L K 
 
 OF MORTALIli: 
 
 Bxe. 
 
 (>. F. Hardy, r^ A.
 
 Ex Librts 
 C. K. OGDEN 
 
 THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 
 LOS ANGELES
 
 The Theory of the 
 
 Construction of Tables of Mortality 
 
 AND OF 
 
 Similar Statistical Tables in use by the Actuary. 
 
 A COURSE OF LECTURES 
 
 BY 
 
 GEORGE FRANCIS HARDY, 
 
 FELLOW OF THE INSTITUTE OF ACTUARIES. 
 
 DELIVERED AT THE 
 
 Institute of Actuaries, Staple Inn Hall, 
 
 Durina the Session, 1904-5. 
 
 ;^1ublisl^cb for tbc |iTStitiiic of ^tluaiir.'i bn 
 CHARLIES AND ICDWIN LAYTON, 
 
 56, FARKIN(;i)OX STKKIT, LONDON, K.C. 
 
 I 909.
 
 PREFATOPvY NOTE. 
 
 .Lo each set of Lectures delivered before the Institute of 
 Actuaries, when published in book form, there has generally 
 been prefixed a short preface, or introduction, written by the 
 President of the Institute then in office. This course, 
 admirable in itself, cannot well be followed on the present 
 occasion, having regard to the fact that Mr. Hardy has, in the 
 interval between the delivery of the Lectures and their 
 publication, himself been elected to the Presidential chair. 
 It has therefore devolved upon us, as Honorary Secretaries of 
 the Institute, to insert this foreword in explanation of a 
 seeming omission, and to express therein the confidence of the 
 Council that the Lectures will be found to be of the greatest 
 interest and value to the profession, which already owes so 
 deep a debt of gratitude to their author. 
 
 J. E. F. 
 
 W.P. P.
 
 
 u 
 
 hni 
 
 PREFACE. 
 
 JLHE object of the following Lectures was to deal with the 
 theoretical considerations that should govern the selection 
 and treatment of such statistics as form the basis of the 
 various tables of mortality, sickness, secession, marriage, 
 superannuation, etc., which are of use to the Actuary. It 
 should be noted that in nearly all cases where mortality 
 tables are specially referred to what is said may be extended 
 to other types of statistics, though, to avoid repetition, that is 
 not always pointed out. 
 
 Some apology is required for the long delay in the publica- 
 tion of the Lectures. It was intended subsequently to their 
 delivery, to expand them into something like a complete 
 treatment of the subject (from the theoretical point of view), 
 and to add a sufficient series of examples to illustrate the 
 various points of theory. Unfortunately I have not found 
 time to carry out this intention, but as regards that part of 
 the subject dealing Avith the use of the Pearsonian Types of 
 Frequency Curves in Statistics this has been rendered un- 
 necessary by the appearance of Mr. Elderton's admirable 
 book upon "Frequency Curves and Correlation *', published 
 by the Institute of Actuaries in 1906. 
 
 A few additions have, however, been made to the 
 Lectures as originally delivered, and where these appeared 
 to interfere with the continuity of the text they have been 
 relegated to notes placed at the end of the Lectures. 
 
 I have very specially to thank Mr. (1. J. Lidstonk, F.I. A., 
 for several valualjle suggestions, in particular for the con- 
 tribution of Notes, and for assistance in preparing the 
 lectures for the Printers; and also \)y. Jamks Buchanan, 
 M.A., F.I. A., F.F.A., for having kindly i-cvised the proofs 
 and checked the algebra and numerical work. 
 
 G. F. n.
 
 The Theory of the 
 
 Construction of Tables of Mortality 
 
 AND OF 
 
 Similar Statistical Tables in use by the Actuar}^ 
 
 BY 
 
 G. F. HARDY, F.I.A. 
 
 FIEST LECTUEE. 
 
 YV HEN the Council asked me to deliver a series of lectures 
 upon some subject connected with Part III of the Institute 
 Examination I selected the construction of mortality and 
 similar statistical tables, mainly because it seemed to me to He 
 at the basis of our work. Actuarial science, in the modern 
 sense of the term, had its origin in the collection of statistics 
 (however rough and inaccurate these may have been), and their 
 use for the purpose of calculating life contingencies; and 
 although the Actuary has now to take account of a wider rano-e 
 of subjects than formerly, the collection and analysis of past 
 experience and the employment of the results of such analysis 
 to forecast the future is still his most important function. 
 
 The title of the lectures is somewhat wider and more 
 ambitious than the contents may be found to warrant. To 
 justify it fully would involve dealing with many questions of 
 detail relating to the collection and tabulation of data, such, 
 for example, as the various methods for computino- the 
 numbers exposed to risk in a mortality experience, &c., 
 which have been many times discussed in the volumes 
 of the Journal of the Institute of Actuaries and many of 
 which are exhaustively dealt with by Mr. Ackland in the 
 recently published " Account of Princij)les and Methods." It 
 is evident that to deal Avith the subject in such detail, would 
 outrun the limits of the six lectures which I have undertaken 
 to deliver. I propose, therefore, to confine myself mainly to 
 
 B
 
 a consideration of the general principles involved in the 
 collection of statistical data, and in the construction frcnn 
 such data of tables, of which the Mortality Table is the best 
 known and the most important, embodying the results in tlie 
 form required by the Actuary, and, at the same time, to give 
 such examples of the application of these principles as may 
 be necessary to illustrate the subject. 
 
 In this opening lecture in particular, I shall ask your 
 indulgence if occasionally my remarks appear to be of an 
 elementary character, as I think it desirable that we should 
 be perfectly clear as to first principles before going on to 
 more detailed consideration of the subject. 
 
 Statistical tables, in one form or another, are familiar to 
 all of us. At the basis of all such tables, and, indeed, of the 
 whole science of statistics, lies one of the most fundamental 
 facts in nature, namely, that all phenomena of which we 
 have any knowledge fall into certain classes, groups or series, 
 and cluster round certain tj^DCs. But for this fact we should 
 be unable to classify our knowledge, indeed, should never 
 have acquired any to classify. Speaking broadly, then, every 
 object and every event that comes Avithin our observation is 
 one of a group or class of similar but not identical objects or 
 events, which, as a class, is marked off by certain special 
 features from every other class, although the dividing line 
 may not always be sharply drawn. These groups or classes 
 are not arbitrary, but are inherent in the nature of things, 
 although it is true that the particular groups which we employ 
 in classifying our knowledge are chosen with a view to our 
 own convenience and to the limitations of our minds. 
 
 From a consideration of a class of objects as a whole, Ave 
 get a conception of an average, or type,* to which each 
 individual in the class more or less conforms, but from 
 Avhich, notwithstanding, every individual also diverges. Such 
 divergencies or variations of individuals from the average 
 type may be discontinuous, themselves running into types, or 
 they may be continuous. Among the individuals forming 
 together the type mankind, are divergencies such as those 
 due to sex, race, nationality, birthplace, occupation, civil 
 condition, &c., discontinuous variations producing sub- 
 groups, the boundaries of which overlap and interlace, each 
 
 * The type of the class should preferably he cousidered as represented by the 
 " mode " or case of most frequent occurrence rather than by the " average " or 
 " mean ", but this point is not here of importance.
 
 o 
 O 
 
 of these smaller groups again being capable of endless 
 subdivision. These divergencies can be dealt with statistically 
 only by counting the members of the various sub-groups. 
 
 On the other hand, there are divergencies, which Ave may 
 term continuous, such as those due to differences of age, 
 height, weight, income, &c., &c., differing from the former 
 class in that they do not involve the separation of the main 
 group into sub-groups, but relate to qualities, possessed by 
 each member of the group in varying degree, capable of 
 measurement and numerical statement, and involving the idea 
 in each instance of an average. Thus we can speak of the 
 average age, height, or income of a group of persons, not of 
 their average occupation or nationality, although w^e may 
 speak of the average constitution of the group in respect of 
 these latter qualities. 
 
 A statistical table deals Avith some natural group of 
 objects or events and is a numerical statement of the manner 
 in which the members of the particular group differ inter se in 
 respect of some special character or characters. If dealing 
 with discontinuous variations, as for example a table showing 
 the occupations of a group of persons, it will exhibit, implicitly 
 or explicitly, the ratio of the magnitude of each sub-group to 
 the whole, at a given moment or moments or on an average of 
 a given period ; or it may take the form of a statement of the 
 extent to which variations in one respect are affected by 
 variations in another, as, for example, a table showing the 
 proportion of the sexes in different nationalities. If dealing 
 with continuous variations, it will either represent a series of 
 measurements of some quality common to members of the group, 
 showing its average value for the group, and the manner in 
 which individual values are grouped round such average, or it 
 may represent, numerically, the manner in Avhich deviations 
 from the average in respect of some one quality A are corre- 
 lated with the deviations in respect of some other quality B. 
 
 It is mainly with the class of statistical table dealing with 
 continuous variations that the Actuary has to deal ; variations 
 in the ages of lives under ol)servation, their ages, or the 
 periods elapsed since entry, at death, withdrawal, marriage, 
 superannuation, &c. In such tables the grouping of individual 
 measures round the average will, in general, but not always, 
 be found to follow, approximately, certain well-defined laws. 
 Taking first the tables dealing with a single variable, the 

 
 following may be considered as an example. It is a 
 statement of the heights of 2,192 school children, and is 
 abridged from that given in a paper by Prof. Karl Pearson. 
 
 Table I. 
 Showing lieiglits of 2,192 School Children, aged 12 years. 
 
 
 
 
 Computed- 
 
 -Observed 
 
 Heights in 
 Centimetres 
 
 No. of Children 
 Observed 
 
 Computed Nos. 
 by Curve 
 
 
 
 + 
 
 - 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 139-140 
 
 1 
 
 • ■ ■ 
 
 ... 
 
 1 
 
 135-138 
 
 6 
 
 3 
 
 ... 
 
 3 
 
 131-134 
 
 31 
 
 25 
 
 ... 
 
 6 
 
 127-130 
 
 107 
 
 119 
 
 12 
 
 ... 
 
 123-126 
 
 321 
 
 338 
 
 17 
 
 
 119-122 
 
 585 
 
 577 
 
 ... 
 
 8 
 
 115-118 
 
 618 
 
 596 
 
 . > • 
 
 22 
 
 111-114 
 
 359 
 
 365 
 
 6 
 
 ... 
 
 107-110 
 
 126 
 
 135 
 
 9 
 
 ... 
 
 103-106 
 
 35 
 
 30 
 
 ... 
 
 5 
 
 99-102 
 
 3 
 
 4 
 
 1 
 
 ... 
 
 Total 
 
 2,192 
 
 2,192 
 
 45 
 
 45 
 
 Note. — In the formula (col. 3) x represents the deviation in centimetres 
 
 2192 
 from the average; c = 7*76 and k has such a value — -p^ as to make the area 
 
 c v,r 
 
 of the graduated curve equal to the ungraduated ; that is, to make the totals of 
 
 columns (2) and (3) equal. 
 
 If we consider the progression of the numbers in 
 
 column (2), we shall see that they form a roughly symmetrical 
 
 series, being largest in the neighbourhood of the average 
 
 height and diminishing gradually on either side. It will be 
 
 seen that the average height is about ^^'^\^, the number 
 
 exceeding this height being approximately equal to the 
 
 number falling short of it. In order to bring out the 
 
 approximate law of the series, I have inserted in column (3) 
 
 the computed numbers on the assumption that the frequency 
 
 of a deviation of -Vx centimetres from the average is 
 
 represented by the function A:e~^'''^% where c has the value 7*76 
 
 2192 
 and K the value — j-= • The expression /ce~'=°/'^% represents 
 
 CVTT 
 
 what is usually termed the curve of " facility of error ", 
 or the " normal '' curve of frequency. It will be seen that 
 while the figures in column (2), are as we should expect 
 them to be with such limited data, somewhat irregular, they 
 conform on the whole fairly closely to the normal curve.
 
 The " normal curve " was first used to represent the dis- 
 tribution as to magnitude of errors of observation in ph3-sical 
 measurements. It must not be regarded as representing a law 
 of Nature, but rather an exti*emely convenient and often very 
 close approximation to observation ; experience proving that 
 in many cases errors of observation and the deviations of 
 individuals from the mean of a class do follow very closely 
 the laAv referred to. The formula is therefore empirical and 
 not to be established by a priori reasoning ; at the same time 
 we may, perhaps, see a logical basis in the following 
 consideration. We may suppose that, in any individual 
 measurement, the deviation from the mean of the class (as 
 the difference in the height of any individual among the 
 2,192 in Table I from the average height of the whole 
 group) is the result of an infinity of minute causes as to 
 whose nature we are in ignorance, any one of which may 
 produce a minute positive or negative deviation from the 
 average. These minute superimposed deviations being 
 indefinitely small and indefinitely numerous, we may without 
 loss of generality assume them of equal magnitude. It is 
 then clear that the masrnitude and sisfn of the total resultinsT 
 deviation in any given case Avill depend upon the extent to 
 which the number of these minute positive deviations exceed 
 the negative, or vice versa. 
 
 If the number of possible causes of deviation is 2n, 
 and if the extent of each indefinitely small deviation is k 
 (n being indefinitely large, but hVn finite), then the 
 probability or "frequency" of a total deviation lying between 
 
 X and x + k will depend on our having ('i+.^r) positive 
 
 values of k and f'^— ^w, j negative values. The probability 
 
 of this occurring will be represented by the appropriate term 
 in the expansion of the binomial {\-\-\Y" or 
 
 X \ X 
 
 ''^'■ll>-2k 
 
 It may easily be shown that this expression, n being 
 indefinitely great, takes the form 
 
 1 -(i)' 
 
 ■y 
 
 , i.e. (Constant) x e ~ Ji 
 
 ir.i
 
 6 
 
 i.e., of the curve of the " facility of error." I do not propose to 
 discuss at any length the properties of this particular curve,J 
 but you will notice that the curve being symmetrical with 
 respect to positive and negative values of x, it assumes 
 that positive and negative deviations of a given magnitude 
 are equally frequent^ the average magnitude of such devia- 
 tions being small or large as c is small or large. The maximum 
 ordinate cori-esponds to the value of a; = 0, which is the 
 average value of x ; it therefore passes through the centre 
 of gravity of the area enclosed by the curve and the axis 
 of X, and also divides that area into two equal parts. It 
 assumes that indefinitely large deviations are possible, hence 
 it cannot be rigidly exact, because when dealing with physical 
 measurements of any kind, indefinitely large errors are not 
 possible. This is not a practical objection to the use of the 
 formula, however, as the probability thereunder of deviations 
 of many times the average value is extremely small. 
 
 The following table, showing the number of entrants in 
 various aged groups in the 0^^ Experience, exhibits a quite 
 different distribution of the deviations from the average : 
 
 Table II. 
 Numher of entrants in quinary age groups QP^^ data. 
 
 
 
 
 Computec 
 
 I — Actual 
 
 Central Age 
 of Group 
 
 Actual Entrants 
 in Group* 
 
 Computed No. 
 by f orniulat 
 
 
 
 
 
 X 
 
 
 
 + 
 
 — 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 20 
 
 431 
 
 436 
 
 5 
 
 
 25 
 
 1,273 
 
 1,305 
 
 32 
 
 
 30 
 
 1,526 
 
 1,473 
 
 ... 
 
 53 
 
 35 
 
 1,269 
 
 1,265 
 
 
 4 
 
 40 
 
 914 
 
 930 
 
 16 
 
 
 45 
 
 591 
 
 604 
 
 13 
 
 * . . 
 
 50 
 
 354 
 
 349 
 
 
 5 
 
 55 
 
 182 
 
 178 
 
 
 4 
 
 60 
 
 83 
 
 79 
 
 ■ • • 
 
 4 
 
 65 
 
 26 
 
 29 
 
 3 
 
 
 70 
 
 7 
 
 8 
 
 1 
 
 
 75 
 
 1 
 
 1 
 
 ... 
 
 
 Totals 
 
 6,657 
 
 6,657 
 
 70 
 
 70 
 
 * Omitting huiiclreds. 
 
 t Fonimla representing number of entrants at given age x = K(.r-18'59)i'*'''^ 
 (88-48 -j-)''*'"^; where log k= -9-2360. 
 
 X The student may consult Woolhouse's paper on " The Philosophy of 
 Statistics" {J. I. A., vol. xvii, p. 37), or an exhaustive analysis of the properties 
 of the curve hy Mr. Sheppard {Phil. Trans., vol. 192, p. lOl') ; See also " Bowlcy's 
 Elements of Statistics ", Part II, Sec. II.
 
 Here the numbers also exhibit a ■well-marked law 
 governing- the deviations from the mean, but this laAv is no 
 longer the same as that shown by the "normal" curve of 
 frequency. The maximum ordinate does not coincide 
 either with the averasre ag'e or with the central ao-e 
 of the series; while the number of cases exceeding the 
 average age no longer equals the number falling short of it. 
 Tn other words, the curve is non-symmetrical or skew. It 
 follows very approximately, hoAvever, a certain law, as will 
 be seen by comparing the numbers in column (2) with 
 those in column (3), which represent the computed numbers 
 according to the formula stated. 
 
 Having regard to the fact that the numbers in column (2) 
 represent lOO's and not units, the differences between the 
 actual and computed numbers are somewhat outside the 
 probable errors of observation. There are, that is to say, 
 "systematic" differences between the two curves. These 
 systematic differences are generally to be expected in dealing 
 with age statistics. It will be seen that they are not 
 incompatible with a close agreement in the general features 
 of the two curves, but they serve as a warning that, in 
 statistics of this nature, formulae representing the 
 distribution of deviations from the mean must be regarded 
 as approximations only. 
 
 If we consider the curves exhibited in Tables I and II we 
 see that the general chai-acter of such curves is determined 
 by a few salient features : 
 
 1. The position of the maximum ordinate; that is, the 
 
 value of the variable having maximum frequency. 
 This value is. termed the mode. 
 
 2. The average or mean value of the variable, being the 
 
 arithmetical mean of all iiidividual values. In a 
 symmetrical curve this coincides with the " mode." 
 
 3. The average deviation from the mean, corresponding 
 
 to the closeness with which the individual measures 
 are grouped round their mean value. There is a 
 certain convenience, for analytical reasons, in 
 adopting as our standard in this respect either the 
 mean of the sfpuires of the individual deviations, or 
 the square root of this (juantity. The latter is 
 termed the standard deviation. We may represent 
 the average of the squares of the deviations, or the
 
 8 
 
 i{ ■ 
 
 J 
 
 mean square" deviation by the symbol /jL.2, when 
 the standard deviation becomes \/a2' 
 4. The equality or otherwise of the positive and negative 
 deviations from the mean ; that is, the symmetry or 
 shewness of the curve. The sum of the first powers 
 of the deviations is, of course, always zero. If the 
 curve is symmetrical, the sum of any odd power of the 
 deviations must be zero, but not otherwise. As we 
 have employed the square root of the average 
 square of the deviations as a measure of the 
 diffuseness or spread of the curve, termed the 
 " standard deviation ", so we may take the ratio of 
 the cube root of the average cube deviation to the 
 " standard deviation " as the standard of 
 " skewness." If we represent the average cube 
 deviation by the symbol fia, the skewnessoi the curve 
 
 may then be measured by ^ — . 
 
 The skewness is sometimes taken as the difference between 
 the " mean " and the " mode ", divided by the standard 
 deviation. 
 
 The sums of the successive powers of the deviations 
 of the variable from the mean, the area of curve being 
 taken as unity, are termed the moments of the curve. 
 
 These observed laws of the variation of measurements 
 from their mean are very general, and are usually, though not 
 invariably, associated with what is termed " homogeneous " 
 data. The distinction between "homogeneous " and "hetero- 
 geneous" data is of considerable importance, although not 
 very easy to define. We may perhaps define a homogeneous 
 group as one in which the continuous variations are from a 
 single type only, and are unaffected by any discontinuous 
 variations in the group if these exist. These conditions will 
 hardly ever prevail, but a group may be considered for practical 
 pui'poses as homogeneous if the variations in the particular 
 quality dealt with are not materially affected by any discontinous 
 variations existing in the group. If, however, the group can 
 be split up into two or three distinct series differing markedly 
 in certain qualities, and these differences are found, or ma}' 
 reasonably be supposed, to affect the character under 
 examination, then the series is " heterogeneous." 
 
 Take, for example, the class representing assured lives of
 
 9 
 
 a given age, but of varying duration of assurance, and assume 
 we are investigating the rate of mortality of the class. If it is 
 found on examination that the duration of assurance materially 
 affects the rate of mortality, then the data treated as a whole 
 is heterogeneous. If it is found, however, that the duration 
 of assurance after reaching a certain point has no such 
 iniluence, or an influence that is insignificant, then the data 
 from this point and ' in this respect may be treated as 
 homogeneous. The same considerations apply to distinctions 
 in class of assurance, amount of policy, occupation, &c. 
 
 The laws which appear to govern deviation from the 
 average in homogeneous data are, in general, so uniform in 
 action that a departure therefrom will frequently indicate 
 that data which might be supposed to be homogeneous are not 
 so. An interesting illustration of this may be seen in the 
 case of the Male Annuitants in the New Offices' Annuity 
 Experience. Consider the following table showing the number 
 of entrants for various groups of ages : — 
 
 Table III. 
 
 Male Annuitants 0«™ Data. 
 Number of entrants at various ages, 18G3-1893. 
 
 Ages 
 
 at Entry 
 
 X 
 
 (1) 
 33-37 
 38-42 
 43-47 
 48-52 
 
 Entrants 
 
 Computed 
 Numbers 
 
 /j:-65\2 
 
 Observed — Computed 
 
 + — 
 
 73 
 119 
 207 
 421 
 
 (3) 
 
 5 
 21 
 
 89 
 266 
 
 (4) (5) 
 
 68 
 
 93 
 118 
 155 
 
 53-57 
 
 58-02 
 63-67 
 68-72 
 73-77 
 
 78-82 
 83-87 
 88-92 
 93-98 
 
 599 
 957 
 
 1,147 
 982 
 660 
 
 252 
 72 
 15 
 
 1 
 
 587 
 
 95 1 
 
 1,H2 
 
 1,007 
 
 655 
 
 313 
 
 109 
 
 29 
 
 6 
 
 12 
 3 
 5 
 
 25 
 5 
 
 61 
 
 37 
 
 14 
 
 6 
 
 These particuhir agt' giniips are selected as tlioro appears 
 to be a slig-lit excess in the nunihcr nf entrants at decennial
 
 10 
 
 and quinquennial ages, and by placing these in the middle 
 of the groups we get rid of the disturbance, which would 
 otherwise affect the numbers. 
 
 An examination of the numbers in column (2), between 
 ages 53 and 78, shows that they form a nearly symmetrical 
 curve, as is seen by a comparison with a "normal" curve of 
 frequency given in column (3).* The numbers above age 78, 
 however, are in defect, and those below 53 are considerably in 
 excess of the figures suggested by the normal curve. As 
 regards the falling-off of the numbers at the older ages, it 
 may be conjectured that it is in part due to the fact that many 
 published tables of the cost of annuities cease at age 75 or 
 80. The observed excess in the number of entrants at ages 
 below 50 evidently represents the entrance at these ages of a 
 class of lives differing from those forming the bulk of the 
 data. It may perhaps be conjectured that a number of these 
 cases are counter lives in contingent reversions, or similar 
 securities, upon whose lives annuities have been purchased to 
 secure the payment of annual premiums. Be that as it may, 
 Ave find that while the deficiency of entrants at the older 
 ages does not appear to affect the mortality rates,, the entrants 
 at the younger ages on the contrary show abnormally heavy 
 mortality, the ungraduated values of the expectation of life 
 for entrants under age 55 being relatively Ioav. Hence we 
 may calculate that the male annuitant experience is hetero- 
 geneous, and in using the results as a basis of calculation 
 for the future, the abnormal part of the experience representing 
 the entrants at the younger ages was propei'ly rejected. 
 
 In addition to tables of the kind Ave have been considering, 
 a statistical table may be a numerical statement of the 
 manner in Avhich variation in one particular from the average 
 of the group is accompanied by variation in some other 
 particular. We may, for instance, have a table representing 
 a number of individuals, arranged according to height, the 
 numbers at each height being further arranged according to 
 weight. We should then have a table of double entry, each 
 roAV or column of Avhich Avould represent a statistical table of 
 the form already considered. By means of this table we should 
 be able to " correlate ", as it is termed, variations in respect to 
 
 •■■■ The constants of this curve were only roughly determined, but the 
 agreement with the observed numbers between ages 53 and 78 is sufficiently 
 close to illustrate the point under discussion.
 
 11 
 
 Aveight witli variations in respect to height. Such a table 
 would represent a mass of figures, the bearing of which could 
 not easily be grasped without some further analysis. If, 
 however, we add to the table a column showing the average 
 weight for persons of a given height, we then have a ready 
 means of seeing how this average weight is affected by a 
 change in height. Having inserted the average, Ave have not 
 exhausted the information which the original figures give us. 
 We need also to know to what extent on the average the 
 weight varies when the height remains constant ; that is, we 
 need to insert against each average Aveight AA'hat Ave have 
 termed the " standard deviation." 
 
 A familiar example of such a table is one shoAving the 
 asres of husbands and wives at marriage. Such a table would 
 take the following form — 
 
 Table IV. 
 Showing Ages of Husbands and Wives at date of Marriage. 
 
 Husbands' 
 
 
 
 AVivEs' Ages 
 
 
 
 
 
 
 
 
 
 Ages 
 
 under 
 
 
 
 
 
 
 Mean Ages 
 
 
 20-30 
 
 30-40 
 
 40-50 
 
 50-60 
 
 60-70 
 
 of 
 
 
 20 
 
 
 
 
 
 
 AV'ives 
 
 under 20 
 
 13 
 
 5 
 
 
 
 
 
 17-8 
 
 20-30 
 
 215 
 
 500 
 
 16 
 
 1 
 
 
 
 22-3 
 
 30-40 
 
 14 
 
 107 
 
 39 
 
 4 
 
 . . . 
 
 
 27-0 
 
 40-50 
 
 1 
 
 14 
 
 23 
 
 12 
 
 2 
 
 
 350 
 
 50-60 
 
 
 2 
 
 6 
 
 9 
 
 4 
 
 
 42-1 
 
 60-70 
 
 
 
 1 
 
 3 
 
 4 
 
 » 
 
 520 
 
 70-80 
 
 
 ... 
 
 
 1 
 
 1 
 
 1 
 
 55-0 
 
 Mean 
 
 
 
 
 
 
 
 Ages of 
 
 251 
 
 27-2 
 
 37-6 
 
 49-0 
 
 58-6 
 
 68-3 
 
 — 
 
 Husbands 
 
 
 
 
 
 ■ 
 
 
 
 If there were no correlation between the ages of the 
 husbands and Avives at marriage, the figures shoAving the 
 average ages for the various colmmis Avould (except for 
 accidental fluctuations) be identical, and the same Avould hold 
 for the average ages of the successive roAVS. 
 
 If a line were drawn through the table cutting those 
 points in the rows corresponding to the average ages, and 
 another lino similarly cutting those points in the columnn 
 representing average ages, it AVould bo found that these 
 points could roughly bo represented l)y straight lines, Avliich
 
 12 
 
 in the present example would be nearly coincident, since the 
 spread of the figures, as measured by their standard deviation, 
 is very similar in both rows and columns. 
 
 It is not always the case, however, that the nature of the 
 correlation can be represented by a straight line. In the 
 following example we have a somewhat different class of 
 table showing the proportions for different age groups of 
 wives and widows in an Indian pension fund. 
 
 Table IVa. 
 Showing proportion of Wives and Widows in a Pension Fund. 
 
 Ages 
 
 Number of 
 Wives 
 
 Number of 
 Widows 
 
 Total 
 
 Widows. 
 
 per-cent of 
 
 Total 
 
 under 20 
 20-30 
 30-40 
 40-50 
 50-60 
 60-70 
 70-80 
 80-90 
 
 19 
 
 1,430 
 
 3,366 
 
 3,329 
 
 1,653 
 
 476 
 
 63 
 
 6 
 
 "so 
 
 355 
 
 1,018 
 
 1,312 
 
 933 
 
 330 
 
 46 
 
 19 
 
 1,480 
 3,721 
 4,347 
 2,965 
 1,409 
 393 
 52 
 
 00 
 3-4 
 9-5 
 23-4 
 44-2 
 66-2 
 84-0 
 88-5 
 
 Here it will be seen, from the run of the figures in the last 
 column, that they cannot be well represented by a straight 
 
 line, being somewhat in the form of the curve of J e ~ ''^' dx, 
 
 a 
 
 or of the curve , witli values of and 1 respectivelv at 
 
 the limits. 
 
 Such a table of correlation has an analogy with the table 
 of the " Exposed to Risk " and " Died ", which ordinarily 
 forms the basis of our Mortality Tables. This table is 
 virtually in the following form — column (4) representing the 
 number of annual survivors being usually omitted as being 
 implicitly contained in columns (2) and (3) — 
 
 Table of Exposed to Risk and Died. 
 
 Age 
 
 Exposed to Risk 
 
 Died 
 
 Survived 
 
 (]) 
 
 (2) 
 
 (3) 
 
 (4)
 
 13 
 
 We have here the ages of the persons observed; the 
 numbers under observation, or " Exposed to Eisk ", Avhich, 
 for the sake of simplicitj^, Ave will suppose to remain under 
 observation for the entire year of age; the number of those Avho 
 die during the year, and of those surviving. If we represent 
 the rate of mortality by q^; then in all cases in column (3) 
 gt^ = l, and in all cases in column (4) qj^ = 0, and we have a 
 table which is analogous to the table of the weights of 
 indi\aduals of respective heights, only that instead of having 
 various values of qjc, we have in the nature of things only 
 two possible values and 1, the average value for each group 
 representing the observed "rate of mortality." This table 
 differs from that correlating weights and heights, or ages of 
 husbands or waves at marriage, agreeing with that correlating 
 age and civil condition, in the fact that a certain quality 
 or characteristic, in this case death during a given year 
 of age, is not present in varying proportions, but is 
 either present or entirely absent. We are thus introduced 
 to the conception of probability, the proportion of any 
 group surviving or dying representing the " probability " 
 of survival or death for any individual of the group taken 
 at random. The idea of probability is also present in 
 the supposed table of weights, although not so obviously. 
 That table would inform us, for example, of the probability 
 of a person of given height exceeding or falling short of 
 a certain fixed standard weight, and Ave should then have 
 a table identical in form Avith the table of Exposed to 
 Risk and Died. 
 
 This conception of probability is important to the Actuary, 
 because his object in collecting statistics is the distinctly 
 practical one of measuring the probability of the happening 
 of certain contingencies. It is necessary to realise clearly 
 what is meant by the statement that the probability of a 
 particular event has this or that value. Laplace pointed 
 out that Avhcn Ave speak of the probability of the happening 
 of a given event, Ave do so only on account of our ignorance 
 of the antecedents of the event, or our inability to completely 
 analyze them. If avc entirely kncAv the antecedents, and if 
 our poAvers of analysis Avere equal to the task, avo could 
 predict the event. In many cases Ave are able to do tins 
 approximately, but Avhere the effective causes at Avork 
 are numerous and obscure, and the result in individual
 
 14 
 
 (apparently similar) cases is very variable, as in all questions 
 affecting life contingencies, we are unable to forecast the 
 event in a given case, and must fall back upon the average 
 result deduced from the examination of a large number of 
 similar cases. In other words, we treat the particular case in 
 question as one of an indefinitely large class of similar cases, 
 a sample of which we have already had under examination. 
 From the results of such examination we infer the composition 
 of the class as a whole, and hence the " probability " or 
 average event in an individual case. If, in the sample 
 observed, a given character is present in a certain proportion 
 of cases, as for instance, where out of a number of persons of 
 given age under observation, a certain proportion have died 
 within the year of age, then we estimate" the probability of 
 the event happening in a particular instance, by the ratio 
 which the number of cases in which the event has occurred 
 bears to the entire number of cases observed.^ To determine 
 the probability of a given event is therefore to assign the 
 case to the natural group or series to which it properly 
 belongs and to pass under examination a sample of the group 
 sufficiently large to enable us to determine approximately the 
 average character of the whole as regards the particular 
 quality in question. We are here speaking of simple events ; 
 the probability of a complex event, such as the survival of 
 one life by another, is, of course, not determined directly by 
 past observations. The latter yield the simple probabilities 
 of surviving each year of age, by suitably combining which 
 we arrive at the value of the probability desired. 
 
 The degree of certainty with which we can deduce the 
 properties of an entire class from the part known to us, 
 depends first on our assurance that the class is homogeneous, 
 or at least that the portion observed is representative, such 
 as would result from a selection of cases made at random, and 
 secondly, on the number of cases that have been under 
 
 " The formula deduced by Laplace by wliicli the true probability of an 
 event which has been observed to happen m times out of w + « trials is taken as 
 
 —^ j; is obviously not aj^plicable to such a function as the rate of mortality, 
 
 m + n + ii 
 
 nor to any analogous function. It is sufficient to consider that in tabulating the 
 
 values of the probability of dying in each year of age, we are using an arbitrary 
 
 unit of time which might just as well be a month or day, in which cases we should, 
 
 by use of the above formula, produce quite different mortality tables from the 
 
 same data.
 
 15 
 
 observation. If we examine the figures in tables similar 
 to Tables I and II, we see that, in proportion as the number 
 of cases under observation is small, the figures I'epresentino* 
 the results of the experience are irregular, while, on the other 
 hand, Avhere the number of facts observed is verv laro-e, the 
 irregularities become relatively less. AVe arrive at the same 
 conclusion from theory. If an indefinitely large group N" 
 contains Np objects of class A and N(l— j)) objects not of 
 class A, and if from the group n objects are selected at 
 random, then on the average ii'p of these will be of class A. 
 If we represent the observed number in any given case as 
 nip + z, the average algebraical value of z will be zero, 
 Avhile its average numerical value, irrespective of sign, will 
 be very nearly Y^'/;ip(l— _p).* This latter quantity clearly 
 increases as nio increases, but at the same time its ratio to 
 np diminishes. Thus in a table of exposed to risk and died 
 the actual irregularities in the number of deaths increase 
 with the magnitude of the experience, but the irregularities 
 in the rate of mortality diminish. Hence from theory as from 
 exi^erience we derive the conviction that if instead of the 
 limited number of facts which Ave have been able to examine, 
 we could have examined an indefinitely large number of 
 similar facts, the results would have been relatively free 
 from irregularity, and capable of being expressed by a 
 continuous curve ; without, of course, being sure that any 
 such curve could be expressed algebraically. 
 
 The idea underlying the graduation of the figures of a 
 statistical table, Avhatever be the process employed, is that a 
 continuous curve may be found representing the general trend 
 of the observations freed from irregularities due to paucity 
 of material. This curve, we have reason to believe, Avill 
 correspond more closely than the ungraduated curve to the 
 results obtainable from a much larger body of facts. This is 
 the rationale of the process of graduation and its justification. 
 Such a process cannot deal Avith systematic errors affecting 
 the table as a Avhole and cannot compensate for inadequate 
 data. It adds Aveight to the results, hoAvever, at each 
 individual point of the table, and assists in bringing into 
 relief the true character of tlie curA'c by freeing it, in a 
 large measure, from accidental irregularities. 
 
 • The average value of 2" will be n}q, the average value of z^ will bo 
 "P?''p-3),and the aA-erage A-alue of 2' will be Jip(/[(,3n — 6;/>j + 1]. See Note A,p. 110.
 
 16 
 
 There may be other objects aimed at in a graduation 
 besides that of removing the irregularities from the rough 
 figures, with the view of bringing out more clearly the law 
 underlying them. The Actuary constructs tables not merely 
 to show what has happened in the past, but to enable him to 
 forecast the future, and as he requires these tables as a basis 
 for financial operations, considerations are introduced which 
 do hot arise in the treatment of purely statistical tables. 
 Whatever class of events the Actuary may have to deal with, 
 will be subject to change with the lapse of time. That 
 portion of the class he has been able to observe lies 
 necessarily in the past ; the conclusions he has derived from 
 their study he proposes to extend to the future. He must 
 therefore consider how far the observed characters of the 
 class are changing or permanent, and must endeavour 
 to distinguish between changes representing permanent 
 tendencies and those due merely to temporary fluctuations. 
 In the selection of data suitable for his purpose the Actuary 
 will aim on the one hand at a sufficiently broad basis both in 
 space and time to eliminate the eifects of local and temporary 
 fluctuations, and on the other hand he will aim at obtaining 
 as far as possible a homogeneous group of data. These two 
 aims are more or less in conflict, and he will lean to the one 
 side or the other, according to the object he has in view. 
 Where, for example, that object is to produce a table that 
 may be adopted as a general standard by various institutions, 
 often difl^ering considerably as to their individual experience, 
 he must aim at a correspondingly broad foundation. In 
 these circumstances it will not generally be possible to obtain 
 a really homogeneous experience. If it is a question of the 
 mortality of assured lives, for instance, this will be found to 
 be affected by endless individual variations, age, sex, duration 
 of assurance, occupation, civil condition, class of assurance, 
 character of the insuring office, &c., &c., and from such 
 material approximately homogeneous data could only be 
 obtained by cutting up the experience into comparatively 
 small groups and thus sacrificing all generality. This can be 
 avoided in practice by first excluding all extreme variations. 
 The sexes will be separately treated, lives so impaired as to 
 prospects of longevity by personal health, family history, 
 occupation, or residence in unhealthy districts as to be " rated 
 up" will be excluded, as also classes of assurance that may 
 
 r
 
 17 
 
 be supposed subject to rates of mortality differing from the 
 average. AVhen the data has thus been trimmed of the 
 extreme variations, a body of experience will generally 
 remain not greatly shrunken from its original dimensions 
 and in which the discontinuous variations are sufficiently 
 numerous and individually unimportant to render the data 
 for practical purposes homogeneous. The rates of mortality, 
 or of withdrawal, can then be treated as functions of the two 
 remaining variables of importance, the age and the time 
 elapsed from date of entry ; or as functions of the age only 
 from the point at which the factor of duration may be found 
 to be unimportant. 
 
 On the other hand, the Actuary's object may be precision 
 rather than generality ; he may have to deal with a group, 
 subject to special conditions and presenting special 
 characteristics, as is usual in the case of pension funds and 
 friendly societies. Here, if the data are at all adequate, better 
 results will be obtained therefrom than by having recourse to 
 any general experience. Where it is insufficient by itself as 
 a basis for statistical tables it may serve as an indication as 
 to what standard table is the most suitable to employ and as 
 to how far and in what direction it may be desirable to 
 introduce any modifications therein. In an experience of 
 this character the data may sometimes be very heterogeneous, 
 but there is usually the safeguard that its composition is 
 approximately constant. 
 
 A question of some importance may here be considered, 
 namely, the relative claims of lives, policies, or amounts 
 assured to form the basis of the mortality table. In the 
 17 Offices' data, the number of policies, in the H^ and 0*^ 
 data, the number of lives passing under observation 
 constitute the basis of the experience, while in the American 
 Offices' Experience (1880) the sum assured was the unit. In 
 the instances of the H^^ and 0^' Tables, wherever a life would 
 have been doubly observed the duplicate assurance was 
 eliminated. In justification of the use of the sums assured 
 as the basis of the experience, in lieu of the number of lives, 
 it may be said that in this way wo represent the financial 
 effect of the mortality, as it makes no difference to the 
 insuring company whether one claim arises for £10,000 or 
 one hundred claims for £100 each. There are, however, serious 
 objections to employing the sums assured as a basis for a
 
 18 
 
 mortality table, based upon a general experience. Either the 
 mortality among the lives carrying large sums assured is 
 similar to the average or it is not. If it is similar, the 
 general character of the table will not be affected by the 
 additional weight given to these lives in the experience, but 
 the irregularities in the deduced rates of mortality will be 
 considerably increased. The result, indeed, will be virtually 
 the same as if we had used a part only of the 
 available data, selected at random, instead of the 
 whole. If, on the other hand, the mortality among the 
 lives insured for large sums is materially different from 
 the average, then the experience is not homogeneous. 
 As a matter of fact, these lives of themselves do not form a 
 homogeneous group. In certain societies they appear to give 
 better rates of mortality than the average ; in others, where 
 they are mainly represented by non-profit policies effected for 
 commercial reasons, they are no doubt subject to higher rates 
 of mortality than the average. As in a general experience, 
 combining the individual experience of many offices, these 
 lives will represent an exceptional or abnormal element, 
 which may or may not persist in the future, and will certainly 
 not persist equally in all societies, it is not desirable in 
 deducing a general mortality table to specially " weight up " 
 this part of the data. 
 
 The same considerations apply, but with somewhat 
 less force, to the plan of making policies rather than 
 lives the basis of an experience. Without dogmatizing 
 upon the point, it appears to me that the proper course is, 
 where two or more policies are effected at the same time or 
 at the same age at entry, to treat them as a single risk, 
 but where the subsequent policies are effected at later ages, 
 involving fresh medical selection, to treat them as separate 
 risks. This means the elimination of duplicates in each of 
 the " select '' tables for individual ages at entry, but no 
 further elimination in the resulting aggregate tables, a course 
 which has the advantage of making the aggregate table the 
 true aggregate of the tables for separate ages at entry. 
 Judging by the results of the 0'^ experience, this course is 
 necessary if we are to produce an aggregate table, 
 representing " ultimate " rates of mortality after the lapse of 
 a stated period from entry, which will join on smoothly to 
 the " select " rates.
 
 19 
 
 A detail of less importance, but of considerable interest, 
 is the question of the proper treatment of withdrawals in 
 a, mortality experience. These are usually treated as 
 withdrawing upon the termination of the days of grace in 
 case of lapse by non-payment of premium, and for the 
 purpose of obtaining the true measure of the moi-tality 
 ■experienced this course is the correct one. It should be 
 borne in mind, however, that to arrive at the finaucial effect 
 of the mortality the numbers of the exposed to risk should 
 correspond to the number of annual premiums paid, and from 
 this point of view the life Avithdrawing should not be treated 
 at risk during the days of grace. The differences in the 
 resulting mortality rates according to the two methods is, of 
 .course, very slight.
 
 SECOND LECTURE. 
 
 JlLaVING dealt in the last lecture with the rationale 
 of graduation in general, I now propose to refer more 
 particularly to the principles underlying certain special 
 methods of graduation. We may divide the various methods 
 which are in use into three classes : 
 
 1. Graphic methods. 
 
 2. Methods based upon Intei-polation or Finite 
 
 Difference formulae, such as Mr. Woolhouse's. 
 
 3. Methods which depend upon the use of Frequency 
 
 Curves, in which Ave may include all methods 
 
 based upon the assumption that the series to be 
 
 graduated can be represented as some function of 
 
 the variable. 
 
 Certain general considerations apply to all these methods. 
 
 We may have to deal either with a single series of numbers, 
 
 such as the number, at successive ages, of lives effecting 
 
 assurances, of persons enumerated at a census, or attacks 
 
 from a given disease, &c. ; or, as more often happens in 
 
 actuarial statistics, the fact of importance may be the ratio 
 
 between the corresponding members of two series of numbers 
 
 as in a table of " Exposed to Risk" and "Died^^ forming the 
 
 basis of the Mortality Table, where the fact sought is the rate 
 
 of mortality at each age given by the ratio of the Died to 
 
 the Exposed to Risk, the actual numbers of these being 
 
 of importance mainly as affording a measure of the 
 
 trustworthiness of the deduced ratio. 
 
 Where only a single series of numbers is involved, the 
 problem is comparatively simple, and an accurate solution is 
 not generally of great importance to the actuary. In the 
 more usual case where the ratio of the corresponding 
 members of two series of numbers is in question, the problem 
 is more complicated. We have a choice of procedure : we 
 may either graduate independently the two series of numbers
 
 21 
 
 (in the case supposed the numbers of the " Exposed to 
 Risk" at each age and the numbers of the "Died"), or, 
 disregarding the irregularities in the two series, we may 
 proceed to deal at once with the ratios only. If each series 
 can be satisfactorily graduated, the resulting curves being 
 smooth and fitting the ungraduated sei-ies sufficiently closely — 
 that is to say, within the limits of the errors of observation — 
 we may then assume that the ratios of the corresponding 
 terms (in the case supposed the rates of mortality) will also 
 be within the limits of error. It may also be said that by 
 working with the rough facts themselves, rather than the 
 ratios between the two, we keep in view the weight of the 
 observations at each point of the curve, and are able to see 
 at once how far our graduated numbers vary from the 
 original, and how far that variation is justified by the number 
 of facts at each particular point. There are, however, some 
 important objections to this course. In the first place, the 
 ratio between the corresponding terms in the two series of 
 numbers represents generally a relatively stable quantity, 
 whereas the actual numbers in either series, depending as 
 they do upon the extent of the experience under review at 
 particular ages, are liable to fluctuations of a more or less 
 arbiti-ary character. Further, supposing the graphic method 
 of graduation or the method of finite differences is employed — > 
 in either case the argument is applicable, although specially 
 so in the former — it will be found that each curve will 
 contain certain outstanding irregularities, as it is not possible 
 entirely to remove all irregularities by those methods. Hence 
 in the adjusted ratios two sets of irregularities will be super- 
 imposed and a less satisfactory series of values obtained than 
 if the ratios themselves had been dealt with. 
 
 A stronger objection, when dealing with a mortality 
 experience, to graduating separately the numbers in the two 
 .series of " Exposed to Kisk" and "Died" rather than their 
 ratio, is that we thereby discard our previous knowledge of 
 the nature of tlie curve expressing that ratio — our general 
 knowledge, that is, of the nature of the curve qjc or /i^ — 
 knowledge wh idi is of considerable assistance in graduating the 
 commencement and end of the tablf uIk re; the data are few. 
 
 Wliere a graduation of both series of iiiiiiil)ers is made, it 
 is preferable, indeed necessary if tin- best results are to be 
 obtained, lift (T fii'st gradual Iiil;; tlic scries cni-r('S])nnding to
 
 22 
 
 the " Exposed to Risk"^ to re-compute the numbers of deaths, 
 lapses or marriages, as the case may be, on the basis of the 
 graduated numbers of the Exposures, and to operate upon 
 these adjusted numbers. We are in this way less Hkely to 
 obscure the law of the series representing the required ratios. 
 Notwithstanding any theoretical objections, there may be 
 occasions on which it is more convenient, or even necessary, 
 to deal with the two series separately ; where, for example, 
 as in the Registrar-General's returns of the population and 
 deaths for certain occupations, we have not the facts for 
 individual ages, but only in certain large groups. The 
 ratio of deaths to exposures for each age group are obviously 
 not satisfactory approximations to the rate of mortality for 
 the central age of the group. In these circumstances it 
 appears to be best to adopt a plan similar in principle, 
 though not in detail, to that employed by Milne in graduating 
 the Carlisle Table, and to draw curves respectively through 
 the parallelograms representing the exposures and the deaths, 
 and from these deduce the numbers for individual ao-es. The 
 graphic method, however, is not very suitable for this purpose, 
 and the use of interpolation formulee does not always give 
 good results. It is generally better to make use of suitable 
 frequency curves. It will be seen later that, where the 
 number of groups is rather small, the use of the normal 
 frequency curve, with certain modifications, enables us to 
 re-distribute the numbers representing the groups of 
 " Exposed " and " Died ", and so obtain graduated numbers 
 for each age, and hence from the ratios of these a graduated 
 rate of moi-tality. {8ee the Sixth Lecture, p. 91.) 
 
 We shall now assume that we are dealing, not with the 
 two independent series, but with the ratio between the two ; 
 as, for example, Avith q^, or some analogous function. 
 
 We may consider we have three independent estimates of 
 the value of g^ : — 
 
 1st — That derived from the observed ratio of the 
 
 died to the exposed at age x. 
 2nd — That derived from the data at neighbouring 
 
 ag'es. 
 3rd — That derived from previous experience of more 
 or less similar data.
 
 23 
 
 The first and second should be suitably combined in the 
 process of graduation. The last is, in the nature of things, 
 a very vague estimate, and bears a relation to that derived 
 directly from the observations, if these are numerous, similar 
 to that of a rough measurement by inferior instrumental 
 means to one made by an instrument of precision. In such 
 case no weig-ht attaches to it. 
 
 There are circumstances, however, in which the a 'priori 
 estimate of the values of qx become important, viz., when the 
 observations at our disposal are extremely few. As the 
 exteut of our observations diminish, the numbers of exposures 
 and deaths becoming smaller, the weight to be attached to 
 the deduced values of the rate of mortality become less, and 
 a point is eventually arrived at when we obtain more 
 trustworthy results by considering to Avhat particular class 
 of examined data the experience most nearly conforms in 
 character, and falling back upon the results of such related 
 experience. 
 
 If we have to deal with a large experience, a somewhat 
 similar difficulty arises at the commencement and end of the 
 table. Generally speaking, we then derive more trustworthy 
 values for the rates at these ages from a consideration of the 
 general trend of the curve and our previous approximate 
 knowledge of its character, than by falling back upon any 
 related experience. 
 
 Coming to the principles underlying each of these three 
 methods of graduation, we consider first the graphic method, 
 whether in the form employed by Milne or in the preferable 
 form employed by Dr. Sprague. This method makes no- 
 furtlier assumption than that the series with Avhich we 
 are dealing would, if the observations were sufficiently 
 extensive, form a continuous and regular curve, and that the 
 irregularities actually occurring in the ungraduated values 
 are due to tlie smallness of the data. 
 
 To Dr. Sprague {J. I. A., vol. xxvi, p. 77) we owe the 
 most systematic and satisfactory exposition of the graphic 
 method. An essential feature in his procedure is the 
 preliminary division of llic «I:ita (which we may suppose 
 arranged by years of age) into groups, so selected as to aii'ord 
 a steady progression in the average rates of mortality for
 
 24 
 
 successive groups, due regard being had to the range of 
 these groups. For examples of the method, the student must 
 be referred to Dr. Sprague's original papers. This process of 
 dividing the data into selected groups appears at first sight to be 
 arbitrary, but it may be justified on the grounds : — (1) That 
 in a series of observations such as we are discussing, where 
 at each age the results are affected by irregularities or errors 
 of observation, a successful graduation will reduce the sum 
 of these errors and also the sum of the " accumulated " errors 
 to zero, or nearly so. Hence if we compute at each age the 
 accumulated errors (reckoning from either end of the series) 
 these must, in order that their sum may be approximately 
 zero, change sign, thus passing through zero, fairly frequently. 
 The data will, therefore, be made up of consecutive groups, 
 larger or smaller, in each of which there is an approximate 
 balance of errors, and it may be assumed that, with a sufficient 
 amount of experience and the exercise of some trouble, these 
 groups can be found by inspection and trial. (2) In further 
 justification of this procedure, it is to be noted that the rates 
 of mortality deduced from the average rates in the selected 
 groups are used as a first approximation only, the final rates 
 being arrived at by repeated comparison of the graduated 
 deaths with the actual numbers until a sufficiently smooth 
 curve and a sufficiently close agreement has been obtained. 
 At the same time I am not convinced that the use of these 
 specially selected groups has any real advantage over the use 
 of groups of constant range, as quinquennial or decennial, 
 provided the operator recognizes that he cannot look for an 
 absolute balance of errors in these latter, but must regard 
 them as equally subject to errors of observation with the 
 numbers at individual asfes. 
 
 Assuming it to be practicable to draw a sufficiently 
 smooth curve, free from sudden changes of curvature, and 
 yet representing the observations sufficiently closely with a 
 due regard to their weight in different parts of the table, 
 there would appear to be nothing to object to in the principle 
 of the graphic method of graduation. In practice, however, 
 there are certain difficulties. The first, particularly in the 
 case of a mortality table, is the question of scale. Anyone 
 who has attempted to make graphic graduations will, I think, 
 have met with this practical difficulty. Whether we graduate 
 sepai'ately the " Exposed to Risk " and " Died ", or whether
 
 25 
 
 we graduate a function such as q^., the difficulty equally 
 arises. The values of q^ niay range in practice from about 
 •005 to, say, about '5, and at the older ages increase so 
 rapidly that the eye does not readily grasp the nature of the 
 curve. In order that it may do so, and that the curve may 
 be drawn and read off with sufficient accui-acy, a certain 
 proportion must be maintained between the horizontal and 
 the perpendicular scale, so that the curve shall not cut the 
 ordinates at too acute an angle. It is also necessary to 
 represent the values of q^ in two or three sections, as the 
 scale suitable to the older ages will not permit of the values 
 at the younger ages being represented with sufficient 
 accuracy. 
 
 Instead of operating on the rates of mortality, we may 
 with advantage employ the logarithms of the rates, or the 
 loo-arithms of the central death rates.* We thus obtain a 
 curve which is much more easily dealt Avith. From the fact 
 that the rates of mortality change slowly at the younger 
 ages, and at the older ages generally approximate to a 
 geometrical progression, the logarithms of the rates are 
 nearly in the form of an arithmetical progression, and are 
 represented by a line having very little curvature. At the 
 oldest ages, indeed, it may very conveniently be taken as a 
 .straight line. 
 
 Perhaps the main difficulty in graphic graduation is that 
 it is by no means easy, even with mechanical aids, to draw a 
 sufficiently smooth curve. The curve as drawn may appear 
 to be smooth, but on reading it off and examining the series 
 of values obtained, we find irregularities which, in order to 
 produce a satisfactory graduation, must be removed by a 
 further adjustment. If we are dealing with a relatively 
 small experience — in which cases these practical difficulties 
 are correspondingly increased — they may be overcome to a 
 large extent by using as a base line a well-graduated standard 
 table representing an experience of similar character. Jiy 
 computing the " expected " deaths according to the standard 
 table, and dealing with the ratio of the actual to the 
 "expected" deaths in successive age groups, we avoid the 
 difficulties due to inequality of scale and to the rai)id increase 
 in the value of the ordinates at the extreme ages. The curve 
 
 ^ See, however, Note B, \). I 1 1, an to precautions in dealing with lugs of rates 
 of moiialitv and similar fnnctions.
 
 26 
 
 of ratios, apart from accidental fluctuations, will often be 
 found to approximate to a straight line, tlie departures from 
 which can be, of course, represented on a relatively large 
 scale. In particular, the difficulty arising from the paucity 
 of observations at either end of the table will be avoided by 
 making each extremity of the curve of ratios terminate in 
 a straight line, the locus of which will depend upon the 
 general trend of the curve in the neighbourhood. The 
 resulting values at the extremes of the table obtained in 
 this way will be more trustworthy than those obtained 
 without the aid of the standard base line."'^ 
 
 In finite difference or interpolation methods of graduation 
 (of which we may take Woolhouse's as the best known type) 
 the underlying assumption is virtually the same as in the 
 graphic method, viz., that the curve is of such a nature that 
 the ordinary methods of interpolation can be applied. Put 
 more precisely, Woolhouse's method assumes that for a range 
 of 15 consecutive ages the values of Z.,. can be represented 
 with sufficient accuracy by a curve of the third order, i.e., 
 lj.+fz=lj.+ at-\-ht'^ + ct^ ^\\\e\\ t is not numerically > 7. As this 
 assumes the fourth and higher differences of Ij, to be zero, we 
 may write 
 
 + 7(Z.._3 + U3)+21(Z.,_2 + 7x+2) + 24(Z,_, + Z^+,)+25Z,} 
 
 where I'x may be taken as the graduated value of that 
 function, the quantities on the right-hand side of the equation 
 being the ungraduated values. 
 
 This fornnila, which is that used by Woolhouse in the 
 graduation of the H^^ Table, is of course only one of 
 numerou.s possible formulas deducible from the above expression 
 for Ix+t- Others may be found resulting in a smoother 
 graduated series, but all the formulas since proposed as 
 improvements on his are based upon the same general 
 principle. An indefinite number of such formulse can be 
 found, even when the range is fixed. t In particular may be 
 
 * See Lidstone, J.I.A., xxx, p. 212. Tliese remarks are equally applicable 
 to graduation l)y a finite difference formula {see J I. A., vol. xli, p. 89). 
 -]- See Todlmnter, J.I.A., xxxii, 378 ; G. F. Hardy, J. I. A., xxxii, 371.
 
 27 
 
 mentioned Mr. J. A. Higliara's, Dr. Karup's, and that used 
 by Mr. J. Spencer in the graduation of the " Manchester 
 Unity " mortality experience. See the following table 
 showing the value of ^I'x in terms of the ungraduated w's : — 
 
 Table V. 
 
 Showing the values of^t, where u'o=1t\utX<f>t, hij various well- 
 knoion Graduation Formidce. 
 
 Distance 
 
 
 
 
 
 from Central 
 Term 
 
 Spencer 
 2 1 -term 
 
 Karup 
 
 Higham 
 
 Woolhouse 
 
 t 
 
 Formula 
 
 
 
 
 
 
 •172 
 
 •200 
 
 •200 
 
 ■200 
 
 ± 1 
 
 •163 
 
 •182 
 
 •192 
 
 •192 
 
 ± 2 
 
 •135 
 
 •139 
 
 •144 
 
 •168 
 
 ± 3 
 
 •095 
 
 •085 
 
 •080 
 
 •056 
 
 ± 4 
 
 •052 
 
 ■034 
 
 •024 
 
 ■024 
 
 ± 5 
 
 •017 
 
 ■000 
 
 ■000 
 
 •000 
 
 ± 6 
 
 -005 
 
 -■013 
 
 -■OK) 
 
 -•016 
 
 ± 7 
 
 -■015 
 
 -■014, 
 
 -■016 
 
 -•024 
 
 ± 8 
 
 -•015 
 
 -■010 
 
 -•008 
 
 •000 
 
 ± 9 
 
 -•009 
 
 -■003 
 
 ■000 
 
 
 ±10 
 
 -•003 
 
 ■000 
 
 
 
 ±11 
 
 •000 
 
 
 
 
 It is clear that no such formula will entirely remove 
 the irregularities in the series, and in AVoolhouse's graduation 
 of the H'^^ Table the outstanding irregularities were removed 
 by an empirical process similar to that employed for the 
 graduation of the 17 Offices' Table, and described in his 
 paper {J.I. A., vol. xii, p. 140-1). The object aimed at in 
 a formula such as these, should be so to select the coefficients 
 of the terms on the riglit hand that, while giving an 
 expression for the value of the central fmiction correct as far 
 as the order of differences employed, the formula will 
 produce the maximum smoothness in the flow of the 
 graduated values. This may l)e done by simple experiment, 
 or we may adopt some empirical measure or standard of 
 smoothness and thereby compute the most advantageous 
 coefficients. We may, for example, adopt as our standard 
 of smoothness the extent to which the second dilferences 
 of our graduated function are affected by the errors of 
 observation in the original table. 
 
 Applying this standard to Woolhouse's fornnil:!, we have
 
 28 
 
 for the graduated second central difference of l^. (using 
 central differences for the sake of symmetry) — 
 
 + 10ia,.+3+ fca,-+4+ 'a-+5 + 'U.-+6 + 4/^+7 — o/j.^g. 
 
 If Ave assume that on the average each of the ungraduated 
 values of l^ on the right-hand side of this equation is subject 
 to a mean error of + e, and if we assume that these errors 
 may be combined according to the normal law, then the mean 
 error of the entire expression for A^Z'j._i will be found by 
 multiplying e by the square root of the sum of the squares of 
 the coefficients, giving 
 
 (^/32_|.42 4.12_|.12_^2=^+ &C.) g^ v/5l0 g^.JO 
 
 125 T25~ ^ ^" 
 
 In the same way it may be shoAvn that in Karup's formula 
 the mean error in A^«x-i is about "OGSe, Avhere e is the 
 mean error of a single value of Uj. . It must not be supposed 
 from these results that the mean errors in the graduated 
 values of Ij; or ^tj. are proportionately reduced. The mean 
 errors in the graduated functions when Woolhouse's formula 
 is employed are reduced to about '42 of the mean errors in 
 the ungraduated functions, or are about equivalent to the 
 mean errors of the ungraduated values corresponding to an 
 experience 5 4 times larger. The graduated table based on 
 the smaller data would, however, be smoother than the 
 ungraduated table based upon the larger data. (See J.I.A., 
 xxxii, pp. 376-7.) 
 
 Taking a generalized formula, such as 
 
 u 
 
 'a;=««.f + ^(«.f-l + ".r + l) + c(Ku,-_2 + 'i'u- + 2)+&C. . . T^iUx-t + njc + t) 
 
 where u' x represents the graduated value of xij,, and 
 assuming that each of the ungraduated values Uj;, &c., 
 are affected by the same mean error + e, it is of course 
 possible to determine the values of a, h, c, &c., so that the 
 mean error in, say, ^^u' x-\ shall be a minimum. Noting that 
 a=l— 26 — 2c — &c., and that h-\--ic + 9rf-i-&c. = 0, in order that 
 the foi'mnla may be correct to 3rd differences, an expression
 
 29 
 
 may be found for A-it'^-i in terms of %._#_,, Ux-t, &c., with 
 coefficients involving c, d, . . . k. If the coefficients of each 
 term are now equated to zero, there will be (2^ + 3) equations 
 of condition with (^ — 1) unknowns, which may be solved 
 by the usual method of least squares. 
 
 This is somewhat theoretical, however, as the values we 
 should obtain for the coefficients would be generally 
 fractional, and the resulting graduation foi*mula would not 
 lend itself to any continuous method of computation, as is 
 the case with Woolhouse's and other similar formulae. A.n 
 alternative would be to fix upon a convenient set of 
 summations, and then to determine the function summed 
 (called by Mr, Lidstone the " operand") so that (1) first and 
 second differences may vanish — see J.I. A., xxxii, 371, &c. ; 
 (2) The range of the formula may be what we require ; 
 and (3) that subject to (1) and (2) the coefficients shall be 
 such as to make the mean error in A- or A^ a minimum. 
 This might give a fairly convenient working formula, as 
 when once the operand was formed the ordinary convenient 
 method of summation would apply. 
 
 If we consider the effect of such a formula of graduation 
 upon the outstanding or unbalanced errors of observation in 
 a small group of ages, we shall see that they are not very 
 materially diminished. If, for example, we express the sum 
 of five consecutive graduated values in terms of the 
 ungraduated values, we shall have, in the case of Woolhouse's 
 formula, 
 
 + 115/^+101Z^+, + 80^,.+2) 
 
 + terms involving other values of /. 
 
 Here it is obvious that any systematic or unbalanced error in 
 the original group will not be greatly reduced (probably 
 to about three-fourths of its amount) in the graduated table. 
 While, therefore, finite difference formuho of graduation 
 yield, generally, a smooth curve as regards the pi'ogression of 
 the graduated values from age to age, they have a tendency 
 to reproduce any waviness in the original, due* to the 
 unbalanced errors affecting small groujis of four or five 
 consecutive ages.
 
 30 
 
 A question arises in connection Avitli this method as to 
 what particular function should be selected for graduation. 
 In the case of Woolhouse's original formula the function 
 operated upon was l^ . Practically speaking, except for the 
 latter portion of the table, this approximates in result to a 
 graduation of the rates of mortality. This may be seen from 
 the following relations. Any adjustment of the Ix column by 
 a finite difference formula has, of course, the same effect as 
 a similar graduation of the cZ.,. column. Since dj.:=lj.qj., and 
 since for the range of ages included in the formula (fifteen 
 in Woolhouse's formula, of which, however, only the five 
 central ages are heavily weighted] the values of l^ are not 
 in general widely different, the graduation of the Ij. or dj; 
 column should give results not materially different from those 
 obtained by graduating q^.. At the older ages, however, 
 there may be significant differences in the results, and I must 
 express my preference for the rate of moi-tality as the more 
 suitable function to graduate if the observations are duly 
 weighted or if proper precautions are taken to avoid 
 anomalous results at either end of the table where data are 
 scanty. 
 
 An objection to the principle of the finite diff'erence 
 methods of graduation is that the weight of the observations 
 is not allowed for at various ages. This objection is not very 
 serious, howevei', as at the commencement and end of the 
 table, where it would be chiefly felt, the method is usually not 
 strictly applied. It may be noted that if the l^ function be 
 graduated, then its rapid decrease in value at the oldest ages 
 in the table gives automatically a diminishing weight to the 
 observations with increasing- age, but at the same time yields 
 somewhat irregular graduated values. The objection may, of 
 course, be got rid of by first applying a smooth series of 
 weights to the function to be graduated, prior to graduation, 
 .and eliminating these factors afterwards. 
 
 A difficulty arises in the use of finite difference formulae 
 from the smallness of the data at the extremes of the table 
 and from the fact that the first 7 or 8 values of the 
 graduated function cannot be obtained from the formula. In 
 the case of a mortality table there is not so much difficulty 
 in dealing with extreme old age, because there, as 
 Woolhouse points out, if we are dealing with the function Ij. it 
 may be taken =0 beyond the limiting age of the table, or if
 
 81 
 
 we are graduating tlie rate of mortality^ ^j. may be put down 
 as equal to unity. As regards the earlier ages, Woolhouse's 
 method is to obtain from the formula the graduated values 
 of Ij; so far as this can be done, that is, to within 7 years of 
 the initial age, and to compute the values for the first seven 
 ages of the table from the values of /q, /;, I3 and Zg 
 {If, representing the value of Z^ for the initial age) on the 
 assumption of a constant third difference. This method may 
 in certain cases lead to anomalous results, even negative 
 rates of mortality. Mr. Ackland has given an alternative 
 method of considerable ingenuity (/.J.^.,vol.xxiii,p. 357). The 
 diflBculty may be avoided by assuming values for the initial 
 ages, as, for example, a constant average value of q^. or d^, or 
 other arbitrary values deducible from the general character 
 of the experience. A more satisfactory method would be to 
 determine q^ for the first 10 or 15 ages, by the method of 
 moments or least squares, on the assumption that it could 
 he represented by a first or second difference function. All 
 these methods, however, are expedients more or less 
 empirical, though they may in practice lead to sufficiently 
 satisfactory results. 
 
 The Finite Difference methods of graduation all 
 assume that the functions to be graduated may be repre- 
 sented for successive small tracts of ages b}" a parabolic curve 
 of the form — 
 
 u_f = a + bx + cx^+ &c. 
 
 We are not bound to assume this particular form of 
 function. We can employ the jorinciple of the Interpolation 
 method, representing our function by some other form, 
 as, for example, mj; = a + hc^ corresponding to Makeham's 
 formula. 
 
 The principle of the methods of graduation we have been 
 discussing, of whicli Woolhouse's is a typo, must not be 
 confounded with that used by Davies in graduating the 
 Equitable experience, nor with that used by Mr. Berridge 
 in graduating the Peerage mortality. These latter are more 
 nearly allied to graduation l)y frequency curves than to 
 Woolhouse's method. In Davies' Equitable graduation, 
 curves of the third order are actually fitted to successive 
 sections of tlic I,r fohuiin, tlie values of Z^ from 10 to 40 being 
 virtually found by a third difference interpolation from tlio
 
 32 
 
 values lio, ho, I30, ho, those from 1^^ to Z70 similarly from the 
 values of Uo, ho, ho, ho, ^^^^ so on. Mr. Berridge's graduation 
 of the Peerage mortality followed a similar principle, except 
 that he represented the entire series of values of log Ix from 
 15 to 75 by means of a single curve of the sixth order, based 
 upon the values of that function for decennial intervals of age. 
 As to the relative merits of graphic and finite difference 
 methods of graduation, the former has an undoubted advantage 
 when the number of facts at our disposal are few. In these 
 cases formulae of the type of Woolhouse's cannot be expected 
 to produce very satisfactory results, as in the comparatively 
 small section of the curve embraced by the formula the true 
 character of the curve will frequently be obscured by the 
 errors of observation. These formulas are at their best when 
 applied to a table based upon fairly extensive data, and 
 presenting a curve without any rapid change of character. 
 The advantages possessed by the graphic method in dealing 
 with a small experience, owing to its flexibility and its power 
 of bringing under contribution large sections of the curve at 
 once, are, however, still more noticeable when frequency 
 curves can be suitably employed. 
 
 We have already spoken of the success or sufficiency of a 
 graduation, but we have not said anything as to Avhat is the 
 proper test of a successful graduation. Before dealing with 
 the general principle of graduation by means of frequency 
 curves, it will be useful to consider this question. There 
 are obviously two conditions that should be fulfilled by a 
 graduation. In the first place, a smooth and continuous 
 progression in the graduated values. This is required because 
 we have good reason for believing that if the true values were 
 ascertainable, they would exhibit this property. In the 
 second place we require an adherence to the original data, 
 sufficiently close to be fairly within what we may conveniently 
 term the errors of observation. 
 
 The standard of smoothness is not easy to define. If a 
 formula is adopted representing the ultimate values of 
 ^x, 9x, or fi-c as a function of the age, this in itself secures 
 a smooth series. In other cases the sufficiency or otherwise 
 of the graduation in this respect must be left to individual 
 judgment. The advantages of a really smooth curve are
 
 33 
 
 mainly found where it is necessary to resort to interpolation 
 or to the use of summation formulse ; and, further, in the 
 practical consideration that with a really smooth curve nearly 
 all tables calculated therefrom can be sufficiently checked by 
 differencing. 
 
 As regards the second requirement, that of adherence to 
 the general features of the ungraduated experience, it is 
 easier to set up a criterion. We have already seen that if 
 the true value of the probability of an event happening at a 
 single trial is p, the event will, on the average, happen np 
 times in n trials, and if there are series of Hi, 112, na, &c., 
 trials in which the probabilities of the respective events are 
 Pi, T'l, "P-i, &c., then on the average the total number of 
 occurrences in such a series of trials will be iiipx + ')i2p2 + 
 ''^3i?3 + , &c. That is to say, if the observed occurrences 
 are 6^, Oo, d-s, &c., then the average value of each term 
 (^Q^—mpi), [6.2— n^p^, &c., and consequently of the sum of 
 such terms, Avill be zero.* It is also obvious that the average 
 value of the sum of the series (^1— 711^1) + 2 (^2— '^2^2) + 
 ^{Bz—nzp'^+, &c., and generally of the series whose rth 
 
 term is 
 
 \r 
 
 ---- {dr — nrPr) 
 
 \t\t 
 
 will be zero. In the case of a mortality experience these 
 quantities {Oi—nipx), &c., represent the deviations of the 
 observed deaths at each age from the " Expected Deaths ", 
 as computed by the true rates of mortality, supposing these 
 to be known. It follows, therefore, that we should expect 
 the total of such deviations on the average to be zero, and 
 in the same way the average value of the successive sums 
 of the accumulated deviations should be zero. Generally, 
 if we put 
 
 S?i = ?lo + ?ll + ??2 + «3 + , &C. 
 
 Sin = t^n = ni + 2n2 + 3?i3 + , &c. 
 %lSn = i^n = 72^ + 37Z3 + 6n4 + , &c. ; 
 we shall have on the average 
 
 t'{dr-n,.pr) = 0. 
 
 • This is not tlio moMi prohahlp. valiu! of tlicso terms, altliongli in general 
 it will bo very close thereto. The Actuarj', however, requirca to consider 
 the averarfe result, not the most probable.
 
 .34 
 
 We should not expect (assuming the true values of p^ to 
 be known) that these sums of the deviations of the actual 
 from the expected numbers would actually be equal to zero 
 in any given case, but we should expect in a long series of 
 cases that the positive values would approximately balance 
 the negative. We do not expect to obtain exactly 1,000 
 heads in a series of 2,000 tossings of a coin, but we should 
 expect to find that the average number of heads over a great 
 number of such series of tossings would be very close to that 
 figure. This reasoning leads us to the conclusion that, 
 given a successful graduation, we should not only have 
 obtained a smooth series, but that the sum of the deviations 
 between the computed events (deaths or otherwise) and 
 the observed numbers, would be nearly zero, and that 
 the successive sums of the accumulated deviations would 
 be small. 
 
 It is not neicessary in practice that this test should be 
 pushed too far. We may be satisfied if the sum of the 
 deviations and the sum of the accumulated deviations are 
 practically zero ; if the total deviations in successive sections 
 of the table {e.g., in quinquennial or decennial groups) appear 
 to be, on the whole, within the limits of the errors of 
 observation ; and if the total of the accumulated deviations 
 changes sign fairly frequently. On the other hand we should 
 expect that the total deviations irrespective of sign should 
 not be materially less than their theoretical amount. 
 Otherwise we should conclude that the series was under- 
 adjusted and that accidental fluctuations in the curve had 
 been incorporated as inherent characteristics. 
 
 These tests of a graduation are Avell known to Actuaries, 
 and, indeed, have been very generally employed by them. 
 So far as they go, they correspond to the method of moments 
 which Prof. Karl Pearson has elaborated and employed with 
 such success in the fitting of frequency curves to statistical 
 data. It is clear, however, that they can only be employed 
 systematically in conjunction with those or other curves 
 capable of analytical expression. Using methods of gradua- 
 tion, based upon Finite Difference formulas, such as 
 Woolhouse's, we cannot secure that the successive sums of 
 the deviations shall vanish, though in general we may expect 
 them to be small. Using the graphic method, we can, by a 
 gradual process of hand-polishing the curve, reduce the
 
 35 
 
 accumulated deviations and their sum to as small a value as 
 we please,^ but the process is a tedious one. 
 
 A second test that has occasionally been applied when the 
 graduation has been effected by means of a fox-mula, is that of 
 making the sums of the squares of the deviations a minimum, 
 the deviations being either in respect of the graduated and 
 observed deaths at each age or those of the graduated and 
 ungraduated values of some function such as 4 or log. 4, 
 This method, known as the method of "Least Squares", is 
 used very generally in connection with measurements in 
 astronomy and other physical sciences and has given rise to 
 a quite extensive literature. It is based upon the assumption 
 that if in a given series of observations the relative frequency 
 of an error x at each observation is represented by the 
 function ke~'^'''^' , then the probability of a conjunction of any 
 
 * It may, perhaps, be worth poiutiiig out that if we have obtained a 
 smooth cui-ve with a general conformity to the original facts, but not making the 
 2 (deviations) or 5- (deviations) vanish, this may be done by the following plan. 
 Assume, for the sake of illustration, that the function graduated is the central 
 death rate mx . Representing by ntx the graduated values of that function by 
 El the "Exposed to Risk" in the middle of the year of age and by ^* the 
 .observed deaths, let 
 
 2(«»^E^-ex) = A 
 
 'S.%mxEx-ex) = B 
 
 then, if m'x = a + (1 + h)mx be the modified rates required, 
 
 «.2(Ex) + i2(E^W;c)=-A 
 
 a.X-CEx) + br-(Exnh)=-'B 
 
 ■whence a and b are detennined. 
 
 If the table on the whole follows Makeham's law the use of this form of 
 correction enables us to neglect all orders of diiferences in the preliminaiy 
 adjustment of nix or ^^. Fornmlfe may thus be employed (as for example, a 
 simple double summation in groups of 10 values, or, still better, successive 
 summations in lO's, ij's and 2's) giving a nuich smoother curve than when 
 account has to be taken of second differences, the resulting systematic error of 
 this first graduation being corrected as above. 
 
 In the alternative, if m'j: = mx + a + bx, 
 
 a2(E^) + 62x(E,)=-A 
 
 b-S.-(E^) + br-Jc{Ex)=-li. 
 
 This method may bo employed in conjunction willi .Mr. Lidslonu's plan oi using 
 JO, standard lahle as a baa3 liiio for jjurposes of graduation. 
 
 D 2
 
 36 
 
 set of errors Xi, a'2, X3, &c., will be proportional to the value 
 of tlie product 
 
 = e~\ c" / 
 
 which clearly has a maximum value when the index of e is 
 numerically a minimum, i.e., when the sum of the squares 
 of the errors {xl + xl + xl + &c.) is the least possible. This 
 expression assumes that the average error, and therefore the 
 probability of a unit error, in each observation is the same, 
 an assumption Avhich may often be fairly made in respect to 
 independent measurements of a physical quantity. If the 
 observations are not of the same weight, so that the 
 pi'obability of the errors of Xi, x-i, Xi, &c., in the respective 
 measures are 
 
 then the most probable solution will evidently be that which 
 makes the sum of these exponents the least possible.* 
 
 The assumptions upon which this method is based are not 
 strictly in accord with the conditions of a mortality experience 
 or similar statistical observation. If the method is applied to 
 the deviations between the observed and graduated deaths, 
 the objection may be raised that the observations at different 
 ages are not of equal weight, and that the probability of a 
 unit error varies at each successive age, while in each case 
 the probability of a given error can only be approximately 
 expressed by the normal function A;e~*"*'', positive and 
 negative errors not being equally probable. It is, of course, 
 possible suitably to weight the observations, so that a 
 unit error is made equally probable. For example, if at 
 any given age there are n "exposures", and if the true 
 probability of death is q, then the " standard deviation " or 
 ^average square deviation =nq{l — q), and the probability 
 of a difference of x between the expected and observed 
 deaths is approximately Ke~'''^'^'^^^~'^^ ; the error in the formula 
 when X is positive nearly compensating the error when x is 
 negative. Hence, if the " Exposed to Risk " and " Died " at 
 each age are multiplied by the factor [nq{l — q)^~^^, where q 
 
 * See Note C, p. 117.
 
 37 
 
 is to he taken at its true or graduated value,'^ then the 
 observations may be considered to be properly weighted for 
 the application of the method of least squares. 
 
 We shall see in the following lectures that there is an 
 intimate relation between the criteria of least squares and 
 moments. This will be better discussed after considering the 
 question of frequency curves and the process of fitting them 
 to a set of statistical observations. 
 
 * The imgraduated values of q cannot be used, as this would result in undue 
 ■weight being given at all ages where the observed mortality was in excess of the 
 average, and insufficient weight where it was in defect. Consequently, the 
 mortality table resulting from this process would on the whole overestimate the 
 mortality throughout. Li other words, the use of the unadjusted values of q 
 introduces a systematic or "biassed" error into the calculations. If this is 
 avoided, however, a verj- rough approximation to the graduated cun-e of q wiU 
 give weights sufficiently near the truth for practical purposes, as a slight change 
 in the relative weights of a given series of obsei-vations produces but little result 
 upon the final solution.
 
 THIRD LECTURE. 
 
 JL PROPOSE in tlie present lecture to consider generally the 
 use of frequency curves in relation to actuarial statistics. 
 We have seen that the graphic method of dealing with these 
 statistics, as also methods based upon finite difference formulae, 
 assume only that the true law of the series, if known, would 
 be found to be represented by a continuous curve amenable to 
 the ordinary processes of interpolation. It is often possible, 
 however, to see that the ungraduated series can be well 
 represented by a curve of a certain distinct character, and 
 when this is found to be the case more satisfactory results are 
 obtained, particularly where the data are few, by fitting to the 
 original series a curve corresponding to its observed general 
 character, so determining the constants in the equation of the 
 curve as to secure the closest agreement with the ungraduated 
 curve. If for example we turn to the series in column (2) of 
 Table I, it will be at once seen that the general character of the 
 series accords very closely to the " normal " frequency curve, 
 or to some curve having the same general features. When 
 we find that, by giving suitable values to the constants, a 
 frequency curve can be made to fit the observations within 
 the limits of the errors of observation we may be satisfied that 
 the graduated curve thus j^roduced is probably a better 
 representation of the original than any that Avould result from 
 a graphic or finite diiference method of graduation. 
 
 Any curve which exhibits the law of variation in a 
 particular function, svich as a table of l^, d^ or /x^.., may be 
 considered for our purpose as a frequency curve. The 
 expression is usually, however, confined to that class of curves 
 which experience seems to show to be specially applicable to 
 the observed distributions of deviations from mean values in 
 statistical tables. We have already seen examples of such
 
 39 
 
 tables where the frequency of the deviations of measures from 
 their mean value follows certain comparatively simple laws. 
 Professor Karl Pearson has examined a considerable variety 
 of statistical data (mainl}^ but not entirely, biological) and 
 finds that in practically all the cases examined the distribution 
 of the various measurements may be represented fairly closely 
 by one or other of the class of curves derived from the 
 differential equation 
 
 l,dy^_Jx-^ H) 
 
 y dx a — hx — cx^ • • \ J 
 
 where x represents the magnitude of a given deviation 
 from the mean of a series of measures and y the frequency 
 of such deviation. 
 
 As this group of curves is of considerable importance, 
 though less so perhaps in relation to actuarial than in relation 
 to some other classes of statistics, it is convenient to consider 
 them first. It is not necessary here to discuss these 
 curves analytically ; the student may be referred to the 
 original papers of Professor Karl Pearson"^, or to all 
 admirably condensed resume by Mr. Robert Henderson in 
 the Journal of the Actuarial Society of America, reprinted 
 J.I.A., xli, 429-442; and to Mr. W. Palin Elderton's treatise on 
 " Frequency Curves and Correlation " in which Professor 
 Pearson's methods are fully described. 'J"'he table at the end 
 of these lectures, which gives a sufficiently complete summary 
 of such of the algebraical properties of these curves as 
 are mo.st useful in practice, is, with some unimportant 
 modifications, based upon that given by Mr. Henderson in 
 his paper. It will be sufficient for our present purpose- 
 to give a brief general description of these curves and of 
 their use in connection with actuarial data. 
 
 We have already seen that the general character of curves, 
 such as those of Taljles I and II, is approximately detennined 
 by the average value of the squares and cubes of t}\& 
 deviations of the variable from its mean value; the former- 
 giving a measure of the compactness or diffuseness of the 
 curve that is of the average extent of the deviations from the 
 mean irrespective of their direction ; the latter a measure of 
 their departure from symmetry, or of the " skewness ", of the 
 curve. It will be useful at this point somewhat to extend 
 
 * PhU. Tram<, vol. ISd, p. 343 ; :yoU197, p. 413, &c.
 
 area= vdx. 
 
 40 
 
 this general statement, and, before proceeding to a description 
 of particular curves, to explain more in detail what is 
 meant by the " moments " of a curve. 
 
 If we suppose y =f(x) to represent the equation to a 
 given curve, x varying between the limits h and k, the 
 total area of the curve will be represented by the 
 expression : 
 
 .= ydi 
 
 Jk 
 
 We may suppose, for instance, to give definiteness to our 
 ideas, that the function y represents the numbers under 
 observation between age x and x + dx, the number of "years 
 of life" observed between these ages being ydx, and the area 
 of the curve, the sum of all these quantities, being the total 
 years of life observed at all ages. If we now multiply 
 each value of ydx by the corresponding age x and divide the 
 total of these products by the total number of the " exposed ", 
 we shall have the average age of the Avhole. Put into 
 symbols : 
 
 rh rk 
 
 xydx-^ ydx = a:Yera,ge value of a;. . . . (2) 
 
 Jk J/i 
 
 = lst moment of the curve round 
 the ordinate for which x = 0. 
 
 = mj, say. 
 Similarly, 
 
 rh rh 
 
 I x^''y.dx-^\ y.dx = average value of a;'* 
 
 = 7ith moment round ordinate for 
 which a; = 0. 
 
 = m„. 
 
 The moments of the curve may be taken round any 
 
 ordinate we please. If, for example, the average value 
 
 of X as found by equation (2), is Xi, then the ordinate 
 
 corresponding to this value of x passes through the centre 
 
 of gravity of the curve, and is termed the "centroid vertical." 
 
 In general it is most convenient to take the value of the 
 
 moments of the curve round this centroid vertical, for Avhich 
 
 obviously the first moment vanishes. The expression for the 
 
 nth. moment round this ordinate then becomes : 
 
 rh rh 
 
 {x—xi^ydx^^ ydx = fjLn . . • • • (3)
 
 41 
 
 the average value of the nth. power of the deviations (^x — Xi) 
 between the values of x and the mean value. When the 
 moments of a curve are spoken of without qualification, 
 it will be understood that they are the moments round 
 the " centroid vertical." These moments are, of course, 
 those already referred to in Lecture I., p. 7, as representing 
 the sums of the powers of the deviations of x from its 
 mean value. 
 
 The following formulae, which maybe readily demonstrated,^ 
 connect the values of the moments round the " centroid 
 vertical" with the moments round the ordinate for which 
 x = 0. Using the same notation as above, we have 
 
 IM3 = ma— Smima -f 2 (mj)^ 
 
 yLt4 = m4 — 4mim3+6(mi)2m2— 3(mi)^ 
 
 . (4) 
 
 where the law of the coefficients is sufficiently obvious. 
 
 For the particular family of curves arising from the 
 differential equation (1) formulaj may readily be found 
 for the moments involving the various constants of the 
 curves, and inversely, the values of the constants can be 
 expressed in terms of the moments. The formulae for 
 the higher moments being sometimes complicated, it 
 is more convenient to tabulate certain functions of the 
 moments, e.(j. : 
 
 /3i 
 
 _W 
 
 /^4 
 
 if^r ^'- i^r' 
 
 7= 
 
 ye, + 4 
 /32 + 3 
 
 from whicli the constants of the curves may be obtained more 
 readily, wliicli are also useful in discriminating between the 
 curves apj)licable to a given set of observations. 
 
 See Eldertoii, ]>. 17-1'J ; Hciidc rsdii, J.I.A., xli, ■K$l-2.
 
 42 
 
 The various curves arising from the differential equatixjn 
 (1) may, for our present purpose, be conveniently classified 
 as under : — 
 
 Class I. Symmetrical curves. Range limited. 
 „ II. „ „ „ unlimited. 
 
 „ III. Skew curves. Range limited in both 
 
 directions ; 
 „ IV. Skew curves. Range limited in one 
 
 direction ; 
 „ V. Skew curves. Range unlimited in either 
 direction ; 
 the various types of curve being as follow. It will be seen 
 that some of these Classes are repesented only by a single 
 type of curve : 
 
 Class I. Symmetrical curves of limited range. — In this 
 class we have only the single curve. 
 
 2\m 
 
 Type I. ^ y = «(l-y 
 
 The values of x range from +a to —a, for either of 
 which values of the variable y becomes zero. 
 
 The average value of x is obviously zero, the corresponding* 
 ordinate y is a maximum, and clearly bisects the area enclosed 
 between the curve and the axis of x. In other words, the 
 "mean", "mode", and "median " of the curve all coincide, 
 as in all symmetrical curves. 
 
 The second moment of the curve 
 
 _ _ a^ 
 
 2m + 3 
 
 and the "standard deviation" 
 
 a 
 
 y2m + 3 
 The fourth moment 
 
 oa 
 
 Im + b 
 
 The value of w will usually be positive when y equals zero at 
 both limits. If m>0<l the curve cuts the base-line at an 
 angle. If m is negative the value of y becomes infinite at 
 both limits, and m is always > — 1. - 
 
 This curve has a close relationship with the symmetrical 
 point binomial curve, whose terms are proportional to the
 
 43 
 
 terms in the expansion of (4 + ^)", the general term of which 
 
 may be written ^ 
 
 y= ■. 
 
 ^ n \n 
 
 [It will, of course, be understood that the ac's in these 
 formulse, and in others, are not identical, but 
 simply stand for some constant in each case, the 
 numerical value of which is determined by the 
 area of the curve.] 
 The binomial curve, however, can be conveniently used 
 only to represent the definite points corresponding to integral 
 
 values of - +x, whereas Type 1 represents a continuous 
 
 curve (Note D, p. 122), The data with which an actuary has 
 to deal are generally in the latter form, for example, the 
 numbers living, the number of deaths, withdrawals, &c., 
 between the ages x and x+l, and although usually the 
 number of terms in the series is so considerable that the 
 curve may be treated as a series of points, on the other hand, 
 a binomial having so many terms will not generally be found 
 a suitable curve to employ. In most instances where a series 
 can be fairly represented by the symmetrical binomial, it can 
 also be fairly represented by Type 1, with possibly some 
 slight difference in range, as will be seen later. 
 
 There are other symmetrical curves of limited range, 
 which are in the nature of frequency curves, but which do 
 not belong to the family of curves derived from equation 
 (1) : such, e.g., as the curve 
 
 whicli, however, we need not discuss here. 
 
 C/a.V6' //. Symmefrical curves of unlimited range. — In this 
 class are two curves lielonging to the family with which we 
 are dealing. 
 
 Tyj>e 2. 7y = /ce-'^=/'^= (5) 
 
 This is the curve of "facility of error", or the " normal " 
 frequency curve. 
 
 The average value of x is clearlj' zero, corresponding to 
 the "mode" or the maximum value of y, and to the median. 
 
 (,•2 
 
 The second moment =^=— , and the standard 
 
 deviation = —7 . 
 
 v/2
 
 44 
 
 ■ Type 1 evidently transposes into this curve when the 
 vahie of a, and hence the range of the curve, is made 
 
 a^ 
 
 indefinitely great. If we put - = c^, making both a^ and m 
 indefinitely great, but their ratio finite, we have 
 
 , ^2v TO 
 
 Limit (l-y =e-^l^' (6) 
 
 Even when the range of the curve is not great, that is 
 when TO and a^ are not large numbers, there is a fairly close 
 agreement between curves of Types 1 and 2 and the symmetrical 
 binomial. 
 
 This may be seen by a numerical example, the folloAving 
 table showing 
 
 1 . The values oi y= ' ' for integral values of 
 
 X, these values being proportionate to the terms in 
 
 /I 1\^ 
 the expansion of the binomial ( 9 + o ) • 
 
 2. The values of y = ^Q^(l-^^^\ 
 
 8. The values of y = l,026e-^=/-., 
 
 the constants in the two latter curves being chosen to give 
 as good general agreement as practicable with the binomial 
 curve. 
 
 Table VI. 
 
 Shoioing Similarity of Types 1 and 2 to the Symmetrical Point 
 
 Binomial. 
 
 Values of 
 
 Binomial curve 
 
 Type 1 
 
 Type 2 
 
 Variable 
 
 36000 
 
 y-993(l-|-J 
 
 (3) 
 
 
 X 
 
 V= 1 
 
 ^ |3+a-|3-a; 
 
 .V = 1026e--^".-'-i 
 
 (1) 
 
 (••;) 
 
 (4) 
 
 -4 
 
 
 
 2 
 
 6 
 
 -3 
 
 50 
 
 47 
 
 56 
 
 -2 
 
 300 
 
 303 
 
 282 
 
 -1 
 
 750 
 
 752 
 
 743 
 
 
 
 1,000 
 
 993 
 
 1,026 
 
 1 
 
 750 
 
 752 
 
 743 
 
 2 
 
 300 
 
 303 
 
 282 
 
 3 
 
 50 
 
 47 
 
 56 
 
 4 
 
 
 
 2 
 
 3,201 
 
 6 
 
 Totals ... 
 
 3,200 
 
 3,200
 
 45 
 
 Had tlie range of the curves been gi-eater, the binomial 
 being taken to a higher power, and the values o£ the 
 constants a^ and m in col. (3) and of c- in col. (4) been 
 larger, the agreement of the three curves Avould have been 
 correspondingly closer. As it is, the two first curves are 
 very nearly identical, while the " normal " curve, although 
 theoretically of unlimited range, is fairly close to the 
 binomial, the terms corresponding to values of x numerically 
 greater than 4, amounting to less than 1 in the aggregate. 
 It will be noticed that the values of y in the limited curves 
 necessarily diminish more rapidly as the limiting values of x 
 are approached, while the normal curve is less flat in the 
 centre. 
 
 1 + ^J 0) 
 
 This curve, Avhich is also symmetrical and unlimited in 
 range, diverges from the normal curve in a direction opposite 
 to Type 1, the values of y diminishing, when x is large, more 
 slowly than in the normal curve. The curve transposes into 
 
 the latter (Type 2) when a^ and m are indefinitely large, ^ =c^ 
 
 being, however, finite. We then have 
 
 Lt. «(l + U =/ce-^=/<==. 
 
 The average value of x in the curve y = K(a- + x^) -'^ is 
 zero, corresponding again to the "mode"; the second 
 
 moment = ii., = t: ^ and the " standard deviation " 
 
 ^' 2m — 'S 
 
 a 3a'^ -, ., . 
 
 = — ^=:z= . The fourth moment =Ui= ^ u-2 and, it is 
 
 y2m-3 2m-o^ 
 
 clear, becomes infinite unless «i > o • Indeed, the higher 
 
 moments of the curve must become infinite whatever be the 
 value of m. 
 
 The classes of symmetrical curves are of somewhat limited 
 application to actuarial statistics, although there are certain 
 cases in which they represent the observations fairly well. 
 
 Class III. Skew curves. Range limited in both directions. — 
 There is only a single curve of this class in the family of 
 curves we are considering, namely : 
 
 Tyi.e4. !/ = <!- ;0 + J W
 
 46 
 
 The values of x, range from —a to +«; the "mode" is 
 for which vahie y is a maximum ; the 
 
 ^ ' " The expressions for the 
 
 a- 
 
 mo — mi 
 
 at a; = ^H '^ 
 
 nil + f^2 
 
 mean value of a; is ^ 
 
 mi + m2 + ^ 
 
 moments of the curve are simplified by putting it into the 
 form given in the table on p. 140. If we write mi = np — 1, and 
 m2 = nq — l (where 'p + q = l), the equation to the curve 
 (which does not, of course, change in character with this 
 
 transposition) becomes 
 
 ^=<-r""(inr" («) 
 
 the variable having the same range of values —a to +a, the 
 
 "mode" being at x = ^(3— p)a; the average value of 
 
 r^—i^q^__rp^(i. the second moment =1x2— ,-i '^> ^^^ ^he 
 
 " standard deviation " the square root of this quantity. 
 
 When mi = m2 = 7n, this Type evidently transposes into 
 Type 1, and thence into Type 2 when w is infinite. 
 
 This curve is related to the skew point binomial arising 
 
 from the expansion of {p + qY', where p and q have 
 
 approximately the same values as in equation (9), and 
 
 where the index of the binomial is not too small, there is 
 
 a fair numerical agreement, as may be seen in the following 
 
 table, where the figures given in col. (2) are proportional to 
 
 fl 2Y 
 the terms in the binomial expansion of ( ^ + -:,) : — 
 
 Table VII. 
 
 Showing Numerical Similarity of the Curve of Type 4 with the 
 
 Skew Sinomial. 
 
 Value of 
 Variable 
 
 X 
 
 Binomial curve 
 ,_ 5760 ,, 
 
 Type 4 
 ^ = K(4.-75 -.r)5-5i(6-25 + a:)"-5« 
 
 ^ 3+a:|3-.r 
 
 -4 
 
 -3 
 
 -2 
 
 -1 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 (a) 
 
 
 
 1 
 
 12 
 
 60 
 
 160 
 
 2i0 
 
 192 
 
 64 
 
 
 
 (3) 
 
 
 
 1 
 
 13 
 
 61 
 
 159 
 
 240 
 
 194 
 
 60 
 
 1 
 
 Totals ... 
 
 729 
 
 729
 
 -17 
 
 It will be seen that for so small a value of n as 6 the 
 binomial curve can be closely represented by means of 
 selected points in the continuous curve of Type 4. When the 
 value of n is large, a much closer agreement is obtainable. 
 
 The skew binomial is of importance to the actuary as 
 representing the law of the deviations between the actual 
 number of events observed in a given series of trials and the 
 " expected " number when comjDuted by the true value 
 of the probabilities. There are very many statistical 
 distributions capable of being well represented by the 
 binomial curve if the latter is treated as a continuous curve. 
 This procedure is not, however, convenient in practice, as it 
 rarely happens that the given ordinates coincide Avith the 
 
 integral values of x in the general term , — - — (-), and, 
 
 \x n—x\qj ' 
 
 moreover, the analysis, when the curve is treated as 
 continuous, is not very simple. {See Note D, p. 122.) 
 
 The form of curve corresponding to Type 4 varies very 
 considerably with certain changes in the values of the 
 constants iiii and vu. In its more usual form, Avhen both 
 rtii and iiu are >1, as in Table VII, the curve bears a 
 general resemblance to the age distribution of the 
 " entrants " in a mortality, or similar experience {see 
 Table II), also to the numbers of the exposed to risk; to 
 the number of marriages, or to the rate of marriage at 
 various ages ; to the average number of children under age, 
 or to the cost of their pensions at the death of the father, a 
 function of use in pension fund valuations; to the number 
 of retirements in such funds where superannuation occurs 
 on invalidity and not at a specified age ; to the incidence 
 of attacks, or of mortality, from certain diseases, &c. Owing 
 to the number of constants involved (as the increment of x 
 may represent any period of time, there are virtually five), 
 the curve is very adaptable. 
 
 It will be readily seen that if the values of both m^ and 
 ?% in equation (8) are high the curve makes very close 
 contact with the axis of x at cither limit; if Wi or ?7i2 lies 
 between and 1, the curve meets the axis of x at an angle ; 
 whereas, if either or both of them are negative, the expression 
 Ixjcomes infinite at one or both limits. The area of the curve 
 and the moments do not, however, become infinite if both ?;i, 
 and in2 are greater than —1.
 
 48 
 
 Class IV. Skew curves. Range limited in one direction. — 
 There are two curves of this class. 
 
 Tyiie 5. ;?/ = /<;a;"»e-*/» (10) 
 
 which is a limiting form of curve No. 4, the values of x 
 ranging from and oo . 
 
 The "mode" is at x = 'ma; the mean value of x is 
 (m + l)a; the second moment (m + l)a^; and the third 
 moment 2(m+l)a^; these being sufficient to determine the 
 constants. 
 
 In the usual form of the curve, that is when m>l, this 
 curve represents fairly well some of the statistical distributions 
 represented by curve No. 4. Owing to the feature that as x 
 becomes large the successive terms have a tendency to run 
 into a geometrical progression, it is not so well suited to such 
 distributions as that of the " exposed to risk ", where the effect 
 of the rapid rise in the rate of mortality at the older ages 
 makes itself felt in an increasingly rapid diminution in the 
 values of y. This is somewhat unfortunate, as the curve is a 
 simple one, determined by the values of its first three 
 moments, and except for the reason stated, well suited for use 
 in connection with Makeham's formula for the force of 
 mortality. 
 
 As in Type 4, the character of this curve may be entirely 
 chansred bv an alteration in the values of the constant m. If 
 this constant vanishes the curve becomes a dimmishmg 
 geometrical progression ; while for negative values of m the 
 curve becomes infinite at the lower limiting value of x. The 
 value of m must in any case > — 1. 
 
 The actuary has to deal with several distributions roughly 
 similar to a diminishing geometrical progression as, for 
 example, the curve of infant mortality, the rate of withdrawal 
 in successive policy years, or the difference between the select 
 and ultimate mortality rates in a select mortality table. Other 
 expressions giving a similar form of curve may be employed to 
 represent these distributions as, for example, y — K (a + 6-"'^), 
 with a minimum value of /ca when x is very large; or 
 y = K{x + a)~^, where if a is small we have a curve again 
 similar to that of infant mortality, x representing the age. 
 
 Types. , = «(?-l)""(|+l)""' (11) 
 
 where the limiting values of x are a and oo , with an
 
 a"v 
 
 49 
 
 ,^erao:e value of x= pr-^j the '' mode " occurring 
 
 ^ ??i2 — mj — 2 ° 
 
 at a;=: — ^a. The expressions for the moments are much 
 
 7712 — 771 1 
 
 simplified by writing the equation to the curve in the form 
 given in the Table on pp. 140-1. 
 
 Type 7. 7/ = A;a;-'»e-"/^ (12) 
 
 Where x varies between and oo , having an average value of 
 , with the "mode" at x= — . The second moment 
 
 m — 2' m 
 
 0.2 
 
 /i2= ; r^, — :^ and the "standard deviation" 
 
 consequently = 
 
 {m-2) y/m-'6 
 
 Here m must be > 3, or the second moment becomes oo , 
 and the fourth moment becomes infinite unless m is greater 
 than 5. 
 
 Neither this nor the preceding curve are of any wide 
 application in actuarial statistics, owing to the fact that the 
 values of y for large values of x diminish with increasing 
 slowness ; a feature not often met with in practice except in 
 such a function as the " rate of withdrawal." The same 
 remark holds good of the single curve constituting Class V. 
 
 Class V. Shew curves. Range unlimited in either direction. 
 
 2\ -m 
 
 
 w 
 
 This is the only skew curve of this family having 
 unlimited range. The average value of x— cy, y. a; the 
 
 " mode " is at ,t — a. 
 2m 
 
 The expressions for the moments and their i'lmctions are 
 
 simplified Ijy writing (+ Ij for m in equation (13), as in 
 
 the Table on pp. 140-1. For the reason stated above, the curve 
 is not specially useful to Actuaries. 
 
 E
 
 50 
 
 Assuming tliat a given statistical series can be represented 
 by one or other of the curves above described, the appropriate 
 curve can be found by means of certain criteria based upon 
 an examination of the " moments " of the curve ; that is to 
 say, the sums of the powers of the deviations from the mean 
 value. These criteria are furnished by the table on pp. 140-1, 
 above referred to. 
 
 As the calculation of the criterion is somewhat lengthy, 
 it may be noted that if the logarithms of y are tabulated 
 for equal intervals of the variable x, and the values of 
 A- log y taken out, these give us information as to the 
 nature of the curve. The value of AMogy will be 
 constant and negative for the " normal " curve Type 2 ; 
 negative and symmetrical with a minimum numerical value 
 in the centre of the range, for Type 1, or for any binomial 
 curve; uniformly negative, non-symmetrical, and with a 
 numerical minimum in the case of Type 4 (where this 
 curve vanishes at the limits) ; and uniformly negative and 
 continuously decreasing towards the upper limit of x in the 
 case of Type 5, where this curve vanishes at the limits. 
 
 In the case, therefore, of those curves most useful to the 
 Actuary the function A^ logy, computed for the ungraduated 
 curve, enables us to select generally the formula most suited 
 to the series. For this purpose if the data are grouped it 
 will generally be better to compute the approximate values 
 f)f the central ordinates of each group by an interpolation 
 formula, such as that given on p. 57. 
 
 Other types of curves will sometimes be found useful 
 besides those arising from the differential equation on p. 39 ; 
 but they do not generally lend themselves so readily to the 
 method of moments. 
 
 If, for example, we write 
 
 / m n \ 
 = /c-e~Va+^ t'+=^'' (14) 
 
 we obtain, when m and n are numerically unequal, a skew 
 curve vanishing when x=—a or —b. We may deal with this 
 curve in practice by determining the values of equidistant 
 ordinates as shown on pp. 57-8. Thus 
 
 logw = /c — -T— — =u; .... (lo 
 
 ° ^ a + x b + x 
 
 y
 
 51 
 
 As log y becomes — x at tlie limits^ we multiply both sides 
 by {a + x){h + x), thence 
 
 w \_ah + (a + l)x + x"-'] 
 
 = K \_ah + (a + 'b)x + x-'] — m{h + x) —n (a + x) 
 
 = A + Bx+Cx'~ {sRj) (16) 
 
 where the unknowns are a, b, A, B and C. 
 
 If we difference three times the right hand side vanishes 
 and we have a series of expressions involving {ab) and [a + h) 
 equated to zero and by suitably grouping these, or by using 
 the method of moments a and h, and thence the remaining 
 constants, may be evaluated. 
 
 A similar process may be employed with advantage with a 
 curve such as the usual form of exposed to risk or died, when 
 the data are in large age groups. We inay then take w in 
 equation (15) to represent the common log of the ratio of the 
 numbers above age x to the numbers below age x in the series. 
 That is, if the total number in the series =N, the number 
 above age a; =Y, we may Avrite 
 
 \N-YJ a + x b + x 
 
 o 
 
 In many cases the constant K' may be omitted if the 
 number of groups is small; in this case C in equation (16) 
 becomes zero. On the other hand it may sometimes be found 
 necessary to add a term to the right hand of equation (16) 
 involving x^. 
 
 I'. w
 
 FOURTH LECTURE. 
 
 W E shall now consider very shortly the problem of fitting 
 frequency curves to statistical data. To do this at length 
 would be impossible in the time at our disposal^ and the 
 student who wishes to pursue the subject in detail may read the 
 original papers, already referred to (p. 39), of Professor Karl 
 Pearson, to whom the development of the subject is due, 
 or Mr. Elderton's book. There are certain general principles 
 however, which may be usefully considered. The method 
 usually employed in fitting these curves is by making the 
 moments of the graduated equal to those of the ungraduated 
 curve, which is equivalent to making the quantities 
 S (deviations), t"^ (deviations), &c., as far as S"* or X^ equal 
 to zero. This method may not always be the most convenient 
 or the best for the purpose of the Actuary, but it is so 
 for most statistical purposes, and has come much into use 
 accordingly. 
 
 We have already seen that, in the case of the curves 
 arising from the differential equation on p. 39, expressions 
 for the moments may be obtained in terms of the constants 
 which will enable us to determine the value of the constants, 
 when the numerical value of the moments is known. For the 
 purpose of fitting the appropriate curve to any given series 
 of observations it is only necessary to determine the value 
 of the moments as given by the observations, that is, the 
 value of the sum of the squares, cubes, &c., of the deviations 
 from the mean value of the variable. 
 
 It will be useful to consider shortly the calculation of the 
 numerical value of the moments in a given instance. Take 
 first the simplest possible case where we have to do not with 
 a continuous curve, but Avith a series of points representing 
 isolated ordinates, where in consequence we replace Integra-
 
 53 
 
 tions by summations. In the following- table, the first column 
 contains the values of the independent variable x, the range 
 of values being from to 6. The second column contains 
 the values of its function y, which are proportionate to the 
 
 successive terms in the expansion of the binomial f -f ^ j , 
 
 the constant multiplier 729 being introduced merely to avoid 
 fractions. The remaining columns, in which the average 
 value of X and the values of the successive moments are 
 worked out, explain themselves. It may be remarked that in 
 this example the average value of x, and the deviations from 
 the average, are all integral, and it is therefore convenient 
 to calculate at once the moments round the average value 
 (" centroid vertical"). In most cases, however, the average 
 and the deviations will not be integral, and then it will 
 be more convenient to calculate the moments round the 
 origin or some selected middle value of the variable, 
 afterwards transferring the moments to the mean by the 
 formulae given on p. 41. 
 
 Table YIII. 
 Moments of the Point Binomial Curve. 
 
 729. - (-) (^) = .(2Y. 
 
 X 
 
 y 
 
 xy 
 
 {x-^)y 
 
 {x-^fy 
 
 (^-4)V 
 
 (.r-4)V/ 
 
 
 
 1 
 
 
 
 - 4 
 
 16 
 
 - 64 
 
 256 
 
 1 
 
 12 
 
 12 
 
 - 36 
 
 108 
 
 -324 
 
 972 
 
 2 
 
 60 
 
 120 
 
 -120 
 
 240 
 
 -480 
 
 960 
 
 3 
 
 160 
 
 480 
 
 -160 
 
 160 
 
 -190 
 
 160 
 
 4 
 
 240 
 
 960 
 
 
 
 
 
 
 
 () 
 
 5 
 
 192 
 
 960 
 
 192 
 
 192 
 
 + 192 
 
 192 
 
 6 
 
 64 
 
 384 
 
 128 
 
 256 
 
 + 512 
 
 1,024 
 
 Totals 
 
 Totals 
 
 729 
 
 1 
 
 2,916 
 4 
 
 
 
 072 
 
 -324 
 
 3,564 
 
 
 
 4 
 
 4 
 
 41, 
 
 
 
 mean vnlne 
 
 
 3 
 
 9 
 
 •J 
 
 + 729 
 
 . 
 
 < ) 1 X 
 
 =/J. 
 
 = M.' 
 
 = M.-. 
 
 = /*♦ 
 
 Obviously, when llio iiioiueiits arc calcuhited al)(»ut the moan 
 the first moment is zero (because it represents the average
 
 54 
 
 deviation from the average value). The even moments are 
 always positive^ because each term is of the form yx^^^'', 
 i.e., essentially positive ; and if the curve is symmetrical the 
 odd moments vanish, because each term of the form y^x^^"^^ is 
 cancelled by a term (equidistant from the mean) of the 
 form y-x{ — a;)^"'*''. In general, where the curve is not 
 symmetrical, the third, fifth, &c., moments will not be zero. 
 
 In the above illustration, we have considered x to have 
 integral values only. This may be said to approximate to 
 the conditions of many statistical tables used by the Actuary 
 where x represents the year of age under observation, and 
 where it is indifferent whether the observations are supposed 
 to be spread over the year in the form of a continuous curve, 
 or whether we consider them all to have reference to the 
 central point of the year. In these cases, however, x will 
 generally have a large range of values, amounting possibly 
 to 60 or 80, and the labour of computing the numerical 
 value of the moments is then much lessened by grouping 
 the facts in larger sections, though we cannot then safely 
 assume the totals of each group to be concentrated at the 
 middle ordinate. 
 
 Take the set of observations in Table IX representing 
 for decennial age groups numbers exposed to risk in the 
 
 middle of each year of age, i.e., ^x=^x— q^x, in the recent 
 
 mortality experience of lives assured by ascending premium 
 policies,"^ excluding the first ten years from entry. Here we 
 have no longer the values of equidistant ordinates of the 
 curve, but the area of the curve enclosed between successive 
 ordinates. To obtain the moments of the curve with any 
 degree of accuracy, we cannot treat these areas as 
 proportional to their central ordinate. 
 
 It will be noticed that the particular curve we are dealing 
 with becomes gradually zero at either extremity,t and we may 
 assume, without serious error, that it makes "close contact" 
 at either end with the axis of x, that is to say, is 
 asymptotic thereto. In these cases, Mr. Sheppard has shoAvn J 
 that very approximate values for the moments may be found 
 
 * See Unadjusted Data, Minor Classes of Assurances, p. 191. 
 
 + We omit the numbers at risk under age 25 (arising from entrants under 
 age 15), amounting to only 25 in all. 
 
 J An elementary demonstration is given in Elderton's Treatise, p. 28-29.
 
 55 
 
 by treating the area of each successive section of the curve 
 as concentrated in the middle ordinate of the section j in 
 other wordsj treating the values of y as representing isolated 
 ordinates exactly as Avas done in Table VIII ; and then 
 applying to the values of the moments so found (denoted by 
 the symbol m') the following adjustments leading to the 
 corrected moments denoted by the symbol m : — 
 
 , 1 
 W2=m2— j2 
 
 , I , , \ 
 
 7??3=i?i 3— -m =m 3 — 7 mi 
 4 4 
 
 ,1,^7 ,1 1 
 
 For moments round the centroid vertical these become^ 
 remembering that /ii = 0, 
 
 , 1 
 
 /*2 = /i2 — 
 
 /^ = /ti3 
 
 12 
 
 A<-.=/i4— :;/^i 
 
 80* 
 
 Table IX. 
 
 Ascending Premium Assurances — Experience 1863-1893. 
 
 Duration 10 years and upwards. 
 Calculation of Moments of ^'' Exposed to Hisk^' Curve. 
 
 
 ExpoHed to 
 
 
 
 
 
 
 Ages 
 
 Kisk 
 
 X 
 
 Xl/ 
 
 X't/ 
 
 x'^t/ 
 
 x\i/ 
 
 25-35 
 
 2,874 
 
 -2 
 
 - 5,748 
 
 11,496 
 
 -22,992 
 
 45,984 
 
 35-45 
 
 22,020 
 
 -1 
 
 -22,020 
 
 22,020 
 
 -22,020 
 
 22,020 
 
 45-55 
 
 26,164 
 
 
 
 
 
 
 
 55-65 
 
 17,391 
 
 1 
 
 17,391 
 
 17,391 
 
 17,391 
 
 17,391 
 
 65-75 
 
 7.845 
 
 2 
 
 15,(;90 
 
 31,380 
 
 62,760 
 
 125,520 
 
 75-85 
 
 1,761 
 
 3 
 
 5,283 
 
 15,849 
 
 47,547 
 
 142,641 
 
 85-95 
 
 81 
 
 4 
 
 324 
 
 1,296 
 
 5,184 
 
 20,736 
 
 Totals 
 
 78,136 
 
 
 10,920 
 
 99,432 
 
 87,870 
 
 374,292 
 
 Uoducpd 
 
 1 
 
 
 •13976 
 
 1-2725 
 
 11246 
 
 4-7903 
 
 to unit area 
 
 
 = ,r/, 
 
 = ««', 
 
 ^m'. 
 
 = m\
 
 56 
 
 From these results we obtain by means of the corrections 
 above stated — 
 
 m, = -13976; m2= 1-1892; m3= 1-0897; w, = 4-1832. 
 
 Whence, by the equations on p. 41. 
 
 /i2= 1-1697; /^3=-5965; /X4=3-7122. 
 
 If quinquennial age groups had been used, making due 
 allowance for the unit of time still being taken as ten years, 
 the corresponding values would have been 
 
 TO, = -13848; /i2=l-1741; ya3 = -5869; yL64=3-7160. 
 
 using these latter values, as the more accurate, we obtain for 
 the values of the functions /5i , ^-2 and 7. 
 
 A=/^V/^'2=-21283; /32=/.V/^\=2-6957; 7= |i±| =-7397 ; 
 
 2 
 As /X3 does not vanish, and 7 is > .3, we see from the table 
 
 on pp. 140-1, that if the series can be represented by any of 
 the curves there given, it must be by No. 4, excluding the skew 
 binomial as unsuitable for reasons already given. It is also 
 obvious from the run of the figures in Table IX, that the 
 curve is limited in both directions. Equating the expressions 
 in Table IX with the above numerical values, we have 
 
 7=1 •''^=•7397; whence 71 = 7-13 
 6 n + 2 
 
 ^- '^(^] + ^\(r-9r =-21283; 
 
 whence (p — qY='b4<6opq 
 
 (j) + g)'-=:4-5453jjg' =1 (since ^ + 2 = 1) 
 
 (^-2^=V4^3="^^^^ 
 giving ^ = -6732; g = -3268 
 
 fM2=^^-a^=V17U; whence a = 3-293 
 11+1 
 
 thus giving a range of 32-93 years on either side of the age 
 for which the value of x in the formula = 0. This has nothing 
 to do with the zero point (age 50) in Table IX. The mean 
 age as is seen from that table is 50 + 1-385 = 51 '385. The 
 value of m, the mean as computed by the above formula, is 
 
 Wi = (<j'— p)a= — 1'1407
 
 57 
 
 that is, 11-407 years earlier than the central point of the range, 
 giving for the latter, 51-385+ 11-407 = 62-79, say. The range 
 of the curve is therefore from age 29-86 to age 95-72 ; and, 
 computing the values of np—1 and iiq — l, ^ve have, for the 
 final form of the equation of the curve, when .c = the age : 
 
 It is often a convenience, however, to have the values 
 of the central ordinates of the groups, which may be 
 approximately obtained by interpolation. If the numbers m 
 any group are represented by the symbol Ux, the number of 
 years in each group being t, the value of the central ordinate 
 of the group (that is to say, the numbers under observation 
 exactly at the central age of the group) will be approximately 
 
 V^^— ^ "^-' V. As, however, it is convenient to treat the 
 interval t as the unit, for the time being, we may write as the 
 
 values of the central ordinates w^- 2^^ ^^^^^ original 
 
 numbers for each group less ^th of their respective central 
 second differences). In the class of curves we are discussing, 
 namely, those having close contact at both ends with the axis 
 of x, the numerical values of the moments as deduced from 
 these ordinates will be very nearly the values for the 
 continuous curve, unless the number of groups is very 
 
 small. Thus the values of ydx, and of the functions 
 
 f xydx, x'^ydx, will be found by taking the sum of the 
 
 ordinates of y, computed as above, and the sum of the 
 products xy, x'^y. 
 
 An advantage attaching to the use of ordinates in lieu of 
 areas is that, in the class of curves we are dealing with, we 
 can, by examination of the differences of the logarithms of 
 the ordinates, gain a better idea of the nature of the curve 
 than can be obtained from the grouped figures. [See Third 
 Lecture, p. 50.) It is also easier to compare the graduated 
 fif-ures as given by the frequency cui-ve by means of isolated 
 ordinates than by means of groups or areas. 
 
 • The formula to Uli ilifTcreiiccs is v^ "^ + - .^[.^ "' nearly, iuiil in 
 
 order that the resultinp 4th nioincnt slioiild a;,'rce exactly with that ohtainctl 
 from the use of the t'TOoi'^-'l fijrnrrB, or areas, with Shcpiianl's corrections, the 
 4tl» diflcreuce is re(|uirt(l, but for pniclital iiurjioscs it is not often needed.
 
 58 
 
 The use of the central ordinates of the groups has 
 the incidental advantage, which is very considerable 
 in the case of a mortality or similar experience, of 
 giving trustworthy values of the force of mortality, or 
 corresponding function, for the ages corresponding to the 
 position of the ordinates. In the usual plan of summarizing 
 a mortality table by giving the numbers at risk and deaths 
 in consecutive age groups, the ratio of the deaths to the 
 numbers at risk in each group is not a useful function, as it 
 does not correctly represent the mortality for the central age, 
 except near the middle of the table, where the numbers under 
 observation in successive years is nearly constant. 
 
 We may apply this method to the example already dealt 
 with on p. 55, viz., the experience of ascending premium 
 policies. The calculations as set out in the following tabular 
 form are sufficiently clear : 
 
 Table X. 
 
 Mortality experience of lives assured ly ascending Premiums, 
 1863-1893. Duration 10 years and upwards. 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 •Estimated Central 
 
 
 
 Central 
 
 
 
 Ordi 
 
 NATES 
 
 
 Ages 
 
 age 
 
 of group 
 
 {X) 
 
 Exposed to 
 Kisk 
 
 Died 
 
 
 
 Central Age 
 
 Exposed 
 to Risk 
 
 Died 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 25-30 
 
 27-5 
 
 266 
 
 2 
 
 168 
 
 •8 
 
 •0048 
 
 30-35 
 
 32-5 
 
 2,607-5 
 
 31 
 
 2,448 
 
 29-2 
 
 -0119 
 
 35-40 
 
 37-5 
 
 8,788 
 
 102 
 
 8,860 
 
 102-0 
 
 •0115 
 
 40-45 
 
 42-5 
 
 13,232-5 
 
 173 
 
 13,389 
 
 175-2 
 
 •0131 
 
 45-50 
 
 47-5 
 
 13,910 
 
 192 
 
 14,007 
 
 191-7 
 
 •0137 
 
 50-55 
 
 52-5 
 
 12,254 
 
 218 
 
 12,284 
 
 218-6 
 
 •0178 
 
 55-60 
 
 57-5 
 
 9,878-5 
 
 229 
 
 9,878 
 
 228-4 
 
 •0232 
 
 60-65 
 
 62-5 
 
 7,512-5 
 
 255 
 
 7,518 
 
 255-4 
 
 •0340 
 
 65-70 
 
 67-5 
 
 5,007-5 
 
 271 
 
 4,994 
 
 274-4 
 
 •0549 
 
 ■ 70-75 
 
 72-5 
 
 2,837 
 
 206 
 
 2,809 
 
 205-6 
 
 •0732 
 
 75-80 
 
 77-5 
 
 1,347-5 
 
 151 
 
 1,324 
 
 151-4 
 
 •1144 
 
 80-85 
 
 82-5 
 
 413-0 
 
 85 
 
 389 
 
 84-8 
 
 •2180 
 
 85-90 
 
 87-5 
 
 77 
 
 24 
 
 66 
 
 22-3 
 
 •3379 
 
 90-95 
 
 . 92-5 
 
 4-5 
 
 3 
 
 2 
 
 2-2 
 
 1^1000 
 
 Totals... 
 
 ... 
 
 78,136 
 
 1,942 
 
 78,136 
 
 1,9420 
 
 ... 
 
 * Taking 5 years as the unit, computing by formula Ux — 
 
 24 ' 
 
 where 
 
 Ux represents the number in columns (3) and (4). By this formula there are 
 — 11 persons exposed to risk at age 22^5; these have been included in the 
 group 25-30.
 
 59 
 
 If the values of the moments are computed from columns 
 exactly as was done with the Binomial Curve (Table VIII, p. 53) 
 they will be found to be pi-actically identical with those found 
 above. The estimated values of /u.^ for the central ages of the 
 group are inserted as they will be used later. 
 
 In many cases the principle of the method of moments 
 may be used to fit a curve to a series of observations without 
 actually computing the numerical values of the moments 
 themselves, using instead the successive summations of the 
 ordinates, or areas, from which the moments can be readily 
 obtained if required. This method is also useful if one or 
 both limits to the range of the curve can be assumed. 
 
 Consider a scheme such as the following, in which, with a 
 view to clearness, we use actual numbers of the series, given 
 on p. 53, instead of symbols : — 
 
 X 
 
 Vx 
 
 2«2 
 
 2^-«x 
 
 23«;. 
 
 2^/<x 
 
 :s,'u^ 
 
 UxXX^ 
 
 
 
 1 
 
 729 
 
 
 ... 
 
 
 ... 
 
 
 
 1 
 
 2 
 3 
 
 12 
 60 
 
 160 
 
 728 
 716 
 656 
 
 2,916 
 2,188 
 1,472 
 
 7.776 
 (6,318) 
 
 4,860 
 
 2,672 
 
 9,180 
 4,320 
 
 15,660 
 (11,070) 
 
 6,480 
 
 12 
 
 960 
 
 12,960 
 
 4 
 
 210 
 
 496 
 
 816 
 
 1,200 
 
 1,648 
 
 2,160 
 
 61,440 
 
 5 
 
 192 
 
 256 
 
 320 
 
 384 
 
 448 
 
 512 
 
 120,000 
 
 6 
 
 64 
 
 64 
 
 64 
 
 64 
 
 64 
 
 64 
 
 82,94 1 
 
 278,316 
 
 In this scheme, each column is formed from the preceding 
 by successive addition from the bottom, in the same way 
 that the M^ column is formed from C, , and Rj. from Mx- 
 
 If we take the value against x = in the column 'Zux, say 
 luo, we see that each value of tij; occurs once only in that 
 total. In the total appearing against x=l in the second 
 summation, say S^u,, eacli value of t(j. occurs x times; 
 siiuihirly the total against x = 2 in the column S^Ux, say 
 
 thoi, represents the sum of the products ^ — '^^'' ''"'^ *^'"
 
 60 
 
 total against x = S in the column %hije, say Xhi^, represents 
 
 the total of the products ~ |f~ ^tx, and so on, the 
 
 coefficients following- the Binomial law. It is evident from 
 this that the sums of the products cc^Ux, xhi^., &c., are 
 implicitly contained in these totals ; and that if these sums 
 of the graduated and ungraduated values are in agreement, 
 the moments of the two curves will also agree. Writing m„ 
 as the value of the nth moment round the ordinate of x = 0, 
 we shall find :* 
 
 mi = ~. — 
 
 7n.2= - 
 
 7ni = 
 
 
 Xuo 
 
 1714 = =^ ■ 
 
 Zuo 
 
 These formulje may be simplified if we write them in a 
 form analogous to central difference formulae — writing, for 
 example : 
 
 these average values being shown in antique type in the 
 Scheme. We then have, omitting the common divisor Xuq : 
 
 m2=2S^UiL 
 
 nii = 24tXhi2^^ + Wo 
 
 The equivalence of the above foi-mulfe may be illustrated by 
 the following numerical examples based on the above scheme. 
 
 * See the demonstration in Note E, p. 124.
 
 61 
 
 Using N as an abbreviation of 3!;7o = tlie total number of 
 observations, we have 
 
 N.m,= 2916 = S2m, 
 
 N.m2= 12636 = 22:3^2 +%~u, =2x4860 + 2916 
 = 2thiri =2x6318 
 
 N.'m3= 57996 = 6S^M3 +6^^., +S%i = 6 x 4320+ 6 x 4860 + 2916 
 = 6S'u.2 +^2,^^ = 6x9180 +2916 
 
 N.7n4=278316 = 24S5it4 + 36S'i(3+14^3^^, + S2ji,=:24x2160 
 
 + 36 X 4320 + 14 X 4860 + 291 6 
 
 = 24>tHi^i +2S^x =24x11070 + 2x6318 
 
 The last may be compared with the direct calculation of 
 x*Uj: given in the last column of the scheme. The values of 
 the moments through the centroid vertical may be obtained if 
 required by the formulas : 
 
 IJUi=m.2—{miY 
 
 /i3 = Wis — 3 (wi)/i2 — (wi) 3 
 
 Ai4 = »l4 — 4(7?li)yti3 — 6(??li)^/i2— (?/ii)^ 
 
 Where the number of terms in the series is few, there is no 
 special advantage in this method ; but if the number of terms 
 is considerable it effects a saving of time, more particularly 
 if the calculation of the moments round the centroid vertical 
 is not needed by the conditions of the problem, as in the case 
 of the graduation of rates of mortality by Makeham^s or 
 any similar frequency formula. 
 
 The case of curves not making close contact with the axis 
 of X at both ends requires to be considered separately, but tlio 
 results obtained are not altogether satisfactory, see Elderton, 
 pages 29-30. TIk; difficulty can, however, to a great extent 
 be avoided in most cases arising in actuarial work by using 
 very small groups, or even individual values for each year of
 
 62 
 
 age, &c., in calculating the moments. The labour although 
 increased is by no means prohibitive if the summation method, 
 above described, be adopted. 
 
 Professor Karl Pearson has shown'^ that the method of 
 fitting a curve by computing its moments should lead to 
 nearly the same results as the method of least squares. If we 
 are fitting to a given sot of observations an ordinary parabolic 
 curve, represented by the equation 7/ = a + 6a; + caj^ + &c., then 
 the method of moments and the method of least squares are 
 identical. t He infers from this fact that, even if y is 
 represented by a more complex expression, the numerical 
 results from the method will be nearly the same as with the 
 method of least squares. It would appear at first sight that 
 the effect of the method of moments is to give equal weight 
 to each observation or group of observations, in spite of their 
 having unequal average errors ; whereas the method of 
 least squares shovild, strictly speaking, be applied only when 
 the average error of each observation is nearly equal. J In a 
 mortality table, where the number of persons under observation 
 and the number of deaths are relatively large in the middle 
 of the table and fall off to zero at the beginning and end, the 
 probability of a given error in the value of g is very much 
 ■smaller at the central ages ; while, on the other hand, the 
 probability of a deviation of a unit in the number of deaths is 
 ■correspondingly greater. The same applies to most tables of 
 statistics, as they usually present a series starting from zero, 
 rising to a maximum, and diminishing to zero again, the 
 weight of the observations being in the middle of the curve, 
 where, however, the probability of a given numerical deviation 
 in the actual numbers is also greater. 
 
 We have seen that in a series of numbers representing the 
 distribution of a group into sub-groups the average error in any 
 
 VfYVi (72. — 771 1 
 — ^ ~ , where n is the 
 
 number in the group and m the (graduated) number in the sub- 
 group. If, as is generally the case, n is large compared to m, 
 
 * Biometrika, vol. i, p. 266-271. 
 
 t This assumes that the unadjusted moments {m not m') are used, i.e., that 
 the niunbers represent ordinates and not areas. If the moments are assumed to 
 represent areas and the corresponding corrections are introduced, the method of 
 moments no longer gives precisely the same results as the method of least 
 •squares : see examples given by Todhunter, J. I. A., xli, 444. 
 
 I See Note C, p. 117.
 
 63 
 
 this expression may be taken as equal to "8 v^w, the average 
 error in the ratio ~~ being approximately '8 Thus, if 
 
 the number at risk at a given age equals n and the true 
 probabilities of death and survivorship, are q and p, then 
 •8 '/njiq'^ (which as p is nearly unity for the greater number 
 of ages may be roughly taken as •SvWmber of deaths), 
 is an approximate expression for the average deviation from 
 the expected number of deaths. The method of moments, 
 if employed to represent a given series by a parabolic curve, 
 assumes an equal probability of unit error in each term of 
 the series. If, thex-efore, the series is of such a character 
 that the extreme values are relatively small, these parts of 
 the data will have somewhat less than their due weight in the 
 fitting process. If, hoAvever, the formula to be fitted does 
 not represent a parabolic curve, but a curve analogous to the 
 normal curve he~^' >'^', say a curve of the form e«+6a;+ei'+&c. 
 then it will be found that, on the assumption that the moan 
 error in any value y is equal to v/?/i (wjiero i/, represents the 
 graduated value of y) the metliod of moments gives the same 
 result as the method of least squares when the observations 
 are duly weighted (see Note F, p. 129). 
 
 We come now to the class of curves representing not 
 the actual numbers in statistical tables, but the ratios of the 
 corresponding numbers in the double series, such as those of 
 tables of " Exposed to Risk " and " Died ", curves, that is, 
 representing such functions as rates of mortality, of marriage, 
 of lapse, of superannuation, &c. The most interesting and 
 important of these is the curve due to Makeham's development 
 of Gompertz's hypothesis, in which the force of mortality at a 
 given age x is represented by the expression 
 
 /ii,^ = A + 13c^ 
 leading to the equation 
 
 log,o/x=K + A'.« + B'c^. 
 
 This curve lias a double value as, apart from its use in 
 graduating a mortality table, it lias the valuable property 
 
 • See Note A, p. 110; .J.I.A., xxvii, 214.
 
 64 
 
 that the values of annuities on n joint lives of various ages 
 can be found from a table of single entry showing the values 
 of annuities on n lives of equal age. Owing to its importance 
 it will be useful to give some attention to the problem of 
 fitting this curve to a mortality experience. We will first 
 consider the case of an aggregate or non-select table, that is, 
 a table in which the rate of mortality is a function of the age 
 alone. 
 
 Various methods have been employed to obtain the values 
 of the constants A, B, c, corresponding to a given experience. 
 That used by Makeham, and subsequently in a modified form 
 by Woolhouse, is based on selected values of log Ix taken from 
 a table already graduated by a finite difference formula. 
 Four values of log Ix may be taken, covering practically the 
 whole of adult life, say the values at ages 20, 40, 60, and 80, 
 or 25, 45, 65, 85. Either set are sufficient to determine 
 the four constants, K, A', B' and c, as above. In Woolhouse's 
 graduation of the H^ Table, both of these sets of ages were 
 employed, the most advantageous values of the constants 
 being found by comparing the deviations between the 
 graduated and ungraduated values of Ix at quinquennial 
 ages according to the two preliminary graduations. If a 
 single set of four values of Ix is taken as the basis of the 
 graduation, the effect is the same as employing the sums of 
 the forces of mortality (/ix+0 between the selected ages, 
 giving equal weight to the values at each age. 
 
 The method employed by Mr. King in the Institute of 
 Actuaries' Text-Book, Part II., substitutes for graduated 
 values of log Ix at isolated ages, the sum of certain 
 groups of the ungraduated values of log Ix . The effect 
 of this method would appear to be to give a diminishing 
 weight to the values of ^x for the ages at the commencement 
 and end of the table, which is so far in accordance with 
 theory, and to eliminate the effect of errors in isolated 
 values of Ix- In Biometrika (vol. i.,p. 298-303) Prof. Pearson 
 has dealt with the same problem, basing the values of the 
 constant upon the successive summations of log Ix . 
 
 It is, perhaps, preferable to deal directly "with the actual 
 exposures and deaths in a manner similar to that first 
 described by Makeham (/.7.^., vol. xvi, p. 344). This can 
 be readily done, and the same method of summations or 
 moments applied as in the case of any other frequency curve.
 
 65 
 
 Tabulate E^+i, that is, the number exposed to risk in the 
 middle of the year of age x, and ^^ representing the deaths 
 occurring between ages x and x+l. Assuming, as we may with 
 sufficient accuracy for ordinary purposes,* that the force of 
 mortality at age x+^, or the function colog ePx, is equal 
 
 to mx the " central death rate " = ^^^ , we have 
 
 If we knew the value of c, we could then tabulate the values 
 of Ea.+4, Ejj+ic^^"*"^, 0x respectively, and summing these values 
 continuously to the end of the table, and again taking the 
 total of these sums, we should obtain equations in this 
 form : — 
 
 (22E,+4)A+ {ttEx^iC-^^)B = {-$t0x) 
 
 a simple simultaneous equation for determining A and B. 
 As a matter of fact, the value of logioC does not usually 
 differ very much from "04, and in general it will be found 
 that a small change in the value of log c does not involve a 
 serious change in the general character of the table. In an 
 important series of observations, however, we cannot assume 
 the value of c. Either we must determine c by a method 
 such as that used by Mr. Woolhouse or Mr. King, Avhich will 
 give a sufficient approximate value, or we may adopt two or 
 more alternative values of c, which appear likely to contain 
 between them the true value. Having obtained the values 
 of constants A and B for each given value of c, set out 
 the expected or graduated deaths, and compare them with 
 the actual numbers in suitable age groups. If the 
 third summation of the differences of the graduated and 
 ungraduated deaths is computed, it will be possible by 
 
 •Assniniiif,' tlio usual table of Kj- and 6^. to represent accurately the facts 
 and to be undisturbed at the older ages (where alone the point is of any 
 
 
 importance) by entrances or by exits other than by death, then =yj; 
 
 a 
 
 acurately; and colog ePi= z-, . —, very nearly, where Wj; is the 
 
 a 
 
 " central death rate " :n — Ta" • '^^^ error caused by omitting the small term 
 
 a 
 
 in the denominator and taking colog «pj;= ^ - is only apjircciable at the 
 
 "X ~\"x 
 
 older ages, amounting to 1 per-cent in the rate of mortality where 2x = '3 
 or about age 90.
 
 66 
 
 interpolation to obtain a value of log c, making these nearly 
 equal to zero. Putting the matter into the language of 
 moments, we shall then have made the first, second and 
 third moments of the graduated and ungraduated curves 
 equal, and in that way we shall have selected what may be 
 considered the best values of the constants A, B and c* 
 
 It may be objected that the use of this particular method 
 is open to the same implication of giving equal weight to all 
 the observations, as in the case of the values of Ix. We can 
 avoid that objection by duly weighting the observations at 
 each age by multiplying the " exposed " and " died " at 
 each age by the approximately graduated values of {dx)~^- 
 But although this would give suitable weights to the 
 observations, if the curve of mortality were a parabolic 
 curve, or if it were known to follow accurately Makeham's 
 Law, it is not quite clear that it would do so in practice. 
 It may be assumed that (when the constants are formed 
 by reproducing the moments of the deaths) in not 
 weighting the observations, Ave give less weight to those at 
 the commencement and at the end of the table than they are 
 theoretically entitled to. But this is not a serious practical 
 objection, Makeham's law is only approximately correct, 
 and as we reach younger adult ages it begins to diverge from 
 the facts of observation ; on the other hand, as we reach the 
 older ages the actual importance of the observations is less 
 than the weight to which they are theoretically entitled, as 
 estimated by the number of deaths, owing to the fact that 
 the actual mortality at those ages does not materially affect 
 financial questions such as rates of premium and reserves. 
 
 Beyond this consideration there is also a degree of doubt 
 attaching to the rates of mortality at extreme ages in any 
 table.t Indeed, we may go further, and say that in all 
 considerable tables of statistics the numbers at the extremes 
 of the table are proportionately more affected by sporadic or 
 accidental errors of observation than those in the body of the 
 table. If we suppose that in a very small percentage of 
 cases the ages of the " Exposed to Eisk " and " Died " are 
 affected by errors of calculation, clerical errors in tran- 
 scribing the data, &c. — these cases being removed from their 
 true position and scattered at random over the table — the 
 
 * See Note G, p. 131. 
 
 + See my notes on this suLject in " PrincijDles and Methods ", p. 1-48.
 
 >67 
 
 efPect upon the data over the great bulk of tlie table will be 
 insignificant owing to the large numbers under observation 
 and to a balance of errors, but the effect upon the experience 
 at the extremes of the table, where the actual numbers under 
 observation are very small, may well be appreciable. 
 
 Reverting to the problem of obtaining the value of c in 
 Makeham's formula directly from the observations, we may 
 endeavour to represent the curve of the " Exposed to Risk " 
 by some frequency curve which can be suitably combined 
 with the formula for jXx to represent the deaths— such, for 
 example, as the normal curve y = ke-'=''''\ or the curve 
 No. 5, y = 'kx''ey'', or by the terms of a binomial expansion 
 {see Calderon, J.I.A., vol. xxxv, p. 157). Unfortunately none 
 of these curves give a very satisfactory representation 
 of the average form of the "Exposed to Risk" curve. 
 In the case of the binomial, in order to get a tolerable 
 fit, it will be generally found that the value of n in the 
 
 expression — — (representing the general term of the 
 
 binomial) must be taken small ; that is to say, the data must 
 be arranged in somewhat large groups of not less than about 
 10 ages to a group. In either case it Avill be necessary, after 
 obtaining a frequency curve fitting the numbers of the 
 "Exposed to Risk," to re-compute the deaths on the basis 
 of these graduated numbers. 
 
 Thus, Avhile it is possible to determine the values of c 
 directly from the observations, the process is laborious. In 
 my opinion, it is preferable to use certain trial values of c 
 which we know to lie near the truth, and, by a comparison of 
 the resulting graduated deaths with the original facts, to 
 select a value which appears to give the best general 
 agreement, which may not always be that making the third 
 summation of the deviations zero."^ 
 
 There is a further point to be considered with respect to 
 the nature of the differences between the original numbers, 
 whether of deaths or of other observations, and the numbers 
 obtained by a graduation following a formula such as that of 
 Makeham. These divergences between the ungraduated 
 and graduated numbers will in part arise from the smallness 
 of the numbers under observation, and may in part arise 
 from the fact that tlie formula does not accurately represent 
 
 «• ,S'tfe Note G, p. 133. 
 
 V 2
 
 68 
 
 the true curve of mortality. For the majority of mortality 
 tables, for male lives at the adult ages, Makeham's formula 
 is so near the truth that we may in practice neglect the 
 systematic errors and assume that the formula represents 
 the true curve of mortality, determining our constants as 
 though the whole of the deviations in the graduated and 
 ungraduated curves are accidental and due to the smallness 
 of the data, but for some tables, notably those representing 
 the mortality of females, this will not be the case. 
 
 Other expressions may be given representing approximately 
 the curve of ixx, as, for instance, 
 
 lx^ = ma^-\-nh'' (1) 
 
 whence 
 
 logioZx=K+Ma^ + N6^ (2) 
 
 an expression which enables us to represent some mortality 
 tables, such as those arising from tropical experience, that 
 are not very readily represented by Makeham's formula. 
 The values of these constants can be readily obtained either 
 from 5 selected values of log la- or from the sums of the values 
 of selected groups of the same function. 
 
 The above formula for l^ preserves in a modified form the 
 principle of uniform seniority. Not, however, in a very 
 practicable shape as in order to compute values of joint-lives 
 (any number) we require tables of h joint-lives of equal age 
 for various values of h. It is of course evident from general 
 considerations that the force of mortality on any number of 
 joint-lives must consist of two terms, each of which is 
 a member of a geometrical progression, and that if we can 
 find an a^e w where the relative values of these two terms is 
 the same as in the joint-life status, the actual values will 
 be the same when multiplied by some suitable constant ^^ 
 The required joint-life annuity will then be represented by 
 the annuity on fc joint-lives all of-age w. 
 
 Take as an example an annuity on the joint-lives of [x) 
 and (i/). Find h and w so that 
 
 log (rt-^' -f- ay) — Iog(&-^ + &^) 
 
 a^ + aJ/ = ka'^ 
 
 whence w= , it, 
 
 log a — logo 
 
 and k={a^ + ay)--^a'"=[h'' + hy)^l'« 
 
 Then it is obvious that if we replace 'X and y by x -f- 1 and t/ 4- ^ 
 'k will remain unaltered and w will become 10 + 1, so that the
 
 69 
 
 principle of uniform seniority is maintained. Thus, an 
 annuity on x and y will be equal to an annuity on h lives all 
 aged 10 ; or, since h will not generally be integral, it will be 
 more convenient to say that a^y = a to where a' is calculated at 
 forces of mortality always h times the normal force, age for 
 age. Thus, we shall require tables, for various standard 
 values of k, and we shall usually require a double interpolation, 
 since neither ic nor k will usually be integral. 
 
 The principle of employing the sum of two (or more) 
 geometrical series to represent the logarithm of a function 
 such as the number living may also be used Avith advantage, 
 as will be seen later on, for census tables. (See the Sixth 
 Lecture.) 
 
 As an example of this formula, we may apply it to the 
 column of log ?^ in the O^ Table. 
 
 Taking the values of log Zj; for ages 20, 37, 54, 71 and 88, 
 we have the following data : 
 
 log Zao = 4- 98432 = K + Ma^o + N?>2o 
 
 log ?37 =4-94279 = K + Ma37 + m^7 
 
 log l,^ = 4-85300 = K + Ma^^ + N&^* 
 log Z., = 4-58086 = K + Ma"' + N6'i 
 log ^88= 3-47509 = K + Ma^s + N//^ 
 whence differencing, and writing 
 
 Ma^(a'--1) = -M'; X?y-=«(?/--l) = -N' ; a''=a; h'^ = l3; 
 
 we have 
 
 M' + X' = log/2o-logZ37= •04153 = A 
 
 M'a + X'/3 = log/,7-log/54= -08979 = 6 
 
 MV + N';82=log/M-logZ;i= -27214=0 
 
 MV + N'/S3 = logZ;,-log788=l-l()577 = D 
 
 whence, noting that 
 
 BD-C^ ^ AD-BC , ,3
 
 70 
 
 we easily obtain : 
 
 a=5-lU82 ; a= 1-1007 
 
 y3= 1-5243 ; h= 1-0251 
 
 M'= -0073886; M= --00026403 
 
 N'= -0341414; N= --039657 
 
 The following comparison of the values of Z.^ and 
 decrements for quinquennial ages will indicate the approxi- 
 mation of the formula to the 0*^ Table. 
 
 Table XI. 
 
 Values of Ix c^nd of {l.v — lx+i) according to the O*^ Table, as 
 compai'ed toith re-graduation hy formula (2). 
 
 Age 
 
 I 
 
 X 
 
 Quinquennial Deceements 
 
 
 
 
 
 Errors 
 
 I 
 
 By 
 
 Formula 
 
 Original 
 Value 
 
 By 
 
 Formula 
 
 Original 
 Value 
 
 
 1 
 
 + 
 
 — 
 
 20 
 
 96,453 
 
 96,453 
 
 2,129 
 
 2,066 
 
 63 
 
 
 25 
 
 94,324 
 
 94,387 
 
 2,467 
 
 2,445 
 
 22 
 
 • •• 
 
 30 
 
 91,857 
 
 91,942 
 
 2,896 
 
 2,947 
 
 • • • 
 
 51 
 
 35 
 
 88,961 
 
 88,995 
 
 3,443 
 
 3,528 
 
 ■ ■ • 
 
 85 
 
 40 
 
 85,518 
 
 85,467 
 
 4,158 
 
 4,205 
 
 . . > 
 
 47 
 
 45 
 
 81,360 
 
 81,262 
 
 5,108 
 
 5,077 
 
 31 
 
 • .. 
 
 50 
 
 76,252 
 
 76,185 
 
 6,350 
 
 6,266 
 
 84 
 
 ... 
 
 55 
 
 69,902 
 
 69,919 
 
 7,927 
 
 7,846 
 
 81 
 
 
 60 
 
 61,975 
 
 62,073 
 
 9,775 
 
 9,766 
 
 9 
 
 
 65 
 
 52,200 
 
 52,307 
 
 11,606 
 
 11,692 
 
 
 86 
 
 70 
 
 40,594 
 
 40,615 
 
 12,753 
 
 12,863 
 
 
 110 
 
 75 
 
 27,841 
 
 27,752 
 
 12,192 
 
 12,222 
 
 
 30 
 
 80 
 
 15,649 
 
 15,530 
 
 9,244 
 
 9,171 
 
 73 
 
 
 85 
 
 6,405 
 
 6,359 
 
 4,827 
 
 4,763 
 
 64 
 
 
 90 
 
 1,578 
 
 1,596 
 
 1,406 
 
 1,410 
 
 
 "4 
 
 95 
 
 172 
 
 186 
 
 167 
 
 179 
 
 
 12 
 
 100 
 
 5 
 
 7 
 
 5 
 
 7 
 
 ... 
 
 2
 
 FIFTH LECTURE. 
 
 Although m the preceding Lecture the application of 
 Makeham's formula has been considered at some length, its 
 importance is such that we may now touch on some further 
 points, and particularly on the application of the formula to 
 the graduation of select tables. 
 
 The suitability of Makeham's formula to the graduation 
 of mortality tables must be judged as we should judge the 
 applicability of any other frequency curve to a given series of 
 observations. That is to say, we must consider whether the 
 observed differences between the graduated and ungraduated 
 values (the computed and actual deaths) fall within what 
 may be properly considered to be the limits of error. 
 For practical purposes, owing to the great convenience 
 attaching to tlie use of the formula, it is worth while to 
 stretch a point in its favour. Instead, therefore, of merely 
 considering the closeness of the agreement between the 
 actual and computed deaths, we may consider how nearly the 
 ungraduated and graduated monetary functions, such as the 
 values of premiums or annuities, are in agreement. If this 
 agreement is sufficient for our purpose, we are justified 
 in adopting the graduation as given by the fornuila, 
 notwithstanding the fact that at certain groups of ages the 
 divergences between the graduated and ungraduated deaths 
 may be greater than would be expected from the theory of 
 probabilities. In this connection it is to be noted that our 
 observations relate to past time, and that the quantities we 
 are measuring are all liable to change with time. Hence in a 
 graduation intended to form the basis of tables of annuities or 
 premiums it is sufficient if the general character of the 
 experience is retained without insisting too strongly upon a 
 strict adherence to minor features. This is illustrated by the 
 following table from " Principles and Methods" (p. 162), in
 
 72 
 
 which we may anticipate for the moment the question of the 
 application of Makeham's formula to select tables : 
 
 Q[^] Whole-Life Participating — Males. 
 3 per-cent Premiums for £100 Assured. 
 
 Age 
 
 P[.l 
 
 G-U 
 
 Sprague's 
 HtM] Select 
 
 HtMl-Ot^H 
 -(3) 
 
 + 
 
 Ungraduated 
 
 Graduated 
 
 + 
 
 — 
 
 (1) 
 20 
 25 
 30 
 35 
 40 
 45 
 50 
 55 
 60 
 65 
 
 (2) 
 1-379 
 1-535 
 1-779 
 2-086 
 2-453 
 2-952 
 3-571 
 4-338 
 5-413 
 6-872 
 
 (3) 
 1-365 
 1-551 
 1-785 
 2-081 
 2-457 
 2-940 
 3-564 
 4-377 
 5-446 
 6-854 
 
 (4) 
 
 -016 
 -006 
 
 -004 
 
 -039 
 -033 
 
 (5) 
 -014 
 
 -005 
 
 -012 
 -007 
 
 -6i'8 
 
 1-563 
 1-703 
 1-925 
 2-218 
 2-602 
 3106 
 3-755 
 4-635 
 5-827 
 7-433 
 
 (7) 
 -198 
 •152 
 •140 
 •137 
 -145 
 •166 
 •191 
 •258 
 •381 
 -579 
 
 Average 
 
 3-238 
 
 3-222 
 
 -004 
 
 ... 
 
 3-477 
 
 •235 
 
 Here columns (4) and (5) show how far the graduated select 
 annual premium P^^^ for each age at entry, differs from the 
 ungraduated value for the same age, while column (7) shows 
 how far the annual premiums deduced by Dr. Sprague from 
 the H'^^ data [Journal of the Institute of Actiiaries, vol. xxii, 
 p. 391) differ from the premiums deduced from the 0^^^ 
 Experience. The average difference between the graduated 
 and ungraduated premiums (irrespective of sign) amounts to 
 '015 per £100 assured, a quite insignificant amount; whereas 
 the difference between the premiums representing the earlier 
 experience and those of the 0'^^ Table, representing the 
 experience of 30 years later, are all positive and average "235 
 per £100 assured. 
 
 Only a part of the differences shown in columns (4) and (5) 
 are due to any systematic difference between the mortality 
 as shown in the 0^**^ data and that assumed by the formula. 
 Assuming, however, that the entire differences were due to 
 this cause, it will be seen that the changes introduced into 
 the values of the monetary functions by using Makeham's 
 formula are a very small percentage of the actual change 
 that has occurred in the value of these functions during the 
 course of 30 years.
 
 73 
 
 Although, therefore, the differences between the graduated 
 and ungraduated deaths do at certain points somewhat 
 exceed the limits of the errors of observation, we are justified 
 in using the graduated table as a standard for the future. 
 
 Each case must, of course, be decided upon its own merits, 
 and while the H^ Experience and the 0^ Experience have, 
 with other tables, proved to be amenable to Makeham's 
 formula, the latter cannot be treated as a " law of mortality ", 
 to which all tables may be expected to conform. As already 
 stated, its suitability must be tested, as that of any other 
 frequency curve, but with rather more latitude owing to its 
 practical advantages. In particular the formula is not 
 generally suitable for tables representing the mortality of 
 Eemale Lives. 
 
 In the last lecture we considered various methods of 
 determining the constants of Makeham's formula for /i..^, best 
 representing a given mortality experience, in particular 
 that depending upon the agreement between the totals of the 
 graduated and ungraduated deaths and of their successive 
 summations. We have so far, however, considered the force 
 of mortality as a function of the age only, so that our results 
 are applicable only to "mixed" tables of mortality, not to 
 "select" tables in which the mortality is treated as a function 
 both of the age of the life and of the duration of tlie 
 assurance. 
 
 The formula owes its value, beyond the incidental 
 advantage that it gives us a very simple and effective 
 method of graduation, to the relation it establishes between 
 the value of an annuity upon joint lives of any age and that 
 of an annuity upon the same number of joint lives of equal 
 age. From the formula for the force of mortality according 
 to Makeham's hypothesis 
 
 /x^ = A + Bc^ 
 
 it follows that the force of mortality for any number of joint 
 lives, aged, for example, at entry a;, y, z, is given by the 
 formula 
 
 where c'^'= „ (c^ 4- r'' + c-)
 
 74 
 
 where t represents the period elapsed since the date of entry. 
 As a value of w satisfying this equation can always be found, 
 and is independent of t, it follows that 
 
 It is seen that the relation subsisting between the value 
 of w and the values of x, y, z, involves the constant c only, 
 and not the constants A and B ; hence, any variation 
 introduced into the values of the constants A and B, having 
 reference to the time elapsed since selection and depending 
 only on t, will not affect the relation between the age w 
 and the ages x, y, and z. We can, therefore, write the 
 force of mortality at age x + t for a life select at age x as 
 follows : 
 
 /^w+e = A+/(0 + [B + ^(0]c-+^ . . . (1) 
 and still retain the relation 
 
 when c'" = - (c^ + c^' + c=) . 
 
 o 
 
 Equation (1) may obviously be written in the form 
 
 /t[,Ht=A, + B,c*+^ (2) 
 
 or alternatively, if, as is often more convenient, we work with 
 the values of colog^^.i-^ in the form 
 
 cologioi/[,r]+^ = a^ + Ac-'^+^ (3) 
 
 where A, and B^, or at and ^t, may be any functions of t, but 
 are not functions of x. We can thus represent the rate of 
 mortality as a function both of the age and of the time elapsed 
 since selection and so approximate fairly to the rates of 
 mortality shown in an " analyzed " or " select " mortality 
 experience, while retaining most of the advantages arising 
 from the use of Makeham's formula. The two functions of t 
 have probably a tendency to become constant as t increases 
 but do not necessarily become so within any special period 
 from the date of entry ; they may continue to change slowly 
 throughout the whole duration of the table, and in theory, no
 
 
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 in 
 
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 N 
 O 
 
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 CM 
 
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 r 
 
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 TD 
 
 
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 cr> 
 
 
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 m 
 
 
 
 
 ^ C 
 
 j_> 
 
 CD 
 
 
 
 a_> _ 
 
 <D 
 
 CO -^ 
 
 
 
 cu 
 
 00 
 
 CTDCU 
 
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 -LJ Cj_ 
 
 
 OoO 
 
 
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 10 
 
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 CM 
 
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 b 
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 If) 
 
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 C\J 
 
 CVJ 
 
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 10 
 
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 CM 
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 eg 
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 CO 
 
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 O) 
 
 ^ 
 
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 <i_ 
 
 
 cr> 
 
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 c 
 
 C/J 
 
 
 0) 
 
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 ^ 
 
 (0 
 
 
 C/) 
 
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 K 
 
 .<o 
 
 \S) 
 
 CO 
 
 CM 
 
 O 
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 6 
 o 
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 6 
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 6 
 
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 75 
 
 doubt, should do so, but for practical purposes it is convenient 
 to make them constant after a few years (say 5, or at most 10) 
 from the date of entry, beyond which point it is assumed that 
 the eilect of " selection " has worn off. 
 
 If we set out separately the data for each year of assurance, 
 that is, for each value of t so far as we intend to trace selection, 
 we shall have a series of equations (corresponding to those 
 shown on p. 65 for an aggregate table) for determining the 
 numerical values of the functions /(O), /(I), &c., ^(0), <^(1), 
 &c., the value of c being necessarily that determined for 
 the " ultimate '' table. In other words, the data for each 
 year of duration are treated as representing a mortality 
 table complete in itself. We obtain in this way values for A, 
 and B, or for at and fit for each value of t, so far as it is 
 proposed to carry the select tables. Unless, however, the 
 experience is a very large one, these values will be very 
 irregular. Indeed, in the case of the 0^^ data, which repre- 
 sent a large experience, we have somewhat irregular values 
 for a, and fit, even during the first ten years of assurance, 
 where the facts are most numerous. The approxunate values 
 of at and fit for the 0*^ data are given on p. 157 on " Principles 
 and Methods.^' If these values are plotted out, the resulting 
 curves exhibit certain obvious characteristics, as will be 
 seen by the diagrams opposite where the regular lines show 
 the ungraduated, and curved lines graduated values of at 
 and fit, and the horizontal lines after 10 years represent the 
 values for the experience of 10 years' duration and upwards, 
 when they are assumed to be constant. A period of 10 years 
 would appear from the data to be the shortest within which 
 we can effect anything like a smooth junction between the 
 "select" and "ultimate" mortality rates. 
 
 The values of at rise very rapidly in the first few years 
 of assurance, but after about 6 or 7 years they appear to 
 approach nearly their final value. In the case of fit, however, 
 we see that if the graduated curve were drawn as closely as 
 is consistent with smoothness through the ungraduated values, 
 it would probably not reach the level of the ultimate value 
 •0000406 until after 15 years from entry, and even then it 
 would be below the value of fit for durations of 15 years and 
 over. Hence it would seem that the value of fi, does not 
 become constant until about 20 years have elapsed from tliei 
 date of entry. "NVc may almost say that while the effect of
 
 76 
 
 selection as reflected in the values of the a constant disappears 
 after about 7 years, the effect upon the values of /3 probably 
 continues throughout the whole of life. The explanation is, 
 no doubt, that the a or A constant represents mortality from 
 accidental causes and from non-constitutional diseases of short 
 duration, whereas the /3 or B constant represents mortality 
 due to diseases of longer duration and to constitutional 
 defects. 
 
 Having obtained numerical values of a, and /3^ for 
 successive values of t, it remains to represent these values 
 by convenient formula. The fact that the function /S, does 
 not reach its ultimate value at the end of 10 years from 
 entry, involves either some sacrifice of the agreement between 
 the adjusted and unadjusted values of this function, or a 
 continuation of the analyzed mortality rates beyond the period 
 of 10 years, which is not very convenient. In consequence of 
 this fact we cannot apply the method of moments in fitting 
 a graduated curve to these values. Where the fitting of a 
 frequency curve involves any systematic departure from the 
 original facts, the method of moments often gives 
 unsatisfactory results, and a curve may be produced 
 departing more widely from the observations than if derived 
 by a tentative method. 
 
 In selecting formula? for graduating the rough values 
 of at and /?/, there are certain conditions which should be 
 fulfilled : 
 
 1. A smooth junction between the curves representing the 
 select and ultimate tables. 
 
 2. An agreement between the graduated and ungraduated 
 values of at, ^t in year 0, as a special importance attaches to 
 the rate of mortality in the first year of assurance. 
 
 3. An agreement between the aggregate graduated and 
 ungraduated values of these functions during the period 
 between the date of entry and the ultimate table. 
 
 To conform to these conditions as far as possible, we 
 must select a curve for the values of yS^ which, whilst 
 running smoothly into the constant value at the end of ten 
 years, will represent fairly well the distinctly lower values of 
 ^t in the years immediately preceding. This may be done by 
 representing the difference between log l^+t (the value of this 
 function in the "ultimate^' table) and log /[.,]+, (the value in 
 the " select " table) so far as this difference is due to changes
 
 77 
 
 in ^i, by an expression of the form n{10—tY^c^, where ^ is 
 the ultimate value ; whence we have the corresponding- 
 difference : 
 
 = 27i(10-0/3c-^ 
 
 = 2?i(10-Oc-^y3c-^-+' 
 
 so that yS^ = [l-2?i(10-Oc-^]/8. 
 
 The result of this is to eliminate from the j3 constant at 
 the latter durations part of the effect of selection, and 
 somewhat to exaggerate the effect in earlier years. 
 
 We have now to decide as to the curve best representing 
 the values of at. The method employed will depend veiy 
 much on the character of the experience we are treating. In 
 the C^*^^ Experience it was again found convenient to adopt 
 an expression for the difference of logioZaj+j; and logioZ|;^]+i>, so 
 far as this difference was due to change in at, containing a 
 term similar to that due to /3j v/ith the addition of a further 
 term representing a geometrical series rapidly diminishing as 
 t increased. The final form of the equation for the 0^'^'^ 
 Experience was as under — 
 
 Having determined the form of this equation, the simplest 
 method for determining the constants is to express in terms 
 of them the difference between the computed deaths by the 
 ultimate table of mortality, and the actual deaths for each age 
 or each group of ages and each year of assurance. 
 
 We have in that way a series of equations for determining 
 the values of these constants m, m', c , n, and hence of 
 A, and B, for each value of t, similar in principle to the 
 equations used for determining the values of the original 
 constants A and B. The only point that arises is as to Avhat 
 particular way we are to group the observations to determine 
 those values. 
 
 The value of m in the above formula having been 
 ascertained witli a view to representing as nearly as 
 practicable the effect of selection upon the constant B, there 
 remain in all, four unknown quantities in the fonnula 
 to bo determined, and the actual equations used to
 
 78 
 
 determine them were formed by taking the first and second 
 summations by ages of the whole of these expressions, 
 representing the difference between the " select " and 
 " ultimate '' rates, first for year of assurance alone, and then 
 for the whole of the ten years. 
 
 The selection of these particular groups is, of course, not 
 a, question of principle, but of convenience. Each case must 
 be treated with reference to the nature of the curve of 
 selection, as brought out by the statistics, and such a process 
 adopted as appears to be calculated to bring out the best 
 results in the particular case in question. 
 
 It may happen in certain tables that it is inconvenient to 
 trace out the effect of selection for so many years, and in 
 particular this is the case in a table representing the mortality 
 of annuitants. In such a table the effect of selection (which 
 is here the self-selection of the annuitant) persists for a very 
 long time. In a table of insured lives, owing to the cessation 
 of new entrants in middle life, practically at about age 55, 
 the mortality at the older ages is but slightly affected by 
 selection. In the case of annuitants, where there is a 
 constant inflow of fresh lives up to 75 or 80 years of age, 
 the mortality is affected by this cause throughout the Avhole 
 extent of the table. To completely represent the effect of 
 selection in such an experience will require an elaborate series 
 of tables, showing for each entry age the value of annuities 
 for each year elapsed since entry for many years duration. 
 The tables given in "' Principles and Methods", pp. 124, 125, 
 show that as regards the 0"'" and 0"-^ Experience, and doubtless 
 the same feature would be found to be general, the values of 
 the expectation of life ten years after entry are appreciably 
 greater than the values for the same ages derived from the 
 "ultimate" rate of mortality (e[j.]+io>e,i.+io). Consequently, if 
 the graduated rates of mortality for the first five or ten years 
 froin entry are employed in conjunction with i*ates representing 
 the aggregate mortality after five or ten years, as the case 
 may be, the ultimate values of the annuities, and also the 
 values of the date of entry will on the whole be under- 
 estimated. In any table used for the grant of annuities 
 it is, however, most important that annuities at the date of 
 entry shall not be undervalued, and of only less importance 
 that the values in succeeding years shall be such as may 
 be safely employed in estimating reserves. Any method.
 
 79 
 
 therefore, of treating an annuity experience Avhicli tends to 
 underestimate the values of annuities is clearly unsuitable. 
 Full weight must accordingly be given to the effect of 
 selection, but to avoid the heavy Avork involved in a complete 
 analysis, the expedient may be adopted of computing a 
 hypothetical table of mortality Avhich will correspond to the 
 values of the annuities, let us say, five years from the date 
 of entry. If this can be done successfully and the rates of 
 mortality for the first five years joined on smoothly with the 
 rates in such hypothetical table, we shall then have a correct 
 measure of the value of annuities at entry and for the five 
 years following, while thereafter the values will be slightly, 
 but not seriously, overestimated, an error which Avill be on 
 the right side. 
 
 We may take as our basis either the values of the 
 " expectation of life " or of the annuities at a suitable rate 
 of interest. We will assume the former to be adopted. As 
 these values (e[r]+5) will depend upon separate groups of data, 
 \\z., the entrants at individual ages, it will not be practicable 
 to construct an ungraduated table of jp^ from the formula 
 
 g 
 Px=T—r^ the irregularities in the individual values of e^ 
 
 leading to anomalous results. A better plan will be to 
 graduate the table of expectations. For this purpose, we 
 may assume any frequency curve which will represent these 
 expectations satisfactorily, for example, a curve such as 
 \ogxQex=^a + hx + cx^ + da'?-itfx^. We may employ values of Bx 
 deduced from the experience of individual ages at entry, 
 or we may combine the entrants in quinary groups of ages, 
 taking due account of the true average age of each group 
 of entrants. 
 
 The only point of importance where difficulty arises is the 
 weighting of the different equations. These are not of equal 
 weight because the expectations of life, as deduced from the 
 unadjusted experience, are based upon a smaller or larger 
 experience, as they fall at the extremes or in the middle of 
 the table, and some method must be devised for giving due 
 Aveight to this fact. This may be done by simply Avoighting 
 the equations Avith the actual number of entrants at that 
 particular age, and much may be said for this method 
 although it slightly underestimates the Aveights at the 
 extreme ages. If Ave are dealing, for example, Avith the
 
 80 
 
 values of annuities, and approximately the same result will 
 
 be arrived at when working with the expectations of life, the 
 
 plan of weighting the unadjusted values in proportion to the 
 
 number of lives entering at each age, would make the total 
 
 cost of all the annuities by the graduated table the same as 
 
 by the ungraduated, an agreement that would have some 
 
 practical value. In the alternative, we may consider that 
 
 each value of the expectation of life (or of the annuity, as 
 
 the case may be) should be weighted in proportion to the 
 
 reciprocal of its average error. Thus if e[.i]+5 = A + z, where 
 
 A is the observed value and z the average error, we shall have 
 
 e . A 
 
 -M+5 _ ^^ j^ |g difficult to determine satisfactorily the 
 
 z z — '' 
 
 average error in the value of the unadjusted expectation of 
 life,* the problem being complicated by the incompleteness 
 of the observations due to the " existing." A fairly 
 satisfactory method of estimating the average error would 
 be as follows. Taking the series consisting of the values 
 of e.x for all values of x, each of those values depending on 
 a given age at entry only, we may assume that the observed 
 second differences of these quantities e^-x — ^e^ + ex+i, which, 
 in a well graduated table, would be very small, are due to 
 the errors of observation in the values 6^-1, e^, and e^+x. In 
 any particular group of entry ages, we may say that the 
 average of the central second differences (taken irrespective 
 of sign) will be, on the average, proportional to the average 
 error in e^ for that particular group. t Computing the average 
 values of the central second differences (without sign), for 
 various sections of the table, and drawing a smooth curve 
 through them, we should obtain values from which suitable 
 relative weights for the individual observations could be 
 deduced. 
 
 This Avould be a very fair method of determining practically 
 the weight to be attached to the values of e^. in different parts 
 of the table. Or we may proceed, as was actually done in 
 the case of the annuity experience graduation, by assuming 
 the error in the value of e^ to be a function, first, of the total 
 number of deaths in the experience representing the particular 
 entry age, and secondly, of the age x. This method may 
 appear somewhat arbitrary, but as only the relative weights 
 
 * See, however, the Sixth Lecture, pp. 100-104. 
 fThe average value of Cj; - 1 — 2e^ + e^+i ^'iH '-^e '^^^ times the average error in ex.
 
 81 
 
 are in question, it is sufficient for the purpose. It must be 
 understood that the relative weights adopted do not very 
 greatly affect the results. The values of Makeham's constants 
 as deduced, for example, from the values of log l^ for ages 
 25, 45, 65, 85, thus giving equal weight to the observed 
 value of mortality from ages 25 to 85, would not generally 
 differ materially from the values resulting from a careful 
 system of weighting, although, of course, the latter are to 
 be preferred. 
 
 Assuming the " exposed to risk " to remain unchanged, 
 the average error in the observed number of deaths is 
 approximately + *8 Vnq^i — q) where n is the total of the 
 " exposed to risk " and nq the total deaths. The average 
 percentage error in the total deaths will, therefore, be 
 
 proportionate to +^J ^ • If we suppose that this average 
 
 error is distributed uniformly through all ages passed through 
 by the particular group of entrants, we can then arrive at a 
 rough estimate of the average error in the observed value of 
 ^xi by computing the effect of a change of, say, 1 per-cent in 
 the mortality rates throughout. 
 
 The assumptions here are not strictly accurate, as errors 
 in the value of e^, arise not only from the total number 
 of deaths being greater or less than the expected amount, 
 but from the manner in which the excess or defect of 
 mortality is distributed through the table. The neglect of 
 this second source of error will not, however, seriously affect 
 the relative weights arrived at, and for practical purposes the 
 relative average errors in the value of 6^, will bo dependent, 
 first, on the average error in the total deaths observed in the 
 experience from which it is deduced, and second, on the 
 extent to which a given percentage error in the mortality 
 distributed unifori^ily through the table will affect the value 
 of ejc. The product of these two factors may be taken as 
 representing sufficiently approximately the expected error in 
 the value of e^, remembering always that this estimated error 
 is not an absolute, but a relative measure at the various ages. 
 When this is done, we have, by taking the reciprocals of 
 those quantities, the weights which we shall give to the 
 observed values of e^ in order to determine our constants. 
 
 It is necessary to point out that this process, while suitable 
 for expectations calculated from entrants at a particular age
 
 82 
 
 or small groups of ages, will not apply to aggregate tables ; 
 for in their case tlie percentage error in the total deaths 
 above age x steadily increases as x increases, so that this 
 method would produce weights steadily diminishing from the 
 youngest age to the oldest, which would obviously be 
 incorrect. 
 
 Notwithstanding the important effect of selection on 
 mortality, it is frequently ignored, as in the H"^^ and 0^^ 
 Tables. It is important to consider, therefore, what is the 
 net effect in a mortality table of neglecting altogether the 
 factor of selection. Considerable additional labour attaches 
 to the use of select tables for valuation purposes, and the 
 question may be asked what kind of errors do we make if we 
 neglect the fact that mortality is a function not only of the 
 age, but also of the duration of assurance, and treat it simply 
 as a function of the age as it is treated in the 0^^ and H^^ 
 Tables. In a mortality table representing assured lives the 
 effect will be seen if we compare a table like the H^ Table 
 with a table like Dr. Sprague's Select Table, or if we compare 
 a table such as the O^^ Table with a table like the 0^^^' 
 Select Table : 
 
 Comparison of Annual Premiums for the Assurance of 100 
 
 (S per-cent interest.) 
 
 Age 
 
 HM 
 
 H[M] 
 
 OM 
 
 OtM] 
 
 
 
 Sprague 
 
 
 
 20 
 
 1-427 
 
 1-563 
 
 1-306 
 
 1-365 
 
 25 
 
 ]-625 
 
 1-703 
 
 1-524 
 
 1-551 
 
 30 
 
 1-880 
 
 1-925 
 
 1-790 
 
 1-785 
 
 35 
 
 2-193 
 
 2-218 
 
 2-116 
 
 2-081 
 
 40 
 
 2-589 
 
 2-602 
 
 2-524 
 
 2-457 
 
 45 
 
 3-114 
 
 3-106 
 
 3046 
 
 2-940 
 
 50 
 
 3-801 
 
 3-755 
 
 3-730 
 
 3-564 
 
 55 
 
 4-725 
 
 4-635 
 
 4-641 
 
 4-377 
 
 60 
 
 5-987 
 
 5-827 
 
 5-872 
 
 5-444 
 
 65 
 
 7-705 
 
 7-433 
 
 7-557 
 
 6-853 
 
 If we compare, as is most convenient, either annuity or 
 premium- values, we shall find that the effect of ignoring the 
 element of selection and treating the mortality rates as 
 a function of the age alone is that, at the younger entry 
 ao-es, premiums are underestimated and annuity-values are
 
 S3 
 
 overestimated."^ The 0^^^' premiums should^ properly speaking, 
 be compared with those derived from a table representing the 
 true aggregate of the select tables, but no such table is avail- 
 able. There is a point, which is in general somewhat greater 
 than the average age at entry, at which the two curves 
 representing the premium values for the mixed and select 
 data cross each other, and for the older ages the premiums by 
 mixed tables are greater than those by the select table. The 
 extent of the differences in the premiums is sufficient to 
 render it necessary, in adopting a basis for assurance 
 premiums, to take into account the question of selection. The 
 only plan by Avhich the use of select tables can safely be 
 avoided, is either by adopting a special form of loading 
 or by throwing out altogether from the data upon which the 
 premiums are based those years of assurance which are 
 seriously affected by selection, that is to say by employing 
 a table of the H-*^ "'^ or O'"'^ type. We then obtain a table 
 which at all ages overestimates the values of the premiums 
 and underestimates the values of annuities. 
 
 A table representing " ultimate " rates of mortality, that 
 is, of the 11^ ■■^^ or 0^^'"^ type, is therefore a safe one to employ 
 for the grant of assurances, although not for the grant of 
 annuities. There is, indeed, very much to be said for the use 
 of a table of that kind for assurance purposes, but, to 
 discuss that question, we should have to go into the finance of 
 life assurance valuations, which hardly comes within the 
 scope of our subject. 
 
 With a view of avoiding the necessity for select tables, a 
 device was adopted by the American offices in their first 
 experience denominated the "final series" method. Tlie 
 object was to produce a table not entirely unaffected by 
 selection, but in which its influence would be reduced to a 
 minimum ; a table of mortality similar to that which might be 
 supposed to prevail in an office of great age doing a uniform 
 and steady new business. To produce that result the lives 
 
 *This is shown, in the table above, to l)o tlie case both with the IP^ and U-^' 
 Tables. Unfortunately, however, in neithei- case is the coini)ari80u very 
 satisfactory. Dr. Sprague's H'^'i ])remiunis from the method of their calculation 
 are ])robably somewhat hij^her tlian the true values, and in the case of the 
 O*' Table we are comparing select premiums based in part upon the aggregate of 
 the select tables, excluding lirst ten years from entry, with 0^^ premiums based 
 upon an aggregate table from which there had been a further elimination 
 of duplicate assurances. 
 
 G 2
 
 84 
 
 existing at the close of tlie observations were traced out 
 through a hypothetical future in which they were assumed to 
 be subject to rates of mortality and lapse identical with 
 the rates actually observed in the past among lives of similar 
 age and duration. The minor details of the pi-ocess we may 
 pass over. The result from a financial point of view is that 
 the premiums are still underestimated for the younger 
 insuring ages^ although not to the same extent as in a table of 
 the H^ type, and are overestimated at the older ages, the 
 point at which the values cross the true curve being earlier 
 than would have been the case had the "final series" 
 adjustment not been used. There are some practical 
 difficulties in adopting a method of this kind. One of these 
 is that after some 15 or 20 years' duration the observed rates 
 of mortality for individual ages and years of assurance 
 depend on a very few facts. We then have to apply 
 the very irregular rates resulting from those iew facts 
 to much larger numbers, including the existing lives that 
 have been brought back hypothetically under observation ; 
 so that where these irregularities become inconveniently 
 large, the application of the method must cease; or else 
 these irregular rates must be subjected to some process of 
 graduation before being used in the calculations. 
 
 This difficulty could be met by using a species of 
 QMdr.) Qj. QM(2o) rj^^^iQ fQj. j.-g],g Qf 15 Qj. 20 years' duration and 
 
 upwards, instead of the rates of mortality deduced from 
 individual years of assurance. There are, however, other 
 objections to this method as an expedient for counteracting 
 the effect of a too short average duration of assurance. 
 
 As the rate of mortality amongst assured lives cannot 
 strictly be treated as a function of age alone, but is also 
 dependent upon the duration of assurance, so the rates of 
 sickness in a Friendly Society, or of re-marriage in a 
 Widow's Fund, are affected, respectively, by the duration 
 of membership, or of widowhood. Sufficiently approximate 
 results may, however, be generally arrived at in these cases 
 by treating the rate of sickness, or of re-marriage, as a 
 function of the age alone : in the former case because 
 the effect of selection is not very great and is soon exhausted, 
 in the latter case because the average constitution, as regards 
 the duration of widowhood, of a group of lives passing under 
 observation at a given age will be found to remain fairly
 
 85 
 
 constant (unless the Pension Fund is of recent establishment) 
 and the financial effect o£ a marriage when it occurs is a 
 function of the age only. 
 
 Where, hoAvever, we are dealing with rates of dis- 
 continuance or lapse, it is important that these should be 
 analyzed both as respects age and duration. Owing to the 
 fact that the financial effect of a discontinuance is mainly 
 dependent upon the duration of assurance, very erroneous 
 conclusions may be deduced by treating the rates as functions 
 of the age alone as has sometimes been done. If this course 
 is adopted special precautions must be taken, such, for example, 
 as deducing the rates from a body of lives representing the 
 " existing " some 10 or 20 years back, and excluding from the 
 " exposed to risk " all more recent entrants, as proposed by 
 Mr. A. W. Watson {J.I.A., xxxv, 313-4).
 
 SIXTH LECTURE. 
 
 I 
 
 N the concluding Lecture we shall deal ^vith some 
 miscellaneous points of general interest or arising out of the 
 previous Lectures. We have already dealt with the nature 
 of the modifications of Makeham's formula for the force of 
 mortality, necessary to enable us to represent satisfactorily 
 the mortality shown by select tables such as the 0^^^^. 
 These modifications consisted in treating the quantities A 
 and B or a and yS in the formulas 
 
 /am+< = A + B.c-^-+^; cologioPM+« = a + /3.c-^+^ 
 
 which are constants as regards the variable x, as functions of 
 t the time elapsed since the date of selection. 
 
 It is clear that a similar course may be pursued if any 
 other formula than Makeham's is employed in the graduation 
 of the " ultimate " table. Thus we may write 
 
 where A^ and B,; will in general be such functions that, as t 
 reaches a certain value, at which the select and ultimate 
 mortality rates merge. A,; becomes zero and B^ unity. The 
 form of these expressions employed for representing the 
 effect of selection suggests that a similar form may be 
 employed for representing rate of discontinuance, which in 
 general may be taken to be a function of the duration of 
 assurance and of the age at entry. The same remark applies 
 to such a function as the rate of remarriage amongst widows, 
 which is, similarly, a function of the duration of widowhood 
 and of the age.
 
 87 
 
 Altlioucrli we have dealt at considerable leno-tli with the 
 use of Makeham's formula in connection -with mortality 
 tables^ there are some further remarks to be made as to its 
 employment in certain special cases, more particularly in 
 connection with the age statistics at a Census. 
 
 If we suppose a population which is (1) subject to uniform 
 rates of mortality, corresponding* at the adult ages to 
 Makeham's formula, (2) such that the numbers living- 
 represent the survivors from a number of births increasing 
 annually in a geometrical progression, and (3) is subject 
 to a rate of emio-ration or immio-ration uniform at all 
 ages, then if Z'^. represent the numbers in the population, 
 at a given moment of time, passing through the exact age x, 
 obviously the curve of V x will follow Makeham's formula, 
 and if we write 
 
 d , 
 di "^ 
 
 /' 
 
 X 
 
 we shall have a formula similar to the usual formula for the 
 force of mortality, but with the constant A increased by r, 
 the rate per annum at which the population is increasing; 
 that is to say, the " natural " rate of increase less the rate 
 of emigration. It is true that hardly any population 
 will be found to conform very closely to the above 
 assumptions, but nevertheless it will be frequently found 
 that the population curve for the adult ages does conform 
 to Makeham's formula for h-, although in most cases it 
 will be necessary to adopt Makeham's second development 
 of Gompertz, with the additional constant in the expression 
 for /jijc. 
 
 If the population is given, as is usual, for decennial age 
 groups {e.g., 15-25, 25-35, 35-45, &c.), the values of the 
 ordinate for the middle age of each group may be 
 obtained with sufficient approximation by deducting from 
 each term Uj. of the series repi^esenting the numbers in 
 successive age groups one twenty-fourth of the central 
 second difference 
 
 V 24 y
 
 88 
 
 From tlie values of I'x thus obtained, by writing 
 
 log Z'a;=K + s.;r 4-gr.c^, 
 
 or, log l'x = ^ + s.x + h.x^ + g.c^ 
 
 as the case may be, the constants may be determined as for a 
 mortality table. 
 
 Take, for example, the male population of England and 
 Wales, enumerated at the Census of 1901, as under : — 
 
 Table XIL 
 Jifale Population in Age-groups : England and Wales, 1901. 
 
 
 
 Cpiitral 
 
 
 
 
 
 
 Age 
 Group 
 
 Numbers* 
 
 Ordinate 
 "- 24 
 
 log (3) 
 
 Alog(3) 
 
 A- log (3) 
 
 A3 log (3) 
 
 Col. (4) 
 Adjusted 
 
 (1) 
 
 (2) 
 
 (3) 
 
 C4) 
 
 (5) 
 
 (6) 
 
 (7) 
 
 (8) 
 
 15-25 
 
 94,693 
 
 
 
 
 
 
 
 25-35 
 
 76,425 
 
 76,373 
 
 4^8829 
 
 -•1093 
 •1415 
 
 
 
 4-88349 
 
 35-45 
 
 59,394 
 
 59,371 
 
 4-7736 
 
 -•0322 
 
 -•0138 
 - -0485 
 -•0987 
 
 4-77301 
 
 45-55 
 55-65 
 
 42,924 
 27,913 
 
 42,863 
 
 27,838 
 
 4-6321 
 4-4446 
 
 -•1875 
 - ^2820 
 -•4752 
 
 - ^0460 
 -0945 
 
 4-63269 
 4-44401 
 
 65-75 
 
 14,091 
 
 14,541 
 
 41626 
 
 -•1932 
 
 416319 
 
 75-85 
 
 5,080 
 
 4,868 
 
 3-6874 
 
 
 
 3-68681 
 
 85-aud 
 
 552 
 
 
 
 
 
 
 
 OVPV 
 
 
 
 
 
 
 
 
 *To reduce the magnitude of these numbers, the figures used are those 
 corresponding to a total population (M & F) of 1,000,000 as given in the Census 
 Report. This, of course, does not affect their relative value nor the form of the 
 curve. 
 
 Here, evidently. Col. (6) cannot be well represented by a 
 Geometrical Progression, but with Col. (7) this is possible 
 without very serious changes in the values. This would give 
 a formula corresponding to Makeham's second modification of 
 Gompertz, viz., 
 
 log Z'.^. = K + A.T + A' . a;2 + B . c^ 
 
 for the values of the logs of the numbers living at age x, 
 given in Col. (4). As these numbers are only approximate, 
 and our object is merely to show the applicability of the 
 formula as a base line, we may adopt a very simple method 
 of determining the constants, similar to that used by 
 Mr. Makeham in his paper on the Law of Mortality {J. I. A., 
 xiii, p. 338 et seq.). If the terms in Col. (4) are alternately
 
 89 
 
 diminished and increased by a quantity z, the quantities in 
 Col. 7 Avill become 
 
 -•0138 + 8;^ 
 
 -•0485-82 
 
 -•0987 + 83 
 
 These terms can obviously be made to form a geometrical 
 
 progression by suitably determining z, and their common 
 
 ratio, found by dividing the sum of the second and third 
 
 terms by the sum of the first and second, will be equal to 
 
 14.7-^ 
 
 ^—^ = 2-363. 
 
 Dividing the sum of the first two terms by 3-363 we get 
 
 -0623 
 
 Q.Q/^o = —"01853 as the adjusted first term, giving 
 
 82=— 47'3 and 2=— 59. Hence the transformed series for 
 Col. (4) is as shown in Col. (8), where the progression 
 accurately follows Makeham's second development. 
 
 It is on the whole more convenient to deal with the 
 numbers living above age x rather than the numbers for 
 the decennial age groups. 
 
 If we treat the numbers in Table XII in this manner, 
 representing the numbers living above age x by the 
 expression 
 
 log Qx = K + 7?m-^ 4- « &^ 
 
 we shall have the results set out in the following table, where 
 the values of the constants have been determined by ignoring 
 the extreme values of log Q^ at ages 15 and 85, and equating 
 the sums of the values of the above expression to the values 
 of OogQ25 + logQ:«), (logQ35 + logQ,3), &c., by which means 
 we obtain for the values of the constants 
 
 log a = -006420 (ma^^) = -1-0582 
 
 log &= -035184 (n7/") =— -007933 
 
 K = 6-4222 
 
 The five figure logaritlims of Q^. were employed in the 
 calculation, but, owing to the nature of the process, the fifth 
 figure in the graduated column cannot then be relied upon ; 
 the logs have therefore been throughout cut down to four 
 figures in the table, which is quite sufficient for the purpose 
 of illusti'ation.
 
 90 
 
 Table XIII. 
 
 Male Population living ahove the undermentioned ages. — England 
 
 and Wales, 1901. 
 {Based upon figures in preceding Tahle.') 
 
 
 Proportional 
 
 
 
 logQ'x 
 
 
 logQ'«- 
 
 -logQx 
 
 Age 
 
 Numbers 
 
 logQ:c 
 
 AlOg Q;, 
 
 
 AlogQ'^ 
 
 
 
 
 
 X 
 
 Qx 
 
 
 
 K + ma^ + nh^ 
 
 
 + 
 
 — 
 
 15 
 
 321,672 
 
 5-5074 
 
 - -1514 
 
 5-5059 
 
 -•1498 
 
 ... 
 
 •0015 
 
 25 
 
 226,979 
 
 5-3560 
 
 - -1783 
 
 5-3561 
 
 --1785 
 
 •0001 
 
 ... 
 
 35 
 
 150,554 
 
 5-1777 
 
 - -2179 
 
 5-1776 
 
 -•2177 
 
 ... 
 
 •0001 
 
 45 
 
 91,160 
 
 4-9598 
 
 - -2764 
 
 4-9599 
 
 -•2766 
 
 •0001 
 
 ... 
 
 55 
 
 48,236 
 
 4-6834 
 
 - -3754 
 
 4-6833 
 
 --3753 
 
 ... 
 
 •0001 
 
 65 
 
 20,323 
 
 4-3080 
 
 - -5573 
 
 4-3080 
 
 -•5574 
 
 ... 
 
 ... 
 
 75 
 
 5,632 
 
 3-7507 
 
 -1-0088 
 
 3-7506 
 
 -•9218 
 
 ... 
 
 -0001 
 
 85 
 
 552 
 
 2-7419 
 
 
 2-8288 
 
 
 •0869 
 
 ... 
 
 The practical identity of the curves at all ages except 15 
 and 85, -which values were not used in determining the 
 constants, suggests that very accurate results might be 
 obtained by making nse of a curve of the above form for 
 interpolation of intermediate values of Qa?. 
 
 It has been proposed to employ Makeham's formula to 
 represent the curve of sickness rates at successive ages, and 
 this has been done with a certain degree of success, but the 
 practical advantages of the formula as applied to sickness 
 rates are not very apparent, as it is usually necessary to 
 know not merely the total sickness rate at each age but its 
 division into sickness of various durations, as the number of 
 weeks per annum during the first six months of illness, from 
 the sixth to the twelfth month, after the twelfth month, 
 &c. As Makeham shows {J.I. A. xvi, 414), the ratio 
 Weeks sickness experienced in the year of age 
 Exposed to risk in middle of year of age 
 is not a function similar to /i.^ but to q^, since it has a definite 
 limit, namely, 52, or 1 if the sickness is expressed in years in 
 lieu of weeks. Hence if we represent the above ratio by the 
 symbol cS^,, we should write 
 
 log(52-s^)=A + B.c-^.
 
 91 
 
 Where, by the constitution of a societj^ there is no formal 
 superannuation, the sickness benefit continuing throughout 
 life, it is almost invariably the practice of actuaries in using 
 Sickness Tables for the purpose of computing contributions 
 or valuing benefits to assume that the so-called " sickness " 
 will become chronic after a certain age, 70, 75, or 80. In 
 such cases, as the rates of sickness actually employed ^vill 
 generally be much below the maximum of 52 weeks, we may 
 use log (N—Sx) = A + Bc-^ . The value of N must be determined 
 by trial. 
 
 Mr. King has given an example in the graduation of the 
 values in the Text-book mortality table at the youngest ages 
 of a further application of Makeham's formula, the term Bc-^ 
 in the expression for the force of Mortality representing, of 
 course, equally well an increasing rate of mortality as in adult 
 life or a diminishing rate as in infancy and childhood. 
 
 In the common case of an asymmetrical series the terms 
 of which become zero, or very nearly so, at each end, the 
 following method of employing the "normal'^ frequency 
 curve to represent the series will often be foimd convenient 
 and effective, particularly if the data are presented in the 
 form of a few groups. Let the successive ordinates of the 
 curve be represented by the equation y=f(^x); we shall 
 assume the total area of the curve to be unity and the area of 
 curve between the limits cr= 00 and x = t will be {^ydx. Let 
 us write 
 
 Yt=\ ydx=-^A e-''dt 
 
 so that Y„ = 1 = 
 
 " / 
 
 V TT 
 
 e-'^dt 
 
 where z is a function of t, the form of wliich is to be 
 determined by the data. For most purposes it will be 
 sufficient to treat z as a parabolic function of t, but it will be 
 seen later that there are certain cases in which a different 
 hypothesis as to the form of the function z is to be preferred. 
 An example will make plain the method of proceeding. 
 Take the 0^' data as summarized on p. viii of the volume 
 of Unadjusted Data (Whole-life, Males). In the last two 
 columns of the table tliere is given the ''proportionate 
 distribution per-cent " of the exposed to risk and died. 
 Taking tlie figures there given we obtain the following tables.
 
 92 
 
 The values of z are found by entering a table of 
 
 1 
 
 x/ 
 
 TT 
 
 l-'\li 
 
 for + and — arguments with the values in the second 
 columns of Tables (XIV) and (XV) . We may employ a table 
 such as that given by Woolhouse (J. I. A., vol. xvii, p. 50) or 
 that given on pages 138, 139, at the end of these lectures. Note, 
 however, that in each of these tables the function tabulated 
 
 is -y- e~'-''dt, say I^, for + arguments only, so that the total 
 
 area of the curve from -co to +go is 2 instead of 1. Hence, 
 if Y^ is > ^ we must put 
 
 _1 1 
 
 , 2 2 -s/ttJ 
 
 t"- 
 
 dt 
 
 so that z takes the value corresponding to the tabular value 
 lz = 2Yt—l. Similarly, if Y^ is <^ we put 2- negative and 
 numerically equal to the argument, giving l2=l — 2Y^. 
 
 Table XIV. 
 0^^ Data. Exposed to Rislc. 
 
 Age 
 t 
 
 
 10 
 20 
 30 
 40 
 50 
 60 
 70 
 80 
 90 
 
 Proportion 
 
 Exposed to Risk 
 
 above age t 
 
 1 r^ 
 - ^- e-t'-dt 
 
 Values 
 of 
 
 2 
 
 Az 
 
 A-3 
 
 A^z 
 
 A^z 
 
 A^'z 
 
 1-00000 
 •99991 
 •99584 
 •90060 
 •65989 
 •39810 
 •18795 
 •05951 
 •00927 
 •U0039 
 
 00* 
 
 2^6500 
 
 1-8660 
 
 -9086 
 
 •2915 
 
 - ^1826 
 
 - ^6261 
 -11023 
 -1-6650 
 -2-3750 
 
 -•7840 
 
 - -9574 
 --6171 
 -•4741 
 -•4435 
 -•4762 
 
 - -5627 
 -•7100 
 
 •3403 
 
 •1430 
 
 •0306 
 
 -•0327 
 
 - -0865 
 
 -1473 
 
 -•1973 
 -1124 
 -•0633 
 - ^0538 
 -•0608 
 
 •0849 
 
 •0491 
 
 •0095 
 
 --0070 
 
 -•0358 
 -•0396 
 - -0165 
 
 * Theoretically the values of z corresponding to a total frequency of 1 and 
 are respectively ±oo. As however s=±3 corresponds to Y = ^999989 or 
 •000011, «= ±3-5 to Y = ^99999963 or ^00000037, and s= ±4 to Y = '999999992 
 or -000000008, it will he seen that any value of z over 3 will sufficiently represent 
 the complete distribution or the zero value, and in practice it would be quite 
 sufficient to insert at the ends of the table any convenient value of s over 3, 
 and consistent with the general run of the intervening terms.
 
 93 
 
 Table XV. 
 
 0^^ I>ata. Deaths. 
 
 Age 
 t 
 
 
 
 10 
 20 
 30 
 40 
 50 
 60 
 70 
 80 
 90 
 
 Proportion of 
 
 Deaths 
 above age t 
 
 1 Z'' 
 
 Values 
 of 
 
 z 
 
 Mean 
 Error of 
 
 in last 
 
 place 
 
 of 
 
 decimals 
 
 As 
 
 A-s 
 
 A^*- 
 
 A'z 
 
 A'z 
 
 1-00000 
 1-00000 
 •99925 
 -97565 
 •88854 
 •74174 
 -53731 
 -28908 
 -08169 
 -00590 
 
 00 * 
 00 * 
 
 2-2450 
 
 1-3939 
 
 -8618 
 
 -4587 
 
 •0663 
 
 - -3932 
 
 - ^9856 
 - 1-7806 
 
 ±192 
 ± 48 
 ± 28 
 ± 24 
 ± 23 
 ± 24 
 ± 32 
 ± 82 
 
 -•8511 
 --5321 
 
 - -4031 
 --3924 
 
 - -4595 
 --5924 
 --7950 
 
 -3190 
 
 •1290 
 
 ■0107 
 
 -•0671 
 
 --1329 
 
 --2026 
 
 --1900 
 --1183 
 --0778 
 --0658 
 -•0697 
 
 •0717 
 
 •0405 
 
 •0120 
 
 - ^0039 
 
 -0312 
 
 -•0285 
 -0159 
 
 * See note at foot of Table XIV on precediiicr page. It is to te noted, that in 
 lieu of the integral of the normal frequency function, the function e^/(l + e') 
 may be used, leading to a method of procedure similar to that referred to 
 on p. 51. 
 
 The column containing the mean error or standard 
 deviation of z in the table of deaths is computed as follows. 
 If the total of the series (in this case the total deaths) is n, 
 and the total above a given point (in this case the number of 
 deaths above age €) is tn, then the mean error in m is equal 
 
 \in. (?(, — m) -r^ . . ^ 1 T 1 1 
 
 to A/ . J? rom this can be calculated the mean 
 
 errors of the values in column (2) . The change in the value 
 of z corresponding to a given change in the values of 
 
 —1— e~^'dt in column (2) being known from the table of this 
 
 function we obtain the values in column (4). These standard 
 deviations are not inserted in the table of Exposed to Kisk^as 
 the principle upon which the mean errors in the proportionate 
 distribution of the deaths are computed is not strictly 
 applicable to the table of Exposed to Risk^ Avhen the latter 
 represent observations spread over a long and continuous 
 period, although it would be applicable if the numbers dealt 
 with represented the exposures in a single calendar year. 
 
 If we examine the columns of the successive differences 
 of z in the two tables, ignoring the infinite values of z
 
 94 
 
 corresponding to a total distribution of unity we shall see 
 that they exhibit a remarkable similarity in the nature of 
 their progression, especially from the columns Ah onwards. 
 It will also be apparent that a very small alteration of the 
 orio-inal values of z in either table would be needed to make 
 the fifth differences constant ; that is, we may assume without 
 serious error that 
 
 z = a + bt + c. -^ — - +&c. 
 
 In order to obtain the closest agreement with the 
 original facts due regard would have to be taken of the 
 Aveights corresponding to the mean errors in the value of 
 z as given in the table. But we shall obtain results quite 
 good enough for all purposes by the following simple 
 procedure. It will he observed from the values of the 
 mean errors that the values of z for ages 40 to 70 have 
 approximately the same weight, those for ages 30 and 80 
 have somewhat less weight and finally those for ages 20 
 and 90 much less. 
 
 If we combine the values of z in sets, thus, 
 
 Z20+3z3o + 24o; 2^30 + 3^40 + 250; &C., 
 
 with their corresponding numerical values we shall obtain 
 six equations to determine the six coefficients, a, h, c, . . ./. 
 Into these equations the values Zao and z^ will enter once, 
 the values 2:30 and Zgo four times, and the remaining values 
 five times. We need not compute the numerical values 
 for all these equations as it will be evident that if we 
 write them down and difference them we shall arrive at 
 the following : 
 
 5ft + 5& + c— 2-20 +3230 + ^40 = 7-2885 
 5Z; + 5c + cZ = A (200 + 3230 + 240) = - 2-8505 
 5c + 5cZ + e = A2(2oo + 8230 + ^40) = -7167 
 . 5cZ + 5e +/ = A3(z2o + 3.^30 + ^40) =- '6227 
 5*^ + 5/ =A^(,-,^ + 32-3o + ?:4o)= -2052 
 5/ =A?{z^ + ^z^ + z^ = - -1326J 
 
 From these equations the values of /, e, d, &c., can be 
 obtained with great facility.
 
 95 
 
 Having obtained a formula for z in terms of t, we can now 
 obtain any term in the series and can also obtain the 
 value of y, the ordinate representing the number of deaths at 
 age X {i.e., approximately between ages x—^ and x+h) since 
 
 y= 
 
 
 f, dz 
 dt 
 
 and 
 
 — =A^/— ^rzf+ - A^iv— A^^/+ A^^.. 
 dt ^ 2 ^^3 ^ 4 ^^5 ^ 
 
 It will generally be sufficient to compute the values of y 
 for decennial or at most quinquennial intervals and to 
 interpolate the resulting values of qx or vi^ for the inter- 
 mediate ages. 
 
 The values of the quantities a, h, c, &c., satisfying the 
 above equations, are 
 
 a= 2-24374 
 h=- -849365 
 c= -316624 
 
 d= --186796 
 e= -067560 
 /=- -026520 
 
 It may be of interest to give the adjusted values of z and 
 the distribution of deaths corresponding to these which are as 
 under : 
 
 Table XVI. 
 
 0^^ Data. Deaths. 
 Adjusted values of z and adjusted distrihuiion of Deaths. 
 
 
 
 1 2 
 
 Last column more ( + ) or 
 
 Age 
 
 z 
 
 
 less ( — ) than corresponding 
 
 
 
 column in Table (XV). 
 
 
 
 
 + 
 
 
 
 
 6136-15 
 
 1-00000 
 
 
 • »• 
 
 10 
 
 3-69061 
 
 1-00000 
 
 
 ... 
 
 20 
 
 2-24374, 
 
 •99925 
 
 
 
 30 
 
 1-39438 
 
 •97569 
 
 •00004 
 
 ... 
 
 40 
 
 •86166 
 
 •8884.8 
 
 
 •0OOO6 
 
 50 
 
 •45872 
 
 •74174 
 
 •ooooo 
 
 
 60 
 
 •06640 
 
 •53740 
 
 •00009 
 
 ... 
 
 70 
 
 - -39352 
 
 •28893 
 
 ... 
 
 •00015 
 
 80 
 
 - -98173 
 
 •08188 
 
 •00019 
 
 
 90 
 
 -1-78289 
 
 •00585 
 
 ... 
 
 •0OO05 
 
 100 
 
 -2-U022U 
 
 •OO0O2 
 
 ... 
 
 ...
 
 96 
 
 The principal objection to this adjustment^ paradoxical as it 
 may sound, is that it too closely follows the original facts, the 
 deviations being very much smaller than the probable errors 
 of the observations. This is, of course, due to the fact that 
 we have included too many constants in our formula. A 
 constant fourth difference in the values of z, however, mav 
 lead to anomalous results, and a constant third difference 
 makes the errors of adjustment too great. The best plan in 
 such a case would be to adjust the exposures by using a 
 constant third difference, to recompute the deaths to 
 correspond to the adjusted exposures in the 10 year groups 
 and then employ a constant third difference for the graduation 
 of the death curve. Or, as an alternative, an expression for 
 z may be assumed of the form 
 
 z = -k + 
 
 m 
 
 + 
 
 n 
 
 a-\-x b + x 
 
 and the values of k, m, n, a, h, determined by weighting the 
 equations in a manner similar to that shown above for the 
 fifth difference curve. 
 
 We have used the 0^' data to illustrate the above process, 
 but generally speaking the latter will be found more useful 
 where the data are only available in large groups, and, in 
 particular, where the limits of the series are not well defined. 
 
 In the following table we have a statement taken from 
 Supplement to the Registrar-General's 45th Annual Report, 
 p. cxviii, showing the number of Innkeepers, &c., living at 
 or over certain given ages. 
 
 Table XVII. 
 Innkeepers, Publicans, 6fe. (1881). 
 
 Ages 
 t 
 
 Living 
 
 above age 
 
 t 
 
 Proportional 
 numbers 
 
 J GO 
 
 Values of 
 
 15 
 
 232,890 
 
 10000 
 
 00 
 
 20 
 
 230,280 
 
 •9888 
 
 1-6147 
 
 25 
 
 222,213 
 
 •9542 
 
 1-1929 
 
 45 
 
 105,153 
 
 •4515 
 
 - •0862 
 
 65 
 
 14,451 
 
 •0620 
 
 -1-0877 
 
 It will be seen that more than 50 per-cent of the numbers 
 living are in the age-group 25-45, and nearly 40 per-cent in
 
 97 
 
 the group 45-65. In such a series the usual methods of 
 interpolation would probably give unsatisfactory results. 
 
 If we treat the values of z as having constant third 
 differences, we obtain the following equations, taking five 
 years of age as the unit — 
 
 a =1-6147 
 
 a-^h =1-1929 
 
 a + 5/; + 10c + 10cZ= - -0862 
 
 a + 9& + 36c + 84cZ=- 1-0877 
 
 h, c, and d are the values, reckoning from age 20, of the 
 differences of z. The values of a and h are e-iven immediateJv 
 and solving the remaining equations for c and d we obtain — 
 
 «= 1-6147 
 
 c = -04863 
 
 &=--4218 
 
 d=_-00782 
 
 which enable us to form at once the following series of 
 xj^uinquennial age groups. 
 
 Table XVIII. 
 Innkeepers, Publicans, Sfc. (1881 Census). 
 
 Age 
 
 Interpolated 
 Values of 
 
 Corresponding 
 
 Values of 
 1 2 
 
 Proportional 
 Population between 
 
 t 
 
 Z 
 
 * 
 
 Age 
 t and {t + 5) 
 
 15 
 
 2-0930 
 
 •9985 
 
 97 
 
 20 
 
 1-6147 
 
 •9888 
 
 346 
 
 25 
 
 1-1929 
 
 •9542 
 
 774 
 
 30 
 
 •8197 
 
 •8768 
 
 1221 
 
 35 
 
 •4874 
 
 •7547 
 
 1499 
 
 40 
 
 •1880 
 
 •6048 
 
 1533 
 
 45 
 
 - -0862 
 
 •4515 
 
 1376 
 
 50 
 
 - -3429 
 
 •3139 
 
 1119 
 
 55 
 
 - -5904 
 
 •2020 
 
 834 
 
 60 
 
 - -8360 
 
 -1186 
 
 566 
 
 65 
 
 -1-0876 
 
 -0620 
 
 342 
 
 70 
 
 -1-3534 
 
 •0278 
 
 176-7 
 
 75 
 
 -1-6407 
 
 •01017 
 
 73-5 
 
 80 
 
 -1-9577 
 
 -00281 
 
 22-5 
 
 85 
 
 - 2-3121 
 
 -fX)053 
 
 4-7 
 
 90 
 
 -2-7117 
 
 -00006 
 
 •6 j 
 
 RcjjreEcnting the populiitinii living above age x otit of a total j)()imlati()ii of 1. 
 
 H
 
 98 
 
 It will be seen that tins distribution shows a small number 
 of cases below age 15. This may be avoided if it is desired 
 to commence the curve at and not before that age, by 
 writing 
 
 m 
 
 2 = 
 
 t — 15 
 
 + ai-ht-\-ct- 
 
 giving 
 
 = h + 2ct — 
 
 dt ' (i-15)2 
 
 m 
 
 the term ^ — ,^ being introduced in order to give the high 
 
 t — J. O 
 
 values of z required near the origin, or, we may write as 
 suggested above, in connection with the 0^ data. 
 
 z 
 
 , m n 
 
 = k+ - . h 
 
 a + a; h + x 
 
 the value of a being taken iu this case as equal to —15. 
 
 This form for the value of z will be found very convenient 
 where the series is known to be limited in either direction and 
 the number of groups is small. In certain cases either 
 a or h may be known, and we have, then, only four constants, 
 m, n, h, and h or a, to determine, for which four groups will 
 suffice. Or it may be convenient to assume values for both 
 a and h, in which case with four groups we may write 
 
 t + a 1 + 
 determining m, n, k and c from the data. 
 
 In the case of any statistics intended to be used by the 
 actuary, it is important to consider not only how far they are- 
 suitable for the purpose for which they are to be employed, 
 but also whether the data are sufficient to render the 
 conclusions drawn from them safe. We have already referred 
 to this question in general terms, but it is necessary to 
 consider it rather more closely. 
 
 In practice the actuary has to deal either, (1), with tables 
 based upon a large number of observations; for example, 
 tables such as the 0^^, the Government Annuitants, the 
 Manchester Unity Tables of Sickness, &c., where the
 
 99 
 
 accidental errors due to the limited numbers are practically 
 insignificant, but where, on the other hand, there may be 
 uncertainty as to the suitability of the experience for the 
 case in hand ; or, (2), with data of more limited extent but 
 known to be applicable, as in the valuation of a pension 
 fund of a Friendly Society by tables based upon its own 
 experience. 
 
 In the latter case it is important to be able to form some 
 judgment as to the extent of the probable errors involved in 
 the use of the data and their effect upon the financial values 
 deduced therefrom. This is a problem not susceptible of an 
 exact solution. It is true that if the series of numbers 
 representing the deaths, marriages, or retirements, as the 
 case may be, can be represented by a frequency curve, the 
 probable error of the constants may be obtained in the manner 
 shown by Professor Karl Pearson in his paper on this subject. 
 But these results will be little practical use to us, as 
 the manner in which these probable errors, which are not 
 independent, will affect the monetary values deduced from 
 the graduated rates is too complicated. We can only deal 
 with the problem in a very general manner. We . are not 
 even sure that the ordinary theory of errors is applicable ta 
 such functions as rates of mortality, sickness, or superan- 
 nuation ; indeed, we may well suspect that it is not strictly 
 applicable. 
 
 If the probability of throwing head at a single toss of a coin 
 is one-half, and if in 100 throws 54 heads appear to 46 tails, we 
 do not suppose that the probability of the average number of 
 50 heads appearing in the next 100 throws is affected. But 
 in the case of the probabilities of death it may well be that 
 an abnormally high or low rate of mortality in a given year 
 may affect the probable rate in succeeding years, and that 
 there may be a tendency for the deviations from the average 
 result to correct themselves, a low rate in a given year 
 leaving a larger number, and a high rate a smaller number, 
 of impaired lives surviving, and thus changing for the time 
 being the constitution of the group under observation. 
 
 The " standard deviation " in the value of a^ as deduced 
 from a given experience has not, that I am aware of, been 
 estimated. It will be instructive to attempt this, as an 
 example, for the 0*'(^> table. It will be sufficient to use 
 approximate methods, as the results will be quite accurate 
 
 H 2
 
 100 
 enough for our purpose. We shall assume that we may take 
 
 C0\0geP = m= —^~ 
 
 and that if the standard deviation in log y = cr, then the 
 standard deviation in y = cry.* 
 
 Taking the observations at a given age x, let us put 
 
 exposed to risk =^2- 
 
 graduated or " true " rate of mortality = q 
 
 graduated deaths =nq =0 
 
 actual deaths =nq + z = d' 
 
 observed value ox a =q =(]_+- y 
 
 where, as we have seen, the average value of z is zero, the 
 average value of z'^ = nq{l — q), &c. {see p. 110). 
 Then the observed value of m = m' where 
 
 ^^' — — ''^^ + ^ — _ — i [_ (terms in powers of z) 
 
 nq + z , q 
 
 = m+fiz), say 
 
 = colog,p+/(z) 
 
 It will be found that the average value of /(z) is not quite 
 ^ero though very nearly so, being equal to w^^^ + ^A 
 nearly, a quantity that may be neglected; and that the 
 
 vi^ 
 
 average value of [/(z)]^ is — very nearly, and 
 
 m 
 
 . /i 
 
 V average value of [/(z)]^= y 
 
 Hence, the standard deviation in the " central " death (or 
 marriage or secession or any similar) rate is very nearly equal 
 to the rate divided by the square root of the number of deaths 
 (marriages or secessions, &c.). The errors in log^^? are of 
 
 *If logcey liave the small error a, y will be changed to e'o8.j/+o-=^.eO- 
 =_y(l + 0- + . . .), i.e., the corresponding error in y will be ay ncarlj-.
 
 101 
 
 course the same, but of opposite sign to those in colog ep. 
 Let the observed value of loge^.r be logep'x' We will write 
 
 '^Ogep'x = '^OgePx + ^lx 
 
 where u^ is the error of observation whose value in a particular 
 case is fixed but unknown, the average value over a long* 
 series of similar observations being zero, and the average 
 
 1 £ -> 1 • (colog ep^y (wj.)2 , . ^, 
 
 value ot ^rJ. being ^— ^ — — or ' : where no is the 
 
 nqx nq^. 
 
 graduated number of deaths at age x. 
 
 Taking an arbitrary radix for our mortality table, say l-c, 
 
 the values of log l-^+f for ages above a- will be 
 
 logeZ'j. = loge^x 
 
 logel'x+l = '^Ogelx + l + «x 
 
 'i0gel'x+t = '^0geh- + f+{Ux+1lx+l + --- + Ux+f-i) 
 
 similarly, we shall have 
 
 logD'x = logDj: 
 and for higher ages 
 
 loge'D'x+t = 'i^Oge'Dx+f+ (iix+ »x+i+ • • . + n.v+^_i) 
 
 whence, on the principle of approximation laid down above, 
 
 JJ x + t -L'x + C/i 
 
 Summing this for all values of t from 1 to infinity, we shall 
 have 
 
 Here the quantity in the bracket in the numerator is the 
 error in the value of N'^ as deduced from the observations in 
 relation to the value of D'^,. corresponding to the arbitrary 
 radix assumed at that age. The average value of each term 
 in the bracket is zero, and the square root of the sum of the 
 average values of the sfjuares of these terms divided by Dj.
 
 102 
 
 will give the standard deviation in the value of o! x as deduced 
 from the data, which, omitting the suffix x, becomes 
 
 1 . 
 
 5j ^Ju^m^ + ti,2N,2 + u^'&i + , &c. 
 
 If the mortality table be graduated the standard deviation of 
 the graduated values of a! x will be somewhat less than that 
 of the ungraduated values, but not materially less, except at 
 the ends of the table, the principal eifect of the graduation 
 being merely to produce a smooth progression in values. 
 
 We might assume, for example, that the effect of graduation 
 was about equivalent to substituting the average error of five 
 successive values of N'^- for the error of the middle value. 
 This would give (omitting a quite insignificant term) the 
 expression 
 
 for the error in the graduated value of a ^ in lieu of the 
 expression given above. 
 
 If we shorten the expression for the standard deviation 
 
 of a X from 
 
 1 , 
 
 -^^s/xi,m^^-u,m^^ui-^^^ , &c. 
 
 to its approximate equivalent ; 
 
 1 , . 
 
 fr \/hxii . Na^ + bu.^ . N;^ + Sitj./ . N.o^ + , &c. , 
 
 and, further, take 
 
 25[colog,(p,)]2 
 
 Observed deaths between x and {x + 5) 
 we shall considerably shorten the labour of calculation, and 
 at the same time, by slightly underestimating the required 
 value, make a rough allowance for the effect of graduation. 
 
 We are now in a position to compute a table of standard 
 deviations for %. for quinquennial intervals of age, the 
 principal steps of the working being set out in the table 
 following. The final columns showing the mean errors or 
 standard deviations in the value of ax and the corresponding 
 mean errors in P.^ found by dividing the former results by the 
 quantity (l + flx)^-* 
 
 * If rt,r liave ail error 0-^., then Pj- will have the error 
 
 Vl + a^; / M + «a; + (r.T; / 1 + "x l + ax + tTx 
 
 (Tr 
 
 = (^a:-^(l + a;c)- nearly.
 
 103 
 
 Table XIX. 
 
 Computation of the standard deviations (Tx in the deduced values 
 of ax and ctx -^ (l + '^'j;)^ «« the deduced values ofVx- 
 
 Age 
 
 25[colog,ii)x+.2]- 
 xlOO 
 
 Deaths 
 between 
 
 Ages 
 X and 
 
 = 10-=(2)-(3) 
 
 xlO-* 
 
 1 
 
 Sum of 
 last column 
 
 
 <Tx 
 
 Dx 
 
 (l + a.)2 
 
 
 
 i+S 
 
 
 
 
 = (rx 
 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 
 (C) 
 
 (T) 
 
 (S) 
 
 15 
 
 •1020 
 
 10 
 
 1-020 
 
 20244^ 
 
 21570^ 
 
 •2200 
 
 •00038 
 
 . 20 
 
 •1113 
 
 122 
 
 •0912 
 
 1163^ 
 
 1326- 
 
 •0653 
 
 -00012 
 
 25 
 
 •1266 
 
 924 
 
 •01370 
 
 IWO 
 
 162-5 
 
 •0274 
 
 •00006 
 
 30 
 
 •1525 
 
 3,072 
 
 •004966 
 
 24-48 
 
 52-52 
 
 •0187 
 
 ■00004 
 
 35 
 
 •1981 
 
 5,689 
 
 •003482 
 
 1020 
 
 28-04 
 
 •0165 
 
 •00004 
 
 40 
 
 •2813 
 
 8,152 
 
 •003451 
 
 5-758 
 
 17-84 
 
 •0159 
 
 •00005 
 
 45 
 
 •4410 
 
 10,257 
 
 •004295 
 
 3^864 
 
 1208 
 
 •0160 
 
 ■00006 
 
 50 
 
 •7632 
 
 12,620 
 
 •006048 
 
 2726 
 
 8-215 
 
 0164 
 
 •00007 
 
 55 
 
 1^444 
 
 14,903 
 
 •009694 
 
 1-9S6 
 
 5-489 
 
 •0169 
 
 -00010 
 
 60 
 
 2-945 
 
 16,618 
 
 ■01772 
 
 l-4i5 
 
 3-503 
 
 •0177 
 
 -00014 
 
 65 
 
 6-359 
 
 17,455 
 
 ■03644 
 
 •9770 
 
 2-059 
 
 •0187 
 
 -00021 ' 
 
 70 
 
 14-32 
 
 16,042 
 
 •08929 
 
 •6052 
 
 1-082 
 
 •0203 
 
 •00033 
 
 75 
 
 33-20 
 
 12,172 
 
 ■2728 
 
 •3185 
 
 •4764 
 
 •0228 
 
 ■00059 
 
 80 
 
 78-51 
 
 7,317 
 
 1-0-3 
 
 •1227 
 
 •1580 
 
 -0272 
 
 •00116 
 
 85 
 
 188-1 
 
 2,865 
 
 6^o66 
 
 •03151 
 
 •03528 
 
 ■0364 
 
 -00267 
 
 90 
 
 4o4^6 
 
 692 
 
 6571 
 
 •003659 
 
 •003775 
 
 -0550 
 
 •00705 
 
 95 
 
 1105^ 
 
 86 
 
 1285^ 
 
 •000118 
 
 -000118 
 
 -0966 
 
 1 
 
 •02146 
 
 The result we have arrived at shows that the mean error, 
 or standard deviation, in the values of the 3 per-cent 
 Annuities in an aggregate experience such as the 0^'''^ is 
 about one-fiftieth of a year's purchase from about 30 to 65 
 years of age. Owing to the greater number of deaths at the 
 younger ages in the 0^^ experience this would about represent 
 standard deviations for that Table from 25 to 65. 
 
 If we suppose an experience in which the data were 
 one-hundredth of the extent of the 0^'*^'^ but similarly 
 distributed, it is obvious, from a consideration of the process 
 by which the above result was obtained, that the standard 
 deviations or mean errors in the annuity-values would be 
 ten times greater than the values found above. Hence, with 
 an experience including about 1,000 deaths distributed 
 approximately as in the 0^' '^ data the deduced annuity-values 
 between ages 30 and 60 would on the average be uncertain 
 to about +"20, or from 1 per-cent to 1-^ per-cent of their 
 values. The standard deviations above obtained Avould be 
 somewhat reduced in a small experience by graduating the 
 experience by Makeham oi- Ijy a suitable frequency curve, 
 but not very materially. It would occujiy too much time
 
 104 
 
 to investigate this point, but we may easily find a limit to the 
 effect of any possible method of graduation in reducing 
 the standard deviations of the annuities. In any ordinary 
 experience such as the 0^, where the observed deaths are a 
 small fraction of the lives passing under observation, the 
 errors in the annuity-values will be due, l", to the mortality 
 on the whole being above or below normal, 2", to the 
 distribution of the mortality being abnormal. This latter 
 factor can alone be affected by any method of graduation. 
 Assume it to disappear altogether, and consider the standard 
 deviation for say a^o (O**''^^ 3 per-cent) obtained on this 
 hypothesis. There were approximately 100,000 deaths 
 observed above age 50 in this experience. We have 
 \/l00,000 = 316 nearly, and if we assume the mortality 
 
 above 50 to be throughout subject to an error of + ^y^ of 
 
 the observed amount, this will be equivalent to changes 
 
 A B 
 
 of + ^czTTi and + ,.^T7^ in the values of the constants A and B 
 — 816 ~" olb 
 
 respectively, which, taking the value of A = "00589 and 
 log c = '039, are equivalent in their eff'ect upon the annuity- 
 value to a change of '00186 in the rate of interest per-cent 
 and of '0341 years in the age. The combined effect of these 
 changes upon the annuity-value at age 50 is equivalent to 
 + •0148 as compared with the standard deviation of '0185 
 obtained above. The very considerable standard deviations 
 at the ends of the table would, however, be reduced in much 
 greater proportion. 
 
 The problem dealt with above is not the same as that of 
 determining the standard deviation in the estimated value of 
 an annuity on a single life. This problem, which is also of 
 importance, has been dealt with by Dr. Bremiker in his paper 
 " On the Bisk Attaching to the grant of Life Assurances " 
 {J.I.A. xvi, pp. 216, 285). As this paper is not very available 
 for students and the notation is not modern, it may be worth 
 Avhile to give the following short demonstration. For the 
 sake of simplicity " continuous ^^ functions are used. 
 
 If the annuitant, aged x at entry, die at the end of the 
 time t the loss to the company granting the annuity, or the 
 deviation from its mean value, referred to the date of entry 
 
 will be 
 
 . ._A,-e-'«
 
 105 
 
 and the sum of the squares of all values of this quantity, 
 multiplied by the frequency in each case, will be 
 
 -Jo (-^S— J dt'^'P-^^'^^= -Jo P dt ^'P^^^^' 
 
 Noting that 
 
 dt 
 
 {tPx)dt = l 
 
 - e- 
 
 (•00 7 
 
 and — e-2'* V, itPx)dt = A'x (at rate of interest = e2*—l) 
 
 Jo dt ^ 
 
 we obtain from the above, as the value of the standard 
 deviation of djc, and therefore with sufficient accuracy for 
 
 practical purposes of a^^ { = (7^— - nearly) the expression 
 
 the first term in the bracket being computed at the rate of 
 interest e^* — 1, and the second at the rate e^ — 1. It is obvious 
 that the standard deviation for A^ will be the above expression 
 multiplied by 8 ; and for A^; less the capitalised value of the 
 annual premiums (P^;) (which Dr. Bremiker terms the " Risk 
 attaching to the grant of Life Assurances" by annual 
 premiums) the risk will be the above expression multiplied by 
 (Pj + S). The premium is here supposed to be payable 
 continuously ; if an ordinary annual premium is in question, 
 we should multiply the above expression for cr by (Pa- + ti). 
 The arithmetical values of these " risks " attaching to grant 
 of assurances or annuities computed at 4 per-cent, according 
 to Heym's mortality table (General Widows Fund of Berlin) 
 are given in the paper referred to, and show, as is obviousl}' 
 the case from general considerations, that the "risk", or 
 average fluctuation whether profit or loss, attaching to the 
 grant of assurances at annual premiums is considerably 
 greater than that attaching to their grant at single premiums. 
 In practice the important question for a life office, in tliis 
 connection — and the same considerations apply to other 
 classes of insurance — is the average amount of the annual (or 
 quinquennial) fluctuation in profit due to the deviation of the
 
 106 
 
 death strain from its average or normal amount. In a 
 soundly managed office these fluctuations never approach the 
 point at which stability is remotely threatened, but they 
 become of importance when they are sufficient to produce any 
 serious variation in the rate of Bonus. 
 
 The mean square deviation of e.„ will be found by putting 
 B = in the expression for a^, which in that case takes the 
 
 indeterminate form - which must be evaluated, according to 
 
 the rules of the Differential Calculus, by differentiating 
 
 numerator and denominator. The resulting expression takes 
 
 the same form, so that the process must be repeated, and the 
 
 limiting value of the expression for a''^ when 8 = will be 
 
 found to be 
 
 ' d- , d^ - ,~| 
 
 T 1 
 
 1^.5 = 02 
 
 which may easily be reduced to the form 
 
 = -]-ft'dh^t-(e.cy 
 
 = [mean square duration— (mean duration)^] 
 
 This being the mean square deviation the standard deviation 
 will be 
 
 a=^ [mean square duration— (mean duration)^]* 
 
 the mean deviation irrespective of sign is approximately 
 
 •798o- and the probable deviation -6740-, or very nearly 
 
 A 2 
 
 -or and ^a respectively.* [Of. De Morgan, Encycl. Metro- 
 
 O o 
 
 politan. Vol. 11, p. 460, Art. 149]. If instead of a single risk 
 the average of n risks be taken, all the above quantities will 
 be divided b}' vn. -.: 
 
 * The exact values for the mean deviation irrespective of sign of the 
 expectation of life and of the annuity will clearly he t\ex and t\ax respectively. 
 Where t is in the first instance equal to e^- and in the seeond to the term of the 
 continuous annuity certain </«, = dj--
 
 10^ 
 
 xotp: a. 
 
 On the Evaluation of the Successive Moments of the 
 Binomial Expansion of (p + q)'\ 
 
 These important moments may be found very simply in the 
 following manner. The expanded series being 
 
 p- + np-'q+ ^ii^^p---V'+ . . . +npq-'+q'^ 
 
 = '^iix, where the subscript is identical with the exponent of q, 
 
 the successive moments round the origin will be ^Ux, -xux, ~:>riix, 
 Ac. AVe will first find the value of :i«x, ^xicx, ^x{x-l)ux, &c. 
 We have 
 
 :^ux = (p + q)"=l"=l 
 ^mx = Oxp"+l X iip"-\i+-l X '*-^zi)p"-V-+ . . . +nq" 
 
 ^Wilp + qY-'^^nq. 
 Similarly 
 ^dx-\)ux^\ X 2 X ''^-^^p--Y+-2 X 3 X »fa-l)fa-3)^y.-3^^3 
 
 + . . . +«('/i-l)(7" 
 = n{,i-\)q-y-' + {n--2)p''-\+ . . . +./'--] 
 
 = n{n-\)q-[p + qY--^n{n-\)r 
 and similarly wc shall find 
 
 ^c{x - \){x - •2)ux = n{n - l)(n - •l)(j\ and so on. 
 
 "V+. . . +f~^
 
 108 
 
 Hence we shall have 
 
 :Zu^=l 
 
 2."nx = 2.XUX --= nq 
 
 v3,, ^ x{x-\) _ iiin-l) 2 
 ^ Ux — ^ — - Ux — q 
 
 v4„ yx{x-\){x-'l) n{n-\){ji-i) . 
 
 ^ Ux = ^ Ux = ;; q 
 
 b b 
 
 ^5 • s, x(x-l)(x-2){x-3) _ nin-l)(n-2)(n-3) ^i 
 
 Ux = ^ tt:; Ux = ~, q 
 
 24 
 
 24 
 
 by the formula on page 59. Hence we have {see the demonstration 
 in Note E, page 124), using vin for the wth moment round the origin 
 
 711(1 = '^Ux = 1 
 
 uii = 2"»j; = uq 
 
 m.2 = 2^Ux + 2"Hx = «('«. -\)q- + nq 
 
 m^ = Q^Six + Q^^ix + ^'ux = n{n - 1 ) {n - 2 )q^ 
 
 + 3n(n- l)q' + nq 
 
 Wi = 2iy,hix + 362;^«x + li^^ix + 2-7<x = n{n - l){n - 2){n - 3)q^ 
 
 + Qn{n - l){n - 2)q^ + 7n{n - l)g^ + nqJ 
 
 These last equations may be found directly, by means of successive 
 difterentiation, according to a method suggested by Bertrand 
 {Calcul des Prohabilites, Chap. IV, Art. 62). AYe have 
 
 / \n II 11-1 UAn — 1) «-0 '> 
 
 ( P + q) - P + np" 'q + -^- 'f' -q- 
 
 Up + qY = \0xv"+\ 
 do L 
 
 d 
 
 + . . . + npq^ ^ + q'' 
 
 X n2f + 2 X -^^ -p" -q 
 
 2 
 
 and q. -~(p + q)" = 
 dq 
 
 + . . . +{n-l)npq'' - + ??'?" ^ 
 lxnp''-'q + 2x'"^'"--^^p''-'"q^+ ... +wg"] 
 
 1st moment.
 
 109 
 
 Similarly, if we differentiate the last series with respect to q, 
 and multiply the result by q (to restore the power of q which 
 is lost in the differentiation) we shall have 
 
 = 2nd moment ; and so on, so that 
 
 [fth moment] =q-r\it ~ l)th moment.] 
 dq 
 
 Thus, the first moment 
 
 = qUp + qr 
 dq 
 
 = nqip + q)"-'^ 
 Second moment 
 
 = Q T [m(p + ^)" ~^]= nin - 1 )q-(p + '?)""- + nq(p + q)" ~ ^ 
 dq 
 
 Third moment 
 
 = ql[n(n-l)q%p + qr-' + nq(p + q)"-'] 
 dq 
 
 ^n(n-l)(n-2)q%p + qr-' + 2n{n-l)q-ip + qr-' 
 
 + n{n - \)q^{p + q)" ~ " + nqdi + g)" " ^ 
 
 = n{n-\){n-2)q^{p + qY-''-VMn-l)q-{p + qY-- + mip + q)"~^ 
 Fourth moment 
 
 = q— [third moment] 
 dq 
 
 = n(n - l)(n - 2)(n - S)q'{p + q)"-' + 'Mn - l){n - 2)q^p + q)"-^ 
 
 + Sn{n -\){n- 2)q\p + qT'^ + 6n(w - \)q\p + qf ' ~ 
 
 + n{n- \)q-{p + q)'"'~- + nq{p + q)'"-~^ 
 
 = n(n - l){n - 2){n - 3)q'(p + q)>'-' + 6/iU - l)(n - 2)q%p + qY-^ 
 + 7n(n - l)q-(p + qy"- + nqip + g)"-^ 
 
 Putting unity'" for all the powers of (p + q), these expressions 
 are the same as previously found — see eiiuations A. 
 
 * 'I'liis may not be tlone at any earlier stape because the differentiations are 
 with respect to 7, takimi p constant, wluTt-as to substitute p-\-q = \ before 
 f1Ili!^biIlg llie (liffcrentiationB would niakey> vary with q.
 
 110 
 
 We have thus obtained the moments round the origin. Thence 
 the moments round the mean may be found by the formulae on 
 p. 41. Thus 
 
 fX2 = m-i - {mi)'' = n{n- \)q- + nq - trq' = nq - nq' 
 
 = 7iq{\ -q) =7ipq 
 
 pi3 = m-i - 3/«i . [1-2 - mi 
 
 = n{n - 1)(?« - 2)5'^ + 3n(« - l)q' + nq 
 - 3irq~ + 3n'q^ - n^q^ 
 
 = nq - Snq- + 2nq^ = nq{ 1 - 3g + 2q") 
 
 = nq{l -q){l - 2q) = npq(p - q) 
 
 Hi = riu - 4?«i . (x-i - 6m{'.fi2 - r«/ 
 
 = nq[{n^ - 6n' +11%- 6^ + 6(n- - 3% + 2)q^ + 7{n-l)q + l 
 - inqil -3q + 2(f) - Q>n^q-{\ - q) - wV] 
 
 = nq[3{n - 2)q^ - 6(w - 2)g- + {3n - 7)g + l] 
 which reduces to 
 
 nq{\-q)[3{n-2){\-q)q+l] 
 
 = npq\3{ii - 2)pq + l] 
 
 It is evident that all the even moments must involve p and q 
 symmetrically ; while the odd moments will invoh^e a symmetrical 
 function of p and q, together with the factor {p - q), because they 
 must vanish when p = {7 (i.e., when the curve is symmetrical) and 
 must only change sign when p and q are ti'ansposed. 
 
 It may be convenient to repeat here the Author's demonstration 
 given, J. I. A., xxvii, 214, of the value of the average deviation from 
 the mean irre'^pective of sign, that is, treating all the deviations as 
 positive. 
 
 If we suppose the event to happen 111 times in the n trials the 
 deviation from the mean number np will be (»i - np)) which, since 
 p + g is always equal to 1, may be put in the form [inq - {n — m)p\
 
 Ill 
 
 This will be positive or negative as m is > or < np ; and the 
 probability of this particular deviation will be 
 
 71 . . . {in+ 1 ) „, „ - w 
 \n-m 
 
 The gi'eatest positive deviation will be nq (when the e^'ent 
 happens at all the n trials) ; the greatest negative de^"iation - vp 
 (when it fails at every trial). 
 
 Hence, we have the following scheme, in which m is to be taken 
 as the next integer <np. 
 
 Possible Deviations from Mean Besidt np. 
 
 ^ 
 
 
 
 
 fcfi 
 (A 
 
 Magnitude 
 
 Probability 
 
 Mrtgnitude x Probability 
 
 
 nq 
 
 pn 
 
 Mp"5' 
 
 
 {n-l)q-p 
 
 np^^-^q 
 
 n{n — \ )|j» -ig- — Hjo"gr 
 
 
 {n-2)q-2p 
 
 n.(n — V) „ ., .. 
 
 n(n-l)(n-2) „ ., ., , ,^ , „ 
 
 9 
 
 > 
 
 
 
 
 ! 
 
 
 (w + 1)2 — {n—m — l)p 
 
 \n-m-l ^ ^ 
 
 « . . . f HJ + 1 ) 
 
 n — m — \ ^ '■ 
 
 m . . . (w + 2) 
 » — >n— 2 
 
 
 mq — (n — m)p 
 
 n . . . (m+ 1) 
 n — m ^ ^ 
 
 n . . . ni 
 
 
 Vii — m^^ 
 
 > 
 
 
 
 »...(»» + !) 
 
 |» — Wl — 1 ^ ^ 
 
 If 
 
 
 ... ... ... 
 
 
 
 q-{n-\)p 
 
 w^2*^ - 1 
 
 npqn - n . (m — l)^2gn - 1 
 
 
 i 
 
 1 
 
 g» 
 
 — 7ipq'>' 
 
 If the final column of ])roducts is examined it will be seen that 
 each positive term is cancelled by a similar negative term in 
 the succeeding jiroduct. Hence, the total of the products, that 
 is to say, the average deviation, is zero, showing that np is the true 
 mean result, the positive and negative deviations from which
 
 112 
 
 exactly balance each other. Of the terms above the horizontal 
 line, representing the positive deviations, the sum is, of course, 
 
 equal to the only uncancelled term, - ; ' ' ^"'+y p^'^+Y-'", 
 
 \ n - 111 - 1 
 
 and similarly of the terms below the line representing the 
 negative deviations the sum is ^ -— ^p™+^g" '"■. 
 
 n — m—1 
 
 Hence, the average magnitude of the deviations, that is, the total of 
 3very possible deviation multiplied by its probability, regardless of 
 sign, is 
 
 o . 
 
 n. 
 
 .(m+1) 
 
 71-731-1 
 
 vi+l^n-m 
 
 2\n 
 
 ■m n-m— I 
 
 pm+Y-^ri _ _ (^) 
 
 which, since the sum of all the probabilities is necessarily 1, will 
 also be the average or mean deviation. This result is exact, not 
 approximate, but where n and rii are large numbers it is necessary to 
 simplify it by tlie^use of Stirling's formula, which gives for large 
 numbers \n= v27r7i"+*e~" nearly. 
 Put {a) into the equivalent form 
 
 2\n{n-m) „t+i^n-m . 
 'III 71 - m 
 
 using Stirling's approximation to the factorials, we have 
 
 = n''+hii-^"'^Hn - m)-^''-"'+^n - m)p"'+Y~" 
 
 J2 
 
 TT 
 
 Since m is the integer immediately below np, we may write 
 m = np-h ; n- m = iiq + k (where k is a fraction) ; hence, we get 
 
 V- »'"{»Ki - -); 
 
 -lUJ-i+k 
 
 _ \ nq/J 
 
 I, k 
 
 -"5 + 5 
 
 i-k 
 
 -k 
 
 np+l-kq)iq+k 
 
 > TT \ npJ 
 
 V Tiq/ 
 
 1 
 
 
 np 
 
 but where np and nq are large numbers, I: being a proper fraction, 
 the last factor is A^ery nearly equal to 1, and (1 j and 
 
 1 H — -) are very nearly equal to c^' and c~^" respectively ; hence, 
 
 nq/ 
 
 the above expression reduces to 
 
 a/ "■ npq = -79788 Jnpq = ^ Jnpq very nearly.
 
 113 
 
 Although this result has been obtained on the assumption that 
 np and nq are large, it will be found to be very approximate 
 even for small numbers. As an extreme case, suppose 120 lives at 
 risk, the probability of death in each case being -02 ; the 
 "expected" deaths would then be 2-4, and the extent by which 
 the actual deaths would, on the average, exceed or fall short of this 
 number would be given by the formula as 
 
 - n/2-4x -98 = 1-227. 
 5 
 
 The true value of the average deviation given by formula (a) is 
 2 120J2M18(.oo)3(.98)ns 
 
 = 1-243, 
 almost identical with the approximate result above,
 
 114 
 
 NOTE B. 
 
 On the Use of Logarithms of the Unadjusted Terms 
 
 OF A Series. 
 
 Consider the number of cases out of a given series falling into 
 a particular group ; or the number of deaths, or analogous events, 
 at a given age or group of ages, accruing out of a given number at 
 risk. Suppose the series to consist of n cases in all, and let the 
 true probability of any case falling into the particular group be p, 
 and let m = np. Let the observed number of cases in the group be 
 m = m + z, where, as we have seen, ~ has an average value of zero, 
 z^ has an average value of 
 
 n 
 z^ has an average value of 
 
 np{\ -p){l- 2p) ^ ^Il(lLzlllk^M, &c. 
 
 If we operate with the logs of the observed quantities m, we 
 must avoid by arbitrary grouping cases in which m is zero, or m in 
 very small when the logs become infinite or very great ; but when 
 this is done we shall still find the logs of the ungraduated numbers 
 less on the average than the values of the graduated (or true) 
 numbers. This may be easily seen from a simple example. Let 
 11 = 4, and p = l,m which case m = np = 2. The observed values of 
 m may be anything from to 4, and we shall have the following 
 possible cases : 
 
 Values of 
 m' = m + l 
 
 Relative 
 
 frequency 
 
 of these values 
 
 log m' 
 
 Products 
 (2)x{3) 
 
 (1) 
 
 1 
 2 
 3 
 4 
 
 (2) 
 
 
 
 Ti? 
 
 4 
 IS 
 
 1 
 
 (3) 
 
 ^-•097 
 
 •000 ; 
 
 •301 
 •477 
 
 •602 
 
 (4) 
 
 (say) -•030 
 
 •113 
 •119 
 
 •038 
 
 Total 
 
 1 
 
 
 •240
 
 115 
 
 Here, to avoid the cases in which the observed value of m' is 
 zero, we have combined the first two groups, taking four cases in 
 which m' = 1, for one case in which m =0, thus giving an average 
 
 value of m'=-, the logarithm of which is --097. Notwith- 
 5 
 
 standing this device our average value of log m' is only -240 as 
 compared with the value of log m = -301 (where m = 2 is the true 
 value or average value of vi). 
 Assume that on the average 
 
 log[Hi'(l +l')] = \ogm 
 
 ==log[(m + ,~)(l+/.-)] 
 
 = \ogm+ - - -^ + -^, &c., + /.• - ~+ Sco. 
 m 2m-' dm 2 
 
 Whence 
 
 k— — h &c. — — - + -^^, — £-^, &c. 
 
 Insert the average values as given above for z, z^, &c., 
 
 1 Jc^ c n- m (n - m){n - 2m) , „ 
 
 A— - + &c. = - — - ^ -^V^3 + &c., 
 
 2 2nm dn'm" 
 
 or, omitting terms of the second order, 
 
 , n-m , 
 
 A;= ^ nearly, 
 
 2nrn 
 
 which, again omitting terms of the second order, may be written 
 
 n - in 
 
 h = 
 
 2nm 
 
 log [m'il + k)] = log m' + ^~~^ = log 
 
 , 1-;/ 
 
 where p is the observed value of the probability p. 
 
 If this expression be substituted for log m in the example 
 given aljove, we should have as the sum of the products of 
 col. (2) X col. (3) the value -309, which is very much nearer the 
 true value -301 than the uncorrected value in the above table. If 
 we take larger numbers, as n= 100, m = np — 10, we shall find ])y a 
 
 similar process the average value of logiofm'n -^ j is -99987 as 
 
 compared with the true value of log /y« = 1-00000. Where the 
 num1)ers n and m are very large, the correction, of course, Ijecomcs 
 insignificant.
 
 116 
 
 It may be shown in like manner that, if we are dealing with the 
 reciprocals of the observed values, then, on the average, 
 
 111 
 7 = - nearly, 
 
 m +1 -p in 
 and again, on the average, 
 
 III ^ — — = V 
 
 m 
 
 Eeverting to the question of the use of the logs of the 
 ungraduated quantities, it will be found that if the above results 
 are made use of in practice, the logarithms will be over-corrected. 
 The reason for this is that we do not eventually arrive at the true 
 values of log rii and log p, the graduated values being still aflected 
 by an outstanding or unbalanced error. If our series consists of a 
 large number of groups, these outstanding errors will be com- 
 paratively sm.all, and the above correction will not be much in excess ; 
 but if the number of groups is very small, our graduated quantities 
 must necessarily follow rather closely the original values, and the 
 use of the above formula would largely over-correct the series. 
 Suppose, for example, Ave had a series of ten groups. We should 
 require about five groups to obtain the general form of the curve, 
 or to determine the constants of any frequency curve employed,, 
 hence the errors of the gi'oups would only be reduced by the ratio 
 
 of approximately — r- and the correction h as shown above should 
 be reduced by half, and proportionately in other cases.
 
 ll: 
 
 NOTE C, 
 
 Ox THE Rationale of the Method of Least Squares. 
 
 In statistical work it often happens that a number of constants, 
 entering into the known mathematical form of a given function, 
 have to be evaluated from a much greater number of observed 
 values of the function. We may, for example, have three constants, 
 such as X, y, z, in the expression Ix + )ny + nz = F, and fifty observed 
 values of F (embodying different values of the coefficients 
 /, m, n) from which to determine the constants. If the observed 
 values of F were rigidly accurate, any three of them, or any three 
 combinations, would suffice to determine the constants, and it 
 would be immaterial what set of three was selected, since all would 
 lead to the same results. But generally the observed values of F 
 will be affected by errors of observation and hence will not be 
 strictly consistent ; and taking the above example each of the 
 
 ' — ^ — - — =1960 diflerent sets of three individual equations 
 
 would in general produce different values of the constants : so that, 
 apart from the prohiljitive amount of labour required in the solution 
 of so many equations, we should have no means of deciding which 
 was the best or most advantageous solution, or how to combine the 
 solutions in order to ol)tain the best average results. The method 
 of least squares sujjiilies the means of combining the original observa- 
 tions in such a manner as to produce a number of eiiuations, equal 
 to the number of. unknowns (in the above example, three), the 
 solution of which by the usual process leads to the most probable 
 values of the unknown constants. 
 
 Suppose that the observed function F is a linear function of the 
 variables x, y, r, of the form Ir + iiiy + n: . . . , and that the errors 
 in the observed values of F follow the "normal law", so that the 
 l)robability of an error k is projjortional toe~'^^^''^, where the standard 
 
 deviation of F is —7-. We shall furllirr suppose that the equations 
 V2
 
 118 
 
 have been so Aveighted that the vahie of c is the same in each of the 
 observations, or that the " precision " is uniform. Thus, for 
 example, if in a given equation the proljability of an error of k in the 
 observed value of F is proportionate to e~^'^'^^, with a standard 
 
 deviation of -4-, then multiplying the equation by -, we shall 
 
 have an equation with a standard deviation of — 7= as before, and 
 
 n/2 
 the probability of an error k will be proportional to e~^"''^" as 
 required. 
 
 Let there be t equations as follows (where t is supposed greater 
 than the number of unknowns, say s) : 
 
 lix + miV + riiz + . . . -WiF = ki 
 
 lox + vioy + noz + . . . -W2F = h. 
 
 (A) 
 
 Itx + mty + ntz+ . . . -tVtF = kt- 
 
 where F represents the true value of the observed function and 
 A'l, k-2, . . ■ the errors of observation. The chance of the errors being, 
 by hypothesis, respectively proportional to e~'^'^^''\ g-^2^/c- _ _ _ ^j^g 
 chance of the conjunction of these individual errors Avill be propor- 
 tional to e"''''^''''"'*'''"-'''^^"'"- • -^j which will obviously have its greatest 
 value when the quantity in brackets is a minimum. Now, the most 
 probable values of the constants will be those that give the greatest 
 probability of the observed event, i.e., the happening of the given 
 combination of errors. Thus, the most probable values will be those 
 making [jiY/c^ + k.j'^/cr + . . .] or ~kt^/c^ or 2A'(^ a minimum— hence 
 the name " method of least squares." 
 Now we have 
 
 2Ar = 2[(/,7: + mty + ritz + ... - w.F)^] 
 
 and since ,r, y, z . . . are supposed to be independent, the minimum 
 value must correspond to such values of .r, ?/, ~ . . . as will make the 
 partial differential co-efficients of this expression, with respect to 
 X, y, z . . . , all vanish.'"' Hence we must have, omitting a 
 common factor 2, 
 
 ,A 
 
 (B) 
 
 ^[tStX + mty + ntZ+ . . . -mvF)] = 
 ':^[mt{ltx + mty + ntz+ ... - w^F)] = 
 :^[nt(ltx + mty + ntz+ . . . -WtF)] = 
 
 ^c, &c., &c. 
 
 * These conditions, though necessary, are not in general .sufficient to ensure a 
 minimum, but in this ease it is obvious that a minimum exists because high 
 negative values and \\\gh positive values of x, y, e . . . alike give large values to 
 the function.
 
 119 
 
 (the sunmiation extending to all values of /) as the system of 
 equations, s in number, for determining the most probable values of 
 .r, y, z. Hence the rule : 
 
 " First prepare the equation by multiplying each by its proper 
 weight (the reciprocal of the probable error or standard dcA-iation), 
 thus giving a set of equations with a uniform p.e. and s.d. Multiply 
 each equation by the coefficient of x and add all the results together ; 
 next multiply each by the coefficient of y and add all the results 
 together, and so on : the resulting aggregate equations, solved in 
 the usual manner, give the most probable values of the constants." 
 
 It will be seen at once that if there is only one constant to be 
 determined, the method based on the normal law of error gives 
 the weighted aA'erage, i.e. the total of the weighted values divided 
 by the total weights, as the most probable. Conversely, it may be 
 shown that if the Aveighted average is the most probable value, then 
 the facility of error must follow the normal law. Apart, however, 
 from any hj-pothesis as to the law of error, it may be shown 
 mathematically that the method of least squares gives results which 
 become more and more nearly accurate as the number of observations 
 increases. Considerations of a more general kind will also lead to 
 the conclusion that the method must produce very good results. 
 Without giA-ing any definite form to the law of error, it is obvious 
 that large errors are less probable than small, and that the most 
 advantageous system of values for the unknown constants Avill be 
 that which produces, on the whole, the smallest numerical dcA-iations 
 (irrespective of sign) between the adjusted and observed values of 
 the function. Xow, if the law of error is supposed unknown, we 
 cannot investigate mathematically the conditions required to produce 
 a minimum deviation irrespective of sign ; and the simplest function 
 of the errors which is independent of sign is the square of the errors, 
 which will be the same for a positive or negative deviation. 
 and at the same time attributes a rapidly increasing importance, or 
 disadvantage, to the errors as they increase in magnitude. Hence 
 we can see, in a very general way, that a method which gives 
 a minimum value to the sum of the squares of the errors, is likely 
 to lead to satisfactory results consistent with elementary notions as 
 to the nature of the errors. Moreover, in actuarial work we usually 
 have to do with numbers sufficiently large to make the normal law 
 of error very near the truth. 
 
 Reverting to the system of equations (B), it will easily be seen 
 that if F is a parabolic function of the form x + ay ■ira'z + . . . the 
 equations for determining .r, y, z, . . . ^c, are equivalent to 
 reproducing ^Y , laY , ^^Y {^ti^, 'ZwY.a, &c., if the eciuations are 
 weighted), i.e., the successive moments of the observations.
 
 120 
 
 It has, so far, been supposed that the function F is a linear 
 function of the constants x, y , z, . . . If this is not the case, 
 suppose that the equally weighted equations, from which the values 
 of a;, y, z, . . . are to be found, are of the form 
 
 fi{x, y, z . . .)- WiF = Jci 
 foio; y, z . . .) - WoF = h - . 
 &c., &c., &c. 
 
 ...(c) 
 
 where /i, /o ■ . • are known functions of the variables x, y, z . . . 
 By means of t of these equations, or of t combinations from 
 amongst them, or otherwise, find approximate values of x, y, z . . . 
 say x},y^,z^ . . . ; and suppose that x = x^ + 8x, y = y^ + 8y, z = z^ + 8z, 
 &c., where it may be supposed that 8x, 8y, 8z . . . , representing 
 small corrections to be found, are so small that their squares 
 may be neglected. Then if 
 
 dfx 
 
 I 
 
 fi=fiix\ y\ z' . . .) ; fl = f-J,{x\ y\ zK..) &0 
 
 ax ax 
 
 + 
 
 and so on, equation (C) will become 
 
 &c. 
 
 &c. 
 
 . . - tViF = ki 
 &c. J 
 
 y 
 
 (D) 
 
 These equations are linear functions of the small corrections 
 8a;, 8y, 8z . . . which can accoixlingly be found by the rules already 
 derived ; and hence are found the corrected values x = x^ + 8x, 
 y = y^ + 8y, &c. The process can l)e repeated, if greater accuracy is 
 desired, until the corrective terms become insignificant. 
 
 In the important particular case of a graduation by Makeham's 
 formula, the original equations are of the form 
 
 U'_^.^i.(A + B/^-^) 
 
 da- 
 
 W 
 
 ^■+:;E 
 
 
 I -i -2. 
 
 {w being the " weight "). Approximate values of the constants, say
 
 121 
 
 A, B', c, being found, the resulting equations for determining 
 3A, SB, Be are as follow, /x'^ representing A' + B'r;'^ : 
 
 (2^)8A + {^w. c "^ bsB + {^w^c '" L- + l^Bc 
 
 (:Sw . B'x +lc" 2 )SA + (2w . B'x + - c'-'')8B 
 
 + {22r.B'-(.r+l)%'--i|s, 
 
 1 ,a;— 
 
 For an example, see J.I. A., xvii, 161-71.
 
 122 
 
 NOTE D. 
 
 On the Use of the Binomial Curve to Repkesent 
 A Continuous Series. 
 
 If the Binomial curve y = , — j- — pY'' ^ niade high contact with 
 
 \x \n-x 
 
 the axis of x at the points x= -I and x = n+l, where y becomes 
 zero, it could be conveniently employed to represent a continuous 
 curve in lieu of representing merely isolated ordinates ; as in 
 that case the moments of the continuous curve would very 
 closely agree with those of the isolated ordinates. The same 
 would be true of any series of equidistant points on the 
 curve supposing these to be fairly numerous. If, for example, 
 we suppose the values of y tabvilated for every integral value 
 of xh, then the tth. moment of the curve would be increased 
 by multiplication by the factor h\ and from the observed 
 numerical values of the first 4 moments h and the remaining 
 constants could be obtained. As, however, the curve y cuts the 
 axis of X at an angle at both limits, this method of proceeding 
 will lead to approximate results only when n is fairly large. 
 
 The area of y treated as a continuous curve may be approxi- 
 mately determined from the well known approximate formula 
 
 /•"+! ,1 11 rfdv\ fdy\ 1 
 
 j_^ y.dx= -.j-^ + yo^. . . + yn+ -^.A+, + i^LltJ-i -UAJ 
 
 y-i and yn+\ being of course equal to zero and the series yo + yi + ■ • -Vn 
 
 is the expansion of ip + qY^ where we assume p + q=\, and is 
 
 1 1 
 
 therefore also =1. As the factor jT vanishes for a; = - 1 ; and 
 
 \Jj Jv — Jy 
 
 vanishes for x^n + l, we have 
 
 \dx/ n+i \x^ ■'^ dxn-x/n+i n+l
 
 123 
 
 d 1 . , , , 
 
 si^ce — |- as IS known = 1 when x-- -I. Hence the area of the 
 
 curve y becomes 
 
 ydx=l + ^.~^.(^ +£_) 
 
 •'-I 1271+lV pq J 
 
 Analogous expressions can be found for the approximate value of 
 the moments 
 
 fxydx, Jx-ydx 
 
 fydx ' fydx ' '^''- 
 
 but the relations which result do not lead to sufficiently convenient 
 formulae for practical use.
 
 124 
 
 NOTE E. 
 
 On the relations between the Successive Moments and 
 THE Successive Summations of a series. 
 
 The relations given on p. 60 may be systematically demonstrated, 
 and developed to any extent that may be required, by means of the 
 ordinary interpolation formulae combined with a table of the 
 power-differences usually known as the " Differences of Nothing " 
 —see Text-Book, Part II, Ch. xxii. Art. 11 ; Sunderland's "Notes 
 on Finite Differences ", pp. 24-5. 
 
 We have, by the ordinary interpolation formula, 
 
 . x.x-\ .1 
 
 % = Vo -t- a;A?;o + — - — A «;o+ • 
 
 and hence % . ■z^a; = % -^'o + ^^Wa; • ^"^o + ~^ — % • ■^"^'o + • • • 
 
 so that %ia^V:c) = (-»a;)^'o + (^xiQ . Avq + ^i ^'-^' n^j . A'c^ + . . . 
 
 = {-Euoho + {^\) • At'o + (2'«.o) . A\-o + ... 
 using the notation of p. 60. 
 Put Vx = x"^ and we have 
 
 2a;"\ w^ = (2wo)0'" + (2-»i)A0'" + (:S^fo)A-. 0'" -f- . . . 
 
 Putting «i equal successively to 1, 2, 3 ... , taking the differences 
 from the table of the differences of nothing, and noting that the 
 first term vanishes whatever the value of m, we can write down 
 at once — 
 
 ^X.Ux = '^'Ul ^ 
 
 ^x^lx = ^'^h + ^^% 
 
 )..A 
 
 2xhtx - 2-i/,i -I- 6^% + G^% 
 
 Ixhix - 2-«i + ?>0^\i. + 1502^% + 2402'«4 + 1202^i/J 
 
 These equations, divided by Sifo give the expressions for the 
 moments set out on page 60.
 
 125 
 
 Taking next the usual central difference formula, 
 
 — On + — — : Cn ^ ^^ iL 
 
 Vx = Vq + xao + — ^'o + 
 
 , ,r(a:^-l)(a;2_4) 
 
 "^ g '' 
 
 — lQ + XaQ+ 19^ oi ,„ 
 
 CqH r; "0 
 
 + 
 
 {x-l){x-\)x{x+\){x+2) 
 
 Co 
 
 2 i2 [3 
 
 _^ {{x + 2){x + 1K>; - 1)} + {{x + IM^ - l)(a: - 2)} dp ^ 
 ^ 2 ■ 14 "^ 
 
 Thus, 
 
 If, 
 
 1*0 
 
 the law of the terms being manifest ; or, abbreviating the expression 
 
 by the single symbol ^'ux+i, the series may be -written 
 
 ^{uxvx) = {^Uo)vo + {^\)ao + {'^\h) h + i^'u.) A) + . . . 
 
 Putting Vx = x"\ forming the central differences of r/;"* as shown 
 in the scheme below, we write doAvn at once 
 
 2.XIIX = 2 Ml 
 
 ^c'ux = 2l\i 
 
 B
 
 126 
 
 v = 
 
 X2 
 
 
 
 v- 
 
 = a;3 
 
 
 1 
 
 
 
 v=x* 
 
 
 X 
 
 Vj, 
 
 A 
 
 A- 
 
 X 
 
 Vx 
 
 A 
 
 A2 
 
 A=« 
 
 jr 
 
 Vx 
 
 A A- 
 
 A» A^ 
 
 -2 
 
 4 
 
 -3 
 
 
 -3 
 
 -27 
 
 19 
 
 
 
 -3 
 
 81 
 
 -65 
 
 
 -1 
 
 1 
 
 -1 
 
 2 
 
 -2 
 
 - 8 
 
 7 
 
 -12 
 
 6 
 
 -2 
 
 16 
 
 50 
 -15 
 
 -36 
 
 
 
 
 
 (0) 
 1 
 
 2 
 
 -1 
 
 - 1 
 
 1 
 
 - 6 
 
 6 
 
 -I 
 
 1 
 
 14 
 - 1 
 
 24 
 
 -12 
 
 1 
 
 1 
 
 3 
 
 2 
 
 
 
 
 
 (1) 
 
 1 
 
 
 
 (6) 
 6 
 
 
 
 
 
 (0) 2 
 
 1 
 
 (0) 24 
 + 12 
 
 2 
 
 4 
 
 
 
 1 
 2 
 3 
 
 1 
 
 8 
 
 27 
 
 7 
 19 
 
 6 
 
 12 
 
 6 
 
 1 
 2 
 3 
 
 I 
 16 
 81 
 
 14 
 15 
 
 50 
 65 
 
 24 
 + 36 
 
 The simplification in the formulae is, of course, due to the fact 
 that when m is even the odd central differences vanish, and when 
 m is odd the even central differences vanish. 
 
 It is sometimes required to find moments of the form 
 
 For this purpose we may use the formula {see " Sunderland's Notes 
 on Finite Differences," p. 32) — 
 
 r(,r-l) 
 
 xix-l){x-h) 
 
 ■'Ox = i(vo + vi) + (x - I) At'o + '""2 "' 2 A^(^o + v-i)+ "ig"" — - A^'t- - 1 
 
 . x{x-l){x+}){x-2)l 
 
 2 
 
 A*(r-i + r-o) + , 
 
 1 / \ 1 / 1 \ A x(x —1)1.0/ \ 
 
 _, 1 (a-+l).r(.^-l) + a;(^--l)(a;-2 ) . « , 
 
 +- — • , L\ t' - 1 
 
 2 3 
 
 + 
 
 {x+l)x(x-l)(x-2)l 
 
 A\v-i + v-o) 
 
 Avhence we find, in the same manner as before, that commencing 
 with ViWi, we shall have 
 
 2vx.wx = ^wi.'-^' +2Vii.Aro + 2V- ^(A^t'o + A^^-i) 
 
 + SV,iA=^^!- 1 + 3-^3 J (A\'-i + A%-o)+ . . .
 
 127 
 
 Putting ., = ^-^4-.;) =^-^^^ 
 
 the following Table shows the values of (2.r - 1)'" and its differences, 
 whence Vx and its differences will be found by dividing by 2'". 
 
 F 2x-l 
 
 (2:r-l)' 
 
 A 
 
 (2x-l)= 
 
 A 
 
 A- 
 
 (2x-l)^ 
 
 A A"- A3 
 
 (2x-l)^ 
 
 A 
 
 A- 
 
 A=« A^ 
 
 I -5 
 
 — 5 
 
 2 
 
 25 
 
 -16 
 
 
 -125 
 
 98 
 
 625 
 
 -544 
 
 
 
 L -3 
 
 -3 
 
 
 9 
 
 
 8 
 
 - 27 
 
 -72 
 
 81 
 
 
 464 
 
 
 
 
 2 
 
 
 - 8 
 
 
 
 26 48 
 
 
 - 80 
 
 
 -384 
 
 ) -1 
 
 -1 
 
 
 1 
 
 
 8 
 
 - 1 
 
 -24 
 
 1 
 
 
 80 
 
 384 
 
 
 
 
 2 
 
 (1) 
 
 
 
 (8) 
 
 (0) 
 
 2 (0) 48 
 
 (1) 
 
 
 
 (80) 
 
 (384) 
 
 L +1 
 
 + 1 
 
 2 
 
 1 
 
 8 
 
 8 
 
 + 1 
 
 + 24 
 26 48 
 
 1 
 
 80 
 
 80 
 
 384 
 384 
 
 I +3 
 
 + 3 
 
 2 
 
 9 
 
 16 
 
 8 
 
 + 27 
 
 + 72 
 98 
 
 81 
 
 544 
 
 464 
 
 
 5 +5 
 
 + 5 
 
 
 25 
 
 
 
 + 125 
 
 
 625 
 
 
 
 
 Dividing by the appropriate power of 2 and inserting the values 
 of - (^0 + vi), A?\„ ^ {A'Vo + A^v- 1), &c., 
 
 the last formula becomes 
 
 ^ ^/2.r - 1\ 
 
 2.VxWx = 2( — ^ — I 
 
 = (when m = l) -"Wij 
 
 Wx , commencing with ( - ) zvi 
 
 (C) 
 
 = (when m = 2) 2'^Wo + - Smi 
 
 4 
 
 = (when m = 3) 62^tv.,i + - ^rwi^ 
 
 4 
 
 = (when m = 4)242%3 + b^hv. + "^ ^ic^ 
 
 16 
 
 Writing now ui = u-q, and so on, i.e., reckoning the ordinates from 
 
 zero, so that the moments are of the form ^^(7) +"3( ' ) + . . . 
 
 these become 
 
 4 
 
 4 
 
 N in^ = 2i'^'i(.,}. + 5^^Ui),+ — -MA 
 
 16 
 
 (T))
 
 128 
 
 This will be made clearer by a numerical example. Take the 
 
 following series. 
 
 
 
 Distance 
 
 
 
 
 
 
 
 from origin 
 
 
 
 
 
 X 
 
 Ux 
 
 multiplied 
 by 2 
 = d 
 
 Vx X d 
 
 Ux ^ d- 
 
 Vx X d^ 
 
 ttx X <£■* 
 
 -5 
 
 16-74 
 
 1 
 
 16-74 
 
 16-74 
 
 16-74 
 
 16-74 
 
 1-5 
 
 15-69 
 
 3 
 
 47-07 
 
 141-21 
 
 423-63 
 
 1270-89 
 
 2-5 
 
 14-70 
 
 5 
 
 73-50 
 
 367-50 
 
 1837-50 
 
 9187-50 
 
 3-5 
 
 12-99 
 
 7 
 
 90-93 
 
 636-51 
 
 4455-57 
 
 31188-99 
 
 60-12 
 
 228-24 
 
 1161-96 
 
 6733-44 
 
 41664-12 
 
 
 
 
 -^-2 = 
 
 -f-4 = 
 
 ^8 = 
 
 -^16 = 
 
 
 
 
 114-12 
 
 290-49 
 
 841-68 
 
 2604-01 
 
 The alternative method by summation will be as follows : 
 
 X 
 
 Mx 
 
 2zij 
 
 ^-Ux 
 
 2%,: 
 
 2^"x 
 
 25«x 
 
 •5 
 1-5 
 2-5 
 3-5 
 
 16-74 
 15-69 
 14-70 
 12-99 
 
 60-12 
 43-38 
 27-69 
 12-99 
 
 144-18 
 
 (114-12) 
 
 84-06 
 
 40-68 
 
 12-99 
 
 137-73 
 53-67 
 12-99 
 
 204-39 
 
 (135-525) 
 
 66-66 
 
 12-99 
 
 79-65 
 12-99 
 
 
 114-12 
 
 22%i+72:«A = 2x 137-73+^ x 60-12 
 4 ' 4 
 
 = 275-46 + 15-03 = 290-49 
 
 62*M2+- -""1 = 6x135-525+ - x 114-12 
 4 4 
 
 = 813-15 + 28-53 = 841-68 
 
 242^M2i + 5SVi+ A-wi = 24x 79-65 + 5 X 137-73+ — x 60-12 
 16 ' 16 
 
 = 1911-60 + 688-65 + 3-76 = 2604-01 
 
 With a heavy series of terms, the saving of labour by the 
 summation method will, as may easily be seen, be very considerable. 
 A further saving of labour may be obtained by calculating the 
 moments round some convenient central point, and thus breaking 
 up the series into two parts in the manner indicated in 
 Mr. Elderton's treatise, pp. 22-33 ; and any of the formula described 
 in these notes may be applied in this manner.
 
 129 
 
 NOTE F. 
 
 On the Identity of the Method of Moments and Method 
 OF Least Squares in the Case of an Exponential 
 Function. 
 
 Suppose y an exponential function of x so that 
 
 2/ = e«+^^+<^^^+'^^- = e^say. 
 
 Then if y be taken to represent any group in a frequency distribu- 
 tion where the number of groups is large the probable error in y 
 will be approximately Jy. Assume the true values of y, i.e., the 
 true values of a, i, c . . . , to be approximately known, and let the 
 observed values of y be denoted by y. If, then, we weight each 
 
 equation y -(r = Q by the factor — .-, writing 
 
 'Jy 
 
 J-{y'-,^) = Q ....... (1) 
 
 we shall have a series of equations of condition in which the 
 probable error is in each case identical ; that is to say, they will be 
 suitably weighted for the application of the method of least 
 squares {see Note C, p. 117-8). 
 
 Writing y' = {y + U.'^y-+Bh. ^ff + &c.) 
 
 da do 
 
 = 2/ (1 -I- 5a + x. 85 + x^. 8c, &c.) 
 equation (l) becomes 
 
 \[y{l + U + x.^h + &c.)-(f] = .... (2) 
 sly 
 
 and multiplying each C(iuation successively by the coefficients of 
 
 -8«, oh, iK;c., i.e., by-^-, ^-'-x,-^^ x', &c., and taking the sum of each 
 v2/ "Jy "Jy
 
 130 
 
 set of products, according to the rules of the method of least 
 squares, we get 
 
 I[y{ l + 8a + u'.8b + ar . 8c, &c.) - r] = 
 
 2MI + 8a + o:.8b + x\8c, &c.) - o\e'] - 
 &c., &c. 
 
 as the system of equations for determining, according to the 
 method of least squares, the small corrections to be applied to the 
 approximate values a, h, c, . . . used in obtaining the approximate 
 values of y. 
 
 Now, obviously, if y is so taken that 
 
 ^xiy-e'') =0 
 2ar{y - e') = 0, &c. 
 
 i.e., if the values of the constants a, h, c are found by the method 
 of moments, &c., the above equations are satisfied by 8a = 8h = 8c = 0; 
 that is to say, the corrections are zero, or the values found for 
 a, i, c ... by the method of moments are in conformity with the 
 method of least squares on the assumption that the observations are 
 
 properly weighted by multiplying by the factors — ,--, the weights 
 
 \ly 
 
 being assumed invariable. It may, however, be supposed that small 
 
 variations in the constants, a, h, c, . . . would produce slight 
 
 variations in the weights, in which case other solutions may exist 
 
 which would also lead, by the method of least squares, to equations 
 
 satisfied by 8a = 8J = 8c = ; but as it is well known that small 
 
 differences in weights have practically no effect on the results, it is 
 
 evident that any such alternative solution must be very close to 
 
 that already formed.
 
 131 
 
 XOTE G. 
 
 On obtaining the value of Makeham's constant c 
 direct from the exposures and deaths. 
 
 As stated in the text an exact value of this constant is not very 
 important, and this may be illustrated by reference to the data for 
 ascending premium assurances given in Table X. An approximate 
 value for r may readily be found by a process such as the following, 
 which is in principle analogous to the aggregate method employed 
 by Mr. King in the Text-Book, Part II. Take the values of ju, for 
 the central age of each group in Table X. Reject the initial and 
 final values, as depending upon only two and three deaths respectively. 
 Take the six values for central ages 321 to 57 J, weighted respectively 
 by the factors 1, 3, 5, 5, 3, 1 ; weight the six values for central 
 ages 47 i to 7 2 J, and also for 62| to 87-> in the same manner. We 
 shall then ha-\"e the following totals : 
 
 .^jH, xl = -0119 
 
 M47}xl = -0137 
 
 /t62txl = -0340 
 
 ^37jx3 = -0345 
 
 yii5._,jx3 = -0534 
 
 /i67ix3 = -1647 
 
 ^42j X 5 = '0655 
 
 ;U57iX5 = -1160 
 
 ^„tx5 = -3660 
 
 /X47i X 5 = 0685 
 
 \t^i\y. 5 = '1700 
 
 Mm X 5 = '5720 
 
 ;u5.,jx3 = -0534 
 
 ;U67iX3 = -1647 
 
 jU82jx3 = -7540 
 
 /i57} X 1 = -0232 
 
 ^-.,jxl = -0732 
 
 /x87}xl = -3379 
 
 Si = -2570 
 
 82 = -5910 
 
 83 = 2-1286 
 
 If the mortality follows Makeham's law, we shall have 
 8, - S2 1-5376 
 
 Oo — fex 
 
 •3340 
 
 = c 
 
 since 15 years is the interval Ijctwccn the centres of our empirical 
 groups. This gives log c = -0442 nearly. If we take the sum of 
 
 k2
 
 132 
 
 the unweighted values of /x in three groups for ages 32i to 47i, 52^ 
 to 67i, and 721 to 87J, we should obtain in similar manner. 
 
 20 _ ^3 — ^5-2 
 
 c — 
 
 •6136 
 •0797 
 
 giving log c = -0443. 
 
 S.2 — fej 
 
 We may conclude, therefore, that log c probably lies between 
 •044 and -045. The values of /x for ages 27^ and 92|, which we 
 have omitted in the foregoing, are respectively much below and 
 much above the general curve. If these values had been 
 included duly weighted, we should have obtained a slightly larger 
 value of log c, nearer to -045. 
 
 If we adopt -045 as an approximate value, we obtain for the 
 values of the constants A and B, by the process described on p. 65, 
 
 A = ^00950 B- •00003712 
 
 We will call this curve (a), the deviations from the adjusted values 
 of 6 in Table X being shown in the Table below. 
 
 We might 
 
 Ascending Premium Assurance Experience. 
 Deviations in Computed Deaths for Curves (a) and (f3). 
 
 Middle 
 
 
 Deviations 
 
 
 
 
 Age 
 
 Observed Deatlis 
 
 Computed Deaths 
 
 —Observed Deaths | 
 
 of 
 Group 
 
 corrected as 
 per Table (X) 
 
 
 
 
 
 
 
 
 
 Curve (a) 
 
 Curve (;8j ] 
 
 
 
 log c = -045 
 
 logo 
 
 = •046 
 
 
 
 + 
 
 
 + 
 
 
 27^ 
 
 •8 
 
 •9 
 
 ... 
 
 1-0 
 
 
 32i 
 
 29-2 
 
 
 3-5 
 
 f . • 
 
 2-8 
 
 ; 37i 
 
 102-0 
 
 
 1-8 
 
 
 •2 
 
 ; 42i 
 
 175-2 
 
 . • . 
 
 73 
 
 
 5-9 
 
 47i 
 
 191-7 
 
 12-8 
 
 
 130 
 
 
 i 52i 
 
 218-6 
 
 3-3 
 
 ■ . • 
 
 2-0 
 
 • • • 
 
 i 57i 
 
 228-4 
 
 7-3 
 
 
 5-1 
 
 • • • 
 
 62i 
 
 255-4 
 
 ■ • • 
 
 2-7 
 
 • • * 
 
 5-2 
 
 : 67i 
 
 274-4 
 
 . . • 
 
 24-8 
 
 
 26-5 
 
 ! 72i 
 
 205-6 
 
 12-0 
 
 
 12-0 
 
 ... 
 
 ; -.. 77i 
 
 151-5 
 
 121 
 
 
 13-6 
 
 > . • 
 
 1 82i 
 
 84-8 
 
 > • > 
 
 6-6 
 
 
 51 
 
 i 87i 
 
 22-3 
 
 
 -5 
 
 •2 
 
 • • • 
 
 92i 
 
 2-1 
 
 ... 
 
 1-1 
 
 ... 
 
 1-1 
 
 Sum of deviations 
 
 48-4 
 
 1 
 48-3 
 
 46-9 
 
 46-8 
 
 Second sum . . 
 
 38-9 
 
 37-5 
 
 39-3 
 
 39-9 
 
 Third sum . . 
 
 13-7 
 
 71-8 
 
 31-9 
 
 36-9
 
 133 
 
 expect from our first rough approximation to log c that a smaller 
 value than -045, say -Oii, would give better results. We find, 
 however, that the third sum of the errors of the (a) curve is 
 negative, and this indicates an increase in the value of log c. 
 
 Since a higher value of c hollows out the curve at the middle 
 ages, increasing the computed deaths at the extremes of the table, 
 it is clear that the eftect must be to increase the third sum of the 
 graduated deaths. 
 
 The probalility is therefore, that curve (a) will not be much 
 improved by changing the value of c. 
 
 If we take the alternative value log c='04:6 we find the 
 deviations from the adjusted values of B in Table X are given for 
 curve (3 on the previous page- 
 
 There is little to choose between the two graduations, notAdth- 
 standing the smallness of the third sum of the deviations in curve 
 ifS), for against this may be put the fact that the three largest 
 errors in (a) are all increased in (/i). On the whole the curves may 
 be taken as showing that an approximate value of log c is generally 
 sufficient, and that nothing is gained by computing this constant to 
 several places of decimals. 
 
 It may at first sight appear inconsistent with the general theory 
 to adopt values of the three constants which do not make the third 
 sum vanish ; i.e., the third moment of the graduated and ungraduated 
 figures identical. It must, however, be remembered that the method 
 of least squares (and with it the method of moments) assumes that 
 the form of the curve is known a primi, in which case the method 
 gives the means of determining the most probable values of the 
 constants involved. When, however, we are dealing with a 
 mortality experience, wc have no a priori right to assume that 
 Makcham's law is strictly applicable ; and, if it is not, the 
 deA^ations instead of following the normal law as assumed in the 
 theory of least squares. Mill include systematic dcAaations due to 
 departure from the Makeham law. In these circumstances the 
 method of least squares is not strictly applicable, and we are 
 therefore justified in allowing other considerations to guide us in 
 selection of the constants. 
 
 We may here note that if the exposures are represented by a 
 fiequency curve, the deaths being recomputed to correspond to the 
 graduated exposures, then the value of logc may, in general, be 
 calculated from the moments of the exposures and of the recomputed 
 deaths. This can readily be done if the exposures are represented
 
 134 
 
 by a binomial curve {see Calderon, J.I. A., xxxv, 157/, although 
 precautions must be taken so to group data that the number of 
 terms in the binomial is not great — not more, say, than five or six ; 
 or by the normal frequency curve {see Elderton's Frequency 
 Curves", pp. 98-100); or by the curve y = kx"'e-'<'', where, if Eq, Ej, 
 &c., represent the successive moments for the exposures round the 
 origin, and Oq, Oi, &c., the similar moments of the recomputed 
 deaths, 
 
 , „ , y-log^C \Ei Eq/ 
 
 we shall have ' ^^^ = —t= ~ 
 
 y (h _ Ml 
 
 VEo E]/ 
 
 whence, y being known, loggC is easily found. 
 
 The above relation may be thus demonstrated. The force of 
 mortality at age x is assumed to be of the form A + Bc^ = A + Be"'^"^" 
 = A + Be''^, putting A = loggC. Thus the death curve will be of the 
 form K.hx™e~'*'^ + '^.hx^"'e~^'^~^^^ , where the second term is of the same 
 form as the first with 7 - A substituted for 7. But by the well- 
 known properties of the Gamma integral {see Williamson's 
 "Integral Calculus", Art. 120) we have • 
 
 y-CO /-Xi 
 
 Jo zJ 
 
 whence it is easily seen that, writing E'o, E'l . . . for the moments 
 of kx'"e-^y-^>\ 
 
 Eo = Eo ^o = AEo + BE' 
 
 7-A 
 (7 -A) 
 
 E,=.Eox^-^^±i (^, = AE,+BEV"^J 
 
 7 
 
 E, = Eo X ^"' + -^^:^^+^^ e, = AE, + BE'o ^"^ + ^^^^^ + ^> 
 
 7 
 whence 
 
 ^o^Eo = A + b|-0 =A + B' 
 
 Eo 
 
 6^i-Ei = A + b|-0-^ =A + B'^^ 
 
 A = B' ^ 
 
 7 — A 
 
 Eo 7 - A 7 - A 
 
 Ai = B' 
 
 7A 
 
 ^2-^Eo==A + B|-0^^,=A + B'(-^ 
 En (-/ - Ar \y - 
 
 (7-A)^ 
 
 Eo (7 - A)"-^ \7 - A 
 
 so that A -^ Ai 
 
 7-A _ 7— ^Qge '
 
 135 
 
 If the exposures, as often happens, can only be represented by 
 a curve of the form y = t<;*(l - ,c)^ (where x represents a propor- 
 tionate part of the range of the curve so that .'■ ranges between 
 and l) and if, as before, we represent the successive moments for 
 exposures and deaths by jiiq, nii, &c., Mq, Mj, &c., where hiq and Mq are 
 made = l, then writing 
 
 (a+l)-(a + /i + 2)Mi = Eo 
 
 (a + l)Mi - (a + fi + 2)Mo = Ri 
 
 it will be found that, putting /• for the range of the curve in years 
 of age, 
 
 ^ Ro 
 
 '^"^ (Mo - Ml) - A(;»o - wix) 
 
 (Mo - Ml, 
 
 (M3 - Mo) - /i(?»3 - llt.y) 
 
 from which as the numerical value of all the quantities except 
 loge c and h are known, these two may l)e easily found. 
 
 This may be shown as follows : — 
 
 Let the curve of exposed to risk be represented by the type 
 
 y = l:r-{ 1 _ ,-y 
 
 where the entire range of the cur\c is taken as unity, and assume 
 k, a, and /3 to be determined in the usual manner. 
 
 Let the curve of the recomputed deaths be of the form 
 
 Akx''{l-x)^ + 'BhHl-.r)^>>y'=^Ay + Bz . . . (l) 
 
 i.e., we assume that — , (log Ix) = A + Be^^ 
 
 ax 
 
 As regards the curve :', we shall have 
 
 log Z = a log;/,- + /i log ( 1 - ./;) + yX 
 
 dz fa (i 
 
 dx \x \ -X 
 
 or, multiplying 1)0th sides by x''^^{l -,/■), 
 
 (.//+^-.//+'^)t = [«../- (a + iy).V + '+7(./+'-./+^)> . . (2) 
 (IX
 
 136 
 
 Integrating the left-hand side of this equation by parts, and 
 noting that the factor {x^'^^ - a:*"*"^) is zero for the limits 1 and 0, 
 
 Jo .'0 
 
 that is 
 
 (t + '2)m't+i -{f + l)iii't = am't - (a + ft)m,+i + y{mt+\ - ''nt+'i) 
 and 
 
 {a + t+\)m't-{o. + (i + t + 2)int+i + y{iii^t+i-m't+o}^0 . . (3) 
 
 there mt represents the /th moment of the curve z round the 
 ordinate ,'■ = 0. 
 
 If 7 = 0, the curve '; becomes identical with y, and Avriting mt 
 for the t\\\ moment of y round the ordinate x = 0, we have 
 
 {u. + t+\)iiit-{a. + ft + f + 2)mt+i = .... (4) 
 
 Write, as before, the total of the exposed =Eo, and of the 
 deaths = d^^, respectively, and represent the total of the exposed 
 multiplied at each age by the factor c'^^ by E'q. 
 
 Let E< and Ot be the tth. moments of the cur^-e of exposed to 
 risk and of the recomputed deaths, the areas of the curves not 
 being taken as =1, ]>ut having the values Eq and ^o above defined, 
 that is to say, representing the total exposures and the total deaths. 
 And let E'? be the it\\ moment of the curve of exposures multiplied 
 at each age liy v"^'' . 
 
 Then Ave have ^, = AE, + BE'^ ........ (5) 
 
 where 9t and E,> are known, but the remaining quantities unknown. 
 
 From (3) and (4) 
 
 (a + ^+l)E,-(a + /i4-/ + 2)E,+i=0 (6) 
 
 and (a + /+l)E;-(« + /:/ + / + 2)E',+i + 7(E'(+i-E',+2) = . (7) 
 
 Write {a + t + l)et- {o. + li + i + i)et+i = If? , 
 
 from (5) 
 
 (a + / + l)(AE, + BEV)-(a + /3 + / + 2)(AE,+i + BE',+i) = K? 
 
 and from (6) 
 
 (a + / + 1)BE', - (a + /3 + ^ + 2)BE',+i = E, 
 
 and from (7) 
 
 ■ ' By(EV2-E'm) = Rf (8)
 
 137 
 
 Since from (5) 
 
 we have y[(^,+,- ^,+1) - A(E,+o-Em)] = R« • ... (9) 
 
 writing ^ = and ^ = 1 respectively, we get 
 
 y[(^,-^i)-A(E2-Ea)]-Ro 
 
 whence Ei[(^o - ^1) - ACE^ - Ej)] = Ro[(^3 - ^2) - A(E3 - E.,)] 
 
 and A= 3ii^-2_-Lli)-_^d^^_-^-2) (10) 
 
 Ki(E2 — El) — RolEa — Eo) 
 
 also, from (9) ^=^__^-^^^_^ 
 
 The A'ahie of B cannot he determined directly from these equations 
 as it enters symmetrically with the values of E'^. It is therefore 
 necessary, having found the value of y, to compute the Aalue of E'o 
 and thence deduce B from equation (5). 
 
 Unless the mortality follows Makeham's law very closely better 
 results will be obtained by calculating both E',, and E'x and obtaining 
 values of A and B satisfying the equations 
 
 AEo + BE'o-^,1 
 AEi + BE'i = 6^j
 
 138 
 
 Tables of Values of y- 
 
 JttJ 
 
 .dx. 
 
 z 
 
 y 
 
 A 
 
 z 
 
 y 
 
 A 
 
 s 
 
 y 
 
 A i 
 
 •01 
 
 •01128 i 
 
 1128 
 
 •51 
 
 •52924 
 
 866 
 
 1-01 
 
 •84681 
 
 i 
 
 403 
 
 •02 
 
 •02256 1 
 
 1128 
 
 •52 
 
 •53790 
 
 856 
 
 1^02 
 
 •85084 
 
 394 
 
 •03 
 
 •03384 i 
 
 1127 
 
 •53 
 
 •54646 
 
 848 
 
 1-03 
 
 •85478 
 
 387 
 
 •04 
 
 •04511 i 
 
 1126 
 
 •54 
 
 •55494 
 
 838 
 
 1-04 
 
 •85865 
 
 379 
 
 •05 
 
 •05637 1 
 
 1 
 
 1125 
 
 •55 
 
 •56332 
 
 830 
 
 1^05 
 
 •86244 
 
 370 
 
 •06 
 
 •06762 . 
 
 1124 
 
 •56 
 
 •57162 
 
 820 
 
 1-06 
 
 •86614 1 
 
 363 
 
 •07 
 
 •07886 1 
 
 1122 
 
 •57 
 
 •57982 
 
 810 
 
 107 
 
 •86977 
 
 356 
 
 •08 
 
 •09008 
 
 1120 
 
 •58 
 
 •58792 
 
 802 
 
 1-08 
 
 •87333 
 
 347 
 
 •09 
 
 •10128 
 
 1118 
 
 •59 
 
 •59594 
 
 792 
 
 1-09 
 
 •87680 \ 
 
 341 
 
 •10 
 
 •11246 
 
 1116 
 
 •60 
 
 •60386 
 
 782 
 
 1-10 
 
 •88021 
 
 332 
 
 •11 
 
 •12362 1 
 
 1114 
 
 •61 
 
 ■61168 
 
 773 
 
 1-11 
 
 •88353 
 
 326 
 
 •12 
 
 •13476 
 
 1111 
 
 •62 
 
 •61941 
 
 764 
 
 1-12 
 
 •88679 
 
 318 
 
 •13 
 
 •14587 
 
 1108 
 
 •63 
 
 •62705 
 
 754 
 
 113 
 
 •88997 
 
 311 
 
 •14 
 
 •15695 
 
 1105 
 
 •64 
 
 •63459 
 
 744 
 
 114 
 
 •89308 
 
 304 
 
 •15 
 
 •16800 
 
 1101 
 
 •65 
 
 •64203 
 
 735 
 
 115 
 
 •89612 
 
 298 
 
 •16 
 
 •17901 
 
 1098 
 
 •66 
 
 ■64938 
 
 725 
 
 116 
 
 •89910 
 
 290 
 
 •17 
 
 •18999 
 
 1094 
 
 •67 
 
 •65663 
 
 715 
 
 1-17 
 
 •90200 
 
 284 
 
 •18 
 
 •20093 
 
 1091 
 
 •68 
 
 •66378 
 
 706 
 
 1^18 
 
 •90484 
 
 278 
 
 •19 
 
 •21184 
 
 1086 
 
 •69 
 
 ■67084 
 
 696 
 
 1^19 
 
 •90761 
 
 270 
 
 •20 
 
 •22270 
 
 1082 
 
 •70 
 
 ■67780 
 
 687 
 
 1^20 
 
 •91031 
 
 265 
 
 •21 
 
 •23352 
 
 1078 
 
 •71 
 
 ■68467 
 
 676 
 
 121 
 
 •91296 
 
 257 
 
 •22 
 
 •24430 
 
 1072 
 
 •72 
 
 •69143 
 
 667 
 
 1^22 
 
 •91553 
 
 252 
 
 •23 
 
 •25502 
 
 1068 
 
 •73 
 
 •69810 
 
 658 
 
 1-23 
 
 •91805 
 
 246 
 
 •24 
 
 •26570 
 
 1063 
 
 •74 
 
 •70468 
 
 648 
 
 124 
 
 •92051 
 
 239 
 
 •25 
 
 •27633 
 
 1057 
 
 •75 
 
 •71116 
 
 638 
 
 125 
 
 •92290 
 
 234 
 
 •26 
 
 •28690 
 
 1052 
 
 •76 
 
 •71754 
 
 629 
 
 126 
 
 •92524 
 
 227 
 
 •27 
 
 •29742 
 
 1046 
 
 ■77 
 
 •72382 
 
 619 
 
 127 
 
 •92751 
 
 222 
 
 •28 
 
 •30788 
 
 1040 
 
 •78 
 
 ■73001 
 
 609 
 
 128 
 
 •92973 
 
 217 
 
 •29 
 
 ■31828 
 
 1035 
 
 •79 
 
 •73610 
 
 600 
 
 1^29 
 
 •93190 
 
 211 
 
 •30 
 
 •32863 
 
 1028 
 
 ■80 
 
 •74210 
 
 590 
 
 130 
 
 •93401 
 
 ■ 
 
 205 
 
 •31 
 
 •33891 
 
 1022 
 
 •81 
 
 ■74800 
 
 581 
 
 1-31 
 
 •93606 
 
 201 
 
 •32 
 
 •34913 
 
 1015 
 
 •82 
 
 •75381 
 
 571 
 
 132 
 
 •93807 
 
 195 
 
 •33 
 
 •35928 
 
 1008 
 
 •83 
 
 •75952 
 
 562 
 
 133 
 
 1 •94002 
 
 189 
 
 •34 
 
 •36936 
 
 1002 
 
 •84 
 
 •76514 
 
 553 
 
 1^34 
 
 •94191 
 
 185 
 
 •35 
 
 •37938 
 
 995 
 
 •85 
 
 •77067 
 
 543 
 
 135 
 
 •94376 
 
 180 
 
 •36 
 
 •38933 
 
 988 
 
 •86 
 
 •77610 
 
 534 
 
 136 
 
 •94556 
 
 175 
 
 •37 
 
 •39921 
 
 980 
 
 •87 
 
 •78144 
 
 525 
 
 137 
 
 •94731 
 
 171 
 
 •38 
 
 •40901 
 
 973 
 
 •88 
 
 : -78669 
 
 515 
 
 138 
 
 •94902 
 
 165 
 
 •39 
 
 •41874 
 
 965 
 
 •89 
 
 1 -79184 
 
 507 
 
 139 
 
 •95067 
 
 162 
 
 •40 
 
 •42839 
 
 958 
 
 •90 
 
 •79691 
 
 497 
 
 140 
 
 •95229 
 
 156 
 
 •41 
 
 •43797 
 
 950 
 
 •91 
 
 1 •80188 
 
 489 
 
 1-41 
 
 , • 95385 
 
 153 
 
 •42 
 
 •44747 
 
 942 
 
 •92 
 
 •80677 
 
 ' 479 
 
 1-42 
 
 •95538 
 
 148 
 
 •43 
 
 •45689 
 
 934 
 
 •93 
 
 •81156 
 
 471 
 
 143 
 
 •95686 
 
 144 
 
 •44 
 
 •46623 
 
 925 
 
 •94 
 
 •81627 
 
 462 
 
 1^44 
 
 •95830 
 
 140 
 
 •45 
 
 •47548 
 
 918 
 
 •95 
 
 •82089 
 
 453 
 
 145 
 
 j ^95970 
 
 135 
 
 •46 
 
 •48466 
 
 909 
 
 ■96 
 
 •82542 
 
 445 
 
 1^46 
 
 ! 96105 
 
 132 
 
 •47 
 
 •49375 
 
 900 
 
 •97 
 
 •82987 
 
 436 
 
 1^47 
 
 •96237 
 
 128 
 
 •48 
 
 •50275 
 
 892 
 
 •98 
 
 ; ^83423 
 
 428 
 
 1^48 
 
 •96365 
 
 125 
 
 •49 
 
 •51167 
 
 883 
 
 •99 
 
 1 ^83851 
 
 419 
 
 1-49 
 
 •96490 
 
 121 
 
 •50 
 
 •52050 
 
 874 
 
 1-00 
 
 •84270 
 
 411 
 
 150 
 
 •96611 
 
 117
 
 139 
 
 Table 
 
 2 f~ 
 of Values of y= ^j-\ e~^.dx — continued. 
 
 JttJ 
 
 z 
 
 ^ 
 
 A 
 
 1 
 
 y 
 
 A 
 
 <v 
 
 y 
 
 A 
 
 1-51 
 
 •96728 
 
 117 
 
 2^01 
 
 •995525 
 
 195 
 
 251 
 
 •9996143 
 
 202 
 
 1-52 
 
 •96841 
 
 111 
 
 2^02 
 
 •995720 
 
 186 
 
 2^52 : 
 
 •9996345 
 
 192 
 
 1-53 
 
 •96952 
 
 107 
 
 203 
 
 •995906 
 
 180 
 
 253 
 
 •9996537 ' 
 
 183 
 
 1-54 
 
 •97059 
 
 103 
 
 204 
 
 •996086 
 
 172 
 
 254 
 
 •9996720 1 
 
 173 
 
 1-55 
 
 •97162 
 
 101 
 
 205 
 
 •996258 
 
 165 
 
 255 
 
 •9996893 ' 
 
 165 
 
 1-56 
 
 •97263 
 
 97 
 
 206 
 
 •996423 
 
 159 
 
 256 
 
 •9997058 
 
 157 
 
 , 1-57 
 
 •97360 
 
 95 
 
 2-07 
 
 •996582 
 
 152 
 
 2-57 
 
 •9997215 
 
 149 
 
 1-58 
 
 ■97455 
 
 91 
 
 2-08 
 
 •996734 
 
 146 
 
 2^58 ; 
 
 •9997364 
 
 141 
 
 1-59 
 
 •97546 
 
 89 
 
 2 09 
 
 •996880 
 
 141 
 
 2^59 
 
 •9997505 
 
 135 
 
 1-60 
 
 •97635 
 
 86 
 
 2^10 
 
 •997021 
 
 134 
 
 2-60 
 
 •9997640 
 
 127 
 
 1-61 
 
 •97721 
 
 83 
 
 2^11 
 
 •997155 
 
 129 
 
 2^61 
 
 •9997767 
 
 121 
 
 1-62 
 
 •97804 
 
 80 
 
 212 
 
 •997284 
 
 123 
 
 2^62 
 
 •9997888 
 
 115 
 
 1-63 
 
 •97884 
 
 78 
 
 2-13 
 
 •997407 
 
 118 
 
 2^63 
 
 •9998003 
 
 109 
 
 ' 1-64 
 
 •97962 
 
 76 
 
 2^14 
 
 ■997525 
 
 114 
 
 364 
 
 •9998112 
 
 103 
 
 1-65 
 
 •98038 
 
 72 
 
 215 
 
 •997639 
 
 108 
 
 265 
 
 •9998215 
 
 98 
 
 1-66 
 
 •98110 
 
 71 
 
 216 
 
 •997747 
 
 104 
 
 2^66 
 
 •9998313 
 
 93 
 
 1-67 
 
 •98181 
 
 68 
 
 2^17 
 
 ■997851 
 
 100 
 
 267 
 
 •9998406 
 
 88 
 
 1-68 
 
 •98249 
 
 66 
 
 2^18 
 
 •997951 
 
 95 
 
 2^68 
 
 •9998494 
 
 84 
 
 : 1-69 
 
 •98315 
 
 64 
 
 219 
 
 •998046 
 
 91 
 
 2-69 
 
 •9998578 
 
 79 
 
 1-70 
 
 •98379 
 
 62 
 
 2-20 
 
 •998137 
 
 87 
 
 2^70 
 
 •9998657 
 
 75 
 
 ! 1-71 
 
 •98441 
 
 59 
 
 2^21 
 
 •998224 
 
 84 
 
 2-71 
 
 •9998732 
 
 71 
 
 1-72 
 
 •98500 
 
 58 
 
 2^22 
 
 •998308 
 
 80 
 
 2-72 
 
 •9998803 
 
 67 
 
 1-73 
 
 •98558 
 
 55 
 
 223 
 
 •998388 
 
 76 
 
 273 
 
 •9998870 
 
 63 ' 
 
 ■ 1-74 
 
 •98613 
 
 54 
 
 2^24 
 
 •998464 
 
 73 
 
 274 
 
 •9998933 
 
 61 i 
 
 1 1-75 
 
 •98667 
 
 52 
 
 2-25 
 
 •998537 
 
 70 
 
 275 
 
 •9998994 
 
 57 : 
 
 ; 1-76 
 
 •98719 
 
 50 
 
 226 
 
 •998607 
 
 67 
 
 2^76 
 
 •9999051 
 
 54 
 
 ; 1-77 
 
 •98769 
 
 48 
 
 2^27 
 
 •998674 
 
 64 
 
 2^77 
 
 •9999105 
 
 51 
 
 1 1-78 
 
 •98817 
 
 47 
 
 2^28 
 
 •998738 
 
 61 
 
 2-78 
 
 •9999156 
 
 48 ' 
 
 ! 1-79 
 
 •98864 
 
 45 
 
 229 
 
 •998799 
 
 58 
 
 2-79 
 
 •9999204 
 
 46 ; 
 
 1-80 
 
 •98909 
 
 43 
 
 2-30 
 
 •998857 
 
 55 
 
 2-80 
 
 •9999250 
 
 43 \ 
 
 1-81 
 
 •98952 
 
 42 
 
 2-31 
 
 •998912 
 
 53 
 
 2-81 
 
 •9999293 
 
 41 i 
 
 1-82 
 
 •98994 
 
 41 
 
 232 
 
 •998965 
 
 51 
 
 2^82 
 
 •9999334 
 
 38 1 
 
 1-83 
 
 •99035 
 
 39 
 
 2-33 
 
 •999016 
 
 49 
 
 2 83 
 
 •9999372 
 
 37 
 
 1-84 
 
 •99074 
 
 37 
 
 234 
 
 •999065 
 
 46 
 
 2^84 
 
 •9999409 
 
 34 
 
 1-85 
 
 •99111 
 
 36 
 
 2-35 
 
 •999111 
 
 44 
 
 2^85 
 
 •9999443 
 
 33 
 
 1-86 
 
 •99147 
 
 35 
 
 2^36 
 
 •999155 
 
 42 
 
 2^86 
 
 •9999476 
 
 31 
 
 1-87 
 
 •99182 
 
 34 
 
 2-37 
 
 •999197 
 
 40 
 
 2-87 
 
 •9999507 
 
 29 
 
 1-88 
 
 •99216 
 
 32 
 
 2-38 
 
 •999237 
 
 38 
 
 2^88 
 
 •9999536 
 
 27 ' 
 
 1-89 
 
 •99248 
 
 31 
 
 2-39 
 
 •999275 
 
 36 
 
 2^89 
 
 •9999563 
 
 26 
 
 1-90 
 
 •99279 
 
 30 
 
 2^40 
 
 •999311 
 
 35 
 
 290 
 
 •9999589 
 
 109 
 
 1-91 
 
 •99309 
 
 29 
 
 241 
 
 •999346 
 
 33 
 
 295 
 
 •9999698 
 
 81 ' 
 
 1-92 
 
 •99338 
 
 28 
 
 242 
 
 •999379 
 
 32 
 
 300 
 
 •9999779 
 
 60 1 
 
 1-93 
 
 •99366 
 
 26 
 
 243 
 
 •999411 
 
 30 
 
 3-05 
 
 •9999839 
 
 45 
 
 1-94 
 
 •99392 
 
 26 
 
 2^44 
 
 •999411 
 
 28 
 
 3^10 
 
 •9999884 
 
 32 
 
 1-95 
 
 •99418 
 
 25 
 
 2^45 
 
 •999469 
 
 28 
 
 315 
 
 •9999916 
 
 24 
 
 1-96 
 
 •99443 
 
 23 
 
 2 4-6 
 
 •999497 
 
 26 
 
 320 
 
 ■9999940 
 
 29 
 
 1-97 
 
 •99466 
 
 23 
 
 247 
 
 •999523 
 
 24 
 
 330 
 
 •9999969 
 
 16 
 
 1-98 
 
 •99489 
 
 22 
 
 2^4« 
 
 •999547 
 
 24 
 
 340 
 
 •9999985 
 
 8 
 
 1-99 
 
 •99511 
 
 21 
 
 2^49 
 
 •999571 
 
 22 
 
 350 
 
 •9999993 
 
 3 
 
 2-00 
 
 •99532 
 
 20 
 
 2-50 
 
 •999593 
 
 21 
 
 3^60 
 
 •9999996 

 
 140 
 
 Table of 
 
 [The constants are restricted to positive quantities of significant value 
 
 Type 
 
 I 
 
 II 
 
 Character of Curve 
 
 Equation y = 
 
 Limits of x 
 
 Mean 
 
 M2 
 
 
 Shape 
 
 Range 
 
 Limited 
 both 
 ways 
 
 Lower 
 
 Upper 
 + a 
 
 Ms 
 
 Symmetrical 
 
 "-1 
 
 
 
 
 a2 
 
 
 
 lc{a^-^' 
 
 » + l 
 
 Symmetrical 
 
 Un- 
 limited 
 
 Jce ~ ^''' (Normal Curve) 
 
 — 00 
 
 + <X) 1 
 
 
 
 w 
 
 
 
 III 
 
 Symmetrical 
 
 Un- 
 
 j'„2 , „2^ ""I'l "^^1 
 
 1 
 
 
 
 a" 
 
 
 
 limited 
 
 Limited 
 both 
 ways 
 
 k{a~ + X-) \- 1 —00+00 
 (m>3) I 
 
 u-1 
 
 IV 
 V 
 
 Skew 
 
 (p + 2 = l) 
 
 + a 
 
 {q-p)a 
 
 M+1 
 
 ma^ 
 
 16{p-q)pq 
 (M + i)(w + 2) 
 
 Skew 
 
 Limited 
 one way 
 
 m-1 - - 
 kx e 
 
 
 
 -t-oo 
 
 ma 
 
 2ma^ 
 
 VI 
 
 VII 
 
 VIII 
 
 Skew 
 Skew 
 
 Limited 
 oue way 
 
 («>3) 
 
 + a 
 
 + 00 
 
 (p+q)a 
 
 w — 1 
 
 «2 
 
 I6(p + q)pq 
 («-l)(«-2) 
 
 Limited 
 one way 
 
 a 
 (M>3) 
 
 
 
 + 05 
 
 a 
 
 n 
 
 4a-' 
 
 n'{n-l.) 
 
 n\n — l){n- 
 
 Skew 
 
 Un- 
 limited 
 
 • (» > 3) 
 
 — 00 
 
 + 00 
 
 pa 
 n 
 
 n"(n-l) 
 
 i 
 
 Notes :— /3i = /a/-^M/. ^82=^4-^ Ma"' 
 Skewness = (Mean — mode) -h <r 
 /3i 
 
 Criterion = K = 
 
 4(2 -37) (4 -37)
 
 141 
 
 Frequency Curves. 
 
 (i.e., all >0), and in Types III, VI, VII and VIII, n must be >3]. 
 
 M4 
 
 3o^ 
 
 
 (» + l)(?j + 3) 
 
 |o^ = 3mo- 
 
 
 3a^ 
 
 (»-lX»-3) 
 
 ^pq{2 + n-Qpq) ^ 
 
 ■- • .. 3 
 
 ^1=M3'-^M: 
 
 (« + 1)(» + 'l){n + 3) 
 
 3»»(m + 2)a-* 
 
 4(» + l)(;>-g)^ 
 
 48^5'(2 + w + 6pj) ^4 
 (»-l)(7i-2)(«-3) 
 
 Not required 
 
 j1 
 
 {n-2Ypq 
 
 16(»-1) 
 
 )8a + 3 
 
 
 2 w + 3 
 
 3 » + 2 
 
 2 
 
 > _ 
 
 3 
 
 V»-3J 
 
 2 
 3 
 
 2 »-3 
 
 3 n — '2i 
 
 2 
 ^3 
 
 2 » + 3 
 
 3[2 + (?t-6)^g](» + l) 
 (» + 2)(m + 3)^2 
 
 ' 3 n + 2, 
 
 \^pq ) 
 
 
 3 
 
 (negative) 
 
 2 
 3 
 
 CO 
 
 2 ?j-3 
 
 4(m- 1)(p + ?)'- ^ 3[2 + (» + 6)/?7](n-l) | 3 m-2 
 (» — :i)('*~'^)i'2 
 
 I <- 
 3 
 
 2 
 
 w — 
 
 3 
 
 3 
 
 » — 
 
 2 
 
 "3 
 
 2 
 
 2 
 
 M — 
 
 3 
 
 „2^^?)[(„ + 6)(»2 + ;/=)-8«'^] ^ 16(»-1) V- 3(w-l)[(» + 6)(>r + y-)-8»'^] 3 »-2 -pj— 7 
 
 n^(»-l)(»-2X»-3) " (»-2)- 'm-^^- {n-'l){n-Z){n- + v-) \ ^ 2 , > q I 1 
 
 Standard Deviation = '^ jj.> = o- . 
 
 ^gi(/3, + 3) 
 2(5/3.: -«)3i- 9)' 
 
 /3,()3.. + 3)2 
 
 4(2/32-3/3i-6j(4)8,-3/3i)
 
 PRINTED BY 
 
 CHARLES AND EDWIN LAYTON, 
 
 56, FARRINGDON STREET, E.G.
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 
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