l/i./ V 1 A p "blitot,on of Th« Celltgt of Agriculturt 
 I \ \J U N I V E K $ I TY OF CALIFORNIA 
 
 '-"hun " 
 
 The seventh report in a series on 
 
 Efficiency in Fruit Marketing 
 
 ECONOMY AND ACCURACY IN 
 ACCOUNTING TO GROWERS FOR FRUIT 
 RECEIVED AT THE PACKING HOUSE 
 
 B. C. French and R. G. Bressler 
 
 LIBRARY 
 
 UNIVERSITY OF CALIFORNIA 
 DAVIS 
 
 CALIFORNIA AGRICULTURAL EXPERIMENT STATION 
 GIANNINI FOUNDATION OF AGRICULTURAL ECONOMICS 
 
 Mimeograph No. 1 49 
 
 June 1953 
 
1 
 
FOREWORD 
 
 This report is the seventh of a series aimed at improved efficiency 
 and lowered costs in the local marketing and packing of deciduous fruits. 
 The present report deals with only one aspect of packing costs — the costs 
 of accounting to growers for fruit received at the packing house. Two 
 commonly used methods of accounting — the separate-lot system and the sam- 
 pling system — are considered as to their costs and accuracy. 
 
 Studies in sample plants indicate that the costs of separate-lot 
 systems vary from $0.13 to $1.09 per 1,000 pounds of total plant volume. 
 Improved procedures could result in about a i*0-per cent reduction in this 
 cost for the average plant. Sampling systems also show wide variations 
 in cost, depending mainly on the desired degree of accuracy and the size 
 and number of growers. The problems and procedures for efficient sample 
 design are discussed, and the costs of sampling systems are compared with 
 the costs of separate-lot procedures for varying conditions. 
 
 These studies were made cooperatively by the Giannini Foundation of 
 Agricultural Economics, California Agricultural Experiment Station, and 
 the Bureau of Agricultural Economics, U. S. Department of Agriculture. 
 They were made under the authority of the Research and Marketing Act of 
 19U6. 
 
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Efficiency in Fruit Marketing 
 
 ECONOMY AND ACCURACY IN ACCOUNTING TO GROWERS 
 FOR FRUIT RECEIVED AT THE PACKING HOUSE 
 
 B. C, Frenchi/ and R. G. Bressleri/ 
 
 In the operation of packing and processing plants for many types of 
 fruits and vegetables, field or orchard -run products are received at the 
 plant from a number of growers. To account accurately to the growers for 
 the products received requires determination of the weight or count for 
 each of several important grade and size classifications. Two methods are 
 commonly used to obtain such information: (1) the separate-lot system, 
 where each producer's product is kept separate from all others as it is 
 handled in the plant and (2) the sampling system, where a small portion 
 of each lot is selected and the distribution of the entire lot estimated 
 from the proportions observed in the sample. With the sampling procedure, 
 the fruit of different growers may be comingled in plant operations. 
 
 The design of such systems has an important influence on the accuracy 
 of accounting to growers for fruit received. In addition, there are im- 
 portant effects on plant operating costs. The first section of this report 
 discusses accuracy and costs for the separate-lot system. Following 
 sections deal with sampling systems and with the comparative plant costs 
 under the two methods of accounting to growers. 
 
 THE SEPARATE-LOT SYSTEM 
 
 The essential characteristic of the separate-lot system is that the 
 individual identity of each grower's product must be maintained until it 
 is sorted, graded, sized, and the amount in each size-grade determined. 
 In order to prevent the mixing of products from several growers, each lot 
 is run separately with a delay or break in plant operations at the end of 
 
 1/ Cooperative Agent of the University of California Experiment Station 
 and the Bureau of Agricultural Economics, U. S. Department of Agriculture. 
 
 2/ Professor of Agricultural Economics and Agricultural Economist in 
 the Experiment Station and Director of the Giannini Foundation. 
 
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each lot. During this delay period, the amounts of product in each of the 
 several classifications are determined and recorded. In fresh fruit pack- 
 ing houses, the delay also permits the packing of products remaining in 
 the bins or packing belts so that the actual packcut can be determined. 
 
 Ideally, such a system should provide a very accurate accounting for 
 each grower subject only to such minor errors as result from weighing 
 procedures and the estimates of the tare weight of containers. But a 
 complete separation of lots would usually require relatively long break 
 or delay periods, and this would greatly increase plant operating costs. 
 In order to increase efficiency and reduce costs, various short-cut pro- 
 cedures are commonly used. These serve to reduce the break-for-lot period, 
 but they also introduce opportunities for error. Most of the possibilities 
 for error stem from the fact that short-cut methods usually do not separate 
 each lot completely, and the amount of product in the overlap is obtained 
 by estimate rather than actual measurement. In addition, errors may result 
 from estimating the weight of fruit in some grades— by applying conventional 
 standard weights to the number of boxes where the standards may differ 
 somewhat from actual weights. 
 
 In spite of these opportunities for error, practices observed in Cali- 
 fornia fruit packing houses appear to be designed to give reasonable 
 accuracy and to protect the interests and equities of both grower and 
 packing house. 
 
 Idle Time With Separate-Lot Systems 
 
 The direct cost of a separate-lot system includes the cost of weighing 
 and tallying the amount of fruit in the several grades, but a more important 
 effect is the impact of the plant delays or breaks on over-all output and 
 efficiency. The interruption of the flow of fruit through the plant means 
 that some workers will be completely idle during this period while others 
 are forced to work at lowered rates of activity. The loss of effective 
 working time depends on such factors as the plant volume per hour, the num- 
 ber of dumping units and sorting-packing lines, the average size of lot for 
 individual growers, and the length of the delay or break-for-lot period. 
 
 Table 1 summarizes the results of studies of separate-lot systems in 
 a number of plants packing apples and pears or receiving olives for process- 
 ing. The factors most subject to control or change by the plant manager 
 
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3 
 
 TABLE 1 
 
 The Effects of the Separate-Lot System on Effective Working Time 
 in California Apple, Pear, and Olive Plants 
 
 
 Volume of fruil 
 
 
 Lots 
 
 Break 
 
 Loss of effective 
 
 
 Per 
 
 Per 
 
 Per 
 
 run per 
 
 time 
 
 work in 
 
 5 time 
 
 PI ant 
 
 plant-hour 
 
 line -hour 
 
 lot 
 
 line-hour 
 
 per lot 
 
 Per hour 
 
 Per cent 
 
 
 1, 
 
 300 pounds 
 
 
 minutes 
 
 
 
 Ai 
 
 sole and pear packing houses 
 
 
 
 A 
 
 20.U 
 
 10.2 
 
 10.2 
 
 1.0 
 
 3.0 
 
 3.0 
 
 5.0 
 
 B 
 
 18.0 
 
 9.0 
 
 
 
 
 >Jl 
 
 5.6 
 
 L 
 
 19.8 
 
 9.9 
 
 a. 3 
 
 2.3 
 
 3.7 
 
 8.6 
 
 1U.U 
 
 M 
 
 2U.8 
 
 2U.8 
 
 19.8 
 
 1.3 
 
 3.1 
 
 luO 
 
 6.7 
 
 N 
 
 38.U 
 
 12.8 
 
 8.5 
 
 1.5 
 
 11.3 
 
 16.9 
 
 28.1 
 
 R 
 
 U5.o 
 
 15.0 
 
 
 
 
 tu7 
 
 7.8 
 
 S 
 
 U6.0 
 
 15.3 
 
 8.1 
 
 1.9 
 
 0.9 
 
 1.8 
 
 3.0 
 
 T 
 
 31.2 
 
 31.2 
 
 
 
 
 h.2 
 
 7.0 
 
 U 
 
 60.6 
 
 30.3 
 
 23.3 
 
 1.3 
 
 2.8 
 
 3.6 
 
 6.0 
 
 
 
 
 o: 
 
 ivfi plants 
 
 I 
 
 3.1 
 
 3.1 
 
 2.0 
 
 1.5 
 
 2.9 
 
 U.5 
 
 7.5 
 
 II 
 
 13.3 
 
 13.3 
 
 U.6 
 
 2.9 
 
 2.7 
 
 7.9 
 
 13.2 
 
 III 
 
 10.0 
 
 10.0 
 
 1.8 
 
 5.U 
 
 2.0 
 
 10.9 
 
 18.2 
 
 IV 
 
 11.8 
 
 11.8 
 
 2.8 
 
 U.2 
 
 3.U 
 
 Hi. 2 
 
 23.7 
 
 V 
 
 lf)& 
 
 10.5 
 
 5.3 
 
 2.0 
 
 2.3 
 
 U.6 
 
 7.7 
 
 VI 
 
 7.2 
 
 7.2 
 
 2.9 
 
 2.5 
 
 U.2 
 
 10.6 
 
 17.7 
 
 VIII 
 
 5.2 
 
 5.2 
 
 2.8 
 
 1.9 
 
 U.9 
 
 9.1 
 
 15.2 
 
 IX 
 
 12.3 
 
 12.3 
 
 3.2 
 
 3.9 
 
 3.6 
 
 lU.O 
 
 23.2 
 
 Dashes indicate data not available. 
 
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 ■ 
 
 
 
 
 
 
 
 
 
 
 
 
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 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 ■-€.2 
 
 
 
 
 
 
 
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 r 
 
are the average size of lot and the length of the break period. In apple 
 and pear plants, the average break-for-lot period ranged from less than 
 one minute in Plant S to more than eleven minutes in Plant N. Individual 
 break-for-lot periods ranged even more widely — from less than one-half 
 minute to more than twenty minutes. In the olive plants studied, the break- 
 for-lots averaged from two to five minutes, while breaks for individual 
 lots ranged from one to ten minutes. In both the olive and the apple and 
 pear plants, the average break time was about three minutes. 
 
 Size of lot is determined primarily by the particular size and pro- 
 duction characteristics of the individual grower. It can be controlled 
 somewhat by plant managers, however, in several ways: (1) scheduling 
 picking and hauling operations from particular orchards so as to result 
 in larger average deliveries at the plant; (2) consolidating several truck- 
 loads from a single grower into a single lot at the plant; and (3) dis- 
 couraging very small growers or by combining the fruit from several small 
 growers into a single lot for plant accounting purposes. The table 
 indicates that average volume per lot ranged from U,000 to 23,000 pounds 
 for apple and pear plants. Individual lots handled at these plants during 
 the period of study ranged from less than 1,000 to more than 70,000 pounds. 
 Considering all plants, the average size of lot for apples and pears was 
 about 10,000 pounds. 
 
 Lots were much smaller in olive plants, averaging only about 3,000 
 pounds. Individual plants had averages from 2,000 to more than 5,000 pounds 
 during the periods studied. These averages were greatly influenced by a 
 few relatively large cases, however, and many lots were of very small size. 
 Individual lots observed during the study ranged from U00 to 26,000 pounds; 
 25 per cent of all lots were less than 1,000 pounds and 32 per cent fell 
 between 1,000 and 2,000 pounds. 
 
 The table also indicates the effects of the separate-lot system on 
 effective working time. As mentioned above, these effects depend on such 
 factors as the plant volume per hour as well as on average size of lot and 
 the average length of the break-for-lot period, 
 
 In apple and pear plants, the loss of time ranged from 3 per cent in 
 Plant S to 28 per cent in Plant N. This extreme variation is due primarily 
 to the difference in the length of the break period— less than one minute 
 for Plant S and more than eleven minutes for Plant N. The lU-per cent loss 
 in Plant L, on the other hand, is primarily the result of unusually low 
 
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5. 
 
 average lot size and the consequent increased number of break periods per 
 hour. Plant U has a break time per lot that is about average, but with lots 
 averaging more than 23,000 pounds, the time loss is only 6 per cent. 
 
 With relatively smaller lots and typically more break-f or-lot periods 
 per hour, the loss in effective working time is typically greater in olive 
 plants than in apple and pear plants. The table indicates that the time 
 loss ranged from 7 per cent in Plant I to 2h per cent in Plant IV. The 
 large losses in Plants IV and IX are due to small lot size relative to plant 
 volume and so to relatively large numbers of breaks per hour. Plant I has 
 the lowest average lot size of all the plants studied, yet, has the lowest 
 time loss. This is due to the very low plant volume per hour and the re- 
 lated small number of lots run per hour. 
 
 The effects of size of lot and break time on effective working time 
 are indicated in Figures 1 and 2 for more or less typical plants. The 
 diagrams are based on typical values for plant volume, lot size, and average 
 break time. In individual plants these factors may differ considerably from 
 typical values. In such cases the time loss would differ from that indi- 
 cated by the diagrams. 
 
 With these limitations in mind, the first diagram (Figure 1) indicates 
 how increases in the average break-f or-lot period cause corresponding in- 
 creases in the time loss. In general, each increase of one minute in break 
 time is associated with an increase of 2.5 per cent in the loss of effective 
 working time in apple and pear plants. Because of typically smaller lot 
 sizes, the effect of increased break tjme is more pronounced in olive 
 plants— each one-minute increase typically results in a 5-per cent loss in 
 effective work time. 
 
 Figure 2 shows the general effects of changes in average lot size on 
 effective working time. Since increases in lot size usually mean fewer 
 lots per hour and so less delay time, this diagram indicates that the loss 
 of effective working time becomes smaller and smaller as lot size is in- 
 creased. In apple and pear plants, lots as small as 2,000 pounds would 
 mean an average loss of nearly ho per cent in effective working time. With 
 lots of 10,000 pounds (approximately average), the time loss would be re- 
 duced to 7.5 per cent while larger lots would bring further reductions, 
 approaching a minimum time loss of 3 per cent. In olive plants, lot sizes 
 of less than 1,000 pounds—and 25 per cent of actual lots fell in this 
 category —would mean time losses of well over h0 per cent in typical plants. 
 
0 4 8 12 16 20 24 
 
 Size of lot in thousand pounds 
 
 Figure 2. Effect of size of lot on the utilization of plant time. 
 
7. 
 
 Even average lots of 3,000 pounds are associated with an average loss of 
 15 per cent. Very large lots can be handled quite efficiently, however, 
 with average time losses approaching 2 per cent. 
 
 Costs of the Separate-Lot System 
 
 As explained above, the major impact of the separate-lot system on 
 plant operating costs is in terms of the loss of effective working time 
 and so in the reduced volume of fruit handled per hour by the plant. In 
 most plants, the elimination of the separate-lot system would permit only 
 minor changes in the working forces — grower tally girls for packed fruit 
 could be eliminated in fresh packing houses, and the number of men weighing 
 and handling graded and sized olives could be reduced in some olive plants. 
 
 Table 2 summarizes data on plant volumes and estimated direct labor 
 costs for the apple, pear, and olive plants included in the study. This 
 table shows that the elimination of the separate-lot system would result 
 in increases in the potential plant volume per hour, with small increases 
 where the present system results in small reductions in effective working 
 time and large volume increases where present time losses are large. Most 
 of the plants would be able to reduce the direct labor payroll per hour, 
 although these changes would be relatively minor. The combined influence 
 of direct labor reductions and increased volume per hour would be reduc- 
 tions in average direct labor costs (exclusive of packing labor and other 
 piece-rate workers) ranging from $0.13 to $1.09 per thousand pounds of 
 apples and pears and from $0.30 to $0.80 per thousand pounds of olives. 
 While these costs of the separate-lot system may not seem large, they 
 become significant in terms of the total volume of fruit handled by a 
 plant in any season. Moreover, the range in costs emphasizes that many 
 plants can improve efficiency and reduce costs by adjusting their separate- 
 lot systems in order to minimize the loss in effective working time. 
 
 These adjustments may include the installation of automatic counting 
 and weighing devices, better coordination of tallying activities in the 
 various parts of the plant, and improved systems of record keeping. The 
 potential costs of the separate-lot systems in the sample plants with the 
 break period between lots reduced to an average of two minutes are shown 
 in the last column of Table 2, No improvement is indicated for Plants S 
 and III since the break time per lot is already two minutes or less. 
 
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8 
 
 TABLE 2 
 
 The Effects of the Separate-Lot System on Potential 
 Plant Volume and on Direct Labor Costs, 
 California Apple, Pear, and Olive Plants 
 
 
 Present — with separate lots 
 
 PotentiaJ 
 
 . — no separate lots 
 
 Labor cost per 1,000 
 
 
 Plant 
 vo_Lume 
 
 
 Labor 
 
 Plant 
 volume 
 
 Direct 
 
 T.a Vinv 
 
 XJGL D Ul 
 
 pounds for the 
 separate-lot system 
 
 Plant 
 
 1,000 
 pounds 
 per nour 
 
 plant 
 payroll 
 per nour 
 
 cost per 
 1,000 
 pounds 
 
 1,000 
 pounds 
 per hour 
 
 plant 
 payroll 
 per hour 
 
 cost per 
 1,000 
 
 Present 
 
 Improved : two- 
 minute breaks- 
 for-lots 
 
 
 
 
 Apple and pear packing 
 
 houses 
 
 
 
 A 
 
 20.U 
 
 $ 62. U0 
 
 $3.06 
 
 21.5 
 
 $ 61. U0 
 
 $2.85 
 
 $0.21 
 
 $0.15 
 
 B 
 
 18.0 
 
 73.90 
 
 U.11 
 
 19.1 
 
 7-2.80 
 
 3.81 
 
 0.30 
 
 
 L 
 
 19.8 
 
 82.80 
 
 U.18 
 
 23.1 
 
 81.70 
 
 3.5U 
 
 0.6U 
 
 0.35 
 
 M 
 
 2U.8 
 
 58.70 
 
 2.37 
 
 26.6 
 
 57.70 
 
 2.13 
 
 0.20 
 
 0.13 
 
 N 
 
 38 .U 
 
 1U0.00 
 
 3.65 
 
 53. U 
 
 136.80 
 
 2.56 
 
 1.09 
 
 0.20 
 
 R 
 
 U5.0 
 
 119.70 
 
 2.66 
 
 U8.8 
 
 117.60 
 
 2.U1 
 
 0.25 
 
 
 S 
 
 U6.0 
 
 13U.20 
 
 2.92 
 
 U7.U 
 
 132.10 
 
 2.79 
 
 0.13 
 
 0.13 
 
 T 
 
 31.2 
 
 103.90 
 
 3.33 
 
 33.5 
 
 102.80 
 
 3.07 
 
 0.26 
 
 
 U 
 
 60.6 
 
 120. Uo 
 
 1.99 
 
 6U.5 
 
 117.30 
 
 1.82 
 
 0.17 
 
 0.13 
 
 
 
 Olive plants 
 
 I 
 
 3.1 
 
 11 .Uo 
 
 3.69 
 
 3.U 
 
 11.U0 
 
 3.37 
 
 0.32 
 
 0.19 
 
 II 
 
 13.3 
 
 2U.90 
 
 1.87 
 
 15.3 
 
 23.60 
 
 1.55 
 
 0.32 
 
 0.25 
 
 III 
 
 10.0 
 
 28.70 
 
 2.87 
 
 12.2 
 
 25.30 
 
 2.07 
 
 0.80 
 
 0.80 
 
 IV 
 
 11.8 
 
 21.00 
 
 1.78 
 
 15.5 
 
 19.10 
 
 1.23 
 
 o.55 
 
 0.35 
 
 V 
 
 10.5 
 
 25.20 
 
 2.U0 
 
 11. u 
 
 2U.00 
 
 2.10 
 
 0.30 
 
 0.28 
 
 VI 
 
 7.2 
 
 17.80 
 
 2.50 
 
 8.7 
 
 16.60 
 
 1.91 
 
 0.57 
 
 0.32 
 
 VIII 
 
 5.2 
 
 17 .Uo 
 
 3.35 
 
 6.1 
 
 16.30 
 
 2.67 
 
 0.68 
 
 0.38 
 
 IX 
 
 12.3 
 
 21.30 
 
 1.73 
 
 16.0 
 
 18.80 
 
 1.18 
 
 0.55 
 
 0.35 
 
 a/Dashes indicate data not available. 
 
1*0 
 
 
 
 
 
 
 
 
 
 
 : 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <y.*»» 0 
 
 
 
 
 ■ 
 
 
 
 
 
 
 
 
 06,501 
 
 
 
 
 
 
 
 - i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■ 
 
 • 
 
 
 
 
 
 
 
 
 : 
 
 ! 
 
 
 
 
 
 
 ! HI 1 
 
 
 
 * 
 
 ■ oi,-?i 
 
 
 : 
 
 
 
 
 ... 
 
 . 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 : 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Oft *\T 
 
 
 
 
 £.S1 
 
 b i 
 
9. 
 
 However, several of the plants show important potential gains in efficiency 
 with the average reduction in cost amounting to about UO per cent. 
 
 The above cost estimates refer only to direct payroll costs and do not 
 include possibilities of cost reduction from the spreading of overhead costs 
 to a larger volume. In order for such reductions in overhead costs to 
 materialize, however, the plant volume per season would have to be increased. 
 While the elimination of the separate-lot system might make such increases 
 possible, the direct effect is in terms of volume per hour . For this reason, 
 overhead costs have not been included in the above calculations. 
 
 Finally, it should be noted that these labor cost estimates for the 
 separate-lot systems are useful primarily in making comparisons among plants 
 using this system. They do not mean that a plant could completely eliminate 
 the costs of accounting to producers. Costs with sampling systems will be 
 described in the following section of this report and these costs then com- 
 pared with the costs for separate-lot systems to determine the particular 
 method most economical under varying conditions. 
 
 SAMPLE GRADING SYSTEMS 
 
 Sample grading systems are used in many fruit packing houses as a means 
 of accounting to growers for the amounts of their fruit falling in each 
 grade or size classification. In designing such a system, two important 
 items must be considered: (1) the selection of the sample and (2) the size 
 of the sample. 
 
 Sample Selection 
 
 Ideally, the sample should be selected at random throughout the entire 
 lot so that the sample will be representative of the actual distribution of 
 grades in the lot. The methods that are commonly used in a number of pack- 
 ing houses may fall a little short of this ideal. In apple and pear plants, 
 for example, the usual procedure is to select several full lugs of fruit 
 at random from each lot, the contents of these boxes serving as a sample. 
 If the fruit in each box is a random representation of all of the fruit in 
 the lot, this procedure will give satisfactory results. However, if, in 
 the picking and field handling operations, fruit of particular grades becomes 
 fairly concentrated in certain boxes, the sample that is obtained may not 
 
■ 
 
 - . ■ : . 
 
 • ■ ; 
 
 ■ 
 
 ■ XiiJ- to 
 
10. 
 
 be representative. Estimates based on these samples will be free from any- 
 consistent bias but will be subject to a wider margin of error than would 
 otherwise be the case. 
 
 Representativeness may be improved somewhat by deriving the sample 
 from a part of each of a number of lugs. For example, if h per cent of 
 a lot of 100 lugs is to be sampled, the sample may be obtained by taking 
 one-half of the fruit from each of eight randomly selected lugs being 
 certain that the selection is not influenced by size or quality character- 
 istics. 
 
 A method used in olive processing plants is to take a handful or 
 scoopful of fruit from each lug as it is received and unloaded, thus, 
 spreading the sample over the entire lot. This practice may give biased 
 results, however, if fruit of particular sizes tend to be sifted out of 
 the hand or scoop during the sampling process. Moreover, this method 
 may conflict with the use of efficient unloading and receiving practices. 
 The magnitude of the possible bias probably will not be large, however, 
 and may be offset to a large extent by the better distribution of the 
 sample throughout the lot. 
 
 Mechanical sampling devices which provide convenient and unbiased 
 methods of sampling the entire lot at random are to be preferred. These 
 devices generally consist of shear boards or drops in a conveyor leading 
 from the dumping station, which can be adjusted to collect a sample of the 
 desired size. The shear boards are merely adjustable boards placed on a 
 conveyor belt which "shear off" a portion of the fruit as it passes by, 
 this portion serving as a sample. Since it is difficult to regulate very 
 precisely the size of the sample obtained by this method, the first sample 
 may be divided or "subsampled" to obtain a final sample of the appropriate 
 size. No attempt has yet been made to study the representativeness of 
 samples selected by this method. The method of sampling by drops in a 
 slot conveyor requires more expensive equipment than the shear board method. 
 This device, which has been used primarily in citrus packing houses, is 
 more precise and has been subjected to a considerable amount of statistical 
 testing. It has not yet been used for deciduous fruits but presumably 
 could be adapted for this type of product .3/ 
 
 3/ For additional information, see Roy J. Smith, "Accuracy Improved on 
 Growers' Returns by New Sampling Machine." Fruit and Vegetable Review , 
 vol. 9, no. 10, January, 19H9. 
 
0<t £o»£<fiH 
 
 mc xnd 
 
 ri4 yd psfur^do 
 
 i . 
 
 3 srkr p^U 4j 
 
 -'am jri'j 
 
 -VI 39b 
 
 's3 nshi-io 
 
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 ,Xi zjaij io 1 b>. 
 
11. 
 
 Sample Size 
 
 The second important aspect of sample grading, and the one of primary 
 concern in this report, is the size of the sample to be selected. With 
 any sampling system, the estimates of the proportions of the fruit in each 
 grade will be subject to some error because the distribution of grades in 
 the small part of the fruit that is sampled may differ somewhat from the 
 true distribution of grades for all the fruit delivered by the grower. The 
 magnitude of such differences can be reduced, however, by increasing the 
 size of the sample. 
 
 The objective in designing a sample grading system is to provide a 
 basis for making accurate payments to growers for their fruit — to limit 
 the possible error in these payments to some specified amount with a small 
 probability of exceeding this limit. The practical sampling procedure to 
 follow depends on the amount of advance information available concerning 
 the total season or pool period deliveries for each grower, the expected 
 proportions of fruit falling in particular grades, and the price for each 
 grade. As knowledge about these conditions increases, sampling systems 
 can be designed that are more efficient in achieving any desired degree of 
 accuracy. 
 
 Four cases regarding the amount of advance information may be distin- 
 guished: Case I — nothing is known about any of the indicated conditions; 
 Case II — reasonable estimates can be made concerning the expected minimum 
 total quantity of fruit to be delivered by each grower during the season; 
 Case III — advance estimates are available concerning the proportions of 
 fruit falling in particular grade classifications as well as total deliver- 
 ies; and Case IV — estimates are available concerning the prices of each 
 grade of fruit as well as the proportions and total deliveries. Under 
 Cases I, II, and III, it is necessary to express accuracy in terms of the 
 weight or proportion of fruit in a particular grade rather than in terms 
 of payments. In Case IV, however, accuracy may be expressed directly in 
 terms of total payments or average prices. 
 
 Accuracy in Estimating the Proportion of Fruit in Each Grade . — Growers 
 and plant operators will ordinarily be interested in obtaining some degree 
 of accuracy relative to the total quantity of fruit delivered by the grower 
 in a season rather than for a particular lot of fruit. Under the conditions 
 of Case I, however, the best that can be done is to sample so as to obtain 
 
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12. 
 
 a given amount of accuracy for each lot that is delivered realizing that, 
 if the grower does deliver more fruit, and if additional samples are taken, 
 the over-all accuracy of estimate will be somewhat better than indicated 
 for the particular lot. On the other hand, if past experience plus the 
 personal knowledge of the plant manager and fieldmen permit a reasonably 
 close estimate of minimum deliveries for each grower (Case II), the sam- 
 pling procedure can be much more efficient. 
 
 For either Case I or II, the quantity of fruit to be sampled depends 
 on (1) the desired degree of accuracy and (2) the quantity of fruit in 
 the lot or the expected total quantity of fruit that the producer will 
 deliver during the season. 
 
 The accompanying diagram (Figure 3), which may be used for both 
 Cases I and II, indicates how the percentage of fruit to be sampled from 
 each lot changes with the size of lot or total deliveries per season for 
 several designated degrees of accuracy. For Case II, a total sample of 
 the required size could be obtained by taking the entire quantity of one 
 or two of the lots of fruit delivered by a grower as a sample. For 
 adequate representation, however, the sample should be spread throughout 
 all of the grower's lots. Therefore, the total sample is obtained by 
 sampling a percentage of each lot of fruit delivered. It is assumed that 
 each lot sample is a random representation of the total of the fruit in 
 the lot. Accuracy is expressed as a per cent of the total weight or number 
 of fruit of all grades. For example, if the admissible error is to be 
 limited to 1 per cent and the true proportion for the particular grade to 
 which the error limit refers is 50 per cent, the estimated proportion for 
 this grade based on the sample may be expected to fall within k9 and 51 
 per cent ($0 t 1) in the great majority of the samples. 
 
 The basic calculations are made in terms of numbers of individual 
 fruit and are generally applicable to most fruits. Since the quantity 
 of fruit is usually measured in terms of weight, an appropriate ratio 
 must be used to convert to numbers. Gravenstein apples and Bartlett 
 pears for example, average about three fruit per pound of field run produce, 
 while Sevillano olives average from forty to sixty fruit per pound. The 
 lower of the two olive figures should be used in estimating the number of 
 fruit delivered by a grower to be more certain of limiting the error of 
 the desired range. 
 
Figure 3. 
 
 Effect of the desired degree of accuracy and total quantity of fruit per lot or per season on the 
 percent to be sampled from each lot for conditions specified under sampling Cases I and II. 
 
To illustrate the use of the diagram, suppose that under Case I con- 
 ditions in which no advance estimates are available, a grower delivers a 
 single lot of fruit consisting of 10,000 pounds of Gravenstein apples or 
 Bartlett pears (30,000 fruit). To be reasonably certain that the error in 
 estimating the proportion of fruit in any grade will be less than 1 per 
 cent of the total weight of fruit in all grades would require a sample of 
 2U.3 per cent of the lot (see point A in the diagram). On the other hand, 
 if minimum total season deliveries for this grower can be estimated in ad- 
 vance (Case II), the per cent to be sampled from each lot can be greatly 
 reduced while still obtaining the desired over-all degree of accuracy. 
 For example, if expected total deliveries were at least 30,000 pounds of 
 apples or pears (90,000 fruit), 1-per cent accuracy could be obtained with 
 a sample of 9.6 per cent of each lot as indicated by point B. For growers 
 with deliveries of at least 100,000 pounds (300,000 fruit), the same 
 accuracy could be obtained with a sample of only 3.1 per cent (see point C). 
 For growers with deliveries above U 50, 000 fruit, use the scale in the upper 
 right-hand corner of the diagram. 
 
 In addition, it is clear that, for any size of delivery, as the limit 
 of admissible error is increased (accuracy reduced), the per cent to be 
 samples from each lot decreases. For example, to limit the admissible 
 error to one-half of 1 per cent for a grower delivering 300,000 fruit would 
 require a sample of 11. h per cent of each lot, but to limit the error to 
 only 1 per cent would require a sample of only 3.1 per cent of each lot. 
 
 It should be emphasized that the limits of admissible error shown in 
 Figure 3 do not represent the amount of error that will occur in each esti- 
 mate but rather upper limits to the error that will be reached or exceeded 
 only infrequently. The limits of admissible error in Figure 3 are based 
 on the probability of obtaining an error of estimate as large as, or larger 
 than, the amount specified only once out of twenty times. This ratio, 
 though arbitrarily selected, is commonly used in statistical analysis. 
 The great bulk of estimates would be subject to errors considerably less 
 than the limits stated. With an error limit of 1 per cent, for example, 
 the average error would be about four-tenths of 1 per cent. Whatever the 
 limit of admissible error, the actual error can be expected to be approxi- 
 mately one-half that amount in about two-thirds of the cases. Moreover, 
 the error for any grade having a true proportion other than $0 per cent 
 would be somewhat below the amounts discussed above. Thus, these curves 
 
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15 
 
 do not indicate the typical amount of error in the estimates of the pro- 
 portions of various grades but upper limits to tbe expected error that 
 would be exceeded only in relatively rare cases. 
 
 If none of the grades approach the proportion of £0 per cent, the 
 procedures outlined for Cases I and II will involve larger samples than 
 actually required to attain the specified degree of accuracy. If there is 
 no way to foretell the approximate values of the proportions in the class 
 or grade containing the largest quantity of fruit, however, these procedures 
 will remain the best available guides.^/ On the other hand, if, from past 
 experience and observations of fieldmen, the plant manager can make a 
 reasonable forecast of the proportion in the largest grade for a given 
 grower and of the approximate amount of fruit that the grower will deliver 
 in the season (Case III), more precise and economical procedures are 
 possible. 
 
 For Case III conditions, the per cent to be sampled from each lot, as 
 determined in Figure 3 (Cases I and II), may be modified according to the 
 following general rules: 
 
 (1) If the expected per cent of fruit in the largest grade or class 
 is between 31 and 69 per cent, use Figure 3 without modification. 
 
 (2) If the expected per cent of fruit in the largest grade or class 
 
 is between 70 and 79 per cent, or 21 and 30 per cent, multiply the per cent 
 to be sampled as determined in Figure 3 by: 
 
 0.9 if the expected deliveries are less than 90,000 fruit per season; 
 
 0.85 if the expected deliveries are 90,000 or more fruit per season. 
 
 (3) If the expected per cent of fruit in the largest grade or class 
 
 h/ Proportions should be calculated in terms of numbers of individual 
 fruit . If the average size of fruit is the same for all grades, proportions 
 of weight will be the same as proportions of individual fruit, but if fruit 
 size varies among classes, the weight and numbers proportions will differ. 
 For example, a grade containing very large fruit might contain a high pro- 
 portion of the total quantity of fruit in terms of weight, but a small pro- 
 portion in terms of numbers. 
 
 5/ These rules are simplified for practical application. See the 
 Appendix for more accurate expressions. 
 
 6/ Use minimum proportions if the expected proportion in the largest 
 grade is greater than 50 per centj maximum expected proportions if less than 
 50 per cent. For example, the expected proportion in the largest grade 
 might be at least 70 per cent or not more than kO per cent. 
 
"CIO 
 
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16. 
 
 is 80 per cent and above, or 20 per cent and below, multiply the per cent 
 to be sampled as determined in Figure 3 by: 
 
 0.8 if the expected deliveries are less than 90,000 fruit per season; 
 
 0.65 if the expected deliveries are 90,000 or more fruit per season. 
 
 As indicated previously, the real objective in designing a sample 
 grading system is to provide estimates that will limit the possible error 
 in final payments to growers to some specified amount. While the procedures 
 outlined for Cases I, II, and III achieve this result indirectly, the mag- 
 nitude of the limit of payment error is influenced by prices. If advance 
 information is available concerning the prices of the various grades of 
 fruit (Case IV), the sampling system may be designed directly in terms of 
 some specified degree of accuracy in the final payments. The calculation 
 of sample size for this case, however, while not difficult, is too complex 
 to present in simple form. Therefore, the procedures outlined for Cases I, 
 II, and III will probably prove most useful for the majority of packing 
 house operators.!/ 
 
 Sampling Cost 
 
 The costs of sample grading include the costs for workers who collect, 
 sort, weigh, tally, and transport the fruit that is sampled and the costs 
 of sample grading equipment ..§/ The sampling equipment consists of items 
 such as grading tables for apples and pears, sizing-sorting equipment for 
 olives, and scales for weighing the fruit. Mechanical sampling devices 
 for sample selection, used in some plants, are not included here. 
 
 Estimates of these costs, based on studies of packing and processing 
 plant operations, are summarized in Table 3. Two levels of equipment cost 
 are indicated for apple and pear plants. For small plants — volume below 
 20,000 pounds per hour — a small grading table with a capacity for one or 
 two sorters would probably be adequate. Larger plants, or plants with a 
 large volume of fruit sampled, may require a larger table with three or 
 four sorter capacity and with power-driven belts. The equipment for olives 
 would be about the same for plants of nearly all sizes. 
 
 7/ A simplified mathematical and graphical treatment of the sampling 
 problem for Case IV is given in Appendix A. 
 
 8/ Since the fruit that is sample graded does not have to be run over the 
 main graders, only part of the sample sorting labor has been treated as a 
 sampling cost. 
 
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17. 
 
 TABLE 3 
 
 Estimated Labor and Equipment Costs for Sample Grading 
 
 m. 
 
 Apple and 
 
 pear plants 
 
 
 
 Volume less 
 than 20,000 
 pounds per hour 
 
 Volume more 
 than 20,000 
 pounds per hour 
 
 Olive 
 plants 
 
 Estimated installed cost of sample 
 grading equipment, 1952a/ 
 
 
 
 
 Grading tableb/ 
 
 
 $ 65o 
 
 $ 700 
 
 Bench scales — dial type 
 
 560 
 $3Io 
 
 560 
 $1,110 
 
 U80 
 $1,180 
 
 Estimated annual cost of equipmentc/ 
 
 $ 70 
 
 $ 160 
 
 $ 160 
 
 Typical labor hours per 1,000 pounds 
 of fruit in samples 
 
 
 1.9 
 
 20 
 
 Typical labor cost per 1,000 pounds 
 of fruit in samplesd/ 
 
 $ 2.28 
 
 $ 2.28 
 
 $ 2h 
 
 a/ Does not include costs of mechanical devices for sample selection. 
 
 b/ Table for small apple or pear plants has capacity for two sorters. Table for 
 large plants has powered belts and capacity for four sample sorters. The 
 olive sizing-sorting table has a capacity of about 150 pounds per hour. 
 
 c/ Based on a standardized set of annual charges for depreciation, repairs, insur- 
 ance, interest, and taxes; grading tables, lh.7 per cent; scales, 10 per cent 
 plus $7.50 per year for reoairs and maintenance. 
 
 d/ Based on typical average wage of $1.20 per hour for sample grading labor. 
 
18. 
 
 The costs given in Table 3 represent only average annual costs of 
 equipment and the labor cost per 1 J 000 pounds of fruit in the samples. 
 Total costs of sample grading depend on the quantity or per cent of fruit 
 received that is sample graded which, in turn, depends on the desired de- 
 gree of accuracy, the total volume handled per season in the packing house, 
 and the distribution of total deliveries among the individual growers. 
 These factors vary from plant to plant so any simple statement of sampling 
 costs is impossible. However, the general nature of these costs and their 
 relation to sampling accuracy may be illustrated with reference to the 
 conditions in a specific plant. Costs for a typical apple or pear packing 
 house and a typical olive processing plant are given in Table h* In these 
 examples it is assumed that estimates are available concerning expected 
 total deliveries per grower but not concerning expected proportions in the 
 various grades — i.e., Case II conditions exist. The percentages to be 
 sampled are thus read directly from Figure 3« 
 
 The size and number of growers are given in columns 1 and 2 of the 
 table where, for simplicity, they have been grouped in several convenient 
 sizes. This is a practical sampling procedure that might be followed in 
 most packing houses. The per cent to be sampled from each lot for each 
 size group was determined from Figure 3 for several levels of accuracy. 
 These figures have been applied to the total quantities of fruit to be 
 sampled for each size group. The columns are then added to indicate the 
 total quantities of fruit that would have to be sampled by this plant in 
 order to obtain various degrees of accuracy. These calculations, which a 
 manager may make for his own plant, indicate a rapidly increasing quantity 
 of fruit sampled — and, thus, an increasing sampling cost — with increases 
 in accuracy, especially for the higher levels of accuracy. 
 
 Note that for any level of accuracy a much higher per cent of fruit 
 is sampled for small growers than for those with large total deliveries. 
 In the illustrative pear plant, for example, to limit the admissible error 
 to 1 per cent for all growers requires that 9,6 per cent of each lot be 
 sampled for small producers and only eight-tenths of 1 per cent for the 
 largest producers. On the average, 2.6 per cent of the fruit would he 
 sample graded. While a uniform sample of this size would give the same 
 total sampling cost and approximately the same degree of accuracy (1 per 
 cent) for the average-size grower, the estimates will tend to be less 
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TABLE 4 
 
 The Effect of the Desired Degree of Accuracy on the Volume cf Fruit Sampled and 
 the Cost cf Sample Grading for Typical Apple, Pear, and Clive Packing Plants 
 
 Deliveries 
 
 Number 
 
 Total 
 
 Total sample 
 
 per grower 
 
 of 
 
 deliveries 
 
 
 Limit of 
 
 admissible error 
 
 — per cent of total weight 
 
 
 per season 
 
 growers 
 
 per season 
 
 5.0 
 
 3-0 
 
 2.0 
 
 1.0 
 
 0. 
 
 5 
 
 1,000 
 pounds^./ 
 
 
 1,000 
 pounds 
 
 per 
 cent 
 
 pounds 
 
 per 
 cent 
 
 pounds 
 
 per 
 
 cent 
 
 pounds 
 
 per 
 cent 
 
 pounds 
 
 per 
 cent 
 
 pounds 
 
 
 
 
 
 Apple or pear plant 
 
 
 
 
 
 
 
 30 
 50 
 
 100 
 
 150 
 
 200 
 400 
 
 4 
 8 
 10 
 
 6 
 
 5 
 2 
 
 120 
 400 
 
 1,000 
 900 
 
 1,000 
 
 800 
 
 0.4 
 
 0-3 
 0.2 
 0.1 
 0.1 
 0.1 
 
 480 
 1,200 
 2,000 
 
 900 
 1,000 
 
 800 
 
 1.2 
 
 0.8 
 0.4 
 0.2 
 0.2 
 0.1 
 
 1,440 
 3,200 
 4,000 
 1,800 
 2,000 
 800 
 
 2.6 
 1.6 
 0.8 
 
 0.5 
 0.4 
 0.2 
 
 3,120 
 6,4oo 
 8,000 
 4,500 
 4,000 
 1,600 
 
 9.6 
 6.0 
 
 3-1 
 2.1 
 1.6 
 0.6 
 
 11,520 
 24,000 
 31,000 
 18, 900 
 16,000 
 6,400 
 
 30.0 
 20.4 
 11.4 
 
 7.9 
 6.0 
 3.0 
 
 36,000 
 81,600 
 114,000 
 71,100 
 60,000 
 24,000 
 
 
 35 
 
 4,220 
 
 0.2 
 
 6,380 
 
 0.3 
 
 13,240 
 
 0.7 
 
 27,620 
 
 2.6 
 
 107,820 
 
 9.2 
 
 386,700 
 
 Total sampling cost*?/ 
 
 
 $175 
 
 $190 
 
 $223 
 
 $4o6 
 
 $1,042 
 
 
 
 
 
 
 Olive p 
 
 Lant 
 
 5 
 10 
 20 
 50 
 100 
 200 
 
 5 
 
 5 
 
 0.9 
 
 45 
 
 2.5 
 
 125 
 
 5.6 
 
 280 
 
 19.3 
 
 965 
 
 49.0 
 
 2,450 
 
 7 
 
 35 
 
 0.2 
 
 70 
 
 0.6 
 
 210 
 
 l.l 
 
 385 
 
 4.7 
 
 1,645 
 
 16.2 
 
 5,670 
 
 10 
 
 100 
 
 0.1 
 
 100 
 
 0.3 
 
 300 
 
 0.5 
 
 500 
 
 2.4 
 
 2,400 
 
 8.9 
 
 8,900 
 
 8 
 
 160 
 
 0.1 
 
 160 
 
 0.1 
 
 160 
 
 0.3 
 
 48o 
 
 1.2 
 
 1,920 
 
 4.6 
 
 7,360 
 
 5 
 
 250 
 
 0.1 
 
 250 
 
 0.1 
 
 250 
 
 0.1 
 
 250 
 
 0.5 
 
 1,250 
 
 1.9 
 
 4,750 
 
 5 
 
 500 
 
 0.1 
 
 500 
 
 0.1 
 
 500 
 
 0.1 
 
 500 
 
 0.3 
 
 1,500 
 
 1.0 
 
 5,000 
 
 1 
 
 200 
 
 0.1 
 
 200 
 
 0.1 
 
 200 
 
 0.1 
 
 200 
 
 0.1 
 
 200 
 
 0.5 
 
 1,000 
 
 41 
 
 1,250 
 
 0.1 
 
 1,325 
 
 0.1 
 
 1,745 
 
 0.2 
 
 2,595 
 
 0.8 
 
 9,880 
 
 2.8 
 
 35,130 
 
 Total sampling cost 
 
 $192 
 
 $202 
 
 $222 
 
 $397 
 
 $1,003 
 
 a/ Average of three fruit per pound used to calculate number of fruit with which to enter Figure 1. 
 b/ Costs based on larger size sample grading table. 
 
 C/ Average of 40 Sevillano olives used to calculate the number of fruit with which to enter Figure 1. Field run 
 fruit vary from 40 to 60 per pound. The lower figure is used to be more certain of limiting the possible error 
 to the desired range. 
 
20 
 
 accurate for large producers. From this standpoint, then, differential 
 sampling — varying the per cent sampled according to grower size — is a 
 more equitable procedure. 
 
 Following procedures similar to those indicated in Table h, costs 
 of sample grading have been estimated for a number of plants now using 
 separate-lot systems. These costs are given in Table £ for several de- 
 grees of accuracy. Details as to the season deliveries for each grower 
 were not readily available for every plant. Therefore, the sampling 
 costs have been estimated by using the average volume of deliveries per 
 grower to determine the average per cent of fruit to be included in sam- 
 ples. The cost estimates will be about the same as with the more detailed 
 procedures. 
 
 As in the typical plants represented in Table k f sampling costs per 
 thousand pounds of fruit received increase with increases in the degree 
 of accuracy. The costs also increase with decreases in the average volume 
 of deliveries per grower and, due to the fixed costs for equipment, de- 
 crease with increases in the total plant volume per season. 
 
 SAMPLING VERSUS SEPARATE-LOT SYSTEMS 
 
 The circumstances under which sampling or separate-lot procedures 
 would be more economical as a means of accounting to growers for the 
 amounts of their fruit falling in each grade or size classification 
 cannot be described in any simple form. As has been indicated, the cost 
 of a separate-lot system depends on such factors as the length of break 
 periods between lots, the average size of lot, rates of plant output, 
 and total direct hourly payroll. At the same time, sampling costs are 
 influenced by the desired degree of accuracy, the amount of advance 
 sampling information available, and the distribution of total deliv- 
 eries among individual growers. The possible sets of conditions for 
 which cost comparisons could be made are innumerable. 
 
 The procedures for calculating the costs of separate-lot systems 
 have been indicated in Tables 1 and 2. Costs of the alternative system — 
 sample grading — may be estimated by following the procedures used in 
 Table 1|, These calculations may be carried out by a manager for his own 
 plant to ascertain which system is likely to be most economical. 
 
cfflT tit 
 
 D30 ~}tCS 
 
 lot OR! 
 
TABLE 5 
 
 Costs cf Sample Grading Systems in Relation to Degree of Accuracy for 
 California Apple, Pear, and Olive Packing Plants 
 
 
 
 
 
 
 Average cost of sample grading 
 
 
 
 
 
 
 (dollars per 1,000 pounds of fruit received^' 
 
 
 
 
 
 Limit of admissible error for the average-size grower — 
 
 
 Volume per 
 
 Average volume cf 
 
 per cent of the total weight delivered 
 
 Plant 
 
 season 
 
 deliveries 
 
 per grower 
 
 3-0 
 
 2.0 
 
 1.0 
 
 0.5 
 
 
 
 
 estimated 
 
 
 
 
 
 
 1,000 pounds 
 
 1,000 pounds 
 
 1,000 fruit 
 
 
 
 
 
 
 Apple and pear 
 
 packing houses 
 
 
 
 
 A 
 
 3,520 
 
 90 
 
 270 
 
 0.06 
 
 0.06 
 
 0.13 
 
 0.33 
 
 B 
 
 2,331 
 
 70 
 
 210 
 
 0.08 
 
 0.09 
 
 0.17 
 
 0.42 
 
 L 
 
 4,524 
 
 115 
 
 3^5 
 
 0.04 
 
 0.05 
 
 0.10 
 
 0.27 
 
 M 
 
 6,385 
 
 320 
 
 960 
 
 0.03 
 
 0.03 
 
 0.05 
 
 0.11 
 
 N 
 
 5,695 
 
 240 
 
 720 
 
 0.03 
 
 0.04 
 
 0.06 
 
 0.14 
 
 R 
 
 9,000 
 
 320 
 
 960 
 
 0.02 
 
 0.02 
 
 0.04 
 
 0.10 
 
 S 
 
 10,300 
 
 115 
 
 345 
 
 0.02 
 
 0.03 
 
 0.08 
 
 0.25 
 
 T 
 
 8,125 
 
 580 
 
 1,7^0 
 
 0.02 
 
 0.02 
 
 0.03 
 
 0.07 
 
 U 
 
 5,745 
 
 190 
 
 570 
 
 0.03 
 
 0.04 
 
 0.07 
 
 0.17 
 
 
 Olive plants 
 
 I 
 
 487 
 
 35 
 
 1,400 
 
 0.35 
 
 0.40 
 
 0.47 
 
 0.95 
 
 II 
 
 1,713 
 
 42 
 
 1,680 
 
 0.12 
 
 0.12 
 
 0.21 
 
 0.62 
 
 III 
 
 3,100 
 
 18 
 
 720 
 
 0.08 
 
 0.12 
 
 O.36 
 
 1.25 
 
 rv 
 
 1,527 
 
 26 
 
 1,040 
 
 0.13 
 
 0.15 
 
 0.32 
 
 0.92 
 
 V 
 
 2,098 
 
 46 
 
 1,840 
 
 0.10 
 
 0.10 
 
 0.20 
 
 0.56 
 
 VI 
 
 1,218 
 
 Ik 
 
 560 
 
 0.18 
 
 0.23 
 
 0.54 
 
 1.67 
 
 VIII 
 
 824 
 
 16 
 
 640 
 
 0.24 
 
 0.29 
 
 0.56 
 
 1.56 
 
 IX 
 
 2,1)09 
 
 56 
 
 2,240 
 
 0.09 
 
 0.09 
 
 0.16 
 
 0.52 
 
 a/ Costs for apples and pears based on large size sample grading table. 
 
 H 
 
22. 
 
 Some general conclusions are possible, however, by comparing the costs 
 of separate-lot systems given in Table 2 with the estimated costs of sam- 
 pling systems given in Table $ for these same plants. The comparative 
 costs are illustrated by the bars in Figure k* For each plant, the bar 
 on the left represents the cost of the separate-lot system. The total area 
 of the bar indicates the present cost of the separate-lot system and the 
 darker stippled area represents the estimated cost with standard two-minute 
 average break time between lots. The present break time in Plants S and 
 III was less than or equal to two minutes so no reduction is indicated. 
 Data on costs with two-minute breaks were not available in Plants B, R, 
 and T. The bars on the right represent the estimated cost of a sample 
 grading system in each of these plants. The total area of the bars shows 
 the estimated cost of sampling with an admissible error of one-half of 
 1 per cent of the total weight of fruit for the average-size grower. The 
 cross-hatched area indicates the cost of sampling with a 1-per cent limit 
 of admissible error. 
 
 For apple and pear plants, the estimated cost of the sampling system 
 with a limit of admissible error of 1 per cent for the average grower is, 
 in all cases, less than the cost of separate-lot systems — even where the 
 break period is reduced to two minutes. If it is desired to limit the 
 admissible error to one-half of 1 per cent, the cost of sampling is sharply 
 increased. However, even in this case sampling costs are less than separate- 
 lot costs in five of the nine plants. Thus, for the majority of these 
 plants, the sampling system tends to be the most economical procedure. 
 
 The case for sampling is less clear in olive plants. While the costs 
 of the separate-lot system per 1,000 pounds of fruit received are gener- 
 ally higher in these plants than in apple and pear plants, so also are 
 sampling costs. With 1-per cent accuracy, estimated sampling costs are 
 less than present costs of separate-lot systems in seven of the eight 
 olive plants and are less than separate-lot costs with two-minute breaks 
 in five of the eight plants. However, if a limit of admissible error of 
 only one-half of 1 per cent is desired, sampling costs are greater than 
 present separate-lot costs in all but one plant. Thus, with a limit of 
 admissible error of 1 per cent or more, the sampling system has a cost 
 advantage in the majority of the plants. If a higher degree of accuracy 
 is desired, either the present or the improved separate-lot system is 
 generally less costly. 
 
■ • • ■ ■ . ■ ■ 
 
 . :■ . 
 
 1 
 
i 
 
 1.00 
 
 to -q 
 
 Jo §> 1 
 
 © | 
 o o> 
 
 || 
 
 °-8 
 
 D) C 
 C 3 
 — O 
 
 p. a 
 
 3 o 
 8 o 
 
 -£ Z 
 
 *S a.0. 
 
 «- to 
 </> E 
 O _D 
 
 0)0 
 
 a 
 
 u 
 
 at 
 > 
 < 
 
 Estimated costs of the 
 separate lot system 
 
 Estimated costs of the sampling system 
 
 Present 
 
 Improved — break periods 
 
 reduced to average of 
 
 2 minutes ' 
 
 Present only — data for 
 estimating cost with improved 
 system not available 
 
 Admissible error of 0.5 per cent of 
 the total weight of fruit for the 
 average-size grower 
 
 Admissible error of 1 .0 per cent of 
 the total weight of fruit for the 
 average-size grower 
 
 50 
 
 B L M N R S 
 — Pear and apple plants 
 
 U 
 
 III IV V VI VIII 
 Olive plants 
 
 IX 
 
 Figure 4. Comparative costs of sampling and separate lot systems in Ca 
 packing or processing plants. 
 
 lifornia pear, apple, and olive 
 
2U 
 
 APPENDIX A 
 
 The sampling chart for Case I and II sampling conditions and the 
 rules for its use under Case III, presented in the previous pages of this 
 report, should provide a useful guide to packing house operators who are 
 interested in setting up sampling systems or in checking on their present 
 procedures. For some plants, however, a more detailed and comprehensive 
 treatment may be required or desired. This appendix is included for the 
 benefit of those who are interested in more precise formulations of sam- 
 pling Cases I, II, and III, or in designing a sampling system directly in 
 terms of accuracy of the final payments to producers — Case IV. The pro- 
 cedures for determining the quantity of fruit to be sampled from each lot 
 for each sampling case are presented here in simple algebraic form. Cases 
 III and IV are also treated graphically. The calculations throughout are 
 based on the probability of exceeding the specified limit of admissible 
 error only once out of twenty times. 
 
 1. Accuracy of an estimated proportion 
 
 The general expression for determining the per cent to be sampled 
 from each lot in order to obtain a desired limit of admissible error is 
 
 c _ P (1 - P) 
 
 (1) 
 
 where S = the per cent to be sampled from each lot 
 
 P = the expected proportion in the particular grade or class 
 
 A = the limit of admissible error as a per cent of the total weight 
 of all grades of fruit 
 
 9/ 
 
 N = the expected total number of fruit delivered.-' 
 
 9/ Note that this expression may be written 
 
 /P (1 - P ) ( N - n ) 
 A = 1.96 V n N 
 
 where n is the total size of sample and the expression under the square 
 ro ot is the standard error of a proportion. When multiplied by 1.96 — i.e., 
 V3 • 8Ul6 — the standard error includes 95 per cent of the sample estimates. 
 Equation (1) is obtained by simply writing the above expression in a form 
 that gives n/N, where n/N = S. As was indicated earlier, to obtain a repre- 
 sentative sample requires that a percentage of the fruit in each lot be in- 
 cluded in the total sample. The absolute quantity of fruit sampled, (n), 
 increases slowly with increases in total deliveries (N). Therefore, the per 
 cent to be sampled from each lot, (S), decreases with increases in N. 
 
• .' ' ". ; 4 ". • • • r 
 
 ■ • • • ' • ' ■ 
 
 . ,. . ■ ... . • ■ - 
 
 S sail 
 
25 
 
 A modification of this expression is used for each of the first three 
 sampling cases described previously. 
 
 Case I ; Nothing is known about total deliveries, expected proportions, 
 or prices. 
 
 P is assumed to be 0.5 as a sample that will limit the admis- 
 sible error to the desired amount for this proportion will result 
 in a smaller error for any other proportion. N is the number of 
 fruit in the lot, which can be estimated. Expression (1) becomes 
 
 S = g m2$ (2) 
 
 W N + - 25 
 
 This equation was used to obtain the curves in Figure 3. 
 Case II t Estimates are available concerning expected total deliveries 
 per season. 
 
 Expression (2) still applies but N now becomes the expected 
 deliveries per season rather than the number of fruit per lot. 
 Case III : Estimates are available concerning expected total deliver- 
 ies per season and the expected quantity of fruit in the largest class or 
 grade. 
 
 Return to expression (1). Substitute for P the expected 
 proportion in the largest grade. Use a minimum value if 
 the expected proportion in the largest grade is more than 
 50 per cent, a maximum value if less than 50 per cent. For 
 example, the expected proportion in the largest grade might 
 be at least 70 per cent or not more than k0 per cent. The 
 expected error in estimating any other grade will not be any 
 greater than the error specified for the largest grade. 
 Example ; 
 
 Suppose that: 
 
 (a) It is expected that a particular grower will have at least 
 65 per cent of his fruit in the largest grade or class. 
 
 (b) Total season deliveries will be at least 90,000 fruit. 
 
 (c) The desired limit of admissible error is three-fourths 
 of 1 per cent of the total weight of fruit of all 
 grades. 
 
.52 
 
 ©sat*, a. 
 
 - 
 
 ' ■ 
 
 ■ 
 
 tc lofts a 
 
 I 
 
26. 
 
 Substituting these values in equation (1) 
 
 s = «65 x .35 m .2275 
 
 ^§g|^ (90,000) + (.65 x .35) >0 ^ggf (90,000) ♦ .2275 
 
 = 5' 0625^ = *^ or ^**? per cent 
 
 Thus, lli. 7 per cent of each lot of fruit delivered by this grower 
 should be sampled. 
 
 A graphic solution to the Case III sampling problem is given in Figure 
 5. The per cent to be sampled from each lot is determined by the expected 
 proportion of the fruit in the largest grade, the admissible error and the 
 total amount of fruit delivered by the grower during the season or pool 
 period. As before, the admissible error curves in the chart are based on 
 the probability of obtaining an error of estimate as large as, or larger 
 than, the amount specified only once out of twenty times. 
 
 To illustrate the use of the diagram, assume again the conditions of 
 the example given above. Enter Figure 5 at point A with the proportion 
 0.65$ move vertically to point B on the curve representing an admissible 
 error of 0.75 per cent; and then mcve horizontally to point C on the curve 
 representing deliveries of 90,000 fruit. Finally, reading down to point D 
 indicates that the desired level of accuracy may be obtained by sampling 
 between Ik and 15 per cent of each lot of the fruit delivered. The alge- 
 braic calculation was llu7 per cent. Note that sampling based on a maximum 
 proportion of 50 per cent, as in the previous procedures (Figure 3), would 
 have indicated a sample of approximately 16 per cent and that the differ- 
 ence between the two methods increases with decreases in the admissible 
 error, with decreases in the pool deliveries and with increases in the de- 
 parture of the proportion in the largest grade from 50 per cent. 
 
 A plant manager may also be interested in estimating the limit of error 
 actually obtained for sample proportions after a sample has been selected. 
 For this purpose, equation (1) may be written 
 
 /P (1 - P) ,1 
 A = 1.96 V N K s L) (3) 
 
.b?f.qmi?a eri bii 
 ijxrlor. •jifiqf/i'j A 
 
 ■ode- nevig 
 
Percent in largest grade Percent to be sampled from each lot 
 
 "Weights are for apples and pears only. In terms of numbers of fruit, the diagram has general applicability for most fruits and vegetables. 
 
 Figure 5. Percent to be sampled from each lot as determined by the percent of fruit in the grade of largest 
 quantity, the limit of admissible error, and the total season or pool deliveries per grower. 
 
28. 
 
 Example : 
 
 Suppose that a sample has been drawn and : 
 
 (a) The proportion in the largest grade — as estimated from the 
 sample — is 60 per cent. 
 
 (b) Total deliveries per season are 100,000 fruit. 
 
 (c) Four per cent of this total was sample graded. 
 
 To estimate the limit of error actually obtained: 
 
 a - i qA A 6 x . It ^ 1 ~T7 -i «/ /.2U x 2h 
 
 ~ 1,96 V TooTooo ( ToU " 1} = 1>96 V 100,000 
 
 1.96 
 
 100 
 
 -\^/.5>76 = .015 or 1.5 per cent 
 
 In other words, the manager may assume that the true proportion of 
 fruit in this grade is contained within the range 58.5 to 61.5 per cent 
 (60 * 1.5), with one chance in twenty of being wrong. The historical 
 error may also be approximated by working backwards in Figure 5» Enter 
 the diagram with the actual per cent sampled; read vertically to the line 
 indicating approximately the total deliveries; read horizontally to the 
 left, stopping above the estimated proportion (as actually measured). 
 Reading to the nearest error line will approximate the limit of error 
 actually obtained. 
 
 2. Accuracy in terms of payments to growers — Case IV 
 
 If information is available concerning the prices of fruit in the 
 various grades or classes as well as expected deliveries and proportions, 
 the sampling system may be designed to give, directly, some specified 
 degree of accuracy in the payments to growers. The general expression 
 for determining the per cent to be sampled from each lot under these 
 conditions is 
 
 (P A 2 + P 2 m/ + P 3 M 3 2 + . ..) - (P^ ♦ P 2 M 2 ♦ P 3 M 3 + ...) 2 
 
 S = — * = ^ * (it) 
 
 (P,1L + PJ/L + P M, + ...) (AN ,) + (P,BL + P 0 M 0 + P 0 M^ + ...) 
 1 1 33 (33UIo" " 1 ) in " 2 2 3 3 
 
 where S = the per cent to be sampled from each lot 
 
 P's are the proportions of fruit in each grade 
 
 M's are the relative or actual prices per fruit for each grade 
 
 A = the limit of admissible error as a per cent of the total 
 
 payment to the grower 
 N = the expected total deliveries (number of fruit). 
 
.Bit Isi; 
 
 : 
 
 yjjHI rap. 
 
 ; , ■.. . 
 
29. 
 
 Note: Where accuracy is expressed as a percentage, it will be 
 convenient to use relative rather than actual prices per 
 fruit. Let the price of the highest grade be 1.00, with 
 all other prices as a fraction of the highest price. 
 
 It will, of course, be impossible to estimate in advance the exact 
 proportion of fruit falling in each grade. Otherwise, there would be no 
 need for sampling. Normally, however, the plant manager will have some 
 idea that (say) at least 50 per cent of the fruit will fall in one grade, 
 not more than 10 per cent in another grade, and so on. To be more certain 
 of limiting the admissible error to the desired range, the advance esti- 
 mates of proportions should be conservative. If there are doubts, the 
 estimates of proportions for the highest price grade should be in the direc- 
 tion of 50 per cent, and the estimates for other grades should be in the 
 direction of increasing the proportions in the very lowest price grades. 
 If the total of the two highest grades is less than 50 per cent, the esti- 
 mates for both of these grades should be in the direction of bringing their 
 total to 50 per cent. 
 Example ; 
 
 Suppose that in a pear packing plant it is expected that: 
 
 (a) A particular grower will have at least 60 per cent of his fruit 
 in the highest price grade and not more than 10 per cent in the 
 lowest price grade. The other 30 per cent falls in intermediate 
 grades. Note that these proportions are in terms of numbers of 
 fruit rather than weight. 
 
 (b) The relative prices for the three grades are: No. 1 = 1.0, 
 No. 2 = 0.3, No. 3 = 0.1. 
 
 (c) The desired limit of admissible error is 1 per cent of the 
 total payment. 
 
 (d) Expected total deliveries are 200,000 fruit. 
 Equation (k) becomes: 
 
 (.6 x l 2 + .3 x .3 2 + .1 x .l 2 ) - (.6 x 1 + .3 x .3 + .1 x .l) 2 
 
 (.6 x 1 + .3 x .3 + .1 x .1) 2 ( .0001 x 200,000 ~) (.6 x l 2 + .3 x .3 2 + .1 x .l 2 ) 
 
 .628 - (-700) 2 .138 
 
 / ?nn s2 ( 2Q~~ + .628 = 20 x .1+9 + 713? = b 
 
 ( * 700) (378TH6- " x) 3.8U6 
 
 Thus, to obtain the desired limit of admissible error, 5.1 per cent 
 of each lot should be sampled. 
 
"ib ' 
 
 rii 
 
 Sfim 539 i. 
 
 l ib s>riJ- nj 
 
 T^)3t ano,Ljloqf>*iq In 8#\l'8fltx*S9 
 artf bus <cfns»o isq 05 So noirf 
 
 t&c CI nedc 
 
 ? wot 3 -isIuoiiiBq A (£ 
 
 riq tes<rf^iri M$ n * 
 •fcBTy eoriq ^3i>woJ 
 
 n& rlcf 
 
 b -'tis so en'T 
 
30. 
 
 Solutions to the Case IV sampling problem are illustrated graphically 
 in Figures 6 and 7 for two sets of assumptions concerning the proportions 
 of produce falling in the various grades. The limits of admissible error 
 are again based on the probability of exceeding that amount only once in 
 twenty times. The first graph is drawn on the assumption that all of the 
 fruit not in the first grade has the price of the lowest grade. In this 
 situation, if the prices of the intermediate grades are higher than that of 
 the lowest grade, the probability of exceeding the limit of error in esti- 
 mating the payments to growers will be even less than the amount specified. 
 To illustrate the use of this diagram, suppose that: 
 
 (a) It is estimated that a particular grower will have about 
 65 per cent of his fruit in the No. 1 grade. 
 
 (b) The expected price of the lowest grade relative to the 
 highest grade is 0.3. 
 
 (c) Total season deliveries are expected to be at least 90,000 
 fruit. 
 
 (d) It is desired to limit the admissible error to three-fourths 
 of 1 per cent (0.75) or less in the total payments. 
 
 Enter Figure 6 at point A with the proportion, 65 per cent (0.65) j 
 move horizontally to point B on the curve representing a relative price 
 of 0.3; then move down to point C on the line representing an admissible 
 error of 0.75 per cent; from C, move across to point D on the curve repre- 
 senting total deliveries of 90,000 fruit. Finally, reading down to point 
 E indicates that the desired level of accuracy may be obtained by sampling 
 13.3 per cent of each lot of fruit delivered. In this example, if the 
 relative prices had been 0.2. the required sample would have been 18 per 
 cent; if 0.5, about 6 per cent. Note that it is also possible to inter- 
 polate between the curves representing various amounts of total deliveries. 
 In the example above, if deliveries had been 120,000 fruit (ijO,000 pounds 
 of apples or pears) instead of 90,000, the required sample would have been 
 about 10.5 per cent. 
 
 If the price of the lowest grade of fruit (culls) is zero and the 
 proportion of the fruit falling in this grade quite small, the per cent 
 to be sampled from each lot as determined from Figure 6 will be larger 
 than required to give the desired degree of accuracy. To avoid this 
 situation, the alternative diagram (Figure 7) may be used under these 
 conditions. This chart is based on the assumption that the proportion 
 
■ 
 
 - 
 
 "is j as 
 
50 
 
 60 
 
 ® 
 70 
 
 80 
 
 S. 90 
 
 e 
 
 Percent to be sampled from each lot 
 15 © 10 5 
 
 d of 
 
 ioi 
 o'l 
 
 07 
 
 30. 
 
 I0 > 
 
 -1000 Fruit 
 
 -1000 Lbs. 
 
 60, 
 20 
 
 TOTAL SEASON 
 OR POOL PERIOD 
 DELIVERIES PER. 
 .GROWER 
 
 225 
 75 
 
 300 
 100 
 
 450 600 
 150 200 
 
 oZ 
 
 © 
 
 Weights are for apples 
 and peors only. In terms\ 
 of numbers of fruit, the 
 diagram has general ap- 
 plicability for most fruits 
 and vegetables. 
 
 900 1200 
 300 400 
 
 1000 Fruit 
 1000 Lbs* 
 
 0.50 
 
 O 
 
 PRICE OF LOWEST GRADE 
 RELATIVE TO HIGHEST GRADE 
 
 •2.00 
 
 LIMIT OF ADMISSIBLE ERROR 
 PERCENT OF WEIGHTED 
 AVERAGE PRICE 
 
 0.75 
 
 1.50 
 
 1.25 
 
 .00 
 
 Figure 6. Percent to be sampled from each lot as determined by the percent of fruit in the No. 1 grade, price 
 of the lowest grade relative to No. 1, the limit of admissible error and the total pool deliveries 
 per grower. 
 
Figure 7. Percent to be sampled from each lot as determined by the percent of fruit in the No. 1 grade, pi 
 of the next to lowest grade relative to No. I, the limit of admissible error and total season de- 
 liveries per grower. The quantity in the lowest grade (culls) is not more than 10 percent and the 
 price is zero. 
 
33. 
 
 of fruit falling in the lowest grade will be no larger than 10 per cent 
 and that the price for this grade will be zero. The fruit not falling in 
 the first grade is assumed to have a price equal to the next to the lowest 
 grade. Again, if the prices of the intermediate grades are greater than 
 that of the next to the lowest, the probability of exceeding the limit of 
 error will be less than the amount specified. Each curve in the upper 
 right-hand section of the chart here represents the price of the next to 
 the lowest grade of fruit relative to the highest grade rather than the 
 lowest grade relative to the highest. The diagram is similar to and used 
 in the same way as Figure 6. 
 
 As before, this equation (U) can be turned around and used to estimate 
 the limit of error for a set of sample estimates already obtained. For 
 this purpose, equation (k) may be written: 
 
 A * l*96yVfr^f + P 2 M 2 2 + ...) - (P^ + P 2 M 2 + ...)f7(| - 1) 
 
 (5) 
 
 N 
 
 Example ; 
 
 Suppose that a sample has been taken and: 
 
 (a) The estimated proportions for four grades of fruit as determined 
 by the sample are No. 1 = .0$, No. 2 = 0.3, No. 3 = 0.1, and 
 No. k = 0.1. 
 
 (b) The prices for the various grades are No. 1 = 1.0, No. 2 = 0.6, 
 No. 3 = 0.3, and No. h = 0.1. 
 
 (c) A 5-per cent sample was taken. 
 
 (d) Total deliveries for this grower were 100,000 fruit. 
 
 (e) The desired limit of admissible error is 1 per cent of the 
 total payment. 
 
 To estimate the limit of error, equation ($) becomes 
 
 K*\.%x/$^**- + -l*-^ + + .lx.l 2 )-(.$xl + .3x.6 + .lx.3 + .lx.l)f7(-^ - 1) 
 
 N 
 
 =1 • 96 -^T.618)- (.?20)f7 (19) 
 100,000 
 
 ~i,ooo, V 19 = ,009 or ' 9 per cent 
 
 Thus, the estimated limit of error is nine-tenths of 1 per cent of the 
 total payment. 
 
Previous Reports in This Series on 
 EFFICIENCY IN FRUIT MARKETING 
 
 Marketing Costs for Deciduous Fruits 
 
 Grading Costs for Apples and Pears 
 
 Orchard-to-Plant Transportation 
 
 Packing Costs for California Apples and Pears 
 
 Building and Equipment Costs, Apple and Pear Packing 
 
 In-Plant Transportation Costs as Related to Materials 
 Handling Methods — Apple and Pear Packing 
 
' ■