •376 
 
 •437 
 
 
 500 
 
 •570 
 
 •643 
 
 •720 
 
 
 801 
 
 •885 
 
 •973 
 
 
 16 
 
 
 181 
 
 •217 
 
 •256 
 
 
 298 
 
 •345 
 
 •394 
 
 •448 
 
 
 504 
 
 •564 
 
 •626 
 
 16 
 
 
 100 
 
 •122 
 
 •147 
 
 
 173 
 
 •204 
 
 •237 
 
 •272 
 
 
 311 
 
 •352 
 
 •395 
 
 "3«t5 
 
 17 
 
 
 054 
 
 •067 
 
 ■082 
 
 
 098 
 
 •118 
 
 •139 
 
 •162 
 
 
 188 
 
 •215 
 
 •245 
 
 t» 
 
 18 
 
 
 028 
 
 •036 
 
 •045 
 
 
 055 
 
 •067 
 
 •080 
 
 •095 
 
 
 111 
 
 •129 
 
 •149 
 
 *o 
 
 19 
 
 
 015 
 
 •019 
 
 •024 
 
 
 030 
 
 ■037 
 
 •045 
 
 •054 
 
 
 065 
 
 •076 
 
 •089 
 
 s 
 
 20 
 
 
 007 
 
 010 
 
 •013 
 
 
 016 
 
 ■020 
 
 •025 
 
 •031 
 
 
 037 
 
 •044 
 
 •052 
 
 a 
 
 21 
 
 
 004 
 
 •005 
 
 •007 
 
 
 009 
 
 •Oil 
 
 •014 
 
 •017 
 
 
 021 
 
 •025 
 
 •030 
 
 | 
 
 22 
 
 
 002 
 
 •002 
 
 •004 
 
 
 005 
 
 •006 
 
 •007 
 
 •009 
 
 
 Oil 
 
 ■014 
 
 •017 
 
 g 
 
 28 
 
 
 001 
 
 •001 
 
 •002 
 
 
 003 
 
 •003 
 
 •004 
 
 •005 
 
 
 006 
 
 •007 
 
 •010 
 
 5 
 
 U 
 
 
 001 
 
 •001 
 
 •001 
 
 
 002 
 
 •002 
 
 •002 
 
 •003 
 
 
 003 
 
 •004 
 
 •006 
 
 a 
 
 25 
 
 
 000 
 
 •000 
 
 •001 
 
 
 001 
 
 •001 
 
 •001 
 
 •001 
 
 
 002 
 
 •002 
 
 •003 
 
 1 
 
 26 
 
 __ 
 
 
 
 •000 
 
 
 000 
 
 •000 
 
 •001 
 
 •001 
 
 
 001 
 
 •001 
 
 •002 
 
 S3 
 
 27 
 
 
 
 
 
 
 
 
 
 
 
 •000 
 
 •ooo 
 
 •000 
 
 •000 
 
 ■001 
 
 &H 
 
 28 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 ~ 
 
 •ooo 
 
 1 
 
TABLE LIII. Angles, Arcs and Decimals of Degrees 125 
 
 Lengths of Ciboclab Ancs 
 
 Deg. 
 
 Arc 
 
 Deg. 
 
 Ave 
 
 Deg. 
 
 Are 
 
 ' 
 
 Deg. 
 
 Are 
 
 n 
 
 Deg. 
 
 Arc 
 
 1 
 
 •017 4533 
 
 61 
 
 1-064 6508 
 
 121 
 
 2-111 8484 
 
 I 
 
 •01667 
 
 •000 2909 
 
 1 
 
 •00028 
 
 •000 0048 
 
 2 
 
 •034 9066 
 
 62 
 
 1082 1041 
 
 12i 
 
 2-129 3017 
 
 2 
 
 •03333 
 
 •000 5818 
 
 2 
 
 •00056 
 
 •000 0097 
 
 S 
 
 •052 3599 
 
 6S 
 
 1-099 5574 
 
 12.i 
 
 2-146 7550 
 
 S 
 
 •05000 
 
 •000 8727 
 
 3 
 
 •00083 
 
 •000 0145 
 
 4 
 
 •069 8132 
 
 64 
 
 1-117 0107 
 
 124 
 
 2-164 2083 
 
 4 
 
 •06667 
 
 001 1636 
 
 4 
 
 •00111 
 
 •0000194 
 
 s 
 
 •087 2665 
 
 65 
 
 1-134 4640 
 
 125 
 
 2-181 6616 
 
 5 
 
 •08333 
 
 •001 4544 
 
 5 
 
 •00139 
 
 •000 0242 
 
 6 
 
 •104 7198 
 
 66 
 
 1-151 9173 
 
 126 
 
 2199 1149 
 
 6 
 
 •10000 
 
 ■001 7453 
 
 a 
 
 •00167 
 
 •0000291 
 
 7 
 
 •122 1730 
 
 67 
 
 1-169 3706 
 
 127 
 
 2-216 5682 
 
 7 
 
 • -11667 
 
 •002 0362 
 
 7 
 
 •00194 
 
 000 0339 
 
 8 
 
 ■139 6263 
 
 68 
 
 1-186 8239 
 
 128 
 
 2-234 0214 
 
 8 
 
 •13333 
 
 ■002 3271 
 
 8 
 
 ■00222 
 
 •000 0388 
 
 y 
 
 •157 0796 
 
 69 
 
 1-204 2772 
 
 129 
 
 2-251 4747 
 
 9 
 
 •15000 
 
 ■002 6180 
 
 9 
 
 ■00250 
 
 •0000436 
 
 10 
 
 •174 5329 
 
 70 
 
 1-221 7305 
 
 130 
 
 2-268 9280 
 
 10 
 
 •16667 
 
 ■002 9089 
 
 10 
 
 •00278 
 
 ■0000485 
 
 u 
 
 •191 9862 
 
 71 
 
 1-2391838 
 
 1S1 
 
 2-286 3813 
 
 11 
 
 •18333 
 
 •003 1998 
 
 11 
 
 •00306 
 
 •0000533 
 
 12 
 
 •209 4395 
 
 72 
 
 1-256 6371 
 
 1S2 
 
 2-3038346 
 
 12 
 
 •20000 
 
 •003 4907 
 
 12 
 
 •00333 
 
 •000 0582 
 
 13 
 
 •226 8928 
 
 7S 
 
 1-274 0904 
 
 1S3 
 
 2-321 2879 
 
 IS 
 
 •21667 
 
 •003 7815 
 
 IS 
 
 •00361 
 
 •0000630 
 
 U 
 
 •244 3461 
 
 74 
 
 1-291 5436 
 
 134 
 
 2-338 7412 
 
 14 
 
 •23333 
 
 •004 0724 
 
 14 
 
 •00389 
 
 •0000679 
 
 15 
 
 •261 7994 
 
 75 
 
 1-308 9969 
 
 185 
 
 2-356 1946 
 
 15 
 
 •25000 
 
 •004 3633 
 
 15 
 
 ■00417 
 
 •0000727 
 
 1C 
 
 •2792527 
 
 76 
 
 1-326 4502 
 
 lm 
 
 2-373 6478 
 
 16 
 
 •26667 
 
 004 6542 
 
 16 
 
 ■00444 
 
 ■0000776 
 
 17 
 
 •296 7060 
 
 77 
 
 1-343 9035 
 
 137 
 
 2-391 1011 
 
 17 
 
 •28333 
 
 004 9451 
 
 17 
 
 •00472 
 
 •0000824 
 
 18 
 
 •3141593 
 
 78 
 
 1-361 3568 
 
 138 
 
 2-408 5544 
 
 18 
 
 •30000 
 
 •005 2360 
 
 18 
 
 •00500 
 
 •0000873 
 
 19 
 
 •331 6126 
 
 79 
 
 1-378 8101 
 
 139 
 
 2-4260077 
 
 19 
 
 •31667 
 
 •005 5269 
 
 19 
 
 •00528 
 
 •0000921 
 
 20 
 
 •3490659 
 
 80 
 
 1-396 2634 
 
 140 
 
 2-443 4610 
 
 20 
 
 •33333 
 
 005 8178 
 
 20 
 
 •00556 
 
 •0000970 
 
 21 
 
 •366 5191 
 
 81 
 
 1-413 7167 
 
 141 
 
 2-460 9142 
 
 21 
 
 •35000 
 
 •006 1087 
 
 21 
 
 •00583 
 
 •000 1018 
 
 22 
 
 •383 9724 
 
 82 
 
 1-431 1700 
 
 142 
 
 2-478 3675 
 
 22 
 
 •36667 
 
 •006 3995 
 
 22 
 
 •00611 
 
 •000 1067 
 
 23 
 
 •401 4257 
 
 83 
 
 1-448 6233 
 
 143 
 
 2-495 8208 
 
 23 
 
 •38333 
 
 •006 6904 
 
 2S 
 
 •00639 
 
 0001115 
 
 ** 
 
 •418 8790 
 
 84 
 
 1-466 0766 
 
 144 
 
 2-513 2741 
 
 24 
 
 •40000 
 
 •006 9813 
 
 24 
 
 •00667 
 
 •0001164 
 
 25 
 
 •436 3323 
 
 85 
 
 1-483 5299 
 
 145 
 
 2-530 7274 
 
 25 
 
 •41667 
 
 •007 2722 
 
 25 
 
 •00694 
 
 •0001212 
 
 26 
 
 •453 7856 
 
 86 
 
 1-500 9832 
 
 146 
 
 2-548 1807 
 
 26 
 
 •43333 
 
 •007 5631 
 
 26 
 
 •00722 
 
 •000 1261 
 
 27 
 
 •471 2389 
 
 87 
 
 1-518 4364 
 
 147 
 
 2-565 6340 
 
 27 
 
 •45000 
 
 •007 8540 
 
 27 
 
 •00750 
 
 •000 1309 
 
 28 
 
 ■488 6922 
 
 88 
 
 1-535 8897 
 
 148 
 
 2-5830873 
 
 28 
 
 •46667 
 
 •008 1449 
 
 28 
 
 •00778 
 
 •0001357 
 
 29 
 
 •506 1455 
 
 89 
 
 1-553 3430 
 
 149 
 
 2-600 5406 
 
 29 
 
 •48333 
 
 •008 4358 
 
 29 
 
 •00806 
 
 ■000 1406 
 
 SO 
 
 •523 5988 
 
 90 
 
 1-570 7963 
 
 150 
 
 2-617 9939 
 
 SO 
 
 •50000 
 
 008 7266 
 
 SO 
 
 •00833 
 
 000 1454 
 
 SI 
 
 •541 0521 
 
 91 
 
 1-588 2496 
 
 151 
 
 2-635 4472 
 
 SI 
 
 •51667 
 
 •009 0175 
 
 SI 
 
 •00861 
 
 •000 1503 
 
 S2 
 
 •558 5054 
 
 92 
 
 1-605 7029 
 
 152 
 
 2-652 9005 
 
 32 
 
 •53333 
 
 •0093084 
 
 32 
 
 ■00889 
 
 •000 1551 
 
 33 
 
 •575 9587 
 
 9S 
 
 1-6231562 
 
 153 
 
 2-6703538 
 
 SS 
 
 •55000 
 
 •0095993 
 
 S3 
 
 00917 
 
 0001600 
 
 84 
 
 •593 4119 
 
 94 
 
 1-6406095 
 
 154 
 
 2-687 8070 
 
 34 
 
 •56667 
 
 •009 8902 
 
 S4 
 
 ■00944 
 
 000 1648 
 
 35 
 
 •610 8652 
 
 95 
 
 1-658 0628 
 
 155 
 
 2-705 2603 
 
 35 
 
 •58333 
 
 •0101811 
 
 35 
 
 •00972 
 
 •000 1697 
 
 36 
 
 •628 3185 
 
 96 
 
 1-675 5161 
 
 156 
 
 2-722 7136 
 
 S6 
 
 •60000 
 
 •0104720 
 
 86 
 
 •01000 
 
 •000 1745 
 
 37 
 
 •645 7718 
 
 97 
 
 1-692 9694 
 
 157 
 
 2-740 1669 
 
 37 
 
 •61667 
 
 •010 7629 
 
 S7 
 
 •01028 
 
 •000 1794 
 
 S8 
 
 •663 2251 
 
 98 
 
 1-710 4227 
 
 158 
 
 2-757 6202 
 
 38 
 
 •63333 
 
 •0110538 
 
 38 
 
 •01056 
 
 •000 1842 
 
 39 
 
 •680 6784 
 
 99 
 
 1-727 8760 
 
 159 
 
 2-775 0735 
 
 39 
 
 •65000 
 
 •Oil 3446 
 
 S9 
 
 •01083 
 
 •000 1891 
 
 40 
 
 •698 1317 
 
 100 
 
 1-745 3293 
 
 160 
 
 2-792 5268 
 
 40 
 
 •66667 
 
 ■Oil 6355 
 
 40 
 
 01111 
 
 •000 1939 
 
 41 
 
 •715 5850 
 
 101 
 
 1-762 7825 
 
 161 
 
 2-809 9801 
 
 41 
 
 •68333 
 
 •Oil 9264 
 
 41 
 
 •01139 
 
 000 1988 
 
 42 
 
 •7330383 
 
 102 
 
 1-780 2358 
 
 162 
 
 2-827 4334 
 
 42 
 
 •70000 
 
 •012 2173 
 
 42 
 
 •01167 
 
 •000 2036 
 
 43 
 
 •750 4916 
 
 10S 
 
 1-797 6891 
 
 163 
 
 2-844 8867 
 
 43 
 
 •71667 
 
 •012 5082 
 
 43 
 
 •01194 
 
 •000 2085 
 
 44 
 
 •767 9449 
 
 104 
 
 1-8151424 
 
 164 
 
 2-862 3400 
 
 u 
 
 •73333 
 
 •012 7991 
 
 44 
 
 •01222 
 
 ■000 2133 
 
 45 
 
 •785 3982 
 
 105 
 
 1-832 5957 
 
 165 
 
 2-879 7933 
 
 45 
 
 •75000 
 
 •013 0900 
 
 45 
 
 •01250 
 
 •000 2182 
 
 46 
 
 •802 8515 
 
 106 
 
 1-8500490 
 
 166 
 
 2-897 2466 
 
 46 
 
 ■76667 
 
 •0133809 
 
 46 
 
 •01278 
 
 •000 2230 
 
 47 
 
 •820 3047 
 
 107 
 
 1-867 5023 
 
 167 
 
 2-914 6999 
 
 47 
 
 •78333 
 
 •0136717 
 
 47 
 
 •01306 
 
 •000 2279 
 
 48 
 
 ■837 7580 
 
 108 
 
 1-884 9556 
 
 168 
 
 2-932 1531 
 
 48 
 
 •80000 
 
 •013 9626 
 
 48 
 
 01333 
 
 •000 2327 
 
 49 
 
 •855 2113 
 
 109 
 
 1-902 4089 
 
 169 
 
 2-949 6064 
 
 49 
 
 ■81667 
 
 •014 2535 
 
 49 
 
 •01361 
 
 •000 2376 
 
 50 
 
 •872 6646 
 
 110 
 
 1-919 8622 
 
 170 
 
 2-967 0597 
 
 50 
 
 •83333 
 
 014 5444 
 
 50 
 
 •01389 
 
 •000 2424 
 
 51 
 
 •8901179 
 
 111 
 
 1-937 3155 
 
 171 
 
 2-984 5130 
 
 51 
 
 •85000 
 
 •014 8353 
 
 51 
 
 •01417 
 
 •000 2473 
 
 52 
 
 •907 5712 
 
 112 
 
 1-954 7688 
 
 172 
 
 3-001 9663 
 
 52 
 
 •86667 
 
 •015 1262 
 
 52 
 
 •01444 
 
 •000 2621 
 
 53 
 
 •925 0245 
 
 US 
 
 1-972 2221 
 
 178 
 
 3019 4196 
 
 5S 
 
 •88333 
 
 •015 4171 
 
 53 
 
 •01472 
 
 •000 2570 
 
 54 
 
 ■942 4778 
 
 114 
 
 1-989 6753 
 
 174 
 
 3-036 8729 
 
 54 
 
 •90000 
 
 •015 7080 
 
 54 
 
 •01500 
 
 •000 2618 
 
 55 
 
 •959 9311 
 
 115 
 
 2007 1286 
 
 175 
 
 3054 3262 
 
 55 
 
 •91667 
 
 •015 9989 
 
 55 
 
 •01528 
 
 •000 2666 
 
 56 
 
 •977 3844 
 
 116 
 
 2024 5819 
 
 176 
 
 3-071 7795 
 
 56 
 
 •93333 
 
 •016 2897 
 
 56 
 
 •01556 
 
 •000 2715 
 
 57 
 
 •994 8377 
 
 117 
 
 2-042 0352 
 
 177 
 
 3089 2328 
 
 57 
 
 •95000 
 
 •016 5806 
 
 57 
 
 •01583 
 
 •000 2763 
 
 58 
 
 1-012 2910 
 
 118 
 
 2-059 4885 
 
 178 
 
 3-106 6861 
 
 58 
 
 •96667 
 
 ■016 8715 
 
 58 
 
 •01611 
 
 •000 2812 
 
 59 
 
 1029 7443 
 
 119 
 
 2-076 9418 
 
 179 
 
 3-124 1394 
 
 59 
 
 •98333 
 
 •017 1624 
 
 59 
 
 •01639 
 
 •000 2860 
 
 60 
 
 1047 1976 
 
 120 
 
 2-094 3951 
 
 180 
 
 3141 5927 
 
 60 
 
 1-00000 
 
 017 4533 
 
 60 
 
 ■01667 
 
 •000 2909 
 
126 
 
 Tables for Statisticians and Biometriciaim 
 TABLE LIV. The G(r, v)-Integrals. 
 
 6 
 
 7 
 8 
 9 
 
 10 
 
 11 
 12 
 13 
 
 u 
 
 15 
 
 16 
 17 
 18 
 19 
 
 21 
 
 25 
 
 SO 
 
 SI 
 32 
 3S 
 
 U 
 35 
 
 36 
 37 
 38 
 39 
 40 
 
 42 
 43 
 U 
 45 
 
 r=l 
 
 logF(r, v) 
 
 0-301 0300 
 •301 0609 
 •301 1538 
 •301 3087 
 •301 5262 
 •301 8067 
 
 •302 1508 
 •302 5594 
 •303 0335 
 ■303 5741 
 •304 1825 
 
 ■304 8603 
 ■305 6091 
 •306 4307 
 •307 3271 
 •308 3006 
 
 •309 3538 
 •310 4892 
 •311 7098 
 •313 0189 
 •314 4200 
 
 •315 9169 
 •317 5137 
 •319 2150 
 •321 0256 
 •322 9507 
 
 •324 9963 
 •327 1685 
 •329 4740 
 •331 9202 
 ■334 5150 
 
 ■337 2672 
 •340 1860 
 •343 2818 
 •346 5656 
 •350 0496 
 
 •353 7469 
 •357 6722 
 •361 8410 
 •366 2708 
 •370 9805 
 
 •375 9908 
 •381 3246 
 •387 0070 
 •393 0658 
 •399 5316 
 
 309 
 
 928 
 
 1550 
 
 2175 
 
 2805 
 
 3441 
 4086 
 4740 
 5406 
 6085 
 
 6778 
 7488 
 8216 
 8964 
 9735 
 
 10532 
 11353 
 12206 
 13091 
 14011 
 
 14969 
 15968 
 17013 
 18106 
 19252 
 
 20456 
 21722 
 23055 
 24462 
 25948 
 
 27521 
 29188 
 30958 
 32838 
 34840 
 
 36974 
 39252 
 41689 
 44298 
 47096 
 
 50103 
 53338 
 
 56824 
 60588 
 64058 
 
 A* 
 
 619 
 621 
 625 
 630 
 636 
 
 645 
 654 
 666 
 679 
 693 
 
 710 
 
 728 
 749 
 771 
 796 
 
 822 
 853 
 885 
 919 
 958 
 
 999 
 1045 
 1093 
 1146 
 1204 
 
 1266 
 1334 
 1407 
 1486 
 1673 
 
 1667 
 1769 
 1881 
 2002 
 2134 
 
 2279 
 2436 
 2609 
 2799 
 3007 
 
 3235 
 3486 
 3764 
 4069 
 
 r=2 
 
 logF(r, v) 
 
 0196 1199 
 •196 2052 
 •196 4614 
 •196 8890 
 •197 4890 
 •198 2627 
 
 •199 2118 
 •200 3385 
 •201 6452 
 •203 1349 
 •204 8110 
 
 •206 6774 
 •208 7382 
 •210 9985 
 •213 4631 
 •216 1383 
 
 •219 0303 
 •222 1462 
 •225 4936 
 •229 0807 
 •232 9167 
 
 •237 0114 
 •241 3755 
 ■246 0203 
 •250 9584 
 •256 2034 
 
 •261 7697 
 •267 6733 
 •273 9311 
 •280 5618 
 •287 5852 
 
 •295 0232 
 ■302 8992 
 •311 2388 
 •320 0695 
 •329 4214 
 
 •339 3271 
 •349 8221 
 •360 9451 
 •372 7382 
 •385 2475 
 
 •398 5232 
 •412 6205 
 •427 5995 
 •443 5266 
 •460 4745 
 
 logfffr, u) 
 
 196 1199 
 196 1391 
 196 1966 
 196 2924 
 196 4264 
 106 5985 
 
 196 8087 
 
 197 0567 
 197 3424 
 
 197 6655 
 
 198 0260 
 
 198 4234 
 
 198 8574 
 
 199 3280 
 
 199 8344 
 
 200 3764 
 
 200 9537 
 
 201 5657 
 
 202 2120 
 
 202 8921 
 
 203 6054 
 
 204 3514 
 
 205 1294 
 
 205 9387 
 
 206 7787 
 
 207 6487 
 
 208 5478 
 
 209 4753 
 
 210 4302 
 
 211 4118 
 
 212 4190 
 
 213 4509 
 
 214 5064 
 
 215 5846 
 
 216 6842 
 
 217 8041 
 
 218 9431 
 
 220 1000 
 
 221 2734 
 
 222 4621 
 
 223 6644 
 
 224 8791 
 
 226 1048 
 
 227 3397 
 
 228 5824 
 
 229 8313 
 
 192 
 
 575 
 
 958 
 
 1340 
 
 1722 
 
 2102 
 2480 
 2857 
 3231 
 3604 
 
 3974 
 4341 
 4705 
 5064 
 5420 
 
 5773 
 6120 
 6463 
 6801 
 7133 
 
 7460 
 7780 
 8093 
 8400 
 8700 
 
 8991 
 9275 
 9549 
 9816 
 10072 
 
 10319 
 10556 
 1 0782 
 10996 
 1 1199 
 
 1 1390 
 1 1569 
 1 1734 
 1 1886 
 1 2023 
 
 1 2148 
 1 2256 
 1 2350 
 1 2427 
 12489 
 
 A* 
 
 383 
 383 
 382 
 381 
 380 
 
 379 
 377 
 375 
 373 
 370 
 
 367 
 364 
 360 
 356 
 352 
 
 347 
 343 
 338 
 332 
 327 
 
 320 
 313 
 
 307 
 300 
 291 
 
 284 
 275 
 266 
 256 
 
 247 
 
 236 
 226 
 213 
 203 
 191 
 
 178 
 165 
 152 
 138 
 125 
 
 109 
 94 
 
 77 
 62 
 
Tables of the G (r, v)-Integrals 
 TABLE LIV— {continued). 
 
 127 
 
 9 
 10 
 
 11 
 
 IS 
 IS 
 
 lit 
 
 IB 
 16 
 
 n 
 
 18 
 19 
 
 21 
 
 25 
 
 26 
 27 
 28 
 29 
 SO 
 
 SI 
 
 S2 
 SS 
 Sit 
 S5 
 
 36 
 87 
 38 
 39 
 ¥> 
 
 41 
 
 43 
 44 
 45 
 
 r=3 
 
 log F(r,*) 
 
 0-124 9387 
 •125 0847 
 •125 5230 
 •126 2545 
 •127 2807 
 •128 6039 
 
 •130 2270 
 •132 1533 
 ■134 3870 
 •136 9331 
 •139 7969 
 
 •142 9850 
 •146 5043 
 •150 3626 
 •154 5688 
 •159 1322 
 
 •164 0636 
 •169 3743 
 
 •175 0768 
 •181 1848 
 •187 7130 
 
 •194 6774 
 •202 0955 
 •209 9858 
 •218 3688 
 •227 2664 
 
 •236 7023 
 •246 7020 
 •257 2933 
 •268 5060 
 '280 3725 
 
 •292 9278 
 •306 2096 
 •320 2589 
 •335 1201 
 •350 8413 
 
 •367 4747 
 •385 0770 
 •403 7099 
 •423 4403 
 •444 3416 
 
 •466 4933 
 
 •489 9829 
 •514 9055 
 •541 3658 
 •569 4783 
 
 log H (r,v) [ A 
 
 0-275 4537 
 •275 4674 
 •275 5084 
 •275 5767 
 •275 6721 
 •275 7948 
 
 •275 9444 
 •276 1209 
 •276 3242 
 •276 5540 
 •276 8101 
 
 •277 0923 
 •277 4003 
 •277 7338 
 •278 0926 
 •278 4762 
 
 •278 8844 
 •279 3167 
 •279 7726 
 •280 2519 
 •280 7539 
 
 •281 2782 
 •281 8243 
 •282 3915 
 •282 9794 
 •283 5873 
 
 •284 2145 
 •284 8604 
 •285 5244 
 •286 2057 
 •286 9035 
 
 •287 6170 
 •288 3456 
 •289 0883 
 •289 8443 
 •290 6127 
 
 •291 3927 
 •292 1832 
 •292 9834 
 •293 7923 
 •294 6089 
 
 ■295 4321 
 •296 2610 
 •297 0945 
 •297 9316 
 •298 7710 
 
 137 
 
 410 
 683 
 955 
 
 1226 
 
 1496 
 1766 
 2032 
 2298 
 2561 
 
 2822 
 3080 
 3336 
 3588 
 3837 
 
 4082 
 4323 
 4560 
 4793 
 5020 
 
 5243 
 5460 
 5673 
 5879 
 6079 
 
 6272 
 6460 
 6640 
 6813 
 6978 
 
 7136 
 
 7286 
 7427 
 7560 
 7684 
 
 7800 
 7905 
 8002 
 
 8166 
 
 8232 
 8289 
 8335 
 8371 
 8394 
 
 A- 
 
 r=4 
 
 273 
 
 273 
 272 
 271 
 270 
 
 269 
 267 
 266 
 263 
 261 
 
 258 
 256 
 252 
 249 
 245 
 
 241 
 237 
 233 
 
 228 
 223 
 
 217 
 212 
 206 
 200 
 194 
 
 188 
 180 
 173 
 165 
 158 
 
 150 
 141 
 133 
 124 
 115 
 
 106 
 97 
 87 
 77 
 67 
 
 57 
 
 46 
 36 
 23 
 
 log F (r, v) 
 
 0071 1811 
 •071 3902 
 •072 0177 
 •073 0650 
 •074 5342 
 •076 4285 
 
 •078 7517 
 •081 5088 
 •084 7055 
 •088 3486 
 •092 4458 
 
 •097 0060 
 •102 0399 
 •107 5555 
 ■113 5680 
 •120 0895 
 
 •127 1349 
 •134 7199 
 ■142 8621 
 •151 5802 
 •160 8948 
 
 •170 8281 
 •181 4042 
 •192 6491 
 •204 5907 
 ■217 2596 
 
 •230 6885 
 •244 9127 
 •259 9707 
 •275 9034 
 ■292 7555 
 
 •310 5754 
 •329 4149 
 •349 3304 
 •370 3832 
 •392 6390 
 
 •416 1697 
 •441 0529 
 •467 3733 
 •495 2227 
 •524 7011 
 
 •555 9177 
 •588 9916 
 •624 0530 
 ■061 2446 
 •700 7225 
 
 logff(r,x) 
 
 0-309 7418 
 •309 7523 
 •309 7840 
 •309 8366 
 •309 9103 
 •310 0049 
 
 •310 1204 
 •310 2565 
 •310 4132 
 •310 5904 
 •310 7877 
 
 •311 0051 
 •311 2423 
 •311 4990 
 •311 7751 
 •312 0701 
 
 •312 3838 
 •312 7159 
 •313 0660 
 •313 4337 
 •313 8186 
 
 •314 2204 
 •314 6385 
 •315 0726 
 •315 5222 
 •315 9866 
 
 ■316 4655 
 •316 9583 
 •317 4644 
 •317 9833 
 •318 5143 
 
 •319 0569 
 •319 6104 
 •320 1741 
 •320 7474 
 •321 3297 
 
 •321 9201 
 •322 5181 
 •323 1228 
 ■323 7335 
 •324 3495 
 
 •324 9700 
 •325 5943 
 •326 2216 
 •326 8510 
 •327 4817 
 
 105 
 316 
 527 
 737 
 946 
 
 1154 
 1361 
 1567 
 1771 
 1973 
 
 2174 
 2372 
 2567 
 2760 
 2950 
 
 3137 
 3321 
 
 3501 
 3677 
 3849 
 
 4018 
 4181 
 4341 
 4495 
 4645 
 
 4789 
 4927 
 5062 
 5189 
 5310 
 
 5426 
 5535 
 5637 
 5733 
 
 5822 
 
 5904 
 5980 
 6047 
 6107 
 6160 
 
 6205 
 6243 
 6272 
 6295 
 6307 
 
 A-' 
 
 211 
 211 
 210 
 
 209 
 208 
 
 207 
 206 
 204 
 202 
 201 
 
 198 
 196 
 193 
 190 
 187 
 
 184 
 180 
 176 
 172 
 168 
 
 164 
 160 
 154 
 150 
 144 
 
 138 
 134 
 
 127 
 121 
 116 
 
 109 
 
 102 
 
 96 
 
 89 
 
 82 
 
 75 
 
 68 
 60 
 53 
 45 
 
 38 
 30 
 22 
 12 
 
 r=5 
 
 logf(r, ») 
 
 0028 0289 
 •028 3019 
 •029 1221 
 •030 4908 
 •032 4110 
 •034 8865 
 
 •037 9224 
 •041 5250 
 •045 7016 
 •050 4609 
 055 8130 
 
 •061 7690 
 •068 3415 
 •075 5446 
 •083 3937 
 •091 9061 
 
 •101 1002 
 •110 9967 
 •121 6176 
 •132 9872 
 •145 1317 
 
 •158 0795 
 •171 8614 
 •186 5105 
 •202 0627 
 •218 5568 
 
 •236 0346 
 •254 5413 
 •274 1255 
 •294 8399 
 •316 7413 
 
 •339 8909 
 •364 3553 
 •390 2059 
 •417 5203 
 •446 3827 
 
 •476 8841 
 •509 1232 
 •543 2072 
 •579 2529 
 •617 3872 
 
 •657 7483 
 •700 4872 
 •745 7688 
 •793 7739 
 •844 6999 
 
 log H ()-, v) A 
 
 0-329 0589 
 •329 0673 
 •329 0930 
 •329 1357 
 •329 1956 
 •329 2723 
 
 •329 3661 
 •329 4765 
 •329 6037 
 •329 7474 
 •329 9075 
 
 •330 0838 
 •330 2761 
 •330 4842 
 •330 7079 
 •330 9469 
 
 •331 2010 
 •331 4700 
 •331 7532 
 •332 0508 
 •332 3621 
 
 •332 6870 
 •333 0250 
 ■333 3757 
 •333 7387 
 •334 1137 
 
 •334 5001 
 •334 8976 
 •335 3057 
 •335 7239 
 •336 1516 
 
 •336 5884 
 •337 0339 
 •337 4874 
 •337 9485 
 •338 4164 
 
 •338 8908 
 •339 3709 
 •339 8563 
 •340 3463 
 •340 8404 
 
 •341 3378 
 •341 8381 
 •342 3405 
 •342 8446 
 •343 3495 
 
 84 
 257 
 428 
 598 
 768 
 
 937 
 1105 
 1272 
 1437 
 1601 
 
 1763 
 1923 
 2081 
 2237 
 2390 
 
 2541 
 2689 
 2833 
 2975 
 3114 
 
 3249 
 3380 
 3507 
 3630 
 3750 
 
 3864 
 3975 
 4081 
 4182 
 4277 
 
 4368 
 4455 
 4535 
 4610 
 4680 
 
 4744 
 4802 
 4854 
 4900 
 4940 
 
 4975 
 5003 
 5024 
 5040 
 5049 
 
 173 
 171 
 171 
 170 
 169 
 
 168 
 167 
 165 
 164 
 162 
 
 160 
 158 
 156 
 153 
 151 
 
 148 
 144 
 142 
 138 
 135 
 
 131 
 127 
 123 
 119 
 115 
 
 110 
 
 106 
 
 101 
 
 96 
 
 91 
 
 87 
 80 
 75 
 70 
 64 
 
 58 
 52 
 46 
 40 
 34 
 
 28 
 
 22 
 
 16 
 
 9 
 
128 Tables for Statisticians and Biometricians 
 
 TABLE UV— (continued). 
 
 r=6 
 
 -4 
 5 
 
 6 
 7 
 8 
 9 
 10 
 
 11 
 12 
 13 
 
 U 
 
 15 
 
 16 
 17 
 18 
 19 
 20 
 
 21 
 22 
 23 
 2A 
 
 26 
 27 
 28 
 29 
 30 
 
 31 
 32 
 33 
 34 
 35 
 
 36 
 37 
 38 
 39 
 40 
 
 4* 
 
 43 
 
 44 
 45 
 
 log F(r, v) 
 
 991 9999 
 
 992 3379 
 
 993 3526 
 
 995 0459 
 997 4213 
 000 4836 
 
 004 2390 
 008 6950 
 013 8607 
 019 7468 
 026 3653 
 
 033 7300 
 041 8562 
 050 7609 
 060 4633 
 070 9839 
 
 082 3457 
 094 5734 
 107 6941 
 121 7375 
 136 7352 
 
 152 7219 
 169 7350 
 187 8149 
 207 0053 
 227 3532 
 
 248 9095 
 271 7291 
 295 8713 
 321 3998 
 348 3836 
 
 376 8974 
 407 0214 
 '438 8428 
 472 4556 
 507 9618 
 
 545 4718 
 585 1052 
 626 9922 
 671 2739 
 718 1040 
 
 767 6502 
 820 0951 
 875 6383 
 934 4983 
 
 996 9145 
 
 log H (r,r) 
 
 0-341 4849 
 •341 4921 
 •341 5137 
 •341 5496 
 •341 5999 
 ■341 6644 
 
 •341 7432 
 •341 8360 
 •341 9428 
 ■342 0635 
 •342 1980 
 
 •342 3461 
 •342 5075 
 •342 6823 
 •342 8701 
 •343 0707 
 
 •343 2839 
 •343 5095 
 •343 7472 
 •343 9968 
 •344 2579 
 
 •344 5302 
 ■344 8135 
 •345 1074 
 •345 4115 
 •345 7256 
 
 •346 0492 
 •346 3819 
 •346 7235 
 •347 0734 
 •347 4311 
 
 •347 7965 
 •348 1689 
 •348 5480 
 •348 9332 
 •349 3242 
 
 •349 7204 
 •350 1213 
 •350 5265 
 •350 9354 
 •351 3477 
 
 •351 7627 
 •352 1799 
 •352 5989 
 •353 0191 
 •353 4399 
 
 72 
 216 
 359 
 503 
 645 
 
 787 
 
 928 
 
 1068 
 
 1207 
 
 1345 
 
 1481 
 1615 
 
 1747 
 1878 
 2006 
 
 2133 
 
 2256 
 2377 
 2496 
 2611 
 
 2723 
 2833 
 2939 
 3041 
 3141 
 
 3236 
 3328 
 3415 
 3499 
 3577 
 
 3653 
 3724 
 3791 
 3852 
 3910 
 
 3962 
 4009 
 4052 
 4090 
 4122 
 
 4150 
 4172 
 4190 
 4202 
 4209 
 
 6? 
 
 144 
 144 
 143 
 143 
 
 142 
 
 141 
 140 
 139 
 137 
 136 
 
 134 
 133 
 
 131 
 128 
 127 
 
 123 
 121 
 119 
 115 
 112 
 
 109 
 
 106 
 
 103 
 
 99 
 
 95 
 
 92 
 88 
 84 
 78 
 76 
 
 71 
 66 
 62 
 57 
 52 
 
 47 
 43 
 38 
 33 
 
 28 
 
 22 
 
 17 
 
 12 
 
 7 
 
 r=7 
 
 logF(r, ») 
 
 1-961 0819 
 •961 4851 
 •962 6953 
 •964 7151 
 •967 5483 
 •971 2008 
 
 •975 6796 
 •980 9940 
 •987 1544 
 •994 1735 
 0-002 0658 
 
 •010 8465 
 ■020 5347 
 •031 1503 
 •042 7157 
 ■055 2551 
 
 •068 7956 
 •083 3665 
 •098 9997 
 •115 7298 
 •133 5946 
 
 •152 6347 
 •172 8941 
 •194 4206 
 •217 2653 
 •241 4839 
 
 •267 1362 
 •294 2868 
 •323 0053 
 •353 3670 
 •385 4529 
 
 •419 3506 
 •455 1549 
 •492 9680 
 •532 9005 
 •575 0721 
 
 •619 6127 
 •666 6629 
 •716 3753 
 ■768 9159 
 •824 4653 
 
 •883 2199 
 ■945 3944 
 I -Oil 2229 
 •080 9618 
 •154 8920 
 
 logff(M') 
 
 350 1576 
 350 1638 
 350 1824 
 350 2134 
 350 2567 
 350 3123 
 
 350 3801 
 350 4601 
 350 5522 
 350 6562 
 350 7720 
 
 350 8995 
 
 351 0386 
 351 1891 
 351 3508 
 351 5235 
 
 351 7071 
 
 351 9012 
 
 352 1058 
 352 3206 
 352 5452 
 
 352 7795 
 
 353 0232 
 353 2760 
 353 5375 
 
 353 8075 
 
 354 0857 
 354 3717 
 354 6652 
 
 354 9659 
 
 355 2732 
 
 355 5871 
 
 355 9070 
 
 356 2324 
 356 5632 
 
 356 8988 
 
 357 2388 
 357 5829 
 
 357 9306 
 
 358 2814 
 358 6350 
 
 358 9910 
 
 359 3488 
 
 359 7080 
 
 360 0682 
 360 4290 
 
 62 
 186 
 310 
 433 
 556 
 
 678 
 
 800 
 
 921 
 
 1040 
 
 1158 
 
 1275 
 1391 
 
 1505 
 1617 
 
 1727 
 
 1836 
 1942 
 2046 
 2148 
 2247 
 
 2343 
 
 2437 
 2528 
 2615 
 2700 
 
 2782 
 2860 
 2935 
 3007 
 3074 
 
 3138 
 3199 
 3255 
 3308 
 3356 
 
 3400 
 3441 
 3477 
 3509 
 3536 
 
 3559 
 
 3578 
 3592 
 3602 
 3608 
 
 tf 
 
 124 
 124 
 123 
 123 
 122 
 
 122 
 121 
 120 
 118 
 117 
 
 116 
 
 114 
 112 
 110 
 109 
 
 106 
 
 104 
 
 102 
 
 99 
 
 97 
 
 94 
 91 
 
 88 
 85 
 82 
 
 79 
 75 
 72 
 67 
 64 
 
 56 
 53 
 
 49 
 44 
 
 4(1 
 86 
 83 
 88 
 81 
 
 I!) 
 14 
 10 
 
 e 
 
 r=8 
 
 log F ()-,./) 
 
 1-934 0080 
 •934 4765 
 •935 8831 
 •938 2304 
 •941 5232 
 ■945 7679 
 
 •950 9729 
 •957 1486 
 •964 3073 
 •972 4634 
 •981 6333 
 
 •991 8357 
 0003 0914 
 •015 4237 
 •028 8583 
 •043 4233 
 
 •059 1498 
 •076 0714 
 •094 2250 
 •113 6505 
 •134 3912 
 
 •156 4939 
 •180 0093 
 •204 9923 
 •231 5019 
 •259 6019 
 
 •289 3613 
 ■320 8543 
 •354 1610 
 •389 3678 
 •426 5682 
 
 ■465 8626 
 •507 3601 
 •551 1780 
 •597 4436 
 •646 2944 
 
 •697 8795 
 •752 3605 
 •809 9127 
 •870 7267 
 •935 0097 
 
 1-002 9876 
 •074 9063 
 •151 0352 
 •231 6680 
 •317 1271 
 
 ',H(r,') 
 
 0-356 5570 
 •356 5624 
 •356 5788 
 •356 6059 
 •356 6440 
 •356 6928 
 
 •356 7524 
 •356 8226 
 •356 9034 
 •356 9947 
 •357 0964 
 
 •357 2083 
 •357 3304 
 •357 4625 
 •357 6044 
 •357 7560 
 
 •357 9170 
 •358 0874 
 •358 2669 
 •358 4553 
 
 •358 6524 
 
 .358 8579 
 •359 0716 
 ■359 2933 
 •359 5226 
 ■359 7594 
 
 •360 0033 
 •360 2540 
 •360 5112 
 •360 7747 
 ■361 0441 
 
 •361 3191 
 ■361 5993 
 •361 8844 
 ■362 1741 
 •362 4681 
 
 •362 7659 
 •363 0671 
 •363 3715 
 •363 6787 
 •363 9883 
 
 ■364 2998 
 •364 6130 
 •364 9274 
 •365 2427 
 •365 5584 
 
 54 
 163 
 
 272 
 380 
 488 
 
 596 
 702 
 808 
 913 
 1017 
 
 1120 
 1221 
 1321 
 1419 
 1516 
 
 1611 
 
 1704 
 1795 
 1884 
 1971 
 
 2055 
 2137 
 2217 
 2293 
 2368 
 
 2439 
 
 2507 
 2572 
 2635 
 2694 
 
 2750 
 2802 
 2851 
 2897 
 2939 
 
 2978 
 3013 
 3044 
 3072 
 3096 
 
 3116 
 3132 
 3144 
 3153 
 3157 
 
 A2 
 
 109 
 109 
 108 
 108 
 107 
 
 107 
 106 
 105 
 104 
 103 
 
 101 
 
 100 
 
 98 
 
 97 
 
 95 
 
 93 
 91 
 
 89 
 
 87 
 84 
 
 82 
 80 
 77 
 74 
 71 
 
 68 
 65 
 62 
 59 
 56 
 
 52 
 49 
 46 
 42 
 39 
 
 35 
 31 
 
 28 
 24 
 
 16 
 
 12 
 9 
 5 
 
Tables of the G (r, v)- Integrals 
 TABLE LIV— {continued). 
 
 129 
 
 o 
 1 
 
 2 
 9 
 
 It 
 5 
 
 6 
 7 
 8 
 9 
 10 
 
 11 
 
 13 
 
 15 
 
 16 
 17 
 
 18 
 19 
 
 SI 
 22 
 23 
 21f 
 
 26 
 27 
 28 
 29 
 SO 
 
 31 
 32 
 33 
 Sk 
 35 
 
 36 
 37 
 38 
 39 
 
 40 
 
 41 
 
 42 
 43 
 
 U 
 45 
 
 r = 9 
 
 log F (r, v) log II (r, v) 
 
 1-909 9294 
 ■910 4635 
 •912 0669 
 •914 7427 
 •918 4961 
 •923 3346 
 
 •929 2675 
 •936 3067 
 •944 4661 
 •953 7619 
 •964 2128 
 
 •975 8398 
 •988 6667 
 0002 7197 
 •018 0278 
 •034 6230 
 
 •052 5403 
 •071 8179 
 •092 4974 
 •114 6239 
 •138 2465 
 
 •163 4180 
 •190 1960 
 •218 6422 
 
 ■248 8237 
 •280 8124 
 
 •314 6864 
 •350 5295 
 •388 4323 
 •428 4925 
 •470 8156 
 
 •515 5153 
 •562 7146 
 •612 5463 
 
 •665 1541 
 ■720 6932 
 
 •779 3322 
 •841 2534 
 •906 6549 
 •975 7519 
 1-048 7784 
 
 •125 9892 
 •207 6624 
 •294 1013 
 •385 6379 
 •482 6360 
 
 B. 
 
 0-361 4744 
 •361 4793 
 ■361 4938 
 ■361 5180 
 •361 5520 
 •361 5955 
 
 •361 6485 
 •361 7111 
 •361 7831 
 •361 8644 
 •361 9550 
 
 •362 0547 
 •362 1635 
 •362 2812 
 •362 4075 
 •362 5426 
 
 •362 6860 
 •362 8378 
 •362 9976 
 •363 1654 
 •363 3409 
 
 •363 5239 
 •363 7142 
 •363 9115 
 •364 1157 
 •364 3264 
 
 ■364 5435 
 •364 7666 
 •364 9955 
 •365 2300 
 •365 4697 
 
 •365 7144 
 •365 9636 
 •366 2173 
 •366 4750 
 •366 7365 
 
 •367 0013 
 •367 2693 
 •367 5400 
 •367 8131 
 •368 0884 
 
 ■368 3654 
 •368 6438 
 •368 9233 
 •369 2036 
 ■369 4842 
 
 49 
 146 
 242 
 339 
 435 
 
 531 
 626 
 720 
 813 
 906 
 
 997 
 1088 
 1177 
 1264 
 1350 
 
 1435 
 1518 
 1599 
 1678 
 1755 
 
 1830 
 1903 
 1973 
 2042 
 2107 
 
 2171 
 2231 
 2290 
 2345 
 2397 
 
 2447 
 2493 
 2537 
 
 2577 
 2615 
 
 2649 
 2680 
 2707 
 2731 
 2752 
 
 2770 
 2784 
 2795 
 2803 
 2806 
 
 A> 
 
 r = 10 
 
 log F(r,v) logH(r,y) A A 2 
 
 1-888 2505 
 •888 8502 
 •890 6508 
 ■893 6556 
 •897 8705 
 •903 3037 
 
 •909 9658 
 •917 8700 
 •927 0317 
 ■937 4692 
 •949 2033 
 
 •962 2575 
 •976 6581 
 •992 4345 
 0009 6193 
 •028 2479 
 
 •048 3595 
 •069 9966 
 •093 2058 
 •118 0374 
 •144 5461 
 
 •172 7910 
 •202 8360 
 •234 7504 
 •268 6087 
 ■304 4913 
 
 •342 4851 
 •382 6838 
 •425 1882 
 •470 1076 
 •517 5593 
 
 •567 6703 
 •620 5776 
 •676 4293 
 •735 3855 
 •797 6194 
 
 •863 3188 
 •932 6869 
 1-005 9444 
 •083 3313 
 •165 1080 
 
 •251 5588 
 •342 9931 
 •439 7491 
 •542 1966 
 •C50 7407 
 
 0-365 3717 
 •365 3761 
 •365 3892 
 •365 4111 
 •365 4416 
 •365 4808 
 
 •365 5287 
 •365 5851 
 •365 6500 
 •365 7233 
 •365 8050 
 
 •365 8949 
 •365 9929 
 •366 0990 
 •366 2129 
 •366 3346 
 
 •366 4639 
 ■366 6007 
 •366 7447 
 •366 8960 
 •367 0541 
 
 •367 2190 
 •367 3904 
 •367 5682 
 •367 7522 
 •367 9420 
 
 •368 1376 
 •368 3386 
 •368 5448 
 •368 7559 
 •368 9718 
 
 ■369 1922 
 •369 4167 
 •369 6451 
 •369 8772 
 •370 1126 
 
 ■370 3510 
 •370 5923 
 •370 8360 
 •371 0819 
 •371 3297 
 
 •371 5790 
 •371 8296 
 •372 0813 
 •372 3335 
 •372 5861 
 
 44 
 131 
 
 219 
 306 
 392 
 
 479 
 564 
 649 
 733 
 817 
 
 899 
 
 981 
 
 1061 
 
 1139 
 
 1217 
 
 1293 
 1368 
 1441 
 1512 
 1581 
 
 1649 
 1714 
 
 1778 
 1839 
 1899 
 
 1956 
 2010 
 2062 
 2112 
 2159 
 
 2203 
 2245 
 2284 
 2321 
 2354 
 
 2385 
 2412 
 2437 
 2459 
 2478 
 
 2494 
 2506 
 2516 
 2522 
 2526 
 
 r=ll 
 
 lo% F(r,v) 
 
 1-868 5367 
 •869 2023 
 •871 2002 
 •874 5345 
 •879 2114 
 •885 2402 
 
 •892 6324 
 ■901 4027 
 •911 5681 
 ■923 1487 
 •936 1674 
 
 •950 6505 
 •966 6268 
 •984 1288 
 0-003 1923 
 •023 8567 
 
 •046 1651 
 •070 1646 
 ■095 9063 
 •123 4460 
 •152 8439 
 
 ■184 1654 
 •217 4809 
 ■252 8669 
 ■290 4057 
 •330 1857 
 
 •372 3033 
 •416 8615 
 •463 9718 
 •513 7544 
 •566 3390 
 
 •621 8657 
 •680 4854 
 •742 3617 
 •807 6710 
 •876 6045 
 
 •949 3690 
 1-026 1888 
 ■107 3073 
 •192 9888 
 •283 5209 
 
 •379 2165 
 ■480 4171 
 •587 4953 
 •700 8587 
 •820 9540 
 
 logff(r, y) 
 
 0-368 5367 
 •368 5407 
 •368 5527 
 •368 5725 
 ■368 6004 
 •368 6361 
 
 •368 6796 
 •368 7310 
 •368 7900 
 ■368 8568 
 •368 9311 
 
 •369 0129 
 •369 1022 
 ■369 1987 
 •369 3024 
 •369 4132 
 
 •369 5308 
 •369 6553 
 •369 7864 
 •369 9240 
 •370 0679 
 
 •370 2179 
 •370 3739 
 •370 5357 
 •370 7030 
 •370 8757 
 
 •371 0536 
 •371 2365 
 •371 4241 
 •371 6162 
 •371 8125 
 
 •372 0129 
 •372 2171 
 ■372 4249 
 ■372 6359 
 •372 8500 
 
 •373 0669 
 •373 2862 
 •373 5078 
 •373 7314 
 •373 9567 
 
 •374 1834 
 •374 4113 
 •374 6400 
 •374 8693 
 •375 0990 
 
 A ! A* 
 
 40 
 120 
 199 
 278 
 357 
 
 435 
 514 
 591 
 668 
 743 
 
 818 
 
 892 
 
 965 
 
 1037 
 
 1108 
 
 1177 
 1245 
 1311 
 1376 
 1439 
 
 1500 
 1560 
 1618 
 1673 
 1727 
 
 1779 
 1829 
 1876 
 1921 
 1964 
 
 2004 
 2042 
 2078 
 2110 
 2141 
 
 2169 
 2194 
 2216 
 2236 
 2253 
 
 2267 
 2279 
 2287 
 2293 
 2296 
 
 17 
 
 80 
 79 
 79 
 79 
 78 
 
 78 
 77 
 77 
 76 
 75 
 
 74 
 73 
 72 
 71 
 69 
 
 68 
 66 
 65 
 63 
 62 
 
 60 
 58 
 56 
 54 
 52 
 
 50 
 47 
 45 
 43 
 
 40 
 
 38 
 36 
 33 
 
 30 
 
 28 
 
 25 
 23 
 20 
 17 
 14 
 
 12 
 9 
 6 
 3 
 
130 
 
 Tables for Statisticians and Biometricians 
 TABLE UN— (continued). 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 is 
 U 
 
 15 
 16 
 
 n 
 
 18 
 19 
 
 n 
 
 25 
 
 26 
 27 
 28 
 29 
 SO 
 
 SI 
 
 Si 
 35 
 
 S6 
 37 
 38 
 39 
 k0 
 
 u 
 
 J,5 
 
 r = V2. 
 
 log F(r,v) 
 
 1-850 4619 
 •851 1933 
 •853 3889 
 •857 0528 
 •862 1923 
 •868 8171 
 
 ■876 9402 
 •886 5774 
 •897 7474 
 •910 4722 
 •924 7770 
 
 •940 6901 
 •958 2436 
 •977 4727 
 •998 4167 
 0021 1187 
 
 •045 6258 
 •071 9896 
 •100 2660 
 •130 5159 
 •162 8054 
 
 •197 2059 
 •233 7945 
 •272 6547 
 •313 8765 
 •357 5571 
 
 •403 8013 
 •452 7221 
 •504 4412 
 •559 0903 
 •616 8111 
 
 •677 7567 
 •742 0923 
 •809 9965 
 ■881 6625 
 •957 2989 
 
 1037 1322 
 •121 4073 
 •210 3905 
 •304 3704 
 •403 6615 
 
 •508 6058 
 •619 5764 
 •736 9806 
 •861 2638 
 •992 9140 
 
 log H (r, v) A 
 
 0371 1582 
 •371 1619 
 •371 1729 
 •371 1912 
 •371 2166 
 •371 2494 
 
 •371 2893 
 •371 3364 
 •371 3907 
 •371 4519 
 •371 5201 
 
 •371 5952 
 •371 6771 
 •371 7656 
 •371 8608 
 •371 9624 
 
 •372 0703 
 •372 1845 
 •372 3048 
 •372 4310 
 •372 5630 
 
 ■372 7006 
 •372 8437 
 •372 9921 
 •373 1456 
 •373 3040 
 
 •373 4672 
 ■373 6349 
 •373 8069 
 •373 9831 
 •374 1631 
 
 •374 3469 
 •374 5341 
 •374 7246 
 •374 9182 
 •375 1145 
 
 •375 3133 
 
 •375 5144 
 •375 7176 
 •375 9226 
 •376 1291 
 
 •376 3370 
 •376 5458 
 •376 7555 
 •376 9657 
 •377 1762 
 
 36 
 110 
 183 
 255 
 328 
 
 400 
 471 
 542 
 613 
 682 
 
 751 
 819 
 886 
 951 
 1016 
 
 1080 
 1142 
 1203 
 1262 
 1320 
 
 1376 
 1431 
 
 1484 
 1535 
 1584 
 
 1632 
 1677 
 1720 
 1762 
 1801 
 
 1838 
 1873 
 1905 
 1935 
 1963 
 
 1988 
 2011 
 2032 
 2050 
 2065 
 
 2078 
 2089 
 2097 
 2102 
 2105 
 
 A« 
 
 73 
 78 
 
 73 
 
 72 
 72 
 
 71 
 71 
 70 
 89 
 SO 
 
 68 
 87 
 86 
 65 
 64 
 
 02 
 61 
 
 ;-.!) 
 58 
 56 
 
 55 
 53 
 51 
 19 
 47 
 
 15 
 43 
 41 
 3!) 
 37 
 
 35 
 33 
 30 
 
 28 
 25 
 
 23 
 21 
 
 18 
 10 
 13 
 
 11 
 8 
 5 
 3 
 
 r=13 
 
 log F (r, v) 
 
 1-833 7746 
 •834 5719 
 ■836 9652 
 •840 9592 
 •846 5615 
 •853 7829 
 
 •862 6374 
 •873 1421 
 •885 3174 
 •899 1873 
 •914 7789 
 
 ■932 1233 
 •951 2549 
 ■972 2124 
 •995 0382 
 0019 7792 
 
 •046 4865 
 ■075 2161 
 •106 0288 
 •138 9908 
 •174 1737 
 
 •211 6551 
 •251 5187 
 •293 8552 
 •338 7623 
 •386 3455 
 
 •436 7186 
 •490 0042 
 •546 3346 
 •605 8526 
 •668 7120 
 
 •735 0790 
 •805 1331 
 •879 0680 
 •957 0932 
 1-039 4354 
 
 •126 3402 
 •218 0735 
 •314 9240 
 •417 2053 
 •525 2853 
 
 •639 4542 
 •760 1977 
 •887 9308 
 2023 1367 
 •166 3448 
 
 log H(r,v) 
 
 0-373 3653 
 •373 3686 
 •373 3787 
 •373 3956 
 •373 4192 
 •373 4494 
 
 •373 4864 
 •373 5299 
 •373 5800 
 ■373 6365 
 •373 6995 
 
 •373 7689 
 •373 8445 
 •373 9263 
 •374 0142 
 •374 1081 
 
 •374 2078 
 •374 3132 
 •374 4243 
 •374 5409 
 •374 6628 
 
 •374 7899 
 •374 9221 
 •375 0591 
 •375 2008 
 •375 3472 
 
 •375 4978 
 •375 6527 
 •375 8115 
 •375 9742 
 •376 1405 
 
 •376 3102 
 •376 4831 
 •376 6589 
 •376 8376 
 •377 0188 
 
 •377 2024 
 •377 3881 
 •377 5757 
 •377 7649 
 •377 9556 
 
 •378 1475 
 •378 3403 
 •378 5338 
 •378 7279 
 •378 9222 
 
 6? 
 
 34 
 101 
 169 
 236 
 303 
 
 369 
 435 
 501 
 566 
 630 
 
 694 
 756 
 818 
 879 
 938 
 
 997 
 1055 
 1111 
 1166 
 1219 
 
 1271 
 1322 
 1370 
 1417 
 1464 
 
 1506 
 1549 
 1589 
 1627 
 1663 
 
 1697 
 
 1729 
 1759 
 1787 
 1812 
 
 1836 
 1857 
 1876 
 1893 
 1907 
 
 1919 
 1928 
 1936 
 1941 
 1943 
 
 (57 
 67 
 
 07 
 07 
 
 00 
 00 
 05 
 05 
 04 
 
 03 
 
 02 
 01 
 00 
 59 
 
 58 
 
 50 
 55 
 53 
 
 52 
 
 51 
 
 48 
 
 47 
 40 
 
 48 
 49 
 
 40 
 38 
 30 
 34 
 
 32 
 
 3D 
 
 88 
 
 20 
 23 
 
 21 
 
 19 
 
 17 
 14 
 12 
 
 10 
 7 
 5 
 2 
 
 r=li 
 
 log F (r, v) 
 
 1-818 2772 
 •819 1404 
 •821 7316 
 •826 0558 
 •832 1212 
 •839 9395 
 
 •849 5258 
 •860 8985 
 •874 0798 
 •889 0954 
 •905 9746 
 
 •924 7510 
 •945 4618 
 •968 1485 
 •992 8571 
 0019 6382 
 
 •048 5469 
 ■079 6435 
 •112 9939 
 •148 6693 
 •186 7470 
 
 •227 3107 
 •270 4509 
 •316 2653 
 •364 8594 
 •416 3469 
 
 •470 8506 
 •528 5029 
 •589 4465 
 •653 8352 
 •721 8353 
 
 •793 6258 
 •869 4003 
 •949 3679 
 1-033 7545 
 •122 8047 
 
 ■216 7831 
 •315 9768 
 •420 6970 
 •531 2818 
 •648 0989 
 
 •771 5487 
 •902 0674 
 2010 1318 
 •186 2627 
 •341 0309 
 
 log H{r,p) 
 
 0-375 2489 
 ■375 2520 
 •375 2614 
 •375 2771 
 ■375 2990 
 •375 3271 
 
 •375 3614 
 •375 4019 
 •375 4484 
 •375 5010 
 •375 5595 
 
 •375 6239 
 •375 6942 
 •375 7702 
 •375 8518 
 •375 9390 
 
 •376 0317 
 •376 1297 
 •376 2329 
 •376 3412 
 •376 4544 
 
 •376 5725 
 •376 6952 
 •376 8225 
 •376 9542 
 •377 0901 
 
 •377 2301 
 •377 3739 
 •377 5215 
 •377 6726 
 •377 8270 
 
 ■377 9846 
 •378 1452 
 •378 3085 
 •378 4745 
 •378 6428 
 
 •378 8133 
 •378 9857 
 •379 1599 
 •379 3356 
 •379 5127 
 
 •379 6909 
 •379 8699 
 •380 0497 
 •380 2299 
 •380 4103 
 
 31 
 
 94 
 157 
 219 
 
 281 
 
 343 
 
 404 
 465 
 526 
 585 
 
 644 
 703 
 760 
 816 
 
 872 
 
 926 
 
 980 
 
 1032 
 
 1083 
 
 1133 
 
 1181 
 1228 
 1273 
 1317 
 1359 
 
 1399 
 1439 
 1476 
 1511 
 1544 
 
 1576 
 1606 
 1634 
 1659 
 1683 
 
 1705 
 1724 
 1742 
 1757 
 1771 
 
 1782 
 1791 
 1797 
 1802 
 1804 
 
 A* 
 
 03 
 63 
 62 
 62 
 62 
 
 61 
 61 
 60 
 60 
 59 
 
 58 
 57 
 56 
 55 
 54 
 
 53 
 
 52 
 51 
 50 
 48 
 
 47 
 45 
 44 
 42 
 40 
 
 39 
 37 
 35 
 33 
 32 
 
 30 
 
 28 
 26 
 24 
 22 
 
 19 
 
 18 
 15 
 13 
 11 
 
 9 
 
 7 
 5 
 2 
 
Tables of the G (r, v)-Tntegrals 
 TABLE LIV— (continued). 
 
 131 
 
 r«=15 
 
 r = 16 
 
 r=17 
 
 
 1 
 2 
 S 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 IS 
 13 
 
 u 
 
 10 
 17 
 18 
 19 
 20 
 
 21 
 
 25 
 
 20 
 21 
 28 
 29 
 SO 
 
 SI 
 
 34 
 35 
 
 S6 
 37 
 38 
 S9 
 ¥> 
 
 41 
 49 
 
 4S 
 U 
 
 log F (r, ») 
 
 803 8114 
 
 804 7405 
 807 5297 
 812 1842 
 818 7130 
 827 1285 
 
 837 44G9 
 849 6881 
 863 8758 
 880 0376 
 898 2051 
 
 •918 4141 
 •940 7047 
 •965 1215 
 •991 7138 
 0-020 5357 
 
 •051 6467 
 •085 1115 
 •121 0005 
 •159 3904 
 •200 3641 
 
 •244 0113 
 •290 4293 
 •339 7229 
 •392 0053 
 •447 3985 
 
 •506 0343 
 •568 0547 
 •633 6129 
 •702 8740 
 •776 0162 
 
 •853 2318 
 ■934 7285 
 1-020 7304 
 •111 4801 
 •207 2399 
 
 •308 2938 
 •414 9495 
 •527 5412 
 •646 4313 
 •772 0144 
 
 •904 7199 
 2045 0157 
 •193 4131 
 ■350 4709 
 •516 8011 
 
 logff(r,0 
 
 0-376 8754 
 •376 8783 
 •376 8871 
 •376 9018 
 •376 9222 
 •376 9485 
 
 •376 9805 
 •377 0183 
 •377 0617 
 •377 1108 
 •377 1655 
 
 •377 2256 
 •377 2912 
 •377 3622 
 •377 4384 
 •377 5199 
 
 •377 6064 
 •377 6979 
 •377 7942 
 •377 8953 
 •378 0011 
 
 ■378 1113 
 •378 2259 
 •378 3448 
 •378 4677 
 •378 5946 
 
 •378 7252 
 •378 8595 
 ■378 9973 
 •379 1383 
 •379 2825 
 
 •379 4296 
 •379 5795 
 •379 7320 
 •379 8869 
 •380 0440 
 
 •380 2031 
 •380 3641 
 •380 5267 
 •380 6907 
 •380 8560 
 
 •381 0223 
 •381 1894 
 •381 3572 
 •381 5254 
 •381 6938 
 
 29 
 
 88 
 
 146 
 
 205 
 
 262 
 
 321 
 
 378 
 435 
 491 
 547 
 
 602 
 656 
 710 
 762 
 814 
 
 915 
 
 964 
 
 1011 
 
 1057 
 
 1102 
 1146 
 1189 
 1229 
 1269 
 
 1307 
 1343 
 1377 
 1411 
 1442 
 
 1471 
 1499 
 1525 
 1549 
 1571 
 
 1591 
 1610 
 1626 
 1640 
 1653 
 
 1663 
 1671 
 
 1678 
 1682 
 1684 
 
 A2 
 
 log F (r, *) 
 
 59 
 66 
 58 
 58 
 58 
 
 57 
 57 
 M 
 
 50 
 55 
 
 54 
 54 
 53 
 
 52 
 51 
 
 50 
 49 
 48 
 
 40 
 
 45 
 
 41 
 43 
 41 
 39 
 38 
 
 3G 
 31 
 33 
 
 31 
 30 
 
 -28 
 86 
 
 24 
 
 22 
 20 
 
 18 
 17 
 14 
 12 
 11 
 
 8 
 6 
 4 
 
 2 
 
 1-790 2485 
 •791 2436 
 •794 2308 
 •799 2158 
 •806 2081 
 ■815 2211 
 
 •826 2719 
 •839 3819 
 •854 5764 
 •871 8848 
 •891 3410 
 
 •912 9831 
 •936 8541 
 •963 0016 
 •991 4782 
 0022 3418 
 
 •055 6559 
 •091 4895 
 •129 9181 
 •171 0234 
 •214 8939 
 
 •261 6258 
 •311 3226 
 •364 0964 
 •420 0681 
 •479 3681 
 
 •542 1372 
 •608 5269 
 •678 7007 
 •752 8356 
 •831 1213 
 
 •913 7633 
 1-000 9834 
 093 0211 
 •190 1352 
 •292 6060 
 
 •400 7368 
 •514 8561 
 •635 3206 
 •762 5175 
 •896 8681 
 
 2038 8307 
 •188 9051 
 •347 6370 
 ■515 6232 
 •693 5170 
 
 log H(t, v) 
 
 0-378 2941 
 •378 2969 
 •378 3051 
 •378 3188 
 •378 3380 
 •378 3626 
 
 •378 3927 
 •378 4281 
 •378 4688 
 ■378 5149 
 •378 5661 
 
 •378 6226 
 •378 6841 
 •378 7507 
 •378 8222 
 •378 8985 
 
 •378 9796 
 •379 0654 
 •379 1558 
 ■379 2506 
 •379 3498 
 
 •379 4532 
 •379 5606 
 •379 6721 
 •379 7874 
 •379 9063 
 
 •380 0289 
 •380 1548 
 •380 2839 
 •380 4162 
 •380 5514 
 
 •380 6893 
 ■380 8299 
 •380 9729 
 •381 1181 
 •381 2654 
 
 •381 4146 
 •381 5655 
 •381 7180 
 •381 8718 
 •382 0267 
 
 ■382 1826 
 •382 3393 
 •382 4966 
 •382 6543 
 •382 8122 
 
 82 
 137 
 192 
 246 
 
 300 
 354 
 408 
 460 
 513 
 
 564 
 615 
 666 
 715 
 764 
 
 811 
 
 858 
 904 
 948 
 992 
 
 1034 
 1075 
 1114 
 1153 
 1190 
 
 1225 
 1259 
 1292 
 1323 
 
 1352 
 
 1380 
 1405 
 1430 
 1452 
 1473 
 
 1492 
 1509 
 1525 
 1538 
 1549 
 
 1559 
 1567 
 1573 
 1577 
 1579 
 
 A 2 
 
 log F (r, y) 
 
 55 
 55 
 55 
 54 
 54 
 
 54 
 53 
 53 
 
 52 
 52 
 
 51 
 50 
 49 
 
 49 
 48 
 
 47 
 
 40 
 45 
 43 
 42 
 
 41 
 
 10 
 
 38 
 
 37 
 35 
 
 34 
 
 33 
 31 
 29 
 28 
 
 20 
 2 1 
 23 
 21 
 19 
 
 17 
 15 
 13 
 12 
 10 
 
 8 
 6 
 4 
 2 
 
 777 4825 
 
 778 5436 
 781 7289 
 787 0445 
 794 5004 
 804 1110 
 
 815 8945 
 829 8736 
 846 0751 
 864 5306 
 
 885 2758 
 
 908 3516 
 933 8035 
 961 6821 
 992 0436 
 024 9495 
 
 060 4672 
 098 6704 
 139 6392 
 183 4606 
 230 2288 
 
 280 0460 
 333 0225 
 389 2774 
 448 9393 
 512 1471 
 
 579 0504 
 649 8104 
 724 6013 
 8036106 
 
 887 0408 
 
 975 1103 
 068 0549 
 166 1294 
 269 6092 
 378 7922 
 
 494 0009 
 
 615 5849 
 
 43 9235 
 
 879 4285 
 
 2022 5477 
 
 •173 7686 
 •333 6229 
 •502 6906 
 •681 6063 
 •871 0649 
 
 log H (r, ») 
 
 0379 5425 
 •379 5450 
 •379 5528 
 •379 5657 
 •379 5838 
 •379 6070 
 
 •379 6352 
 •379 6686 
 •379 7070 
 •379 7503 
 •379 7986 
 
 •379 8517 
 •379 9096 
 •379 9723 
 •380 0396 
 •380 1115 
 
 •380 1878 
 •380 2686 
 •380 3537 
 •380 4430 
 •380 5363 
 
 •380 6336 
 •380 7348 
 •380 8397 
 •380 9482 
 •381 0602 
 
 •381 1756 
 •381 2941 
 •381 4157 
 •381 5402 
 •381 6674 
 
 •381 7973 
 •381 9296 
 •382 0642 
 •382 2009 
 •382 3395 
 
 •382 4800 
 •382 6220 
 •382 7655 
 ■382 9102 
 •383 0561 
 
 •383 2028 
 •383 3503 
 •383 4984 
 •383 6468 
 •383 7953 
 
 26 
 
 78 
 
 129 
 
 181 
 
 232 
 
 283 
 333 
 
 384 
 433 
 483 
 
 531 
 
 579 
 627 
 673 
 719 
 
 764 
 808 
 851 
 893 
 933 
 
 973 
 1012 
 1049 
 1085 
 1120 
 
 1153 
 1185 
 1216 
 1245 
 1273 
 
 1299 
 1323 
 1346 
 1367 
 1387 
 
 1404 
 1420 
 1435 
 1448 
 1458 
 
 1468 
 1475 
 
 1480 
 1484 
 1486 
 
 52 
 52 
 51 
 51 
 51 
 
 51 
 
 50 
 50 
 49 
 49 
 
 48 
 47 
 47 
 46 
 45 
 
 44 
 43 
 42 
 41 
 40 
 
 39 
 37 
 36 
 35 
 33 
 
 32 
 31 
 29 
 
 28 
 26 
 
 24 
 23 
 21 
 20 
 18 
 
 16 
 14 
 13 
 11 
 9 
 
 7 
 6 
 4 
 
 17-2 
 
132 
 
 Tables for Statisticians and Biometrieians 
 TABLE LIV— (continued). 
 
 r=18 
 
 r=19 
 
 r = 20 
 
 4 
 
 5 
 
 6 
 7 
 8 
 9 
 10 
 
 11 
 12 
 IS 
 
 U 
 15 
 
 16 
 
 17 
 IS 
 19 
 SO 
 
 SI 
 
 24 
 25 
 
 26 
 
 27 
 
 SO 
 
 SI 
 
 82 
 S3 
 34 
 S5 
 
 36 
 37 
 38 
 39 
 40 
 
 41 
 42 
 4» 
 
 U 
 45 
 
 log F (r, *) 
 
 1-7G5 4249 
 •766 5520 
 •769 9355 
 •775 5818 
 •783 5015 
 •793 7099 
 
 •806 2262 
 •821 0746 
 •838 2835 
 •857 8863 
 •879 9209 
 
 •904 4307 
 •931 4638 
 •961 0741 
 •993 3209 
 0-028 2696 
 
 •065 9915 
 •106 5648 
 •150 0744 
 •196 6125 
 •246 2790 
 
 •299 1823 
 •355 4391 
 •415 1758 
 •478 5288 
 •545 6451 
 
 •616 6833 
 •691 8145 
 •771 2230 
 •855 1078 
 ■943 6834 
 
 1037 1813 
 •135 8513 
 •239 9636 
 ■349 8099 
 •465 7060 
 
 ■587 9937 
 •717 0435 
 •853 2571 
 •997 0711 
 2-148 9600 
 
 •309 4403 
 •479 0754 
 •658 4799 
 •848 3263 
 3-049 3507 
 
 log // (r, c) A 
 
 0380 6494 
 •380 6518 
 •380 6591 
 •380 6713 
 •380 6884 
 •380 7103 
 
 •380 7370 
 •380 7685 
 •380 8048 
 •380 8457 
 •380 8913 
 
 •380 9415 
 •380 9962 
 •381 0554 
 •381 1190 
 •381 1869 
 
 •381 2591 
 •381 3354 
 •381 4157 
 •381 5001 
 •381 5882 
 
 •381 6802 
 •381 7757 
 •381 8749 
 •381 9774 
 ■382 0832 
 
 •382 1921 
 •382 3041 
 •382 4189 
 ■382 5365 
 •382 6567 
 
 •382 7794 
 •382 9043 
 •383 0314 
 •383 1605 
 •383 2915 
 
 •383 4241 
 •383 5583 
 •383 6938 
 •383 8305 
 •383 9683 
 
 •384 1069 
 •384 2462 
 •384 3860 
 ■384 5262 
 •384 6665 
 
 24 
 
 73 
 
 122 
 
 171 
 
 219 
 
 267 
 315 
 362 
 409 
 456 
 
 502 
 547 
 592 
 636 
 679 
 
 722 
 763 
 804 
 843 
 
 882 
 
 919 
 
 956 
 
 991 
 
 1025 
 
 1058 
 
 1089 
 1120 
 1148 
 1176 
 1202 
 
 1227 
 1249 
 1271 
 1291 
 1310 
 
 1326 
 1342 
 1355 
 1367 
 1377 
 
 1386 
 1393 
 1398 
 1402 
 1403 
 
 A D 
 
 log /<>,..) 
 
 1-754 0014 
 •755 1945 
 •758 7762 
 •764 7532 
 •773 1369 
 •783 9431 
 
 •797 1925 
 •812 9104 
 •831 1269 
 •851 8772 
 •875 2015 
 
 •901 1455 
 •929 7C03 
 •961 1026 
 •995 2351 
 032 2269 
 
 •072 1535 
 •115 0974 
 •161 1483 
 •210 4036 
 •262 9690 
 
 •318 9588 
 •378 4966 
 •441 7157 
 •508 7603 
 ■579 7858 
 
 •654 9597 
 •734 4627 
 •818 4896 
 •907 2506 
 1-000 9723 
 
 •099 8993 
 •204 2956 
 ■314 4464 
 •430 6601 
 •553 2702 
 
 •682 6377 
 •819 1539 
 •963 2435 
 2-115 3673 
 •276 0267 
 
 •445 7673 
 •625 1841 
 •814 9263 
 3 015 7041 
 •228 2952 
 
 log H{r,v)\ A 
 
 0-381 6376 
 •381 6399 
 •381 6469 
 •381 6584 
 •381 6746 
 •381 6954 
 
 •381 7207 
 •381 7505 
 •381-7849 
 •381 8237 
 •381 8669 
 
 •381 9144 
 •381 9663 
 •382 0224 
 •382 0826 
 •382 1470 
 
 •382 2154 
 •382 2877 
 •382 3638 
 •382 4437 
 •382 5273 
 
 •382 6144 
 •382 7049 
 •382 7988 
 •382 8960 
 •382 9962 
 
 •383 0994 
 •383 2055 
 •383 3143 
 •383 4257 
 •383 5396 
 
 •383 6558 
 •383 7742 
 •383 8946 
 •384 0170 
 •384 1411 
 
 •384 2667 
 •384 3938 
 •384 5222 
 •384 6518 
 •384 7823 
 
 •384 9136 
 •385 0455 
 •385 1780 
 •385 3108 
 •385 4437 
 
 23 
 
 70 
 
 116 
 
 162 
 
 208 
 
 253 
 299 
 343 
 
 388 
 432 
 
 476 
 519 
 561 
 603 
 643 
 
 684 
 723 
 761 
 799 
 836 
 
 871 
 906 
 939 
 971 
 1002 
 
 1032 
 1061 
 1088 
 1114 
 1139 
 
 1162 
 1184 
 1204 
 1223 
 1241 
 
 1257 
 1271 
 1284 
 1295 
 1305 
 
 1313 
 1320 
 1324 
 1328 
 1330 
 
 A- 
 
 logf(r.K) 
 
 tf 
 48 
 46 
 M 
 
 40 
 
 45 
 
 45 
 45 
 41 
 
 44 
 
 43 
 49 
 
 ■1-2. 
 41 
 10 
 
 aa 
 
 aa 
 
 38 
 37 
 M 
 
 35 
 33 
 
 89 
 
 31 
 30 
 
 2f) 
 27 
 96 
 98 
 
 ^3 
 
 22 
 90 
 19 
 
 17 
 10 
 
 II 
 13 
 11 
 10 
 8 
 
 7 
 6 
 3 
 
 2 
 
 1-743 1485 
 •744 4077 
 •748 1876 
 •754 4955 
 •763 3431 
 •774 7474 
 
 •788 7299 
 •805 3174 
 •824 5417 
 ■846 4398 
 •871 0541 
 
 •898 4326 
 •928 6293 
 •961 7039 
 •997 7224 
 0036 7574 
 
 •078 8894 
 •124 2043 
 •172 7969 
 •224 7699 
 •280 2346 
 
 ■339 3115 
 
 •402 1306 
 •468 8327 
 •539 5095 
 •614 5047 
 
 •693 8148 
 •777 6902 
 •866 3362 
 •959 9740 
 1-058 8424 
 
 •163 1992 
 •273 3224 
 •339 5124 
 •512 0941 
 •641 4188 
 
 •777 8668 
 •921 8503 
 2073 8165 
 •234 2509 
 •403 6815 
 
 •582 6831 
 
 •771 8822 
 
 •971 9629 
 
 3183 6730 
 
 •407 8314 
 
 log H (>;v) A 
 
 0-382 5253 
 •382 5275 
 •382 5341 
 •382 5451 
 •382 5605 
 •382 5802 
 
 •382 6042 
 •382 6326 
 •382 6652 
 ■382 7021 
 •382 7432 
 
 •382 7883 
 •382 8376 
 •382 8909 
 •382 9482 
 •383 0093 
 
 •383 0743 
 •383 1430 
 •383 2153 
 •383 2912 
 •383 3706 
 
 •383 4534 
 •383 5394 
 •383 6286 
 ■383 7209 
 •383 8162 
 
 •383 9142 
 •384 0150 
 •384 1184 
 •384 2243 
 •384 3325 
 
 •384 4429 
 •384 5554 
 •384 6698 
 •384 7860 
 •384 9039 
 
 •385 0233 
 •385 1440 
 •385 2660 
 •385 3891 
 •385 5130 
 
 •385 6378 
 •385 7632 
 •385 8890 
 ■386 0151 
 •386 1414 
 
 22 
 
 66 
 
 110 
 
 154 
 
 197 
 
 241 
 284 
 327 
 369 
 411 
 
 452 
 493 
 533 
 573 
 611 
 
 650 
 687 
 724 
 759 
 794 
 
 828 
 860 
 892 
 923 
 952 
 
 981 
 1008 
 1034 
 1059 
 
 1082 
 
 1104 
 1125 
 1144 
 1162 
 1179 
 
 1194 
 1208 
 1220 
 1231 
 1240 
 
 1247 
 1254 
 1258 
 1261 
 1263 
 
 A* 
 
 44 
 44 
 44 
 44 
 43 
 
 43 
 43 
 
 42 
 42 
 41 
 
 41 
 
 40 
 40 
 39 
 39 
 
 37 
 37 
 36 
 35 
 34 
 
 33 
 
 32 
 31 
 
 30 
 28 
 
 27 
 26 
 25 
 23 
 22 
 
 21 
 
 19 
 18 
 17 
 15 
 
 14 
 
 12 
 
 11 
 
 9 
 
 8 
 
 I 
 
Tables of the G (r, v)-lntegrals 
 TABLE LIV— (continued). 
 
 133 
 
 r=21 
 
 r=22 
 
 • = 23 
 
 6 
 7 
 8 
 9 
 10 
 
 11 
 
 12 
 
 13 
 
 U 
 15 
 
 16 
 
 17 
 18 
 19 
 
 20 
 
 25 
 
 27 
 
 SO 
 
 SI 
 
 82 
 S3 
 
 35 
 
 36 
 37 
 38 
 89 
 40 
 
 41 
 42 
 43 
 44 
 45 
 
 log F (r, x) log H (r, v) 
 
 1-732 8121 
 734 1352 
 •738 1155 
 •744 7542 
 •754 0660 
 •766 0683 
 
 •780 7641 
 ■798 2414 
 •818 4736 
 •841 5196 
 •867 4241 
 
 •896 2374 
 •928 0162 
 •962 8233 
 0000 7283 
 •041 8074 
 
 •086 1444 
 •133 8306 
 •184 9653 
 •239 6563 
 •298 0208 
 
 •360 1851 
 •426 2861 
 •496 4716 
 •570 9011 
 •649 7464 
 
 •733 1933 
 
 ■821 4416 
 
 •914 7070 
 
 1013 2222 
 
 •117 2380 
 
 •227 0250 
 •342 8756 
 •465 1055 
 •594 0558 
 •730 0957 
 
 •873 6248 
 2-025 0761 
 •184 9195 
 •353 6651 
 •531 8676 
 
 •720 1308 
 •919 1130 
 3-129 5327 
 •352 1756 
 •587 9021 
 
 383 3271 
 383 3292 
 383 3354 
 383 3459 
 383 3606 
 383 3793 
 
 383 4022 
 383 4293 
 383 4604 
 383 4955 
 383 5346 
 
 383 5776 
 383 6245 
 383 6753 
 383 7299 
 383 7881 
 
 383 8500 
 383 9154 
 
 383 9843 
 
 384 0566 
 384 1322 
 
 384 2111 
 384 2930 
 384 3780 
 384 4659 
 384 5566 
 
 384 6500 
 384 7460 
 384 8445 
 
 384 9453 
 
 385 0484 
 
 385 1535 
 385 2607 
 385 3696 
 385 4803 
 385 5926 
 
 385 7063 
 385 8213 
 
 385 9375 
 
 386 0547 
 386 1728 
 
 386 2916 
 386 4110 
 386 5308 
 386 6510 
 386 7712 
 
 & 
 
 log F (r,v) 
 
 21 
 
 63 
 
 105 
 
 146 
 
 188 
 
 229 
 270 
 311 
 351 
 391 
 
 430 
 469 
 508 
 545 
 582 
 
 619 
 654 
 689 
 723 
 756 
 
 788 
 820 
 850 
 879 
 907 
 
 934 
 
 960 
 
 985 
 
 1008 
 
 1031 
 
 1052 
 1071 
 1090 
 1107 
 1123 
 
 1137 
 
 1150 
 1162 
 1172 
 1181 
 
 1188 
 1194 
 1198 
 1201 
 1203 
 
 1722 9451 
 •724 3364 
 •728 5130 
 •735 4826 
 •745 2584 
 •757 8590 
 
 •773 3082 
 •791 6354 
 •812 8757 
 •837 0698 
 •864 2646 
 
 •894 5129 
 •927 8740 
 •964 4139 
 0-004 2054 
 •047 3287 
 
 •093 8713 
 •143 9291 
 •197 6061 
 •255 0156 
 •316 2801 
 
 •381 5322 
 •450 9155 
 •524 5848 
 •602 7073 
 •685 4632 
 
 •773 0472 
 •865 6689 
 •963 5543 
 1-066 9472 
 •176 1108 
 
 •291 3286 
 •412 9072 
 •541 1773 
 •670 4967 
 •819 2523 
 
 •969 8630 
 2-128 7827 
 •296 5039 
 •473 5612 
 •660 5361 
 
 •858 0615 
 3066 8272 
 •287 5865 
 ■521 1629 
 •768 4580 
 
 log #(»•,«) 
 
 384 0548 
 384 0568 
 384 0628 
 384 0728 
 384 0867 
 384 1047 
 
 384 1266 
 384 1523 
 384 1820 
 384 2155 
 384 2529 
 
 384 2940 
 384 3388 
 384 3872 
 384 4393 
 384 4949 
 
 384 5540 
 384 6164 
 384 6822 
 384 7513 
 384 8235 
 
 384 8987 
 
 384 9770 
 
 385 0581 
 385 1420 
 385 2286 
 
 385 3178 
 385 4094 
 385 5034 
 385 5996 
 385 6980 
 
 385 7984 
 
 385 9007 
 
 386 0047 
 386 1104 
 386 2175 
 
 386 3261 
 386 4359 
 386 5468 
 386 6586 
 386 7713 
 
 386 8848 
 
 386 9987 
 
 387 1131 
 387 2278 
 387 3426 
 
 log F (»-,*) 
 
 20 
 
 60 
 
 100 
 
 140 
 
 179 
 
 219 
 258 
 297 
 335 
 373 
 
 411 
 
 448 
 485 
 521 
 556 
 
 591 
 
 625 
 658 
 690 
 722 
 
 753 
 782 
 811 
 839 
 866 
 
 892 
 916 
 940 
 962 
 984 
 
 1004 
 1023 
 1040 
 1057 
 1072 
 
 1085 
 1098 
 1109 
 1119 
 1127 
 
 1134 
 1140 
 1144 
 1147 
 1148 
 
 40 
 40 
 40 
 40 
 
 aa 
 
 39 
 39 
 
 38 
 38 
 38 
 
 37 
 37 
 
 aa 
 
 35 
 35 
 
 34 
 33 
 32 
 31 
 31 
 
 30 
 
 aa 
 
 28 
 27 
 26 
 
 25 
 24 
 22 
 
 21 
 
 20 
 
 19 
 18 
 16 
 15 
 14 
 
 12 
 11 
 
 10 
 8 
 7 
 
 (i 
 4 
 3 
 1 
 
 1-713 5069 
 •714 9643 
 •719 3391 
 •726 6397 
 •736 8798 
 •750 0786 
 
 •766 2613 
 •785 4585 
 •807 7070 
 •833 0493 
 •861 5346 
 
 •893 2180 
 •928 1617 
 •966 4346 
 0-008 1129 
 •053 2804 
 
 ■102 0289 
 •154 4586 
 •210 6783 
 •270 8064 
 •334 9713 
 
 •403 3116 
 •475 9774 
 •553 1308 
 •634 9466 
 •721 6136 
 
 •813 3351 
 •910 3304 
 1-012 8362 
 •121 1074 
 •235 4191 
 
 •356 0681 
 •483 3750 
 •617 6859 
 •759 3748 
 •908 8466 
 
 2-066 5394 
 •232 9279 
 ■408 5272 
 •593 8968 
 •789 6446 
 
 •996 4325 
 3-214 9823 
 •446 0817 
 •690 5919 
 •949 4561 
 
 logff(r,x) 
 
 384 7182 
 384 7202 
 384 7259 
 384 7355 
 384 7488 
 384 7660 
 
 384 7869 
 384 8116 
 384 8400 
 384 8721 
 384 9078 
 
 384 9471 
 
 384 9899 
 
 385 0363 
 385 0861 
 385 1393 
 
 385 1958 
 385 2556 
 385 3185 
 385 3845 
 385 4536 
 
 385 5256 
 385 6004 
 385 6780 
 385 7583 
 385 8411 
 
 385 9264 
 
 386 0141 
 386 1040 
 386 1961 
 386 2902 
 
 386 3862 
 386 4840 
 386 5836 
 386 6846 
 386 7871 
 
 386 8910 
 
 386 9960 
 
 387 1021 
 387 2091 
 387 3169 
 
 387 4254 
 387 5344 
 387 6438 
 387 7535 
 387 8633 
 
 19 
 
 57 
 
 96 
 
 134 
 
 172 
 
 209 
 247 
 284 
 321 
 357 
 
 393 
 
 429 
 464 
 498 
 532 
 
 565 
 598 
 629 
 660 
 691 
 
 720 
 748 
 776 
 803 
 828 
 
 853 
 877 
 899 
 921 
 941 
 
 960 
 
 978 
 
 995 
 
 1011 
 
 1025 
 
 1038 
 1050 
 1061 
 1070 
 1078 
 
 1085 
 1090 
 1094 
 1097 
 1098 
 
 A2 
 
 38 
 38 
 38 
 38 
 38 
 
 37 
 
 37 
 37 
 36 
 36 
 
 36 
 35 
 34 
 34 
 33 
 
 33 
 
 32 
 31 
 30 
 29 
 
 28 
 28 
 27 
 26 
 25 
 
 24 
 23 
 22 
 20 
 19 
 
 18 
 17 
 16 
 14 
 13 
 
 12 
 
 11 
 10 
 
 8 
 
 7 
 
134 Tables for Statisticians and Biometricians 
 
 TABLE LIV— (continued). 
 
 o 
 i 
 2 
 s 
 4 
 
 5 
 
 6 
 7 
 8 
 9 
 10 
 
 11 
 12 
 IS 
 
 U 
 IB 
 
 1G 
 17 
 18 
 19 
 
 SI 
 
 ss 
 
 2S 
 
 n 
 
 27 
 28 
 29 
 SO 
 
 SI 
 
 32 
 S3 
 Sit 
 S5 
 
 36 
 37 
 S8 
 S9 
 40 
 
 41 
 
 U 
 
 45 
 
 r = 24 
 
 Jagfta ») 
 
 1-704 4818 
 ■705 9862 
 •710 5584 
 •718 1899 
 •728 8942 
 •742 6914 
 
 •759 6077 
 •779 6750 
 •802 9317 
 •829 4225 
 •859 1983 
 
 •892 3170 
 •928 8434 
 •968 8495 
 0-012 4148 
 •059 6268 
 
 •110 5814 
 •165 3831 
 
 •224 1458 
 •286 9928 
 •354 0583 
 
 •425 4870 
 •501 4356 
 ■582 0734 
 ■667 5830 
 •758 1612 
 
 •854 0205 
 •955 3899 
 1-062 5163 
 •175 6661 
 •295 1263 
 
 •421 2069 
 ■554 2426 
 ■094 5945 
 •842 6534 
 •998 8417 
 
 2163 6169 
 •337 4747 
 ■520 9526 
 •714 6347 
 •919 1558 
 
 3-135 2068 
 •363 5411 
 •604 9809 
 •860 4254 
 
 4-130 8591 
 
 log H (r, v) 
 
 0-385 3256 
 •385 3275 
 •385 3330 
 •385 3421 
 •385 3549 
 •385 3714 
 
 •385 3915 
 •385 4151 
 •385 4423 
 •385 4730 
 ■385 5073 
 
 •385 5450 
 •385 5860 
 •385 6305 
 •385 6782 
 •385 7292 
 
 •385 7834 
 •385 8406 
 •385 9009 
 •385 9642 
 •386 0304 
 
 •386 0994 
 •386 1711 
 •386 2455 
 ■386 3225 
 •386 4018 
 
 •386 4836 
 •386 5676 
 •386 6538 
 •386 7420 
 •386 8322 
 
 •386 9242 
 •387 0180 
 •387 1134 
 ■387 2102 
 •387 3085 
 
 •387 4080 
 •387 5086 
 •387 6103 
 •387 7128 
 •387 8162 
 
 •387 9201 
 •388 0246 
 •388 1295 
 •388 2346 
 •388 3398 
 
 18 
 
 55 
 
 92 
 
 128 
 
 164 
 
 201 
 236 
 272 
 307 
 342 
 
 377 
 411 
 
 444 
 477 
 510 
 
 542 
 573 
 
 603 I 
 633 ' 
 662 
 
 690 
 717 
 744 
 769 
 794 
 
 818 
 840 
 862 
 882 
 902 
 
 920 
 938 
 954 
 969 
 982 
 
 995 
 1006 
 1017 
 1026 
 1033 
 
 1040 
 1045 
 1048 
 1051 
 1052 
 
 A- 
 
 37 
 37 
 86 
 86 
 36 
 
 86 
 86 
 
 35 
 
 80 
 
 34 
 
 34 
 31 
 83 
 83 
 
 32 
 
 31 
 30 
 80 
 
 as 
 
 28 
 
 27 
 26 
 26 
 25 
 21 
 
 23 
 
 22 
 21 
 
 18 
 
 18 
 
 17 
 16 
 
 15 
 14 
 13 
 
 11 
 
 10 
 
 9 
 
 8 
 
 6 
 
 6 
 
 3 
 3 
 
 1 
 
 r = 25 
 
 log F (r, ») 
 
 1-695 7781 
 •697 3569 
 •702 1393 
 ■710 1019 
 •721 2704 
 •735 6660 
 
 •753 3158 
 •774 2534 
 •798 5185 
 ■826 1578 
 •857 2243 
 
 ■891 7784 
 •929 8876 
 •971 6270 
 0-017 0795 
 •066 3362 
 
 •119 4971 
 •176 6711 
 •237 9768 
 •303 5430 
 •373 5093 
 
 •448 0267 
 •527 2584 
 •611 3809 
 •700 5843 
 •795 0741 
 
 •895 0715 
 1-000 8153 
 •112 5627 
 •230 5913 
 •355 2004 
 
 •486 7129 
 •625 4776 
 •771 8709 
 •926 3001 
 2 '089 2053 
 
 •261 0633 
 •442 3906 
 •633 7476 
 •835 7426 
 3-049 0373 
 
 •274 3517 
 •512 4708 
 •764 2515 
 4030 6307 
 •312 6342 
 
 log H {>; v) 
 
 0-385 8838 
 •385 8855 
 •385 8908 
 •385 8996 
 •385 9119 
 •385 9277 
 
 ■385 9470 
 •385 9697 
 •385 9958 
 •386 0253 
 •386 0582 
 
 ■386 0943 
 •386 1338 
 •386 1764 
 ■386 2223 
 •386 2712 
 
 •386 3232 
 ■386 3782 
 •386 4361 
 •386 4969 
 •386 5604 
 
 •386 6267 
 •386 6955 
 •386 7669 
 •386 8408 
 •386 9170 
 
 •386 9955 
 •387 0762 
 •387 1589 
 •387 2436 
 •387 3302 
 
 •387 4185 
 •387 5085 
 •387 6001 
 •387 6931 
 •387 7874 
 
 ■387 8829 
 •387 9795 
 •388 0771 
 •388 1756 
 •388 2748 
 
 •388 3746 
 •388 4749 
 •388 5766 
 •388 6765 
 •388 7775 
 
 A A 2 
 
 18 
 
 53 
 
 88 
 
 123 
 
 158 
 
 193 
 
 227 
 261 
 295 
 329 
 
 362 
 394 
 
 427 
 458 
 489 
 
 520 
 550 
 579 
 608 
 635 
 
 662 
 689 
 714 
 739 
 762 
 
 785 
 807 
 827 
 847 
 866 
 
 883 
 900 
 916 
 930 
 943 
 
 955 
 966 
 976 
 985 
 992 
 
 998 
 1003 
 1007 
 1009 
 1010 
 
 35 
 35 
 35 
 35 
 35 
 
 35 
 3-1 
 34 
 34 
 
 88 
 
 33 
 32 
 32 
 31 
 31 
 
 30 
 
 29 
 
 89 
 
 28 
 27 
 
 26 
 
 25 
 25 
 21 
 23 
 
 22 
 81 
 
 20 
 19 
 18 
 
 17 
 16 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 9 
 7 
 6 
 
 5 
 4 
 
 2 
 1 
 
 r = 26 
 
 \ogF(r,v) 
 
 1-687 4284 
 •689 0840 
 •694 0540 
 ■702 3476 
 ■713 9805 
 ■728 9746 
 
 •747 3581 
 •769 1658 
 •794 4395 
 •823 2272 
 •855 5847 
 
 •891 5743 
 •931 2665 
 •974 7393 
 0-022 0791 
 ■073 3806 
 
 •128 7480 
 •188 2945 
 •252 1435 
 •320 4290 
 ■393 2963 
 
 •470 9025 
 •553 4176 
 •641 0250 
 •733 9226 
 ■832 3242 
 
 •936 4600 
 1-046 5783 
 •162 9470 
 •285 8547 
 •415 6129 
 
 •552 5576 
 •697 0516 
 •849 4867 
 2-010 2864 
 •179 9088 
 
 •358 8499 
 •547 6471 
 •746 8833 
 •957 1916 
 3179 2602 
 
 •413 8384 
 •661 7426 
 •923 8647 
 4-201 1786 
 •494 7524 
 
 log H(r,v) A A* 
 
 0-386 3984 
 •386 4001 
 •386 4052 
 •386 4136 
 ■386 4254 
 •386 4406 
 
 •386 4591 
 •386 4810 
 •386 5061 
 •386 5345 
 •386 5661 
 
 ■386 6009 
 •386 6388 
 •386 6798 
 •386 7239 
 •386 7710 
 
 •386 8210 
 •386 8738 
 •386 9295 
 •386 9880 
 •387 0491 
 
 •387 1128 
 •387 1790 
 •387 2477 
 •387 3187 
 •387 3920 
 
 •387 4674 
 •387 5450 
 •387 6246 
 ■387 7060 
 •387 7893 
 
 •387 8742 
 •387 9608 
 •388 0488 
 •388 1382 
 •388 2289 
 
 •388 3208 
 •388 4137 
 •388 5075 
 •388 6022 
 •388 6975 
 
 •388 7935 
 •388 8900 
 •388 9868 
 •389 0838 
 •389 1810 
 
 17 
 51 
 
 85 
 118 
 152 
 
 185 
 218 
 251 
 284 
 316 
 
 348 
 379 
 411 
 440 
 471 
 
 500 
 529 
 557 
 584 
 611 
 
 637 
 662 
 687 
 710 
 733 
 
 755 
 776 
 796 
 815 
 833 
 
 850 
 865 
 880 
 894 
 907 
 
 919 
 929 
 938 
 947 
 954 
 
 960 
 964 
 968 
 970 
 972 
 
Tables of the G (r, v)-Integrals 
 TABLE LIV— (continued). 
 
 135 
 
 f" 
 
 r=27 
 
 r=28 
 
 r=29 
 
 \og F(r,v) 
 
 log B(r,v) 
 
 A 
 
 A 8 
 
 logf(r,») 
 
 log H(r,v) 
 
 A 
 
 A a 
 
 log F(r,y) 
 
 log If (r, ») 
 
 A 
 
 A s 
 
 
 
 1-679 3877 
 
 0-386 8744 
 
 16 
 
 49 
 
 81 
 
 114 
 
 146 
 
 
 1-671 6341 
 
 0-387 3160 
 
 16 
 
 47 
 
 79 
 
 110 
 
 141 
 
 
 T-664 1478 
 
 0-387 7268 
 
 15 
 
 46 
 
 76 
 
 106 
 
 136 
 
 
 1 
 
 •681 1094 
 
 •386 8760 
 
 33 
 
 •673 4219 
 
 •387 3176 
 
 31 
 
 •666 0017 
 
 •387 7283 
 
 30 
 
 2 
 
 •686 2778 
 
 •386 8809 
 
 33 
 
 •678 7887 
 
 •387 3223 
 
 31 
 
 •671 5669 
 
 •387 7328 
 
 30 
 
 3 
 
 •694 9025 
 
 •386 8891 
 
 32 
 
 •687 7445 
 
 •387 3301 
 
 31 
 
 •680 8538 
 
 •387 7404 
 
 30 
 
 4 
 
 •706 9998 
 
 •386 9005 
 
 32 
 
 •700 3062 
 
 •387 3411 
 
 31 
 
 •693 8799 
 
 •387 7510 
 
 30 
 
 5 
 
 •722 5923 
 
 •386 9151 
 
 32 
 
 •716 4973 
 
 •387 3552 
 
 31 
 
 •710 6695 
 
 •387 7647 
 
 30 
 
 6 
 
 •741 7096 
 
 •386 9329 
 
 178 
 210 
 242 
 273 
 304 
 
 32 
 
 •736 3483 
 
 •387 3724 
 
 172 
 203 
 233 
 264 
 293 
 
 31 
 
 •731 2544 
 
 •387 7813 
 
 166 
 
 196 
 225 
 254 
 283 
 
 30 
 
 7 
 
 •764 3877 
 
 ■386 9539 
 
 32 
 
 •759 8968 
 
 •387 3927 
 
 30 
 
 •755 6734 
 
 •387 8008 
 
 30 
 
 8 
 
 •790 6698 
 
 •386 9781 
 
 31 
 
 •787 1876 
 
 •387 4160 
 
 30 
 
 •783 9728 
 
 •387 8234 
 
 29 
 
 9 
 
 •820 6063 
 
 •387 0055 
 
 31 
 
 •818 2728 
 
 •387 4424 
 
 30 
 
 •816 2068 
 
 •387 8488 
 
 29 
 
 10 
 
 •854 2546 
 
 •387 0359 
 
 31 
 
 •853 2121 
 
 •387 4717 
 
 30 
 
 •852 4372 
 
 •387 8772 
 
 29 
 
 11 
 
 •891 6799 
 
 ■387 0694 
 
 335 
 
 365 
 395 
 424 
 453 
 
 30 
 
 ■892 0731 
 
 •387 5041 
 
 323 
 
 352 
 381 
 409 
 437 
 
 29 
 
 •892 7340 
 
 •387 9084 
 
 312 
 
 340 
 368 
 395 
 422 
 
 28 
 
 12 
 
 •932 9552 
 
 •387 1059 
 
 30 
 
 •934 9315 
 
 •387 5393 
 
 29 
 
 •937 1757 
 
 •387 9424 
 
 28 
 
 is 
 
 •978 1615 
 
 •387 1454 
 
 29 
 
 '981 8715 
 
 •387 5774 
 
 28 
 
 •985 8495 
 
 •387 9792 
 
 27 
 
 U 
 
 0-027 3887 
 
 •387 1879 
 
 29 
 
 0-032 9863 
 
 •387 6183 
 
 28 
 
 0-038 8519 
 
 •388 0187 
 
 27 
 
 IS 
 
 •080 7353 
 
 •387 2332 
 
 28 
 
 •088 3780 
 
 •387 6620 
 
 27 
 
 •096 2888 
 
 •388 0609 
 
 26 
 
 16 
 
 ■138 3092 
 
 •387 2814 
 
 482 
 509 
 536 
 563 
 588 
 
 28 
 
 •148 1586 
 
 •387 7084 
 
 464 
 491 
 517 
 543 
 567 
 
 27 
 
 •158 2763 
 
 •388 1057 
 
 448 
 474 
 499 
 524 
 548 
 
 26 
 
 17 
 
 •200 2283 
 
 •387 3323 
 
 27 
 
 •212 4505 
 
 •387 7576 
 
 26 
 
 •224 9410 
 
 •388 1531 
 
 25 
 
 18 
 
 •266 6208 
 
 •387 3859 
 
 26 
 
 •281 3865 
 
 •387 8093 
 
 25 
 
 •296 4208 
 
 •388 2031 
 
 25 
 
 19 
 
 •337 6258 
 
 •387 4422 
 
 26 
 
 •355 1112 
 
 •387 8635 
 
 25 
 
 •372 8653 
 
 •388 2555 
 
 24 
 
 SO 
 
 •413 3944 
 
 •387 5010 
 
 25 
 
 •433 7811 
 
 •387 9203 
 
 24 
 
 •454 4367 
 
 •388 3103 
 
 23 
 
 SI 
 
 •494 0896 
 
 •387 5624 
 
 613 
 
 638 
 661 
 684 
 706 
 
 24 
 
 •517 5657 
 
 •387 9794 
 
 592 
 615 
 638 
 660 
 681 
 
 23 
 
 •541 3106 
 
 •388 3674 
 
 571 
 
 594 
 616 
 637 
 657 
 
 23 
 
 22 
 
 •579 8882 
 
 •387 6261 
 
 24 
 
 •606 6479 
 
 •388 0409 
 
 23 
 
 ■633 6767 
 
 •388 4268 
 
 22 
 
 23 
 
 •670 9806 
 
 •387 6923 
 
 23 
 
 •701 2256 
 
 •388 1047 
 
 22 
 
 •731 7398 
 
 •388 4883 
 
 21 
 
 n 
 
 •767 5727 
 
 •387 7607 
 
 22 
 
 •801 5122 
 
 •388 1706 
 
 21 
 
 •835 7212 
 
 •388 5520 
 
 20 
 
 25 
 
 ■869 8863 
 
 •387 8312 
 
 21 
 
 ■907 7380 
 
 •388 2387 
 
 20 
 
 •945 8594 
 
 •388 6177 
 
 20 J 
 
 26 
 
 •978 1616 
 
 •387 9039 
 
 727 
 747 
 766 
 784 
 802 
 
 20 
 
 1-020 1512 
 
 •388 3088 
 
 701 
 720 
 739 
 756 
 773 
 
 19 
 
 1-062 4115 
 
 •388 6854 
 
 677 
 696 
 713 
 730 
 
 747 
 
 19 
 
 27 
 
 1-092 6538 
 
 •387 9786 
 
 19 
 
 •139 0194 
 
 •388 3808 
 
 18 
 
 •185 6549 
 
 •388 7550 
 
 17 
 
 28 
 
 •213 6439 
 
 •388 0552 
 
 18 
 
 •264 6312 
 
 •388 4547 
 
 18 
 
 •315 8886 
 
 •388 8263 
 
 17 
 
 29 
 
 •341 4310 
 
 •388 1337 
 
 17 
 
 •397 2979 
 
 •388 5303 
 
 17 
 
 •453 4351 
 
 •388 8993 
 
 16 
 
 SO 
 
 •476 3386 
 
 •388 2138 
 
 16 
 
 •537 3550 
 
 •388 6077 
 
 16 
 
 •598 6420 
 
 •388 9740 
 
 15 
 
 31 
 
 •618 7157 
 
 •388 2956 
 
 818 
 833 
 
 848 
 861 
 
 873 
 
 15 
 
 •685 1648 
 
 •388 6865 
 
 789 
 804 
 818 
 830 
 
 842 
 
 15 
 
 •751 8846 
 
 •389 0501 
 
 762 
 776 
 789 
 802 
 813 
 
 14 
 
 32 
 
 •768 9393 
 
 •388 3790 
 
 14 
 
 •841 1182 
 
 •388 7669 
 
 14 
 
 •913 5680 
 
 •389 1277 
 
 13 
 
 S3 
 
 ■927 4164 
 
 •388 4638 
 
 13 
 
 2-005 6375 
 
 •388 8487 
 
 13 
 
 2-084 1297 
 
 •389 2067 
 
 12 
 
 34 
 
 2-094 5869 
 
 •388 5499 
 
 12 
 
 •179 1790 
 
 •388 9317 
 
 12 
 
 ■264 0426 
 
 •389 2868 
 
 11 
 
 35 
 
 •270 9268 
 
 •388 6372 
 
 11 
 
 •362 2366 
 
 •389 0159 
 
 11 
 
 •453 8181 
 
 •389 3682 
 
 10 
 
 36 
 
 •456 9512 
 
 •388 7257 
 
 885 
 895 
 904 
 912 
 918 
 
 10 
 
 •555 3447 
 
 •389 1012 
 
 853 
 863 
 872 
 879 
 886 
 
 10 
 
 •654 0100 
 
 •389 4505 
 
 824 
 833 
 841 
 849 
 855 
 
 9 
 
 37 
 
 •653 2186 
 
 •388 8151 
 
 9 
 
 •759 0825 
 
 •389 1875 
 
 9 
 
 •865 2184 
 
 •389 5338 
 
 8 
 
 38 
 
 •860 3344 
 
 ■388 9055 
 
 8 
 
 •974 0781 
 
 •389 2746 
 
 7 
 
 3-088 0940 
 
 •389 6179 
 
 7 
 
 39 
 
 3 '078 9562 
 
 ■388 9967 
 
 7 
 
 3-201 0137 
 
 •389 3625 
 
 7 
 
 •323 3436 
 
 •389 7028 
 
 6 
 
 40 
 
 •309 7991 
 
 •389 0885 
 
 6 
 
 •440 6310 
 
 •389 4511 
 
 5 
 
 •571 7357 
 
 •389 7883 
 
 5 
 
 41 
 
 •553 6412 
 
 •389 1809 
 
 924 
 929 
 932 
 934 
 936 
 
 5 
 
 •693 7374 
 
 •389 5402 
 
 891 
 896 
 899 
 901 
 902 
 
 4 
 
 •834 1065 
 
 •389 8744 
 
 860 
 865 
 868 
 870 
 871 
 
 4 
 
 42 
 
 •811 3309 
 
 •389 2738 
 
 3 
 
 •961 2127 
 
 •389 6298 
 
 3 
 
 4-111 3678 
 
 •389 9608 
 
 3 
 
 43 
 
 4-083 7948 
 
 •389 3670 
 
 2 
 
 4-244 0178 
 
 •389 7197 
 
 2 
 
 •404 5148 
 
 •390 0476 
 
 2 
 
 44 
 
 •372 0436 
 
 •389 4604 
 
 1 
 
 •543 2028 
 
 •389 8098 
 
 1 
 
 •714 6355 
 
 •390 1346 
 
 1 
 
 45 
 
 •677 1878 
 
 •389 5540 
 
 
 •859 9176 
 
 •389 9000 
 
 
 5-042 9213 
 
 •390 2217 
 
 
136 
 
 Tables for Statisticians and Biometriciam 
 TABLE LIV—(cuntinued). 
 
 9 
 10 
 
 11 
 12 
 13 
 14 
 
 in 
 
 16 
 17 
 
 18 
 19 
 
 20 
 
 21 
 
 22 
 ..'.: 
 24 
 
 27 
 
 29 
 
 SO 
 
 37 
 38 
 
 .10 
 
 40 
 
 41 
 42 
 
 U 
 46 
 
 r=30 
 
 log F(r, p) 
 
 1 656 9109 
 •058 8309 
 064 5945 
 •074 2126 
 087 7031 
 ■705 0914 
 
 ■726 4101 
 ■751 6995 
 •781 0077 
 •814 3906 
 •851 9121 
 
 •893 6447 
 •939 6098 
 •990 0775 
 0-044 9676 
 •104 4498 
 
 •168 6443 
 •237 6820 
 •311 7056 
 •390 8701 
 •475 3432 
 
 •565 3065 
 •660 9566 
 •762 5053 
 •870 1810 
 ■984 2323 
 
 1-104 9235 
 •232 5 123 
 •367 3981 
 •509 8245 
 •660 1814 
 
 •818-8570 
 •986 2707 
 2-162 8749 
 •349 1593 
 •545 6529 
 
 •752 9288 
 ■971 6080 
 3 202 3639 
 •445 9277 
 ■703 0947 
 
 •974 7302 
 
 4-261 7776 
 
 •505 2668 
 
 •886 3234 
 
 5-220 1801 
 
 log // (r, v) 
 
 0-388 1099 
 •388 1113 
 •388 1157 
 •388 1231 
 •388 1333 
 •388 1465 
 
 ■388 1625 
 ■388 1815 
 •388 2032 
 •388 2278 
 •388 2552 
 
 •388 2854 
 •388 3183 
 •388 3538 
 •388 3920 
 •388 4328 
 
 •388 4762 
 •388 5220 
 •388 5703 
 •388 6209 
 •388 6739 
 
 •388 7291 
 •388 7865 
 •388 8460 
 •388 9076 
 •388 9711 
 
 •389 0366 
 •389 1038 
 •389 1727 
 •389 2433 
 •389 3155 
 
 •389 3891 
 •389 4642 
 ■389 5405 
 •389 6180 
 •389 6966 
 
 •389 7762 
 •389 8567 
 •389 9380 
 •390 0201 
 •390 1028 
 
 •390 1859 
 •390 2695 
 •390 3534 
 •390 4375 
 •390 5217 
 
 A- 
 
 15 
 
 44 
 
 73 
 
 103 
 
 132 
 
 161 
 
 189 
 218 
 246 
 
 274 
 
 302 
 
 329 
 356 
 382 
 408 
 
 433 
 
 458 
 483 
 507 
 530 
 
 552 
 574 
 595 
 616 
 635 
 
 654 
 672 
 690 
 700 
 722 
 
 736 
 750 
 763 
 
 775 
 786 
 
 796 
 805 
 813 
 
 820 
 827 
 
 832 
 836 
 839 
 841 
 842 
 
 r=31 
 
 log F(r,v) 
 
 649 9073 
 651 8935 
 657 8555 
 667 8048 
 681 7598 
 699 7466 
 
 721 7993 
 747 9592 
 778 2762 
 812 8080 
 
 851 6207 
 
 894 7893 
 942 3977 
 994 5394 
 051 3173 
 112 8450 
 
 179 2464 
 250 6573 
 327 2249 
 409 1093 
 496 4842 
 
 589 5372 
 688 4714 
 793 5058 
 904 8771 
 022 8405 
 
 147 6710 
 279 6653 
 419 1433 
 566 4498 
 721 9568 
 
 886 0657 
 059 2096 
 241 8567 
 434 5127 
 037 7246 
 
 852 0847 
 078 2349 
 316 8712 
 568 7494 
 834 6915 
 
 115 5919 
 412 4256 
 726 2571 
 058 2499 
 
 409 0782 
 
 log//(r, v) 
 
 0-388 4679 
 •388 4694 
 •388 4736 
 •388 4808 
 •388 4906 
 •388 5034 
 
 •388 5189 
 •388 5372 
 •388 5583 
 •388 5822 
 •388 6086 
 
 •388 6378 
 •388 6696 
 •388 7041 
 •388 7410 
 •388 7805 
 
 •388 8225 
 •388 8668 
 •388 9136 
 •388 9626 
 •389 0138 
 
 •389 0673 
 •389 1228 
 •389 1804 
 •389 2400 
 •389 3015 
 
 •389 3648 
 •389 4299 
 •389 4966 
 •389 5649 
 •389 6348 
 
 •389 7060 
 •389 7786 
 •389 8525 
 •389 9275 
 •390 0035 
 
 •390 0806 
 •390 1585 
 •390 2372 
 •390 3100 
 •390 3960 
 
 •390 4771 
 •390 5580 
 •390 0392 
 •390 7206 
 ■390 8021 
 
 14 
 43 
 71 
 99 
 127 
 
 155 
 
 183 
 211 
 238 
 265 
 
 292 
 318 
 344 
 370 
 395 
 
 419 
 444 
 467 
 490 
 513 
 
 534 
 556 
 570 
 596 
 615 
 
 633 
 650 
 667 
 683 
 698 
 
 713 
 726 
 738 
 750 
 761 
 
 770 
 779 
 787 
 794 
 800 
 
 805 
 809 
 812 
 814 
 815 
 
 A' 
 
 )- = 32 
 
 log *'(<-, i>) 
 
 1-643 1220 
 •045 1748 
 •051 3354 
 ■061 6158 
 •676 0352 
 •694 6208 
 
 •717 4073 
 •744 4379 
 •775 7637 
 •811 4444 
 ■851 5483 
 
 •896 1530 
 •945 3449 
 •999 2205 
 0-057 8864 
 •121 4595 
 
 •190 0681 
 •263 8522 
 •342 9638 
 •427 5684 
 •517 8452 
 
 •613 9879 
 •716 2063 
 ■824 7266 
 •939 7929 
 1-061 6692 
 
 •190 6391 
 •327 0091 
 •471 1094 
 •623 2962 
 •783 9534 
 
 •953 4956 
 2-132 3701 
 ■321 0601 
 •520 0879 
 •730 0183 
 
 •951 4628 
 3-185 0841 
 •431 6009 
 •691 7938 
 •366 5111 
 
 4-256 6765 
 •563 2967 
 •887 4707 
 
 5-230 3999 
 •593 3997 
 
 log // (r, •>) 
 
 A A- 
 
 0-388 8034 
 •388 8048 
 •388 8089 
 ■388 8158 
 •388 8254 
 •388 8378 
 
 •388 8528 
 ■388 8706 
 •388 8910 
 •388 9140 
 •388 9397 
 
 •388 9680 
 •388 9988 
 •389 0322 
 •389 0680 
 •389 1062 
 
 ■389 1469 
 •389 1899 
 •389 2351 
 •389 2826 
 •389 3323 
 
 •389 3840 
 ■389 4379 
 •389 4937 
 •389 5514 
 •389 6109 
 
 •389 0723 
 •389 7353 
 •389 8000 
 •389 8001 
 •389 9338 
 
 ■390 0028 
 •390 0732 
 •390 1447 
 •390 2174 
 •390 2911 
 
 •390 3657 
 •390 4412 
 •390 517 1 
 •390 5944 
 •390 6719 
 
 •390 7498 
 •390 8282 
 •390 9069 
 •390 9857 
 •391 0646 
 
 14 
 41 
 69 
 90 
 123 
 
 151 
 177 
 204 
 231 
 257 
 
 283 
 308 
 333 
 358 
 383 
 
 406 
 430 
 453 
 475 
 497 
 
 518 
 538 
 
 558 
 577 
 596 
 
 613 
 630 
 646 
 662 
 677 
 
 090 
 703 
 715 
 727 
 737 
 
 746 
 755 
 763 
 769 
 775 
 
 780 
 784 
 787 
 
 789 
 789 
 
 28 
 27 
 27 
 27 
 
 27 
 
 27 
 27 
 26 
 26 
 26 
 
 26 
 25 
 25 
 
 24 
 24 
 
 23 
 23 
 
 22 
 22 
 21 
 
 20 
 20 
 19 
 19 
 
 18 
 
 17 
 16 
 15 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 9 
 
 9 
 
 8 
 7 
 6 
 5 
 
 4 
 3 
 
 2 
 
 1 
 
Tables of the G (r, v)-Inteyrals 
 TABLE IAY— (continued). 
 
 137 
 
 o 
 1 
 
 2 
 S 
 
 4 
 5 
 
 6 
 7 
 8 
 9 
 10 
 
 11 
 
 12 
 13 
 
 '4 
 15 
 
 16 
 
 n 
 
 18 
 19 
 
 20 
 
 21 
 
 9.9. 
 
 25 
 
 '26 
 27 
 28 
 29 
 90 
 
 SI 
 S2 
 
 3b 
 
 36 
 87 
 38 
 39 
 
 ItO 
 
 hi 
 
 m 
 
 43 
 44 
 45 
 
 r = 33 
 
 log F(r, v) 
 
 1-636 5434 
 •638 6617 
 ■645 0208 
 •655 6323 
 •670 5163 
 •689 7005 
 
 ■713 2210 
 •741 1222 
 ■773 4568 
 •810 2865 
 ■851 6818 
 
 •897 7225 
 •948 4980 
 0004 1077 
 •064 6615 
 •130 2802 
 
 •201 0960 
 •277 2533 
 •358 9091 
 •446 2340 
 ■539 4127 
 
 •638 6453 
 •744 1480 
 •856 1542 
 •974 9160 
 1100 7050 
 
 •233 8145 
 •374 5602 
 •523 2830 
 ■680 3501 
 •846 1578 
 
 2021 1335 
 •205 7387 
 •400 4717 
 •605 8714 
 •822 5204 
 
 3-051 0494 
 •292 1420 
 •546 5396 
 •815 0471 
 
 4-098 5399 
 
 •397 9704 
 •714 3773 
 5-048 3939 
 •402 7696 
 •777 3311 
 
 log H (r, v) 
 
 0-389 1183 
 •389 1197 
 •389 1237 
 •389 1304 
 •389 1397 
 •389 1516 
 
 •389 1662 
 ■389 1835 
 ■389 2033 
 •389 2256 
 •389 2505 
 
 •389 2780 
 •389 3078 
 •389 3402 
 •389 3749 
 •389 4120 
 
 •389 4514 
 ■389 4931 
 ■389 5370 
 •389 5830 
 •389 6312 
 
 •389 6814 
 •389 7336 
 ■389 7877 
 •389 8437 
 •389 9014 
 
 •389 9609 
 •390 0220 
 ■390 0847 
 •390 1489 
 •390 2145 
 
 •390 2815 
 •390 3497 
 •390 4190 
 •390 4895 
 •390 5610 
 
 •390 6333 
 •390 7065 
 •390 7805 
 •390 8551 
 •390 9302 
 
 •391 0058 
 •391 0818 
 •391 1581 
 •391 2345 
 •391 3111 
 
 13 
 
 40 
 
 67 
 
 93 
 
 120 
 
 146 
 172 
 198 
 224 
 249 
 
 274 
 299 
 323 
 347 
 371 
 
 394 
 417 
 439 
 461 
 
 482 
 
 502 
 522 
 541 
 560 
 578 
 
 595 
 611 
 627 
 642 
 656 
 
 669 
 682 
 693 
 705 
 715 
 
 724 
 732 
 739 
 746 
 751 
 
 756 
 760 
 763 
 765 
 766 
 
 if 
 
 r=34 
 
 log*' (r, v) 
 
 1-630 1576 
 •632 3421 
 •638 8996 
 ■649 8424 
 •665 1908 
 •684 9738 
 
 •709 2283 
 •738 0001 
 •771 3436 
 •809 3225 
 •852 0090 
 
 •899 4858 
 •951 8449 
 0-009 1887 
 •071 6306 
 •139 2949 
 
 •212 3180 
 •290 8487 
 ■375 0487 
 •465 0938 
 •561 1746 
 
 •663 4971 
 •772 2842 
 •887 7765 
 1010 2336 
 •139 9357 
 
 •277 1848 
 •422 3065 
 •575 6518 
 •737 5995 
 •908 5576 
 
 2088 9669 
 •279 3029 
 •480 0791 
 •691 8509 
 •915 2186 
 
 3150 8322 
 •399 3963 
 •661 6747 
 •938 4971 
 
 4230 7654 
 
 •539 4613 
 •865 6549 
 5-210 5144 
 •575 3166 
 •961 4600 
 
 logff(r, v) 
 
 0-389 4146 
 •389 4159 
 •389 4198 
 •389 4262 
 •389 4353 
 •389 4469 
 
 •389 4611 
 •389 4778 
 •389 4970 
 •389 5187 
 •389 5429 
 
 •389 5695 
 •389 5985 
 •389 6299 
 •389 6636 
 •389 6996 
 
 •389 7379 
 •389 7783 
 •389 8209 
 •389 8656 
 •389 9123 
 
 •389 9611 
 •390 0117 
 •390 0643 
 •390 1186 
 •390 1746 
 
 •390 2324 
 •390 2917 
 •390 3525 
 •390 4148 
 •390 4785 
 
 •390 5435 
 •390 6097 
 •390 6770 
 •390 7454 
 •390 8148 
 
 •390 8850 
 •390 9561 
 •391 0278 
 •391 1002 
 •391 1732 
 
 ■391 2466 
 •391 3203 
 •391 3943 
 •391 4686 
 •391 5429 
 
 13 
 39 
 65 
 91 
 116 
 
 142 
 167 
 192 
 217 
 242 
 
 266 
 290 
 314 
 337 
 360 
 
 382 
 405 
 426 
 447 
 467 
 
 487 
 507 
 525 
 543 
 561 
 
 577 
 593 
 609 
 623 
 637 
 
 650 
 662 
 673 
 684 
 694 
 
 703 
 711 
 718 
 724 
 729 
 
 734 
 738 
 
 740 
 742 
 743 
 
 A'-' 
 
 r=35 
 
 log.F(r, v) 
 
 1-623 9542 
 •626 2048 
 •632 9608 
 ■644 2348 
 •660 0478 
 •680 4295 
 
 ■705 4180 
 ■735 0604 
 •769 4129 
 •808 5407 
 •852 5188 
 
 •901 4317 
 •955 3744 
 0-014 4525 
 •078 7824 
 •148 4925 
 
 •223 7229 
 •304 6270 
 •391 3713 
 •484 1369 
 •583 1198 
 
 •688 5323 
 ■800 6039 
 ■919 5823 
 1-045 7350 
 •179 3502 
 
 •320 7390 
 •470 2366 
 •628 2047 
 •795 0329 
 •971 1417 
 
 2-156 9847 
 ■353 0516 
 ■559 8711 
 •778 0150 
 
 3-008 1016 
 
 •250 8000 
 ■506 8356 
 •776 9950 
 4-062 1324 
 ■363 1764 
 
 •681 1377 
 
 5-017 1183 
 
 •372 3207 
 
 ■748 0596 
 
 6-145 7749 
 
 log U ('■.») 
 
 389 6937 
 389 6949 
 389 6987 
 389 7050 
 389 7138 
 389 7251 
 
 389 7389 
 389 7551 
 389 7738 
 389 7948 
 389 8183 
 
 389 8442 
 389 8724 
 389 9029 
 389 9356 
 
 389 97U0 
 
 390 0078 
 390 047 1 
 390 0884 
 390 1319 
 390 1773 
 
 390 2246 
 390 2738 
 390 3248 
 390 3776 
 390 4321 
 
 390 4881 
 390 5458 
 390 6049 
 390 6654 
 '390 7273 
 
 390 7904 
 390 8547 
 390 9201 
 
 390 9865 
 
 391 0539 
 
 391 1222 
 391 1912 
 391 2609 
 391 3312 
 391 4021 
 
 391 4734 
 391 5450 
 391 6169 
 391 6890 
 391 7612 
 
 13 
 38 
 63 
 
 88 
 113 
 
 138 
 162 
 187 
 211 
 235 
 
 259 
 
 282 
 305 
 328 
 350 
 
 372 
 393 
 414 
 434 
 
 454 
 
 473 
 492 
 510 
 528 
 545 
 
 561 
 576 
 591 
 605 
 619 
 
 631 
 643 
 654 
 664 
 674 
 
 682 
 690 
 697 
 703 
 709 
 
 713 
 
 716 
 719 
 721 
 722 
 
 A a 
 
 25 
 25 
 25 
 25 
 25 
 
 25 
 24 
 24 
 24 
 24 
 
 23 
 23 
 23 
 22 
 22 
 
 21 
 21 
 20 
 20 
 
 19 
 
 19 
 18 
 17 
 17 
 16 
 
 16 
 15 
 14 
 13 
 13 
 
 12 
 11 
 10 
 
 18 
 
138 
 
 Tables for Statisticians and Biometriciaus 
 TABLE LIV— (continued). 
 
 6 
 7 
 8 
 9 
 10 
 
 11 
 
 12 
 IS 
 
 U 
 15 
 
 16 
 
 17 
 18 
 19 
 
 20 
 
 28 
 
 24 
 
 25 
 
 26 
 
 29 
 30 
 
 SI 
 
 S2 
 S3 
 81* 
 85 
 
 S6 
 
 S7 
 S8 
 SO 
 
 ¥> 
 
 42 
 43 
 44 
 45 
 
 r =36 
 
 log F(r, v) 
 
 1-617 9231 
 •620 2399 
 •627 1943 
 •638 7995 
 ■655 0771 
 •676 0575 
 
 ■701 7800 
 •732 2932 
 •767 6546 
 •807 9316 
 •853 2010 
 
 •903 5502 
 •959 0766 
 0-019 8889 
 •086 1070 
 •157 8628 
 
 •235 3007 
 •318 5782 
 ■407 8669 
 •503 3529 
 •605 2380 
 
 •713 7407 
 •829 0968 
 •951 5615 
 1-081 4096 
 •218 9381 
 
 •364 4667 
 •518 3405 
 •680 9313 
 •852 6402 
 2033 8997 
 
 •225 1766 
 •426 9744 
 •639 8374 
 ■864 3536 
 3101 1591 
 
 •350 9424 
 •614 4497 
 •892 4902 
 4-185 9427 
 •495 7624 
 
 •822 9894 
 
 5-168 7569 
 
 •534 3023 
 
 •920 9782 
 
 6330 2655 
 
 log H{r, v) 
 
 389 9572 
 389 9584 
 389 9620 
 389 9682 
 389 9767 
 
 389 9877 
 
 390 0011 
 390 0168 
 390 0350 
 390 0555 
 390 0783 
 
 390 1035 
 390 1309 
 390 1605 
 390 1924 
 390 2264 
 
 390 2625 
 390 3007 
 390 3409 
 390 3832 
 390 4273 
 
 390 4733 
 
 390 5212 
 390 5708 
 390 6221 
 390 6750 
 
 390 7296 
 390 7856 
 390 8431 
 390 9019 
 
 390 9621 
 
 391 0234 
 391 0859 
 391 1495 
 391 2141 
 391 2796 
 
 391 3460 
 391 4131 
 391 4809 
 391 5492 
 391 6181 
 
 391 6874 
 391 7571 
 391 8270 
 391 8971 
 391 9673 
 
 12 
 37 
 61 
 
 85 
 110 
 
 134 
 158 
 182 
 205 
 228 
 
 251 
 274 
 296 
 318 
 340 
 
 361 
 
 382 
 402 
 422 
 441 
 
 460 
 478 
 496 
 513 
 529 
 
 545 
 560 
 575 
 588 
 601 
 
 614 
 625 
 636 
 646 
 655 
 
 664 
 671 
 678 
 684 
 689 
 
 693 
 697 
 699 
 701 
 702 
 
 if 
 
 r=37 
 
 log F(r, v) 
 
 1-612 0550 
 •614 4378 
 •621 5908 
 •633 5272 
 •650 2694 
 •671 8485 
 
 •698 3051 
 •729 6890 
 •766 0594 
 •807 4855 
 •854 0464 
 
 •905 8318 
 •962 9420 
 0025 4886 
 •093 5948 
 •167 3965 
 
 •247 0418 
 •332 6929 
 •424 5260 
 •522 7325 
 •627 5199 
 
 •7391128 
 •857 7536 
 •983 7045 
 1-117 2483 
 •258 6900 
 
 •408 3585 
 •566 6085 
 •733 8222 
 •910 4119 
 2-096 8223 
 
 •293 5330 
 ■501 0620 
 •719 9685 
 •950 8571 
 3-194 3816 
 
 •451 2499 
 •722 2290 
 4008 1507 
 •309 9183 
 ■628 5140 
 
 •965 0066 
 5320 5613 
 
 •696 4499 
 6094 0627 
 
 •514 9222 
 
 log II (r, v) 
 
 •0390 2063 
 ■390 2074 
 •390 2110 
 •390 2170 
 •390 2253 
 •390 2360 
 
 ■390 2490 
 •390 2643 
 •390 2820 
 •390 3020 
 •390 3242 
 
 •390 3486 
 •390 3753 
 •390 4041 
 •390 4351 
 •390 4682 
 
 •390 5034 
 •390 5405 
 •390 5797 
 •390 6208 
 •390 6637 
 
 •390 7085 
 •390 7551 
 ■390 8033 
 •390 8532 
 •390 9048 
 
 •390 9578 
 •391 0123 
 •391 0682 
 ■391 1255 
 ■391 1840 
 
 •391 2437 
 •391 3046 
 ■391 3664 
 •391 4293 
 •391 4930 
 
 •391 5576 
 ■391 6229 
 •391 6888 
 •391 7553 
 •391 8224 
 
 •391 8898 
 •391 9576 
 •392 0256 
 •392 0938 
 ■392 1621 
 
 12 
 36 
 60 
 83 
 107 
 
 130 
 154 
 177 
 200 
 222 
 
 245 
 267 
 288 
 310 
 331 
 
 352 
 372 
 391 
 411 
 430 
 
 448 
 466 
 483 
 499 
 515 
 
 530 
 545 
 559 
 573 
 585 
 
 597 
 608 
 619 
 628 
 637 
 
 646 
 653 
 659 
 665 
 670 
 
 674 
 678 
 680 
 682 
 683 
 
 &« 
 
 r = 38 
 
 log F(r,v) 
 
 1-606 3413 
 •608 7902 
 •616 1417 
 •628 4093 
 •645 6161 
 •667 7940 
 
 •694 9847 
 •727 2393 
 •764 6187 
 •807 1940 
 •855 0464 
 
 •908 2680 
 ■966 9620 
 0-031 2429 
 •101 2374 
 •177 0849 
 
 •258 9378 
 •346 9624 
 •441 3400 
 ■542 2671 
 ■649 9569 
 
 •764 6400 
 •886 5655 
 1-016 0028 
 •153 2423 
 •298 5974 
 
 •452 4059 
 •615 0321 
 •786 8688 
 •968 3394 
 2-159 9006 
 
 •362 0454 
 •575 3055 
 •800 2556 
 3037 5167 
 ■287 7604 
 
 •551 7138 
 •830 1647 
 4-123 9677 
 •434 0507 
 •761 4223 
 
 5-107 1807 
 •472 5225 
 •858 7544 
 
 6 267 3043 
 •699 7361 
 
 log H{r,v)\ A A* 
 
 I 
 
 0.390 4421 
 •390 4433 
 •390 4468 
 •390 4526 
 •390 4607 
 •390 4711 
 
 •390 4837 
 •390 4987 
 •390 5159 
 •390 5353 
 •390 5569 
 
 •390 5808 
 •390 6067 
 •390 6348 
 •390 6650 
 •390 6972 
 
 •390 7314 
 
 •390 7676 
 •390 8057 
 •390 8457 
 •390 8876 
 
 •390 9312 
 •390 9765 
 •391 0235 
 •391 0721 
 •391 1223 
 
 •391 1739 
 •391 2270 
 •391 2815 
 •391 3372 
 •391 3942 
 
 •391 4523 
 •391 5115 
 ■391 5718 
 •391 6330 
 •391 6950 
 
 •391 7579 
 ■391 8215 
 •391 8857 
 •391 9505 
 •392 0157 
 
 •392 0814 
 ■392 1474 
 •392 2136 
 •392 2800 
 •392 3465 
 
 12 
 35 
 58 
 81 
 104 
 
 127 
 149 
 172 
 194 
 216 
 
 238 
 260 
 281 
 302 
 322 
 
 342 
 362 
 381 
 400 
 418 
 
 436 
 453 
 
 470 
 486 
 502 
 
 517 
 531 
 545 
 557 
 570 
 
 581 
 592 
 603 
 612 
 621 
 
 629 
 636 
 642 
 648 
 653 
 
 657 
 660 
 662 
 664 
 665 
 
 23 
 23 
 23 
 23 
 23 
 
 23 
 23 
 
 22 
 22 
 22 
 
 22 
 21 
 21 
 20 
 20 
 
 20 
 19 
 19 
 18 
 18 
 
 17 
 17 
 16 
 16 
 15 
 
 14 
 14 
 13 
 12 
 12 
 
 11 
 
 10 
 
 9 
 
 9 
 
 8 
 
 7 
 6 
 6 
 5 
 
 4 
 
Tables of the G (r, v)- Integrals 
 TABLE LI V— (continued). 
 
 139 
 
 s 
 
 9 
 10 
 
 11 
 12 
 13 
 
 U 
 15 
 
 16 
 17 
 18 
 19 
 
 20 
 
 21 
 
 ** 
 
 25 
 
 26 
 27 
 28 
 29 
 SO 
 
 SI 
 
 32 
 S3 
 34 
 35 
 
 36 
 37 
 38 
 39 
 40 
 
 41 
 
 42 
 4S 
 44 
 45 
 
 r=39 
 
 logF(r, v) 
 
 1-600 7740 
 603 2891 
 ■610 8391 
 ■623 4380 
 •641 1093 
 •663 8860 
 
 •691 8107 
 ■724 9361 
 •763 3246 
 •807 0490 
 •856 1930 
 
 •910 8510 
 •971 1287 
 0-037 1440 
 •109 0268 
 •186 9202 
 
 •270 9806 
 •361 3789 
 ■458 3010 
 •561 9488 
 •672 5410 
 
 •790 3143 
 •915 5247 
 1-048 4484 
 •189 3837 
 •338 6522 
 
 •496 6007 
 •663 6033 
 •840 0630 
 2 026 4145 
 •223 1268 
 
 •430 7056 
 •649 6970 
 •880 6908 
 3-124 3244 
 •381 2874 
 
 •652 3259 
 •938 2488 
 4-239 9332 
 •558 3315 
 •894 4793 
 
 5-249 5035 
 •624 6328 
 
 6-021 2079 
 •440 6950 
 •884 6991 
 
 log H (r,y) 
 
 03a0 6658 
 •390 6669 
 •390 6703 
 •390 6760 
 •390 6838 
 ■390 6940 
 
 •390 7063 
 •390 7209 
 •390 7377 
 •390 7566 
 •390 7777 
 
 •390 8009 
 •390 8262 
 ■390 8535 
 ■390 8829 
 •390 9143 
 
 •390 9477 
 •390 9829 
 ■391 0201 
 •391 0591 
 •391 0998 
 
 •391 1423 
 •391 1865 
 •391 2323 
 •391 2796 
 •391 3285 
 
 •391 3788 
 •391 4306 
 •391 4836 
 •391 5379 
 •391 5935 
 
 •391 6501 
 •391 7078 
 •391 7665 
 •391 8261 
 •391 8866 
 
 •391 9479 
 •392 0098 
 •392 0724 
 •392 1355 
 •392 1991 
 
 •392 2631 
 •392 3274 
 •392 3919 
 •392 4566 
 •392 5213 
 
 A A2 
 
 11 
 34 
 
 56 
 
 79 
 
 101 
 
 124 
 146 
 168 
 189 
 211 
 
 232 
 253 
 274 
 294 
 314 
 
 334 
 353 
 371 
 390 
 
 408 
 
 425 
 442 
 458 
 474 
 489 
 
 503 
 517 
 530 
 543 
 555 
 
 566 
 
 577 
 587 
 596 
 605 
 
 612 
 619 
 626 
 631 
 636 
 
 640 
 643 
 645 
 647 
 648 
 
 r=40 
 
 logF (r, *) 
 
 1-595 3459 
 ■597 9271 
 •605 6756 
 •618 6058 
 •636 7416 
 ■660 1171 
 
 •688 7760 
 •722 7721 
 ■762 1697 
 •807 0434 
 •857 4789 
 
 •913 5732 
 •975 4348 
 0043 1845 
 •116 9556 
 •196 8949 
 
 •283 1630 
 •375 9350 
 •475 4017 
 •581 7701 
 •695 2648 
 
 •816 1285 
 •944 6237 
 1081 0339 
 •225 6650 
 •378 8470 
 
 •540 9357 
 •712 3146 
 ■893 3974 
 2-084 6300 
 •286 4933 
 
 •499 5062 
 •724 2290 
 ■961 2666 
 321 1 2729 
 •474 9551 
 
 •753 0789 
 
 4-046 4738 
 
 •356 0397 
 
 •682 7535 
 
 5-027 6774 
 
 •391 9676 
 •776 8842 
 
 6-183 8028 
 •614 2272 
 
 7-069 8037 
 
 log H (r, v) 
 
 0-390 8782 
 •390 8793 
 •390 8826 
 •390 8881 
 •390 8958 
 •390 9057 
 
 •390 9177 
 •390 9319 
 •390 9483 
 •390 9667 
 ■390 9873 
 
 ■391 0099 
 •391 0346 
 •391 0612 
 •391 0899 
 •391 1205 
 
 •391 1530 
 •391 1874 
 •391 2236 
 •391 2616 
 •391 3014 
 
 •391 3428 
 •391 3859 
 •391 4305 
 •391 4767 
 •391 5243 
 
 •391 5734 
 •391 6238 
 •391 6756 
 •391 7285 
 •391 7827 
 
 •391 8379 
 ■391 8942 
 ■391 9514 
 •392 0095 
 ■392 0685 
 
 •392 1282 
 •392 1886 
 •392 2496 
 •392 3112 
 •392 3732 
 
 •392 4355 
 •392 4982 
 •392 5612 
 •392 6242 
 •392 6874 
 
 11 
 33 
 
 55 
 77 
 99 
 
 120 
 142 
 163 
 185 
 206 
 
 226 
 247 
 267 
 287 
 306 
 
 325 
 
 344 
 362 
 380 
 397 
 
 414 
 431 
 
 447 
 462 
 
 477 
 
 491 
 504 
 517 
 530 
 541 
 
 552 
 563 
 572 
 581 
 590 
 
 597 
 604 
 610 
 615 
 620 
 
 624 
 627 
 629 
 630 
 631 
 
 A- 
 
 r = 41 
 
 logf (r, v) 
 
 1-590 0501 
 •592 6975 
 •600 6445 
 •613 9059 
 •632 5064 
 •656 4807 
 
 •685 8737 
 •720 7406 
 •761 1472 
 ■807 1702 
 ■858 8973 
 
 •916 4280 
 ■979 8734 
 0-049 3576 
 •125 0171 
 •207 0023 
 
 •295 4780 
 •390 6238 
 •492 6351 
 •601 7243 
 •718 1215 
 
 •842 0755 
 •973 8556 
 1-113 7524 
 •262 0794 
 •419 1750 
 
 •585 4038 
 •761 1593 
 •946 8652 
 2-142 9789 
 •349 9933 
 
 •568 4405 
 •798 8947 
 3041 9761 
 •298 3551 
 •568 7567 
 
 •853 9658 
 
 4'154 8329 
 
 •472 2803 
 
 ■807 3096 
 
 5-161 0098 
 
 •534 5660 
 •929 2701 
 
 6346 5322 
 •787 8938 
 
 7 255 0429 
 
 log H (r, v) 
 
 A A 2 
 
 0-391 0801 
 •391 0812 
 ■391 0844 
 •391 0898 
 •391 0973 
 •391 1069 
 
 •391 1187 
 •391 1325 
 •391 1485 
 •391 1665 
 •391 1865 
 
 •391 2086 
 •391 2327 
 •391 2587 
 •391 2867 
 •391 3165 
 
 •391 3483 
 •391 3818 
 •391 4172 
 •391 4542 
 •391 4930 
 
 •391 5334 
 •391 5754 
 •391 6190 
 •391 6640 
 •391 7105 
 
 •391 7584 
 •391 8076 
 •391 8581 
 •391 9097 
 •391 9626 
 
 •392 0164 
 •392 0713 
 •392 1272 
 •392 1839 
 •392 2414 
 
 •392 2997 
 •392 3586 
 •392 4181 
 •392 4782 
 •392 5386 
 
 •392 5995 
 •392 6607 
 •392 7221 
 •392 7836 
 •392 8452 
 
 18—2 
 
 11 
 32 
 54 
 75 
 96 
 
 118 
 139 
 159 
 180 
 200 
 
 221 
 241 
 260 
 280 
 299 
 
 317 
 335 
 353 
 371 
 
 388 
 
 404 
 420 
 436 
 451 
 465 
 
 479 
 492 
 505 
 517 
 528 
 
 539 
 549 
 558 
 567 
 575 
 
 583 
 589 
 595 
 600 
 605 
 
 609 
 612 
 614 
 615 
 616 
 
 21 
 21 
 21 
 21 
 21 
 
 21 
 21 
 21 
 20 
 20 
 
 20 
 20 
 19 
 19 
 19 
 
 18 
 18 
 17 
 17 
 16 
 
 16 
 16 
 15 
 14 
 14 
 
 13 
 13 
 
 12 
 11 
 11 
 
 10 
 9 
 9 
 
140 
 
 Tables for Statisticians and Biometricians 
 TABLE LIV— (continued). 
 
 6 
 7 
 S 
 9 
 10 
 
 11 
 12 
 IS 
 
 H 
 15 
 
 16 
 
 n 
 
 18 
 19 
 80 
 
 21 
 22 
 23 
 
 26 
 27 
 28 
 29 
 SO 
 
 SI 
 
 S2 
 S3 
 SI, 
 35 
 
 86 
 37 
 
 ¥> 
 V- 
 
 4* 
 
 43 
 
 U 
 45 
 
 ■/■=42 
 
 log F (r, v) 
 
 1-584 8804 
 •587 5939 
 •595 7394 
 •609 3321 
 •628 3972 
 •652 9704 
 
 •683 0975 
 •718 8352 
 •760 2510 
 •807 4232 
 •860 4419 
 
 •919 4089 
 •984 4383 
 0-055 6569 
 •133 2048 
 •217 2361 
 
 •307 9195 
 •405 4390 
 •509 9950 
 •621 8050 
 •741 1047 
 
 •868 1492 
 1-003 2143 
 •146 5976 
 •298 6206 
 •459 6298 
 
 •629 9989 
 •810 1308 
 2-000 4600 
 •201 4548 
 •413 6204 
 
 •637 5019 
 •873 6876 
 3122 8129 
 •385 5647 
 •662 6857 
 
 •954 9802 
 
 4263 3195 
 
 •588 6485 
 
 •931 9935 
 
 5-294 4700 
 
 •677 2923 
 
 6-081 7839 
 
 •509 3896 
 
 •961 6887 
 
 7 440 4103 
 
 logflfr, v) A 
 
 0-391 2724 
 •391 2734 
 •391 2766 
 •391 2818 
 •391 2891 
 •391 2985 
 
 •391 3100 
 •391 3235 
 ■391 3391 
 •391 3567 
 ■391 3763 
 
 •391 3978 
 ■391 4213 
 •391 4467 
 •391 4740 
 •391 5032 
 
 •391 5341 
 •391 5669 
 •391 6014 
 •391 6376 
 •391 6754 
 
 •391 7149 
 ■391 7559 
 •391 7984 
 •391 8424 
 •391 8878 
 
 ■391 9345 
 •391 9826 
 •392 0318 
 •392 0823 
 •392 1338 
 
 •392 1864 
 •392 2400 
 •392 2945 
 •392 3499 
 •392 4060 
 
 •392 4629 
 ■392 5204 
 ■392 5785 
 •392 6371 
 ■392 6962 
 
 •392 7556 
 •392 8153 
 •392 8752 
 •392 9353 
 •392 9954 
 
 10 
 31 
 52 
 73 
 
 94 
 
 115 
 135 
 156 
 176 
 196 
 
 216 
 235 
 254 
 273 
 
 292 
 
 310 
 328 
 345 
 362 
 
 378 
 
 395 
 410 
 425 
 440 
 454 
 
 467 
 480 
 493 
 504 
 516 
 
 526 
 536 
 545 
 554 
 562 
 
 569 
 575 
 581 
 586 
 590 
 
 594 
 597 
 599 
 601 
 601 
 
 r = 43 
 
 \F (>■.") 
 
 1-579 8310 
 •582 6106 
 •590 9546 
 •604 8785 
 •624 4083 
 •649 5803 
 
 •680 4416 
 •717 0501 
 •759 4750 
 •807 7965 
 ■862 1068 
 
 •922 5103 
 ■989 1236 
 0062 0767 
 •141 5130 
 •227 5903 
 
 •320 4814 
 ■420 3748 
 •527 4755 
 ■642 0063 
 •764 2086 
 
 •894 3438 
 1032 6937 
 •179 5637 
 •335 2826 
 •500 2054 
 
 •674 7149 
 •859 2234 
 2054 1759 
 •260 0519 
 ■477 3687 
 
 •706 6845 
 •948 6018 
 3203 7711 
 ■472 8957 
 •756 7363 
 
 4-056 1162 
 •371 9277 
 •705 1384 
 
 5056 7991 
 •428 0520 
 
 •820 1404 
 6-234 4197 
 
 •672 3690 
 7 135 6055 
 
 •025 8999 
 
 log H (r,v) 
 
 391 4556 
 391 4566 
 391 4597 
 391 4648 
 391 4720 
 391 4812 
 
 391 4924 
 391 5056 
 391 5208 
 391 5380 
 391 5571 
 
 391 5781 
 391 6011 
 391 6259 
 391 6526 
 391 6810 
 
 391 7113 
 391 7433 
 
 391 7770 
 391 8123 
 391 8493 
 
 391 8878 
 391 9279 
 
 391 9694 
 
 392 0124 
 392 0567 
 
 392 1024 
 392 1493 
 392 1974 
 392 2467 
 392 2970 
 
 392 3484 
 392 4007 
 392 4540 
 392 5081 
 392 5629 
 
 392 6185 
 392 6747 
 392 7314 
 392 7886 
 392 8463 
 
 392 9044 
 
 392 9627 
 
 393 0212 
 393 0799 
 393 1386 
 
 10 
 31 
 51 
 72 
 92 
 
 112 
 
 132 
 152 
 172 
 191 
 
 210 
 230 
 248 
 267 
 285 
 
 303 
 320 
 337 
 353 
 370 
 
 385 
 401 
 415 
 430 
 443 
 
 457 
 469 
 481 
 493 
 504 
 
 514 
 523 
 
 533 
 541 
 
 548 
 
 556 
 562 
 567 
 572 
 577 
 
 580 
 583 
 585 
 587 
 587 
 
 if 
 
 r = 44 
 
 log F (r, v) 
 
 1-574 8962 
 •577 7420 
 ■586 2845 
 •600 5397 
 •620 5341 
 •646 3049 
 
 •677 9004 
 •715 3798 
 •758 8138 
 •808 2846 
 •863 8866 
 
 •925 7265 
 •993 9238 
 0068 6113 
 •149 9361 
 •238 0595 
 
 •333 1584 
 •435 4256 
 •545 0710 
 •662 3227 
 •787 4277 
 
 •920 6532 
 I 062 2883 
 •212 6450 
 •372 0600 
 •540 8966 
 
 •719 5463 
 •908 4315 
 2-108 0073 
 •318 7645 
 ■541 2327 
 
 ■775 982!) 
 3023 6317 
 ■284 8450 
 ■560 3426 
 •850 9027 
 
 4-157 3682 
 •480 6520 
 •821 7444 
 
 5-181 7208 
 •561 7502 
 
 ■963 1049 
 6-387 1719 
 
 •835 4649 
 7-309 6389 
 
 •811 5060 
 
 logH>, k) 
 
 391 6305 
 391 6315 
 391 6345 
 391 6395 
 391 6465 
 391 6554 
 
 391 6664 
 391 6793 
 391 6912 
 391 7109 
 
 391 7296 
 
 391 7502 
 391 7726 
 391 7769 
 391 8229 
 391 8508 
 
 391 8803 
 391 9116 
 391 9445 
 
 391 9791 
 
 392 0152 
 
 392 0528 
 392 0920 
 392 1326 
 392 1746 
 392 2179 
 
 392 2625 
 392 3084 
 392 3554 
 392 4035 
 392 4528 
 
 392 5030 
 392 5541 
 392 6062 
 392 6590 
 392 7126 
 
 392 7669 
 392 8218 
 392 8773 
 392 9332 
 
 392 9896 
 
 393 0463 
 393 1033 
 393 1006 
 
 393 2178 
 
 393 2752 
 
 10 
 30 
 50 
 
 70 
 90 
 
 109 
 129 
 149 
 168 
 187 
 
 206 
 
 224 
 243 
 261 
 278 
 
 296 
 313 
 329 
 346 
 361 
 
 377 
 
 391 
 406 
 420 
 433 
 
 446 
 459 
 470 
 482 
 492 
 
 502 
 512 
 
 520 
 528 
 536 
 
 543 
 549 
 
 555 
 559 
 564 
 
 567 
 
 670 
 572 
 573 
 
 574 
 
 A 2 
 
 20 
 20 
 20 
 20 
 20 
 
 20 
 20 
 19 
 19 
 19 
 
 19 
 18 
 18 
 18 
 17 
 
 17 
 17 
 1G 
 16 
 15 
 
 15 
 14 
 14 
 13 
 13 
 
 12 
 12 
 11 
 11 
 10 
 
 9 
 9 
 
 8 
 8 
 7 
 
 6 
 6 
 5 
 4 
 4 
 
Tables of the G (r, v)- Integrals 
 
 TABLE LIV— {continued). 
 
 141 
 
 <t>° 
 
 8 
 
 y 
 
 10 
 
 a 
 12 
 is 
 U 
 
 15 
 
 10 
 17 
 18 
 19 
 20 
 
 m 
 
 22 
 28 
 
 u 
 
 35 
 
 80 
 
 31 
 
 82 
 
 85 
 
 36 
 
 37 
 
 lfi 
 
 41 
 
 4S 
 
 U 
 
 r = 45 
 
 log F (r, v) 
 
 1-570 0711 
 •572 9830 
 •581 7241 
 •596 3106 
 ■616 7696 
 ■643 1393 
 
 •675 4689 
 •713 8192 
 •758 2623 
 •808 8824 
 •865 7761 
 
 ■929 0524 
 ■998 8337 
 0-075 2558 
 •158 4691 
 •248 6386 
 
 •345 9453 
 •450 5863 
 •562 7765 
 •682 7492 
 •810 7568 
 
 •947 0729 
 1091 9931 
 •245 8365 
 ■408 9476 
 •581 6979 
 
 •764 4880 
 •957 7499 
 2161 9490 
 •377 5876 
 •605 2071 
 
 •845 3918 
 3098 7723 
 •366 0297 
 •647 9002 
 •945 1800 
 
 4258 7310 
 •589 4872 
 •938 4614 
 
 5-306 7534 
 •695 5595 
 
 6-106 1805 
 •540 0352 
 ■998 6720 
 ■483 7836 
 •997 2234 
 
 Iok if (r, i.) 
 
 0-391 7975 
 •391 7984 
 •391 8014 
 •391 8063 
 •391 8131 
 •391 8219 
 
 •391 8326 
 •391 8452 
 •391 8598 
 •391 8762 
 •391 8944 
 
 •391 9145 
 •391 9365 
 •391 9602 
 •391 9857 
 •392 0129 
 
 •392 0418 
 •392 0724 
 •392 1046 
 •392 1383 
 •392 1737 
 
 •392 2105 
 •392 2488 
 •392 2885 
 •392 3295 
 ■392 3719 
 
 •392 4155 
 •392 4603 
 •392 5063 
 •392 5534 
 •392 6015 
 
 •392 6506 
 •392 7006 
 •392 7515 
 •392 8032 
 •392 8556 
 
 •392 9087 
 •392 9624 
 •393 0166 
 •393 0713 
 •393 1264 
 
 •393 1818 
 •393 2376 
 •393 2935 
 •393 3496 
 •393 4057 
 
 10 
 29 
 49 
 68 
 
 88 
 
 107 
 126 
 145 
 164 
 183 
 
 201 
 219 
 237 
 255 
 272 
 
 289 
 306 
 322 
 338 
 353 
 
 368 
 383 
 397 
 411 
 424 
 
 436 
 448 
 460 
 471 
 481 
 
 491 
 500 
 509 
 517 
 524 
 
 531 
 537 
 542 
 547 
 551 
 
 555 
 
 557 
 559 
 
 Ml 
 
 561 
 
 r = 46 
 
 \oa,F(r, v) 
 
 1-565 3509 
 •568 3289 
 •577 2688 
 •592 1863 
 •613 1100 
 •640 0785 
 
 •673 1423 
 •712 3634 
 •757 8157 
 •809 5852 
 •867 7706 
 
 •932 4834 
 0-003 8487 
 •082 0054 
 •167 1071 
 •259 3228 
 
 •358 8372 
 . "465 8522 
 •580 5873 
 •703 2808 
 •834 1911 
 
 •973 5979 
 1-121 8031 
 •279 1333 
 •445 9406 
 •622 6047 
 
 •809 5352 
 2-007 1738 
 •215 9964 
 •436 5163 
 •669 2873 
 
 •914 9064 
 
 3174 0186 
 
 •447 3201 
 
 •735 5636 
 
 4-039 5631 
 
 •360 1998 
 •698 4283 
 5-055 2844 
 •431 8925 
 ■829 4749 
 
 6-249 3623 
 •693 0048 
 
 7-161 9854 
 •658 0347 
 
 8-183 0474 
 
 Ior II (r,,) 
 
 0-391 9572 
 •391 9581 
 •391 9610 
 •391 9658 
 •391 9725 
 •391 9811 
 
 ■391 9915 
 •392 0039 
 •392 0181 
 •392 0342 
 •392 0520 
 
 •392 0717 
 •392 0932 
 ■392 1164 
 •392 1413 
 •392 1679 
 
 •392 1962 
 •392 2261 
 •392 2576 
 •392 2906 
 •392 3252 
 
 •392 3612 
 •392 3987 
 •392 4375 
 ■392 4776 
 •392 5191 
 
 •392 5618 
 •392 6056 
 •392 6506 
 •392 6967 
 •392 7437 
 
 •392 7918 
 •392 8407 
 •392 8905 
 •392 9410 
 •392 9923 
 
 •393 0442 
 •393 0967 
 •393 1498 
 •393 2033 
 •393 2572 
 
 •393 3115 
 •393 3660 
 •393 4207 
 •393 4755 
 •393 5304 
 
 10 
 29 
 48 
 67 
 86 
 
 105 
 124 
 142 
 161 
 179 
 
 197 
 215 
 232 
 249 
 266 
 
 283 
 299 
 315 
 331 
 346 
 
 360 
 375 
 
 388 
 402 
 414 
 
 427 
 438 
 450 
 461 
 471 
 
 480 
 489 
 498 
 505 
 513 
 
 519 
 525 
 530 
 535 
 539 
 
 543 
 
 545 
 547 
 548 
 549 
 
 a? 
 
 logP(r, i.) 
 
 1-560 7311 
 •563 7753 
 •572 9134 
 •588 1625 
 •609 5508 
 •637 1182 
 
 ■670 9162 
 •711 0082 
 •757 4696 
 ■810 3885 
 •869 8656 
 
 ■936 0149 
 0-008 9642 
 •088 8555 
 •175 8458 
 •270 1076 
 
 ■371 8299 
 •481 2188 
 •598 4987 
 •723 9132 
 •857 7263 
 
 1-000 2236 
 •151 7141 
 •312 5311 
 •483 0345 
 •663 6124 
 
 •854 6834 
 2-056 6988 
 •270 1448 
 •495 5462 
 •733 4685 
 
 •984 5222 
 
 3-249 3662 
 
 •528 7119 
 
 •823 3284 
 
 4-134 0476 
 
 •461 7700 
 •807 4710 
 5-172 2090 
 •557 1330 
 •963 4920 
 
 6-392 6458 
 •846 0761 
 
 7-325 4006 
 •832 3876 
 
 8-368 9732 
 
 log if (r, ») 
 
 0-392 1100 
 •392 1110 
 •392 1138 
 •392 1185 
 •392 1250 
 •392 1334 
 
 •392 1437 
 •392 1557 
 •392 1697 
 ■392 1854 
 •392 2029 
 
 •392 2221 
 •392 2431 
 •392 2658 
 •392 2902 
 ■392 3163 
 
 •392 3440 
 •392 3732 
 •392 4041 
 •392 4364 
 •392 4702 
 
 •392 5055 
 •392 5421 
 •392 5801 
 •392 6194 
 •392 6600 
 
 •392 7018 
 •392 7447 
 •392 7887 
 ■392 8338 
 •392 8799 
 
 •392 9269 
 ■392 9748 
 •393 0235 
 •393 0729 
 •393 1231 
 
 •393 1740 
 •393 2254 
 •393 2773 
 •393 3297 
 ■393 3824 
 
 •393 4355 
 •393 4889 
 •393 5424 
 •393 5961 
 •393 6498 
 
 A A* 
 
 9 
 28 
 47 
 66 
 
 84 
 
 103 
 121 
 139 
 157 
 175 
 
 193 
 210 
 227 
 244 
 261 
 
 277 
 293 
 308 
 323 
 338 
 
 353 
 367 
 
 380 
 393 
 406 
 
 418 
 429 
 440 
 451 
 461 
 
 470 
 479 
 487 
 495 
 502 
 
 508 
 514 
 
 519 
 524 
 
 528 
 
 531 
 534 
 536 
 537 
 537 
 
 19 
 19 
 19 
 19 
 
 18 
 
 18 
 18 
 18 
 18 
 18 
 
 17 
 17 
 17 
 17 
 16 
 
 16 
 16 
 15 
 15 
 14 
 
 14 
 13 
 13 
 
 13 
 12 
 
 12 
 11 
 11 
 10 
 9 
 
 9 
 
 8 
 8 
 7 
 6 
 
 6 
 5 
 5 
 4 
 3 
 
 3 
 
 2 
 1 
 
 
142 
 
 Tables for Statisticians and Biometricians 
 TABLE LIV— (continued). 
 
 9 
 
 10 
 
 11 
 
 12 
 13 
 
 n 
 
 15 
 16 
 
 n 
 
 18 
 19 
 
 20 
 
 21 
 
 27 
 28 
 29 
 SO 
 
 SI 
 
 Sit 
 35 
 
 37 
 38 
 S9 
 
 40 
 41 
 
 42 
 48 
 
 44 
 45 
 
 r=48 
 
 log F (r, ») 
 
 1-556 2075 
 •559 3178 
 •568 6545 
 •584 2348 
 •606 0878 
 •634 2541 
 
 •668 7863 
 •709 7492 
 •757 2198 
 •811 2881 
 •872 0569 
 
 •939 6427 
 0-014 1760 
 •095 8020 
 •184 6808 
 •280 9888 
 
 •384 9189 
 •496 6818 
 •616 5066 
 •744 6421 
 •881 3579 
 
 1026 9460 
 •181 7216 
 •346 0254 
 •520 2250 
 •704 7169 
 
 •899 9284 
 2106 3205 
 •324 3901 
 ■554 6729 
 ■797 7467 
 
 3-054 2350 
 •324 8107 
 •610 2007 
 •911 1903 
 
 4-228 6293 
 
 •563 4374 
 •916 6109 
 
 5289 2309 
 •682 4708 
 
 6-097 6055 
 
 •536 0267 
 
 •999 2449 
 
 7-488 9134 
 
 8-006 8381 
 
 •554 9966 
 
 log // (r, v) 
 
 392 2565 
 392 2574 
 392 2601 
 392 2647 
 392 2711 
 392 2794 
 
 392 2894 
 392 3012 
 392 3149 
 392 3302 
 392 3474 
 
 392 3662 
 392 3868 
 392 4090 
 392 4329 
 392 4584 
 
 392 4855 
 392 5142 
 392 5444 
 392 5760 
 392 6092 
 
 392 6437 
 392 6796 
 392 7168 
 392 7553 
 392 7950 
 
 392 8359 
 392 8779 
 392 9210 
 
 392 9652 
 
 393 0103 
 
 393 0563 
 393 1032 
 393 1509 
 393 1993 
 393 2485 
 
 393 2982 
 393 3486 
 393 3994 
 393 4507 
 393 5024 
 
 393 5543 
 393 6066 
 393 6590 
 393 7116 
 393 7642 
 
 A2 
 
 9 
 28 
 46 
 64 
 82 
 
 100 
 118 
 136 
 154 
 171 
 
 189 
 206 
 222 
 239 
 255 
 
 271 
 287 
 302 
 317 
 331 
 
 345 
 359 
 
 372 
 385 
 397 
 
 409 
 420 
 431 
 441 
 451 
 
 460 
 469 
 
 477 
 485 
 491 
 
 498 
 503 
 508 
 513 
 517 
 
 520 
 522 
 524 
 526 
 526 
 
 r=49 
 
 log F (r,v) 
 
 1-551 7763 
 •554 9527 
 •564 4879 
 •580 3995 
 •602 7172 
 •631 4824 
 
 ■666 7488 
 •708 5826 
 •757 0624 
 ■812 2801 
 •874 3406 
 
 •943 3629 
 0-019 4803 
 •102 8409 
 •193 6083 
 ■291 9625 
 
 •398 1005 
 •512 2374 
 •634 6070 
 •765 4636 
 •905 0822 
 
 1-053 7610 
 •211 8218 
 •379 6125 
 •557 5084 
 •745 9141 
 
 •945 2662 
 2-156 0351 
 •378 7283 
 •613 8925 
 •862 1178 
 
 3-124 0408 
 •400 3485 
 •691 7826 
 •999 1454 
 
 4-323 3042 
 
 •665 1980 
 
 5-025 8441 
 
 •406 3461 
 
 •807 9020 
 
 6-231 8144 
 
 •679 5011 
 7-152 5073 
 
 •652 5197 
 8-181 3822 
 
 •741 1137 
 
 log H (r, v) A A2 i og p (r _ „) 
 
 0-392 3969 
 •392 3978 
 •392 4005 
 •392 4050 
 •392 4113 
 •392 4193 
 
 •392 4291 
 •392 4407 
 •392 4541 
 •392 4692 
 •392 4859 
 
 •392 5044 
 •392 5245 
 •392 5463 
 •392 5697 
 •392 5947 
 
 •392 6213 
 •392 6493 
 •392 6789 
 •392 7099 
 ■392 7424 
 
 ■392 7762 
 •392 8114 
 ■392 8478 
 •392 8855 
 •392 9244 
 
 •392 9645 
 •393 0057 
 •393 0479 
 •393 0911 
 •393 1353 
 
 •393 1804 
 ■393 2263 
 •393 2731 
 •393 3205 
 •393 3686 
 
 •393 4174 
 •393 4667 
 •393 5165 
 •393 5667 
 •393 6174 
 
 •393 6683 
 •393 7195 
 •393 7708 
 •393 8223 
 •393 8739 
 
 9 
 27 
 45 
 63 
 
 81 
 
 98 
 116 
 133 
 151 
 168 
 
 185 
 201 
 218 
 234 
 250 
 
 266 
 281 
 296 
 310 
 324 
 
 338 
 
 352 
 365 
 377 
 389 
 
 401 
 412 
 422 
 432 
 442 
 
 451 
 
 459 
 467 
 474 
 481 
 
 488 
 493 
 498 
 502 
 506 
 
 509 
 512 
 514 
 515 
 516 
 
 = 50 
 
 1-547 4336 
 •550 6762 
 •560 4099 
 •576 6529 
 •599 4352 
 •628 7993 
 
 •664 7999 
 •707 5046 
 •756 9936 
 •813 3607 
 •876 7129 
 
 •947 1718 
 0-024 8733 
 •109 9685 
 •202 6245 
 •303 0249 
 
 •411 3709 
 •527 8818 
 •652 7964 
 •786 3739 
 •928 8954 
 
 1-080 6649 
 •242 0110 
 •413 2886 
 •594 8807 
 •787 2004 
 
 •990 6930 
 2-205 8389 
 •433 1556 
 •673 2014 
 •926 5783 
 
 3193 9359 
 •475 9753 
 •773 4538 
 
 4087 1898 
 •418 0685 
 
 •767 0481 
 
 5'135 1668 
 
 •523 5508 
 
 •933 4228 
 
 6-366 1119 
 
 •823 0651 
 7305 8593 
 
 •816 2157 
 8-356 0160 
 
 •927 3206 
 
 •H(r,v) 
 
 0392 5316 
 •392 5325 
 •392 5352 
 •392 5396 
 •932 5457 
 •392 5536 
 
 •392 5633 
 •392 5746 
 •392 5877 
 ■392 6025 
 ■392 6189 
 
 •392 6370 
 •392 6568 
 •392 6781 
 •392 7010 
 •392 7255 
 
 •392 7516 
 •392 7791 
 •392 8080 
 •392 8384 
 •392 8702 
 
 •392 9034 
 •392 9378 
 ■392 9736 
 •393 0105 
 •393 0486 
 
 •393 0879 
 •393 1282 
 •393 1696 
 •393 2120 
 •393 2553 
 
 •933 2995 
 •393 3445 
 •393 3903 
 •393 4368 
 •393 4840 
 
 •393 5318 
 •393 5801 
 •393 6289 
 •393 6781 
 •393 7277 
 
 •393 7776 
 •393 8278 
 •393 8781 
 •393 9286 
 •393 9791 
 
 9 
 26 
 
 44 
 62 
 79 
 
 96 
 114 
 131 
 148 
 164 
 
 181 
 197 
 213 
 229 
 245 
 
 260 
 275 
 290 
 304 
 318 
 
 331 
 
 345 
 357 
 369 
 381 
 
 393 
 
 404 
 414 
 424 
 433 
 
 442 
 450 
 458 
 465 
 472 
 
 478 
 483 
 488 
 492 
 496 
 
 499 
 502 
 504 
 505 
 505 
 
 A2 
 
 18 
 18 
 18 
 
 17 
 17 
 
 17 
 17 
 17 
 17 
 
 17 
 
 16 
 16 
 16 
 16 
 15 
 
 15 
 15 
 14 
 14 
 14 
 
 14 
 13 
 
 12 
 12 
 11 
 
 11 
 
 10 
 
 10 
 
 9 
 
 9 
 
Miscellaneous Constants in Frequent Use 143 
 
 TABLE LV. 
 
 Miscellaneous Constants. 
 
 n 3-141 5926 54 
 log 7T -497 1499 
 log 2tt -798 1799 
 
 log i- 1-201 8201 
 
 log 4= 1'600 9100 
 e 2-718 2818 28 
 
 - -367 8794 41 
 e 
 
 log e -434 2944 82 
 
 loge 1 * -036 1912 07 
 
 log log e -637 7799 16 
 
 1 centimetre = -393 70432 ins. 
 
 1 inch = 2-539 9772 cm. 
 
 1 square cm. = -155 00309 sq. ins. 
 
 1 square inch— 6451 4842 sq. cms. 
 
 1 cubic cm. = -061 025386 cub. ins. 
 
 1 cubic inch =16-386 623 cub. cm. 
 
 1 kilogram = 2-204 6212 lbs. avoir. 
 
 1 lb. avoir. = -453 59265 kg. 
 
 1 radian =57295 7795 degrees. 
 
 1 degree = -017 4532 925 radians. 
 
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