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Digitized by the Internet Archive 
 in 2020 with funding from 
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ELEMENTARY MATHEMATICAL 
 STATISTICS 
 
ELEMENTARY MATHEMATICAL 
 STATISTICS 
 
 BY 
 
 WILLIAM DOWELL BATEN, Ph.D. 
 
 ASSOCIATE PROFESSOR OF MATHEMATICS AND RESEARCH ASSOCIATE IN STATISTICS 
 OF THE MICHIGAN AGRICULTURAL EXPERIMENT STATION 
 MICHIGAN STATE COLLEGE 
 
 NEW YORK 
 
 JOHN WILEY & SONS, Inc. 
 
 London: CHAPMAN & HALL, Limited 
 1938 
 
311.2 
 
 232SZ 
 
 Copyright, 1938 
 
 BY 
 
 William Dowell Baten 
 
 All Rights Reserved 
 
 This book or any part thereof must not 
 be reproduced in any form without 
 the written permission of the publisher. 
 
 PRINTED IN U. S. A. 
 
 PRESS OF 
 
 BRAUNWORTH & CO.. INC. 
 BUILDERS OF BOOKS 
 BRIDGEPORT. CONN. 
 
Dedicated to My Wife 
 
 ALLIE 
 
 WHO HAS ENCOURAGED ME TO PRESS 
 FORWARD TOWARD HIGHER GOALS 
 
 36695 
 

PREFACE 
 
 This book was written primarily for students interested in 
 statistics who have not studied differential and integral calculus. 
 It attempts to develop formulas and fundamental relations by the 
 use of very simple algebra, trigonometry, and analytical geometry. 
 Each bit of theory is illustrated by an example. The book contains 
 many problems for students to solve, because the author believes 
 that one learns statistics to a great extent by solving problems 
 designed to bring out the meaning of theory. 
 
 Where theory requires advanced mathematics, examples are 
 given to point out the plausibleness of the theory and to furnish 
 information which should make the theory more acceptable. 
 These examples should help students bridge difficult gaps between 
 theory and application. , 
 
 Comparisons are presented for distinguishing differences be¬ 
 tween the concepts of linear correlation, non-linear correlation, 
 and correlation based on the correlation ratio. Details have not 
 been spared in the presentation of partial, multiple, and tetra- 
 choric correlation. 
 
 Ideas concerning sampling are developed by sampling from a 
 finite parent population before going to the infinite; this method 
 of approach should clarify a great deal of the theory with regard to 
 the characteristics of distributions of averages, variances, and 
 other statistics. This manner of introducing the most important 
 concepts in this field should make them easier to understand and 
 apply. 
 
 This book can be used by those interested in business and 
 economic statistics, for it contains, in addition to needed subject 
 matter in these fields, chapters on index numbers, trend analysis, 
 analysis of time series, analysis of variance, and the methods of 
 presenting data by graphs and charts. 
 
 Significance between statistics is introduced in the material 
 pertaining to sampling and later discussed in an entire chapter. 
 
 Near the end of the book is a short chapter on the analysis of 
 variance, which is the most recent development in applied statis¬ 
 tics, and which is being used extensively throughout the world in 
 
 vU 
 
 36695 
 
Vlll 
 
 PREFACE 
 
 the various experimental stations and laboratories. The intro¬ 
 duction of this rapidly developing subject should stimulate desires 
 to do further study concerning this valuable technique for analyzing 
 experimental data. 
 
 It is hoped that answers to the problems will make the book 
 more usable as a textbook and as a reference book. 
 
 William Dowell Baten. 
 
 Michigan State College 
 March, 1938. 
 
CONTENTS 
 
 CHAPTER PAGE 
 
 1 Summations, Charts, and Graphs. 1 
 
 Summation symbols, line charts, bar diagrams, pie charts, 
 three-dimensional graphs, Zee charts, belt charts, geographic 
 charts, logarithmic charts. 
 
 2 Statistical Averages. 20 
 
 Arithmetic mean, arithmetic mean of frequency distributions, 
 indirect method of computing the mean, the mean for the com¬ 
 bination of sets, the mean of grouped data, frequency polygons, 
 histograms, ogive, the median, the mode, geometric mean, har¬ 
 monic mean. 
 
 3 Measures of Dispersion . 53 
 
 Mean deviation, indirect method of calculating the mean devia¬ 
 tion, standard deviation, moments of a distribution, standard 
 deviation of grouped data, standard deviation for the com¬ 
 bination of sets, quartiles, coefficient of variation. 
 
 4 The Normal Curve. 83 
 
 Properties of the normal curve, graduation by areas and ordi¬ 
 nates of the normal curve. 
 
 5 Skewness and Kurtosis. 96 
 
 Skewness, kurtosis. 
 
 6 Permutations, Combinations, and Probability. 108 
 
 Permutations, combinations, probability, probability in re¬ 
 peated trials. 
 
 7 Bernoulli Distribution. 120 
 
 Fraction frequencies, moments of the Bernoulli distribution, 
 application. 
 
 8 Index Numbers. 130 
 
 Relatives, aggregative index numbers, various index numbers, 
 the time reversal test, the factor reversal test, splicing. 
 
 9 Observational Equations. 145 
 
 How observational equations arise, least squares method, 
 observational equations with unknown constants, solution of 
 the normal equations and standard error of prediction in terms 
 of fundamental summations, predicting equations in terms of 
 deviations from the mean, computations shortened by use of a 
 provisional mean, predicting equations in more than two 
 unknowns, solution of simultaneous equations. 
 
 IX 
 
CONTENTS 
 
 x 
 
 CHAPTER PAGE 
 
 10 Correlation Coefficient. 168 
 
 Linear correlation, rank correlation, graphic meaning of the 
 correlation coefficient, a method for computing the correlation 
 coefficient when there is a large number of items, the value of 
 the correlation coefficient in predicting, multiple correlation, 
 partial correlation coefficient, tetrachoric correlation. 
 
 11 Sampling. 201 
 
 Average of sample averages, standard deviation of the dis¬ 
 tribution of sample averages, sampling from an unknown popu¬ 
 lation, standard deviation of a sum, significance between two 
 means. 
 
 12 Non-linear Regression. 223 
 
 Predicting equations, non-linear correlation, correlation ratio. 
 
 13 The Analysis of Time Series. 233 
 
 Secular trend, seasonal variations, link relative method of find¬ 
 ing seasonals, the moving average method of finding seasonals, 
 indexes of seasonal variations from trend line, cycle fluctua¬ 
 
 tions. 
 
 14 Analysis of Variance. 249 
 
 Sums of squares, randomized blocks, the Latin square. 
 
 15 Standard Errors of Certain Statistics. 263 
 
 The standard error of the mean, standard error of a percentage, 
 the standard error of a product, the standard error of a quotient, 
 the standard error of certain other statistics, tests of significance 
 between means of small samples, significance of a correlation 
 coefficient. 
 
 16 Tables. 282 
 
 The area under the normal curve, ordinates of the normal 
 curve, values of F and t, squares and square roots of numbers, 
 reciprocals of numbers, logarithms of numbers. 
 
 Answers. 325 
 
 Index. 333 
 
ELEMENTARY MATHEMATICAL STATISTICS 
 
 CHAPTER 1 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 In the study of statistics, symbols are used which enable one 
 to shorten the writing of long expressions, certain series and sum¬ 
 mations. The following example will illustrate the meaning of a 
 much-used symbol. 
 
 (1.1) The sum of the first nine positive integers 
 
 = 1 + 2+3 + 44-5 + 6 + 7 + 8-1-9. 
 
 The left-hand member of the above equation may be designated 
 by a symbol which will shorten the writing of this summation. 
 This sum may be expressed by the symbol 
 
 9 
 
 where it is understood that v takes on integral values from 1 to 9 
 inclusive. This is read “the sum of the v’s from 1 to 9 inclusive,” 
 or “the sum of the positive integers from 1 to 9 inclusive.” The 
 letter v is called the variable of summation and takes values as 
 indicated. Equation (1.1) may be written as: 
 
 9 
 
 y^ v = l-f-2 + 3 + 4 + 5 + 6 + 7 + 8 + 9, 
 
 or 
 
 9 
 
 ^ \ = 45. 
 
 The symbol 2 is the Greek letter, capital sigma, and will be 
 used to represent a sum or summation. It is sometimes read 
 
2 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 “the sigma of the v’s,” or “the summation of the v’s,” or “the 
 sum of the r’s.” 
 
 The equation v = 1 below the symbol 2 designates where the 
 summation begins, and the number above sigma indicates where 
 the summation stops. In the above example the sum begins 
 with 1 and ends with 9. 
 
 The sum of the squares of the integers from 4 to 11 may be 
 written as: 
 n 
 
 ^2 v 2 = 4 2 + 5 2 + 6 2 + 7 2 + 8 2 + 9 2 + 10 2 + ll 2 . 
 
 V = 4 
 
 Numbers in the column below represent heights of 10 men 
 
 Heights of Men 
 v 
 
 ri 
 
 = 
 
 68.2 in. 
 
 t’2 
 
 = 
 
 67.4 
 
 t’3 
 
 = 
 
 69.3 
 
 Vi 
 
 = 
 
 72.0 
 
 Vi 
 
 = 
 
 71.2 
 
 Ve 
 
 = 
 
 66.9 
 
 Vl 
 
 = 
 
 68.8 
 
 t'a 
 
 = 
 
 69.0 
 
 1’9 
 
 = 
 
 70.2 
 
 t’lO 
 
 = 
 
 67.6 
 
 Zv = 690.6 in. 
 
 The sum of the heights in this column may be represented as a 
 summation as follows: 
 
 ( 1 . 2 ) 
 
 10 
 
 i= 1 
 
 = v ! + V2 + 1'3 + V 4 + V5 + V 6 + V 1 + Vg + 1'9 + ^10 
 
 = 68.2 in. + 67.4 in. + 69.3 in. + . . . + 70.2 in. 
 
 or 
 
 + 67.6 in. = 690.6 in., 
 
 690.6 in. 
 
 The first item in the column, 68.2 inches, is represented by Vi ; 
 the seventh item is represented by vj, etc. In this case the 
 
SUMMATIONS, CHARTS AND GRAPHS 
 
 3 
 
 variable of summation is i; it runs from 1 to 10. The left-hand 
 member of equation (1.2) is read “the sum of the v’s from v\ to v\q 
 inclusive,” or “the sum of the v’s” or “the sum of the heights of 
 the 10 men.” Many times the numbers below and above sigma 
 are omitted; the above summation may be written as: 
 
 - 690.6 in. 
 
 Summations which follow also point out the use of 2. 
 
 1 JL 1 1 1 A. 
 
 1 = 2 
 
 D 
 
 =/(l) +/(2) +/(3) +/(4) +/(5). 
 
 P = 1 
 
 The variable of summation in the first sum is k; the variable of 
 summation in the next is p. 
 
 To expand a sigma means to write out the summation in full; for 
 
 7 
 
 example, ^ ^ u 4 expanded is 
 
 u= 2 
 
 2>=' 
 
 2 4 + 3 4 + 4 4 + 5 4 + 6 4 + 7 4 . 
 
 Let it be required to write the following series by use of sigma: 
 4-6 + 5-7 + 6-8 + 7-9 + 8-10 + 9-11 -f 10-12. 
 
 The first numbers in the terms of this series begin with 4 and increase by 
 unity up to and including 10. The second numbers in the terms begin 
 with 6, increase from term to term by unity, and run to 12 inclusive. 
 The series can be written as a sigma thus: 
 
 10 
 
 Y. x(x + 2) = 4-6 + 5-7 + . . . + 10-12. 
 
 x = 4 
 
 The following two theorems are indispensable. 
 
 Theorem 1.1. The summation or the sigma of a constant 
 times a function involving a variable of summation is equal to the 
 constant times the sigma of the function, or 
 
 n n 
 
 Y a '/( x ») = a-Yj( x i)> 
 
 4-1 4-1 
 
 where a is a constant and /(x) is a function of x. 
 
4 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 Pkoof: 
 
 7* 
 
 ^ af{Xi) = a-f{x. i) + a-f(x 2 ) + . . . + a-f(x n ) 
 
 1 = 1 « 
 
 = a- i) + fix 2 ) + . . . +/(x»)] = q T> 0. 
 
 <= 1 
 
 As an illustration 
 
 ^ 3(xi 3 + a\ 2 - 2x0 = 3^^ (a:, 3 + aq 2 - 2x0. 
 
 1=1 
 
 <=i 
 
 Theorem 1.2. The summation or the sigma of the sum of two 
 functions involving a variable of summation is equal to the sum 
 of the sigmas of the functions, or 
 
 n n 
 
 T [f(Xi) + g(xt )] = yy f(xi) + y^gixi), 
 
 1=1 1=1 1=1 
 
 where f(x ) and g{x) are functions of x. 
 
 Proof: 
 
 n 
 
 ^ [/(a:.) + g(x x )] = fix 1 ) + gixi) + /(x 2 ) + g(x 2 ) + . . . 
 
 + /(*») + gix n ) = [fix 1 ) + fix 2 )'+ . . . + /(*„)] 
 + [^(a-i) + gix 2 ) + . . . + 5 f(x„)] 
 
 = ^3 
 
 1 = 1 
 
 1 = 1 
 
 
 
 PROBLEMS 
 
 
 Expand the following summations. 
 
 
 , £,. 
 
 1 
 
 «• tr 
 
 r = 0 
 
 24 
 
 9- £><>. 
 
 i= 17 
 
 9 
 
 2. » +1). 
 
 3 
 
 d= -11 
 
 6 
 
 10. y>. 
 
 s = 3 
 
 11 
 
 3. ^VifiVi). 
 
 i=5 
 
 62 
 
 7. ][>«. 
 
 1=51 
 
 13 
 
 11. J2 v ^ vi) - 
 
 i= 1 
 
 16 
 
 4. £log (3x + 4). 
 
 x = 11 
 
 6 
 
 8. ^iv + 3)fiv). 
 
 v= -4 
 
 12 
 
 12 . ^ ^vjWj. 
 
 j=5 
 
CHARTS AND GRAPHS 
 
 5 
 
 n 
 
 Given that ^ ^ v 
 
 = ”(» + !); 7> = 
 
 n(n + l)(2n+l) 
 
 6 ’ 
 
 v= 1 
 
 n 2 (n -4- l) 2 
 
 V=1 
 
 V =1 
 
 , find the following: 
 
 
 n 
 
 - D- 
 
 n 
 
 14. ^2 ](6y 2 — 2v). 
 
 n 
 
 15. ^~\(r+l)(y+2). 
 
 !/=l 
 
 v= 1 
 
 v= l 
 
 n 
 
 
 20 
 
 22 ( 8k 3 - 12« 2 + 4»). 
 
 »=i 
 
 17. £>. 
 
 V= 11 
 
 Write the following summations by use of sigma: 
 
 18. 7-10 + 8-11 + 9-12 + 10-13 + . . . + 15-18. 
 
 19. 15/(15) + 16/(16) + . . . + 42/(42). 
 
 20. 4 3 -5 + 5 3 -6 + . . . + 12 3 -13. 
 
 21 . 
 
 22 . 
 
 23. 
 
 2 >io v n Vu Vu 
 
 v 6 f(v e ) + ^ 7 /(^ 7 ) + . . . + v 3 of(v 3 o). 
 
 3• 7 3 + 5-7 2 - 4-7 + 3-8 3 + 5-8 2 - 4-8 + 3-9 3 + 5-9 2 - 4-9 + . . . 
 . . . + 3-11 3 + 5-11 2 - 4-11. 
 
 CHARTS AND GRAPHS 
 
 In analyzing statistical data it is often beneficial to exhibit 
 results by means of graphs and charts, which enable one to grasp 
 quickly the information contained in a mass of measurements, 
 computations, and long columns of figures. Several types of charts 
 and graphs which are used in many leading newspapers, scientific 
 journals, and important reports will be explained in the remainder 
 of this chapter. 
 
 Chart 1.1 contains 3 line charts pertaining to prices of wheat, 
 corn, and oats since 1919. These broken lines allow the eye to 
 see at once trends of prices of these essential commodities before, 
 during, and after the depression. It is interesting to note that 
 decreases in the price of one often accompanied decreases in the 
 other two. Wheat prices were relatively higher in 1920 than prices 
 of corn and oats. On examining these line diagrams it will be seen 
 that prices for these grains started to drop as early as 1924. A 
 marked increase began in 1933. 
 
6 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 1920 1925 1930 1935 
 
 Chakt 1.1.—Line charts showing average prices of wheat, corn and oats from 
 1920 to 1935. (Agriculture Statistics of U. S. A., 1937.) 
 
 Bar diagrams often furnish adequate methods of presenting 
 facts which enable one to make comparisons immediately. Charts 
 1.2 and 1.3 show production of cars and trucks and potatoes since 
 1928 and 1927 respectively. There was a drop in production of 
 cars and trucks from 1929 to 1932 and an increase from 1932 to 
 1936. Production of potatoes (as chart 1.3 shows) did not have 
 such great decreases and increases, as the heights of the bars show. 
 The depression did not affect the number of bushels of potatoes 
 grown as much as the number of cars and trucks manufactured. 
 The peak for the automobile industry was in 1929; that for pota- 
 
CHARTS AND GRAPHS 
 
 7 
 
 toes was in 1928. The space between bars should be about one- 
 half the width of the bar. 
 
 1.000,000 
 
 Chart 1.2.—Production of cars and trucks in United States and Canada. 
 (28th Annual Report of General Motors Corporation.) 
 
 Chart 1.3. — Bar diagram showing average production of potatoes for U. S. A. 
 from 1927 to 1936. ( Agriculture Statistics of U. S. A., 1937.) 
 
 Pie charts are useful for comparing percentages and the differ¬ 
 ent parts of a whole. Chart 1.4 exhibits percentages of cotton 
 production for 1935-36 and enables one to make rapid compari¬ 
 sons. Egypt and Brazil produced that year the same amount of 
 cotton; China ranked third. The United States of America pro¬ 
 duced 44.3 per cent of all cotton grown in that period. 
 
 Chart 1.5 presents percentages of the divisions of the student 
 body at one of the large agricultural colleges in America. This 
 type of chart is easy to construct and can be interpreted readily. 
 
 Three-dimensional graphs can be used effectively to portray 
 contrasts. Charts 1.0 and 1.7 are self-explanatory. Often pic- 
 
8 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 Chart 1.4.—Pie chart showing percentages of production 
 of cotton during 1935-36. 
 
 Chart 1.5.—Percentages of students in Michigan State College in 1935-36. 
 
CHARTS AND GRAPHS 
 
 9 
 
 tures of objects will catch the eye where other types of charts will 
 not. Many interesting bits of information have been given to the 
 public through three-dimensional diagrams. 
 
 $4,320,000 
 
 Chart 1.6. —Three-dimensional chart showing amount of 
 building in a certain city. 
 
 AZ 
 
 1 
 
 1 
 
 7 
 
 
 (~7\ 
 
 
 7 
 
 r~. 
 
 / ) j 
 
 1934 1935 1936 1937 
 
 Chart 1.7. —Relative amounts of ice sold by all ice companies 
 in a certain city from 1934 to 1937. 
 
 Zee charts set out in graphic form as a rule 3 phases of a busi¬ 
 ness or project. This type is shown in chart 1.8; it contains 
 monthly sales, cumulative sales, and 12-month moving total sales. 
 The three lines look like the letter Z. The cumulative line is 
 formed by plotting sales for January; January and February; Jan¬ 
 uary, February, and March; etc. The moving 12-month total sales 
 line is formed by plotting the sales for the past 12 months. The 
 managers can see three important phases of their business for 
 the year. In this chart the moving 12-month total sales increased 
 from December to June. The graph of cumulative sales is nearly 
 a straight line. Monthly sales fell off considerably during July, 
 August, and September. 
 
 Belt or strip charts can be employed when it is desirous to pre¬ 
 sent several phases of a business or project. Chart 1.9 shows relief 
 cases in Michigan for 1936 and part of 1937. The various belts or 
 strips make vivid the relief program in Michigan. Aid to depen- 
 
10 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 dent children and the blind started in October, as is seen in the 
 chart. Sometimes different colors are used for the different strips. 
 
 There are many kinds of geographic charts. One is shown in 
 chart 1.10, which indicates the percentages of the population of 
 
 2,000,000 
 
 in 
 
 1,875,000 ^ 
 
 1,750,000 £ 
 
 1,625,000 o 
 z 
 
 1,500,000 « 
 1,375,000 $ 
 1,250,000 .1 
 
 4 -* 
 
 1,125,000 "5 
 E 
 
 1,000,000 3 
 
 875,000 'g 
 750,000 
 
 4 -* 
 
 625,000 ° 
 
 00 
 
 500,000 c 
 > 
 o 
 
 250,000 
 
 125,000 
 
 c 
 
 (0 
 
 n 
 
 0 ) 
 
 a. 
 
 < 
 
 00 
 
 a 
 
 
 > 
 
 o 
 
 D 
 
 0) 
 
 o 
 
 o 
 
 a> 
 
 < 
 
 c/) 
 
 O 
 
 Z 
 
 a 
 
 Chart 1.8.—Zee chart showing total unit sales of General Motors cars and 
 trucks in 1936. ( 28th Annual Report of General Motors Corporation.) 
 
 Michigan by counties on relief. Well-constructed maps showing 
 routes and all towns in which deliveries are made can be of much 
 help to the superintendent of a business firm in assisting him to 
 visualize his territory and the extent of his business. Sales for 
 different districts can be shown on such maps. This type of chart 
 is difficult to construct but when well drawn reveals a great deal 
 of information. 
 
TOTAL RELIEF CASES IN MICHIGAN 
 
 STATE AND FEDERAL PROGRAMS 
 
 CHARTS AND GRAPHS 
 
 11 
 
 a 
 
 ci 
 
 tc 
 
 IS 
 
 o 
 
 CO 
 
 o 
 
 hJO 
 
 P 
 
 e 
 
 ■ <s> 
 
 as 
 
 fO 
 
 As 
 
 £ 
 
 o 
 
 3 
 
 CQ 
 
 rS 5 
 
 £ 
 
 o 
 
 p 
 
 c3 
 
 bJO 
 
 c3 
 
 o 
 
 tJD 
 
 a 
 
 $■ 
 
 o 
 
 rP 
 
 CQ 
 
 t-i 
 
 c3 
 
 rP 
 
 <v 
 
 PQ 
 
 H 
 
 tf 
 
 <5 
 
 a 
 
 O 
 
 Emergency Relief Administration.) 
 
12 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 OCT. 1936 
 
 0.0% 
 
 TO 
 
 4 
 
 9% 
 
 5 0% 
 
 TO 
 
 9 
 
 9% 
 
 10 0% 
 
 TO 
 
 14 
 
 9% 
 
 15 0% 
 
 TO 
 
 19 
 
 9% 
 
 20 0% 
 
 TO 
 
 29 
 
 9% 
 
 30 0% 
 
 TO 
 
 39 
 
 9% 
 
 400% 
 
 TO 
 
 49 9% 
 
 500% 
 
 & < 
 
 OVER 
 
 Chart 1.10.—Geographic chart showing per cent of population in Michigan 
 on relief — cases on state and Federal relief programs. (Monthly Bulletin on 
 Public Relief Statistics, Oct., 1936.) 
 
PROBLEMS 
 
 13 
 
 PROBLEMS 
 
 1. Construct line charts for the following data: 
 
 Number of Hogs Number of Cattle and 
 
 
 on Farms, 
 
 Calves on F 
 
 
 Millions 
 
 Millions 
 
 1920 
 
 60.2 
 
 70.4 
 
 1921 
 
 58.9 
 
 68.8 
 
 1922 
 
 59.8 
 
 68.8 
 
 1923 
 
 69.3 
 
 67.5 
 
 1924 
 
 66.6 
 
 66.0 
 
 1925 
 
 55.8 
 
 63.4 
 
 1926 
 
 52.1 
 
 60.6 
 
 1927 
 
 55.5 
 
 58.2 
 
 1928 
 
 61.9 
 
 57.3 
 
 1929 
 
 59.0 
 
 58.9 
 
 1930 
 
 55.7 
 
 61.0 
 
 1931 
 
 54.8 
 
 63.0 
 
 1932 
 
 59.3 
 
 65.8 
 
 1933 
 
 62.1 
 
 70.2 
 
 1934 
 
 58.6 
 
 74.3 
 
 1935 
 
 39.0 
 
 68.5 
 
 1936 
 
 42.8 
 
 68.2 
 
 This material was taken from the Agriculture Statistics of U. S. A., 1937. 
 
 2. Construct line charts on one page for the following materials taken 
 from the Agriculture Statistics of U. S. A., 1937. 
 
 
 Average Price of 
 
 Average Price of 
 
 Average Price of 
 
 
 Eggs Per Dozen, 
 
 Butter per Pound, 
 
 Bread per Pound, 
 
 
 Cents 
 
 Cents 
 
 Cents 
 
 1925 
 
 30.4 
 
 45.0 
 
 9.4 
 
 1926 
 
 28.8 
 
 44.4 
 
 9.4 
 
 1927 
 
 25.0 
 
 47.3 
 
 9.2 
 
 1928 
 
 28.0 
 
 47.4 
 
 9.1 
 
 1929 
 
 29.9 
 
 45.0 
 
 8.9 
 
 1930 
 
 23.7 
 
 36.5 
 
 8.2 
 
 1931 
 
 17.5 
 
 28.3 
 
 7.2 
 
 1932 
 
 14.2 
 
 21.0 
 
 6.6 
 
 1933 
 
 13.8 
 
 21.7 
 
 7.8 
 
 1934 
 
 17.0 
 
 25.7 
 
 8.3 
 
 1935 
 
 33.4 
 
 29.8 
 
 8.3 
 
 1936 
 
 21.7 
 
 33.1 
 
 8.2 
 
14 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 3. Construct a bar chart for the following data: 
 
 Farm Wage Rates Per Month Without Board 
 
 Year 
 
 1927 
 
 1928 
 
 1929 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 1935 
 
 1936 
 
 $48.63 
 
 48.65 
 
 49.08 
 
 44.59 
 
 35.03 
 
 26.67 
 
 24.51 
 
 27.17 
 
 29.48 
 
 31.82 
 
 4. Compare the production of potatoes for Michigan and Maine by a 
 bar chart. 
 
 
 Michigan 
 
 Maine 
 
 Year 
 
 Millions of Bushels 
 
 1927 
 
 23.1 
 
 37.4 
 
 1928 
 
 31.4 
 
 39.4 
 
 1929 
 
 15.9 
 
 49.9 
 
 1930 
 
 14.3 
 
 45.3 
 
 1931 
 
 23.8 
 
 50.3 
 
 1932 
 
 29.9 
 
 39.5 
 
 1933 
 
 20.7 
 
 42.0 
 
 1934 
 
 34.3 
 
 55.2 
 
 1935 
 
 28.1 
 
 38.4 
 
 1936 
 
 26.1 
 
 44.0 
 
 5. Construct pie charts for the following data: 
 
 Principal Exporting Countries of Cotton (1000 Bales) 1935-36 
 
 U. S. A. British India Egypt Brazil Argentina 
 
 6,397 2,999 1,688 650 202 
 
 6. Construct a pie chart for the following data pertaining to how a 
 certain family spent its income. Salary $3,600. 
 
 Payments on home. $900.00 
 
 Gas, light, water, fuel, telephone. 270.00 
 
 Taxes, upkeep on home, insurance. 540.00 
 
 Food. 600.00 
 
PROBLEMS 
 
 15 
 
 Clothes. $150.00 
 
 Upkeep of car, trips. 400.00 
 
 Installments on furniture. 300.00 
 
 Church, charity. 150.00 
 
 Pleasure, journals, books. 200.00 
 
 Miscellaneous. 90.00 
 
 7 . Construct a Zee chart for the following data: 
 
 Michigan Emergency Relief Statistics 
 Number of People on Relief * 
 
 
 1935 
 
 1936 
 
 Jan. 
 
 237,231 
 
 82,764 
 
 Feb. 
 
 216,634 
 
 88,862 
 
 Mar. 
 
 200,984 
 
 88,753 
 
 Apr. 
 
 190,600 
 
 81,075 
 
 May 
 
 181,295 
 
 71,195 
 
 June 
 
 183,468 
 
 64,146 
 
 July 
 
 183,732 
 
 61,451 
 
 Aug. 
 
 180,725 
 
 62,922 
 
 Sep. 
 
 175,284 
 
 65,191 
 
 Oct. 
 
 163,867 
 
 64,336 
 
 Nov. 
 
 142,635 
 
 65,931 
 
 Dec. 
 
 107,494 
 
 70,277 
 
 * This material was taken from Monthly Bulletin on Public Relief Statistics published by 
 Michigan Emergency Relief Administration, Vol. IV, No. 9, 1937. 
 
 8. Construct a belt or strip chart for the following sales for a book 
 
 store: 
 
 No. Bibles 
 
 Fiction 
 
 No. Textbooks 
 
 All Others 
 
 Jan. 
 
 37 
 
 20 
 
 55 
 
 61 
 
 Feb. 
 
 23 
 
 16 
 
 31 
 
 38 
 
 Mar. 
 
 28 
 
 30 
 
 48 
 
 56 
 
 Apr. 
 
 35 
 
 27 
 
 36 
 
 70 
 
 May 
 
 30 
 
 22 
 
 16 
 
 53 
 
 June 
 
 23 
 
 15 
 
 62 
 
 39 
 
 July 
 
 20 
 
 11 
 
 19 
 
 32 
 
 Aug. 
 
 14 
 
 9 
 
 8 
 
 26 
 
 Sep. 
 
 18 
 
 24 
 
 91 
 
 72 
 
 Oct. 
 
 29 
 
 38 
 
 63 
 
 81 
 
 Nov. 
 
 48 
 
 36 
 
 31 
 
 90 
 
 Dec. 
 
 63 
 
 50 
 
 13 
 
 134 
 
 9 . Bring to class two geographic charts which you have seen in the 
 local newspapers or current journals. 
 
16 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 LOGARITHMIC GRAPHS 
 
 Sometimes it becomes necessary to employ logarithmic paper 
 for presenting data in graphic form. For semilogarithmic paper 
 one of the axes is divided into equal units while the other axis is 
 
 Years 
 
 Chart 1.11.—Population of Michigan from 1810 to 1930 plotted on semi¬ 
 logarithmic paper. 
 
 divided into logarithm units. The horizontal axis in chart 1.11 
 represents years; the vertical axis represents the logarithms of the 
 population figures for Michigan from 1810 to 1930. The graph is 
 nearly a straight line, showing that the rate of increase is nearly 
 a constant for decades. It is difficult to place on the same page 
 census figures for Michigan in 1810 and those for the last two dec¬ 
 ades, when plotted on an arithmetic scale. 
 
 Chart 1.12 shows the compound amount on $10 compounded 
 
LOGARITHMIC GRAPHS 
 
 17 
 
 annually at 10 per cent plotted on semilogarithmic paper. The 
 line is straight when plotted on this kind of paper. 
 
 Years 
 
 Chart 1.12.—Compound amount on $10.00 compounded annually at 10 per 
 cent plotted on semilogarithmic paper. 
 
 Relations similar to the following can be represented by straight 
 lines if drawn on semilogarithmic paper. 
 
 y = a-b x , 
 or 
 
 log y = log a -f- x log b, 
 
 where a and b are positive constants. The constant a may be 
 equal to 1. The constant a is equal to y when x is equal to zero. 
 I he compound amount law, represented in chart 1.12, is 
 
 y = 10(1.10)-, 
 or 
 
 log y = log 10 + 2 log (1.10); 
 
 here a = 10, b = 1.10, y = the compound amount and x = the 
 length of time in years. 
 
18 
 
 SUMMATIONS, CHARTS AND GRAPHS 
 
 Double logarithmic paper can be used to plot relations similar 
 to the following 
 
 y = a-x b , 
 or 
 
 log y = log a + b log x, 
 
 where a and b are constants. The constant a must be positive; it 
 is equal to y when x = 1. 
 
 Chart 1.13 shows the speed of trackmen for various distances 
 plotted on double logarithmic paper. The line is nearly straight 
 with a slight break at 880 yards. 
 
 «-T ci oo into co o” 
 
 H 
 
 Distance in Yards 
 
 Chart 1.13.—Speed of trackmen at Michigan State College for various 
 distances plotted on double logarithmic paper. 
 
PROBLEMS 
 
 19 
 
 PROBLEMS* 
 
 1. Plot on semilogarithmic paper the following census figures for the 
 U. S. A. Try to predict what the population will be in 1950. 
 
 Year 
 
 Population 
 
 1790 
 
 3,929,000 
 
 1800 
 
 5,308,000 
 
 1810 
 
 7,240,000 
 
 1820 
 
 19,638,000 
 
 1830 
 
 12,866,000 
 
 1840 
 
 17,069,000 
 
 1850 
 
 23,192,000 
 
 1860 
 
 31,443,000 
 
 1870 
 
 38,558,000 
 
 1880 
 
 50,156,000 
 
 1890 
 
 62,948,000 
 
 1900 
 
 75,995,000 
 
 1910 
 
 91,972,000 
 
 1920 
 
 105,711,000 
 
 1930 
 
 122,775,000 
 
 2 . Plot on single logarithmic paper the following data: 
 
 Angle of Contact between Belt and Pullet from Pull 
 
 Angle, radians. 1 2 3 4 5 
 
 P, pounds. 4.5 6.5 9.5 14 20 
 
 3 . Plot on double logarithmic paper the following data: 
 
 Track and Field Records of Olympic Men 
 Distance, meters.. 100 200 400 
 
 Time. 10.3 sec. 21.2 sec. 46.2 sec. 
 
 Distance, meters.. 800 1,500 5,000 
 
 Time. 1 min. 49.8 sec. 3 min. 51.2 sec. 14 min. 30 sec. 
 
 4 . Plot the following data on double logarithmic paper: 
 
 Age of tree, years.. 5 10 25 50 75 100 
 
 Height, feet. 6.5 11 22 38 50 64 
 
 * Problems 1, 2, 3, and 4 were taken by permission from “Statistical Data” 
 by S. E. Crowe. 
 
CHAPTER 2 
 
 STATISTICAL AVERAGES 
 ARITHMETIC MEAN 
 
 Since averages play important roles in statistical work, it is 
 essential to be familiar with some of the more common ones and 
 to know how to obtain them by the shortest methods. Five aver¬ 
 ages are introduced in this chapter together with methods for 
 securing them. The first to be considered is the arithmetic aver¬ 
 age or the arithmetic mean. 
 
 The following column contains the number of children in families 
 living in a certain section. 
 
 No. of Children 
 per Family, 
 v 
 
 Vi 
 
 = 
 
 1 
 
 v 2 
 
 = 
 
 3 
 
 Vi 
 
 = 
 
 5 
 
 Vi 
 
 = 
 
 8 
 
 Vt 
 
 = 
 
 4 
 
 V 6 
 
 = 
 
 2 
 
 V? 
 
 = 
 
 6 
 
 Vs 
 
 = 
 
 5 
 
 v 9 
 
 = 
 
 4 
 
 Via 
 
 = 
 
 9 
 
 Vn 
 
 = 
 
 11 * 
 
 Vl2 
 
 = 
 
 6 
 
 Viz 
 
 = 
 
 4 
 
 Vu 
 
 = 
 
 5 
 
 Vlb 
 
 = 
 
 2 
 
 
 = 
 
 75 
 
 20 
 
ARITHMETIC MEAN 
 
 21 
 
 The average number of children per family for this section is equal to 
 
 Mean = M v 
 
 1 + 3 + 5 + ... + 5 + 2 
 15 
 
 — = 5 children. 
 
 15 
 
 Five is the average number of children per family and is a good repre¬ 
 sentative of the number of children on the average for each family for 
 this group of families. The mean of a set of items in many instances 
 is a good representative of the items and shows a central tendency of 
 the items. 
 
 Let the quantity represent the fth item or variate in a set of 
 variates. In the above example vi = 1, V 2 = 3, ^8 = 5, etc. The 
 quantity v is a variable, and the particular values of v are called 
 variates. In the above illustration the number of children per 
 family is a variable; the actual values observed in the families are 
 variates. 
 
 The arithmetic mean of a set of variates is equal to the sum of 
 the variates divided by the number of variates, or 
 
 (2.1) 
 
 Mean = M v = 
 
 V\ + Vo + . . . + v n hv 
 
 n 
 
 n 
 
 where n is the number of variates in the set. 
 
22 
 
 STATISTICAL AVERAGES 
 
 PROBLEMS 
 
 1. Find the mean of the following figures, which are the number of 
 seeds in pods of a certain plant. 7, 11, 14, 9, 10, 12, 8, 11, 10, 13, 12, 
 12, 14, 10, 11, 9, 13, 11, 12, 11. 
 
 2. The next table exhibits data pertaining to rainfall at Grand Rapids 
 and Marquette, Michigan, from 1912 to 1935 during December. (Taken 
 from the Year Book of the U.S.A. Dept, of Agri., 1935.) 
 
 Precipitation in Inches 
 
 Year 
 
 Grand Rapids 
 
 Marquette 
 
 1912 
 
 1.32 
 
 2.42 
 
 1913 
 
 0.31 
 
 0.94 
 
 1914 
 
 1.89 
 
 0.85 
 
 1915 
 
 1.22 
 
 2.17 
 
 1916 
 
 3.81 
 
 3.09 
 
 1917 
 
 0.82 
 
 3.86 
 
 1918 
 
 4.02 
 
 2.94 
 
 1919 
 
 1.19 
 
 1.89 
 
 1920 
 
 4.19 
 
 2.27 
 
 1921 
 
 4.14 
 
 2.04 
 
 1922 
 
 1.40 
 
 1.14 
 
 1923 
 
 2.18 
 
 1.52 
 
 1924 
 
 1.66 
 
 1.53 
 
 1925 
 
 1.52 
 
 2.80 
 
 1926 
 
 1.86 
 
 2.36 
 
 1927 
 
 3.16 
 
 3.19 
 
 1928 
 
 2.79 
 
 1.13 
 
 1929 
 
 2.71 
 
 3.47 
 
 1930 
 
 1.34 
 
 1.97 
 
 1931 
 
 2.64 
 
 1.09 
 
 1932 
 
 2.68 
 
 1.60 
 
 1933 
 
 1.74 
 
 2.58 
 
 1934 
 
 1.51 
 
 2.30 
 
 1935 
 
 1.65 
 
 2.91 
 
 Find the average amount of rainfall during December for the two cities, 
 and compare results. Locate the cities on a map, and discuss results. 
 
 3. Gallons of gasoline sold by a certain filling station during June: 
 
 480, 459, 468, 477, 453, 490, 450, 441, 484, 472, 460, 449, 502, 465, 468, 
 
 481, 430, 463, 498, 478, 405, 446, 473, 453, 469, 438, 458, 467, 420, 439. 
 Find the average output of gasoline per day during June for this station. 
 Plot these data on a vertical scale, draw a line representing the mean, 
 and find the number of items below and above this line. 
 
ARITHMETIC AVERAGES FOR FREQUENCY DISTRIBUTION 23 
 
 4 . Number of eggs laid by 40 hens during 20 weeks. 
 
 Weeks 
 
 No. of Eggs 
 
 Weeks 
 
 No. of Eggs 
 
 1 
 
 180 
 
 11 
 
 210 
 
 2 
 
 176 
 
 12 
 
 240 
 
 3 
 
 168 
 
 13 
 
 209 
 
 4 
 
 164 
 
 14 
 
 256 
 
 5 
 
 180 
 
 15 
 
 236 
 
 6 
 
 177 
 
 16 
 
 227 
 
 7 
 
 193 
 
 17 
 
 240 
 
 8 
 
 196 
 
 18 
 
 220 
 
 9 
 
 212 
 
 19 
 
 213 
 
 10 
 
 200 
 
 20 
 
 190 
 
 Find the average number of eggs laid by the hens per week and 
 the average number laid per hen per week. 
 
 ARITHMETIC AVERAGE FOR FREQUENCY DISTRIBUTIONS 
 
 Data in problem 1 of the last section can be arranged into a 
 frequency distribution by collecting the items. This is how the 
 first two columns in the next table arose. 
 
 No. of Seeds 
 per Pod 
 v 
 
 7 
 
 8 
 9 
 
 10 . 
 
 11 
 
 12 
 
 13 
 
 14 
 
 No. of Pods or 
 Frequency 
 /(») 
 
 1 
 
 1 
 
 2 
 
 3 
 5 
 
 4 
 2 
 2 
 
 v-f(v) 
 
 7 
 
 8 
 18 
 30 
 55 
 48 
 26 
 28 
 
 Total 2/(t;) = 20 2 »•/(») = 220 
 
 This distribution of the items is called a frequency distribution 
 because the items are listed with their respective frequencies. 
 Number 11 appears 5 times, 12 appears 4 times, etc. Frequencies 
 arc designated by f(v) and are found in that column. The column 
 headed by v-f(v) contains products of the variates by their respec¬ 
 tive frequencies. For example, 12 occurred 4 times, hence the 
 sum of these four 12’s is 48; this 48 appears in the v-f(v) column. 
 The sum of the numbers in the v-f(v) column is the sum of all items 
 
24 
 
 STATISTICAL AVERAGES 
 
 and is the same as was obtained in the first problem of the last 
 section. The mean of the items can be written in terms of fre¬ 
 quencies as follows: 
 
 Mean = M v 
 
 Sum of the items 
 No. of items 
 
 2v-M 
 
 mv) { 
 
 The total number of items, n, is equal to the sum of the frequencies, 
 or 
 
 n = m«i) = mv). 
 
 The mean for the above example is 
 
 M 
 
 v 
 
 mf(v) 
 
 mv) 
 
 220 
 
 20 
 
 11 seeds. 
 
 A frequency distribution enables one to put a great deal in a 
 little space and aids one in calculating the average if several of the 
 variates are repeated. 
 
 PROBLEMS 
 
 1. The following data give the number of ligulate flowers or ray flowers 
 
 on heads of oxeye daisies growing on a certain plot: 12, 16, 15, 20, 18, 
 17, 18, 21, 18, 15, 17, 19, 20, 14, 18, 16, 15, 13, 19, 18, 17, 14, 21, 17, 
 
 16, 14, 18, 19, 20, 13, 15, 16, 17, 19, 16, 18, 19, 17, 16, 19, 18, 18, 18, 
 
 17, 14, 16, 18, 17, 16, 18, 19, 15, 20, 19, 18, 17, 16, 15, 18, 19, 19, 17, 
 
 16, 16, 15, 18, 19, 20, 18, 20, 15, 16, 19, 18, 18, 17, 17, 17, 16, 19, 16, 
 
 18, 17, 18, 21, 22, 20, 19, 16, 18, 15, 21, 14, 19, 19, 19, 17, 18, 16, 17, 
 
 16, 17, 15. Arrange the items in a frequency distribution, and find the 
 average number of ligulate flowers per daisy. 
 
 2. Below is a frequency distribution of the number of loaves of bread 
 sold by a grocery store for 92 days. 
 
 No. or Loaves No. of 
 Sold per Day Days 
 v f(v) 
 
 175 1 
 
 182 4 
 
 186 9 
 
 193 17 
 
 197 22 
 
 201 14 
 
 204 10 
 
 No. of Loaves No. of 
 Sold per Day Days 
 v f(v) 
 
 219 5 
 
 222 3 
 
 228 3 
 
 231 2 
 
 235 1 
 
 237 1 
 
 Find the average number of loaves sold per day. 
 
INDIRECT METHOD OF COMPUTING THE MEAN 25 
 
 3 . The next table contains scores made by seventh-grade students on 
 a certain test. Find the average score. 
 
 
 No. OF 
 
 
 No. OF 
 
 
 No. OF 
 
 Scores Students Scores 
 
 Students 
 
 Scores 
 
 Students 
 
 V 
 
 m 
 
 V 
 
 m 
 
 V 
 
 f(v ) 
 
 49 
 
 2 
 
 65 
 
 42 
 
 81 
 
 68 
 
 50 
 
 3 
 
 66 
 
 45 
 
 82 
 
 60 
 
 51 
 
 3 
 
 67 
 
 52 
 
 83 
 
 53 
 
 52 
 
 2 
 
 68 
 
 59 
 
 84 
 
 46 
 
 53 
 
 3 
 
 69 
 
 65 
 
 85 
 
 38 
 
 54 
 
 5 
 
 70 
 
 66 
 
 86 
 
 25 
 
 55 
 
 8 
 
 71 
 
 72 
 
 87 
 
 17 
 
 56 
 
 10 
 
 72 
 
 75 
 
 88 
 
 8 
 
 57 
 
 14 
 
 73 
 
 79 
 
 89 
 
 3 
 
 58 
 
 16 
 
 74 
 
 83 
 
 90 
 
 4 
 
 59 
 
 19 
 
 75 
 
 90 
 
 91 
 
 3 
 
 60 
 
 23 
 
 76 
 
 88 
 
 92 
 
 2 
 
 61 
 
 25 
 
 77 
 
 91 
 
 93 
 
 3 
 
 62 
 
 28 
 
 78 
 
 90 
 
 94 
 
 2 
 
 63 
 
 32 
 
 79 
 
 84 
 
 95 
 
 1 
 
 64 
 
 37 
 
 80 
 
 75 
 
 96 
 
 1 
 
 If A is 
 
 considered to be from 90 to 
 
 100, B from 80 to 90, C from 70 to 
 
 80, D from 60 to 70, and E 
 
 any score below 60, 
 
 find the percentages of 
 
 A’s, B’s 
 
 , C’s, D’s, and E’s. 
 
 
 
 
 
 
 INDIRECT METHOD OF 
 
 COMPUTING 
 
 THE MEAN 
 
 Figures in the first column of the next table are the number of 
 
 gallons of gasoline sold each day for 10 days. 
 
 
 
 
 No. of Gallons of 
 
 
 
 
 
 Gas Sold per Day 
 
 d 
 
 d 
 
 
 
 V 
 
 v — 460 
 
 v —455 
 
 
 
 450 
 
 
 -10 
 
 - 5 
 
 
 
 445 
 
 
 —15 
 
 -10 
 
 
 
 462 
 
 
 2 
 
 7 
 
 
 
 470 
 
 
 10 
 
 15 
 
 
 
 449 
 
 
 -11 
 
 - 6 
 
 
 
 455 
 
 
 — 5 
 
 0 
 
 
 
 458 
 
 
 - 2 
 
 3 
 
 
 
 460 
 
 
 0 
 
 5 
 
 
 
 464 
 
 
 4 
 
 9 
 
 
 
 447 
 
 
 -13 
 
 - 8 
 
 
 
 2w = 4,560 
 
 2d = 
 
 -40 
 
 2 d = 10 
 
 
26 
 
 STATISTICAL AVERAGES 
 
 From each item in the first column subtract a provisional or 
 guessed mean, 460, and place these differences in the first d column. 
 This column gives deviations of the variates from this provisional 
 mean. The average of the deviations of the variates from 460 is 
 — 4 gallons, hence the average number of gallons sold per day is 
 
 2d 
 
 M v = 460 + — = 460 — 4 = 456 gallons. 
 
 Numbers in the third column are deviations of the variates from 
 another provisional mean, 455 gallons; the average number of gal¬ 
 lons sold per day is 
 
 2 d 
 
 M v — 455 + — = 455 + 1 = 456 gallons, 
 
 as was obtained when 460 gallons was used as provisional mean. 
 
 The object of this method of finding the mean is to reduce the 
 size of the variates and hence to reduce the amount of computing. 
 Numbers in the second and third columns are smaller than those 
 in the first. If the provisional mean is chosen as a number near 
 the middle of the range of the items, in most cases, deviations of 
 the items from this provisional mean will be small compared to the 
 original items. The next theorem shows that this method holds 
 for any provisional mean. 
 
 Theorem 2.1. The mean of a set of variates is equal to the 
 provisional mean, h, plus the average of deviations of the variates 
 from the provisional mean, or 
 
 ( 2 . 2 ) M v = h T Md- 
 
 Proof: By definition, deviations of the variates from the pro¬ 
 visional mean, h, are 
 
 di = vi — h 
 
 C?2 = V2 — h 
 
 dz = Vz — h 
 
 d n = v n — h 
 
 Total 2 d = 2v — n-h 
 
PROBLEM 
 
 27 
 
 Divide the above totals by n, the number of variates; this becomes 
 
 or 
 
 2d 
 
 n 
 
 U ' h t . r ,, , 
 
 -— , or Md = M v — h, 
 
 n n 
 
 M v — h + AIj. 
 
 PROBLEM 
 
 1. The following table contains attendance at a Sunday school for 
 one quarter. By use of a provisional mean find the average attendance. 
 
 No. of Students 
 per Sunday 
 137 
 130 
 146 
 129 
 143 
 146 
 149 
 
 No. of Students 
 per Sunday 
 158 
 149 
 151 
 158 
 155 
 147 
 
 The indirect method of finding the mean can be used in finding 
 the mean of a frequency distribution, as the next illustration will 
 show. The table below gives the number of eggs laid by 267 
 
 pheasant hens. 
 
 No. OF 
 Eggs 
 
 V 
 
 No. OF 
 Pheasants 
 
 m 
 
 d 
 
 v— 13 
 
 d-m 
 
 7 
 
 i 
 
 -6 
 
 -6 
 
 8 
 
 7 
 
 -5 
 
 -35 
 
 9 
 
 13 
 
 -4 
 
 -52 
 
 10 
 
 25 
 
 -3 
 
 -75 
 
 11 
 
 34 
 
 -2 
 
 -68 
 
 12 
 
 59 
 
 -1 
 
 -59 
 
 13 
 
 55 
 
 0 
 
 -295 
 
 14 
 
 41 
 
 1 
 
 41 
 
 15 
 
 23 
 
 2 
 
 46 
 
 16 
 
 6 
 
 3 
 
 18 
 
 17 
 
 2 
 
 4 
 
 8 
 
 18 
 
 1 
 
 5 
 
 5 
 
 
 267 
 
 
 + 118 
 
 -177 
 
28 
 
 STATISTICAL AVERAGES 
 
 Let the provisional mean, h, be equal to 13 eggs; deviations of the 
 variates from this provisional mean are listed in the third column. 
 Products of these deviations by their respective frequencies are 
 listed in the last column. The mean, according to formula (2.2), is 
 
 M v = h + M d = 13 - ~ = 13 - 0.66 = 12.34 eggs. 
 
 Theorem 2.1 is one of the most important theorems in statistics. 
 
 PROBLEMS 
 
 1. By use of a provisional mean compute the mean of the following 
 distribution of petals on the ordinary buttercup (Ranunculus bulbosus). 
 
 fa 
 
 O 
 
 6 
 
 & 
 
 No. OF 
 
 fa 
 
 O 
 
 6 
 
 y 
 
 No. OF 
 
 Petals 
 
 F LOWERS 
 
 Petals 
 
 Flowers 
 
 V 
 
 m 
 
 V 
 
 f(v) 
 
 5 
 
 40 
 
 11 
 
 177 
 
 6 
 
 52 
 
 12 
 
 104 
 
 7 
 
 126 
 
 13 
 
 35 
 
 8 
 
 165 
 
 14 
 
 8 
 
 9 
 
 204 
 
 15 
 
 4 
 
 10 
 
 215 
 
 
 1,130 
 
 2. The table below 
 
 contains the number of strokes used by a golfer 
 
 for 100 days on a certain course. 
 
 Find by use of a 
 
 provisional mean his 
 
 average score. 
 
 
 
 
 No. of Strokes No. of 
 
 No. of Strokes No. of 
 
 for 18 Holes 
 
 Days 
 
 for 18 Holes Days 
 
 80 
 
 1 
 
 91 
 
 12 
 
 83 
 
 3 
 
 92 
 
 8 
 
 85 
 
 10 
 
 93 
 
 5 
 
 86 
 
 17 
 
 94 
 
 3 
 
 88 
 
 26 
 
 95 
 
 1 
 
 89 
 
 14 
 
 
 
 3. If the mean of 179 weights is 138.3 pounds, what is the sum of the 
 weights? 
 
 4. If the sum of lung capacities of pupils in a class is 5,625 cubic 
 inches and the mean lung capacity for these pupils is 225 cubic inches, 
 find the number of pupils in the class. 
 
 5. If the mean of 75 items is 40.4 quarts and the mean of 25 itemsjs 
 39.1 quarts, find the mean of the 100 items. 
 
THE MEAN FOR THE COMBINATION OF SETS 
 
 29 
 
 6. The table below contains the distribution of stigmatic rays on seed 
 
 capsules of Shirley poppies. (See Biometrika, II, 
 
 1902, p. 89.) 
 
 No. OF 
 
 No. OF 
 
 3 
 
 0 
 
 0 
 
 *1 
 
 No. OF 
 
 Rats 
 
 Capsules 
 
 Rays 
 
 Capsules 
 
 6 
 
 3 
 
 14 
 
 302 
 
 7 
 
 11 
 
 15 
 
 234 
 
 8 
 
 38 
 
 16 
 
 128 
 
 9 
 
 106 
 
 17 
 
 50 
 
 10 
 
 152 
 
 18 
 
 19 
 
 11 
 
 238 
 
 19 
 
 3 
 
 12 
 
 305 
 
 20 
 
 1 
 
 13 
 
 315 
 
 
 1,905 
 
 Find the average number of stigmatic rays per poppy seed capsule by use 
 of a provisional mean. 
 
 THE MEAN FOR THE COMBINATION OF SETS 
 
 Theorem 2.2. If the mean of n\ items is M\ and the mean of 
 712 items is M 2 , then the mean of the combined set of n\ + no 
 items is 
 
 (2.3) 
 
 M 1+2 
 
 n\M 1 + 712 M 2 
 71 x + 772 
 
 Proof: Since the numerator in the above equation is the sum 
 of the items in both sets and the denominator is the total number 
 of items, the mean of the combined sets is equal to the right-hand 
 member of the equation. 
 
 In general, if the mean of 711 items is Mi, the mean of 772 items 
 is M 2 , the mean of 773 items is M 3 , . . . , and the mean of n a items is 
 M a , then the mean of the items in all the sets is 
 
 (2.4) Mi + 2+ ... + 
 
 n\M\ + 772 M 2 + 773 M 3 + . . . + n a M s 
 
 77 1 + 772 + 773 + . . . 77„ 
 
 PROBLEMS 
 
 1. The mean score on an arithmetic test of 
 
 55 students in Ward 1 is 76.2, 
 
 69 
 
 “ “ 2 “ 74.3, 
 
 30 
 
 “ “ 3 “ 78.1, 
 
 44 
 
 “ “ 4 “ 75.8. 
 
 Find the average score on this test for the city. 
 
30 
 
 STATISTICAL AVERAGES 
 
 2. The average weight of the 35 men in dormitory A is 140.4 pounds, 
 
 “ 139.3 “ , 
 
 “ 138.6 “ 
 
 tt tt 
 
 a a 
 
 67 
 
 96 
 
 a a 
 
 a a 
 
 B 
 
 C 
 
 Dormitory C contains 22 Japanese men whose average weight is 136.3 
 pounds. Find the average weight of the men in all dormitories without 
 the Japanese. 
 
 3. Average amount of gas sold per day by Station 1 was 430 gallons, 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 2 
 
 t l 
 
 362 
 
 It 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 3 
 
 tt 
 
 284 
 
 tt 
 
 (( 
 
 tt 
 
 it 
 
 a 
 
 it 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 4 
 
 tt 
 
 685 
 
 tt 
 
 ti 
 
 tt 
 
 it 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 5 
 
 it 
 
 598 
 
 tt 
 
 u 
 
 tt 
 
 ti 
 
 a 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 a 
 
 6 
 
 ti 
 
 507 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 7 
 
 tt 
 
 706 
 
 tt 
 
 a 
 
 tt 
 
 tt 
 
 u 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 a 
 
 CO 
 
 tt 
 
 843 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 a 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 9 
 
 tt 
 
 389 
 
 tt 
 
 If these data were for the month of May, find the average number of 
 gallons of gas sold by the company owning these stations per day. Find 
 the amount sold during the month. 
 
 4. How should formula (2.3) be written if n 2 is a subset of ru, and it 
 is required to find the mean of the ni — n 2 items? 
 
 5. Express n in terms of the sum of the variates and the mean. 
 
 6. The following data relate to sales of stamps in a post office for 
 one day. 
 
 Denomination 
 
 No. 
 
 Denomination 
 
 No. 
 
 of Stamps 
 
 Sold 
 
 of Stamps 
 
 Sold 
 
 \ cent 
 
 648 
 
 6 cents 
 
 89 
 
 1 “ 
 
 1409 
 
 10 “ 
 
 94 
 
 2 cents 
 
 1278 
 
 16 “ 
 
 59 
 
 3 “ 
 
 2941 
 
 25 “ 
 
 37 
 
 5 “ 
 
 703 
 
 
 
 Find the amount received from stamps. 
 
 THE MEAN OF GROUPED DATA 
 
 Many times, labor is saved in computing when variates are 
 divided into groups or classes. This will be illustrated by an 
 example. The table below contains the distribution of weights of 
 100 male students. The original measurements were made to the 
 nearest half pound. These weights have been grouped into classes 
 of 10 pounds. 
 
THE MEAN OF GROUPED DATA 31 
 
 Weight of 
 
 Class 
 
 Fre- 
 
 
 
 Students 
 
 Marks 
 
 QUENCY 
 
 z-134.75 
 
 
 V 
 
 X 
 
 f(v) 
 
 d 
 
 d-m 
 
 99.75-109.75 
 
 104.75 
 
 1 
 
 -3 
 
 - 3 
 
 109.75-119.75 
 
 114.75 
 
 9 
 
 -2 
 
 -18 
 
 119.75-129.75 
 
 124.75 
 
 23 
 
 -1 
 
 -23 
 
 129.75-139.75 
 
 134.75 
 
 25 
 
 0 
 
 -44 
 
 139-75-149.75 
 
 144.75 
 
 19 
 
 1 
 
 19 
 
 149.75-159.75 
 
 154.75 
 
 15 
 
 2 
 
 30 
 
 159.75-169.75 
 
 164.75 
 
 4 
 
 3 
 
 12 
 
 169.75-179.75 
 
 174.75 
 
 4 
 
 100 
 
 4 
 
 16 
 
 +77 
 
 +33 
 
 The lower closed limit of the second class is 109.75 pounds, 
 because every weight which was less than this and greater than 
 99.75 pounds was put into the first class and all above this and less 
 than 119.75 pounds was put into the second class. The person 
 who weighed the students wanted weights to the nearest half- 
 pound. If the scales showed a weight to be 109.83 pounds it was 
 recorded as 110 pounds, as this was the weight to the nearest half 
 pound; had the weight been 109.65 pounds it would have been 
 recorded as 109.5 pounds. Any weight between 109.75 pounds 
 and 110.25 pounds was recorded as 110 pounds; any weight 
 between 137.25 pounds and 137.75 pounds was recorded as 137.5 
 pounds. After the weights were recorded they were grouped into 
 classes of 10 pounds. 
 
 The class mark of a class is the average of the upper and lower 
 closed limits of the class. The class mark of the third class is 
 
 119.75 + 129.75 
 2 
 
 124.75 pounds. Class marks are written in the 
 
 second column. Class marks are representatives of the items in 
 the respective classes. Weight, 124.75 pounds, represents the 
 weight of each item in the third class. Some weights in the third 
 class were less than this class mark, and some were greater. In 
 grouping variates into classes it is assumed that the class mark is 
 the average of the variates in the class. Of course, the class mark 
 is the average of the variates in the class if there arc the same num¬ 
 ber in the class above the class mark as below it, and equally 
 
32 
 
 STATISTICAL AVERAGES 
 
 spaced. This happens often when there are large numbers of items 
 in the classes and the classes have been made small. When vari¬ 
 ates have been grouped into classes the individual variates are not 
 known; the class mark is the representative of all variates in the 
 class. There are 9 weights in the second class; the original mea¬ 
 surements of these are not known now. The class mark, 114.75 
 pounds, now represents these 9 measurements; this is the same as 
 saying that each of the 9 measurements is 114.75 pounds. The 
 distribution is now made up of class marks and their frequencies, 
 which are the numbers of measurements in the respective classes. 
 The frequency of class mark 114.75 pounds is 9 because there are 
 9 measurements in this class. 
 
 The mean of the distribution made up of class marks and their 
 frequencies can be found by applying formula (2.2). Choose the 
 class mark of the fourth class as provisional mean and place the 
 deviations of class marks from this provisional mean in column 
 four. Deviations, d, are now in terms of class units; that is, the 
 difference between two consecutive class marks is 10 pounds or 1 
 class unit. The mean of the distribution made up of the d’s is 
 
 M d 
 
 Hd-f _ +33 
 
 2 / “ 100 
 
 0.33 class unit. 
 
 This 0.33 class unit is converted into pounds by multiplying by 10, 
 the size of the class; hence the mean of the distribution made up of 
 class marks is 
 
 (2.5) M x = h + M d -w = 134.75 + (0.33)10 = 138.05 lb., 
 
 where w is the width of the class. The mean of the z’s, 138.05 
 pounds, is considered to be the mean of the grouped data, or the 
 mean of the v’s. It is not always the same as would be obtained 
 by taking the average of the measurements before they were 
 thrown into classes. Work in more advanced statistics shows 
 that the mean of class marks and the mean of original measure¬ 
 ments before they are grouped differ by a negligible quantity, 
 provided the classes are not too large. Quantity w is the width of 
 the class; it is 10 pounds in this case. Formula (2.5) is the same 
 as formula (2.2) with w — 1. The object of grouping is to reduce 
 the amount of labor in computing. 
 
THE MEAN OF GROUPED DATA 
 
 33 
 
 The degree of accuracy of measurements is very important, for 
 this enables one to obtain the class limits. Suppose that measure¬ 
 ments in the above illustration had been made to the nearest 
 pound; then the classes would have been written as follows: 
 
 99.5-109.5 
 
 109.5- 119.5 
 
 119.5- 129.5 
 etc. 
 
 Had measurements been made to the nearest 0.1 pound the classes 
 would have been written as: 
 
 99.95-109.95 
 
 109.95-119.95 
 
 etc. 
 
 It is very important to get the class limits correct, for they deter¬ 
 mine class marks which determine the mean of the distribution. 
 
 One cannot measure anything to the nearest-of 
 
 J & 1 , 000 , 000 , 000,000 
 
 a unit or to the nearest infinitesimal of a unit; hence the degree of 
 
 accuracy of measurements must be given so that anyone may be 
 
 able to group the data. An incorrect way of writing classes in the 
 
 above example where measurements were made to the nearest 0.1 
 
 pound is as follows: 
 
 100-110 
 
 110-120 
 
 etc. 
 
 Classes in this case may be written in open limits as follows: 
 
 100-109.9 
 
 110-119.9 
 
 120-129.9 
 
 etc. 
 
 To find the closed limits from the open limits one decreases the 
 lower open limit of the class by one-half of the degree of accuracy 
 of the measurements and adds one-half of the degree of accuracy 
 to the upper open limit. The lower closed limit of a class is the 
 same as the upper closed limit of the preceding class; this is not 
 true for open limits. 
 
34 
 
 STATISTICAL AVERAGES 
 
 The following figures represent weights of 100 men. The 
 original measurements were made to the nearest pound. 
 
 Weights 
 
 Fre¬ 
 
 Weights 
 
 Fre¬ 
 
 Weights 
 
 Fre¬ 
 
 Weights 
 
 Fre¬ 
 
 
 quency 
 
 
 quency 
 
 
 quency 
 
 
 quency 
 
 111 
 
 1 
 
 126 
 
 3 
 
 139 
 
 1 
 
 154 
 
 1 
 
 112 
 
 1 
 
 127 
 
 2 
 
 140 
 
 5 
 
 155 
 
 3 
 
 113 
 
 2 
 
 128 
 
 1 
 
 141 
 
 2 
 
 156 
 
 1 
 
 114 
 
 1 
 
 129 
 
 1 
 
 143 
 
 6 
 
 158 
 
 1 
 
 115 
 
 1 
 
 130 
 
 4 
 
 144 
 
 3 
 
 159 
 
 1 
 
 117 
 
 2 
 
 131 
 
 2 
 
 145 
 
 1 
 
 160 
 
 1 
 
 118 
 
 2 
 
 132 
 
 1 
 
 146 
 
 2 
 
 162 
 
 1 
 
 119 
 
 1 
 
 133 
 
 2 
 
 147 
 
 2 
 
 163 
 
 1 
 
 120 
 
 1 
 
 134 
 
 3 
 
 148 
 
 2 
 
 164 
 
 1 
 
 121 
 
 3 
 
 135 
 
 1 
 
 149 
 
 1 
 
 168 
 
 1 
 
 122 
 
 4 
 
 136 
 
 4 
 
 150 
 
 5 
 
 169 
 
 1 
 
 124 
 
 1 
 
 137 
 
 1 
 
 152 
 
 1 
 
 175 
 
 1 
 
 125 
 
 2 
 
 138 
 
 5 
 
 153 
 
 2 
 
 179 
 
 1 
 
 The above data were grouped in 2’s, 3’s, and so on, up through 
 10’s. The lower limit of the first class was always 109.5 pounds. 
 The means for the different ways of grouping are: 
 
 Mi = 138.16 lb., 
 
 M 6 
 
 = 138.12 lb., 
 
 M 2 = 138.28 “ , 
 
 Mn 
 
 = 138.42 “ , 
 
 M 3 = 138.24 “ , 
 
 00 
 
 = 138.38 “ , 
 
 CO 
 
 00 
 
 CO 
 
 r-H 
 
 II 
 
 m 9 
 
 = 138.12 “ , 
 
 M 6 = 138.25 “ , 
 
 Mio 
 
 = 138.50 “ , 
 
 where Mg represents the mean when the size of the class was 6 
 pounds, etc. The mean of the ungrouped data is M\ = 138.16 
 pounds, and is the true mean, provided the original measurements 
 are considered exact. Means for grouping in 6’s and 9’s are nearer 
 the “true” mean than means for other groupings. 
 
 This example shows that means obtained from grouped data 
 give results which differ very little from the true mean; in this case 
 they differ by less than 0.35 pound from the “real” mean. Had 
 there been more items in the distribution there would have been 
 better results. This example shows that in grouping one does not 
 always secure the true average but an average which is approxi¬ 
 mately equal to the true average. Of course, the true mean may 
 never be obtained because the measurements were made only to 
 the nearest pound. 
 
PROBLEMS 
 
 35 
 
 PROBLEMS 
 
 1. Group the following freshmen mathematics averages in groups of 
 
 5, and find the mean of the distribution. Use 49.5 as the lower limit 
 of the first class: 62, 82, 78, 78, 82, 78, 92, 68, 85, 75, 60, 85, 50, 92, 72, 
 
 88, 72, 70, 75, 82, 92, 60, 72, 95, 95, 95, 75, 78, 75, 95, 83, 85, 75, 70, 68, 
 
 85, 82, 65, 88, 78, 82, 92, 95, 92, 82, 78, 82, 82, 74, 75, 82, 55, 68, 92, 68, 
 
 92, 72, 75, 78, 62, 95, 72, 83, 83, 65, 75, 82, 88, 72, 78, 78, 95, 77. 
 
 2. The following table gives the frequency distribution of rays per 
 inflorescence of Zizia aurea (Golden Alexander). Find the mean number 
 of rays per cluster. 
 
 No. of Rays 
 
 Fre¬ 
 
 No. of Rays 
 
 Fre¬ 
 
 per Cluster 
 
 quency 
 
 per Cluster 
 
 quency 
 
 10-11 
 
 21 
 
 20-21 
 
 65 
 
 12-13 
 
 39 
 
 22-23 
 
 22 
 
 14-15 
 
 98 
 
 24-25 
 
 6 
 
 16-17 
 
 127 
 
 26-27 
 
 11 
 
 18-19 
 
 111 
 
 
 500 
 
 3. The following table contains lung capacities, to the nearest cubic 
 inch, of 536 women students at the University of Michigan. Find the 
 mean, and plot the distribution. 
 
 Lung 
 
 No. OF 
 
 Lung 
 
 No. OF 
 
 Capacity 
 
 Women 
 
 Capacity 
 
 Women 
 
 79.5- 99.5 
 
 2 
 
 179.5-199.5 
 
 114 
 
 99.5-119.5 
 
 14 
 
 199.5-219.5 
 
 53 
 
 119.5-139.5 
 
 78 
 
 219.5-239.5 
 
 30 
 
 139.5-159.5 
 
 78 
 
 239.5-259.5 
 
 9 
 
 159.5-179.5 
 
 158 
 
 
 536 
 
 4 . The table below gives weights of 194 women students at the Uni¬ 
 versity of Michigan, measurements being made to the nearest 0.1 pound. 
 Find the mean. 
 
 Weights Frequency 
 
 79.45- 89.45 2 
 
 89.45- 99.45 18 
 
 99.45- 109.45 28 
 
 109.45- 119.45 55 
 
 119.45- 129.45 49 
 
 129.45- 139.45 19 
 
 Weights Frequency 
 
 139.45- 149.45 11 
 
 149.45- 159.45 6 
 
 159.45- 169.45 3 
 
 169.45- 179.45 2 
 
 179.45- 189.45 1 
 
 194 
 
36 
 
 STATISTICAL AVERAGES 
 
 5. The following is a distribution of pedicels per inflorescence of 
 Daucus carota (wild carrot) from Michigan. Find the mean. Make the 
 
 classes twice as large and find the 
 
 mean. Compare this with the mean 
 
 found before the classes were doubled in size. Which is nearer the true 
 
 mean? 
 
 No. of Pedicels 
 
 No. OF 
 
 No. of Pedicels 
 
 No. of 
 
 per Cluster 
 
 Clusters 
 
 per Cluster 
 
 Clusters 
 
 25-29 
 
 1 
 
 60- 64 
 
 119 
 
 30-34 
 
 19 
 
 65- 69 
 
 74 
 
 35-39 
 
 44 
 
 70- 74 
 
 34 
 
 40-44 
 
 133 
 
 75- 79 
 
 13 
 
 45-49 
 
 152 
 
 80- 84 
 
 15 
 
 50-54 
 
 230 
 
 So— 89 
 
 1 
 
 55-59 
 
 164 
 
 105-109 
 
 1 
 
 1,000 
 
 6. Find the mean of the distribution of wages per day for employees 
 
 of a certain factory. 
 
 Wages per Day 
 
 No. OF 
 
 Wages per Day 
 
 No. OF 
 
 Open limits 
 
 Employees 
 
 Open Limits 
 
 Employees 
 
 $2.50-2.99 
 
 3 
 
 $6.50-6.99 
 
 97 
 
 3.00-3.49 
 
 7 
 
 7.00-7.49 
 
 38 
 
 3.50-3.99 
 
 16 
 
 7.50-7.99 
 
 11 
 
 4.00-4.49 
 
 46 
 
 8.00-8.49 
 
 4 
 
 4.50-4.99 
 
 81 
 
 8.50-8.99 
 
 2 
 
 5.00-5.49 
 
 143 
 
 9.00-9.49 
 
 1 
 
 5.50-5.99 
 
 239 
 
 9.50-9.99 
 
 1 
 
 6.00-6.49 
 
 176 
 
 865 
 
 GRAPHICAL REPRESENTATION OF FREQUENCY DISTRIBUTIONS 
 
 The frequency distribution of petals on buttercups, as given on 
 page 28, is plotted in Fig. 2.1. The horizontal axis represents the 
 number of petals; the vertical, the number of flowers or frequencies. 
 
 FREQUENCY POLYGONS AND HISTOGRAMS 
 
 Figure 2.1 graphically represents the distribution of petals on 
 these buttercups. Points b, c, d, etc., have been joined in Fig. 2.2 
 to make a frequency polygon abodefghijklma. The object of this 
 polygon is to enable one to see at once the nature of the frequen¬ 
 cies. There were no buttercups with 8§ petals, hence there is no 
 
FREQUENCY POLYGONS AND HISTOGRAMS 
 
 37 
 
 frequency at this point. Line fg excluding / and g represents no 
 frequency. This polygon is used extensively to help the eye grasp 
 the situation quickly, although it has no meaning for points 
 between points plotted in Fig. 2.1. 
 
 Petals 
 
 Fig. 2.1—Frequency distribution of petals on buttercups. 
 
 When data are grouped into classes, areas of rectangles are used 
 to represent frequencies instead of vertical lines as in Fig. 2.1. 
 The bases of these rectangles are equal to the width of the classes; 
 the altitudes are equal to the respective frequencies. Each base 
 
 Petals 
 
 Fig. 2.2.—Frequency polygon of the distribution of petals on buttercups. 
 
 is equal to 1 class unit; hence the area is equal to the frequency. 
 The representation of frequencies of the distribution of weights 
 on page 31 is shown in Fig. 2.3. 
 
 Bases of the rectangles begin and end at the lower and upper 
 closed class limits, respectively. Class marks are represented by 
 points at the middle of the bases of the rectangles. The graph in 
 which frequencies are represented by rectangles is called a histo¬ 
 gram of the distribution. Figure ABCDEFGHIJKA is the his- 
 
38 
 
 STATISTICAL AVERAGES 
 
 togram of the distribution mentioned. Rectangle CY represents 
 the frequency or the number of students whose weights were 
 between 109.75 pounds and 119.75 pounds. Rectangle PH repre¬ 
 sents the number whose weights were between 149.75 pounds and 
 159.75 pounds, etc. A histogram helps one to grasp immediately 
 the nature of the frequencies. The vertical lines should always 
 be drawn in, for they make the rectangles complete. These rect- 
 
 Weights 
 
 Fig. 2.3.—Histogram of the distribution of weights of male students. 
 
 angles are part of the histogram which represent frequencies; they 
 should always be complete. 
 
 PROBLEMS 
 
 1. Represent graphically the distribution in problem 6 on page 29. 
 
 2. Represent graphically the distribution hi problem 1 on page 24. 
 
 3. Represent graphically distributions in problems 4, 5, 6 on pages 35 
 and 36. Draw a vertical line at the mean of each. How much area is 
 to the left of this line? How much is to the right of this line? 
 
 4. If a person picked at random a cluster from the plot from which 
 the distribution in problem 5 on page 36 came, what is the probability of 
 getting a cluster which had pedicels between 60 and 64 inclusive? 
 
 OGIVE 
 
 Figure 2.4 exhibits the total frequencies of Fig. 2.1 up to and 
 including certain points. For example, the vertical line od in 
 Fig. 2.4 gives the number of buttercups which had 5, 6, and 7 
 petals; line gp gives the number of buttercups which had 5, 6, 7, 8, 
 9, and 10 petals; etc. This graph of the cumulative frequencies is 
 called an ogive or a cumulative frequency polygon. At the point 
 
OGIVE 
 
 39 
 
 4 the ogive is zero, at 16 it is equal to 1,130. To the left of 5 it is 
 zero, and to the right of 15 it is equal to 1,130. The ogive rises 
 slowly at the left end of the range, rises rapidly for values in the 
 
 Fig. 2.4.—Ogive of the distribution of petals on buttercups. 
 
 middle of the range, and rises slowly at values near the right end 
 of the range. 
 
 The ogive for the distribution in Fig. 2.3 is plotted in Fig. 2.5. 
 Vertical line CX represents the total frequencies for students who 
 
 Weights 
 
 Fig. 2.5.—Ogive for the distribution of weights of male students. 
 
 weighed less than 120 pounds; line PG represents the number of 
 students who weighed less than 160 pounds, etc. This ogive is 
 equal to zero to the left of 99.75 pounds and equal to 100 to the 
 right of 179.75 pounds. The difference between hues PG and QF 
 gives the frequency for the class 149.75-159.75. 
 
40 
 
 STATISTICAL AVERAGES 
 
 PROBLEMS 
 
 1. Construct the ogive for the distribution in problem 3 on page 25. 
 
 2. Construct the ogive for the distribution in the example concerning 
 eggs laid by pheasants on page 27. 
 
 3. Construct the ogives for distributions in problems 4, 5, and 6 on 
 pages 35 and 36. 
 
 4. From the ogive for the distribution in problem 5 on page 36 esti¬ 
 mate the number of clusters which had 42 and 43 pedicels, the number 
 which had 73, 74, 75, and 76 pedicels, and the number which had 51 
 pedicels. 
 
 5. From the ogive for the distribution in problem 6 on page 36 esti¬ 
 mate the number of employees who received $5.25-5.65, also the number 
 who received $6.15-6.40, also the number who received $5.75. 
 
 THE MEDIAN 
 
 The median of a set of variates is the variate or quantity which 
 has the same number of variates less than it as the number greater 
 than it. The median of the variates, 4, 5, 9, 10, 11 is 9, as there 
 are 2 variates less than 9 and 2 greater than 9. The median of the 
 set 6, 8, 11, 12, 13, 16 is equal to any number between 11 and 12; 
 it is customary to take the average of 11 and 12, or 11.5, as the 
 median. In this case, the median is not one of the items. The 
 median always divides the distribution into equal parts. A rule 
 for finding the median for data which are not grouped is as follows: 
 
 ( 2 . 6 ) 
 
 m = the 
 
 th item, w T hen the items are arranged in 
 
 ascending or descending order. 
 
 In the first example the th item is the 
 
 item, or the third item. In the second example the 
 
 
 th 
 
 th 
 
 ( 6 | \ ^ / 
 
 —-—)th item, or the 3|th item. There is no 3^th 
 
 item; it is considered to be the quantity half way between the 
 third and fourth item, wdiich is 11.5, as found before. Formula 
 (2.6) always gives the median of a set of variates which are not 
 grouped into classes. 
 
MEDIAN FOR GROUPED DATA 
 
 41 
 
 PROBLEMS 
 
 1. Find the median of the distribution of the following scores made on 
 a certain test: 80, 81, 76, 59, 55, 91, 63, 58, 82, 95, 79, 85, 83, 61, 58, 78, 
 82, 90, 86, 73, 59, 62, 71, 84, 87, 97, 67, 75, 86. 
 
 2. Find the median age of students in a class if the ages are: 19, 20, 
 18, 22, 25, 27, 21, 23, 24, 23, 26, 20^ 
 
 MEDIAN FOR GROUPED DATA 
 
 Let it be required to find the median of the following distribu¬ 
 tion of weights of 123 men of age 20, where original measurements 
 were made to the nearest 0.5 pound. 
 
 Weights, Classes Frequency 
 
 99.75-109.75 3 
 
 109.75- 119.75 11 
 
 119.75- 129.75 27 
 
 129.75- 139.75 <32 
 
 139.75- 149.75 23 
 
 149.75- 159.75 17 
 
 159.75- 169.75 6 
 
 169.75- 179.75 4 
 
 123 
 
 The median is the value which divides the area of the histo¬ 
 gram into two equal parts; it is found by the formula 
 
 (2.7) 
 
 n 
 
 where C 2 is the lower closed limit of the class in which the -th item 
 
 2 
 
 lies, 712 is the sum of all frequencies below C 2 , and fc is the number 
 
 n 
 
 of items in the class containing the -th item and w is the width of 
 
 Li 
 
 71 
 
 this class. The median usually lies in the class in which the -th 
 item lies. 
 
 The n/2th item is the 6l|th item and is located in the fourth 
 class. The lower limit of this class is C 2 = 129.75 pounds, 112 = 41, 
 /a = 32, and w = 10 pounds. The median is 
 
 m 
 
 ( 61.5 - 41 \ 
 
 = 129.75 + (-— - ) 10 = 129.75 + 6.41 = 
 
 136.161b. 
 
42 
 
 STATISTICAL AVERAGES 
 
 Line RS in Fig. 2.6 drawn through the point 136.16 pounds, 
 perpendicular to the horizontal axis, has one half of the area to 
 the left and one half to the right. For a grouped distribution the 
 median is the value on the base line of the histogram, such that a 
 perpendicular line to the base line through a point representing 
 this value divides the area of the histogram into equal parts. 
 
 The median, in many cases, is a good representative of the 
 items and is used by many investigators. Sizes of the items below 
 and above the median do not affect its value, as the following 
 
 Weights 
 
 Fig. 2.6. —Distribution of weights of men 20 years of age, showing that the 
 median divides the distribution into two equal parts. 
 
 illustration shows. The median of 3, 5, 6, 7, 9 is 6, and so is the 
 mean. If 9 were changed to 14 the median would still be 6, 
 while the mean would be changed to 7. 
 
 The median for grouped data cannot be obtained by formula 
 (2.6) because a line perpendicular to the horizontal axis at the 
 point obtained by this formula will not divide the area into two 
 equal parts. This will be shown by finding the median as is done 
 by some authors for the set of above weights. The formula which 
 is given by some authors is 
 
 n + 1 
 2 ~ 712 
 
 m = C 2 +---• w. (Not correct for grouped data.) 
 
 h 
 
 When this is used the median becomes 
 62 — 41 
 
 m = 129.75 + -- 10 = 129.75 + 6.56 = 136.31 lb. 
 
 32 
 
 If a line is drawn through this point perpendicular to the base line 
 
MEDIAN FOR GROUPED DATA 
 
 43 
 
 in the histogram it does not divide the area into two equal parts. 
 The area to the left is 62 and the area to the right is 61, which 
 shows that the above formula is not the correct formula for finding 
 the median of a grouped distribution. Formula (2.7) will give 
 the median when the sum of frequencies is an odd or an even 
 number. 
 
 The median is used as a representative of wages of employees 
 of a factory when the mean would have no meaning. Let the 
 wages per month of the employees be as follows: 
 
 Wages per Month 
 $ 75 
 
 85 
 105 
 115 
 125 
 130 
 140 
 200 
 600 
 
 $1,575 
 
 The mean is $175, and the median is $125. Here the median is a 
 better representative of the wages than the mean. Values of 
 extreme items greatly affect the mean but do not influence the 
 median. 
 
 Let it be required to find the median of the distribution of 
 children per family for a certain district. 
 
 No. of Children No. of 
 per Family Families 
 
 0 4 
 
 1 17 
 
 2 35 
 
 3 80 
 
 4 63 
 
 5 20 
 
 6 6 
 
 7 2 
 
 8 1 
 
 228 
 
44 
 
 STATISTICAL AVERAGES 
 
 Formula (2.7) must be used in this case to find the median, although 
 the class width is unity. The median is 
 
 m = 2.5 + 
 
 114 - 56 
 80 
 
 1 - 2.5 + 0.725 = 3.225 children. 
 
 PROBLEMS 
 
 1. Find the median score made on a test in arithmetic if the scores 
 were: 
 
 
 No. OF 
 
 
 No. of 
 
 Scores 
 
 Students 
 
 Scores 
 
 Students 
 
 Below 39.5 
 
 7 
 
 69.5-74.5 
 
 189 
 
 39.5-44.5 
 
 17 
 
 74.5-79.5 
 
 261 
 
 44.5-49.5 
 
 26 
 
 79.5-84.5 
 
 119 
 
 49.5-54.5 
 
 38 
 
 84 . 5-89.5 
 
 80 
 
 54.5-59.5 
 
 53 
 
 89.5-94.5 
 
 18 
 
 59.5-64 . 5 
 
 79 
 
 94.5-99.5 
 
 3 
 
 64.5-69.5 
 
 104 
 
 
 994 
 
 Plot the data and erect a perpendicular at the median, 
 the left and also to the right of this line. 
 
 Find the area 
 
 2. Find the median age for 
 
 the city whose age 
 
 distribution is 
 
 follows: 
 
 
 
 
 Age 
 
 No. OF 
 
 Age 
 
 No. OF 
 
 Group 
 
 People 
 
 Group 
 
 People 
 
 Under 1 year 
 
 2,190 
 
 25.29 
 
 9,833 
 
 1- 2 
 
 2,164 
 
 30-34 
 
 9,120 
 
 2- 3 
 
 2,326 
 
 35—44 
 
 17,198 
 
 3- 4 
 
 2,394 
 
 45-54 
 
 13,017 
 
 4- 5 
 
 2,368 
 
 55-64 
 
 8,396 
 
 5- 9 
 
 12,607 
 
 65-74 
 
 4,720 
 
 10-14 
 
 12,005 
 
 Over 75 years 
 
 1,911 
 
 15-19 
 
 11,552 
 
 
 122,671 
 
 20-24 
 
 10,870 
 
 
 Can one find the mean of this set of data? 
 
THE MODE 
 
 45 
 
 3. Find the median wage if wages per day are given below. 
 
 
 No. OF 
 
 
 No. OF 
 
 Wages 
 
 Employees 
 
 Wages 
 
 Employees 
 
 $2.00-2.24 
 
 4 
 
 $4.75-4.99 
 
 400 
 
 2.25-2.49 
 
 7 
 
 5.00-5.24 
 
 384 
 
 2.50-2.74 
 
 13 
 
 5.25-5.49 
 
 218 
 
 2.75-2.99 
 
 26 
 
 5.50-5.74 
 
 167 
 
 3.00-3.24 
 
 33 
 
 5.75-5.99 
 
 92 
 
 3.25-3.49 
 
 80 
 
 6.00-6.24 
 
 28 
 
 3.50-3.74 
 
 CO 
 
 OO 
 
 6.25-6.49 
 
 11 
 
 3.75-3.99 
 
 4.00-4.24 
 
 IH 374 
 
 178 
 
 6.50-6.74 
 
 4 
 
 4.25-4.49 
 
 4.50-4.74 
 
 259 
 
 331 
 
 
 2,444 
 
 THE MODE 
 
 The mode of a set of variates is the variate which occurs the 
 most frequently. In Fig. 2.1, 10 petals appeared more often than 
 any other. In Fig. 2.3, the class mark of the fourth class might 
 be taken as the approximate mode of weights; however, this class 
 mark might not be the real mode. The mode of a grouped distri¬ 
 bution is sometimes defined as the value of the variate at which 
 the smooth curve which fits the data best has a maximum or great¬ 
 est frequency. In some distributions there may not be a mode, 
 for there may be two values at which the frequencies are greater 
 than the others and are equal. The mode is sometimes spoken 
 of as the typical value. Size 37 in men’s suits is the mode, for 
 more suits of this size are sold than of any other size. 
 
 There are various formulas for finding the approximate mode 
 for grouped data. The following formula is used by many statis¬ 
 ticians and is useful for distributions which have one class which 
 has a frequency greater than any of the frequencies of the other 
 classes; it is 
 
 (2.8) Mode = L + - ^ . • w, 
 
 Ji +J2 
 
 where L is the lower limit of the modal class or the class in which 
 the mode falls, /i is the frequency of the class just below the modal 
 class, ft is the frequency of the class just above the modal class, 
 amd w is the width of the class. 
 
46 
 
 STATISTICAL AVERAGES 
 
 Grouping has much to do with the value of the mode. Extreme 
 values of the variates do not affect the value of the mode. The 
 mode for the distribution on page 41 is 
 
 23 
 
 Mode = 129.75 + — • 10 = 129.75 + 4.60 = 134.35 lb. 
 
 £ i “f - Ao 
 
 PROBLEMS 
 
 1. Find the mode of the following distribution of bracts per cluster of 
 Daucus carota (wild carrot). Plot the data, and compare the mean and 
 median with the mode. 
 
 No. of Bracts No. of 
 
 No. of Bracts No. of 
 
 per Cluster Clusters 
 
 per Cluster Clusters 
 
 8 98 
 
 12 201 
 
 9 143 
 
 13 189 
 
 10 159 
 
 14 3 
 
 11 205 
 
 15 2 
 
 2. Find the mode of the heights of trees growing on a study plot if the 
 following information was found by measurements after a certain lapse of 
 
 time. 
 
 
 Height of Trees 
 
 No. of Trees 
 
 Under 5 ft. 
 
 49 
 
 5- 9 
 
 75 
 
 10-14 
 
 91 
 
 15-19 
 
 146 
 
 20-24 
 
 102 
 
 25-29 
 
 51 
 
 30-39 
 
 11 
 
 Over 40 ft. 
 
 3 
 
 
 528 
 
 Compare the mode with the median. 
 
 
 3. Find the mode of the distribution of young per litter in mice 
 {Peromyscus truei truei from Deadman Flat, near Flagstaff, Arizona): 
 
 No. in Litter 
 
 No. of Mother Mice 
 
 1 
 
 6 
 
 2 
 
 11 
 
 3 
 
 15 
 
 4 
 
 8 
 
 5 
 
 2 
 
 6 
 
 1 
 
 43 
 
THE GEOMETRIC MEAN 
 
 47 
 
 THE GEOMETRIC MEAN 
 
 The geometric mean of a set of n positive variates is the nth 
 root of their product. If the variates are v\, V 2 , v%, . . . , v n , the 
 geometric mean is by definition 
 
 ( 2 . 9 ) G = '\ / vi-V 2 -V 3 - . . . v n - 
 The logarithm of G is 
 
 (2.10) log G = - 2 log 
 
 n 
 
 If the frequencies of the variates are respectively 
 
 h, h, h, • • • , fn, 
 
 the geometric mean is 
 
 One of the objects of the geometric mean is to enable one to 
 find averages of ratios and averages of rates of increase and 
 decrease. Consider the following example. A certain city of 
 100,000 inhabitants increased from 1900 to 1910 by 6 per cent, 
 from 1910 to 1920 by 9 per cent, and from 1920 to 1930 by 12 per 
 cent; find the average rate of increase per decade. The 1930 cen¬ 
 sus gave the population as 129,407. The required rate is the rate 
 which will accumulate 100,000 into 129,407 in 3 conversion periods; 
 that is, 100,000 (1 + r ) 3 = 129,407, which is the ordinary com¬ 
 pound interest law. The rate r must be the same for each decade. 
 
 Let us take the arithmetic average of the rates and determine 
 whether it is the required rate. The average of the rates is 0.09. 
 Using this rate for the average per decade, the population for 1910 
 is 109,000, that for 1920 is 118,810, and that for 1930 is 129,503, 
 which is not equal to the population 129,407. This shows that the 
 arithmetic mean of the rates is not the required rate. 
 
 The population in 1910 was 106 per cent of what it was in 1900; 
 in 1920 it was 109 per cent of what it was in 1910, and in 1930 it 
 was 112 per cent of what it was in 1920. The geometric mean of 
 1.06, 1.09, and 1.12 is 
 
 G = \/ (1.06)(1.09)(1.12) = 1.08973, 
 
 and the average rate per 10-year period is 0.08973. According to 
 this rate the population in 1910 should have been 108,973, in 1920 
 it should have been 118,751, and in 1930 it should have been 
 129,407, as it was. 
 
48 
 
 STATISTICAL AVERAGES 
 
 The following data relate to values of an automobile. 
 
 Year 
 
 Value 
 
 1930 
 
 $720 
 
 1931 
 
 609 
 
 1932 
 
 500 
 
 1933 
 
 375 
 
 1934 
 
 227 
 
 1935 
 
 107 
 
 Find the average rate of decrease. The ratios of the value for 
 1 year divided by the value for the preceding year are: 
 
 10 
 10 
 10 
 10 
 10 
 
 50 
 
 G = 0.682984 = 1 - r, 
 
 where r is the rate of decrease. The average rate of decrease is 
 0.317016, for 
 
 720(1 - 0.317016) 5 = 107. 
 
 PROBLEMS 
 
 1. The following figures are quotations on a certain bond for 12 days: 
 54§, 54f, 55, 55 b, 55f, 56 j, 56, 56|, 55|, 56f, 57, 57j. Find the average 
 rate of increase per day. Plot these data. 
 
 2 . The following table gives the number of children under 1 year of 
 age out of 100,000 who w T ere living during the first year for each month. 
 
 
 No. Living 
 
 
 No. Living 
 
 Months 
 
 Out of 100,000 
 
 Months 
 
 Out of 100,000 
 
 0-1 
 
 100,000 
 
 6 - 7 
 
 92,245 
 
 1-2 
 
 96,106 
 
 7- 8 
 
 91,701 
 
 2-3 
 
 95,089 
 
 8 - 9 
 
 91,204 
 
 3-4 
 
 94,241 
 
 9-10 
 
 90,747 
 
 4-5 
 
 93,500 
 
 10-11 
 
 90,320 
 
 5-6 
 
 92,842 
 
 11-12 
 
 89,919 
 
 »i = 609/720 = 0.84583; 
 Vi = 500/609 = 0.82102; 
 v 3 = 375/500 = 0.75000; 
 Vi = 227/375 = 0.60533; 
 Vi = 107/227 = 0.47137; 
 
 log (0.84583) = 
 log (0.82102) = 
 log (0.75000) = 
 log (0.60533) = 
 log (0.47137) = 
 
 9.927,2831 - 
 9.914,3537 - 
 9.875,0613 - 
 9.781,9922 - 
 9.673,3619 - 
 
 49.172,0522 - 
 
 , „ 49.172,0522 
 
 log G =--= 9.834,4105 - 10 
 
THE GEOMETRIC MEAN 
 
 49 
 
 Find the average rate of decrease for the first 6 months and also for the 
 last 6 months. Discuss the results. 
 
 3. The figures below are the population of whites and negroes of the 
 United States of America from 1900 to 1930, inclusive. 
 
 Year 
 
 No. of Whites 
 
 
 No. of Negroes 
 
 1900 
 
 66,809,196 
 
 
 8,833,994 
 
 1910 
 
 81,364,447 
 
 
 9,827,763 
 
 1920 
 
 94,820,915 
 
 
 10,463,131 
 
 1930 
 
 108,864,207 
 
 
 11,891,143 
 
 Compare the average rate of increase per decade for whites and negroes. 
 
 4. Find the 
 
 average rate of increase 
 
 of the population of Canada. 
 
 The population figures are: 
 
 
 
 Year 
 
 Population 
 
 Year 
 
 Population 
 
 1881 
 
 4,266,041 
 
 1911 
 
 7,169,960 
 
 1891 
 
 4,770,123 
 
 1921 
 
 8,767,206 
 
 1901 
 
 5,322,238 
 
 1931 
 
 10,376,786 
 
 5. Data below relate to exports from the United States before and 
 
 during the depression: 
 
 
 
 Before Depression 
 
 
 During Depression 
 
 Year 
 
 Exports 
 
 Year 
 
 Exports 
 
 1926 
 
 4,808,660,000 
 
 1930 
 
 3,843,181,000 
 
 1927 
 
 4,865,375,000 
 
 1931 
 
 2,424,289,000 
 
 1928 
 
 5,128,356,000 
 
 1932 
 
 1,611,016,000 
 
 1929 
 
 5,240,995,000 
 
 1933 
 
 1,674,975,000 
 
 Compare the average rates of increase or decrease for the two periods. 
 
 6 . The geometric mean of 11 variates is 1.05; find the product of the 
 variates. 
 
 7. The geometric mean of a set of variates is 1.079548, and the product 
 of the variates is 2.1499. Find the number of variates in the set. 
 
 8 . Plot: A = 100(1 + r) 4 . 
 
 9. Plot: A = 100(1 - r)\ 
 
 THE HARMONIC MEAN 
 
 The harmonic mean of a set of variates, Vi, V 2 , vz, . . . , v n , is 
 equal to the number of the variates divided by the sum of the 
 reciprocals of the variates, or 
 
 ( 2 . 12 ) 
 
 H = 
 
 n 
 
 n 
 
 1.1 1 
 
 I —■ + • • . H — 
 
 Vl V2 V n 
 
 where v> ^ 0 
 
 v 
 
50 
 
 STATISTICAL AVERAGES 
 
 If the variates have respectively the frequencies /i, / 2 , . . . 
 then the harmonic mean of the set of variates is 
 
 (2.13) 
 
 2 /.~ _ 2 / 
 
 1>1 l»2 V n V 
 
 The following example illustrates the use of the harmonic mean. 
 
 Mr. Jones buys the articles listed below: 
 
 8 lb. of sugar for $1.00, 
 
 10 “ of rice “ 1.00, 
 
 16 “ of beans “ 1.00, 
 
 6 “ of lard “ 1.00. 
 
 Find the average price per pound and the number of pounds he can buy 
 at this average for $1.00. 
 
 1 lb. of sugar costs $|, 
 
 “ “ “ rice “ $A, 
 
 “ “ “ beans “ 
 
 “ “ “ lard “ $i, 
 
 hence the average price per pound is 
 
 I + 1 X Q + 1^6 + i 
 
 4 
 
 $0.1136. 
 
 The number of pounds that can be bought at this average price is the 
 harmonic mean of the number of pounds purchased, and this is 
 
 f T~ to + Tg 4~ 6 0.1136 
 
 4 
 
 The harmonic mean in the above example gave the number of 
 pounds which could be purchased at the average price for $ 1 . 00 . 
 
 Consider that a man travels in his car for 30 minutes at 40 miles per 
 hour, during the next 20 minutes at 30 miles per hour, and during the 
 following 10 minutes at 20 miles per hour. What was his rate per hour? 
 
 He goes (30/60)40 = 20 mi. during the first 30 min., 
 
 “ “ (20/60)30 = 10 “ “ “ next 20 “ , 
 
 “ “ (10/60)20 = 3.33 “ “ “ “ 10 “ , 
 
PROBLEMS 
 
 51 
 
 hence in the hour he traveled 33.33 miles, and this is his rate per hour. 
 This is a harmonic mean, for his rate is distance divided by time and is 
 
 20 + 10 + 3.33 
 20 10 T33 
 
 40 + 30 + 20 
 
 33.33 mi. per hr. 
 
 The following was observed with regard to mowing a large university 
 campus: 
 
 5 men mowed the campus in 3 days, 
 
 2 CC CC CC CC CC 0 cc 
 
 y cc cc cc cc cc 2 iC 
 
 On the average how long would it take 1 man to mow the campus? 
 
 Each of the first 5 men mowed -rg-th of the campus per day, 
 
 3 “ “ T Vth “ “ 
 
 7 “ “ xjth “ “ 
 
 CC CC 
 CC CC 
 
 CC 
 
 CC 
 
 The average amount of the campus that was mowed per day by 1 man was 
 
 5 I 3 I 7 
 T5 -r T5 -T- li 
 
 15 
 
 0.067th of the campus per day. 
 
 The length of time it would take 1 man on the average to mow the lawn 
 would be the harmonic mean 
 
 15 
 
 'A + Ta + TT 
 
 15 days. 
 
 PROBLEMS 
 
 1. The following is the record of an airplane for 3 hours: 
 
 For the first 80 min. it traveled at the rate of 150 mi. per hr., 
 
 “ . “ next 40 “ “ “ “ “ “ “ 130 “ “ “ , 
 
 cc cc cc 2 Q u (( u it u (( (( 2 QQ u cc cc 
 
 “ “ “ 20 “ “ “ “ “ “ “ 60 “ “ “ ] 
 
 CC 
 
 10 
 
 50 
 
 Find the average speed per hour. 
 
 2 . Mr. Henry purchased the following groceries: 
 
 7 lb. of lard for $1.00, 14 lb. of bread for $1.00, 
 
 4“ “ butter for $1.00, 180 “ “salt “$2.00, 
 
 5 “ “ coffee “ $1.00, 25 “ “ sugar “ $1.25. 
 
 Find the average cost per pound and the average number of pounds at 
 this average that can be purchased for $1.00. 
 
52 
 
 STATISTICAL AVERAGES 
 
 3. If 9 men can do a piece of work in 6 days, 
 
 4 “ “ “ the same piece of work in 10 days, 
 
 g <i u u ii ii ii ii ii it 7 a 
 
 12 “ “ “ “ “ “ “ “ “4 “ . 
 
 Find the time it takes on the average for 5 men to do the piece of work 
 
 and the length of time it takes the average man to do the work. 
 
 4. A store has: 
 
 50 Latin 
 
 books at... 
 
 .$ 2.10 per 
 
 book, 
 
 25 Greek 
 
 (( 
 
 u 
 
 1.85 “ 
 
 ll 
 
 35 German 
 
 U 
 
 u 
 
 1.25 “ 
 
 ll 
 
 17 Spanish 
 
 U 
 
 u 
 
 1.65 “ 
 
 It 
 
 11 English 
 
 u 
 
 ll 
 
 . .95 “ 
 
 ll 
 
 43 French 
 
 u 
 
 ll 
 
 1.30 “ 
 
 It 
 
 27 Russian 
 
 u 
 
 ll 
 
 2.80 “ 
 
 ll 
 
 Find the average price per book. 
 
 5. A machine requiring 3 men to operate makes 50 articles in 40 min., 
 
 Another “ 
 
 ll U 
 
 Find the average time it takes the machines to make the article. 
 
 How many men are required to make 1 article per minute? 
 
 6 . For a set of positive variates prove that 
 
 M v ^ G v ^ II v . 
 
 7. If the harmonic mean of n variates is 240/57 and the sum of the 
 reciprocals of the variates is 57/60, find n. 
 
 8 . If the harmonic mean of 10 items is 25, find the sum of the recipro¬ 
 cals of the variates. 
 
 5 
 
 2 
 
 ll (( 
 
 U (( 
 
 62 
 
 30 
 
 “ 30 
 “ 20 
 
CHAPTER 3 
 
 MEASURES OF DISPERSION 
 MEAN DEVIATION 
 
 Each of two marksmen A and B stands at the same place and 
 shoots 10 shots at a horizontal line CE drawn on the target wall. 
 Figures 3.1 and 3.2 exhibit their records. 
 
 1 t 
 
 
 T r t T 
 
 ill 
 
 1 E c 
 
 ‘ 1 •* 1 l 
 
 Fig. 3.1. — A’s record. Fig. 3.2. — B’s record. 
 
 The average of the distances of the shots from line CE gives a 
 good idea how close the shots came to the line or how far the shots 
 fell from the line. The average of the deviations all taken positive 
 for A is 8.5 mm.; this average for B is 3.9 mm. Clearly B is the 
 better marksman. If the average of the deviations of the shots 
 from CE is small the shots fell close to the line; if this average is 
 large the shots fell farther from the line. Marksmen can be com¬ 
 pared by comparing the averages of the deviations of the shots 
 from the line at which they shoot. 
 
 In a similar way the average of the deviations of variates from 
 the mean of the variates all taken positive shows how near the 
 items are to the mean. If the average of the absolute values 
 (absolute value means the value taken positive) of the deviations 
 of the variates from the mean is large then there is a wide scatter¬ 
 ing of the variates about the mean; if this value is small the vari¬ 
 ates differ very little from the mean and the mean is a better rep¬ 
 resentative of the central tendency of the variates than when the 
 average of the deviations is large. 
 
 The average of the absolute values of the deviations of the 
 
 53 
 
54 
 
 MEASURES OF DISPERSION 
 
 variates v\, V 2 , vz, . . . , v n from the mean of the variates is called 
 the mean deviation of the set of variates and is 
 
 (3.1) 
 
 M.D. 
 
 Vi ~ M v | 
 n 
 
 ) 
 
 where Vi represents the deviation of the zth item from the mean. 
 If the items form a frequency distribution, the mean deviation of 
 the items is 
 
 (3.2) 
 
 M.D. 
 
 Vi - M \ •/(»,-) 
 
 m*) 
 
 s|M-/0h) 
 
 mt 
 
 ; 
 
 =fm 
 
 The following measurements are lengths of feet of men 20 years 
 of age. 
 
 Length 
 of Feet 
 
 
 in Inches 
 
 
 V 
 
 V 
 
 11.8 
 
 +0.04 
 
 11.3 
 
 -0.46 
 
 11.9 
 
 +0.14 
 
 12.4 
 
 +0.64 
 
 111 
 
 -0.66 
 
 12.6 
 
 +0.84 
 
 10.9 
 
 -0.86 
 
 11.5 
 
 -0.26 
 
 12.0 
 
 +0.24 
 
 12.1 
 
 +0.34 
 
 117.6 
 
 0.00 
 
 V 
 
 1\ T TI W 
 
 | 5 | 4.48 
 
 
 10 ~ 10 
 
 
 Length 
 
 
 of Feet 
 
 
 in Inches 
 
 v 2 
 
 V 
 
 0.0016 
 
 11.6 
 
 0.2116 
 
 11.4 
 
 0.0196 
 
 11.5 
 
 0.4096 
 
 11.1 
 
 0.4356 
 
 11.6 
 
 0.7056 
 
 11.3 
 
 0.7396 
 
 12.0 
 
 0.0676 
 
 11.6 
 
 0.0576 
 
 11.5 
 
 0.1156 
 
 11.9 
 
 2.7640 
 
 115.5 
 
 0.448in.; M.D. = - 
 
 V 
 
 v 2 
 
 +0.05 
 
 0.0025 
 
 —0.15 
 
 0.0225 
 
 -0.05 
 
 0.0025 
 
 -0.45 
 
 0.2025 
 
 +0.05 
 
 0.0025 
 
 -0.25 
 
 0.0625 
 
 +0.45 
 
 0.2025 
 
 +0.05 
 
 0.0025 
 
 -0.05 
 
 0.0025 
 
 +0.35 
 
 0.1225 
 
 0.00 
 
 0.6250 
 
 | 1.90 
 
 = 0.19 in. 
 
 10 
 
 Items in the second set of measurements lie closer to their mean 
 than items in the first set to their mean. The mean deviation is 
 a measure of dispersion, for it shows how the items are dispersed 
 or scattered about the mean. A small mean deviation signifies 
 little scattering of the variates about the mean; a large mean 
 deviation denotes extensive scattering of the variates about the 
 mean. 
 
 Consider the distribution of terminal inflorescences of water 
 hemlock (Cicuta maculata). 
 
MEAN DEVIATION 
 
 55 
 
 No. of Terminal No. of 
 
 Inflorescences 
 per Plant v 
 
 Plants 
 
 f(v) 
 
 vf(y) 
 
 V 
 
 1 v 1/00 * 
 
 vf(v) 
 
 3 
 
 8 
 
 24 
 
 -2 
 
 16 
 
 -16 
 
 4 
 
 19 
 
 76 
 
 -1 
 
 19 
 
 -19 
 
 5 
 
 44 
 
 220 
 
 0 
 
 0 
 
 0 
 
 6 
 
 24 
 
 144 
 
 + 1 
 
 24 
 
 24 
 
 7 
 
 4 
 
 28 
 
 +2 
 
 8 
 
 8 
 
 8 
 
 1 
 
 8 
 
 +3 
 
 3 
 
 3 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 
 * 
 
 II 
 
 100 
 
 500 
 
 
 70 
 
 0 
 
 The mean and mean deviation are respectively: 
 
 M v — 5 terminal inflorescences per plant, 
 
 2 | v | f(v) 70 
 
 M.D.„ = — ——— = —— = 0.70 terminal inflorescence. 
 2 f(v) 100 
 
 The mean deviation can be found without finding the deviations 
 of the variates from the mean. If a deviation of a variate Vi from 
 the mean, that is, Vi — M, is negative, then — (v { — M) is positive. 
 According to this the sum of all possible deviations minus the sum 
 of all negative deviations divided by n is M.D., or 
 
 + ~ 
 
 2(®i - M) - 2 (pi - M ) 
 
 (3.3) M.D. = —---—-- , 
 
 n 
 
 + 
 
 where 2 designates the sum of all positive deviations from the mean 
 
 and 2 designates the sum of all negative deviations from the mean. 
 Equation (3.3) may be written as: 
 
 (3.4) M.D. 
 
 + — + — 
 
 2i>,-— n\M — 2i\— 2i> — (jii — 712) • M 
 
 n 
 
 n 
 
 where n\ is the number of variates greater than the mean and ri 2 
 is the number of variates less than the mean. The number n is 
 not always equal to nj + 712 because some of the deviations from 
 the mean may be zero. 
 
 Formula (3.4) enables one to find the mean deviation without 
 finding deviations from the mean, for it is in terms of the original 
 variates, the number ni of variates greater than the mean, n-i the 
 number less than the mean, M the mean, and n the number of 
 
56 
 
 MEASURES OF DISPERSION 
 
 variates. According to the above formula the mean deviation of 
 the first set of feet lengths of page 54 is 
 
 M.D. 
 
 72.8 - 44.8 - (6 - 4)11.76 
 10 
 
 28.0 - 23.52 
 10 
 
 0.448 in. 
 
 When the variates are in a frequency distribution the formula 
 for the mean deviation is 
 
 (3.5) M.D. 
 
 Zvjfivi) ~ - [2/(t>Q - mv,)]-M v 
 
 2 /(») 
 
 + 
 
 where the summation 2 i; ! /(j> 1 ) denotes the sum of all variates 
 
 greater than the mean and 2 vjiy,) denotes the sum of those less 
 than the mean. If a computing machine is available, it is easy 
 to find the mean deviation, for one turns the crank forward for 
 variates greater than the mean and backward for those less than 
 the mean; this gives the difference of the first two terms in (3.4) 
 and (3.5). By noticing the counter in the upper dial of numbers 
 on the machine, the value of the quantity in parentheses or brackets 
 in these formulas is at once obtained. From these values one can 
 readily find M.D. This is by far the easiest method, for it is not 
 necessary to use paper and pencil for any of the computations. 
 
 In all these examples the algebraic sum of the deviations of the 
 items from the mean is zero. This is always true, as the next 
 theorem will show. 
 
 Theorem 3.1. The algebraic sum of the deviations of the 
 variates from the mean of the set of variates is zero. 
 
 Proof: By definition the deviations from the mean are: 
 
 vi = v\ — M v 
 V2 = V2 — M v 
 Vs = 1'3 — M v 
 
 Vn V n •! I V 
 
 Adding: 2w — Zv — nM v - 2r — n(2r/n) = Sv — 2v = 0. 
 
 If one finds M.D. by calculating the deviations from the mean, 
 a check on the computations is that the sum of these deviations 
 
INDIRECT METHOD OF CALCULATING M.D. 
 
 57 
 
 should be zero, or approximately zero, if deviations are correct to 
 certain decimal places. 
 
 INDIRECT METHOD OF CALCULATING M.D. 
 
 Suppose h (= 11.5 inches) is subtracted from each item of the 
 first set on page 54. Results from these subtractions are listed in 
 the next table. Column d contains a set of items much smaller 
 than the original items. It has been shown that the mean of the 
 Fs can be found by using the d’s. It will now be shown that the 
 M.D. of the v’s can be found by using the d’s. Set up the d — Md 
 
 Length 
 
 
 
 
 of Feet 
 
 v — h 
 
 
 
 V 
 
 d 
 
 v — M v 
 
 d - M d 
 
 11.8 
 
 +0.3 
 
 +0.04 
 
 +0.04 
 
 11.3 
 
 -0.2 
 
 -0.46 
 
 -0.46 
 
 11.9 
 
 +0.4 
 
 +0.14 
 
 +0.14 
 
 12.4 
 
 +0.9 
 
 +0.64 
 
 +0.64 
 
 11.1 
 
 -0.4 
 
 -0.66 
 
 -0.66 
 
 12.6 
 
 + 1.1 
 
 +0.84 
 
 +0.84 
 
 10.9 
 
 -0.6 
 
 -0.86 
 
 -0.86 
 
 11.5 
 
 0.0 
 
 -0.26 
 
 -0.26 
 
 12.0 
 
 +0.5 
 
 +0.24 
 
 +0.24 
 
 12.1 
 
 +0.6 
 
 +0.34 
 
 +0.34 
 
 M v = 11.76 in.; M d = 0.26. 
 
 column, or the set of deviations of the d’s from the mean of the 
 d’s. By comparing columns v — M v and d — M d it is seen that 
 the deviations of the v’s from the mean of the v’s are identically 
 equal to the corresponding deviations of the d’s from the mean of 
 the d’s. This example shows that for this case M.D.„ = M.D. d . 
 
 Theorem 3.2. From each variate of a set of variates subtract 
 a constant h, thus forming a new set of variates. The deviation 
 of the fth variate in the original set from the mean of the original 
 variates is equal to the deviation of the fth variate in the new set 
 from the mean of the new set of variates, or 
 
 Vi — M v = di — M d , or i)i = di. 
 
 Proof: By definition 
 
 di = Vi — h. 
 
58 
 
 MEASURES OF DISPERSION 
 
 By formula (2.2), h = M v — M d . Substituting this value for h in 
 the above equation gives 
 
 3. = Vi — M v + M d , or di — M d = v { - M v , or 3, = Vi, 
 which proves the theorem. 
 
 This theorem enables one to set up a new set of variates which 
 are smaller than the original set and enables one to find the mean 
 deviation of the original set by finding the mean deviation of the 
 new set of smaller variates. 
 
 The mean deviation of a grouped distribution is approximately 
 equal to the mean deviation of the distribution made up of class 
 marks. 
 
 PROBLEMS 
 
 1. Find the mean deviation of the following set of items by finding 
 each deviation from the mean. 
 
 No. of Auto Tires 
 Sold per Day 
 
 No. of Auto Tires 
 Sold per Day 
 
 21 
 
 23 
 
 19 
 
 20 
 18 
 
 26 
 
 17 
 
 24 
 
 22 
 
 16 
 
 How many items are within 1 M.D. of the mean? 
 
 2. Find the mean deviation by use of a provisional mean and the 
 number of variates within 1 M.D. of the mean of the distribution of rows 
 of kernels on ears of corn: 
 
 No. of Rows of Kernels 
 on Ears of Corn 
 
 No. of Ears of Corn, 
 Frequency 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 2 
 
 32 
 
 218 
 
 482 
 
 470 
 
 232 
 
 82 
 
 20 
 
 1,538 
 
THE STANDARD DEVIATION 
 
 59 
 
 3 . Find the mean deviation, the percentage of items within 1 M.D. 
 of the mean, and the percentage within 2 M.D. of the mean, of the follow¬ 
 ing distribution: 
 
 No. of Alpha Particles 
 Radiates From a Disk in 
 
 Minute * 
 
 Frequency 
 
 0 
 
 57 
 
 1 
 
 203 
 
 2 
 
 382 
 
 3 
 
 525 
 
 4 
 
 532 
 
 5 
 
 408 
 
 6 
 
 273 
 
 7 
 
 139 
 
 8 
 
 45 
 
 9 
 
 27 
 
 10 
 
 10 
 
 11 
 
 4 
 
 12 
 
 0 
 
 13 
 
 1 
 
 14 
 
 1 
 
 2,607 
 
 THE STANDARD DEVIATION 
 
 Square roots of the averages of the squares of deviations of shots 
 from line CE in Figs. 3.1 and 3.2 also measure the amount of scat¬ 
 tering or dispersion of the shots about CE. The square root of 
 the average of the squares of the deviations for marksman A is 
 about 9.1 mm.; that for marksman B is about 4.4 mm. The larger 
 value indicates the greater scattering of shots about CE and hence 
 reveals the poorer marksman. 
 
 In the same way the square root of the average of the squares of 
 deviations of variates from the mean of the variates is a measure 
 of dispersion or a measure of the scattering of the items about the 
 mean. The smaller this quantity is, the less amount of scatter¬ 
 ing; if there is little scattering then the mean is a good representa¬ 
 tive of the central tendency of the items. 
 
 The standard deviation of a set of variates is the square root of 
 
 * Rutherford and Geiger, “The Probability Variation in the Distribution 
 of Alpha Particles,” Phil. Mag., Scries 6, Vol. 20 (1910), pp. 698-701. 
 
60 
 
 MEASURES OF DISPERSION 
 
 the average of the squares of the deviations of the variates from 
 the mean of the variates, or 
 
 (3-6) 
 
 S.D. 
 
 <Tv 
 
 p - M v 
 n 
 
 ) 2 
 
 where <r r is the symbol for the standard deviation, and v { = v> — M v . 
 
 When variates form a frequency distribution the standard 
 deviation is 
 
 (3.7) 
 
 S.D. = c v 
 
 0 ,- ~ MYfipj) 
 
 2/OO 
 
 Consider again the length of feet given on page 54. The sum of the 
 squares of the deviations of the lengths from the means are given on 
 page 54. From these values the standard deviations are respectively 
 
 <j\ = V 7 0.2764 - 0.53 in, = 0.0625 = 0.25 in. 
 
 The standard deviation of the first set of lengths is 0.53 inch and is larger 
 than the standard deviation of the second set, because there is more 
 scattering of the items in the first set about the mean than there is in the 
 second. The standard deviation is used more often in statistics than 
 any other measure of dispersion. 
 
 The standard deviation is roughly about 125 per cent of the 
 mean deviation. 
 
 Formula (3.6) can be written in terms of the original variates 
 instead of the deviations from the mean as : 
 
 (3.8) 
 
 /Zp - M ) 2 /S 
 
 ^=v —-— =v- 
 
 0 2 -2 v-M + M 2 ) 
 
 n 
 
 = J— - - 2M ' 2 - V + — = J— - 2 -M°- + M 2 
 
 * n n n v n 
 
 » n y n \ n / 
 
 In a similar way formula (3.7) may be written in terms of fre¬ 
 quencies 
 
 _ f 2Q - MYfjv) / Zv 2 f(v) /Siftioy 
 (3.9) a v yj 2/0) V 2f(v) \ Sffo) ) ' 
 
 2/0) V 2/0) / 
 
THE STANDARD DEVIATION 
 
 61 
 
 Formulas (3.8) and (3.9) express the standard deviation in 
 terms of the average of the squares of the original items and the 
 average of the items; hence it is not necessary to determine devia¬ 
 tions from the mean. 
 
 Consider the distribution of petals of buttercups: 
 
 No. of Frequency 
 Petals 
 
 V 
 
 f(v) 
 
 v-f(v) 
 
 v 2 f(v) 
 
 (v + 1 Yf{v) 
 
 5 
 
 40 
 
 200 
 
 1,000 
 
 1,440 
 
 6 
 
 52 
 
 312 
 
 1,872 
 
 2,058 
 
 7 
 
 126 
 
 882 
 
 6,174 
 
 8,064 
 
 8 
 
 165 
 
 1,320 
 
 10,560 
 
 13,385 
 
 9 
 
 204 
 
 1,836 
 
 16,524 
 
 20,400 
 
 10 
 
 215 
 
 2,150 
 
 21,500 
 
 26,015 
 
 11 
 
 177 
 
 1,947 
 
 21,417 
 
 25,488 
 
 12 
 
 104 
 
 1,248 
 
 14,976 
 
 17,576 
 
 13 
 
 35 
 
 455 
 
 5,915 
 
 6,860 
 
 14 
 
 8 
 
 112 
 
 1,568 
 
 1,800 
 
 15 
 
 4 
 
 60 
 
 900 
 
 1,024 
 
 
 1,130 
 
 10,522 
 
 102,406 
 
 124,580 
 
 M v = 9.3115 + petals. 
 
 _ JijgW _ (WWV _ y/ 90.6248 - 86.7040 
 
 N z /(») V z/« ) 
 
 = V3.9208 = 1.9801 petals. 
 
 It is essential to check all computations to avoid errors which 
 might arise. The last column in the above table enables one to 
 check computations. The summation of the last column expanded is 
 
 2(t> + 1 ) 2 f(v) = 2(w 2 + 2v + l)/(t;) = 2 v 2 f(v) + 22 vf(v) + n. 
 
 This means that the sum of the quantities in the last column 
 should be equal to the sum of the fourth column plus twice the 
 
62 
 
 MEASURES OF DISPERSION 
 
 sum of the third column plus the sum of the second. For the 
 above table these values are 
 
 124,580 = 102,406 + 2(10,522) + 1,130 = 124,580, 
 
 which check the computations. 
 
 It is of great importance to know the number of items within 1, 
 2, 3, and 4 standard deviations of the mean. For the distribution 
 of petals these values are as follows: 
 
 Number of items within 1 a of the mean is 761 or 67.3%, 
 
 “ “ “ “ 2<t “ “ “ “ 1,078 or 95.4%, 
 
 “ “ “ “ 3 cr “ “ “ “ 1,130 or 100%. 
 
 Nearly tw T o-thirds of the variates are within 1 a of the mean. All 
 
 the variates are within 3 <r’s of the mean. In many distributions 
 found from observations, about two-thirds of the variates lie 
 within 1 a of the mean; about 95 per cent lie within 2 a’s of the 
 mean, and practically all lie within 3 a’s of the mean. In a large 
 majority of distributions very few variates lie beyond 4 <r’s of the 
 mean. These facts indicate the importance of the mean and 
 standard deviation of a distribution. 
 
 Since the standard deviation is the square root of the average 
 of the squares of deviations of variates from the mean it is the 
 same as the standard deviation of a set of variates formed by sub¬ 
 tracting a provisional mean from each variate in the distribution. 
 See theorem 3.2. 
 
 Formula (3.9) can now be written 
 
 (3.10) 
 
 This will be illustrated by the distribution of petals of buttercups 
 on page 61. Let the provisional mean h = 10 petals. Subtract 10 
 from each variate, and place these values in column 3 of the fol¬ 
 lowing table. 
 
THE STANDARD DEVIATION 
 
 63 
 
 No. OF 
 
 Petals 
 
 f(v) 
 
 d 
 
 dm 
 
 dW) 
 
 5 
 
 40 
 
 -5 
 
 - 200 
 
 1,000 
 
 6 
 
 52 
 
 -4 
 
 - 208 
 
 832 
 
 7 
 
 126 
 
 -3 
 
 - 378 
 
 1,134 
 
 8 
 
 165 
 
 -2 
 
 - 330 
 
 660 
 
 9 
 
 204 
 
 -1 
 
 - 204 
 
 204 
 
 10 
 
 215 
 
 0 
 
 -1,320 
 
 000 
 
 11 
 
 177 
 
 + 1 
 
 177 
 
 177 
 
 12 
 
 104 
 
 2 
 
 208 
 
 416 
 
 13 
 
 35 
 
 3 
 
 105 
 
 315 
 
 14 
 
 8 
 
 4 
 
 32 
 
 128 
 
 15 
 
 4 
 
 5 
 
 20 
 
 100 
 
 
 1,130 
 
 
 + 542 
 
 4,966 
 
 - 778 
 
 (d + W(d) 
 
 4,540 
 
 Md = — 0.6885 _ petal. 
 
 From (3.10) 
 
 c v = v'4.3947 - 0.4740 = 1.9801 petals. 
 
 As a check on computations the sum of the sixth column should 
 be equal to the sum of the fifth column plus twice the sum of the 
 fourth plus the sum of the second column. These values are 
 
 4,540 = 4,966 + 2(-778) + 1,130 = 4,540, 
 
 which check computations in the table. 
 
 Formula (3.8) may now be written as 
 
 (3.11) 
 
 ' n ’ 
 
 = _ (™Y 
 
 * n \n / 
 
 2 (d - M d ) 2 
 
 n 
 
 It is not necessary to write quantities in columns 4, 5, and 6 if 
 one has access to a computing machine, for one can readily sum 
 these columns on the machine by the cumulating process. 
 
 Formulas (3.10) and (3.11) are employed by most workers in 
 statistics because they allow the use of a provisional mean, which 
 enables one to reduce the size of the variates and greatly diminish 
 the amount of computing. 
 
64 
 
 MEASURES OF DISPERSION 
 
 Formulas (3.10) and (3.11) may be written as follows: 
 (3.12) S.D. = a = yi 
 
 
 n 
 
 - ( —V = - Vn2d 2 - (2d) 2 ; 
 
 \ n / n 
 
 (3.13) S. 
 
 2d 2 /(d) 
 
 2d/(d) ] 2 
 
 2 m 
 
 s 
 
 i 
 
 = ~ V2/(d)-2d 2 /(d) - [2d-/(d)] 2 . 
 
 H. C. Carver gave the above formulas for the first time and likes 
 to use them in preference to others. (See “Elementary Mathe¬ 
 matical Statistics,” by H. C. Carver.) A disadvantage of formulas 
 (3.12) and (3.13) is that they involve such large numbers. Accord¬ 
 ing to formula (3.13) the standard deviation of the distribution 
 on page 63 is 
 
 = ^ ^1,130(4,966) - (778) 2 = yi- V5,006,296 
 
 2,237.4754 
 
 1,130 
 
 = 1.9801 petals, as found before. 
 
 MOMENTS OF A DISTRIBUTION 
 
 The nth moment of a frequency distribution about a fixed point* 
 (usually about zero) is equal to the average of the nth powers of 
 the variates measured from this fixed point, or 
 
 (3.14) 
 
 M n-.v 
 
 hv n 2 v n f(v) 
 
 V 2 f{v) ‘ 
 
 The nth moment of a distribution of variates about the mean 
 of the variates is the average of the nth powers of the deviations 
 of the variates from the mean, or 
 
 (3.15) 
 
 S(f - M v y = 2(t> - MMv) = 2j?"/(t>) 
 n 2 f(v) 2 f(v) 
 
 According to these definitions the mean of a set of variates is 
 the first moment, the average of the squares is the second moment, 
 etc. The first two moments in both cases are 
 
 2(r — a) n 
 
 * The moment about a fixed point a is n' n ■ v -- 
 
PROBLEMS 
 
 65 
 
 First moment 
 
 2»/(») , 2 v 2 f{v) 
 
 n'i. v = M v = M = ; second moment / 2 :,= 
 
 2 / 0 ) 
 
 First moment about the mean is 
 
 20 - M)/0) 2D-/0) 
 
 Ml = 
 
 2 / 0 ) ' 
 
 2 / 0 ) 
 
 Second moment about the mean — 
 
 = 0. 
 
 m) 
 
 20 - M) 2 /0) 2D 2 -/0) 
 
 2/0) “ 2/0) 
 
 The second moment about the mean may be written as follows: 
 
 v 0 - M) 2 f(v) 20 2 - 2vAf + M 2 )f(v) 
 
 M2 : p 
 
 2 / 0 ) 
 
 2 / 0 ) 
 
 2» 2 /0) 22f(v)v-M 2/0) -M 2 
 
 ~^ 0 ) 2/(0 + 2 / 0 ) 
 
 = M2:. - 2M 2 + M 2 = m' 2:0 - Tf 2 , 
 
 or 
 
 (3.16) 
 
 M2:< 
 
 t „ = M'2:»-M 2 . 
 
 PROBLEMS 
 
 1. A stem of a certain water plant was broken off and inverted in 
 water. Tiny bubbles passed out of the end that was cut. Layers of 
 waxed paper were held between the sun and the plant. The following 
 table gives the distribution of the number of bubbles per minute. 
 
 Layers 
 
 No. OF 
 
 Layers 
 
 No. OF 
 
 Layers 
 
 No. of 
 
 of 
 
 Bubbles 
 
 of 
 
 Bubbles 
 
 of 
 
 Bubbles 
 
 Wax 
 
 per 
 
 Wax 
 
 per 
 
 Wax 
 
 per 
 
 Paper 
 
 Minute 
 
 Paper 
 
 Minute 
 
 Paper 
 
 Minute 
 
 0 
 
 36 
 
 9 
 
 56 
 
 18 
 
 32 
 
 1 
 
 38 
 
 10 
 
 58 
 
 19 
 
 26 
 
 2 
 
 39 
 
 11 
 
 57 
 
 20 
 
 21 
 
 3 
 
 41 
 
 12 
 
 55 
 
 21 
 
 16 
 
 4 
 
 43 
 
 13 
 
 54 
 
 22 
 
 9 
 
 5 
 
 46 
 
 14 
 
 51 
 
 23 
 
 5 
 
 6 
 
 47 
 
 15 
 
 47 
 
 24 
 
 2 
 
 7 
 
 49 
 
 16 
 
 44 
 
 25 
 
 1 
 
 8 
 
 53 
 
 17 
 
 38 
 
 
 964 
 
66 
 
 MEASURES OF DISPERSION 
 
 Find the number of layers or the light intensity which produced the most 
 bubbles and the standard deviation of the distribution. 
 
 2. Find the number of items within 1, 2, and 3 standard deviations 
 of the mean of the distribution in problem 2 on page 28 and express in 
 percentages. 
 
 3. Find the percentage of the items within 1, 2, and 3 <r’s of the mean 
 of the distribution in problem 1 on page 65. 
 
 4. The standard deviation of a distribution is 1.63 rays, and the mean 
 is 9.93 rays; find the second moment of the distribution. 
 
 5. The second moment of a set of variates is 24.6 square gallons; the 
 standard deviation is 0.47 gallon; find the mean of the distribution. 
 
 6. Prove that 2 (v — M ) 2 = 2 v 2 — (2 v) 2 /n. 
 
 7. The mean and standard deviation of 100 items are respectively 
 80 people and 5 people. If the 100th item, 86 people, is omitted from the 
 data, find the sum of the squares of the 99 items. 
 
 8. Using the data in problem 7 find the standard deviation of the 
 99 items. 
 
 9. If the mean deviation of a set of items is 7.8 balls and the sum of 
 the deviations of the items from the mean taken positive is 3,900 balls, 
 find the number of items. 
 
 THE STANDARD DEVIATION OF GROUPED DATA 
 
 When variates are grouped into classes, individual variates 
 going into a class are represented by the class mark. If the 
 variates are uniformly distributed, this class mark is a good repre¬ 
 sentative of the variates in the class, and the sum of the variates 
 in the class is equal to the class mark times the frequency (number 
 in the class) of the class. The sum of the squares of the variates 
 in a class is not equal to the class mark squared multiplied by the 
 number of variates in the class. Suppose that a class from 6.5 
 
 had the following variates before the 
 
 grouping was made. 
 
 
 V 
 
 f(v) 
 
 »/(») 
 
 v 2 f(v) 
 
 
 ' 7 
 
 1 
 
 7 
 
 49 
 
 Class {6.5 — 9.5} 
 
 8 
 
 2 
 
 16 
 
 128 
 
 
 9 
 
 1 
 
 9 
 
 81 
 
 Class mark 8 
 
 
 4 
 
 32 
 
 258 
 
 The mean of the items in this class is equal to the class mark 8, 
 and the sum of the items in this class, 32, is equal to the class mark, 
 
THE STANDARD DEVIATION OF GROUPED DATA 67 
 
 8, times the frequency 4 of the class; but the sum of the squares, 
 258, is not equal to the class mark squared, times the frequency of 
 the class. In the above class the sum of the squares of the items 
 is 258, while the square of the class mark times the frequency, or 
 number in the class, is 256, a difference of 2 square units. To com¬ 
 pensate for similar differences throughout the distribution for all 
 classes, a correction is made, which enables one to find the standard 
 deviation approximately for the grouped data. In advanced 
 courses in statistics it has been shown that the approximate stand¬ 
 ard deviation for a grouped distribution of discrete variates is 
 
 (3.17) 
 
 ^adj. 
 
 ■W, 
 
 where k is the number of values of the variates that can go into a 
 class, and where discrete variates are variates which differ by whole 
 units.* Things which can be counted are discrete variables, for 
 example, the number of children in a family, the number of petals 
 on flowers, the number of stamps sold by a post office, the number 
 of cents paid employees, etc. In the above class, k = 3, for 7, 8, 
 and 9 are the only discrete variates that can fall in this class, 
 6.5-9.5. In problem 2 on page 35, A; = 2; in problem 5 on page 36, 
 k = 5. 
 
 The following distribution of pedicels per inflorescence of a 
 certain plant will be used to show that it is necessary to make 
 adjustments in order to obtain approximately the correct standard 
 deviation of the distribution of grouped data. The mean and 
 standard deviation of the distribution were obtained before the 
 variates were grouped into classes. For each grouping the mean, 
 unadjusted a and the adjusted a were obtained. The adjusted 
 standard deviations were determined from formula (3.17). The 
 distribution is given below for the data before grouping. 
 
 * °ndj. = adjusted standard deviation. 
 
12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 26 
 
 27 
 
 28 
 
 29 
 
 30 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 37 
 
 38 
 
 39 
 
 40 
 
 J. V 
 
 JST 
 
 62 
 
 87 
 
 59 
 
 47 
 
 52 
 
 26 
 
 33 
 
 37 
 
 24 
 
 37 
 
 18 
 
 20 
 
 17 
 
 16 
 
 7 
 
 8 
 
 8 
 
 6 
 
 6 
 
 4 
 
 3 
 
 2 
 
 2 
 
 6 
 
 3 
 
 1 
 
 2 
 
 1 
 
 MEASURES OF DISPERSION 
 
 Pedicels of Daucus carota (Wild Carrot) 
 
 \ t O. OF 
 
 Pedicels 
 
 per 
 
 iCSTERS 
 
 Cluster 
 
 2 
 
 41 
 
 3 
 
 42 
 
 4 
 
 43 
 
 6 
 
 44 
 
 8 
 
 45 
 
 8 
 
 46 
 
 9 
 
 47 
 
 28 
 
 48 
 
 15 
 
 49 
 
 19 
 
 50 
 
 43 
 
 51 
 
 54 
 
 52 
 
 57 
 
 53 
 
 74 
 
 54 
 
 97 
 
 55 
 
 121 
 
 56 
 
 156 
 
 57 
 
 178 
 
 58 
 
 203 
 
 59 
 
 221 
 
 60 
 
 243 
 
 61 
 
 285 
 
 62 
 
 247 
 
 63 
 
 296 
 
 64 
 
 288 
 
 65 
 
 305 
 
 66 
 
 350 
 
 67 
 
 288 
 
 68 
 
 408 
 
 69 
 
 Pedicels 
 
 No. OF 
 
 PER 
 
 Clusters 
 
 Cluster 
 
 386 
 
 70 
 
 380 
 
 71 
 
 360 
 
 72 
 
 380 
 
 73 
 
 349 
 
 74 
 
 367 
 
 75 
 
 332 
 
 76 
 
 355 
 
 77 
 
 390 
 
 78 
 
 340 
 
 79 
 
 423 
 
 80 
 
 360 
 
 81 
 
 367 
 
 82 
 
 362 
 
 83 
 
 325 
 
 84 
 
 309 
 
 85 
 
 275 
 
 86 
 
 236 
 
 87 
 
 258 
 
 88 
 
 216 
 
 89 
 
 211 
 
 90 
 
 188 
 
 91 
 
 169 
 
 92 
 
 141 
 
 93 
 
 113 
 
 94 
 
 128 
 
 95 
 
 129 
 
 96 
 
 115 
 
 97 
 
 126 
 
 
THE STANDARD DEVIATION OF GROUPED DATA 69 
 
 Correct values for M and a are those that were found before 
 grouping; these values are listed in the first line of the table below. 
 Results for various ways of grouping are given also in the table. 
 The first group in every case had 11.5 as lower limit. 
 
 Groups of 
 
 Means 
 
 Unadjusted a 
 
 Adjusted a 
 
 1 
 
 47.3060 
 
 12.6711 
 
 12.6711 
 
 2 
 
 47.2984 
 
 12.6755 
 
 12.6656 
 
 3 
 
 47.3104 
 
 12.6972 
 
 12.6710 
 
 4 
 
 47.2972 
 
 12.7164 
 
 12.6671 
 
 5 
 
 47.2977 
 
 12.7473 
 
 12.6686 
 
 6 
 
 47.3003 
 
 12.8006 
 
 12.6860 
 
 7 
 
 47.3121 
 
 12.8212 
 
 12.6644 
 
 8 
 
 47.3044 
 
 12.9018 
 
 12.6967 
 
 9 
 
 47.3559 
 
 12.9137 
 
 12.6529 
 
 10 
 
 47.2555 
 
 12.9107 
 
 12.6268 
 
 11 
 
 47.2380 
 
 13.1401 
 
 12.6538 
 
 12 
 
 47.3041 
 
 13.0615 
 
 12.5971 
 
 In all these 
 
 cases the means 
 
 are about the 
 
 same as the correct 
 
 mean, showing that a correction is not needed for the mean when 
 data are grouped. All the means are nearly equal to the true 
 mean, 47.3060 pedicels. Sometimes these means are a little larger 
 than the true mean and sometimes a little smaller. The mean of 
 the distribution for groups of 3 is 47.3104 pedicels, and the mean 
 for the distribution for groups of 8 is 47.3044 pedicels. For this 
 example it is not necessary to make adjustments for the arithmetic 
 average when the data are grouped. It is shown in more advanced 
 work in statistics that it is not necessary to make corrections on 
 the arithmetic mean when the items are grouped into classes.* 
 
 This is not true of standard deviations. The unadjusted stand¬ 
 ard deviations get larger as classes are made larger, which is seen 
 for the example under consideration by examining the third column 
 in the above table. The unadjusted a arising from groupings of 3 
 is smaller than that obtained from groupings of 8 and 9, etc. To 
 bring the unadjusted standard deviations nearer to the true value, 
 adjustments are necessary. These adjusted or corrected c’s are 
 found in the fourth column of the last table and were determined 
 by using formula (3.17). In all cases adjusted o-’s are about equal 
 
 * H. C. Carver, “Editorial,” The Annals of Mathematics, Vol. I, 1930, 
 page 111. 
 
70 
 
 MEASURES OF DISPERSION 
 
 to the correct a = 12.6711 pedicels, which is the a obtained before 
 the data were grouped. 
 
 This illustration clearly shows, for this example, that it is nec¬ 
 essary to make adjustments in order to reduce the standard devia¬ 
 tion of the distribution of class marks of the grouped data. The 
 mean and a obtained from the distribution of class marks may 
 never be equal to the true mean and standard deviation; they are 
 considered to be approximately equal to them. The differences 
 between the true mean and standard deviation and those obtained 
 from the distribution of class marks, after a has been adjusted, are 
 very small; the values of M and a obtained from the grouped data, 
 after a has been adjusted, are used as estimates of the true values. 
 Formula (3.17) is generally used when distributions have high 
 contact at both ends of the range, that is, distributions which rise 
 gradually from the base line at both ends of the range and have 
 one maximum. Details will now be given for the above distribu¬ 
 tion for groups of eights. 
 
 No. OF 
 
 No. OF 
 
 
 
 
 Pedicels 
 
 Pedicels 
 
 
 
 
 per 
 
 per 
 
 
 
 
 Cluster 
 
 Cluster 
 
 
 
 
 Open 
 
 Closed 
 
 Class 
 
 Fre- 
 
 
 Limits 
 
 Limits 
 
 Marks 
 
 QUENCY 
 
 d 
 
 12-19 
 
 11.5-19.5 
 
 15.5 
 
 88 
 
 -5 
 
 20-27 
 
 19.5-27.5 
 
 23.5 
 
 480 
 
 -4 
 
 28-35 
 
 27.5-35.5 
 
 31.5 
 
 1,829 
 
 -3 
 
 36-43 
 
 35.5-43.5 
 
 39.5 
 
 2,865 
 
 -2 
 
 44-51 
 
 43.5-51.5 
 
 47.5 
 
 2,936 
 
 -1 
 
 52-59 
 
 51.5-59.5 
 
 55.5 
 
 2,492 
 
 0 
 
 60-67 
 
 59.5-67.5 
 
 63.5 
 
 1,295 
 
 + 1 
 
 68-75 
 
 67.5-75.5 
 
 71.5 
 
 574 
 
 2 
 
 76-83 
 
 75.5-83.5 
 
 79.5 
 
 202 
 
 3 
 
 84-91 
 
 83.5-91.5 
 
 87.5 
 
 44 
 
 4 
 
 92-99 
 
 91.5-99.5 
 
 95.5 
 
 15 
 
 5 
 
 
 
 
 12,800 
 
 
 df d'f 
 -440 2,200 
 
 -13,113 46,725 
 
 M d = — 13,113/12,800 = 1.02445“ class units. 
 M v = 55.5 + (-1.02445)8 = 47.3044 pedicels. 
 
THE STANDARD DEVIATION OF GROUPED DATA 71 
 
 The unadjusted standard deviation is 
 
 a = (V M ' 2;d - M d 2 )8 = (V3.65039 - 1.04950)-8 
 = 12.9018 pedicels. 
 
 The adjusted standard deviation is 
 
 1 - 
 
 O'adj. — 
 
 M 2 : d ~ M d 2 - 
 
 k 2 
 
 12 
 
 ■w 
 
 = \ 3.65039 - 1.04950 
 
 
 = V2.51886-8 = 12.6967 pedicels. 
 
 The correction is made on the quantity under the square-root 
 
 sign. 
 
 Continuous Variates 
 
 When distributions are composed of continuous variates, that 
 is, variates which may differ by infinitesimal amounts, quantity k 
 which represents the number of variates which can be put into a 
 class is infinite, as there can be an infinite number of values between 
 the upper and lower class limits. For example, if weights of men 
 are grouped into classes of 10 pounds, the class 89.5-99.5 might 
 have any weight between these two values, and surely this is 
 infinite in number. In the case of continuous variates formula 
 
 (3.17) becomes 
 
 (3.18) cr adJ = (a/ h' 2 : a — Md 2 — yj) • w , 
 since now 1/k 2 = 0. 
 
 The following example will illustrate steps in finding <r for a grouped 
 set of continuous variates. 
 
72 
 
 MEASURES OF DISPERSION 
 
 Weights of Freshmen at the University of Michigan. 
 Measurements Made to the Nearest Pound 
 Class 
 
 Classes Mark Frequency d df(d ) d 2 f (d+l)' : f 
 
 89.5- 99.5 
 
 94.5 
 
 1 
 
 99.5-109.5 
 
 104.5 
 
 19 
 
 109.5-119.5 
 
 114.5 
 
 80 
 
 119.5-129.5 
 
 124.5 
 
 179 
 
 129.5-139.5 
 
 134.5 
 
 260 
 
 139.5-149.5 
 
 144.5 
 
 210 
 
 149.5-159.5 
 
 154.5 
 
 140 
 
 159.5-169.5 
 
 164.5 
 
 62 
 
 169.5-179.5 
 
 174.5 
 
 28 
 
 179.5-189.5 
 
 184.5 
 
 11 
 
 189.5-199.5 
 
 194.5 
 
 4 
 
 199.5-209.5 
 
 204.5 
 
 4 
 
 209.5-219.5 
 
 214.5 
 
 2 
 
 1,000 
 
 Md = — 0.489 class unit. 
 
 M v = 144.5 1b. + (-0.489)10 lb. 
 
 —5 
 -4 
 -3 
 -2 
 -1 
 0 
 
 + 1 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 7 
 
 -489 3,183 2,205 
 
 139.61 lb. 
 
 By formula (3 .18) the adjusted standard deviation is 
 
 cr a dj. = V3.183 -0.23912 -0.08333 • 10 = (V2.86055-)10 
 
 = 16.9132 lb. 
 
 PROBLEMS 
 
 1. Find the number of items within 1, 2, and 3 cr’s of the mean of the 
 following distribution of ray flowers per head of oxeye daisies taken from 
 a plot near Ann Arbor, Michigan. Plot the data, and show the frequen¬ 
 cies in a histogram. 
 
 o 
 
 o 
 
 ”3 
 
 Frequency 
 
 No. OF 
 
 Frequency 
 
 Ray Flowers 
 
 
 Ray Flowers 
 
 
 per Head 
 
 
 per Head 
 
 
 11-12 
 
 3 
 
 21-22 
 
 391 
 
 13-14 
 
 428 
 
 23-24 
 
 34 
 
 15-16 
 
 722 
 
 25-26 
 
 9 
 
 17-18 
 
 741 
 
 27-28 
 
 4 
 
 19-20 
 
 667 
 
 29-30 
 
 1 
 
 3,000 
 
STANDARD DEVIATION OF THE COMBINATION OF SETS 73 
 
 2. Construct a histogram of the following distribution of the pulse 
 beats per minute in English convicts. (See Biometrika, Vol. II, page 21.) 
 
 Pulse Beats 
 
 No. OF 
 
 Pulse Beats 
 
 No. OF 
 
 per Minute 
 
 Convicts 
 
 per Minute 
 
 Convicts 
 
 45-48 
 
 2 
 
 81- 84 
 
 86 
 
 49-52 
 
 5 
 
 85- 88 
 
 62 
 
 53-56 
 
 17 
 
 89- 92 
 
 42 
 
 57-60 
 
 57 
 
 93- 96 
 
 15 
 
 61-64 
 
 90 
 
 97-100 
 
 18 
 
 65-68 
 
 150 
 
 101-104 
 
 9 
 
 69-72 
 
 120 
 
 105-108 
 
 5 
 
 73-76 
 
 131 
 
 109-112 
 
 3 
 
 77-80 
 
 109 
 
 113-117 
 
 3 
 
 924 
 
 On the graph show the percentage of items within 1, 2, 
 
 and 3 standard 
 
 deviations of the mean. 
 
 
 
 
 3. Make the classes twice as large 
 
 in the above problem, and compare 
 
 the mean and standard deviation with those found before 
 
 
 4. The next table contains lung capacities of women 
 
 students at the 
 
 University of Michigan. 
 
 Capacities given to the nearest cubic inch. 
 
 No. of Frequency 
 
 No. OF 
 
 Frequency 
 
 Cubic Inches 
 
 
 Cubic Inches 
 
 
 79.5- 99.5 
 
 2 
 
 179.5-199.5 
 
 114 
 
 99.5-119.5 
 
 14 
 
 199.5-219.5 
 
 53 
 
 119.5-139.5 
 
 78 
 
 219.5-239.5 
 
 30 
 
 139.5-159.5 
 
 122 
 
 239 . 5-259.5 
 
 9 
 
 159.5-179.5 
 
 158 
 
 
 580 
 
 Find the percentage of items within 1, 2, 3, and 4 standard deviations of 
 the mean. If a student is picked at random from this group, what is the 
 probability that her lung capacity will be between 1 and 2 c’s to the right 
 of the mean? 
 
 5. If M v and M w are the means of 2 sets of positive variates, prove 
 that the mean of the items in both sets lies between the two means if 
 M v 7*^ M w . 
 
 THE STANDARD DEVIATION OF THE COMBINATION OF SETS 
 
 The mean and standard deviation of 100 test scores are respec¬ 
 tively M v = 74.3 and a u = 4; the mean and standard deviation of 
 
74 
 
 MEASURES OF DISPERSION 
 
 80 other scores on the same test are respectively M w = 75.2 and 
 <j u . = 3.6. Let it be required to find the mean and standard devia¬ 
 tion of the 180 scores. The mean of the entire set of scores accord¬ 
 ing to formula (2.3) is 
 
 M v 
 
 + W 
 
 100(74.3) + 80(75.2) 
 180 
 
 74.70. 
 
 To find the standard deviation one must use a formula similar 
 to (3.8) for the combined sets of scores. One must find the fol¬ 
 lowing 
 
 (3.19) 
 
 &v + w 
 
 = 4 
 
 Sum of squares of all scores 
 
 180 
 
 - {M v+W y. 
 
 It is necessary to find the sum of the squares of all the scores. 
 According to formula (3.8) 
 
 j) v 2 -y v 2 
 
 <j „ 2 = —- M v 2 , or - —- = o\, 2 + M v 2 , or 
 
 n n 
 
 (3.20) 2t’ 2 = nW + M 2 ). 
 
 In a similar manner the sum of the square of the scores of the w 
 variates is 
 
 (3.21) 2tc 2 = m{(j w 2 + M w 2 ), 
 
 where m is the number of variates. According to the last two 
 formulas the sum of the squares for the first set of 100 scores is 
 2h’ 2 = 100(4 2 + 74.3 2 ), and the sum of the squares of the 80 test 
 scores is 2ic 2 = 80(3.6 2 + 75.2 2 ). Substituting these in formula 
 (3.19) gives _ 
 
 7v+w "Y 
 
 100 (4 2 + 84.3 2 ) + 80 (3.6 2 + 75.2 2 ) 
 
 180 
 
 - (74.7) 2 
 
 -4 
 
 1,007,089 
 180 
 
 - 5,580.09 = a/ 14.85 = 3.854. 
 
 Formulas (3.20) and (3.21) enable one to obtain the general 
 formula for determining the standard deviation for the combina¬ 
 tion of two sets of data in terms of means, standard deviations, 
 
STANDARD DEVIATION OF THE COMBINATION OF SETS 75 
 
 and numbers of items in the two sets. The formula for the stand¬ 
 ard deviation of the items in both sets is 
 
 (3.22) 
 
 @v -f- ^ 
 
 -4 
 
 ni( a„ 2 + M v 2 ) + n 2 (o' i« 2 + M w 2 ) 
 ni + n 2 
 
 ( M v+v ) 2 , 
 
 where n\ and n 2 are the number of items in the sets respectively. 
 
 This formula can be extended to the standard deviation for the 
 combination of the items in several sets. This formula is 
 
 (3.23) 
 
 p l + °2+ • • • + d 8 
 
 4 
 
 Wi(cri,| 2 + M V1 2 ) + ity)+ . . . + n s (<r Vs 2 + M v 2 ) 
 
 n\ + n 2 + • • • + n s 
 
 (3/pj-j- P2 -r... Vg) 
 
 where w* represents the number of items in the fth set of variates, 
 M V{ and a Vi are respectively the mean and standard deviation of 
 the ith set, and M n+V2+ ... +„ s is the mean of the items in all sets. 
 
 Suppose that the mean and standard deviation of 20 items are respec¬ 
 tively 21 inches and 2 inches, and that the 20th item is 24 inches. Let it 
 be required to find the mean and standard deviation after the 20th item 
 has been omitted. The mean of the 19 items is 
 
 20(21) - 24 
 
 M i 9 = -= 396/19 = 20.84 in. 
 
 19 
 
 The following formula for the standard deviation of a difference can be 
 derived like (3.22). 
 
 (3.24) 
 
 n i(a v 2 
 
 &v —w \l 
 
 T M v 2 ) — Ti2((T w - T M w 2 ) 
 
 n i — n 2 
 
 - (AT,_)*- 
 
 According to (3.24) the standard deviation of the 19 items is 
 
 I 
 
 a 19 — AT 
 
 20(2 2 + 2 1 2 ) - 1(0* + 24 2 ) 
 19 
 
 - (20.84) 2 
 
 = V438.ll - 434.31 = VT80 = 1.95“ in. 
 
 since the standard deviation of one item is zero. 
 
 Formula (3.24) is very important, for many times it becomes 
 necessary to eliminate items after the mean and standard deviation 
 have been found. Observations which differ from the mean by 
 more than 4 standard deviations are rare and do not really belong 
 to the set. Sometimes items which arc more than 3 cr’s from the 
 
76 
 
 MEASURES OF DISPERSION 
 
 mean are omitted. If for any reason variates are found to be 
 inconsistent with the other items in the set they are omitted from 
 the data. Many times one does not know they are inconsistent 
 until compared with the M v and a v . The following example illus¬ 
 trates this point. 
 
 Thirteen students were requested to measure the length of a 
 blackboard 10 times and to find the mean and standard deviation 
 of their measurements. The 13 means and <r’s were used to find 
 the mean and a of the 130 measurements. After this was found, 
 the mean of one student’s measurements was 9 standard deviations 
 from the mean of all measurements. After his measurements were 
 omitted his mean was off more than 9 cr’s. On talking to the 
 student it was discovered that he had measured the top of the 
 board 5 times and the bottom of the board 5 times. It was dis¬ 
 covered that the top of the board was f inches longer than the 
 bottom, which was measured by the other students. Formula 
 (3.23) was used for finding the a for the combined set of items; 
 formula (3.24) was used to eliminate the inconsistent measurements 
 of the student who measured the top and bottom of the blackboard. 
 
 PROBLEMS 
 
 1. The means and standard deviations of heights of men in five 
 dormitories are as follows: 
 
 Dormitory 
 
 Mean 
 
 S. D. 
 
 No. of Men 
 
 A 
 
 67.6 
 
 2.6 
 
 65 
 
 B 
 
 67.3 
 
 2.4 
 
 47 
 
 C 
 
 68.1 
 
 2.7 
 
 34 
 
 D 
 
 66.8 
 
 2.5 
 
 149 
 
 E 
 
 65.4 
 
 3.2 
 
 178 
 
 
 
 
 473 
 
 Find the mean and standard deviation of the heights of all men in the 
 five dormitories. 
 
 2. In problem 1, Dormitory E has 27 Japanese students. The mean 
 and standard deviation of these heights are respectively 64.1 inches and 
 2.3 inches. Find the mean and standard deviation of the heights in 
 dormitory E after heights of the Japanese have been omitted. 
 
 3. If a v and a w are the standard deviations of two sets and a„ > a w , 
 prove that <t v + w > 
 
QUARTILES 
 
 77 
 
 4. The following data give means and standard deviations of scores 
 made on a certain test for seventh-grade students in wards of a certain 
 city. 
 
 Ward 
 
 Mean 
 
 S. D. 
 
 No. of Students 
 
 1 
 
 69.8 
 
 4.2 
 
 42 
 
 2 
 
 70.4 
 
 4.7 
 
 65 
 
 3 
 
 73.4 
 
 4.4 
 
 24 
 
 4 
 
 71.6 
 
 3.9 
 
 54 
 
 5 
 
 74.3 
 
 4.0 
 
 93 
 
 6 
 
 72.0 
 
 5.1 
 
 17 
 
 These results 
 
 were turned in at the superintendent’ 
 
 s office. Find the 
 
 mean and standard deviation for the entire city. Discuss the records 
 
 for each ward 
 
 in comparison with results for the city. 
 
 
 5. The following data pertain to weights of freshmen in three large 
 
 universities. 
 
 
 
 No. OF 
 
 University Mean 
 
 S.D. 
 
 Freshmen 
 
 1 
 
 138.6 lb. 
 
 16.7 lb. 
 
 907 
 
 2 
 
 139.2 “ 
 
 16.4 “ 
 
 1,102 
 
 3 
 
 138.9 “ 
 
 17.3 “ 
 
 841 
 
 
 
 
 2,850 
 
 Find the average weight of freshmen and the standard deviation of all 
 these weights. 
 
 6. The mean and standard deviation of 83 measurements are respec¬ 
 tively 17.3 gallons and 1.3 gallons. It is found that one of the measure¬ 
 ments, 22.3 gallons, was incorrectly made. Omit this item and find M 
 and a for the others. 
 
 QUARTILES 
 
 Quartiles of a distribution of variates are quantities which 
 divide the distribution into 4 equal parts. The median is some¬ 
 times spoken of as the second quartile, Q 2 . The first or lower 
 quartile of a grouped frequency distribution is found by a formula 
 similar to that for the median. This formula for the first or lower 
 quartile is 
 
 n 
 
 4 
 
 ni 
 
 (3.25) 
 
 Qi — Ci -f- 
 
 h 
 
 w, 
 
78 
 
 MEASURES OF DISPERSION 
 
 where Ci is the lower limit of the class in which the (n/4)th item 
 falls when the items are arranged in order; n\ is the sum of the 
 frequencies of the classes below Ci;/i is the frequency of the class 
 in which the (n/4)th item falls; n is the sum of the frequencies in 
 all classes; and w is the width of the class in which the (n/4)th 
 item falls. 
 
 The third quartile is found by the formula 
 
 (7- 
 
 (3.26) Q 3 = C 3 + 
 
 where C 3 , fz, and w are respectively the lower limit, the frequency 
 and the width of the class in which the (fn)th item falls, 713 is the 
 sum of all frequencies below this class, and n is the total frequency. 
 
 One-half of the items of a frequency distribution falls between 
 the first and third quartiles. 
 
 Another measure of dispersion is 
 
 (3.27) 
 
 Q = 
 
 Q:i — Qi 
 
 2 
 
 
 and is called the quartile deviation or the semi-interquartile range. 
 
 The following example will illustrate the steps in finding the semi- 
 interquartile range. 
 
 Lung Capacity 
 
 No. OF 
 
 of Women 
 
 Women 
 
 79.5- 99.5 
 
 2 
 
 99.5-119.5 
 
 14 
 
 119.5-139.5 
 
 78 
 
 139.5-159.5 
 
 122 
 
 159.5-179.5 
 
 158 
 
 179.5-199.5 
 
 114 
 
 199.5-219.5 
 
 53 
 
 219.5-239.5 
 
 30 
 
 239.5-259.5 
 
 9 
 
 
 580 
 
 Qi = 139.5 + 
 Q 3 = 179.5 + 
 
 ( 580 ^ 2 ~ 94 ) 20 
 / 3 • 580/4 -374 \ 
 
 V 114 ) 
 
 = 139.5 + 8.36 = 147.86 cu. in. 
 
 20 = 179.5 + 10.70 = 190.20 cu. in. 
 
THE COEFFICIENT OF VARIATION 
 
 79 
 
 from which 
 
 Q = 
 
 Q 3 ~ Q i 
 
 2 
 
 190.20 - 147.86 
 2 
 
 21.17 cu. in. 
 
 Fifty per cent of the distribution lies between Qz and Qi, and 
 about 50 per cent lies within a distance of Q from the median. 
 For symmetrical distributions there is exactly 25 per cent of the 
 distribution within a distance of Q to the right of the median and 
 25 per cent within a distance of Q to the left. The above distri- 
 
 50 % 
 
 /-*-s 
 
 Fig. 3.3. — Distribution of lung capacities of women students, 
 showing the quartiles. 
 
 bution is exhibited in Fig. 3.3 by a histogram, together with the 
 quartiles. 
 
 THE COEFFICIENT OF VARIATION 
 
 The coefficient of variation is equal to the ratio of the standard 
 deviation to the mean multiplied by 100 and is 
 
 (3.28) 
 
 100cr„ 
 
 This is an abstract number and shows the variability of the items 
 in the distribution. It is a very good quantity for comparing two 
 distributions. If the mean and standard deviations of one set of 
 variates are respectively 37 inches and 2 inches, and the mean 
 and standard deviations of another set of variates are respectively 
 
80 
 
 MEASURES OF DISPERSION 
 
 37 inches and 3.2 inches, the coefficient of variation enables one to 
 compare the variabilities of the two sets. These coefficients are 
 
 Vi = 100(2/37) = 5.4 and V 2 = 8.6, 
 
 which show that there is much more variability in the second set 
 than in the first.* 
 
 If standard deviations of two distributions are equal and the 
 means are not the same, the coefficients of variation may not give 
 any information concerning the variabilities of the distributions. 
 Let both <r’s be equal to 10 gallons. Mi = 40 gallons, and Mo = 60 
 gallons. The coefficients of variation are, respectively, Vi = 25, 
 Vo = 17. These values would indicate that the first was more 
 variable or showed more variability than the second. There is no 
 difference between the amount of variability of the two distribu¬ 
 tions. The main use of V is for comparison of distributions which 
 have means which are about the same and for comparisons of dis¬ 
 tribution which are expressed in different units. Heights of girls 
 6 years of age and of girls 19 years furnish an example of this point. 
 The mean and standard deviations of heights of 6-year-old girls are 
 respectively 45.7 inches and 2 inches, while these for 19-year-old 
 girls are respectively 62 inches and 2 inches. One cannot say that 
 the heights of the 6-year-old girls are more variable than the 
 heights of 19-year-old girls because V is greater in the former. 
 The percentages of items within 1, 2, and 3 o-’s of the mean indicate 
 how the items are scattered about the means and would show 
 variability much better than V, for this situation. The quantity 
 V is useful, but care should be taken when interpreting results. 
 
 The coefficient of variation is the percentage the standard devi¬ 
 ation is of the mean. 
 
 PROBLEMS 
 
 1. If the first quartile is 142 and the semi-interquartile is 18, what is 
 the third quartile? 
 
 2 . The coefficient of variation is 40 and the mean is 30; find <r. 
 
 3. The following frequency distributions pertain to weights of light 
 and heavy men. Find the coefficients of variation for the two distribu¬ 
 tions and discuss the results. 
 
 For test of significance see chapter on Standard Errors of Statistics. 
 
PROBLEMS 
 
 81 
 
 Weights of Men No. of Men Weights of Men No. of Men 
 
 89.5- 99.5 
 
 2 
 
 199.5-209.5 
 
 8 
 
 99.5-109.5 
 
 11 
 
 209.5-219.5 
 
 30 
 
 109.5-119.5 
 
 36 
 
 219.5-229.5 
 
 57 
 
 119.5-129.5 
 
 71 
 
 229.5-239.5 
 
 69 
 
 129.5-139.5 
 
 96 
 
 239.5-249.5 
 
 74 
 
 139.5-149.5 
 
 49 
 
 249.5-259.5 
 
 39 
 
 149.5-159.5 
 
 21 
 
 259.5-269.5 
 
 17 
 
 159.5-169.5 
 
 9 
 
 269.5-279.5 
 
 8 
 
 169.5-179.5 
 
 3 
 
 279.5-289.5 
 
 3 
 
 179.5-189.5 
 
 1 
 
 289.5-299.5 
 
 2 
 
 — 299.5-309.5 _L 
 
 308 
 
 4. Find the semi-interquartile range for the scores made at a certain 
 
 university by students in the mathematics department. 
 
 Scores No. of Students Scores No. 
 
 of Students 
 
 Under 50 
 
 55 
 
 75-79 
 
 129 
 
 50-54 
 
 31 
 
 80-84 
 
 95 
 
 55-59 
 
 49 
 
 85-89 
 
 78 
 
 60-64 
 
 73 
 
 90-94 
 
 20 
 
 65-69 
 
 102 
 
 95 and above 
 
 9 
 
 70-74 144 
 
 How many items are within 1 semi- 
 
 785 
 
 •interquartile range of the median? 
 
 5. The following distributions 
 
 of ligulate flowers of 
 
 oxeye daisies 
 
 were taken respectively at the beginning and end of the flowering season. 
 
 Compare the coefficient of variation for the two samples. 
 
 No. of Ligulate No. of Ligulate 
 
 Flowers per Head No. of Flowers per Head 
 
 No. OF 
 
 First of Season 
 
 Heads 
 
 Last of Season 
 
 Heads 
 
 11-12 
 
 0 
 
 11-12 
 
 3 
 
 13-14 
 
 131 
 
 13-14 
 
 139 
 
 15-16 
 
 242 
 
 15-16 
 
 241 
 
 17-18 
 
 238 
 
 17-18 
 
 240 
 
 19-20 
 
 229 
 
 19-20 
 
 232 
 
 21-22 
 
 146 
 
 21-22 
 
 128 
 
 23-24 
 
 10 
 
 23-24 
 
 17 
 
 25-26 
 
 3 
 
 25-26 
 
 0 
 
 27-28 
 
 0 
 
 27-28 
 
 0 
 
 29-30 
 
 1 
 
 29-30 
 
 0 
 
 1,000 
 
 1,000 
 
82 
 
 MEASURES OF DISPERSION 
 
 6. Use an ogive to find the quartiles of the first distribution in problem 
 No. 3. 
 
 7. Set up formulas for finding the percentiles of a frequency distri¬ 
 bution. 
 
 8. Use these formulas for finding the percentiles of the distribution 
 in No. 4. 
 
 9 . Set up formulas for finding the deciles of a frequency distribution. 
 
CHAPTER 4 
 
 THE NORMAL CURVE 
 
 PROPERTIES OF THE NORMAL CURVE 
 
 One of the most important frequency distributions in statistics 
 is the normal distribution. Since it plays such an important role 
 it is essential to know some of its properties. The equation of the 
 normal curve is 
 
 -i _ (V-M r)2 
 
 (4.1) Y = - 7 =—e~ 2ff * 2 , 
 
 V 2 7r a v 
 
 where e = 2.718 and -k = 3.1416, these being approximate values 
 of 7r and e. If M v — 0 and a v = 1, the above equation becomes 
 
 (4.2) Y = —y= e -’ 2/2 = 0.3989 e ~ cV2 . 
 
 v 2ir 
 
 Several tables have been constructed for values of e +s and 
 
 _ V 1 
 
 e~ s for various values of s. The table below contains values of e 2 
 and of Y for various values of v. Values of Y are plotted in Fig. 4.1. 
 
 V 
 
 e -’ 2/2 
 
 Y 
 
 V 
 
 e -1,2/2 
 
 Y 
 
 -3.0 
 
 0.0111 
 
 0.0044 
 
 +0.2 
 
 0.9802 
 
 0.3910 
 
 -2.5 
 
 0.2437 
 
 0.0175 
 
 +0.5 
 
 0.8781 
 
 0.3521 
 
 -2.0 
 
 0.1353 
 
 0.0540 
 
 + 1.0 
 
 0.6065 
 
 0.2420 
 
 -1.5 
 
 0.3230 
 
 0.1295 
 
 + 1.5 
 
 0.3230 
 
 0.1295 
 
 -1.0 
 
 0.6065 
 
 0.2420 
 
 +2.0 
 
 0.1353 
 
 0.0540 
 
 -0.5 
 
 0.8781 
 
 0.3521 
 
 +2.5 
 
 0.0437 
 
 0.0175 
 
 -0.2 
 
 0.9802 
 
 0.3910 
 
 +3.0 
 
 0.0111 
 
 0.0044 
 
 On examining the equation of the curve in Fig. 4.1 it is seen 
 that the curve approaches the horizontal axis very rapidly to the 
 left of —3 and to the right of +3, that the curve is symmetrical 
 and has one maximum, which is at the origin. The area under 
 the curve above the horizontal axis is unity; the area within 1 unit 
 of the origin is about 68 per cent of the total area, or the area 
 within 1 standard deviation (cr = 1 here) of the origin or the mean 
 
 83 
 
84 
 
 THE NORMAL CURVE 
 
 is about 68 per cent of the entire area. About 95 per cent of the 
 area is within 2 units or standard deviations of the origin; about 
 99 per cent of the area is within 3 a’s of the origin and about 
 99.999 per cent of the area is within 4 units or 4 cr’s of the origin, 
 which is the mean of the distribution. The curve has inflection 
 points at +1 and —1. One half of the area is within 0.6745 unit 
 of the origin or the mean. The curve decreases very rapidly to 
 
 y 
 
 Fig. 4.1. 
 
 the right of +0.5 and to the left of —0.5. The slope of the curve 
 to the left of —3 and to the right of +3 is nearly zero. 
 
 The ogive of this normal curve is plotted in Fig. 4.2 and shows 
 the amount of area under the curve in Fig. 4.1 from — oo to various 
 values of v. The ogive starts at zero at — oo and rises very slowly 
 until v = — 2 is reached, where it rises more rapidly until v = + 2 
 is reached; beyond +2 the ogive gradually approaches the line 
 Y = 1. At + oo it is equal to +1. At the origin the height of 
 the ogive above the horizontal axis is +0.5, as it should be, since 
 the area under the curve in Fig. 4.1 from — oo to 0 is one-half of the 
 entire area under the curve. The line AB in Fig. 4.2 gives the 
 area under the curve in Fig. 4.1 to the left of ab, while the length 
 of the line STF in Fig. 4.2 gives the area under the curve in Fig. 4.1 
 to the left of sw. 
 
 Table I gives values of the ogive curve for the normal curve 
 for various values of t, where the equation of the normal curve is 
 
 1 - 
 
 various values of t. 
 
 Table II gives ordinates of the normal curve for 
 
PROPERTIES OF THE NORMAL CURVE 
 
 85 
 
 Let it be required to find the area under the normal curve 
 between 0.3 and 0.9. From the ogive curve in Table I the area 
 to the left of 0.3 under the normal curve is 0.6179, while the area 
 to the left of 0.9 and under the normal curve is 0.8159; hence the 
 area between 0.3 and 0.9 is 0.8159 — 0.6179 = 0.1980. The area 
 between —0.67 and +0.67 is equal to 0.7486 — 0.2514 = 0.4972, 
 which is nearly one-half of the area under the curve Y. 
 
 Y=1 
 
 Fig. 4.2. — Ogive of the normal curve. 
 
 Suppose that a distribution of variates is normal with a = 1 
 and that there are 20,000 variates in the distribution. Let it 
 be required to find the number between 0.4 and 0.8 unit to the 
 right of the mean of the distribution. Imagine that the origin 
 of a normal distribution coincides with the mean of the given 
 distribution. The area between 0.4 and 0.8 from Table I is 
 0.7881 — 0.6554 = 0.1327, which is 13.27% of the entire area; 
 hence the number of variates of the 20,000 between these limits 
 is 20,000(0.1327) = 2,654. The number between —2 and —1.6 
 is equal to 20,000(0.03205) = 641. Since areas under the normal 
 curve from — oo to t are given in percentages it is easy to find the 
 number of variates in any normal distribution between two limits 
 or within any class. 
 
 Many teachers believe that grades for a large number of stu¬ 
 dents are normally distributed. Assuming this to be true, let it 
 be required to find the percentages of A’s, B’s, C’s, D’s, and E’s. 
 Assume that the normal curve extends for practical purposes from 
 — 3 to +3, or extends over a range of 6 units. Since there are 
 
86 
 
 THE NORMAL CURVE 
 
 5 grade groups each will cover 6/5 units on the horizontal axis. 
 The values of these limits for the grade groups in terms of the V s 
 
 are -3, -1.8, -0.6, +0.6, +1.8, +3. 
 
 By utilizing Table I the following percentages are found for these 
 grades: 
 
 Percentage of A’s = 3.5, percentage of B’s = 23.8, 
 
 “ “ C’s = 45.1, “ “ D’s = 23.8, 
 
 “ “ E’s = 3.5, “ beyond —3 and +3 = 0.3. 
 
 These are plotted in Fig. 4.3. 
 
 Grade Curve 
 
 Fig. 4.3.—Percentages of grades on the assumption that the grade curve 
 
 is normal. 
 
 GRADUATION 
 
 In general there are very few distributions with unity as stand¬ 
 ard deviation. To be able to use Tables I and II it is necessary 
 to change the original units into standard units or to change v’s 
 into V s. Tables I and II are in terms of the standard unit t. To 
 change from any unit to standard units it is necessary to use t 
 equal to 
 
 (4.3) 
 
 Vi 
 
 (7y 
 
 The mean of the V s is zero, as can be seen by finding the average 
 of the V s from (4.3); the standard deviation of the t's is unity, as 
 can be seen by finding the average of the squares of the t’s. The 
 normal curve in terms of t is 
 
 1 
 
 y/ 2 -kcti 
 
 (t- w 1 
 
 1 
 
 \^2tt 
 
 ft 
 
 > 
 
 e 
 
GRADUATION 
 
 87 
 
 since M t = 0 and a t = 1. The following illustration wall show how 
 a distribution can be compared with a normal distribution, or how 
 a distribution can be graduated by a normal curve. The first two 
 columns in the following table contain the distribution which is 
 to be graduated. To be able to compare frequencies of this set 
 of variates with those of the normal, it is necessary to change the 
 units from inches to standard units by use of formula (4.3). 
 
 Heights of Adult Males Born in England, Ireland, Scotland 
 
 and Wales 
 
 (The Anthropometric Committee to the British Association, Report 
 18,893, p. 256. Original measurements made to the nearest ■§• inch.) 
 
 Classes 56^ to 57*, 57* to 58*, etc. 
 
 
 
 
 
 Differ- 
 
 Expected 
 
 
 Fre- 
 
 
 Area 
 
 ence 
 
 Fre- 
 
 Heights 
 
 QUENCY 
 
 t 
 
 Below t 
 
 of Areas 
 
 QUENCIES 
 
 57* 
 
 2 
 
 -4.09 
 
 0.00002 
 
 0.00009 
 
 1 
 
 58* 
 
 4 
 
 -3.70 
 
 .00011 
 
 .00036 
 
 3 
 
 59* 
 
 14 
 
 -3.31 
 
 .00047 
 
 .00128 
 
 11 
 
 60* 
 
 41 
 
 -2.92 
 
 .00175 
 
 .00379 
 
 33 
 
 61* 
 
 83 
 
 -2.54 
 
 .00554 
 
 .01024 
 
 88 
 
 62* 
 
 169 
 
 -2.15 
 
 .01578 
 
 .02342 
 
 201 
 
 63* 
 
 394 
 
 -1.76 
 
 .03920 
 
 .04614 
 
 396 
 
 64* 
 
 669 
 
 -1.37 
 
 .08534 
 
 .07820 
 
 671 
 
 65* 
 
 990 
 
 - .98 
 
 .16354 
 
 .11405 
 
 979 
 
 66* 
 
 1,223 
 
 - .59 
 
 .27759 
 
 .14315 
 
 1,229 
 
 67* 
 
 1,329 
 
 - .20 
 
 .42074 
 
 .15460 
 
 1,327 
 
 68* 
 
 1,230 
 
 + .19 
 
 .57534 
 
 .14360 
 
 1,234 
 
 69* 
 
 1,063 
 
 + .58 
 
 .71904 
 
 .11243 
 
 965 
 
 70* 
 
 646 
 
 + .96 
 
 .83147 
 
 .08002 
 
 687 
 
 71* 
 
 392 
 
 + 1.35 
 
 .91149 
 
 .04758 
 
 409 
 
 72* 
 
 202 
 
 + 1.74 
 
 .95907 
 
 .02434 
 
 209 
 
 73* 
 
 79 
 
 2.13 
 
 .98341 
 
 .01072 
 
 92 
 
 74* 
 
 32 
 
 2.52 
 
 .99413 
 
 .00406 
 
 35 
 
 75* 
 
 16 
 
 2.91 
 
 .99819 
 
 .00132 
 
 11 
 
 76* 
 
 5 
 
 3.30 
 
 .99951 
 
 .00038 
 
 3 
 
 77* 
 
 2 
 
 3.69 
 
 .99989 
 
 .00009 
 
 1 
 
 78* 
 
 0 
 
 4.07 
 
 .99998 
 
 .000016 
 
 0 
 
 79* 
 
 0 
 
 4.46 
 
 .999996 
 
 
 
 8,585 8,585 
 
 M v = 67.4584 in.; <r v = 2.5723 in. 
 
88 
 
 THE NORMAL CURVE 
 
 The t’s which correspond to the v’s were found in the above 
 table by formula (4.3) after the mean and standard deviation were 
 obtained. The t corresponding to the lower limit of the first 
 class is 
 
 vi - M v _ 56.9375 - 67.4584 
 c v 2.5723 
 
 1 
 
 2.5723 
 
 (-10.5209) = - 4.09. 
 
 The value of t corresponding to the lower limit of the second class is 
 
 t‘2 = 
 
 57.9375 - 67.4584 
 2.5723 
 
 (-9.5209) =- 3.70, 
 
 2.5723 
 
 where the quantity in the last parentheses is greater than the 
 quantity in the parentheses above by 1 unit. If one has access to 
 a computing machine the t’s can be obtained by turning the crank 
 handle of the machine backward once for each t, since the quan¬ 
 tities in the parentheses for successive t’s differ by unity. These 
 t values are listed in column 3 in the above table. 
 
 The fourth column in the above table contains areas under this 
 normal curve to the left of the various values of t; these values 
 were taken from Table I. The first value in column 4 is the area 
 under the normal curve to the left of t = — 4.09; the next value is 
 the area to the left of —3.70, etc. Quantities in column 5 are 
 areas under this normal curve between successive values of t. The 
 fourth quantity in this column is the area between t = — 2.54 and 
 t = — 2.92, and is found by subtracting the two quantities 0.00175 
 and 0.00554 in column 4. Quantities in column 5 are found by 
 subtracting successive quantities in column 4. Quantities in col¬ 
 umn 5 are the percentages of frequencies for classes of a distribu¬ 
 tion which is normal. Expected frequencies for classes of the dis¬ 
 tribution of 8,585 heights are found by multiplying the percentages 
 for the various classes by 8,585. These frequencies, called expected 
 frequencies, are found in the last column in the table. Values in 
 the last column are sometimes called graduated values of the origi¬ 
 nal frequencies. The sum of the expected frequencies is 8,585, 
 which is the sum of all frequencies in the original distribution. 
 The original distribution is not a normal distribution but approxi¬ 
 mates one. 
 
 The percentage of heights within 1 a of the mean is found by 
 adding the frequencies between —1 and +1 and is about 68 per 
 cent of the total distribution. 
 
PROBLEMS 
 
 89 
 
 PROBLEMS 
 
 1. Graduate the following distribution by the normal distribution,* 
 if the mean = 7.0055 and the standard deviation = 0.121. 
 
 Classes 
 Closed Limits 
 
 Frequency 
 
 Classes 
 Closed Limits 
 
 Frequency 
 
 6.625-6.675 
 
 0 
 
 7.025-7.075 
 
 55 
 
 6.675-6.725 
 
 2 
 
 7.075-7.125 
 
 50 
 
 6.725-6.775 
 
 12 
 
 7.125-7.175 
 
 40 
 
 6.775-6.825 
 
 20 
 
 7.175-7.225 
 
 15 
 
 6.825-6.875 
 
 22 
 
 7.225-7.275 
 
 11 
 
 6.875-6.925 
 
 46 
 
 7.275-7.325 
 
 3 
 
 6.925-6.975 
 
 57 
 
 7.325-7.375 
 
 1 
 
 6.975-7.025 
 
 66 
 
 
 400 
 
 2. By using percentages of grades on page 86 find the number of 
 students making A’s, B’s, C’s, D’s, and E’s respectively if there are 300 
 students. 
 
 3. Assume that the range of the normal curve for practical purposes 
 is 7 instead of 6; find the percentages of A’s, B’s, C’s, D’s, and E’s for 
 the grade curve if it is normal. 
 
 4. Assume that the grade curve is normal and that each group is 
 divided into three equal parts, C —, C, C+, B —, B, B+, etc. Find the 
 percentages of the different grades. Assume that the range is from 
 -3 to +3. 
 
 5. The following scores were made on an intelligence test. Assume 
 that these scores are distributed normally; find the expected frequencies 
 for the various score groups. What can be said about the test as measur¬ 
 ing intelligence? 
 
 Scores 
 
 Frequency 
 
 Scores 
 
 Frequency 
 
 34.5-39.5 
 
 3 
 
 69.5-74.5 
 
 109 
 
 39.5-44.5 
 
 17 
 
 74.5-79.5 
 
 78 
 
 44.5-49.5 
 
 28 
 
 79.5-84.5 
 
 46 
 
 49.5-54.5 
 
 49 
 
 84.5-89.5 
 
 31 
 
 54.5-59.5 
 
 74 
 
 89.5-94.5 
 
 15 
 
 59.5-64.5 
 
 113 
 
 94.5-99.5 
 
 5 
 
 64.5-69.5 
 
 124 
 
 
 
 6. Plot the normal curve from values given in Table II. 
 
 * “An Application of Thiele’s Semi-Invariants to the Sampling Problem” 
 by C. C. Craig, Melron , Vol. 7, No. 4, 1928, pages 3-75. 
 
90 
 
 THE NORMAL CURVE 
 
 7. If ti = 2.4, M v = 89.4 feet, and a v = 3.7 feet, find the v that corre¬ 
 sponds to this t. 
 
 8. Plot the ogive of the normal curve from values given in Table I. 
 
 9. If ti = —2.3, Vi = 200 gallons, and <r v = 7.2 gallons, find M v . 
 
 10. If t = —0.72, v = 34.6 cc., M v = 37.8 cc., find a v . 
 
 ORDINATES OF THE NORMAL CURVE 
 
 The histogram of the 
 
 following normal distribution is plotted 
 
 Fig. 4.4. 
 
 
 TABLE 
 
 4.1 
 
 
 V 
 
 f(v) 
 
 V 
 
 /(») 
 
 
 0 
 
 1 
 
 8 
 
 175 
 
 M v - 7.0, 
 
 1 
 
 2 
 
 9 
 
 121 
 
 <j v — 2.0. 
 
 2 
 
 9 
 
 10 
 
 66 
 
 
 3 
 
 28 
 
 11 
 
 28 
 
 
 4 
 
 66 
 
 12 
 
 9 
 
 
 5 
 
 121 
 
 13 
 
 2 
 
 
 6 
 
 175 
 
 14 
 
 1 
 
 
 7 
 
 196 
 
 
 1,000 
 
 
 Let it be required to i 
 
 construct 
 
 a histogram of this distribution 
 
 that the 
 
 variable shall be 
 
 
 
 Vi J\I V 
 ti — , 
 
 Cv 
 
 instead of v it and so that the sum of the frequencies shall be unity 
 instead of 1,000. This shall be done in two steps which shall be 
 described in detail. 
 
 The t variates according to the above value of t run from t\ = 
 (0 — 7)/2 to f 15 = (14 — 7)/2 for class marks. 
 
 Let it first be required to construct a histogram to have the 
 same frequencies as the histogram of data in Table 4.1 but with 
 t as the variable instead of v. From the definition of t it is seen 
 that the bases of the rectangles of the required histogram will be 
 equal to the bases of the histogram in Fig. 4.4 except they will be 
 divided by a = 2. Since the bases of rectangles in the required 
 histogram are equal to 1/tr = \ the altitudes of these rectangles 
 must be equal to <rf(v) so as to contain the same areas as the rect¬ 
 angles in Fig. 4.4, or so as to have the same corresponding fre¬ 
 quencies. The new histogram possesses rectangles with bases 
 
ORDINATES OF THE NORMAL CURVE 
 
 91 
 
 equal to one-half of the base of the rectangles in Fig. 4.4, and 
 altitudes twice as high as those in Fig. 4.4. The area of rectangle 
 APRT in Fig. 4.4 is equal to 1 X 66 = 66, while the area of 
 the corresponding rectangle aprt in Fig. 4.5, which is the first 
 required histogram, is (1/2) X 132 = 66. Since the bases in 
 
 Fig. 4.4. — Histogram of the distribution in Table 4.1. The/(v) distribution. 
 
 Fig. 4.5.—Histogram of the distribution given in the first two columns of 
 Table 4.2. The F(t) distribution. 
 
 Fig. 4.5 are one-half of the size of the bases in Fig. 4.4, the altitude 
 of the former must be twice as great as those in the latter, in order 
 to have corresponding areas the same. Mid-points of rectangles 
 of Fig. 4.5 are given in Table 4.2 in the t column. Altitudes are 
 given in the column headed by F{t) = ov/(«0, where F(t) repre¬ 
 sent the altitudes of the first required histogram. Frequencies of 
 the F(t) distribution or histogram are listed in column 3; they are 
 equal to the respective areas of the rectangles. 
 
92 THE NORMAL CURVE 
 
 
 
 TABLE 4.2 
 
 
 
 
 
 Area of 
 
 Area of 
 
 Ordinates or 
 
 
 
 Rectangles Rectangles 
 
 Altitudes of 
 
 
 
 of F(t) Curve of 
 
 fi(t ) Curve 
 
 Rectangles 
 of fiit) Curve 
 
 t 
 
 F{t) = <rf(v) 
 
 a 
 
 1 o/OO 
 a N 
 
 o/O) 
 
 N 
 
 -3.5 
 
 2 
 
 i 
 
 0.001 
 
 0.002 
 
 -3.0 
 
 4 
 
 2 
 
 .002 
 
 .004 
 
 -2.5 
 
 ■ 18 
 
 9 
 
 .009 
 
 .018 
 
 -2.0 
 
 56 
 
 28 
 
 .028 
 
 .056 
 
 -1.5 
 
 132 
 
 66 
 
 .066 
 
 .132 
 
 -1.0 
 
 242 
 
 121 
 
 .121 
 
 .242 
 
 — 0.5 
 
 350 
 
 175 
 
 . 175 
 
 .350 
 
 0.0 
 
 392 
 
 196 
 
 .196 
 
 .392 
 
 +0.5 
 
 350 
 
 175 
 
 . 175 
 
 .350 
 
 + 1.0 
 
 242 
 
 121 
 
 .121 
 
 .242 
 
 1 .5 
 
 132 
 
 66 
 
 .066 
 
 .132 
 
 2.0 
 
 56 
 
 28 
 
 .028 
 
 .056 
 
 2.5 
 
 18 
 
 9 
 
 .009 
 
 .018 
 
 3.0 
 
 4 
 
 2 
 
 .002 
 
 .004 
 
 3.5 
 
 2 
 
 1 
 
 .001 
 
 .002 
 
 Let it 
 
 now be required to construct 
 
 a histogram which shall 
 
 have bases of the same size as those in the histogram for F(t ) = <rf(v), 
 
 Fig. 4.6. — Histogram of the distribution given in the first and last columns 
 of Table 4.2. The /+) distribution. 
 
ORDINATES OF THE NORMAL CURVE 
 
 93 
 
 as shown in Fig. 4.5, and with rectangles the sum of whose areas 
 shall equal unity. To do this, each altitude of the rectangles in 
 Fig. 4.5 must be divided by IV = 1,000; this makes the altitudes 
 
 0 -/ 0 ) 
 
 of the rectangles in the second required histogram equal to 
 
 N 
 
 The second required histogram is shown in Fig. 4.6; the bases of 
 the rectangles of this histogram are equal to 1/a, and the altitudes 
 
 are equal to hit) = - , which gives area equal to (1 /a)[af(y)/N] = 
 
 f{v)/N. The areas of the rectangles in Fig. 4.6 are the respective 
 areas in Fig. 4.4 expressed as percentages. Now we have a histo¬ 
 gram in Fig. 4.6 whose bases are equal to the bases of the original 
 histogram in Fig. 4.4 divided by a, and whose total frequency is 
 unity. These changes are shown in the above table. 
 
 The above method of going from the f(v) distribution to the 
 flit) distribution was given to show the relation between/(v) and 
 flit), which shows the nature of Table II. Table II contains ordi¬ 
 nates of the/i(i) curve. One can go directly from f(v) tofi(t) by 
 the formula 
 
 (4.4) 
 
 because 
 
 
 hit) 
 
 Fit) afjv) 
 N N ' 
 
 One can go directly from f\(t) to f(v ) by the formula 
 
 (4.5) 
 
 because 
 
 N 
 
 fiv) = -hit), 
 
 a 
 
 m - - m = -mo. 
 
 a a 
 
 Formula (4.5) is of great importance in graduating distribu¬ 
 tions by ordinates of the normal curve given in Table II. Of 
 course, this is for curves which are approximately normal, or for 
 comparing distributions with normal distributions. An example 
 will be given to illustrate the method of graduating distributions 
 by ordinates of the normal curve. 
 
 The function /i(<) is always known, for its ordinates are given 
 
94 
 
 THE NORMAL CURVE 
 
 in Table II and they can be used to graduate any distribution 
 since they are given in percentages. For any particular distribu¬ 
 tion the V s are found from the mean and standard deviation. 
 From values of the t's one finds the ordinates of the desired curve, 
 or the expected frequencies, on the assumption that it is normally 
 distributed. From formula (4.5) the expected values of the v curve 
 are obtained. These are the values of f(v) on the assumption that 
 it is normal. 
 
 The following table contains the distribution of the number of 
 questions answered on an intelligence test by 10,000 students. 
 
 No. OF 
 
 Questions 
 Answered = v 
 
 Fre¬ 
 
 quency 
 
 t 
 
 m 
 
 Expected 
 
 Frequency 
 
 36 
 
 7 
 
 -3.50 
 
 0.00087 
 
 4 
 
 37 
 
 18 
 
 -3.00 
 
 .00443 
 
 22 
 
 38 
 
 80 
 
 -2.50 
 
 .01753 
 
 87 
 
 39 
 
 260 
 
 -2.01 
 
 .05292 
 
 264 
 
 40 
 
 640 
 
 -1.51 
 
 .12758 
 
 635 
 
 41 
 
 1,230 
 
 -1.01 
 
 .23955 
 
 1,193 
 
 42 
 
 1,740 
 
 -0.51 
 
 .35029 
 
 1,745 
 
 43 
 
 1,980 
 
 -0.01 
 
 .39892 
 
 1,987 
 
 44 
 
 1,760 
 
 +0.48 
 
 .35553 
 
 1,771 
 
 45 
 
 1,200 
 
 +0.98 
 
 .24681 
 
 1,229 
 
 46 
 
 680 
 
 1.48 
 
 .13344 
 
 665 
 
 47 
 
 280 
 
 1.98 
 
 .05618 
 
 280 
 
 48 
 
 94 
 
 2.48 
 
 .01842 
 
 92 
 
 49 
 
 22 
 
 2.98 
 
 .00471 
 
 23 
 
 50 
 
 9 
 
 3.47 
 
 .00097 
 
 5 
 
 10,000 10,002* 
 
 M v = 43.0268 questions, 
 36 - 43.0268 
 
 <x v = 2.0077 questions. 
 
 t\ = 
 
 2.0077 
 
 = — 3.50, <2 = 
 
 37 - 43.0268 
 2.0077 
 
 = — 3.00, etc. 
 
 The fourth column in the above table was found from Table II 
 for respective values of t\ these are the ordinates of the normal 
 curve. The fifth column was found from formula (4.5). The first 
 number in this column was found as follows: 
 
 f(vi) = (0.00087) = 4980.82 (0.00087) = 4 students. 
 
 * Because of rounding off decimals. 
 
PROBLEMS 
 
 95 
 
 The fifth number in the last column was found as follows: 
 
 f(v 5 ) = 4980.82(0.12758) = 635 students. 
 
 The other expected frequencies were found similarly. 
 
 PROBLEMS 
 
 1. Graduate the following distribution by use of ordinates of the 
 
 normal curve. 
 
 
 
 
 Time of Running 
 
 No. OF 
 
 Time of Running 
 
 No. OF 
 
 1 Mile 
 
 Days 
 
 1 Mile 
 
 Days 
 
 For 1 Track Man 
 
 
 for 1 Track Man 
 
 
 4 min. 18 sec. 
 
 1 
 
 4 min. 24 sec. 
 
 17 
 
 4 “ 19 “ 
 
 3 
 
 4 “ 25 “ 
 
 13 
 
 4 “ 20 “ 
 
 7 
 
 4 “ 26 “ 
 
 6 
 
 4 “ 21 “ 
 
 12 
 
 4 “ 27 “ 
 
 2 
 
 4 “ 22 “ 
 
 18 
 
 4 “ 28 “ 
 
 1 
 
 4 “ 23 “ 
 
 20 
 
 
 
 2. A distribution is normal with mean = 140.4 pounds and standard 
 deviation is 16.6 pounds. If there are 347 items between 154 pounds and 
 162 pounds, including 154 and 162, find the number of items, N, in the 
 entire distribution. Original measurements made to the nearest \ pound. 
 
 3. In a normal distribution with mean = 37.4 inches and a = 1.8 
 inches the number of items with measurement 35 inches is 509. Find 
 the total number of items in the entire distribution. 
 
 4 . In a normal distribution with mean = 16 grams and a = 1.3 grams, 
 the number of items between 13.6 and 14.2, including 13.6 and 14.2, 
 together with those between 17.3 and 18.6, including 17.3 and 18.6, is 804. 
 Find the number of items in the whole distribution if the original measure¬ 
 ments were made to the nearest 0.1 gram. 
 
 5. In a distribution with M v = 99.7 feet and cr v = 5.4 feet, there are 
 478 items from 106 to some point h not including h. If the distribution is 
 normal, find the value of h, provided there are 2,468 items in the entire 
 distribution, and the original measurements are made to the nearest unit. 
 
 6. A drygoods store finds that the sale of ladies’ hose approximates 
 a normal distribution. Find the number of each size it should have on 
 hand to supply 20,000 ladies if sizes range from 7, 7§, 8, 8§ . . .,11. 
 Assume that the range of the normal curve is from —3.6 to +3.6. 
 
CHAPTER 5 
 
 SKEWNESS AND KURTOSIS 
 SKEWNESS 
 
 In an earlier chapter it was shown that the sum of the devia¬ 
 tions of the variates from the mean was equal to zero, the average 
 of the absolute values of these deviations was equal to the mean 
 deviation, and the square root of the average of the squares of these 
 deviations was defined as the standard deviation. For normal dis¬ 
 tributions and approximately normal distributions there is about 
 68 per cent of the variates within 1 a of the mean, about 95 per 
 cent within 2 o-’s of the mean, and about 99.9 per cent within 3 o-’s. 
 Very few r variates are beyond 3 standard deviations of the mean. 
 It w r as also pointed out that the inflection points of the normal 
 curve w'ere located 1 a to the right of the mean and 1 a to the left. 
 The square root of the average of the squares of the deviations 
 from the mean proved to be a very important characteristic of 
 frequency distributions; it will prove of more importance as a 
 measure of dispersion in material which follows. 
 
 This chapter presents the importance of the averages of the 
 cubes of the deviations of the variates from the mean and also the 
 value of the average of the fourth pow r ers of these deviations. 
 
 Let us consider again the normal distribution given on page 90, 
 which is listed in the next table and graphically shown in Fig. 4.4. 
 
 Deviations of the variates from the real mean are listed in the 
 third column of the above table. Column 4 contains the cubes of 
 these deviations multiplied by their respective frequencies. Since 
 the distribution is symmetrical with respect to the mean, the cubes 
 of the negative deviations are equal to the corresponding cubes of 
 the positive deviations; hence the average of the cubes of these 
 deviations is zero. This is always true of symmetrical frequency 
 distributions and, of course, of normal distributions. 
 
 Figure 5.0 presents graphically the above normal distribution. 
 It has one maximum point and contains as much area to the left 
 
 96 
 
SKEWNESS 
 
 97 
 
 V 
 
 }{v) 
 
 I 
 
 s 
 
 II 
 
 v 3 -J(v) 
 
 v 4 -f(v) 
 
 0 
 
 1 
 
 -7 
 
 - 343 
 
 2,401 
 
 1 
 
 2 
 
 -6 
 
 - 432 
 
 2,592 
 
 2 
 
 9 
 
 -5 
 
 -1,125 
 
 5,625 
 
 3 
 
 28 
 
 -4 
 
 -1,792 
 
 7,168 
 
 4 
 
 66 
 
 -3 
 
 -1,782 
 
 5,346 
 
 5 
 
 121 
 
 -2 
 
 - 968 
 
 1,936 
 
 6 
 
 175 
 
 -1 
 
 - 175 
 
 175 
 
 7 
 
 196 
 
 0 
 
 -6,617 
 
 000 
 
 8 
 
 175 
 
 + 1 
 
 + 175 
 
 175 
 
 9 
 
 121 
 
 +2 
 
 + 968 
 
 1,936 
 
 10 
 
 66 
 
 +3 
 
 + 1,782 
 
 5,346 
 
 11 
 
 28 
 
 4 
 
 1,792 
 
 7,168 
 
 12 
 
 9 
 
 5 
 
 1,125 
 
 5,625 
 
 13 
 
 2 
 
 6 
 
 432 
 
 2,592 
 
 14 
 
 1 
 
 7 
 
 343 
 
 2,401 
 
 
 1,000 
 
 
 + 6,617 
 
 50,486 
 
 
 
 Total 
 
 0,000 
 
 
 mean as 
 
 to the right. The 
 
 curve rises gradually at 3 <r’s to 
 
 the left of the mean, rises rapidly between — (3/2)<r and —(1/2 )a, 
 and reaches its maximum at the origin, in the neighborhood of 
 
 196 
 
 - 3 d - 2 d -Id Id 2 d 3 d 
 
 Fig. 5.0. — Histogram and frequency curve of a normal distribution. 
 
 which it has very little slope. To the right of the mean it does 
 the same things but in reverse order, which makes it a symmetrical 
 curve. It is a curve with zero skewness. 
 
 If the average of the cubes of the deviations of the variates from 
 
98 
 
 SKEWNESS AND KURTOSIS 
 
 the mean is not equal to zero the distribution is not normal or sym¬ 
 metrical; hence this average of the cubes indicates whether or not 
 a frequency distribution is asymmetrical or skew. Various mea¬ 
 sures of skewness have been given by other authors. A measure for 
 skewness, or for the amount of departure from symmetry, which is 
 being adopted by a large number of statisticians, is the average of 
 the cubes of the deviations of the variates from the mean divided 
 by the cube of the standard deviation, or 
 
 (5.1) 
 
 M3 : v 
 
 “3- = TT- 
 
 In other words, skewness is the third moment about the mean 
 divided by the cube of the standard deviation. Dividing by a 3 
 makes the skewness of a distribution an abstract quantity, which 
 does not depend upon the unit of the original variates. 
 
 If the sum of the positive deviations of the variates from the 
 mean is greater than the sum of the negative deviations the skew¬ 
 ness is positive, and vice versa. 
 
 Table 5.1 contains askew frequency distribution and computa¬ 
 tions necessary for finding skewness. 
 
 TABLE 5.1 
 
 Number of Gallons of Gasoline Used on a Certain Route by a Bus 
 
 No. of Gallons 
 
 
 
 
 V 
 
 f(v) 
 
 V 
 
 v°-m 
 
 v 3 f(v) 
 
 35 
 
 5 
 
 -4 
 
 80 
 
 - 320 
 
 36 
 
 42 
 
 -3 
 
 378 
 
 -1,134 
 
 37 
 
 145 
 
 -2 
 
 580 
 
 -1,160 
 
 38 
 
 249 
 
 -1 
 
 249 
 
 - 249 
 
 39 
 
 196 
 
 0 
 
 000 
 
 -2,863 
 
 40 
 
 178 
 
 + 1 
 
 178 
 
 + 178 
 
 41 
 
 101 
 
 2 
 
 404 
 
 + 808 
 
 42 
 
 51 
 
 3 
 
 459 
 
 1,377 
 
 43 
 
 19 
 
 4 
 
 304 
 
 1,216 
 
 44 
 
 9 
 
 5 
 
 225 
 
 1,125 
 
 45 
 
 4 
 
 6 
 
 144 
 
 864 
 
 46 
 
 1 
 
 7 
 
 49 
 
 343 
 
 
 1,000 
 
 
 3,050 
 
 +5,911 
 
 +3,048 
 
SKEWNESS 
 
 99 
 
 n v = 39 gallons, 
 
 H2-.v — 3.05; 
 
 <7 v 
 
 M 3:* = 3.048; 
 
 <*3:v 
 
 1.746 gallons; 
 
 M 3 :v 
 
 3.048 
 
 5.325 
 
 0.5724. 
 
 The distribution in Table 5.1 is plotted in Fig. 5.1 with the plot 
 of a normal distribution. Compare the part of the curve A to the 
 right of the mean with the part to the left. Notice that the maxi¬ 
 mum is to the left of the mean. 
 
 Fig. 5.1. — A distribution with positive skewness versus a normal distribution; 
 
 Fig. 5.2. — A frequency curve with negative skewness versus a normal 
 
 frequency curve. 
 
 Figure 5.2 enables one to compare a normal frequency curve 
 with a frequency curve possessing negative skewness. The skew 
 curve has its maximum to the right of the mean and the tail to the 
 left. Other situations may arise. 
 
 Skewness is often interpreted as the amount of deviation from 
 the normal curve, since the skewness of the normal curve is zero. 
 In reality, skewness measures the amount of asymmetry or the 
 
100 
 
 SKEWNESS AND KURTOSIS 
 
 amount of deviation from symmetry. Distributions of heights of 
 people at various ages possess very little skewness; distributions 
 of weights for various ages possess much skewness. Tails of weight 
 distributions contain measurements for heavy people. Often dis¬ 
 tributions pertaining to number of seeds possess positive skewness 
 because of the many more pods with large numbers of seeds than 
 with small numbers. Frequency distributions with long tails to 
 the left or to the right of the mean differ markedly from normal 
 distributions. 
 
 It is not actually necessary to find deviations from the mean, 
 as was done in Table 5.1, in calculating a 3:v , for it can be found 
 from the variates themselves. The average of the cubes of the 
 deviations from the mean, or the third moment about the mean of 
 the distribution, is 
 
 2(vi - M) 3 _ ZQ, 3 - 3 M-Vj 2 + UP-Vj - M z ) 
 n n 
 
 Zt> 3 _ 3 M-Zv 2 + 3M 2 Zt’ nAP 
 n n n n ’ 
 
 Al 3:v = n' 3:v -3M-/ 2:v + 2M 3 . 
 
 Hence skewness is 
 (5.2) a 3: . 
 
 m ' 3: „ —3MV 2: , + 2M 3 
 
 which enables one to find a 3:r without finding deviations from the 
 mean. 
 
 Since skewness is defined in terms of deviations of the variates 
 from the mean it can be secured from the distribution obtained by 
 subtracting a provisional mean from each variate; see Theorem 3.2. 
 In other words, 
 
 (5.3) a 3;5 = a 3:d = , 
 
 which is easier to calculate since variates in the (/-distribution are 
 smaller than those in the original set. The next table contains a 
 set of grouped variates for which skewness has been found, together 
 with all necessary computations. 
 
SKEWNESS 
 
 101 
 
 Weights of Freshmen at the University of Michigan 
 
 Weights 
 
 f(v) 
 
 d 
 
 dm 
 
 dW) 
 
 dW) 
 
 89.5- 99.5 
 
 1 
 
 -4 
 
 - 4 
 
 16 
 
 - 64 
 
 99.5-109.5 
 
 13 
 
 -3 
 
 - 39 
 
 117 
 
 -351 
 
 109.5-119.5 
 
 64 
 
 -2 
 
 -128 
 
 256 
 
 -512 
 
 119.5-129.5 
 
 128 
 
 -1 
 
 -128 
 
 128 
 
 -128 
 
 129.5-139.5 
 
 175 
 
 0 
 
 000 
 
 000 
 
 000 
 
 139.5-149.5 
 
 155 
 
 + 1 
 
 155 
 
 155 
 
 + 155 
 
 149.5-159.5 
 
 97 
 
 +2 
 
 194 
 
 388 
 
 776 
 
 159.5-169.5 
 
 37 
 
 3 
 
 111 
 
 333 
 
 999 
 
 169.5-179.5 
 
 27 
 
 4 
 
 108 
 
 432 
 
 1,728 
 
 179.5-189.5 
 
 9 
 
 5 
 
 45 
 
 225 
 
 1,125 
 
 189.5-199.5 
 
 7 
 
 6 
 
 42 
 
 252 
 
 1,512 
 
 199.5-209.5 
 
 2 
 
 7 
 
 14 
 
 98 
 
 686 
 
 
 715 
 
 
 +370 
 
 2,400 
 
 5,926 
 
 M v = 134.5 + (0.51748)10 = 139.6748 lb., 
 a v = (V3.35664 - 0.267786 - 0.083333)10 = 17.3364 lb., 
 t*3 : d = M 3 : d ~ 3.M • fx 2 : d + 2M 3 
 
 = 8.28811 - 3(0.51748) (3.35664) + 2(.51748) 3 
 = 3.35417, 
 
 « 3:e 
 
 M3:d-10 3 
 
 (0"aUJ.) 3 
 
 3.35417 
 
 5.21047 
 
 = 0.6437. 
 
 Class marks of the distribution in the above table are exhibited 
 graphically in Fig. 5.3. The part of the range to the right of the 
 
 mean is larger than the part to the left. Notice that the maximum 
 is to the left of the mean. About 69 per cent of this distribution 
 lies within 1 a of the mean. 
 
102 
 
 SKEWNESS AND KURTOSIS 
 
 Grade curves are not normal, as was assumed in Chapter 4, for 
 on examining a large number of grades one will soon discover no 
 little amount of skewness. Percentages of grades at a large uni¬ 
 versity for the year 1933-34 are given in Fig. 5.4. The frequency 
 curve for this distribution is not normal; it possesses a rather large 
 skewness. There are more A and B grades than D and E grades. 
 Of course university students form a special population. One is 
 not justified in using the normal distribution for university grades. 
 
 Fig. 5.4.— Distribution of grades of college students. 
 
 The only adjustment necessary in finding the skewness of a 
 grouped distribution is to use the adjusted standard deviation in 
 the formula for « 3 . The formula is the third moment about the 
 mean of the distribution of class marks divided by the adjusted 
 standard deviation, or 
 
 (5.4) 
 
 <*3 : adj. 
 
 M.3 : class marks _ M3 : d ’ 
 
 ( ff adj.) 3 (°adj.) 3 
 
 PROBLEMS 
 
 1. Using the percentages for grades given in Fig. 5.4, find the number 
 of students in each grade group for 7,486 university students. 
 
 2. If a 3 . „ = 0.67 and g 3 : „ = 5.36 cu. cm., find <r„. 
 
 3. If a 3: „ = 0.82 and <j v = 14.21, find Zv 3 /N. 
 
 4 . If a 3 . c = 0.6, cr v = 5, Sv 3 = 37,500, find N. 
 
 5. Find the skewness of the distribution of rays in tail fins of flounders. 
 See “Biometrika und Variationsstatistik,” by P. Riebesell, page 760. 
 
KURTOSIS 
 
 103 
 
 No. OF 
 
 No. OF 
 
 No. OF 
 
 No. OF 
 
 Rays 
 
 Flounders 
 
 Rays 
 
 Flounders 
 
 47 
 
 5 
 
 55 
 
 111 
 
 48 
 
 2 
 
 56 
 
 74 
 
 49 
 
 13 
 
 57 
 
 37 
 
 50 
 
 23 
 
 58 
 
 16 
 
 51 
 
 58 
 
 59 
 
 4 
 
 52 
 
 96 
 
 60 
 
 2 
 
 53 
 
 134 
 
 61 
 
 1 
 
 54 
 
 127 
 
 
 
 703 
 
 6. Find the skewness of the distribution of ages of women at marriage. 
 Taken from Detroit News during 1936. 
 
 Ages at 
 
 No. OF 
 
 Ages at 
 
 No. of 
 
 Marriage 
 
 Women 
 
 Marriage 
 
 Women 
 
 15-19 
 
 36 
 
 45-49 
 
 100 
 
 20-24 
 
 1,078 
 
 50-54 
 
 52 
 
 25-29 
 
 856 
 
 55-59 
 
 16 
 
 30-34 
 
 442 
 
 60-64 
 
 16 
 
 35-39 
 
 244 
 
 65-69 
 
 6 
 
 40-44 
 
 150 
 
 70-74 
 
 4 
 
 3,000 
 
 Plot the distribution and discuss results. 
 
 7 . Derive a formula for finding a 3 for combining two sets of data in 
 terms of a 3 s, cr's, M’s, and N’s* 
 
 8. Use the formula derived in problem 7 for obtaining the skewness 
 for the combination of the sets A and B if the following is known. 
 
 Set A: M x = 139.55 lb., o x = 17.75 lb., a 3 . * = 0.78, JVi = 718 
 = the number of items. 
 
 Set B: M v = 138.65 lb., <r y = 18.03 lb., a 3:v = 0.94, N 2 = 1,000 
 = the number of items. 
 
 KURTOSIS 
 
 Some frequency curves are symmetrical and yet are not normal. 
 Some possess more flatness at the mode than the normal; others arc 
 more peaked. Consider Figs. 5.5 and 5.6. Frequency curve A 
 
 * W. D. Baten, “A Formula for Finding the Skewness of the Combination 
 of Two or More Examples,” Jour. Am. Slat. Assoc., 1935, pages 95-98. 
 
104 
 
 SKEWNESS AND KURTOSIS 
 
 in Fig. 5.5 deviates markedly from B, a normal curve, because 
 of its flatness at the maximum. Curve C in Fig. 5.6 differs also 
 from the normal B, because of its peak at the mode. Some 
 symmetrical distributions have more items in the neighborhood 
 of the mean than the normal, hence a tall peak at the mode. 
 
 Fig. 5 . 5 .—Normal distribution B and distribution with negative kurtosis A. 
 
 Fig. 5.6.—Normal distribution B and distribution with positive kurtosis C. 
 
 A measure of the amount of flatness or lack of flatness of a dis¬ 
 tribution is called kurtosis and is defined in terms of the average of 
 the fourth powers of the deviations of the variates from the mean. 
 This measure is 
 
 (5.5) K = a i:v - 3=^f-3. 
 
 cr v 
 
 Kurtosis for normal curves is zero. Symmetrical frequency 
 curves which are flatter at the mode than the normal curve have 
 negative kurtosis; those which have a taller peak at the mode than 
 the normal have positive kurtosis. Tables 5.2 and 5.3 contain 
 respectively frequency distributions with negative and positive 
 kurtosis, together with necessary computations for finding K. 
 These distributions are shown in Figs. 5.5 and 5.6. 
 
KURTOSIS 
 
 105 
 
 Table 5.2 Table 5.3 
 
 V 
 
 m 
 
 V 
 
 *+/(«) 
 
 
 V 
 
 /W 
 
 V 
 
 v 2 -f(v) 
 
 v i -f(v) 
 
 0 
 
 1 
 
 -7 
 
 49 
 
 2,401 
 
 0 
 
 1 
 
 -7 
 
 49 
 
 2,401 
 
 1 
 
 2 
 
 -6 
 
 72 
 
 2,592 
 
 1 
 
 2 
 
 -6 
 
 72 
 
 2,592 
 
 2 
 
 11 
 
 -5 
 
 275 
 
 6,875 
 
 2 
 
 8 
 
 -5 
 
 200 
 
 5,000 
 
 3 
 
 30 
 
 -4 
 
 480 
 
 7,680 
 
 3 
 
 22 
 
 -4 
 
 352 
 
 5,632 
 
 4 
 
 73 
 
 -3 
 
 657 
 
 5,913 
 
 4 
 
 54 
 
 -3 
 
 486 
 
 4,374 
 
 5 
 
 132 
 
 -2 
 
 528 
 
 2,112 
 
 5 
 
 111 
 
 -2 
 
 444 
 
 1,776 
 
 6 
 
 166 
 
 -1 
 
 166 
 
 166 
 
 6 
 
 190 
 
 -1 
 
 190 
 
 190 
 
 7 
 
 170 
 
 0 
 
 000 
 
 000 
 
 7 
 
 224 
 
 0 
 
 000 
 
 000 
 
 8 
 
 166 
 
 + 1 
 
 166 
 
 166 
 
 8 
 
 190 
 
 + 1 
 
 190 
 
 190 
 
 9 
 
 132 
 
 2 
 
 528 
 
 2,112 
 
 9 
 
 111 
 
 2 
 
 444 
 
 1,776 
 
 10 
 
 73 
 
 3 
 
 657 
 
 5,913 
 
 10 
 
 54 
 
 3 
 
 486 
 
 4,374 
 
 11 
 
 30 
 
 4 
 
 480 
 
 7,680 
 
 11 
 
 22 
 
 4 
 
 352 
 
 5,632 
 
 12 
 
 11 
 
 5 
 
 275 
 
 6,875 
 
 12 
 
 8 
 
 5 
 
 200 
 
 5,000 
 
 13 
 
 2 
 
 6 
 
 72 
 
 2,592 
 
 13 
 
 2 
 
 6 
 
 72 
 
 2,592 
 
 14 
 
 1 
 
 7 
 
 49 
 
 2,401 
 
 14 
 
 1 
 
 7 
 
 49 
 
 2,401 
 
 
 1,000 
 
 
 4,454 
 
 55,478 
 
 
 1,000 
 
 
 3,586 
 
 43,930 
 
 M v = 7; a v = V+454 = 2.11045, 
 
 Hi . „ = 55.478, 
 
 a 4 . v = ^ = 2.79654, 
 
 0V 1 
 
 K = a i:v - 3 = -0.20346. 
 
 M v = 7; «r v = V 3.586 = 1.89367, 
 M4 :c = 43.93, 
 a 4 :c = 3.416, 
 
 K = a i:v - 3 =+0.416. 
 
 It is not necessary to find the deviations of the variates from 
 the mean in calculating kurtosis, for the fourth moment about the 
 mean of a frequency distribution can be written as follows: 
 
 2 {v - MYf{v) 2 v 4 f(v) 4M-2y 3 /(V) 
 
 M4:c ~ " 2/(y) 2 f(v) 
 
 or 
 
 + 
 
 6M 2 -2+/(w) 
 
 mv) 
 
 - 3 M 4 , 
 
 (5.6) M4:„ = m' 4 :» - 4M-m' 3 : 0 + 6M 2 • + 2 : „ - 3M 4 . 
 
 From . „ and a v> K can be obtained. 
 
 When data are grouped into classes it is necessary to use the 
 following adjustments in calculating kurtosis. 
 
106 
 
 SKEWNESS AND KURTOSIS 
 
 For discrete variates* * 
 
 (5.7) K = 
 
 M4:<*-(^)(l-l/fc 2 )M2: d + 
 
 (1 — l/k 2 )(7 — 3/k 2 ) 
 240 
 
 \k* 
 
 (o'adj .)' 1 
 
 3, 
 
 where k is the number of variates that can be put in a class, and d 
 is the deviation from some provisional mean. For continuous 
 variates 
 
 (5.8) 
 
 2 M 2 : il 2 1 ()]^’ 4 
 
 (•^adj .) 4 
 
 The following example will illustrate computations necessary 
 for obtaining skewness and kurtosis for grouped data. 
 
 Hip Measurements of Freshmen at the University of Michigan. 
 Measurements Made to the Nearest 0.1 Inch 
 
 Hip 
 
 Fre- 
 
 
 Measurement 
 
 quency 
 
 d 
 
 25.95-27.95 
 
 2 
 
 -4 
 
 27.95-29.95 
 
 3 
 
 -3 
 
 29.95-31.95 
 
 7 
 
 -2 
 
 31.95-33.95 
 
 92 
 
 -1 
 
 33.95-35.95 
 
 222 
 
 0 
 
 35.95-37.95 
 
 171 
 
 + 1 
 
 37.95-39.75 
 
 65 
 
 2 
 
 39.95-41.96 
 
 13 
 
 3 
 
 41.95-43.95 
 
 3 
 
 4 
 
 43.95-45.95 
 
 2 
 
 580 
 
 5 
 
 dm 
 
 dW) 
 
 d 3 M 
 
 am 
 
 +239 
 
 825 
 
 + 1,127 
 
 5,241 
 
 M v = h + M a -w = 34.95 + (0.412068)2 = 35.7741 in., 
 <r adj . - (\/1.42241 - 0.16980 - 0.08333)2 = 2.16266 in., 
 
 « 3 :t> = 
 
 M3 :d-2 3 (1.94310 - 1.75839 + 0.13994)8 
 
 (tfadi.) 3 _ (2.16266) 3 
 
 = + 0.25677, 
 
 — (^4:d ~ §M2 :d 4~ gIo)16 _ ^ _ 
 
 (o'adj.) 4 
 
 [(9.03621-3.20275+1.44915-0.086496) - 0.5(1.25261)+ 0.029167]16 
 
 (2.16266) 4 
 
 - 3 
 
 = + 1.85316. 
 
 * H. C. Carver, “Editorial,” Annals of Mathematical Statistics, Vol. I, 
 No. 1, 1930, page 111, and J. R. Abemethy, “On the Elimination of Systematic 
 Errors Due to Grouping,” ibid., Vol. IV, 1933, pages 263-277. 
 
PROBLEMS 
 
 107 
 
 PROBLEMS 
 
 1. Find the skewness and kurtosis of the following distribution of 
 weights of female babies at birth. Measurements made to nearest 
 gram. 
 
 Weight 
 
 No. OF 
 
 Weight 
 
 No. OF 
 
 at Birth 
 
 Babies 
 
 at Birth 
 
 Babies 
 
 1,095-1,324 
 
 3 
 
 3,395-3,624 
 
 82 
 
 1,325-1,554 
 
 5 
 
 3,625-3,854 
 
 44 
 
 1,555-1,784 
 
 5 
 
 3,855-4,084 
 
 15 
 
 1,785-2,014 
 
 6 
 
 4,085-4,314 
 
 13 
 
 2,015-2,244 
 
 13 
 
 4,315-4,544 
 
 5 
 
 2,245-2,474 
 
 20 
 
 4,545-4,774 
 
 2 
 
 2,475-2,704 
 
 43 
 
 4,775-5,004 
 
 1 
 
 2,705-2,934 
 
 60 
 
 
 500 
 
 2,935-3,164 
 
 91 
 
 
 3,165-3,394 
 
 92 
 
 
 
 2. Find the kurtosis of the following distribution of stamens of flowers 
 of May apples {Podophyllum peltatum) near Ann Arbor, Michigan. 
 
 No. OF 
 
 No. OF 
 
 No. OF 
 
 No. OF 
 
 Stamens 
 
 Flowers 
 
 Stamens 
 
 Flowers 
 
 9 
 
 3 
 
 15 
 
 90 
 
 10 
 
 30 
 
 16 
 
 41 
 
 11 
 
 95 
 
 17 
 
 8 
 
 12 
 
 369 
 
 18 
 
 3 
 
 13 
 
 246 
 
 
 1,000 
 
 14 
 
 115 
 
 
 = - 0.12, 
 
 find «4 
 
 
 
 4. Find K for the distribution on page 97. 
 
CHAPTER 6 
 
 PERMUTATIONS, COMBINATIONS, AND PROBABILITY 
 
 INTRODUCTION 
 
 Theorem 6.1. If a task can be done in m ways, and if after 
 it has been done a second task can be done in n ways, then both 
 tasks can be done, in this order, in mn ways. 
 
 Proof: After the first task has been done the second task 
 can be done in any one of the n ways. Since there are m ways of 
 doing the first, there are mn ways of doing both tasks. 
 
 The meaning of this theorem will be illustrated by several 
 examples. 
 
 Example. 1. If there are 3 roads leading up the east side of a moun¬ 
 tain and 5 down the west side, a person can cross the mountain on 3 X 5 = 
 15 different routes. 
 
 Example 2. A coin and a 6-sided die can fall in 12 different ways, for 
 after the coin has fallen the die might fall in 6 different ways. 
 
 Theorem 6.1 is very important, for it is used in much of the 
 theory and in many of the problems to follow. The fundamentals 
 of permutations, combinations, and probability are based upon this 
 theorem. 
 
 Example 3. A combination lock has 3 concentric rings of numbers. 
 The lock is opened when 3 particular numbers, 1 on each ring, are in line; 
 how many different combinations can be formed if the first ring contains 
 the numbers 1, 2, 3, and 4, the second ring contains the numbers 3, 4, 5, 6, 
 and 7 and the third ring contains the numbers 1 and 2? 
 
 According to the above theorem there will be 4 X 5 X 2 = 40 different 
 combinations. 
 
 PROBLEMS 
 
 1. In a classroom where boys and girls sit on opposite sides of the room 
 there are 3 vacant seats on the girls’ side and 5 on the boys’ side. A boy 
 and a girl enter the class late. In how many ways can they take seats? 
 
 108 
 
PERMUTATIONS 
 
 109 
 
 2. In how many different ways can 3 ordinary dice fall? 
 
 3. In how many ways can an ordinary die and a 4-sided die fall? 
 
 4. In how many ways can a student post two letters if there are 
 5 mail boxes? 
 
 5. John Tol enters 4 track events and Ted Tol his brother enters 5 
 other events. How many first places may the Tol boys win? How 
 many first and second places may they w r in? 
 
 6. There are 7 fraternities on a university campus. In how many 
 ways may John and Ted choose fraternities? 
 
 7. There are 4 roads from A to B, 3 roads from B to C, and 5 from 
 C to D. In how many different ways can a person go from A to D by way 
 of B and C? 
 
 8. A student has 3 coats, 5 pairs of trousers, 3 hats, and 2 pairs of 
 shoes. In how many ways can he dress? 
 
 9. A key has 5 teeth of 5 different lengths. How many different keys 
 of 5 teeth can be made? 
 
 PERMUTATIONS 
 
 The different arrangements that can be formed from the letters 
 a, b, and c taken all at a time are 
 
 abc, acb, bac, bca, cab, and cba. 
 
 The arrangement acb is different from the arrangement bca. Order 
 of the letters is important, for different orders give different 
 arrangements. 
 
 Each of the arrangements that can be formed by taking some 
 or all of a group of objects is called a permutation. The arrange¬ 
 ment cab is also a permutation. 
 
 The permutations that can be formed from the letters a, b, and 
 c taken 2 at a time are 
 
 ab, ba, ac, ca, be, cb. 
 
 According to Theorem 6.1, the number of permutations that can 
 be formed from 3 things taken 2 at a time is 3 X 2, since the first 
 place can be filled in 3 ways and after it is filled the second place 
 can be filled in 2 ways. Hence the number of permutations that 
 can be formed from the letters a, b, and c taken 2 at a time is 
 3P2 = 3X2 = 6 permutations. 
 
 Theorem 6.2. The number of permutations that can be 
 formed from n different things taken r at a time is 
 
 (6.1) n P r = n(n — l)(n — 2){n — 3) . . . (n — r + 1). 
 
110 PERMUTATIONS, COMBINATIONS, AND PROBABILITY 
 
 Proof: The first thing can be taken from any one of the n 
 things; after the first has been chosen the second can be taken 
 from any of the n — 1 remaining things; after the second has been 
 chosen the third can be taken from any of the n — 2 remaining 
 things, . . . , after the (r — l)th has been chosen the rth thing 
 may be chosen from any of the n — r + 1 remaining. Hence 
 Theorem (6.2) follows from Theorem (6.1). 
 
 Example 4. How many different 4-letter words can be formed from 
 the letters in the word thing, where the same letter cannot be used twice 
 in the same word? This can be obtained from (6.1); the number is the 
 number of permutations that can be formed from 5 letters taken 4 at a 
 time, and is 
 
 s P 4 = 5X4X3X2 = 120 different ways. 
 
 PROBLEMS 
 
 1. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4, 
 5, 6, 7, 8, 9 if the same digit is not used twice in the number? 
 
 2. How many 3-digit numbers can be formed from the digits 0, 1, 2, 
 3, 4, 5, 6, 7, 8, 9, if the same digit is not used twice in the number? 
 
 3. In how many ways can 6 people take seats in a coach with 9 vacant 
 seats? 
 
 4. In how many ways may 10 different books be placed on a shelf? 
 
 5. How many different signals can be made with 5 different colored 
 flags if there are 4 places for the flags and no color is to be used twice in 
 the same signal? 
 
 6. How many different baseball teams can be formed from 10 men who 
 can play in any position? 
 
 7. How many different baseball teams can be formed from 13 men, 
 2 of whom can only pitch, 3 of whom can only catch, 5 of whom can play 
 infield, and 3 of whom can play outfield? 
 
 8. In how many ways can a man string 10 different colored beads? 
 
 9. A person is allowed to shoot 5 ducks, 3 rabbits, 2 pheasants, and 
 1 deer. How many ways can a person bring home 4 different kinds of 
 game, where the limit was reached for each kind? 
 
 COMBINATIONS 
 
 In permutations, order was taken into consideration. The 
 arrangement or permutation ab was different from the permutation 
 ba. When order is not considered, the number of ways of selecting 
 r things from n different things is called the number of combina¬ 
 tions of n things taken r at a time. The selection or combination 
 
COMBINATIONS 
 
 111 
 
 ab is the same as ba. Permutations of the letters a, b, c taken 
 2 at a time are 
 
 ab, ba, ac, ca, be, cb; 
 
 The combinations of these letters taken 2 at a time are 
 
 ab, ac, be. 
 
 Since order does not enter into the number of combinations, the 
 number of combinations can always be found from the number of 
 permutations by dividing by the number of ways the letters can 
 be interchanged in each arrangement. In the above example the 
 number of combinations of the letters taken 2 at a time is equal to 
 the number of permutations divided by 2!, or 
 
 3C2 = 
 
 3 P 2 3X2 
 
 2 ! 
 
 1 X 2 
 
 = 3. 
 
 Theorem (6.3). The number of combinations of n different 
 things taken rata time is equal to the number of permutations of 
 n things taken r at a time divided by factorial r, or 
 
 Proof: Since order is not considered in finding the number of 
 combinations the number of permutations divided by r! will be 
 equal to the number of combinations since there are r! permuta¬ 
 tions of r things taken rata time. In other words, for each way 
 of selecting r things from n things there are r! ways of arranging 
 these r things. Hence the number of permutations is always r! 
 times as large as the number of combinations. 
 
 Example 5. How many different handshakes can 8 people make 
 with each other? This is a combination problem, for when A shakes 
 hands with B, B also shakes hands with A, and this is only 1 handshake. 
 The answer to the question is the number of ways of taking 2 things from 
 8 different things, or the number of combinations that can be formed from 
 8 things taken 2 at a time; according to Theorem (6.3) it is equal to 
 
 t,Ci = S —" = - - - 28 handshakes. 
 
 2! 1X2 
 
 Example 6. How many different committees of 4 can be chosen 
 from 7 people? 
 
112 PERMUTATIONS, COMBINATIONS, AND PROBABILITY 
 
 Here, again, order does not enter into the problem, for a committee of 
 4 is only 1 committee, regardless of how they sit in the committee room. 
 The number of committees is the number of combinations that can be 
 formed from 7 things taken 4 at a time, or 
 
 vC 4 = 
 
 7-6-5-4 
 
 1-2-3-4 
 
 35 committees. 
 
 Some of the committees will contain 3 men which were on other com¬ 
 mittees, but no 2 of the committees will contain the same 4 men. 
 
 Example 7. How many triangles can be formed from 9 points, no 
 three of which are in a straight line? This is a combination problem, for 
 the triangle ABC is the same as the triangles ACB, CAB, BCA, etc. The 
 answer is the number of combinations of 9 things taken 3 at a time, and is 
 
 9C3 
 
 9-8-7 
 
 1-2-3 
 
 84 different triangles. 
 
 Sometimes a problem will involve permutations and combina¬ 
 tions, as the following example illustrates. 
 
 Example 8 . How many words of 4 letters can be formed from 8 
 different letters so that the word does not have 2 letters alike? 
 
 There are 8 C 4 
 
 8-7-G-5 
 
 1-2-3-4 
 
 = 70 ways of choosing groups of 4 letters 
 
 from the 8 letters. After the 4 letters are chosen there are 4! = 24 dif¬ 
 ferent words that can be formed from each group of 4 letters; hence the 
 number of different words that can be formed from 70 groups of 4 letters 
 is 70 X 24 = 1,680 different words, or 8 C 4 -4 ! = 1,680 different words. 
 
 This example might have been solved by finding the number of permu¬ 
 tations of 8 things taken 4 at a time. 
 
 Example 8 a. There are 6 different consonants and 5 different vowels. 
 How many words of 4 letters can be formed if the word is to have 2 con¬ 
 sonants and 2 vowels and the same letter cannot be used twice in the 
 same word? 
 
 There are e C 2 ways of getting the 2 consonants and 5 C 2 ways of getting 
 the 2 vowels, hence there are eCV&Cb ways of getting the consonants and 
 vowels for the words. After the 2 consonants and the 2 vowels have been 
 chosen one can form 4! different words with them. Hence the number 
 of 4-letter words is equal to 
 
 $C 2 - t>C 2 * 4 ! = —1-2-3-4 = 3,600 different words. 
 
 1-2 1-2 
 
PROBABILITY 
 
 113 
 
 Experience will enable one to know whether problems can be 
 solved by formulas for permutations, combinations, or both. 
 
 PROBLEMS 
 
 1. How many straight lines can be formed from 11 points, no three of 
 which are on a straight line? 
 
 2 . How many ways can 3 white balls be drawn at random from a bag 
 containing 5 white balls and 4 red balls if 3 balls are drawn at random? 
 
 3 . How many committees of 7 students can be formed from 8 sopho¬ 
 mores and 5 freshmen if the committee is to consist of 4 sophomores and 
 3 freshmen? 
 
 4 . How many basketball teams can be formed from 6 men if each can 
 play in any position? 
 
 5. How many different examinations of 6 questions can be formed 
 from 10 different questions? How many of these will contain the first 
 3 questions? 
 
 6 . A bag contains 4 white balls, 5 red balls, and 7 blue balls. A 
 person draws at random 5 balls. How many ways can he draw? 
 
 (а) 2 white and 3 red balls? 
 
 (б) 5 blue balls? 
 
 (c) 1 white, 2 red, and 2 blue balls? 
 
 (d) 5 red balls? 
 
 (e) 2 white, 1 red, and 2 blue balls? 
 
 7 . How many different services can be held with 3 preachers, 6 singers, 
 7 deacons and 2 organists if a service requires 1 preacher, 4 singers, 3 
 deacons, and 1 organist? 
 
 8. How many parallelograms can be formed from 7 vertical parallel 
 lines and 5 horizontal parallel lines? 
 
 9 . How many ways may 13 cards be drawn from a deck of 52 cards? 
 
 10 . A railway line has 12 stations. How many different tickets must 
 be made so that each station can sell a ticket to any other station? 
 
 11 . How many different sums of money can be formed with a penny, 
 a nickel, a dime, a quarter, a half dollar, and a dollar? 
 
 12 . How many of the sums in problem 11 do not contain the dime and 
 half dollar? 
 
 PROBABILITY 
 
 If an event can happen in m different ways and fail in n differ¬ 
 ent ways and all of the ways are equally likely, then the prob¬ 
 ability of the event happening on any trial is 
 
 m 
 
 m + n ’ 
 
 (6.3) 
 
114 PERMUTATIONS, COMBINATIONS, AND PROBABILITY 
 
 the probability of the event not happening is 
 
 (6.4) 
 
 Q = 
 
 n 
 
 m + n ’ 
 
 According to this definition the probability of an event happen¬ 
 ing on any trial is the number of favorable ways to its occurrence 
 divided by the total number of ways in which it can happen and 
 fail. 
 
 Example 9. A man draws at random a ball from a bag containing 6 
 red balls and 4 purple balls, all of which are of the same size. According 
 to the above definition the probability of drawing a purple ball is 
 
 4 _ 
 
 P ~ 4 + 6 “ 10’ 
 
 since the purple ball can be drawn in 4 ways out of 10 possible ways of 
 drawing the ball. 
 
 PROBLEMS 
 
 1. What is the chance of getting the sum of the spots to turn up on 
 
 2 dice to equal 5? To equal 11? 10? 7? 
 
 2. Out of a bag containing 5 red balls and 4 black balls there are drawn 
 at random 3 balls. What is the probability that: 
 
 (а) All will be red? 
 
 (б) 2 will be red and 1 black? 
 
 (c) 2 will be green? 
 
 (d) 2 will be black? 
 
 (e) 1 will be red and 2 black? 
 
 3 . What is the probability of getting a 4-spot on a 4-sided die and a 
 
 5-spot on a 6-sided die if both are thrown? What is the probability of 
 getting the sum of the spots to be equal to 7? 5? 10? 
 
 4 . The probability of house A burning is 0.08, and the probability of 
 house B burning is 0.04. What is the probability of: 
 
 (а) both houses burning? 
 
 (б) A not burning and B burning? 
 (c) A and B not burning? 
 
 5. At a party there are 24 people and 5 prizes. Out of a bag containing 
 slips of paper with the names of the people present 1 slip is drawn at 
 random and is put back in the bag before the next drawing. What is the 
 probability that: 
 
PROBABILITY IN REPEATED TRIALS 
 
 115 
 
 (а) Mr. Jones will get a prize? 
 
 (б) Mr. Jones will receive all the prizes? 
 
 (c) Mr. and Mrs. Jones will receive no prize? 
 
 (d) Mr. and Mrs. Jones will each get a prize? 
 
 (e) Mr. and Mrs. Jones will receive all the prizes? 
 
 6 . The following histogram of a distribution containing 1,000 items is 
 given. 
 
 Fig. 6.0.—Histogram of a distribution of weights of 1,000 men. 
 
 If an item is drawn at random, what is the probability that it will lie 
 between 139.95 and 159.95? between 99.95 and 119.95? between 142 and 
 154? between 163.6 and 170.3? 
 
 7 . On a slot machine there are 3 disks with the 10 digits 0, 1, 2, 3, 4, 
 5, 6, 7, 8, 9, on each. When a coin is inserted the 3 disks revolve several 
 times and come to rest. What is the probability of getting 3 particular 
 numbers, one on each disk, to appear? 
 
 PROBABILITY IN REPEATED TRIALS 
 
 If the probability of success in 1 trial is f and the probability 
 of failure is \, then the probability of 2 successes in 3 trials may 
 be analyzed as follows: 
 
 1. A success on the first and second trials and a failure on the 
 third; the probability of such is (f)(f)(i) = fa. 
 
 2. Successes on the first and third trials and failure on the 
 second; the probability of such is (f)(^)(f) = fa. 
 
 3. Failure on the first trial and success on the last two trials; 
 the probability of such is (i)(f)(f) = - 6 9 y. 
 
 - Hence the probability of 2 successes in 3 trials is the sum of 
 these probabilities, or fa + fa + fa = f-f-. 
 
 This might have been found by taking the third term in the 
 expansion of the binomial (| + f) 3 , for 
 
 (t + !) 3 - (1 + M + M + i), 
 
 (6.5) (i + I) 3 = (I) 3 + 3(i) 2 (f) + 3 (i)(- 3 -) 2 + (I) 3 . 
 
116 PERMUTATIONS, COMBINATIONS, AND PROBABILITY 
 
 The right-hand side of (6.5) was obtained by multiplying the 3 
 factors in the above line together. The third term in the right- 
 hand member of (6.5) was obtained by adding the following 3 
 products of 3 factors: f in the first factor by f in the second factor 
 by \ in the third factor; f in the first factor, by f in the second 
 factor by f in the third factor; f in the first factor, by f in the sec¬ 
 ond factor by f in the third factor. 
 
 The last term in (6.5) is the product of (§)(§)(§) and is the 
 probability that there will be a success on the first, second, and 
 third trials. The second term in (6.5) is the probability that there 
 will be 1 success and 2 failures in the 3 trials, while the first term 
 is the product (j)(^)(^), which is the probability of a failure on 
 each of the 3 trials. 
 
 Suppose that a die is thrown 4 times and a success is considered 
 getting an ace to turn up. The terms in the expansion of (^ + -§-) 4 
 give the probabilities of the various numbers of successes. They 
 may be exhibited as follows: 
 
 Probability of 
 
 (6-6) (f + |) 4 = 
 
 Successes and 
 failures 
 
 (t) 4 + 4c l( t) 3 a)+4 c 2 (t) 2 (D 2 + 4 c 3 (t) i a) 3 + 4 c 4 (!) 4 
 
 0 12 3 4 
 
 4 3 2 1 0 
 
 The first term in the right member of (6.6) is the probability 
 of no successes and 4 failures in the 4 trials, the second term is the 
 probability of 1 success and 3 failures, the third term is the prob¬ 
 ability of 2 successes and 2 failures, the fourth term is the prob¬ 
 ability of 3 successes and 1 failure, and the last term is the prob¬ 
 ability of 4 successes in the 4 trials. The reason the terms of the 
 binomial expansion can be used for the probabilities is that the 
 coefficients of the terms give the number of ways the various 
 successes and failures may occur in the 4 trials. For example, 
 consider the probability of getting 1 success in 4 trials; there may 
 be 1 success on the first trial and failure on the other trials, or 1 
 success on the second trial and failure on the others, or a success 
 on the third trial and failure on the others, or a success on the 
 last trial and failure on the others; this gives 4 C 1 different ways 
 for 1 success in the 4 trials. The sum of probabilities for these 4 
 will give the probability of getting 1 success in the 4 trials. This 
 
 (£)(f) 3 + (!)(t) 3 + (!)(t) 3 + (M) 3 = 4(£)(f) 3 , 
 
 sum is 
 
PROBABILITY IN REPEATED TRIALS 
 
 117 
 
 which is the second term in (6.6). Other terms of (6.6) may also 
 be explained. 
 
 In general, if the probability of occurrence of an event in each 
 trial is p and the probability of its non-occurrence is q and the 
 trials are independent, then the probabilities of the various num¬ 
 bers of successes in n trials are given by the terms in the expansion 
 of the binomial ( q + p) n , or 
 
 (6.7) (q + p) n = n C 0 q n + nCiq n - l p + n C 2 q n ~ 2 p 2 + nCsq n ~ 3 p 3 
 
 + • • • + nC r q n r p T + . . . + nC n -\q-p n 1 + n C n p n , 
 
 where the first term is the probability of no successes, the second 
 term is the probability of 1 success and n — 1 failures, the third 
 term is the probability of 2 successes and n — 2 failures, etc. The 
 (r + l)th term is the probability of r successes and n — r failures 
 in n trials; for the r successes might happen in n C r different ways, 
 that is, there are n C r trials among the n trials in which there may 
 be r success and the rest failures. 
 
 Example 9a. An ordinary die is thrown 5 times, or 5 ordinary dice 
 are thrown once. The probabilities for the various successes are given 
 by terms in the expansion of (f + g-) 6 , since the probability of getting 
 an ace on each trial is g- and the probability of not getting an ace is -f. 
 This expansion is 
 
 (-1 + i) 6 i (I) 5 + 5(tm) + I0(t) 3 (i) 2 + I0($8i(i)* + 5(f) (i) 4 + (i) B . 
 
 The probability of getting 3 aces in the 5 trials is given by the fourth term 
 
 250 
 
 7,776 
 
 in the expansion and is 10[ - 
 The probability of getting exactly 1 ace in 5 throws is given by the second 
 
 3,125 
 
 7,776 
 
 term in the expansion and is 5( - 
 
 The probability of getting at least 4 aces is given by adding the last 2 terms 
 
 _ 1_ = 26 
 
 16 7,776 7,776 
 
 i uc lUUDauiiiij yji at itaot ^ 
 
 andis5 («)® , + ©’’^ 
 
 The probability of getting at least 2 and no more than 3 aces is given by 
 adding the third and fourth terms and is 
 
118 PERMUTATIONS, COMBINATIONS, AND PROBABILITY 
 
 PROBLEMS 
 
 1. A 4-sided die is thrown 6 times. Find the probability of getting: 
 
 (a) exactly 4 aces; 
 
 ( b ) at least 3 aces; 
 
 (c) exactly 6 aces; 
 
 (d) at most 2 aces; 
 
 (e) at least 3 and at most 5 aces. 
 
 2. The probability of an event happening in 1 trial is f. Find the 
 probability, in 7 trials, of getting: 
 
 (а) exactly 4 successes; 
 
 (б) at least 5 and no more than 6 successes; 
 
 (c) less than 3 successes; 
 
 (d) 5 failures; 
 
 (e) all failures; 
 
 (f) not 5 successes. 
 
 3. A coin is tossed 10 times. Find the probability of getting: 
 
 (а) exactly 2 heads; 
 
 (б) exactly 6 tails; 
 
 (c) more than 7 tails. 
 
 (d) 9 heads and 1 tail. 
 
 4. The probability of any ship of a company being destroyed on a 
 certain voyage is 0.02. The company owns 6 ships for this voyage. 
 What is the probability of: (a) losing 1 ship? (6) losing at most 2 ships? 
 (c) losing none? 
 
 5. The probability of its raining \ inches on any of the 3 days before 
 Christmas is 0.08 for a certain town. A merchant takes out rain insurance 
 for these 3 days. Find the probability that (a) it will not rain during the 
 3 days; ( b ) \ inch of ram will fall during 1 of the 3 days; (c) it will rain 
 \ inch only on the last day; (d) it will rain \ inch every day. 
 
 6 . If a man hits a traget 3 times out of 5, find the probability that he 
 will, on 4 shots, hit the target; (a) exactly 3 times; ( b) 1 time only; (c) no 
 time; (d) at least twice. 
 
 7. The mean and standard deviation of a normal distribution of 
 heights of boys are respectively 47.8 inches and 2.1 inches. A boy is 
 picked at random from the distribution of 3,489 heights. (Original 
 measurement made to the nearest 0.25 inch.) Find the probability that 
 his height is (a) greater than 51.5 inches; ( b ) at least 49 inches; (c) between 
 45.25 inches and 50.75 inches and not including either; ( d ) less than 
 44 inches; (e) either between 44.75 inches and 46 inches inclusively or 
 
PROBLEMS 
 
 119 
 
 between 52 inches and 54.5 inches, not to include 52 and 54.5; (/) exactly 
 50 inches—use ordinate table here. 
 
 8 . The mean and standard deviation of a normal distribution of 2,484 
 measurements are respectively 20.6 centimeters and 1.2 centimeters. A 
 measurement is picked at random. The probability of this measurement 
 being between 21 centimeters and w centimeters inclusive is 0.2268. Find 
 the number w if measurements were made to the nearest unit. 
 
 9. For a normal distribution find the probability that an item that is 
 picked at random is within 1 a of the mean. Between +2 o-’s and +3 <r’s 
 of the mean. 
 
 10. For any distribution, what is the probability that an item picked 
 at random is between the first and third quartiles? Between the third 
 quartile and the extreme right end of the range? 
 
CHAPTER 7 
 
 BERNOULLI DISTRIBUTION 
 FRACTION FREQUENCIES 
 
 Let the frequencies of a given distribution be multiplied by a 
 constant H, thus forming a new distribution with total frequency 
 H times the total frequency of the original distribution. The two 
 distributions have variates identically equal, but the frequencies 
 of the new distribution are H times as large as the corresponding 
 frequencies of the original. How are the characteristics, that is, 
 M, a, < 23 , and K, of the new and original distributions related? 
 
 Let the original set of variates be given by the first two columns 
 in the table below, and the variates with new frequencies be given 
 by the third and fourth columns. The characteristics of the new 
 distribution will be found and compared with those of the original. 
 
 v f{v ) 
 
 v Hf(v ) 
 
 vHf(v) 
 
 v-Hf(v) 
 
 v 3 Hf(v) 
 
 Vi f(v i) 
 
 Vi Hf(v 0 
 
 V\Hf(i'\) 
 
 fiWOi) 
 
 t-iWOi) 
 
 t '2 f(v 2 ) 
 
 t'2 HfiVi) 
 
 
 r 2 2 ///(r 2 ) 
 
 v 2 3 Hf(v 2 ) 
 
 v 3 f(v 3 ) 
 
 Vi Hf(vi) 
 
 ViHfivi) 
 
 Vi 2 Hf(v 3 ) 
 
 v 3 3 Hf(v 2 ) 
 
 Vn /On) 
 
 l'n (V-n) 
 
 v n Hf(v „) 
 
 v n -Hf(v n ) 
 
 Vn 3 Hf(v n ) 
 
 The mean of the new distribution is 
 
 ,, _ v\Uf{vi) + V 2 Hf(v 2 ) + . . . + vjlf(vn) _ HZvf{v) = 
 
 " " ew ~ ~ Hffa) + Hf(v 2 ) . . . + Hf{v n ) ~ 2/(t/) ~ 1 v ' 
 
 In other words, the mean of a new distribution formed by mul¬ 
 tiplying the frequencies of a given distribution by a constant is 
 equal to the mean of the original distribution. 
 
 120 
 
FRACTION FREQUENCIES 
 
 121 
 
 The second moment of the data with new frequencies is 
 
 , = Vi 2 Hf( Vl ) + V2 2 Hf{y 2 ) + ■ ■ ■ + Vn 2 Hf(y n ) = H2v 2 f(v) 
 
 M 2 :new Hf(v i) + Hf(v 2 ) + . . . + Hf(v n ) H2f(v) 
 
 — n '2 ■. v ) the second moment of the original distribution. 
 
 The standard deviation of the distribution with frequencies H 
 times as large as the given frequencies is 
 
 O-new = VV 2 : D ew ~ nev,) 2 = V / 2 . 5 - M v 2 = <X V . 
 
 In a similar way it can be shown that <23 and K are respectively 
 equal for the two distributions. 
 
 The above has shown that if two distributions have identical 
 variates, but the frequencies of the second are H times as large as 
 the corresponding frequencies of the first, then the characteristics 
 of the two are respectively equal. 
 
 If H is a fraction the frequencies of the new distribution might 
 be fractions instead of integers; this would not affect M, a, < 23 , and 
 K. If H = l/N, where N is the total frequency of the given set, 
 then the frequencies of the new distribution are in percentages of 
 the total frequencies of the given set. In this case, frequencies 
 are the probabilities of the various variates being drawn when one 
 variate is drawn at random. Tables I and II contain distributions 
 with frequencies expressed in percentages; each frequency is a 
 probability. Consider the set of variates in the first two columns 
 of the following table. Let a second distribution have the same 
 variates and frequencies equal to H = 1/100 times as large as the 
 frequencies in column 2. These are listed in column 3. 
 
 V 
 
 f(v) 
 
 H = 0.01 
 
 0.01 TO) 
 
 67 
 
 2 
 
 0.02 
 
 68 
 
 3 
 
 .03 
 
 69 
 
 6 
 
 .06 
 
 70 
 
 12 
 
 .12 
 
 71 
 
 21 
 
 .21 
 
 72 
 
 17 
 
 .17 
 
 73 
 
 13 
 
 .13 
 
 74 
 
 8 
 
 .08 
 
 75 
 
 5 
 
 .05 
 
 76 
 
 2 
 
 .02 
 
 77 
 
 1 
 
 .01 
 
 
 100 
 
 1.00 
 
122 
 
 BERNOULLI DISTRIBUTION 
 
 Columns 1 and 3 form a new distribution with total frequency 
 equal to unity and with M, a, 0 : 3 , and K equal to those of the dis¬ 
 tribution in columns 1 and 2 . 
 
 Fig. 6.1.—Probability or frequency distribution of the number of successes 
 in nine trials when the probability of success in each trial is equal to 
 
 Fig. 6.2. — Histogram of the probability or frequency distribution of the 
 number of successes in nine trials when the probability of success in each 
 
 trial is } 4 - 
 
 The above shows that frequency distributions may have frac¬ 
 tions for frequencies. 
 
 Let the probability of an event’s happening in each trial be -j 
 and the probability of its not happening in each trial §. In 9 trials 
 the event may happen no times, 1 time, 2 times, . . . , 8 times, or 
 9 times. The probabilities or frequencies of the various numbers 
 of occurrences are given by the terms in the expansion of the 
 
MOMENTS OF BERNOULLI DISTRIBUTION 
 
 123 
 
 binomial (f + -f) 9 ; these are arranged in a frequency distribution 
 in the following table, where v represents the various numbers of 
 occurrences in the 9 trials. 
 
 v Frequency f(v) 
 
 0 (f) 9 = 512/19,683 
 
 1 9®*® = 2,304/19,683 
 
 2 36(f) 7 ® 2 = 4,608/19,683 
 
 3 84(f) 6 ® 3 = 5,376/19,683 
 
 4 126®*® 4 = 4,032/19,683 
 
 5 126(f) 4 ® 6 = 2,016/19,683 
 
 6 84® 3 ® 6 = 672/19,683 
 
 7 36(f) 2 ® 7 = 144/19,683 
 
 8 9(f) ® 8 = 18/19,683 
 
 9 . ...® 9 = 1/19,683 
 
 2 /(») = 1 
 
 The sum of the frequencies in the above table is unity, the 
 mean is 3 occurrences, and the standard deviation is 1.414 occur¬ 
 rences. This distribution is graphically represented in Figs. 6.1 
 and 6.2. 
 
 MOMENTS OF BERNOULLI DISTRIBUTION 
 
 In general, if p is the probability of the occurrence of an event 
 in each trial and q is the probability of the non-occurrence of the 
 event in each trial and the trials are independent, then the fre¬ 
 quencies of the possible numbers of occurrences in n trials are given 
 by the terms of the expansion of the binomial (q + p) n . Such a 
 distribution is called a Bernoulli distribution. This distribution is 
 given in Table 7.1 The characteristics of the general Bernoulli 
 frequency distribution will now be derived. 
 
 The mean of the Bernoulli distribution in Table 7.1 is found by 
 summing the third column; after factoring out of each term np, 
 this sum is 
 
 M v = np 
 
 <r' + (» - 1)8—V + - - - 8-V 
 
 + . . . + (n - 1 )qp n ~ 2 + p n ~ l 
 
 np(q + p) n ~ l = np, since (q + p) n_1 = 1. 
 
 This shows that the mean of a Bernoulli distribution is equal to the 
 
124 
 
 BERNOULLI DISTRIBUTION 
 
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MOMENTS OF BERNOULLI DISTRIBUTION 
 
 125 
 
 number of trials, n, multiplied by p, the probability of the occur¬ 
 rence of the event in each trial. 
 
 The second moment of this distribution is found from column 4. 
 The sum of the quantities in this column is 
 
 hv{v — 1 )f(v) = ’Zv 2 f{v) — 2v/(i>) =n(n— l)p 2 
 (n-2)(n-3) „ 
 
 + 
 
 2! 
 
 q n ~ 2 -\-(n — 2)q n ~ 3 p 1 
 q n - 4 p 2 + . . . +(n —2)gp”- 3 +p”- 2 
 
 = n{n — 1 )p 2 (?+p)" -2 = n(n — 1 )p 2 = n 2 p 2 — np 2 . 
 
 2 v 2 -f(v) = n 2 p 2 — np 2 + 2 vf(v). 
 
 Hence 
 
 P 2 -.v = Sv 2 /(y) = n 2 p 2 — np 2 +'2,vf(v) =n 2 p 2 — np 2 -\-np, 
 since ~Zvf(v) = np as found for the mean. Therefore 
 M2 : t.= P2-.v ~ M v 2 = n 2 p 2 — np 2 + np — n 2 p 2 = np(l — p) = npq, 
 since q + p = 1. The standard deviation is 
 
 = y/ npq, 
 
 which shows that the standard deviation is the square root of the 
 product of n, the number of independent trials; p, the probability 
 of the occurrence of the event in each trial; and q, the probability 
 of failure in each trial. 
 
 The third moment can be obtained from column 5, for the sum 
 of the numbers in this column is 
 
 2i>(» - l)(i> - 2 )f(v) = 2 v 3 f(v) - 32 v 2 f(v) + 22vf{v) 
 
 = n{n — l)(n — 2)p 3 [g n_3 + (n — 3 )q n ~ 4 p 
 (n — 3 ){n — 4) 
 
 + 
 
 ^n-5-,2 
 
 2! 
 
 V + •. • 
 
 + (n — 3 )qp n ~ A + p n-3 ] 
 
 = n(n — l)(n — 2 )p 2 (q + p) n_3 
 = n 3 p 3 — 3 n 2 p 3 + 2 np 3 . 
 
126 
 
 BERNOULLI DISTRIBUTION 
 
 Hence 
 
 p' 3:c = 2i> 3 /( v) = n 3 p 3 — 3 n 2 p 3 + 2 np 3 + 32?> 2 /(p) — 22 vf(v) 
 
 _ n 3p3 _ 3n 2 p 3 +2np 3 +3n 2 p 2 — 3np 2 +3np — 2np 
 
 — n 3p3 _ 3 n 2 p 3 _j_ 2np 3 + 3n 2 p 2 — 3 np 2 + np. 
 
 The third moment about the mean is 
 
 M3 : v = P 3 ; p — 3M» • ^2 : p + 2ilf„ 3 
 
 = n 3 p 3 — 3 n 2 p 3 + 2np 3 + 3n 2 p 2 — 3 np 2 + np 
 — 3 np(n 2 p 2 — np 2 + np) + 2 n 3 p 3 
 
 = np(2p 2 - 3p + 1) = np[2p(p - 1) + (1 - p)], 
 or 
 
 M 3 : v = np(-2pq + q) = npq{q - p). 
 
 The skewness is 
 
 a 3:t = 
 
 M3 
 
 q - p 
 
 vnpq 
 
 a 3 :|1 = 0 according as q = p. 
 
 The mean, standard deviation, and skewness of a Bernoulli dis¬ 
 tribution are respectively 
 
 M v = np, 
 
 «3 
 
 Q ~ P 
 
 'Vnpq 
 
 APPLICATION 
 
 Example. If 720 ordinary dice are thrown on the table, what is the 
 expected number of aces? 
 
 The probability of an ace for each die or trial is £ = p, while q = 
 Hence the expected number of aces or mean number of aces is np = 
 720(g) = 120. This means that on the average 120 aces will be found 
 among the faces to turn up. This does not mean that exactly 120 aces 
 will turn up every time 720 dice are thrown on the table, but that in a 
 long series of throws with 720 dice the average number of aces will be 120. 
 
 When 720 dice are thrown the possible numbers of aces to turn up 
 range from 0 to 720. The frequencies or probabilities of these various 
 numbers of aces are given by the terms in the expansion of the binomial 
 (I + I) 720 - The standard deviation of this Bernoulli distribution is 
 
 5 _ 1 
 
 o' = Vtmm = 10 aces, and the skewness is b 6 = 0.067, which is 
 not far from the skewness of a normal distribution. 
 
APPLICATION 
 
 127 
 
 Example. Let it be required to find the probability of getting more 
 than 140 aces in 1 throw of 720 dice, also the probability of getting at 
 least 112 aces and no more than 133 aces. 
 
 The probability for the first part is the sum of the 580 terms in the 
 expansion of the binomial (£ + £) 720 beyond the term that gives the 
 probability of exactly 140 aces, or is the sum of the terms 
 
 72oC 14 l(t) 679 (i) 141 + 72oC 14 2(-t) 578 (!) 142 + . . . + (f) 72 °. 
 
 It would be very difficult to find this sum by arithmetic or by means of 
 logarithms. An approximation to it can be found by employing the 
 normal curve in Table I, since the skewness of this Bernoulli distribution 
 differs very little from zero. We want to find the area under the normal 
 curve beyond the t that corresponds to 140.5 (we are dealing with discrete 
 variates). The corresponding t is 
 
 140.5 - 120 , „ 
 
 t = - = +2.0o standard units. 
 
 10 
 
 The area to the left of this value of t is found from Table I to be 0.97982; 
 hence the area to the right of this t is 0.02018, which is the approximate 
 probability of getting more than 140 aces on 1 throw of 720 dice. This 
 means that if one threw the 720 dice down 100 times one could expect 
 more than 140 aces to appear 2 times. 
 
 The t's that correspond to the second part of the above example are 
 
 111.5 - 120 
 10 
 
 = -0.85, 
 
 tl — 
 
 133 .5 - 120 
 10 
 
 = 1.35. 
 
 The area between these two f’s under the normal curve is found from 
 Table I to be 0.71383; hence the probability of getting at least 112 and 
 no more than 133 aces is 0.71383, which means that 71 times out of 100 
 throws with 720 dice one can expect between 112 and 133 aces to turn up. 
 
 Example. The probability of a man of age A dying within a year 
 is 0.02. An insurance company has 10,000 men of this age insured with 
 it. Find the probability of having to pay less than 180 death claims; 
 also the probability of having to pay exactly 217 death claims. 
 
 This is solved by means of a Bernoulli distribution, for the event, 
 a man dying within a year, has probability of 0.02 in each “trial and 
 0.98 for failure. The mean, standard deviation, and skewness of the 
 distribution made up of the possible numbers of men dying within the 
 year are: 
 
 M v - 200 men, a v = 14 men, a v = 0.07; call it zero. 
 
128 
 
 BERNOULLI DISTRIBUTION 
 
 The t that corresponds to the lower limit of the rectangle in the histo¬ 
 gram which con tarns the 180th item is 
 
 t = 
 
 179.5 - 200 
 14 
 
 = — 1.46 standard units. 
 
 The area under the normal curve to the left of this t is 0.07215, hence 
 the probability of having to pay less than 180 death claims is 0.07215. 
 This means that if the company had 100 groups of 10,000 men it would 
 expect to have less than 180 death claims to pay in 7 of these groups, 
 or if 100 companies each had 10,000 men of age A, 7 of these companies 
 would on the average pay less than 180 death claims during the year. 
 
 The second part of the above example will be solved by means of the 
 ordinate table for a normal curve. The t corresponds to the class mark 
 of the group that contains 217 is 
 
 217 - 200 „ , , . 
 
 t =- =1.21 standard umts. 
 
 14 
 
 The ordinate of the normal curve with unity frequency is found from 
 Table II to be 0.19186; hence according to formula (4.5) the ordinate of 
 this Bernoulli distribution for the v that corresponds to this t is 
 
 f(v) = -/(f) = — (0.19186) = 0.01370, 
 a 14 
 
 which is the probability of having to pay exactly 217 claims during the 
 year. 
 
 This may also be found by the area Table I for 
 
 t\ — 
 
 216 .5 - 200 
 14 
 
 = 1.18, t 2 = 
 
 217.5 - 200 
 14 
 
 = 1.25. 
 
 The area between these f’s under this normal curve is 0.01335, which is the 
 probability of there being exactly 217 deaths during the year. The result 
 is nearly the same as found by the ordinate table. For finding the proba¬ 
 bility for an exact number of events the ordinate table is preferred; how¬ 
 ever, there is very little difference between them in many cases. 
 
 PROBLEMS 
 
 1. If the probability of an event happening in each trial is f, find 
 the probability of the event happening 3 tunes in 8 trials. If v repre¬ 
 sents the numbers of possible occurrences in 8 trials, plot the histogram 
 and frequency polygon of this distribution. 
 
 2. A coin is thrown 10 tunes. Find the probability of getting: 
 
PROBLEMS 
 
 129 
 
 (a) 5 heads to turn up; ( b ) 7 tails; (c) Plot the frequency polygon and 
 histogram of the distribution. 
 
 3. Twelve hundred 4-sided dice are thrown on the table. Find the 
 expected number of aces to turn down. Find the probability of getting: 
 
 (a) more than 324 aces; 
 
 ( b ) less than 281 aces; 
 
 (c) at least 281 aces and at most 324 aces; 
 
 ( d ) exactly 300 aces. 
 
 (e) express the answer for (d) as a term in a binomial expansion 
 
 and try to find its value. 
 
 4. Forty-nine men of a certain age, out of 1,000 men of this age, died 
 during the last year. An insurance company has 3,482 men of this age 
 insured with it. Find the number of death claims the company expects 
 to pay during the year. Find the probability of paying: 
 
 (a) more than 175 death claims; 
 
 ( b ) less than 166 claims; 
 
 (c) exactly 170 death claims. 
 
 5. If a new distribution is formed by multiplying each variate of a 
 given distribution by a constant H, how are the characteristics of the new 
 distribution related to those of the given distribution? 
 
 6 . Find the kurtosis of the general Bernoulli distribution. What does 
 this approach when n —> cc ? 
 
CHAPTER 8 
 
 INDEX NUMBERS 
 
 RELATIVES 
 
 Consider prices of tobacco from 1924 to 1934 given in Table 8.1. 
 The third column contains relative prices with the price of 1924 as 
 
 TABLE 8.1 
 
 Prices of Tobacco, 1924-34; Relative Prices 
 
 Year 
 
 Price per Pound on 
 Dec. 1, Cents 
 
 Relative Prices 
 Price of 1924 = 100 
 
 Relative Prices 
 Price of 1934 = 100 
 
 1924 
 
 19.0 
 
 100 
 
 86.36 
 
 1925 
 
 16.8 
 
 88 
 
 
 1926 
 
 17.9 
 
 94 
 
 
 1927 
 
 20.7 
 
 109 
 
 
 1928 
 
 20.0 
 
 105 
 
 
 1929 
 
 18.4 
 
 97 
 
 
 1930 
 
 12.8 
 
 67 
 
 
 1931 
 
 8.2 
 
 43 
 
 
 1932 
 
 10.5 
 
 55 
 
 
 1933 
 
 13.0 
 
 68 
 
 
 1934 
 
 22.0 
 
 116 
 
 100 
 
 the base. The relative price for 1925 is 88, which means that the 
 price per pound of tobacco in 1925 was 88 per cent of the price of 
 tobacco in 1924. The relative price for 1928 was 105, which means 
 that the 1928 price per pound of tobacco was 105 per cent of the 
 
 price per pound of tobacco in 1924. 
 
 The relative price is 100 
 
 where po is the price in the year whose price is the base and p i is 
 the price in any other year; in other words, the relative price is the 
 quotient of the price of the commodity in year “1”, and the price of 
 the same commodity in the base year “0” multiplied by 100. These 
 
 130 
 
AGGREGATE INDEX NUMBERS 
 
 131 
 
 relative prices are simple index numbers and enable one readily to 
 compare prices of tobacco for the various years. The simple index 
 number of price for 1930 is 67; the price index number for 1934 is 
 116. These relatives show the relative changes of prices of tobacco 
 as time passes and hence form a time series. In its simplest form 
 an index number is a name given to a term in a time series expressed 
 as a percentage of some base, or expressed as a relative. 
 
 On examining the above relatives of prices it is seen that prices 
 of tobacco rose from 1925 to 1927, decreased from 1927 to 1931, 
 and increased from 1931 to 1934. These relatives or simple index 
 numbers point out general trends of tobacco prices for these 11 
 years. 
 
 AGGREGATE INDEX NUMBERS 
 
 Suppose that one desires to know the relation between the level 
 of prices in 1930 and 1934 of 10 commodities. A simple relative 
 price as was given for each year in the last table is not adequate, 
 as there are 10 prices for each year. The object of an index num¬ 
 ber of prices in this case is to indicate the relation of the level of 
 the prices of these 10 commodities for the two 12-month periods. 
 Many index numbers have been used by economists for exhibit¬ 
 ing this relation; several index numbers will be introduced in this 
 chapter. Table 8 2 contains prices of 10 important commodities 
 for depression years. 
 
 TABLE 8.2 
 
 Prices of Ten Commodities for 1930 to 1934; Dollars 
 
 Crop 
 
 Unit 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Wheat. 
 
 Bu. 
 
 $0,670 
 
 $0,390 
 
 $0,379 
 
 $0,741 
 
 $0,880 
 
 Corn. 
 
 Bu. 
 
 .594 
 
 .321 
 
 .318 
 
 .522 
 
 .847 
 
 Oats. 
 
 Bu. 
 
 .322 
 
 .213 
 
 .157 
 
 .334 
 
 .491 
 
 Cotton. 
 
 Lb. 
 
 .095 
 
 .057 
 
 .065 
 
 .097 
 
 .126 
 
 Sugar, B. 
 
 Lb. 
 
 .0036 
 
 .0030 
 
 .0026 
 
 .0025 
 
 .0025 
 
 Tobacco . 
 
 Lb. 
 
 .128 
 
 .082 
 
 .105 
 
 .130 
 
 .220 
 
 Potatoes . 
 
 Bu. 
 
 .915 
 
 .464 
 
 .395 
 
 .823 
 
 .517 
 
 Barley . 
 
 Bu. 
 
 .404 
 
 .325 
 
 .220 
 
 .433 
 
 .710 
 
 Rye. 
 
 Bu. 
 
 .440 
 
 .336 
 
 .276 
 
 .618 
 
 . 746 
 
 Rice. 
 
 Bu. 
 
 .784 
 
 .494 
 
 .419 
 
 .778 
 
 .775 
 
 Totals . 
 
 
 $4.3556 
 
 $2.6850 
 
 $2.3366 
 
 $4.4785 
 
 $5.3145 
 
132 
 
 INDEX NUMBERS 
 
 Price levels may be measured by comparing the aggregates of 
 prices for various years with the sum of the prices, or the aggregate 
 of the prices for 1930 as base. These index numbers are given in 
 Table 8.3 for prices listed in Table 8.2; they are found by dividing 
 the sum of the prices of a particular year by the sum of the prices 
 of the same commodities in 1930, and multiplying by 100. The 
 
 TABLE 8.3 
 
 Aggregate Index Numbers for Prices of Tobacco 
 From 1930 to 1934 
 
 Year 
 
 Index Numbers 
 Aggregates of 
 Prices 
 
 Index Numbers 
 Relative Aggregates 
 1930 = 100 
 
 Index Numbers 
 Relative Aggregates 
 1934 = 100 
 
 1930 
 
 $4.3556 
 
 100 
 
 81.96 
 
 1931 
 
 2.6850 
 
 62 
 
 
 1932 
 
 2.3366 
 
 54 
 
 
 1933 
 
 4.4785 
 
 103 
 
 
 1934 
 
 5.3145 
 
 122 
 
 100 
 
 last number in the third column of Table 8.3, 122, is a price index 
 number for 1934 and indicates the level of prices of 1934 compared 
 with the level of prices of 1930. This, 122, means that the sum, 
 or aggregate, of the prices of the 10 commodities in 1934 was 122 
 per cent of the aggregate of the prices for these 10 commodities in 
 1930. The level of prices for 1934 was much higher than that for 
 1930. 
 
 Let po (,) represent the price of the fth commodity for the year 
 which is used as base and pi (i) the price of this same commodity in 
 some year “1”, whose price index number is desired. The aggrega¬ 
 tive index number of prices is represented symbolically as 
 
 ( 8 . 1 ) 
 
 I ag. 
 
 = ( 100 ) 
 
 Spi (<> 
 
 2po (i) 
 
 Pi (1) + Pi (2) + Pi (3> + . . . + Pi (n) 
 
 Po a) + Po <2) + Po (3) +. . . + po in) 
 
 which is the sum of the prices for year “1” divided by the sum of the 
 prices of the same commodities for year “0”, the base year, multi¬ 
 plied by 100. 
 
VARIOUS INDEX NUMBERS 
 
 133 
 
 For example, the aggregative index number of prices for 1932 
 is 54, which is the sum of the prices of the 10 commodities men¬ 
 tioned above divided by the aggregate of the prices of these same 
 commodities in 1930 multiplied by 100. This type of index num¬ 
 ber is called aggregative since it is the quotient of aggregates. The 
 index numbers in Table 8.3 reveal a decided drop in the level of 
 prices from 1930 to 1932 and a rapid rise from 1932 to 1934. 
 
 VARIOUS INDEX NUMBERS 
 
 Other index numbers of prices are obtained from relatives of 
 prices of the commodities for various years with prices of 1930 as 
 bases. These relatives are recorded in Table 8.4. 
 
 TABLE 8.4 
 
 Relative Prices. Prices of 1930 as Bases 
 
 Crop 
 
 Unit 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Wheat. 
 
 Bu. 
 
 100 
 
 58.21 
 
 56.57 
 
 110.60 
 
 131.34 
 
 Corn. 
 
 Bu. 
 
 100 
 
 54.04 
 
 53.54 
 
 87.88 
 
 142.59 
 
 Oats. 
 
 Bu. 
 
 100 
 
 66.15 
 
 48.76 
 
 103.73 
 
 152.48 
 
 Cotton. 
 
 Lb. 
 
 100 
 
 60.00 
 
 68.42 
 
 102.11 
 
 132.63 
 
 Sugar, B. 
 
 Lb. 
 
 100 
 
 83.33 
 
 72.22 
 
 69.44 
 
 69.44 
 
 Tobacco. 
 
 Lb. 
 
 100 
 
 64.06 
 
 82.03 
 
 101.56 
 
 171.88 
 
 Potatoes. 
 
 Bu. 
 
 100 
 
 50.71 
 
 43.17 
 
 89.95- 
 
 56.50 
 
 Rice . 
 
 Bu. 
 
 100 
 
 63.01 
 
 53.44 
 
 99.23 
 
 98.85+ 
 
 Barley . 
 
 Bu. 
 
 100 
 
 80.45- 
 
 54.46 
 
 107.18 
 
 175.74 
 
 Rye. 
 
 Bu. 
 
 100 
 
 76.36 
 
 62.73 
 
 140.45+ 
 
 169.55 
 
 Totals. 
 
 
 1,000 
 
 656.32 
 
 595.34 
 
 1,012.13 
 
 1,301.00 
 
 The arithmetic average, the median, and the geometric mean 
 of relative prices for various years have been used as index num¬ 
 bers of prices. These arc given in Table 8.5 for data in Table 8.4. 
 
 Any one of the columns of index numbers in Table 8.5 gives a 
 good idea of the levels of prices during these depression years. 
 Each set of indexes reveals a decrease of the levels of prices from 
 1930 to 1932 and an increase from 1932 to 1934. The purpose of 
 index numbers is to show relative changes with respect to some 
 base. 
 
134 
 
 INDEX NUMBERS 
 
 TABLE 8.5 
 
 Mean, Median, and Geometric Mean as Index Numbers 
 of Prices. Prices of 1930 as Bases 
 
 Year 
 
 Arithmetic 
 Average of 
 Relative 
 Prices 
 
 Median of 
 Relative 
 Prices 
 
 Geometric 
 Mean of 
 Relative 
 Prices 
 
 Prices of 1934 = 100 
 
 Mean 
 
 Median 
 
 Geometric 
 
 Mean 
 
 1930 
 
 100 
 
 100 
 
 100 
 
 88.35 
 
 72.72 
 
 81.76 
 
 1931 
 
 65.63 
 
 63.53 
 
 64.81 
 
 
 
 
 1932 
 
 59.53 
 
 55.51 
 
 58.53 
 
 
 
 
 1933 
 
 101.21 
 
 101.84 
 
 99.75 
 
 
 
 
 1934 
 
 130.10 
 
 137.52 
 
 122.3 
 
 100 
 
 100 
 
 100 
 
 The arithmetic average of relative prices is 
 
 (8.2) V - (100) , 
 
 , = ^ 
 
 mean 
 
 n 
 
 which was used in finding arithmetic averages in column 2 of 
 Table 8.5. 
 
 Since there is an even number (10) of commodity prices the 
 medians of relative prices were found by taking the geometric 
 mean of the two middle relative prices for each year. The reason 
 for taking the geometric mean of the two middle relatives instead 
 of the arithmetic average will be given later. These medians 
 appear in column 3 of Table 8.5. 
 
 The geometric mean of relative prices is written symbolically as 
 
 (8.3) 
 
 »/ pi (1) 
 
 ^po (1) 
 
 Pi (1) x Pi t2) 
 
 P o 
 
 ( 2 ) 
 
 X 
 
 Pi 
 
 (3) 
 
 Vo 
 
 (3) 
 
 x ... X 
 
 Pi 
 
 (n) 
 
 PO 
 
 (n) 
 
 ( 100 ). 
 
 The logarithm of I 0 can be written in terms of the logarithms 
 of the relatives as 
 
 (8.4) 
 
 lOg Ig =~ 
 
 n 
 
 pi (l> pi (2) pi (3) 
 
 l°g — 7U + ^ + log —TV, 
 
 po (1) po 1 -’ po W) 
 
 + • •. + log 
 
 Pl (n) 
 
 po (n) - 
 
 4 
 
VARIOUS INDEX NUMBERS 
 
 135 
 
 which is often used in finding I g . The necessary computations 
 for securing the geometric mean of the relative prices given in 
 Table 8.5 for 1934 are given in Table 8.6. 
 
 TABLE 8.6 
 
 Logarithms of Relative Prices 
 
 Logarithms of Relative Prices 
 
 Logarithms of Relative Prices for 
 
 Given in Column 7 of Table 8.4 
 
 1930 with Prices of 1934 as Bases 
 
 2.118 4079 
 
 1.881 5921 
 
 2.154 0970 
 
 1.845 9030 
 
 2.183 2256 
 
 1.816 7744 
 
 2.122 6469 
 
 1.877 3531 
 
 1.839 6375 
 
 2.160 3625 
 
 2.235 2127 
 
 1.764 7873 
 
 1.752 0694 
 
 1.247 9306 
 
 1.244 8768 
 
 1.755 1231 
 
 2.229 2861 
 
 1.770 7139 
 
 2.994 9956 
 
 1.005 0044 
 
 10)20.874 4555, 
 
 10)19.125 5444, 
 
 log = 2.087 4456, 
 
 -Slog = 1.912 5544, 
 
 n 
 
 antilog = I g , 
 
 antilog = Ig, 
 
 CO 
 
 <M 
 
 CM 
 
 t-H 
 
 II 
 
 Cs 
 
 Ig = 81.76. 
 
 The geometric means for 1931, 1932, 1933 were obtained by 
 finding the averages of the logarithms of relative prices given in 
 Table 8.4 and then finding the antilogarithms of these averages. 
 These geometric means are listed in Table 8.5 and indicate changes 
 in the level of prices as time elapsed. 
 
 When constructing index numbers, which indicate relative 
 changes in business conditions, the importance of certain prices 
 should be taken into consideration. For example, the price of 
 bread is much more important than the price of turkeys and should 
 be given a greater weight. One way of assigning weights to vari¬ 
 ous prices is to multiply prices by the respective quantities pro¬ 
 duced during the base year. In this way each commodity enters 
 
136 
 
 INDEX NUMBERS 
 
 with its price for that year and the amount sold during the base 
 year. If quantities produced in the base year are used as weights 
 the weighted aggregate index number of prices is 
 
 (8.5) I p = I wt = (100) f^-° 
 
 2p 0 <7o 
 
 _ aoo) P1 (1 W U + ?1 (2 W 2) + . . . + Pi M qo M 
 Po (1, 9o (1) + Po (2) ?o (2> + . . . + p 0 M -qo M 
 
 Since any year “1” may be taken as the base year, formula 
 
 (8.5) becomes, if the year “1” is the base year, 
 
 (8.6) I wt = (100) 
 
 2piSi 
 
 Table 8.7 contains the quantities of commodities whose prices 
 are recorded in Table 8.2. 
 
 TABLE 8.7 
 
 Production of Commodities, 1,000 Units 
 
 Crop 
 
 Unit 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Wheat. 
 
 Bu. 
 
 889,702 
 
 932,221 
 
 745,788 
 
 528,975 
 
 469,496 
 
 Com. 
 
 Bu. 
 
 1,733,429 
 
 2,229,088 
 
 2,514,613 
 
 2,038,706 
 
 1,107,887 
 
 Oats. 
 
 Bu. 
 
 1,277,379 
 
 1,126,913 
 
 1,246,548 
 
 731,500 
 
 528,815 
 
 Cotton. 
 
 Lb. 
 
 6,966,000 
 
 8,548,000 
 
 6,501,000 
 
 6,523,500 
 
 4,865,500 
 
 Sugar B. 
 
 Lb. 
 
 18,398,000 
 
 15,806,000 
 
 18,140,000 
 
 22,060,000 
 
 14,962,000 
 
 Tobacco... . 
 
 Lb. 
 
 1,647,377 
 
 1,583,567 
 
 1,106,091 
 
 1,377,639 
 
 1,095,662 
 
 Potatoes... . 
 
 Bu. 
 
 332,693 
 
 372,994 
 
 357,871 
 
 320,203 
 
 385,287 
 
 Rice . 
 
 Bu. 
 
 44,923 
 
 44,873 
 
 41,250 
 
 37,058 
 
 38,296 
 
 Barley. 
 
 Bu. 
 
 303,752 
 
 198,543 
 
 302,042 
 
 155,825 
 
 118,929 
 
 Rye . 
 
 Bu. 
 
 46,275 
 
 32,290 
 
 40,639 
 
 21,150 
 
 16,040 
 
 Totals . 
 
 
 31,639,530 
 
 30,874,489 
 
 30,915,842 
 
 33,794,556 
 
 23,614,885 
 
 The weighted index numbers of prices of these farm crops 
 appear in Table 8.8 together with Fisher’s* ideal index number, 
 which is written in formula (8.7) 
 
 (8.7) /ideal = (100) 
 
 * “The Making of Index Numbers,” by Irving Fisher. 
 
TIME REVERSAL TEST 
 
 137 
 
 TABLE 8.8 
 
 Weighted Index Numbers of Prices * 
 
 Year 
 
 Aggregative Index Num¬ 
 bers. Prices Weighted by 
 Base Year Quantities 
 1930 = Base Year 
 
 Aggregative Index 
 Numbers. Prices 
 Weighted by Quanti¬ 
 ties of Base Year 1934 
 
 Fisher’s 
 Ideal Index 
 Numbers 
 
 1930 
 
 100 
 
 77.51 
 
 100 
 
 1931 
 
 61.54 
 
 
 60.57 
 
 1932 
 
 61.03 
 
 
 59.70 
 
 1933 
 
 101.60 
 
 
 99.44 
 
 1934 
 
 138.88 
 
 100 
 
 133.86 
 
 * Prices used to the nearest penny except for sugar b. and its price was used to the 
 nearest nj mill. 
 
 The numbers given in Table 8.8 reveal a decrease in the level 
 of prices from 1930 to 1932 and an increase in the level from 1932 
 to 1934. 
 
 TIME REVERSAL TEST 
 
 For a number to be a good index number it should meet or 
 satisfy the time reversal test and the factor reversal test. The 
 time reversal test is merely a test to determine whether a given 
 method for finding an index number will work both ways in time, 
 backward and forward. For example, if the price level of 1934 
 is 150 per cent of the 1930 level, then the 1930 level of 1934 should 
 be 66§ per cent of the 1934 level, or 1.50 per cent times 0.66§ per 
 cent should be equal to unity. In other words, when price data 
 for two years are treated by the same method and the bases are 
 interchanged the first index number of prices should be the recip¬ 
 rocal of the second, and vice versa. 
 
 The simple index numbers given in Table 8.1 meet the time 
 reversal test. The aggregative index numbers of prices also meet 
 this test, for the index 122, for 1934, with the aggregate of prices of 
 1930 as bases, multiplied by the index 81.96, for 1930, with the 
 aggregate of prices of 1934 as bases, is equal to 
 
 122 X 81.9 6 = (9 g9g) / 10;000 = unity . 
 
138 
 
 INDEX NUMBERS 
 
 In this case the base years were reversed, and the product of the 
 index numbers expressed in percentages is unity. 
 
 The arithmetic average of relative prices does not meet this 
 time reversal test, for / \ / x 
 
 £©. 2 © 
 
 X 
 
 n n 
 
 is not always equal to unity. In (Table 8.5) the arithmetic aver¬ 
 age of the relative prices for 1934 with the prices of 1930 as bases 
 is 130.1 and the arithmetic average of relative prices for 1930 with 
 prices of 1934 as bases is 88.35. Their product divided by 10,000 
 is (130.10 -88.35)/10,000 = 1.149, which shows that the arithmetic 
 average of relative prices does not always meet the time reversal 
 test. 
 
 The median of relative prices does meet this test when there is 
 an odd number of commodity prices; when there is an even num¬ 
 ber of commodity prices the median also meets this test if the 
 median is considered to be the geometric mean of the two middle 
 relative prices. This is why the geometric mean of the two mid¬ 
 dle relative prices was used for the median of relative prices instead 
 of the arithmetic mean of the two middle relative prices in 
 Table 8.5. The median price index, 137.52, for 1934 with respect 
 to 1930, multiplied by the median price index, 72.72, with respect 
 to 1934, is unity when divided by 10,000, or 
 
 137.52 X 72.72/10,000 = 1. 
 
 Let pi (i) /Po (i) be the middle relative price with regard to the 
 base year “0”; this is the median of the relative prices. The ratio 
 Po U) /Pi U) will be the median of the relative prices with respect to 
 the base year “1”. Their product is of course unity. 
 
 Let there be an even number of commodity prices and let 
 Pi U) /po U) and pi (i+1) /po <!+1) be the middle two relative prices 
 with regard to year “0”. The geometric mean of these two rela¬ 
 tives is 
 
 /pi (i> pi (i+1) _ 
 
 * p ^ ' po (i+1) ' 
 
 this was used as the median of the relative prices. The two mid¬ 
 dle relative prices with regard to year “l” are 
 
 Po (i) /pi (i) and p 0 (i+1, /pi (i+1) , 
 
THE FACTOR REVERSAL TEST 
 
 139 
 
 and their geometric mean is 
 
 Po (i) p 0 n+n . 
 
 pi (i) pi ti4 ' 1) ’ 
 
 this is used as the median of the relatives with respect to year “1”. 
 The product of the two medians when there is an even number of 
 commodities is unity. 
 
 The geometric mean of relative prices as an index number 
 always meets the time reversal test for 
 
 ”/ pi (1) x Pi^ x Pi^ x 
 
 *7 
 
 V 0 
 
 (i) 
 
 Po 
 
 (2) 
 
 Po 
 
 ( 3 ) 
 
 X 
 
 V 1 
 
 (n) 
 
 VO 
 
 (n) 
 
 x njP0 a) x PO^ x P0 (3) 
 
 Pi 
 
 ( 2 ) 
 
 Pi 
 
 ( 2 ) 
 
 Pi 
 
 ( 3 ) 
 
 X . . . X 
 
 Po 
 
 (») 
 
 Pi 
 
 (n) 
 
 = 1. 
 
 The product of these for 1934 with respect to 1930 and for 1930 
 with respect to 1934 is 122.3 X 81.76/10,000 = .9999 =1, approxi¬ 
 mately. 
 
 The weighted aggregative index numbers of prices do not 
 
 always meet the first test, for the product X does not 
 
 Zpoqo Zpiqi 
 
 always equal unity. For data in Tables 8.2 and 8.7 this product is 
 138.88 X 77.51/10,000 = 1.08. 
 
 Example. Prove that Fisher’s ideal index number meets the 
 time reversal test. 
 
 THE FACTOR REVERSAL TEST 
 
 To determine whether the factor reversal test is met another 
 index number is formed by interchanging the factors in an index 
 number. The q’s are changed to p’s and the p’s are changed to q’s; 
 the product of the two index numbers then should give the value 
 
 ratio 
 
 ._ Zpigi 
 
 2pc//o 
 
 . The weighted aggregative index number of prices ■ 
 
 -JPiqo 
 
 Spotfo 
 
 2 ^ ^ 
 
 becomes —- when the factors are interchanged or reversed. The 
 
 2ry 0 po 
 
 price index number I p has been changed to the quantity index 
 number I q . Their product should be equal to the value ratio if 
 the price index meets the factor reversal test. 
 
140 
 
 INDEX NUMBERS 
 
 The index numbers given in this chapter so far which do meet 
 the time reversal test will be examined to determine whether or not 
 they meet the factor reversal test. 
 
 The “ideal” index number meets this test, for 
 
 l lpigo ^ / Zqipo SgiPi _ Spigi 
 
 * Ipoqo Zpoqi ' Zqopo ZqoPi Zpoqo ’ 
 
 that is, the index of price times the index of quantity should give 
 the value ratio, and it does in the case of the “ideal” index num¬ 
 ber. For data in Tables 8.2 and 8.7 the “ideal” index number is 
 133.88 when the prices are weighted by the respective quantities 
 and year “0” is 1930; when the factors are reversed the “ideal” index 
 of quantities is 64.16. Their product divided by 10,000 is 
 
 133.86 X 64.16/10,000 = .86, 
 
 which is equal to —- , the value ratio. Fisher’s “ideal” index 
 
 2p 0 qo 
 
 number of prices meets both tests and is considered a good index 
 number. 
 
 Consider the median of relative prices. To determine whether 
 or not it meets the factor reversal test an index of quantities must 
 be found by changing the p’s to q’s and then finding the median 
 of the relative quantities. Let the median of the relative quanti- 
 
 q i 
 
 U) 
 
 ties with respect to 1930 be and the median of relative prices 
 
 q o 
 
 pi (,) 
 
 be —— . Their product 
 Po (,) 
 
 P i (i) x 
 Po (i) qo U) 
 
 should be equal to the value ratio, , if the index number 
 
 here considered meets this test. The median relative price in 
 Table 8.5 for 1934 with regard to 1930 is 137.52, and the median 
 relative quantity of production for 1934 with regard to 1930 is 
 
 , . 137.52 X 65.197 
 
 65.197. Their product divided by 10 4 is- ^ ^ - = .8966 
 
 which is not equal to the value ratio .86. Hence the median of 
 relative prices as an index number does not meet the factor reversal 
 test. 
 
THE FACTOR REVERSAL TEST 
 
 141 
 
 When the p’s are changed to q’s in the geometric mean of rela¬ 
 tive prices the index becomes the geometric mean of relative quan- 
 
 n /gi (i) (2) g^n) 
 
 tities of production, \— — X — — X ... X —. If the geometric 
 
 x g , o Q) qo L ’ qo M 
 
 mean of relative prices, as an index number, meets the factor 
 reversal test the product 
 
 # 
 
 ™ Pi^ Pi ™ x Pi^ 
 
 P 0 (1) Po (2) Po (3> Po (n) 
 
 X 
 
 \go a) 
 
 Qi 
 
 ( 2 ) 
 
 20 
 
 (2) 
 
 X 
 
 21 
 
 ( 3 ) 
 
 2o 
 
 ( 3 ) 
 
 X . . . X 
 
 2i 
 
 (n) 
 
 2o 
 
 (n) 
 
 should be equal to the value ratio. The geometric mean of rela¬ 
 tive prices for 1934 with respect to 1930 is 122.3, and the geometric 
 mean of relative quantities for 1934 with regard to 1930 is 61.284. 
 Their product divided by 10,000 is not equal to the value ratio .86; 
 hence the geometric mean of relative prices, as an index number, 
 does not always meet the second test. 
 
 The index number 
 
 ( 8 . 8 ) 
 
 I = 
 
 S(go + qi)pi 
 
 S(2o + 2i)Po 
 
 ( 100 ) 
 
 meets the time reversal test but does not meet the factor reversal 
 test. Fisher recommends it as being the most practicable index 
 number. For data in Tables 8.2 and 8.7 it is 133.86. This index 
 nearly meets the second test. Table 8.9 allows one to compare 
 the “ideal” index numbers of prices and the index number given 
 by (8.8). 
 
 TABLE 8.9 
 
 Ideal Index Compared With Index (8.8) 
 
 Year 
 
 “Ideal” Index Number 
 of Prices; 
 
 1930 Equals Base 
 
 Index Number Given by 
 Formula (8.8); 
 
 1930 Equals Base 
 
 1930 
 
 100 
 
 100 
 
 1931 
 
 60.57 
 
 60.52 
 
 1932 
 
 59.70 
 
 59.66 
 
 1933 
 
 99.44 
 
 99.57 
 
 1934 
 
 133.86 
 
 134.94 
 
142 
 
 INDEX NUMBERS 
 
 Values in the above table show that there is not a great differ¬ 
 ence between the “ideal” index number and that given by formula 
 (8.8). In the light of the above discussion it seems best to use the 
 “ideal” index number or that given by (8.8). 
 
 It would prove very helpful to have different index numbers 
 presented in class by different members. For example the Bureau 
 of Labor index numbers, the Federal Reserve Board Index of Fac¬ 
 tory Payrolls. Dow-Jones Stock Price Indexes, Bradstreet Monthly 
 Commodity Index, Export Prices and Import Prices, and others. 
 
 PROBLEMS 
 
 1. Complete column 4 in Table 8.1. 
 
 2. Complete column 4 in Table 8.3. 
 
 3. Complete Table 8.5. 
 
 4. The following table gives prices of maple sirup and numbers of 
 gallons sold from 1924 to 1934. Find the relative quantities and relative 
 prices with the base year equal to 1924. Discuss the trends of quantities 
 and the trends of prices. 
 
 Quantity and Price of Maple Sirup from 1924 to 1934* 
 
 Year 
 
 Quantity Sold, 1,000 Gallons 
 
 Price per Gallon 
 
 1924 
 
 3,574 
 
 $2.00 
 
 1925 
 
 2,817 
 
 2.08 
 
 1926 
 
 3,504 
 
 2.12 
 
 1927 
 
 3,429 
 
 2.05 
 
 1928 
 
 2,782 
 
 2.02 
 
 1929 
 
 2,361 
 
 2.03 
 
 1930 
 
 3,641 
 
 2.03 
 
 1931 
 
 2,213 
 
 1.72 
 
 1932 
 
 2,412 
 
 1.51 
 
 1933 
 
 2,186 
 
 1.18 
 
 1934 
 
 2,395 
 
 1.13 
 
 * These data were taken from the XJ. S. Agricultural Year Book, 1935. 
 
 5. The following table gives prices and quantities sold for 10 fruits 
 and vegetables. 
 
SPLICING 
 
 143 
 
 Quantity and Price of 10 Fruits and Vegetables 
 From 1930 to 1934 
 
 
 Crop 
 
 Unit 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Quantity 
 
 Oranges 
 
 1,000 box 
 
 55,270 
 
 50,166 
 
 51,368 
 
 47,289 
 
 58,351 
 
 Price 
 
 
 Box 
 
 *1.64 
 
 *1.33 
 
 *1.09 
 
 *1.59 
 
 *1.72 
 
 Quantity 
 
 Apples 
 
 1,000 bu. 
 
 102,058 
 
 106,025 
 
 85,575 
 
 74,962 
 
 75,160 
 
 Price 
 
 
 Bu. 
 
 *1.02 
 
 $.65 
 
 *.62 
 
 *.78 
 
 *.91 
 
 Quantity 
 
 Peaches 
 
 1,000 bu. 
 
 54,186 
 
 76,689 
 
 42,443 
 
 44,692 
 
 45,404 
 
 Price 
 
 
 Bu. 
 
 *.88 
 
 $.56 
 
 *.53 
 
 *.76 
 
 *.80 
 
 Quantity 
 
 Pears 
 
 1,000 bu. 
 
 25,664 
 
 23,357 
 
 22,050 
 
 21,192 
 
 23,474 
 
 Price 
 
 
 Bu. 
 
 l.i D 
 
 *.60 
 
 *.39 
 
 $.55 
 
 *.70 
 
 Quantity 
 
 G. Fruit 
 
 1,000 box 
 
 18,934 
 
 15,147 
 
 15,149 
 
 14,243 
 
 18,248 
 
 Price 
 
 
 Box 
 
 *1.21 
 
 *1.06 
 
 *.S4 
 
 *1.12 
 
 *.92 
 
 Quantity 
 
 Grapes 
 
 Short ton 
 
 2,443,042 
 
 1,621,315 
 
 2,203,752 
 
 1,909,581 
 
 1,775,168 
 
 Price 
 
 
 Ton 
 
 *19.33 
 
 *22.39 
 
 *13.16 
 
 $17.75 
 
 *20.01 
 
 Quantity 
 
 Lemons 
 
 1,000 box 
 
 7,950 
 
 7,800 
 
 6,704 
 
 7,295 
 
 7,500 
 
 Price 
 
 
 Box 
 
 *2.35 
 
 *1.95 
 
 *2.10 
 
 *2.35 
 
 *2.30 
 
 Quantity 
 
 Tomatoes 
 
 1,000 lb. 
 
 900,046 
 
 897,343 
 
 954,159 
 
 855,049 
 
 958,240 
 
 Price 
 
 
 Bu. 
 
 *1.61 
 
 *1.10 
 
 *1.03 
 
 *1.03 
 
 *1.30 
 
 Quantity 
 
 S. Pot. 
 
 1,000 bu. 
 
 53,117 
 
 63,043 
 
 78,431 
 
 65,134 
 
 67,400 
 
 Price 
 
 
 Bu. 
 
 *1.082 
 
 *.725 
 
 *.537 
 
 *.697 
 
 *.807 
 
 Quantity 
 
 Beans 
 
 1,000 bag3 
 
 13,900 
 
 12,843 
 
 10,440 
 
 12,338 
 
 10,159 
 
 Price 
 
 
 100-lb. bag 
 
 *4.19 
 
 *2.14 
 
 *2.01 
 
 *2.79 
 
 *3.65 
 
 Find the “ideal - ’ index numbers for these data, with 1930 as base, and 
 discuss the result. 
 
 6. Find the index numbers given in (8.8) for the data in problem 5. 
 
 SPLICING 
 
 Consider the following index numbers of two series. 
 
 Year 
 
 Index Number 
 
 Year 
 
 Index Number 
 
 1930 
 
 100 (base) 
 
 1934 
 
 100 (base) 
 
 1931 
 
 90 
 
 1935 
 
 103 
 
 1932 
 
 82 
 
 1936 
 
 115 
 
 1933 
 
 87 
 
 1937 
 
 120 
 
 1934 
 
 94 
 
 
 
 If all indexes in the first series are divided by 94, the index for 1934 in 
 the first series, we will get one series of index numbers for the entire time 
 1930 to 1937 with 1934 as base. These index numbers are 
 
 Year 
 
 Index Numbers 
 
 1930 
 
 106.1 
 
 1931 
 
 95.7 
 
 1932 
 
 87.2 
 
 1933 
 
 92.6 
 
 1934 
 
 100.0 
 
 1935 
 
 103.0 
 
 1936 
 
 115.0 
 
 1937 
 
 120.0 
 
 This is known as linking, or sometimes as splicing. 
 
144 
 
 INDEX NUMBERS 
 
 PROBLEMS 
 
 1. Given the following index numbers: 
 
 Year 
 
 Index Number 
 
 Year 
 
 Index Number 
 
 1926 
 
 100 
 
 1930 
 
 100 
 
 1927 
 
 104 
 
 1931 
 
 93 
 
 1928 
 
 105 
 
 1932 
 
 78 
 
 1929 
 
 124 
 
 1933 
 
 84 
 
 1930 
 
 198 
 
 1934 
 
 89 
 
 
 
 1935 
 
 95 
 
 
 
 1936 
 
 97 
 
 
 
 1937 
 
 102 
 
 Combine these two series of index numbers with 1926 as base. 
 Combine these two series of index numbers with 1930 as base. 
 
 2. Given the following index numbers of prices: 
 
 Year 
 
 Index 
 
 1930 
 
 100 
 
 1931 
 
 90 
 
 1932 
 
 76 
 
 1933 
 
 80 
 
 1934 
 
 89 
 
 1935 
 
 95 
 
 1936 
 
 98 
 
 1937 
 
 104 
 
 Write the indexes with 1934 as base. 
 
 3. Notice index numbers in the daily or weekly newspapers. 
 
CHAPTER 9 
 
 OBSERVATIONAL EQUATIONS 
 
 HOW OBSERVATIONAL EQUATIONS ARISE 
 
 Equations which arise from observations or direct measure¬ 
 ments are called observational equations. Let x represent the 
 length of an eraser and y the length of a book. Mark off on the 
 blackboard the length of the eraser twice and the length of the 
 book three times and then measure the entire length. This gives 
 for the first observational equation 
 
 2x + 3 y — 38.75 in. 
 
 Cn the blackboard again lay off the length of the eraser and to 
 the left of this mark lay off the length of the book twice. Measure 
 the distance between the starting line and the finishing line; this 
 gives another observational equation 
 
 x — 2y — — 10.25 in. 
 
 Lay the eraser on the board five times and the book once. Mea¬ 
 sure this length. This gives a third observational equation 
 
 5x + y = 41.50 in. 
 
 From direct observation these three equations in two unknowns 
 
 arose 
 
 
 
 (1) 
 
 2x + 3 y = 
 
 38.75, 
 
 (2) 
 
 x - 2 y = - 
 
 10.25, 
 
 (3) 
 
 5x + y = 
 
 41.50. 
 
 The question arises concerning the method of solving these equa¬ 
 tions for x and y. Three methods will be given. 
 
 First Method. Solve for x and y by using the first two equa¬ 
 tions, then by using the first and third equations, and finally by 
 
 145 
 
146 
 
 OBSERVATIONAL EQUATIONS 
 
 using the second and third. Take the average of these solutions. 
 These values are 
 
 from (1) and (2) x\ = 6.68 in., y\ = 8.46 in., 
 
 from (1) and (3) xo = 6.60 in., yz = 8.52 in., 
 
 from (2) and (3) xs = 6.61 in., y% = 8.43 in., 
 
 the average is x — 6.63 in., y = 8.47 in. 
 
 The average of these three values may be considered a solution. 
 
 When these values are substituted in the observational equa¬ 
 tions the following residual errors are obtained: 0.08, 0.06, —0.12. 
 These errors are called residual errors because the real or true errors 
 are not known. The sum of the squares of these residual errors is 
 0.0244. 
 
 Secoxd Method. Consider that the first two observational 
 equations are as reliable as any other two. Solving these for x and 
 y gives x = 6.68 inches and y = 8.46 inches. When these values 
 are used for x and y the residual errors are 0.01, 0.01, —0.36, and 
 the sum of the squares is 0.1298. The first two residual errors 
 would be smaller if the numbers were not rounded off to two deci¬ 
 mal places. The sum of the squares of the residual errors for the 
 second method is larger than that obtained from the first method. 
 
 Third Method, the Least Squares Method. Multiply the 
 first observational equation by the coefficient of x, the second by 
 the coefficient of x in it, and the third equation by the coefficient 
 of x in it, and add. Do the same with the coefficients of y. These 
 calculations give 
 
 4x + 6 y = 77.50, 
 
 x — 2 y = —10.25, 
 25x + 5y = 207.50, 
 
 6x + 9 y 
 -2x+ 4 y 
 5x + 5 y 
 
 116.25, 
 
 20.50, 
 
 41.50, 
 
 30Z + 9y = 274.75, 9 y + 14 y = 178.25. 
 
 T his gives two equations in two unknowns: 
 
 30x +9 y = 274.75, 
 
 9x + 14 y = 178.25, 
 
 which are called normal equations. The following values are 
 obtained by solving the normal equations for x and y, x = 6.61 
 inches, y = 8.48 inches. The residual errors are: 0.09, +0.10, 
 0.03, and the sum of the squares of the residual errors is 0.0190, 
 
LEAST SQUARES METHOD 
 
 147 
 
 which is smaller than each of the results given by the first two 
 methods. The third method will be considered the “best” method 
 for solving observational equations. 
 
 By definition the “best set of values” for the unknowns will be 
 those which make the sum of the squares of the residual error a 
 minimum. 
 
 Lemma. The quadratic expression ax 2 + 2 bx + c is a mini¬ 
 mum if ax + b = 0, provided a > 0, and b and c are fixed real 
 numbers. 
 
 Proof: Multiply the quadratic by a and add and subtract b 2 ; 
 this becomes 
 
 (9.1) a 2 x 2 + 2 abx + b 2 + ac — b 2 = (ax + b) 2 + ac — b 2 . 
 
 This last expression is a minimum when ax + b = 0, for the per¬ 
 fect square of a real number cannot be a negative number. When 
 (9.1) is a minimum, so is the original quadratic expression. Hence, 
 ax 2 + 2 bx + c is a minimum when ax + b = 0. 
 
 PROBLEMS 
 
 1. Find the minimum of the following quadratic expressions and verify 
 your results by graphs: (a) 3x 2 — 24x + 11; ( b ) 2x 2 + lx — 5; (c) 
 7x 2 + 42x + 69. 
 
 2. Find the “best” values for x and y, the residual error, and the sum 
 of the squares of the residual error, if the observational equations are: 
 
 3x + 2y = 23.0, 
 
 x - y = - 0.8, 
 
 -2x + 4y = 12.3. 
 
 LEAST SQUARES METHOD 
 
 Proof will be given showing that the third method will give 
 the “best” values for the unknowns, or will give values for the 
 unknowns which make the sum of the squares of the residual 
 errors a minimum. Let the following be a set of observational 
 equations: 
 
 (9.2) 
 
 a\x + biy = hi, 
 a<zx + boy = h‘2, 
 a 3 X + b.iy = hi, 
 
148 
 
 OBSERVATIONAL EQUATIONS 
 
 where the a’s, 6 ’s, and h 's are known. These equations are as 
 exact as the observations that were made. The normal equations, 
 according to the third method, are 
 
 (ai 2 + a 2 2 + a 3 2 )x + (ai6i + a 2 b 2 + a 3 bz)y 
 
 = a,\hi + a 2 h 2 -f- 0363 , 
 (ai^i + a 2 b 2 + 0363 ) 3 : + ( 61 2 + b 2 2 + bz 2 )y 
 
 = b\h\ + b 2 h 2 + 6363. 
 
 When values, which are yet unknown, for x and y are substi¬ 
 tuted in the observational equations (9.2), residual errors arise. 
 Let ei, e 2 and e 3 represent these errors respectively. They are 
 
 aix + biy - hi = e h 
 a 2 x -f- b 2 y — h 2 = e 2 , 
 a 3 x + b 3 y — 63 = e 3 . 
 
 The sum of the squares of these errors is 
 
 ei 2 + e 2 2 + e 3 2 = (aix + biy - hi) 2 + ( a 2 x + b 2 y - h 2 ) 2 
 
 + (a 3 x + b 2 y - h 3 ) 2 , 
 or 
 
 Ze? = (Zar)x 2 + 2 (Za^y - ZaJn)x 
 
 + [(Zbc)y 2 - 2(2 bihi)y + Zh> 2 ]. 
 
 This is a quadratic expression in x if y is held constant. By the 
 lemma this quadratic expression is a minimum when 
 
 Za 2 -x + Zab-y — Z ah = 0, 
 
 or when 
 
 Za 2 -x + Zab-y = Zah] 
 
 this is the first normal equation in (9.3). 
 
 Write the sum of the squares of the errors as a quadratic expres¬ 
 sion in y\ this becomes 
 
 Ze? = Zb?y + 2(2 abx - 2 bh)y + [( 2 o 2 ) 3: 2 - 2(2 ah)x + 2/i 2 ]. 
 
PROBLEMS 
 
 149 
 
 By the lemma this quadratic expression in y is a minimum, if x is 
 held constant, when 
 
 2 b 2 -y + 2 ab-x — 'Zbh = 0 , 
 
 or when 
 
 2 b 2 -y + 2 ab-x = 2 bh, 
 
 which is the second normal equation in (9.3). 
 
 The question arises as to what the constants are, that is, what 
 is the constant for y when y is held constant in the quadratic expres¬ 
 sion for x and what is the constant for x when x is held constant in 
 the quadratic expression in y. These are obtained by solving the 
 two normal equations: 
 
 (9.4) 
 
 2 a 2 -x + 2 ab-y = 2 ah, 
 2 a 6 -x + 2 b 2 -y = hbh, 
 
 which are the conditions which make the two quadratic expressions 
 minimums. The solution gives values of x and y which make the 
 sum of the squares of the residual errors a minimum, and these 
 values by definition are the “best values” for x and y. 
 
 PROBLEMS 
 
 1. Find the best values for x and y, the residual errors, and the sum 
 of the squares of the errors, if the observational equations are: 
 
 x + y = 11.45 cm., 
 
 3x — y = 3.40 cm., 
 
 — x + 2y = 11.70 cm. 
 
 2. Find the best values for x, y, and z, the residual errors, and the 
 sum of the squares of the errors, if 
 
 2x + y + 2 = 19.2, 
 
 — 2x + y + 2z = 7.2, 
 
 — x — y — z = 0.1, 
 
 3x + 2 y — z = 11.2. 
 
 3. Find the best values of x and y if the observational equations are: 
 
 2x — y = 0 . 1 , 
 
 x + 2 y = 4.9, 
 
150 
 
 OBSERVATIONAL EQUATIONS 
 
 -7x- y =+1.1, 
 
 3x — y = 1.2, 
 
 — 5x + 2 y = —0.9. 
 
 4. If problem 3 were solved by the first method on page 145, how many 
 pairs of simultaneous equations would have to be solved? 
 
 5. How would the lemma on page 147 be affected if a < 0? 
 
 6. A surveying crew measures the distance from A to B three times 
 and the distance from B to C once. Another crew measures the distance 
 from A to B twice and the distance from B to C three times. A third 
 crew measures the first distance once and the second distance twice. 
 The following observational equations were obtained: 3x + y = 33.1 
 miles; 2x + 3 y = 48.3 miles; x + 2y = 29.5 miles. Set up the normal 
 equations, and find the best values for the distance from A to B and 
 the distance from B to C. 
 
 OBSERVATIONAL EQUATIONS WITH UNKNOWN CONSTANTS 
 
 In the observational equations of the preceding section the con¬ 
 stants were known, that is, the coefficients of x, y, and z were 
 known. In the example the eraser was laid down twice and the 
 book three times, giving 2x + 3 y = 38.75 for the first observa¬ 
 tional equation. These coefficients arose from observations. 
 Often problems arise which lead to observational equations in 
 which the coefficients are unknown, while certain values of the 
 variables or “unknowns” are given from observation. 
 
 It has been found by experience that, for fixed ages, weights 
 and heights of men are connected by a linear relation. Let y and 
 x represent respectively -weights and heights of men of a certain 
 age. Let the linear relation which connects weights and heights 
 be: 
 
 (9.5) y = a + bx. 
 
 The weights and heights of 10 men are found by direct measure¬ 
 ments. When these 10 values are substituted in (9.5), 10 equa¬ 
 tions arise which will also be called observational equations. The 
 least squares method can be used to find values of a and b, such 
 that the sum of the squares of the residual errors shall be a mini¬ 
 mum. The following data give the corresponding measurements 
 of weights and heights of 10 men of a certain age: 
 
151 
 
 EQUATIONS WITH UNKNOWN CONSTANTS 
 
 Heights 
 
 Weights 
 
 Heights 
 
 Weights 
 
 in Inches 
 
 in Pounds 
 
 in Inches 
 
 in Pounds 
 
 X 
 
 V 
 
 X 
 
 y 
 
 66.8 
 
 139 
 
 62.9 
 
 119 
 
 66.0 
 
 117 
 
 67.6 
 
 146 
 
 70.1 
 
 150 
 
 68.6 
 
 137 
 
 68.1 
 
 166 
 
 68.4 
 
 174 
 
 64.2 
 
 122 
 
 72.3 
 
 141 
 
 The observational equations are obtained by substituting these 
 values of x and y in the linear relation between x and y,y = a + bx, 
 which will be called the predicting equation. This equation 
 enables one to predict the weight of a man of this age when his 
 height is known. These observational equations are: 
 
 (9.6) 
 
 x y 
 
 a + 66.8 b = 139, 
 a + 66.0 b = 117, 
 a + 70.1 b = 150, 
 a + 68.1 b = 166, 
 a + 64.2 b = 122, 
 a + 62.9 b = 119, 
 a + 67.6 b = 146, 
 a + 68.6 b = 137, 
 a + 68.4 b = 174, 
 a + 72.3 b = 141. 
 
 The normal equations are 
 
 (9.7) 
 
 Na + 2ari& = 2 yi 
 
 2xiO + 2xj 2 6 = 2x; ?/i, 
 
 where i runs from 1 to iV and where N is the number of measure¬ 
 ments. In the case here considered these equations are: 
 
 10a + 675.005 = 1,411, 
 
 675.0a + 45,629.485 = 95,508, 
 
 from which a = — 126.460 and 5 = 3.96385. Hence the predict¬ 
 ing equation is 
 
 y = - 126.460 + 3.96385x, 
 
 which enables one to predict weights when heights arc known. 
 
152 
 
 OBSERVATIONAL EQUATIONS 
 
 The quantity y = a + bx = — 126.46 + 3.96385a; is the theo¬ 
 retical weight ; the quantity yo is the observed value. The differ¬ 
 ence between the observed and the theoretical is called the residual 
 error; hence residual error = e = yo — y = yo — (a + bx) = yo — 
 ( — 126.46 + 3.96385x). Substituting the weight and height of the 
 first man in the predicting equation gives the first residual error: 
 
 ci = 139 - (-126.46 + 3.96385(66.8)) = + 0.675. 
 
 The residual errors are: 
 
 ci = 139 + 126.46 - 3.96385(66.8) =+ 0.675 lb. 
 
 e 2 = 117 + 126.46 - 3.96385(66.0) =-18.154 “ 
 
 e 3 = 150 + 126.46 - 3.96385(70.1) = - 1.406 “ 
 
 c 4 = 166 + 126.46 - 3.96385(68.1) =+22.522 “ 
 
 e 5 = 122 + 126.46 - 3.96385(64.2) = - 6.019 “ 
 
 e 6 = 119 + 126.46 - 3.96385(62.9) = - 3.866 “ 
 
 e 7 = 146 + 126.46 - 3.96385(67.6) = + 4.504 “ 
 
 e 8 = 137 + 126.46 - 3.96385(68.6) = - 8.460 “ 
 
 e 9 = 174 + 126.46 - 3.96385(68.4) =+29.333 “ 
 
 do = 141 + 126.46 - 3.96385(72.3) =-19.126 “ 
 
 The sum and sum of the squares of these errors are respectively: 
 
 2ei = + 0.003 lb., Ze; 2 = 2,208.501 
 
 The standard error of prediction is by definition the square root 
 of the average of the squares of the residual errors and is 
 
 Ce = 
 
 = \/220.8501 = 14.861 lb. 
 
 Examine each of the above errors as to its size in standard errors 
 of prediction. 
 
 The symbol <x e has been used here; later it will be shown that 
 this u e is similar to the a used before, that is, that the predicting 
 equation plays the role of the mean and the residual errors are 
 actually deviations from this “mean.” The sum of these residual 
 errors will be shown to be zero. The a e here used is called the 
 standard error of prediction or the standard error of estimate. 
 
PROBLEMS 
 
 153 
 
 The following graph shows the straight line, y = — 126.46 + 
 3.96385x, and all the residual errors as distances from this line 
 parallel to the vertical line. 
 
 Fig. 9.0.—Predicting line for predicting weights from heights with the standard 
 
 error of estimate. 
 
 PROBLEMS 
 
 1. Given the following measurements for the right thigh and the right 
 calf of 10 male freshmen. Assume that these measurements are con¬ 
 nected linearly. Set up the predicting equation, the observational equa¬ 
 tions, and the normal equations, and find the predicting equation for 
 predicting right thigh measurements from right calf measurements. 
 Find the sum of the residual errors and the standard error of prediction. 
 
 Right Thigh 
 19.0 
 
 18.6 
 21.0 
 
 20.8 
 18.0 
 
 Right Calf 
 
 12.8 
 
 11.7 
 
 14.4 
 
 13.5 
 
 12.5 
 
 Right Thigh 
 
 22.7 
 
 20.9 
 22.1 
 24.0 
 17.5 
 
 Right Calf 
 
 14.7 
 
 13.1 
 
 14.2 
 15.5 
 
 12.2 
 
 Plot the predicting equation and show the errors in the plot. 
 
 2. In what range on the horizontal axis does the predicting line have 
 meaning? 
 
 3. Given a set of items; what is the sum of the deviations of the items 
 from the mean? 
 
 4. How far from zero is the sum of the residual errors in problem 1? 
 What does this suggest about the predicting equation? 
 
OBSERVATIONAL EQUATIONS 
 
 154 
 
 5. On each side of the predicting line draw a line parallel to it at 
 a distance of 1 standard error of prediction away. Count how many of 
 the ordinates for right thigh measurements fall in this band. 
 
 6. Given a set of variates distributed normally. What percentage of 
 the variates falls within 1 standard deviation from the mean of these 
 variates? 
 
 7. What percentage of the measurements falls within the band men¬ 
 tioned in problem 5? 
 
 SOLUTION OF THE NORMAL EQUATIONS AND THE STANDARD 
 ERROR OF PREDICTION IN TERMS OF FUNDAMENTAL 
 SUMMATIONS 
 
 Let the relation between x and y be linear, that is, y = a + bx, 
 and let the following be a set of n observational equations, where 
 a and b are unknowns and the x’s and y's arise from measurements: 
 
 a + bx i = 2 / 1 , 
 a + bx 2 = 2 / 2 , 
 a + bx 3 = yz, 
 
 a + bx n = y n , 
 
 The normal equations are 
 
 na + Zx-b = Zy, 
 
 Zx-a + Zx 2 -b = Zxy, 
 
 from which 
 
 Zy Zx 2 — Zx Zxy n Zxy — Zx Zy 
 
 n Zx 2 — (2a:) 2 ’ n Zx 2 — ( Zx ) 2 
 
 These express the constants a and b in terms of the fundamental 
 summations, n, 2x, 2a; 2 , Zy, Zxy. 
 
 The standard error of prediction can also be expressed in terms 
 of these fundamental summations together with Zy 2 , as: 
 
 ei 2 = [yi — (a+bxy)] 2 = yi 2 +a?+b 2 xi 2 +2abxi — 2ayi — 2bxiyi, 
 e 2 2 = [y -2 — (a+bxo)] 2 = y- 2 2 +a 2 -\-b 2 X 2 2 + 2 abx 2 — 2 ay 2 — 2 bxoy 2 , 
 ez 2 = [yz - (a +6x 3 )] 2 = yz 2 +a 2 +b 2 x 3 2 +2abxz - 2ayz - 2bxzyz, 
 
 e n 2 = [Vn — (a+bx n )] 2 = y n 2 +a 2 +b 2 x n 2 +2abx n — 2ay n — 2bx n y n 
 
 (9.8) 2e 2 = Zy 2 + [na 2 + 2 x 2 -b 2 + 2ab-Zx] — 2a- Zy — 2bZxy. 
 
PROBLEMS 
 
 155 
 
 Multiply the first normal equation by a and the second by b, 
 which gives on adding: 
 
 na 2 + 2ab • Ex + b 2 • Ex 2 = a • Ey + b • 2 xy. 
 
 Substituting this expression in (9.8) for the quantity in brackets 
 gives for the sum of the squares of the residual errors: 
 
 (9.9) 2e 2 = 2 y 2 + a Ey -f- b 2 xy — 2a Ey — 2b Exy 
 
 = 2 y 2 — a-Ey — b-Exy. 
 
 Then 
 
 (9.10) <J e = 
 
 which expresses the standard error of prediction in terms of the 
 fundamental summations, Ex, 2 y, 2 xy, 2a; 2 , Ey 2 , and n. 
 
 When these six summations are known, the predicting equation 
 can be written with little trouble; the standard error of prediction 
 can also be written at once. It is not necessary to w T rite the 
 observational equations, or even the normal equations, if the 
 values of a and b on page 154 are used. 
 
 " n ' 
 
 'Ey 2 — a-Ey — b E-xy 
 
 n 
 
 PROBLEMS 
 
 1. Given the following chest and waist measurements of men 18 years 
 of age. Assume that chest measurements are connected linearly with 
 waist measurements. Find the predicting equation for predicting chest 
 measurements from waist measurements. Find the standard error of 
 prediction. 
 
 Chest 
 
 Waist 
 
 Chest 
 
 Waist 
 
 Measurement 
 
 Measurement 
 
 Measurement 
 
 Measurement 
 
 32.0 
 
 28.0 
 
 31.0 
 
 25.6 
 
 37.0 
 
 34.5 
 
 32.2 
 
 30.0 
 
 31.4 
 
 27.0 
 
 36.1 
 
 32.1 
 
 36.5 
 
 35.8 
 
 29.0 
 
 25.0 
 
 29.5 
 
 26.0 
 
 31.0 
 
 29.0 
 
 Plot the data and draw the line of prediction. 
 
 2. The standard error of prediction is 14.2 cubic feet, and the sum of 
 the squares of the residual errors is 30,850.92. Find the value of n, the 
 number of observations which were used to find a c . 
 
156 
 
 OBSERVATIONAL EQUATIONS 
 
 PREDICTING EQUATION IN TERMS OF DEVIATIONS 
 FROM THE MEAN 
 
 From earlier material it is known that 
 
 (9.11) 2x 2 * = 2(x - M z ) 2 = 2x 2 - (2x) 2 /n 
 and 
 
 2x 2 = 2x 2 + (2x) 2 /n; 
 
 also 
 
 (9.12) 2y 2 = 2y 2 - (Zy) 2 /n and 2y 2 = 2y 2 + (Zy) 2 /n. 
 
 A similar formula will be derived for 2 xy as follows: 
 
 Zxy = 2(x - M x ){y - M y ) = 2(xy - xM y - yM z + MJM y ) 
 
 = 2 xy - M y - 2x - Mx-'Ly + nM x M y 
 
 = Zxy - Zx-Zy/n - Zy-Zx/n + (2x)(2 y)/n 
 or 
 
 (9.13) 2xy=2xy-^=^, 
 
 n 
 
 and 
 
 v v-- , ^x-Zy 
 
 Zxy = 2 xy + - . 
 
 Substituting the values of 2 xy and 2x 2 from the above in the value 
 of b given on page 154 gives: 
 
 k _ n{Zxy 4- 2x2 y/n) — 2x2 y _ 2 xy 
 ~ n[Zx 2 + (2x) 2 /n] - (2x) 2 = Zi 2 ’ 
 
 The value of a found by solving the first normal equation is 
 
 a = M v — bM x . 
 
 The predicting equation is therefore 
 
 (9.14) y = M y - M x (Zxy/Zx 2 ) + {Zxy/Zx 2 )x. 
 The standard error of prediction reduces to 
 
 (9.15) 
 
 = 
 
 (Sxy) 2 
 
 n2x 2 
 
 * Note that x (read bar x) and y are derivations of x and y from their 
 respective means. 
 
PREDICTING EQUATION IN TERMS OF DEVIATIONS 157 
 
 Example. The following data are measurements of 10 men’s weights 
 
 and chest measurements. 
 
 
 
 y 
 
 X 
 
 y 
 
 X 
 
 Wts. Lb. 
 
 Ch. Meas. In. 
 
 Wts. Lb. 
 
 Ch. Meas. In. 
 
 139 
 
 34.5 
 
 119 
 
 34.5 
 
 117 
 
 34.5 
 
 146 
 
 37.0 
 
 150 
 
 37.6 
 
 137 
 
 33.0 
 
 166 
 
 38.0 
 
 174 
 
 38.3 
 
 122 
 
 33.4 
 
 141 
 
 34.0 
 
 It is assumed that weights and chest measurements are connected linearly 
 by the equation y = a + bx. The predicting equation and standard 
 error of prediction will be found by the new formulas. 
 
 According to (9.11), (9.12), and (9.13) 
 
 2x 2 = 2x 2 - (2 xY/n = 12,624.96 - 12,588.30 = 36.66, 
 
 2y 2 = 2 y 2 - (2y) 2 /n = 202,353 - 199,092.1 = 3.260.9, 
 
 2 xy = 2 xy — (2x) (Zy)/n = 50,341.5 — 50,062.3 = 279.2, 
 b = 2xy/2x 2 = 279.2/36.66 = 7.61593, 
 a = M v - M x b = 141.1 - 35.48(7.61593) =- 129.113. 
 
 The predicting equation is therefore 
 
 y = - 129.113 + 7.61593 x. 
 
 The standard error of prediction is <r e = 10.65 + , which is smaller than the 
 standard error of prediction found on page 152 when weights were pre¬ 
 dicted from heights. The smaller a e shows that, for these 10 men, chest 
 measurement is better for predicting weight than height. The illustra¬ 
 tions on page 152 and on page 157 point out the importance of the standard 
 error of prediction. 
 
 The above results can be derived by easier methods. Let the 
 predicting equation be written in terms of deviations from the 
 means, that is 
 
 (9.16) y = a' + bx. 
 
 The normal equations which are obtained from the observational 
 
 equations are: , , . 
 
 na + 2xo = 2y, 
 
 2 xa' + 2 x 2 b = 2 xy. 
 
 The 2x = 0 and 2 y — 0, hence the first normal equation gives 
 a' = 0 and the second normal equation gives 
 
 b = 2x$/2x 2 , 
 
158 
 
 OBSERVATIONAL EQUATIONS 
 
 which is the same as b found on page 156. The predicting equation 
 becomes 
 
 (9.17) y = bx, or y = — • x, 
 
 z*x 
 
 Or y-_ 
 
 y - M v = -=£ (x - M x ), 
 
 2x“ 
 
 0r y = M v - (Zxy/Zx 2 )M x + (2 xy/Zx 2 )x, 
 
 which is the same equation as found in (9.14). Equation (9.16) is 
 the same as equation (9.14). 
 
 Consider the predicting equation in the form y = a + bx. 
 Translate the origin to the point (M x , M y ) by the transformation 
 equations used in analytical geometry. 
 
 y = y' + M y , y' = y - M y = y, and x = x' + M x . 
 
 These used in the above predicting equation give: 
 
 y' -f M y = a + b(x' + M x ), or y = a — M v + bM x + bx, 
 which becomes, if a = M y — bM x , 
 
 y = bx, 
 
 as equation (9.17). 
 
 Substitute the coordinates of the point (M x , My) in the predict¬ 
 ing equation (9.14). These coordinates satisfy this equation, hence 
 the point (M x , M y ) lies on the predicting line. Therefore the a 
 found here is the same as the a found before; hence line (9.14) is 
 the same as line (9.17). 
 
 The graph shows the predicting equation plotted with respect 
 to the two sets of axes. 
 
 Fig. 9.1.—Equation of the predicting line in terms of original units and 
 
 deviations from the means. 
 
COMPUTATIONS BY USE OF PROVISIONAL MEAN 159 
 
 PROBLEMS 
 
 1. Given the figures below for school census for a certain town from 
 1917 to 1936. Find 2x 2 , 2 xy from formulas (9.11) and (9.13), then a and 
 b from formulas on page 156, and the linear predicting equation from 
 (9.14) for predicting census figures from years. Find a e from (9.15). 
 
 Year 
 
 Census 
 
 Year 
 
 Census 
 
 1917 
 
 1,840 
 
 1927 
 
 2,510 
 
 1918 
 
 1,850 
 
 1928 
 
 2,600 
 
 1919 
 
 1,931 
 
 1929 
 
 2,690 
 
 1920 
 
 2,003 
 
 1930 
 
 2,782 
 
 1921 
 
 2,048 
 
 1931 
 
 2,705 
 
 1922 
 
 2,054 
 
 1932 
 
 2,820 
 
 1923 
 
 2,183 
 
 1933 
 
 2,910 
 
 1924 
 
 2,340 
 
 1934 
 
 3,045 
 
 1925 
 
 2,387 
 
 1935 
 
 3,183 
 
 1926 
 
 2,460 
 
 1936 
 
 3,300 
 
 Derive formula 
 
 (9.15). 
 
 
 
 3. About what is the probability that one will predict a value within 
 1 a e of the predicting line? 
 
 4. Does the predicting equation give exact values? 
 
 5. Using the predicting equation found in problem 1 predict the census 
 for 1920 and for 1936. How many standard errors of prediction did you 
 miss the observed census figures? 
 
 COMPUTATIONS SHORTENED BY USE OF THE PROVISIONAL 
 
 MEAN 
 
 The constant b and the standard error of prediction have 
 been developed in terms of the following summations: n, Zx 2 , 2 rj 2 , 
 2 xy. These can be computed by subtracting provisional means 
 from the variates of the original observations. Let the deviation 
 from a provisional mean for the x’s be represented by 
 
 x\ = Xi — h, 
 
 where h is the provisional mean. The deviation of x\ from its 
 mean is equal to the corresponding deviation of x from its mean as 
 shown in an earlier chapter. Hence the deviations from the mean 
 in the new set of variates arc equal respectively to the deviations 
 of the variates in the original set from their mean. If values of 
 the original observations are large numbers they can be made 
 smaller by subtracting a provisional mean from each variate. The 
 
160 
 
 OBSERVATIONAL EQUATIONS 
 
 quantities a, b, and <x e can now be written in terms of the new sum¬ 
 mations : 
 
 as 
 
 (9.18) 
 
 n, 2x', 2 y', 2x' 2 , 2 y'\ 2x'y', 
 
 . _ 2# _ 2 x'y' = 2x'y' - (2x')(2y')/n 
 2x 2x' 2 2x' 2 - (2x') 2 /n ’ 
 
 a — M y — Mxb = My> h y — {Mx> -j- h x )b, 
 
 where h y is the provisional mean taken from each y and h x is the 
 provisional mean taken from each x measurement. 
 
 The standard error of prediction can now be written in terms 
 of the new set of data as follows: 
 
 (9.19) 
 
 (2xy) 2 
 
 n2x 2 
 
 (Sxy ) 2 
 
 n2x' 2 
 
 (2 y') 2 /n 
 n 
 
 [2 x'y' - (2xQ(2 y')/nY 
 n[2x' 2 - (2x') 2 /«] 
 
 These formulas appear to be more complicated than the others, 
 but they are not so complicated as they appear. In analyzing a 
 problem of this nature the following 6 summations must be 
 computed: 
 
 n, 2x', 2x' 2 , 2 y', 2 y'\ 2 x'y' 
 
 The values of a, b, and a e can be immediately obtained. It is best 
 to calculate the following 
 
 (9.20) 
 
 (2 y ') 2 
 
 y ,-/2 _ y ,/2 _ 
 
 "y — 
 
 n 
 
 2x' 2 = 2x' 2 - 
 
 (2x') 2 
 
 9 
 
 n 
 
 2 x'y' = 2 x'y' 
 
 (2xQ(2yQ 
 n 
 
 and then substitute them in (9.19) and the formulas for a and b. 
 
 Example. Given the following weights and corresponding right 
 thigh measurements of the 10 men of the example at the beginning of this 
 chapter. Assume that weights and right thigh measurements are related 
 linearly; find the predicting equation and cr e . 
 
COMPUTATIONS BY USE OF PROVISIONAL MEAN 161 
 
 y 
 
 x 
 
 y 
 
 X 
 
 Weights 
 
 Right 
 
 Weights 
 
 Right 
 
 in Pounds 
 
 Thigh Meas. 
 in Inches 
 
 in Pounds 
 
 Thigh Meas. 
 in Inches 
 
 139 
 
 20.0 
 
 119 
 
 19.5 
 
 117 
 
 19.0 
 
 146 
 
 22.2 
 
 150 
 
 20.4 
 
 137 
 
 21.5 
 
 166 
 
 24.0 
 
 174 
 
 24.2 
 
 122 
 
 19.5 
 
 141 
 
 21.2 
 
 Let 140 be h v and 21 = h x . After these have been subtracted from 
 the above data we get the following 
 
 y' 
 
 Xc 
 
 2/ 
 
 x' 
 
 - l 
 
 -1.0 
 
 -21 
 
 -1.5 
 
 -23 
 
 -2.0 
 
 + 6 
 
 + 1.2 
 
 + 10 
 
 -0.6 
 
 - 3 
 
 +0.5 
 
 +26 
 
 +3.0 
 
 +34 
 
 +3.2 
 
 -18 
 
 -1.5 
 
 + 1 
 
 +0.2 
 
 From these the fundamental summations are found to be 
 
 2*' = 1.5, 22/1= 11, = 30.83, Zy' 2 = 3,273 
 
 Zx'y’ = 292.2, Zx' 2 = 30.605, Zy' 2 = 3,260.9, Zx'y' = 290.55, 
 
 from which a =—59.689, b = 9.49355, a c = 7.0887. The predicting 
 equation is 
 
 y =-59.6885 + 9.4935 x. 
 
 The standard deviation, a e = 7.089 pounds, is much smaller than 
 that found when predicting weights from heights and weights from chest 
 measurements. Hence right thigh measurements, for these 10 men, are 
 better for predicting weights than heights or chest measurements. 
 
 Theorem 9.1. The sum of the residual errors is equal to zero. 
 Proof: Let us use the predicting line in the form y = bx. The 
 residual errors are now 
 
 ei = yi — bx i, 
 
 62 = 2/2 — bx 2 , 
 
 63 = #3 — bx 3, 
 
 6n ]] n bx n 
 
 Add 2e = Zy — bZx = 0 + b-0 = 0. 
 
162 
 
 OBSERVATIONAL EQUATIONS 
 
 This shows that the predicting line is similar to a real mean, since 
 the sum of the deviations from it is zero. The standard error of 
 prediction is very similar to a standard deviation. Sometimes the 
 predicting line is spoken of as a “moving average.” 
 
 PROBLEMS 
 
 1. The following table contains the average length of intestines of 
 birds in centimeters and the average weight of the body in grams. Find 
 the (linear) predicting equation for predicting intestine length from body 
 weight and the standard error of prediction.* 
 
 Average Length of 
 Intestines of 
 Several Birds, 
 Centimeters 
 
 4.3 
 5.8 
 6.5 
 
 7.3 
 
 8.4 
 9.0 
 9.7 
 
 10.2 
 
 11.0 
 
 11.6 
 
 12.4 
 
 12.6 
 
 Average Weight 
 of Several Birds, 
 Grams 
 
 1.5 
 
 2.7 
 
 3.6 
 
 4.2 
 
 5.4 
 5.9 
 
 6.5 
 
 7.3 
 8.1 
 
 8.8 
 
 9.7 
 
 9.8 
 
 2. The following data give the ages in years and heights in inches for a 
 group of girls, where x represents ages and y represents heights: 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 X 
 
 y 
 
 4 
 
 40 
 
 7 
 
 50 
 
 9 
 
 56 
 
 11 
 
 50 
 
 13 
 
 60 
 
 16 
 
 60 
 
 4 
 
 42 
 
 7 
 
 50 
 
 9 
 
 58 
 
 11 
 
 54 
 
 13 
 
 60 
 
 16 
 
 64 
 
 5 
 
 42 
 
 7 
 
 52 
 
 9 
 
 58 
 
 11 
 
 54 
 
 13 
 
 58 
 
 16 
 
 62 
 
 5 
 
 44 
 
 8 
 
 54 
 
 10 
 
 48 
 
 11 
 
 58 
 
 13 
 
 64 
 
 16 
 
 62 
 
 5 
 
 46 
 
 8 
 
 54 
 
 10 
 
 46 
 
 11 
 
 60 
 
 14 
 
 58 
 
 17 
 
 64 
 
 6 
 
 44 
 
 8 
 
 56 
 
 10 
 
 50 
 
 11 
 
 60 
 
 14 
 
 66 
 
 17 
 
 66 
 
 6 
 
 46 
 
 8 
 
 56 
 
 10 
 
 50 
 
 12 
 
 56 
 
 14 
 
 64 
 
 17 
 
 64 
 
 6 
 
 50 
 
 8 
 
 58 
 
 10 
 
 54 
 
 12 
 
 58 
 
 14 
 
 60 
 
 18 
 
 64 
 
 6 
 
 48 
 
 9 
 
 50 
 
 10 
 
 56 
 
 12 
 
 58 
 
 15 
 
 66 
 
 18 
 
 66 
 
 7 
 
 48 
 
 9 
 
 52 
 
 10 
 
 60 
 
 12 
 
 64 
 
 15 
 
 64 
 
 18 
 
 62 
 
 7 
 
 48 
 
 9 
 
 54 
 
 11 
 
 48 
 
 12 
 
 62 
 
 15 
 
 62 
 
 19 
 
 68 
 
 * Taken from “Experiments and the Digestion of Food by Birds,” by 
 James Stevenson, Wilson Bulletin, Vol. XLV., No. 4, 1935, pages 155-167. 
 
EQUATIONS IN MORE THAN TWO UNKNOWNS 163 
 
 Find the predicting equation for predicting heights from ages. Find 
 the average height for each group, and plot these points on the same 
 graph with the predicting line. What does the predicting line actually 
 give for a predicted value? 
 
 PREDICTING EQUATIONS IN MORE THAN TWO UNKNOWNS 
 
 Assume that 3 variables x, y and z are connected linearly as 
 
 (9.21) y = a + bx + cz, 
 
 where y is predicted from the values of x and z. Here there will 
 be n observational equations and three normal equations in three 
 unknowns. The number of normal equations can be reduced to 
 two if the predicting equation is written in terms of the deviations 
 from the means, viz.: 
 
 y — a' + bx + cz. 
 
 There are still 3 unknowns, a', b, and c, and it does not appear as 
 though there will be a reduction of normal equations. Examine 
 the normal equations 
 
 na' + 2 xb + 2zc = 2 y, 
 
 2 x-a' + 2 x 2 -b + 2 xz-c = 2 xy, 
 
 2 z-a' + 2 xz-b + 2 z 2 -c -- ~Ezy, 
 
 The sum of the deviations from the mean is zero; hence 2x, = 2 y = 
 2z = 0, and the first normal equation reduces to na' = 0; hence 
 a' = 0. The above three normal equations become: 
 
 2 x 2 -b + 2 xz-c = 2 xy, 
 
 Hxz-b + 2i 2 -c = 2 zy. 
 
 The predicting equation becomes 
 
 (9.22) y = bx + cz. 
 
 To be able to use this predicting equation for predicting, the 
 deviations from the mean of the x’s and the deviations from the 
 mean of the z’s must be known; this then gives for the predicted 
 value of the y a deviation from the mean of the y’s. Measure¬ 
 ments, as a rule, are not given in terms of deviations from the 
 mean; hence the last predicting equation is impracticable. Pre¬ 
 dicting equation (9.21) should be used for predicting after the 
 
164 
 
 OBSERVATIONAL EQUATIONS 
 
 constants are found by using (9.22). The quantities b and c in 
 
 (9.22) are the same as in (9.21), for y = bx + cz can be written as 
 
 y-M v = b(x-M x )+c(z-M 2 ), or y= (M v —bM x —cMJ+bx+cz, 
 which is (9.21) if 
 
 (9.23) a = M v - bM x - cM z . 
 
 Equation (9.22) is used to reduce the number of normal equa¬ 
 tions by one. This enables one to find b and c much more quickly. 
 When b and c are found, a can be found easily by using (9.23). 
 
 Equation (9.22) can be obtained from (9.21), as was done on 
 page 158. 
 
 The standard error of prediction is for this case: 
 
 (9.24) 
 
 G e 
 
 — b • 2x?/ — c ■ ~Zzy 
 n 
 
 Consider the problem of finding the predicting equation for predicting 
 weights from heights and thigh measurements by using the data on 
 pages 151 and 161. Let the linear relation be 
 
 y = a + bx + cz, or y = bx + cz, 
 
 where y represents weights, x represents heights, and z right thigh measure¬ 
 ments. The normal equations, if the second equation is used, are 
 
 2x 2 -5 + 2xz-c = 2 xy , 
 
 2 zx-b + 2 z 2 -c = 2 zy. 
 
 or 
 
 2 x' 2 b + 2x'z'c = 2 x'y\ 
 Zx'z'b + 2z' 2 c = 2 z’y'. 
 
 Using hy = 140, h x = 67, h 2 = 21, the data reduce to the following: 
 
 y' 
 
 x' 
 
 z! 
 
 y' 
 
 x' 
 
 z' 
 
 - l 
 
 -0.2 
 
 -1.0 
 
 -21 
 
 -4.1 
 
 -1.5 
 
 -23 
 
 -1.0 
 
 -2.0 
 
 + 6 
 
 +0.6 
 
 + 1.2 
 
 + 10 
 
 +3.1 
 
 -0.6 
 
 - 3 
 
 + 1.6 
 
 +0.5 
 
 +26 
 
 + 1.1 
 
 +3.0 
 
 +34 
 
 + 1.4 
 
 +3.2 
 
 -18 
 
 -2.8 
 
 -1.5 
 
 + 1 
 
 +5.3 
 
 +0.2 
 
 
 2 y' = 
 
 11, 2x' = 5.0, 
 
 2 z' = 1.5, 
 
 
 
 
 2 y' 2 = 
 
 3,273, 2x' 2 = 
 
 69.48, 2z' 2 = 
 
 30.83, 
 
 
PROBLEMS 
 
 165 
 
 2y' 2 = 3,260.9, 2x' 2 = 66.98, 2 z' 2 = 30.605, 
 
 2 x'y' = 271.0, 2 x'z' = 21.05, 2 z'y' = 292.2, 
 
 2*y = 265.5, 2x'r = 20.30, 2 z'y' = 290.55. 
 
 The normal equations are 
 
 66.986 + 20.3c = 265.5, 
 
 20.306 + 30.605c = 290.5. 
 
 from which 
 
 6 = 1.362, c = 8.586. 
 
 From (9.23) 
 
 a =-132.429. 
 
 The predicting equation is 
 
 y = -132.429 + 1.362x + 8.586z, 
 and the standard error of prediction is a e = 6.361 pounds. 
 
 The size of the standard error of prediction shows that weights 
 are better predicted from heights and right thigh measurements 
 together than from either one. Compare the <r e ’s on pages 157 and 
 
 161 . 
 
 PROBLEMS 
 
 1. If y — a + bx + cz and there are n measurements for each variable, 
 set up the observational equations and the normal equations for finding 
 of a, 6, and c. 
 
 2. Assume that weights, heights, and chest measurements for men are 
 connected linearly. Find the predicting equation for weights by using 
 data on pages 151 and 157. Find a e . Compare with the above illus¬ 
 tration. 
 
 3. Assume a linear relation between weight, chest measurements, and 
 right thigh measurements. Find the predicting equation for weights by 
 using data on pages 151, 157, and 161. 
 
 4. Compare the sizes of the a e found in problems 2 and 3 and on page 
 165, and determine which two measurements are best for predicting 
 weights. 
 
 6. Derive formula (9.24). 
 
 6. Prove by translating the axes that the predicting plane y = a -+• 
 bx + cz is the same as the predicting plane y = bx + cz. 
 
 7. Prove that the point (M x , M v , M t ) is on the predicting plane 
 y = a + bx + cz. 
 
166 
 
 OBSERVATIONAL EQUATIONS 
 
 SOLUTION OF SIMULTANEOUS EQUATIONS 
 
 Given the following equations in three unknowns, where the 
 coefficients of the unknowns are large numbers: 
 
 (1) 10.0 a + 354.80 6+ 211.50 c = 1,411.00, 
 
 (2) 354.8 a + 12,624.96 6 + 7,527.15 c = 5,034.15, 
 
 (3) 211.5 a + 7,527.15 6 + 4,053.83 c = 30,133.20. 
 
 Divide each by the coefficient of a, getting: 
 
 (1) a + 35.4800 6 + 21.1500 c = 141.100, 
 
 (2) a + 35.5833 6 + 21.2152 c = 141.887, 
 
 (3) a + 35.5894 6 + 21.2947 c = 142.474. 
 
 Subtract (2) from (3) and (1) from (2), and then multiply by 10,000. 
 
 This gives 61 6 + 795 c = 5,870, 
 
 1,094 6 + 1,447 c = 13,740. 
 
 Divide by the coefficients of 6, getting 
 
 6 + 13.033 c = 96.230, 
 
 6 + 1.323 c = 12.559. 
 
 Subtracting we get, 11.710 c = 83.671, 
 
 Therefore c = 7.1671, 6 = 3.051, a = — 119.154. 
 
 It is best to subtract the first two equations, then the next two, 
 a,nd then the next two, etc., in preference to using one as the sub¬ 
 trahend throughout, for this one subtrahend might have been more 
 incorrect than the others, and hence more errors would creep into 
 the results. By using the first two, the second tw T o, etc., there will 
 be a tendency to equalize the errors in the different equations or 
 make all enter in about the same way or to the same weight. 
 
 This method of reducing the coefficients of the first unknown 
 to unity is far easier than solving by determinants or by other 
 algebraic methods, especially when the coefficients are large. 
 
PROBLEMS 
 
 167 
 
 PROBLEMS 
 
 1. Assume that weights, heights, chest measurements, and right thigh 
 measurements are connected linearly. Find the predicting equation for 
 weights, using data on pages 151, 157, and 161. Find <r e . 
 
 2. Use data in problem 1 and those on pages 152, 157, 161, and 165. 
 Do heights, chest measurements, and right thigh measurements enable 
 one to predict weights better than heights alone? Chest measurements? 
 Right thigh measurements? Better than heights and chest measure¬ 
 ments, or heights and thigh measurements or chest and thigh measure¬ 
 ments? What is the determining factor? 
 
 3. Using the predicting equation in problem 1 find the predicted 
 weight of the third individual; the weight of the tenth man. How large 
 are the errors in terms of cr e ? 
 
 4. Prove that, if y = a + bx + cz + du + gw, then 
 
 -4 
 
 hy 2 — ahy — bZxy — chzy — dhuy — ghwy 
 
 and that 
 
 -4 
 
 2y 2 — bhxy — cZzy — dZuy — g~Zwy 
 
 n 
 
 5. Prove that, if the predicting equation is y = a -f bx, then 
 
 <Te 
 
 (Zxy) 2 
 
 2x 2 2 y 2 
 
 6. When will the standard error of prediction be equal to zero? 
 
 7. Discuss the situation when the denominator in a and 6 on page 154 
 is equal to zero. 
 
 8. Set up the normal equations when the predicting equation is 
 y = a + bx + ex 2 , if there are given n corresponding measurements for 
 x and y. 
 
 9. Set up the normal equations when y and x are connected by the 
 relation y — a/x, if n corresponding measurements are given for x and y. 
 
 10. Do the values of a and b, found on page 160, depend on the pro¬ 
 visional means h x and h u 7 
 
CHAPTER 10 
 
 CORRELATION COEFFICIENT 
 
 LINEAR CORRELATION 
 
 Assume that two variables x and y are connected linearly by 
 (10.1) y = a + bx, 
 
 where the x’s and y 's are given from observation or from direct 
 measurements. When these values are substituted in this equa¬ 
 tion, n observational equations arise, from which the following 
 normal equations are obtained: 
 
 n-a + Zx-b = Zy, 
 Zx-a + Zx 2 -b = Zxy. 
 
 The solution of these normal equations gives 
 
 a = M y — M x 
 
 The standard error of prediction is found to be 
 
 ( 10 . 2 ) 
 
 or 
 
 (10.3) 
 where 
 
 (10.4) 
 
 &y ) 2 
 nZx 2 
 
 I (Zxy) 2 
 
 (Zx 2 )(Zy 2 ) ’ 
 
 G e Gy *\/^l T yx ~ , 
 
 _ 
 
 v Vz^W 2 ’ 
 
 which is called the Pearson linear correlation coefficient between 
 x and y, or the linear correlation coefficient, which may be written 
 
 (10.5) 
 
 T yz 
 
 UG X ■ Gy 
 
 168 
 
LINEAR CORRELATION 
 
 169 
 
 The quantity r yx can never be greater than 1 and it can never 
 be less than — 1 , for if \r yx \ > 1 , the standard error of prediction 
 which is positive or zero would be imaginary, as is seen by exam¬ 
 ining (10.3). 
 
 If r =± 1, then <r e — 0, and hence all the residual errors are 
 equal to zero and hence all the values of y will fall on the line of 
 prediction; this means perfect correlation between x and y. 
 
 If r = 0, then <r e = <r y . To be able to understand what this 
 means let us write the predicting equation in terms of the correla¬ 
 tion coefficient, and the quantities a x , a y , M X) M y . This is 
 
 y = M y - M x • • x. [See (9.14).] 
 
 From (10.5) 
 
 2 xy = n<x x -<T y -r xy . 
 
 Therefore the predicting equation is 
 
 (10.6) y = M v — M x • — • r yx + r yx ■ — ■ x, 
 
 (J x G x 
 
 which gives the predicting equation in terms of the means, stand¬ 
 ard deviations, and correlation coefficient; hence, if r yz — 0, the 
 predicting equation becomes 
 
 (10.7) y = M y , 
 
 which shows that for any value of the variable x the best value of 
 y will always be the mean of the y’s. For example, if heights of 
 several people are known, then the predicted weights for each 
 individual would be the average of all weights, that is, if the cor¬ 
 relation coefficient between heights and weight were zero. This 
 average of all weights, M y , would represent the weight of a heavy 
 man as well as a light man. This is the same as saying that for 
 any height there is one and only one weight, which is the average 
 of all weights. Here the predicting curve is a straight line parallel 
 to the horizontal axis. This means that there is no relation be¬ 
 tween x and y, that is, y does not depend upon x. 
 
 If r = — 1, y becomes larger when x becomes smaller, as can 
 be seen from predicting equation (10.6). Correlation here is per¬ 
 fect. If two test scores were connected in such a way as to yield 
 a linear correlation coefficient of —1, then high scores in one test 
 would correspond to low scores in the other, and vice versa. 
 
170 
 
 CORRELATION COEFFICIENT 
 
 Thus the quantity r yz shows the dependence of one variable 
 upon the other. Sometimes the linear correlation coefficient is 
 defined as the amount of dependence one variable has upon 
 another. 
 
 If the variables are not connected by a linear relation, the linear 
 coefficient of correlation should not be used. It has very little 
 meaning w r hen the variables are not connected linearly. 
 
 When the observed values of x and y are not very large and the 
 number of observations does not exceed 50 the best formula for 
 computing the correlations coefficient is 
 
 2x?7 nSxw — 2x-2w 
 
 (10 8 ) r = _ J = — 
 
 V2x 2 2y 2 V>2x 2 - (2x) 2 ][n2y 2 - (2y) 2 ] ' 
 
 If values of x and y are large it will shorten calculations by 
 
 subtracting provisional means, and then using the new set of data. 
 
 The value of r, obtained from (10.8), is 
 
 Hxy Zx'y' 
 
 (10.9) r vz = - - = ■ . . ■ 
 
 V2x 2 -2y 2 V2x' 2 -2y' 2 
 
 nSxV - (Sx'XSyQ 
 
 V[n2x' 2 - (2x') 2 ][nZy' 2 - (2 y') 2 ] ’ 
 
 where the primes represent values after provisional means have 
 been subtracted from the variates. 
 
 Example. Find the coefficient of correlation between weights and 
 right thigh measurements of men if the following measurements are given. 
 
 Weight 
 
 in 
 
 Pounds 
 
 y 
 
 Right 
 
 Thigh, 
 
 Inches 
 
 X 
 
 y — 140 
 
 y’ 
 
 x — 21 
 x' 
 
 x'y’ 
 
 139 
 
 20.0 
 
 - 1 
 
 -1.0 
 
 + 1.0 
 
 117 
 
 19.0 
 
 -23 
 
 -2.0 
 
 + 46.0 
 
 150 
 
 20.4 
 
 + 10 
 
 -0.6 
 
 - 6.0 
 
 166 
 
 24.0 
 
 +26 
 
 +3.0 
 
 + 78.0 
 
 122 
 
 19.5 
 
 -18 
 
 -1.5 
 
 + 27.0 
 
 119 
 
 19.5 
 
 -21 
 
 -1.5 
 
 + 31.5 
 
 146 
 
 22.2 
 
 + 6 
 
 + 1.2 
 
 + 7.2 
 
 137 
 
 21.5 
 
 - 3 
 
 +0.5 
 
 - 1.5 
 
 174 
 
 24.2 
 
 +34 
 
 +3.2 
 
 + 108.8 
 
 141 
 
 21.2 
 
 + 1 
 
 +0.2 
 
 + 0.2 
 
 
 
 + 11 
 
 + 1.5 
 
 +292.2 
 
 30.83 
 
 3,273 
 
PROBLEMS 
 
 171 
 
 Using (10.9) 
 
 _ _ 10(292,2) - (1.5)(11) _ 
 
 "\/[10(30.83) - (1.5)*] [10(3,273) - (ll) 2 ] 
 r vx = 0.9197. 
 
 
 OTHER FORMS OF 1 UX 
 
 The correlation coefficient can be written as 
 
 2 xy — nM x M v 2x'?/ — nM x 'M v > 
 
 T yx = * 
 
 Tiff xffy Tiff xfff 
 
 The standard error of prediction is 
 
 2 _ 2?/ 2 — a'Zy — bZxy 
 
 2 ~ ' 
 
 = a y y / 1 — r 2 ; hence r 2 = 1- — = 1 — 
 
 2£ 2 
 
 which gives r in terms of the summations 2 y, 2 y 2 , 2 y 2 , 2 xy, and 
 the constants a and b. 
 
 G X -v ~ 
 
 2 x — y 2 2(x — y — M x _ v ) 2 2(x — M x — y + M y ) 2 
 
 n 
 
 n 
 
 n 
 
 2 (s - y) 2 2x 2 22xy 2 y 2 22xy , , 
 
 d - = a* 2 -b <r v 2 . 
 
 n n n 
 
 n 
 
 n 
 
 Therefore 
 
 Hence 
 
 22 ^ 2 I _ 2 
 
 n 
 
 = g z 2 + ay — g x _ 2 
 
 Tux 
 
 g 2 + ay — <7 z -v 
 
 2 Ti(J X G y 
 
 PROBLEMS 
 
 1. Find the linear correlation coefficient between weights and heights, 
 using data on page 151. 
 
 2. Do the same for weights and chest measurements, using data on 
 page 157. 
 
 3. How do the standard errors of prediction compare with the different 
 correlation coefficients for problems 1 and 2 and that computed on page 
 152? (Compare pages 157, 161, and 171.) 
 
 4. (liven the following grades for freshmen in the Iowa Placement 
 Test in mathematics and in their first-semester work in college mathe- 
 
172 
 
 CORRELATION COEFFICIENT 
 
 matics. Find the linear correlation coefficient between scores on the 
 test and first-semester grades in mathematics. Choose provisional means. 
 
 Iowa 
 
 First 
 
 Placement 
 
 Semester 
 
 Test 
 
 Mathematics 
 
 
 Grades 
 
 40 
 
 75 
 
 37 
 
 85 
 
 30 
 
 72 
 
 47 
 
 82 
 
 18 
 
 55 
 
 57 
 
 88 
 
 28 
 
 55 
 
 25 
 
 65 
 
 29 
 
 88 
 
 39 
 
 78 
 
 Iowa 
 
 First 
 
 Placement 
 
 Semester 
 
 Test 
 
 Mathematics 
 
 
 Grades 
 
 31 
 
 75 
 
 50 
 
 92 
 
 39 
 
 88 
 
 27 
 
 75 
 
 41 
 
 75 
 
 60 
 
 95 
 
 39 
 
 60 
 
 42 
 
 85 
 
 35 
 
 65 
 
 37 
 
 74 
 
 5. When will r = o represent perfect correlation? 
 
 RANK CORRELATION COEFFICIENT 
 
 Let x be the rank of the variate instead of the actual value, and 
 let y be the corresponding rank of another variate. If x = 1, then 
 the variate corresponding to this x is either the largest or smallest 
 of all the variates. Let us consider rank in descending order. A 
 rank of 2 for a measurement means that the corresponding variate 
 or measurement is second largest. A rank of n means that the 
 corresponding variate is the least in the set of n variates. Here x 
 represents the ranks of one set of variates and y represents the 
 ranks of the corresponding variates in the other set. A man may¬ 
 be the tallest man in a group of men, but his weight may be fourth 
 from the heaviest man in the group. Then x = 1 and y = 4. 
 
 The rank correlation coefficient is the linear correlation coeffi¬ 
 cient between ranks of the corresponding variates. Let the x’s be 
 placed in descending order. The x’s will now be the first n integers. 
 
 x 
 
 1 
 
 2 
 
 3 
 
 4 
 
 s 
 
 n 
 
 n — s 
 
RANK CORRELATION COEFFICIENT 
 
 173 
 
 The sum of the x’s is equal to the sum of the y’s which is also 
 the sum of the first n integers; this is equal to 
 
 2x = 'Ey = ^ (1 + n ); also Ex 2 - Ey 2 
 
 Zi 
 
 n(n -f- l)(2n + 1) 
 6 
 
 = l 2 + 2 2 + . . . + n 2 .* 
 
 The correlation coefficient is, as before, 
 
 = 
 
 VE&Ef 
 
 Everything is known except Exy = Exy — (Ex)(Ey)/n. From 
 the difference D in the ranks of x and y one can secure a value for 
 Exy. The first difference is 
 
 therefore 
 
 Di = x\ — 2 / 1 ; 
 
 D i 2 = xi 2 — 2x1?/! + yi 2 , and 2xiyi = x\ 2 — D i 2 + y i 2 ; 
 hence the sum of all the product terms is 
 2 Exy = Ex 2 - ED 2 + Ey 2 = 2 Ex 2 - ED 2 and Exy = Ex 2 - ED 2 / 2. 
 
 Substitute the values of 2x 2 , Ey 2 , and Exy in the formula for r yx ; 
 this gives 
 
 R _ nSx?/ - (Ex)(Ey) _ 
 
 V[n2x 2 - {Ex) 2 ][nEy 2 - (2?/) 2 ] 
 
 nn(n + l)(2n + l)/6 - n2D 2 /2 - n 2 (n + l) 2 /4 
 y/[nn{n + 1)(2 n + l)/6 — n 2 {n + l) 2 /4] 2 
 
 __ ED 2 _ 
 
 n{n + 1) [(2n + l)/3 — (n + l)/2] 
 
 _ 62D 2 _ 
 
 n{n + l)(4n + 2 — 3n — 3) 
 
 (10.10) R yx = 1 - 
 
 6 2D 2 
 n(n 2 — 1) 
 
 * This is given in some college algebras. 
 
174 
 
 CORRELATION COEFFICIENT 
 
 The symbol R yx will be used to represent the rank correlation 
 coefficient. The rank correlation coefficient can in many instances 
 be calculated much more quickly than the Pearson linear correla¬ 
 tion coefficient, for the 2Z) 2 can be obtained easily. Differences in 
 the ranks are usually very small and can be squared and summed 
 quickly after the data are arranged in ranks. 
 
 Often, R yz gives as much information about the two correspond¬ 
 ing sets of variates as is desired. 
 
 If there are ties in the ranks, make the ranks so that the sum 
 of the ranks is the same as if there had been no ties. For example, 
 suppose 3 tied for fifth place. The ranks would be 5, 6, and 7, 
 had there been no ties. To make the ranks so that the sum of 
 these ranks is the same as if there had been no ties, make xs = 6, 
 xe = 6, x- = 6. This 6 would be the average of the three ranks 
 if there had been no ties. If xq and £10 tied for ninth place, make 
 the rank of both 9.5. In this way the sum of the ranks is still 
 n(n + l)/2, but the sum of the squares of the ranks has been 
 slightly changed, but not enough to affect the result much. When 
 there are ties the rank correlation coefficient is not exact or exactly 
 correct, yet it is used as though it were correct. The quantities 
 r yx and R yx are about the same. However, this is not always the 
 case, as the following two series will show: 
 
 x: 60, 50, 40, 30, 10, 
 
 y: 100, 98, 97, 3, 1. 
 
 Here the rank correlation coefficient is 1 or perfect, while r yx is not 
 equal to unity. 
 
 Example. Let us find the rank correlation coefficient of the following 
 data. The x’s have been arranged in descending order. 
 
PROBLEMS 
 
 175 
 
 Freshmen 
 
 Mathe- 
 
 SRAGES 
 
 Average 
 
 X 
 
 y 
 
 x - y 
 
 Z> 2 
 
 95 
 
 90 
 
 1.5 
 
 1 
 
 0.5 
 
 0.25 
 
 95 
 
 89 
 
 1.5 
 
 2 
 
 -0.5 
 
 0.25 
 
 92 
 
 82 
 
 3 
 
 6 
 
 3.0 
 
 9.00 
 
 90 
 
 83 
 
 4 
 
 4.5 
 
 -0.5 
 
 0.25 
 
 88 
 
 83 
 
 5 
 
 4.5 
 
 0.5 
 
 0.25 
 
 85 
 
 80 
 
 6 
 
 7.5 
 
 -1.5 
 
 2.25 
 
 82 
 
 77 
 
 7.5 
 
 11 
 
 -3.5 
 
 12.25 
 
 82 
 
 79 
 
 7.5 
 
 9.5 
 
 -2. 
 
 4.00 
 
 78 
 
 70 
 
 10.5 
 
 17.5 
 
 -7. 
 
 49.00 
 
 78 
 
 84 
 
 10.5 
 
 3 
 
 7.5 
 
 56.25 
 
 78 
 
 79 
 
 10.5 
 
 9.5 
 
 1 
 
 1.00 
 
 78 
 
 80 
 
 10.5 
 
 7.5 
 
 3 
 
 9.00 
 
 75 
 
 74 
 
 13.5 
 
 14.5 
 
 -1 
 
 1.00 
 
 75 
 
 76 
 
 13.5 
 
 12.5 
 
 1 
 
 1.00 
 
 72 
 
 76 
 
 15 
 
 12.5 
 
 -2.5 
 
 6.25 
 
 68 
 
 74 
 
 16.5 
 
 14.5 
 
 2 
 
 4.00 
 
 68 
 
 70 
 
 16.5 
 
 17.5 
 
 -1 
 
 1.00 
 
 65 
 
 67 
 
 18 
 
 20 
 
 -2 
 
 4.00 
 
 58 
 
 69 
 
 19 
 
 19 
 
 0 
 
 0.00 
 
 50 
 
 72 
 
 20 
 
 16 
 
 -4 
 
 16.00 
 
 
 
 
 
 
 177.00 
 
 Rxy ‘ 
 
 62 D 2 
 
 
 6(177) 
 
 = 1 - 0.133 
 
 
 20(399) 
 
 = 0.867, 
 
 which shows that there is high correlation between what a person does in 
 freshman mathematics and what he will do during his four years in college. 
 
 PROBLEMS 
 
 1. Find the rank correlation coefficient for the data on page 171 in 
 problem 4. How does the Pearson linear correlation coefficient compare 
 with the rank correlation coefficient? 
 
 2. The following figures are the freshmen’s mathematics average and 
 their averages made on science courses taken after the freshman year. 
 These data were taken from records at the University of Texas. 
 
CORRELATION COEFFICIENT 
 
 176 
 
 Freshmen 
 
 Science 
 
 Freshmen 
 
 Science 
 
 Mathe¬ 
 
 Average 
 
 Mathe¬ 
 
 Average 
 
 matics 
 
 Above 
 
 matics 
 
 Above 
 
 Average 
 
 First Year 
 
 Average 
 
 First Year 
 
 68 
 
 73 
 
 68 
 
 72 
 
 75 
 
 65 
 
 85 
 
 78 
 
 78 
 
 77 
 
 72 
 
 85 
 
 95 
 
 95 
 
 88 
 
 89 
 
 82 
 
 75 
 
 85 
 
 81 
 
 65 
 
 74. 
 
 85 
 
 82 
 
 78 
 
 87 
 
 90 
 
 92 
 
 85 
 
 90 
 
 79 
 
 76 
 
 75 
 
 67 
 
 78 
 
 75 
 
 72 
 
 81 
 
 82 
 
 87 
 
 82 
 
 82 
 
 92 
 
 93 
 
 72 
 
 73 
 
 78 
 
 75 
 
 72 
 
 69 
 
 76 
 
 79 
 
 Find the rank 
 
 correlation coefficient 
 
 for these data. 
 
 Does ability in 
 
 mathematics help one in sciences? 
 
 GRAPHIC MEANING OF THE CORRELATION COEFFICIENT 
 
 Fig. 10.1.—Predicting line with respect to the a; and y axes and with respect 
 
 to the x and y axes. 
 
 The above graph, Fig. 10.1, shows the predicting equation 
 plotted with respect to the x and y axes and with respect to the 
 x and y axes. The predicting equation, with the x and y axes as 
 reference lines, is y — a + bx and with the x and y axes, after the 
 
 origin has been changed to (M x , M v ), is y = bx, or y = 
 
 <Tx 
 
 X. 
 
 tx 
 
 X 
 
 0T 
 
 Let 
 
 and t 
 
PROBLEM 
 
 177 
 
 then the equation of this line becomes 
 (10.11) t v = r yx -t x , 
 
 which gives the hne of prediction in terms of standard units and 
 the correlation coefficient. These units are abstract and do not 
 depend upon the units of x and y given originally. In this equa¬ 
 tion the correlation coefficient r yx is the slope of the predicting line 
 or what is sometimes called the line of regression. If 0 < r< 1, the 
 tangent of the angle which this line makes with the horizontal axis 
 is always less than or equal to 1. Call this angle 6. In this case d 
 will be less than 45°. If r is negative, d will satisfy the inequalities, 
 135° 5S 6 ^ 180. The angles d are shown in Figs. 10.2 and 10.3. 
 
 Figures 10.2 and 10.3 show parts of the predicting line plotted 
 in standard units and the values of d when r is positive and negative. 
 
 Fig. 10.2 Fig 10.3 
 
 If x is predicted from y, another predicting equation is obtained. 
 Let it be 
 
 x = c + dy, or x = dy, or t x = r zy t y 
 
 Let B be the angle between the predicting line for y and the 
 predicting line for x. Examine the above figures. If B is zero 
 there is perfect correlation. The size of B gives information con¬ 
 cerning the correlation between the two variables. If r is positive, 
 0 5= B 5= 90. If B is large there is very little correlation between 
 the two variables. 
 
 PROBLEM 
 
 1. Find the predicting equation for predicting heights from weights 
 from data on page 151. Plot the predicting equation for weights from 
 heights and the predicting equation for heights from weights. Plot these 
 two equations on the same graph in terms of the deviations from the 
 means. Plot these two lines on another graph in terms of standard 
 units. Find the size of 6, of \P, and of angle B. 
 
Weight Measurements (y) Pounds 
 
 178 
 
 CORRELATION COEFFICIENT 
 
 Correlation Table of Shoulder and 
 
 Shoulder 
 
 Classes 
 
 
 13.95 
 
 to 
 
 14.45 
 
 14.45 
 
 to 
 
 14.95 
 
 14.95 
 
 to 
 
 15.45 
 
 15.45 
 
 to 
 
 15.95 
 
 
 Class 
 
 Mark 
 
 
 14.20 
 
 14.70 
 
 15.20 
 
 15.70 
 
 
 
 / 
 
 
 6 
 
 15 
 
 49 
 
 68 
 
 
 d 
 
 
 - 5 
 
 - 4 
 
 - 3 
 
 - 2 
 
 
 df 
 
 
 - 30 
 
 - 60 
 
 -147 
 
 -136 
 
 
 <p-f 
 
 150 
 
 240 
 
 441 
 
 272 
 
 209.5 
 to 
 
 219.5 
 
 214.5 
 
 1 
 
 7 
 
 7 
 
 49 
 
 
 
 
 
 199.5 
 to 
 
 209.5 
 
 204.5 
 
 2 
 
 6 
 
 12 
 
 72 
 
 
 
 
 
 189.5 
 to 
 
 199.5 
 
 194.5 
 
 7 
 
 5 
 
 35 
 
 175 
 
 
 
 
 
 179.5 
 to 
 
 189.5 
 
 184.5 
 
 10 
 
 4 
 
 40 
 
 160 
 
 
 
 
 
 169.5 
 to 
 
 179.5 
 
 174.5 
 
 24 
 
 3 
 
 72 
 
 216 
 
 
 
 
 
 159.5 
 to 
 
 169.5 
 
 164.5 
 
 36 
 
 2 
 
 72 
 
 144 
 
 
 -8 
 
 1 
 
 -8 
 
 
 -4 
 
 1 
 
 -4 
 
 149.5 
 to 
 
 159.5 
 
 154.5 
 
 97 
 
 1 
 
 97 
 
 97 
 
 
 
 -3 
 
 1 
 
 -3 
 
 -2 
 
 2 
 
 -4 
 
 139.5 
 to 
 
 149.5 
 
 144.5 
 
 157 
 
 0 
 
 0 
 
 
 
 
 0 
 
 5 
 
 0 
 
 0 
 
 9 
 
 0 
 
 129.5 
 to 
 
 139.5 
 
 134.5 
 
 174 
 
 -1 
 
 -174 
 
 174 
 
 5 
 
 1 
 
 5 
 
 4 
 
 2 
 
 8 
 
 3 
 
 11 
 
 33 
 
 2 
 
 14 
 
 28 
 
 119.5 
 to 
 
 129.5 
 
 124.5 
 
 129 
 
 -2 
 
 -258 
 
 516 
 
 
 8 
 
 5 
 
 40 
 
 6 
 
 17 
 
 102 
 
 4 
 
 23 
 
 92 
 
 109.5 
 to 
 
 119.5 
 
 114.5 
 
 67 
 
 -3 
 
 -201 
 
 603 
 
 15 
 
 1 
 
 15 
 
 12 
 
 4 
 
 48 
 
 9 
 
 11 
 
 99 
 
 6 
 
 17 
 
 102 
 
 99.5 
 
 to 
 
 109.5 
 
 104.5 
 
 13 
 
 -4 
 
 - 52 
 
 208 
 
 20 
 
 3 
 
 60 
 
 16 
 
 3 
 
 48 
 
 12 
 
 4 
 
 48 
 
 8 
 
 2 
 
 16 
 
 89.5 
 to 
 
 99.5 
 
 94.5 
 
 1 
 
 -5 
 
 - 5 
 
 25 
 
 25 
 
 1 
 
 25 
 
 
 
 
 Total 
 
 
 718 
 
 
 — 355 
 
 2V 
 
 2,439 
 
 ry'2 
 
 105 
 
 136 
 
 279 
 
 230 
 
PROBLEM 
 
 179 
 
 Weight Measurements of University Men 
 
 Measurements (x) Inches 
 
 15.95 
 
 to 
 
 16.45 
 
 16.45 
 to 
 
 16.95 
 
 16.95 
 
 to 
 
 17.45 
 
 17.45 
 
 to 
 
 17.95 
 
 17.95 
 
 to 
 
 18.45 
 
 18.45 
 
 to 
 
 18.95 
 
 18.95 
 
 to 
 
 19.45 
 
 19.45 
 
 to 
 
 19.95 
 
 Totals 
 
 16.20 
 
 16.70 
 
 17.20 
 
 17.70 
 
 18.20 
 
 18.70 
 
 19.20 
 
 19.70 
 
 
 170 
 
 129 
 
 141 
 
 74 
 
 39 
 
 14 
 
 12 
 
 1 
 
 
 - 1 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 718 
 
 -170 
 
 0 
 
 141 
 
 148 
 
 117 
 
 56 
 
 60 
 
 6 
 
 -15 Zx' 
 
 170 
 
 
 141 
 
 296 
 
 351 
 
 224 
 
 300 
 
 36 
 
 2,621 Zx'2 
 
 
 
 
 
 
 
 35 
 
 1 
 
 35 
 
 
 35 
 
 
 
 
 
 
 24 
 
 1 
 
 24 
 
 30 
 
 1 
 
 30 
 
 
 54 
 
 
 0 
 
 1 
 
 0 
 
 5 
 
 1 
 
 5 
 
 10 
 
 2 
 
 20 
 
 15 
 
 1 
 
 15 
 
 20 
 
 1 
 
 20 
 
 25 
 
 0 
 
 0 
 
 30 
 
 1 
 
 30 
 
 90 
 
 
 0 
 
 1 
 
 0 
 
 4 
 
 1 
 
 4 
 
 8 
 
 1 
 
 8 
 
 12 
 
 2 
 
 24 
 
 16 
 
 3 
 
 48 
 
 20 
 
 2 
 
 40 
 
 
 124 
 
 -3 
 
 2 
 
 -6 
 
 0 
 
 2 
 
 0 
 
 3 
 
 3 
 
 9 
 
 6 
 
 4 
 
 24 
 
 9 
 
 6 
 
 54 
 
 12 
 
 3 
 
 36 
 
 15 
 
 4 
 
 60 
 
 
 177 
 
 -2 
 
 2 
 
 -4 
 
 0 
 
 4 
 
 0 
 
 2 
 
 11 
 
 22 
 
 4 
 
 9 
 
 36 
 
 6 
 
 6 
 
 36 
 
 8 
 
 1 
 
 8 
 
 10 
 
 1 
 
 10 
 
 
 96 
 
 - 1 
 12 
 
 -12 
 
 0 
 
 8 
 
 0 
 
 1 
 
 20 
 
 20 
 
 2 
 
 30 
 
 60 
 
 3 
 
 17 
 
 51 
 
 4 
 
 4 
 
 16 
 
 5 
 
 3 
 
 15 
 
 
 143 
 
 0 
 
 19 
 
 0 
 
 0 
 
 48 
 
 0 
 
 0 
 
 47 
 
 0 
 
 0 
 
 21 
 
 0 
 
 0 
 
 7 
 
 0 
 
 0 
 
 1 
 
 0 
 
 0 
 
 0 
 
 0 
 
 
 1 
 
 61 
 
 61 
 
 0 
 
 38 
 
 0 
 
 -1 
 
 40 
 
 -40 
 
 -2 
 
 7 
 
 -14 
 
 0 
 
 
 
 
 81 
 
 2 
 
 47 
 
 94 
 
 0 
 
 22 
 
 0 
 
 -2 
 
 15 
 
 -30 
 
 
 
 
 
 
 298 
 
 3 
 
 27 
 
 81 
 
 0 
 
 4 
 
 0 
 
 -3 
 
 3 
 
 -9 
 
 
 
 
 
 
 336 
 
 
 0 
 
 1 
 
 0 
 
 
 
 
 
 
 
 172 
 
 
 
 
 
 
 
 
 
 25 
 
 214 
 
 
 -19 
 
 134 
 
 180 
 
 152 
 
 190 
 
 30 
 
 1,631 Zx'y ' 
 
180 
 
 CORRELATION COEFFICIENT 
 
 A METHOD FOR COMPUTING THE CORRELATION COEFFICIENT 
 WHEN THERE IS A LARGE NUMBER OF ITEMS 
 
 When there is a large number of variates for the variables under 
 consideration it would require a great deal of trouble to find the 
 sum of the product terms by the method given above. One way 
 of reducing the amount of computing is to group the variates for 
 the two sets of data. The classes for the x variates may not be 
 the same size as the classes for the y variates. After the data have 
 been grouped into their respective classes the class marks are then 
 the representatives of the variates falling in the different classes. 
 
 One of the best ways of understanding this method is to go 
 through an example. On page 178 are a two-way correlation 
 table and the necessary calculations for finding the coefficient 
 of correlation. The quantity x represents shoulder measurements 
 of university men; y represents their weights. Shoulder measure¬ 
 ments were put into classes of § inch; the weights were grouped 
 into classes of 10 pounds. The provisional mean for the x’s is 
 16.70; that for the y’s is 144.5 pounds. These are indicated by the 
 heavy lines near the middle of the table. The figures in the center 
 of the squares represent frequencies of the double classes. For 
 example, there are 9 men with shoulder measurements between 
 
 17.45 and 17.95 inches and with weights between 159.5 and 169.5 
 pounds; there are 23 men with shoulder measurements between 
 
 15.45 and 15.95 inches and with weights between 119.5 and 129.5 
 pounds, etc. There are 3 men in the double class 18.45-18.95 
 inches and 169.5-179.5 pounds. This class is 4 units from the 
 provisional mean for the ar’s and 3 units from the provisional mean 
 for the y’s. The product of these is 12 double class units or prod¬ 
 uct units. This 12 appears in the upper right-hand corner of this 
 square. Since the frequency of this double class is 3, there are 
 hence 3 X 12 double units for this class. This 36 is placed in the 
 lower left-hand corner of the square. The double class 15.95-16.45 
 inches and 169.5-179.5 pounds is — 1 class unit from the provisional 
 mean for the z’s and +3 class units from the provisional mean for 
 the y’s, hence this class is —3 double class units from the origin 
 (16.70 inches, 144.5 pounds). Since the frequency of this class is 2, 
 there are —6 double units for this class. This —6 appears in the 
 lower left-hand corner of this square. The sum of all the numbers 
 in the lower left-hand corners of the squares gives the cross product 
 
COMPUTING THE CORRELATION COEFFICIENT 181 
 
 term 2 x'y'. In this case 2 x'y' = 1,631, which is found in the 
 extreme lower right-hand square. It is the sum of the product 
 terms for the squares in the table, and can be found by adding the 
 product terms vertically or horizontally. The entire object of the 
 above correlation table is to find the cross product term 2 x'y'. 
 
 The column and the row d-f enable one to secure the means for 
 the two variables and also the sums 2a:' and 2 y'. The column 
 and row d 2 f enable one to secure the second moments for the two 
 sets and also the sums 2x' 2 and 2 y' 2 , from which the standard 
 deviations can be obtained. 
 
 From the correlation table one gets 
 
 2 y' = - 355, 2zj = - 15, 2 y' 2 = 2,439, 2a;' 2 = 2,621, 
 
 2 x'y' = 1,631, 
 
 from which the following are obtained 
 
 2 y' 2 = 2,263.48, 2x' 2 = 2,620.69, 2x'y' = 1,623.58. 
 
 Since the data are grouped, corrections must be made in finding 
 the standard deviations. 
 
 (Ty 
 
 Ox 
 
 263.48 
 
 718 
 
 620.69 
 
 718 
 
 ) 
 
 - 0.08333 
 
 ) 
 
 - 0.08333 
 
 10 = 17.52, 
 
 0.5 = 0.94. 
 
 Therefore the correlation coefficient is 
 
 2 xy [ 1,623.58] 10(0.5) 8,117.90 
 
 Tvx ~ na x -<j v ~ 718(17.52)(0.94) ~ 11,824.45 
 
182 
 
 CORRELATION COEFFICIENT 
 
 PROBLEMS 
 
 1. Given the following data for freshmen English average and the 
 average made in all subjects during the 4 years in college, for women at 
 the University of Texas. The grades were made to the nearest unit. 
 Group the data in 5’s and find the correlation coefficient between freshmen 
 English average and 4-vear average in all subjects. Use 64.5-69.5 as the 
 first class for English grades and 59.5-64.5 for the first class for the four- 
 year averages. 
 
 English 4-Year English 4-Year English 4-Year English 4-Year English 4-Year 
 
 Aver¬ 
 
 Aver¬ 
 
 Aver¬ 
 
 Aver¬ 
 
 Aver¬ 
 
 Aver¬ 
 
 Aver¬ 
 
 Aver¬ 
 
 Aver¬ 
 
 Aver¬ 
 
 age 
 
 age 
 
 age 
 
 age 
 
 age 
 
 age 
 
 age 
 
 age 
 
 age 
 
 age 
 
 X 
 
 V 
 
 X 
 
 V 
 
 X 
 
 V 
 
 X 
 
 V 
 
 X 
 
 V 
 
 85 
 
 84 
 
 85 
 
 81 
 
 92 
 
 86 
 
 75 
 
 81 
 
 68 
 
 76 
 
 78 
 
 74 
 
 85 
 
 84 
 
 68 
 
 74 
 
 78 
 
 70 
 
 82 
 
 91 
 
 75 
 
 79 
 
 82 
 
 85 
 
 88 
 
 81 
 
 S5 
 
 77 
 
 82 
 
 76 
 
 68 
 
 73 
 
 75 
 
 72 
 
 85 
 
 70 
 
 80 
 
 78 
 
 78 
 
 73 
 
 82 
 
 71 
 
 92 
 
 94 
 
 85 
 
 78 
 
 75 
 
 75 
 
 72 
 
 77 
 
 82 
 
 82 
 
 88 
 
 89 
 
 71 
 
 71 
 
 72 
 
 73 
 
 85 
 
 79 
 
 85 
 
 84 
 
 92 
 
 92 
 
 65 
 
 68 
 
 79 
 
 75 
 
 85 
 
 86 
 
 90 
 
 72 
 
 82 
 
 74 
 
 88 
 
 82 
 
 75 
 
 75 
 
 78 
 
 76 
 
 92 
 
 92 
 
 68 
 
 80 
 
 78 
 
 80 
 
 75 
 
 78 
 
 85 
 
 79 
 
 88 
 
 82 
 
 65 
 
 67 
 
 78 
 
 80 
 
 88 
 
 85 
 
 95 
 
 87 
 
 72 
 
 78 
 
 65 
 
 69 
 
 95 
 
 88 
 
 68 
 
 83 
 
 68 
 
 76 
 
 85 
 
 80 
 
 82 
 
 80 
 
 95 
 
 85 
 
 65 
 
 77 
 
 72 
 
 80 
 
 78 
 
 71 
 
 85 
 
 85 
 
 88 
 
 90 
 
 82 
 
 85 
 
 72 
 
 80 
 
 81 
 
 85 
 
 82 
 
 80 
 
 65 
 
 74 
 
 75 
 
 79 
 
 75 
 
 79 
 
 68 
 
 62 
 
 85 
 
 87 
 
 75 
 
 76 
 
 68 
 
 71 
 
 78 
 
 85 
 
 72 . 
 
 76 
 
 65 
 
 71 
 
 92 
 
 84 
 
 82 
 
 83 
 
 65 
 
 80 
 
 68 
 
 83 
 
 85 
 
 87 
 
 75 
 
 82 
 
 75 
 
 81 
 
 95 
 
 90 
 
 85 
 
 85 
 
 82 
 
 74 
 
 95 
 
 88 
 
 88 
 
 84 
 
 85 
 
 85 
 
 82 
 
 76 
 
 78 
 
 83 
 
 68 
 
 72 
 
 85 
 
 90 
 
 
 
 75 
 
 78 
 
 72 
 
 83 
 
 75 
 
 71 
 
 88 
 
 86 
 
 
 
 2. Find the correlation coefficient between weights and right thigh 
 measurements for the data given in the following table. Assume a linear 
 relation. Find the predicting equation for predicting weights from right 
 thigh measurements. Find the mean weight for each group of thigh 
 measurements. Plot these means on the same plot with the predicting 
 line. Draw lines parallel to the predicting line at a distance of 1 cr e from 
 it. What percentage of these means falls in this band? What value 
 does the predicting line give for any thigh measurement? 
 
Weights and Right Thigh Measurements of Freshmen at University of Michigan 
 
 PROBLEMS 
 
 183 
 
 ipui o[/[ isoirou oq. s^uauwjnsuam PlUiR 
 
184 
 
 CORRELATION COEFFICIENT 
 
 THE VALUE OF THE CORRELATION COEFFICIENT IN PREDICTING 
 
 Consider the following table for various values of the correlation 
 coefficient: 
 
 ryx 
 
 1 - r 2 
 
 C-J 
 
 1 
 
 > 
 
 1 - vT 
 
 0.1 
 
 0.99 
 
 0.995 
 
 0.005 
 
 .2 
 
 .96 
 
 .980 
 
 .020 
 
 .3 
 
 .91 
 
 .954 
 
 .046 
 
 .4 
 
 .84 
 
 .947 
 
 .053 
 
 .5 
 
 .75 
 
 .886 
 
 .134 
 
 .6 
 
 .64 
 
 .800 
 
 .200 
 
 .7 
 
 .51 
 
 .714 
 
 .286 
 
 .8 
 
 .36 
 
 .600 
 
 .400 
 
 .9 
 
 .19 
 
 .436 
 
 .564 
 
 .92 
 
 .15 
 
 .392 
 
 .608 
 
 .94 
 
 .12 
 
 .341 
 
 .659 
 
 .96 
 
 .08 
 
 .280 
 
 .820 
 
 .98 
 
 .04 
 
 .198 
 
 .811 
 
 1.00 
 
 0.00 
 
 0.000 
 
 1.000 
 
 The question arises as to the importance of r yx in predicting 
 values of one variable when values of the others are known. If 
 the correlation coefficient r yx = 0 , the predicting equation reduces 
 to y = M y , which means that for each value of x the predicted 
 value of y is always equal to M y . This will be considered a mere 
 guess for the value of y. In this case the standard error of predic¬ 
 tion is a e = a y . If r yx = 0.5, then a e = 0 . 8660 -^; this value of a e is 
 0.134 of what it was when y was predicted by a mere guess. If all 
 the y ’s had been multiplied by 0 . 866 , then <r e = 0.866 a y and the 
 mean of these y 's would have been 0.8664/,,. From the informa¬ 
 tion available the best predicting equation would be y = 0.8664/„, 
 which is absurd, since y and x are connected by some equation and 
 the value of y cannot be constant for all values of x. This shows 
 that, if one knows that two variables are connected in such a way 
 as to give a linear correlation coefficient of 0.5 and nothing more is 
 known about the two variables, the best predicting value of y would 
 be 0.8664/,,, where M y is the mean of all values of y. This goes to 
 show that the correlation coefficient is not very helpful in predict¬ 
 ing values of one variable from values of the other. * The essentials 
 
 * The value of r can be used in predicting if the variables are in standard 
 units. 
 
PROBLEMS 
 
 185 
 
 in predicting are the predicting equation, y = a + bx, and the 
 standard error of prediction. 
 
 The quantity 1 — V l — r 2 shows how much worse off one is 
 by using values of x and the value of the correlation for predicting 
 values of y. 
 
 If r = 0.6, how much has the standard error of prediction been 
 reduced from that obtained by making a mere guess, that is, when 
 y = My is used as the predicting equation? What is 1 — a/I —r 2 ? 
 
 COEFFICIENT OF DETERMINATION 
 
 It is interesting to note that the equation a e — c/vl 
 may be solved for r, giving 
 
 — r„ 
 
 ( 10 . 12 ) 
 
 -xM- 
 
 This shows that as a e becomes smaller the correlation coefficient r yx 
 becomes larger, and as a e approaches <r y the value of r approaches 0. 
 
 The value of r 2 is sometimes called the coefficient of determina¬ 
 tion for it measures the per cent of the variability of the dependent 
 set of variates determined from the independent. This coefficient 
 of determination is 
 
 It is really the ratio of the difference between a y 2 and a? to a u 2 . If 
 c e 2 represents the variability of the values of the variates of the 
 dependent variable about the predicting equation and a 2 repre¬ 
 sents the variability of these variates about their mean, then 
 a 2 — <y 2 represents the part of the variability of the dependent 
 variable accounted for by the variability in the independent 
 variable. This coefficient, r 2 , is the ratio of the variability of y 
 due to the variability in x and the variability of y, since a y 2 = a v 2 
 (1 — r 2 ) + r 2 a v 2 . 
 
 PROBLEMS 
 
 1. Given that y and x are connected linearly, and the following: 
 
 M x = 34.7, M y = 67.3, a x = 1.9, a u = 2.3, r yx = 0.01. 
 
 Write the predicting equation; discuss it as a predicting equation in com¬ 
 parison with y = M v as a predicting equation. lias the correlation 
 coefficient helped in predicting values of y? Examine b. 
 
 2. Suppose that waist measurements, x, and weights, y, are connected 
 linearly and that r yx = 0.87, <j y = 19.3 pounds, <r x = 1.4 inches. Find the 
 value of y when x is 1.3 standard deviations above the mean of the x’s. 
 
186 
 
 CORRELATION COEFFICIENT 
 
 3. Using the data in problem 2, find the predicted value of y if in 
 addition it is known that M v = 139.6 pounds, M x = 29.8 inches, and x 
 is equal to 30.9 inches. 
 
 4. If a e = 15 pounds, and r = 0.8, find a y . 
 
 MULTIPLE CORRELATION 
 
 Multiple correlation shows in a measure how one variable 
 depends upon two or more variables. The multiple correlation 
 coefficient measures the amount of dependence one variable has 
 upon two or more variables when the variables are connected in 
 pairs by linear relations and a “plane” is the surface which fits the 
 data best. If weights, heights, and chest measurements are con¬ 
 nected linearly, the multiple correlation coefficient gives, to some 
 extent, the amount of dependence of the weight of an individual 
 upon his height and chest measurement. This coefficient will be 
 derived by a method similar to that for obtaining the Pearson 
 linear correlation coefficient. 
 
 Let y, x, and z be connected by the linear relation 
 y = a + bx + cz, or y = bx + cz. 
 The normal equations are 
 
 2x 2 -6 + 2xz-c = 2 xy 
 Zxzb + 2 z 2 -c = 2 zy, 
 
 from which 
 
 2z 2 2 xy — 'ZxzUzy 
 = 2x 2 22 2 - (2xf) 2 ’ 
 
 The linear relation is 
 
 and c = 
 
 2x 2 2 zy — 'Zzx'Zxy 
 2x 2 2z 2 - (2lx) 2 ‘ 
 
 (10.13) y = 
 
 2z 2 2xy — 2lx2zy 
 
 2x 2 2z 2 
 
 x + 
 
 (2xl) 2 
 
 The standard error of prediction is as before 
 
 <7 e = V[^y 2 -bZxy-cZzy]/n, 
 
 f" 2x 2 2 zy — Hzx'Zjxy 
 
 L 2x 2 2z 2 - (2x 
 
 2x 
 (2xz) 2 J 
 
 or 
 
 a.— 
 
 /J„_, [2z 2 2xy — 2xz2zy]2xy 
 
 [2x 2 2zy — 2xz2xy] 2 zy 
 
 / \ y 2x 2 2z 2 — (2xz) 2 
 
 2x 2 2z 2 — (2xz) 2 J 
 
MULTIPLE CORRELATION 
 
 187 
 
 “V 2# 
 
 2z 2 (2xy) 2 — 2xf2z?/2xi!/ + 2x 2 (2z?/) 2 — 2xz Hxyllzy 
 
 >*7 2 — —- 
 
 2x 2 2 Z 2 -(2xz) 2 
 
 W?[> 
 
 2z 2 (2xy) 2 + 2x 2 (2zy) 2 — 22xz2xy2zy 
 [2x 2 2z 2 — (2xz) 2 ]2?/ 2 
 
 — (Xy A/ 1 Ry.xJ^j 
 where 
 
 /2z 2 (2xy) 2 — 22xy2xz2zy + 2x 2 (2zy) 2 
 (10.14) R u .„ =\ [2x 2 2z 2 — (2xz) 2 ]2y 2 
 
 which is called the multiple correlation coefficient. 
 The standard error of estimate may be written as 
 
 I (2 xy) 2 (2zy) 2 2IxzZzy2xy 
 
 I 2x 2 2 y 2 + 2z 2 2?/ 2 2z 2 2y 2 2x 2 
 
 X 2x 2 2z 2 
 
 or 
 
 r 2 TyxT xz r zy T~ Ty? 
 
 (10.15) — O'y'V 1 — 1 _ 2 > 
 
 -*■ ' xz 
 
 hence the multiple correlation coefficient is equal to 
 
 _ ~ C 2‘ r yx r vz r xz ~t~ r zd 
 
 (10.16) Ry.xi — "Y 1 _ r 2 
 
 The multiple correlation coefficient is given in (10.14) in terms 
 of the fundamental summations; it is given in terms of the Pearson 
 linear correlation coefficients in (10.16). Formula (10.14) is the 
 most practical one to use. 
 
 As in the Pearson linear correlation coefficient, the multiple 
 correlation coefficient cannot be greater than 1 or less than — 1. 
 It is always considered to be positive. 
 
 The symbol Ry. xl means the multiple correlation coefficient 
 between y and x and z. It arose from a linear relation connecting 
 y with x and z. The points are scattered about a plane in three 
 dimensions. 
 
1S8 
 
 CORRELATION COEFFICIENT 
 
 Example. The multiple correlation coefficient between weight, and 
 heights, and right thigh measurements, by using data on page 165, and 
 formula (10.14), is 
 
 R 
 
 icl. ht ch 
 
 / ( 30.605) (265.5) 2 - 2(265.5) (20.3) (290.6) + 66.98(290.6) 2 
 
 ” \ [(66.98) (30.605) - (20.3) 2 ] (3,260.9) 
 
 = 0.947. 
 
 PROBLEMS 
 
 1. If the multiple correlation coefficient is 1, and r vx = 0.1, r s „ = 0.2, 
 find r xz , and discuss your results. 
 
 2. (a) If r IV = r 2v = 0, what does R u . xz equal? Discuss this situation. 
 (5) If r xv = R v . X z — 0, what does r zv equal? Discuss this situation. 
 
 3. Discuss the situation when r xv = ?•*„ = 1 in formula (10.15). 
 Make a drawing to illustrate. 
 
 4. Find the multiple correlation coefficients between weights, heights, 
 and chest measurements, if it is assumed that y = a + bx + cz, where 
 y represents weights, x heights, and z chest measurements. Use the data 
 on pages 151 and 157. 
 
 5. Assume that the four-year average grade is linearly connected with 
 freshman mathematics grades and freshman English grades. Find the 
 multiple correlation coefficient between four-year average and freshman 
 mathematics and English grades if the following data are given for men. 
 These data were taken from records at the University of Texas. 
 
 Mathe¬ 
 
 English 
 
 4-Year 
 
 Mathe¬ 
 
 English 
 
 4-Year 
 
 matics 
 
 Averages Averages 
 
 matics 
 
 Averages 
 
 Averages 
 
 Averages 
 
 
 
 Averages 
 
 
 
 68 
 
 75 
 
 74 
 
 79 
 
 68 
 
 72 
 
 75 
 
 82 
 
 74 
 
 68 
 
 72 
 
 68 
 
 78 
 
 82 
 
 84 
 
 85 
 
 75 
 
 79 
 
 95 
 
 88 
 
 90 
 
 72 
 
 82 
 
 78 
 
 82 
 
 72 
 
 79 
 
 88 
 
 68 
 
 77 
 
 65 
 
 68 
 
 67 
 
 85 
 
 86 
 
 84 
 
 78 
 
 92 
 
 80 
 
 85 
 
 85 
 
 85 
 
 75 
 
 68 
 
 76 
 
 90 
 
 88 
 
 90 
 
 85 
 
 81 
 
 80 
 
 79 
 
 82 
 
 79 
 
 72 
 
 75 
 
 76 
 
 78 
 
 68 
 
 70 
 
 82 
 
 82 
 
 78 
 
 50 
 
 65 
 
 72 
 
 72 
 
 60 
 
 72 
 
 82 
 
 75 
 
 85 
 
 82 
 
 78 
 
 80 
 
 
 
 
MULTIPLE CORRELATION—OVER THREE VARIABLES 189 
 
 6. Discuss the situation when r xz = 1, in formula (10.16). 
 
 7. When will r xy = r xz = r zy = R y xz l 
 
 8. Given that r vx = 0.62, r vz — 0.94, r zz = 0.34; find R v _ xz . 
 
 9. Express the predicting equation y = bx + cz in terms of standard 
 deviations, and correlation coefficients. 
 
 10. Find Ry.xz when r xy = r xz = r zy — 0.999. 
 
 MULTIPLE CORRELATION WHEN THERE ARE MORE THAN 
 THREE VARIABLES 
 
 Assume that the variables y, x\, X 2 , . . . , x a are connected by 
 the linear relation 
 
 y = cixi + c 2 x 2 + . . . + c a x a . 
 
 The normal equations are 
 
 ciZxi 2 + c 2 2 xiX 2 + . . . + c s 2* \x a = 2a hy, 
 
 Ci2xjX 2 + C22X2 2 + . . . + C s 2*2*8 = 2x2 y, 
 
 Cl 2*1*3 + C22x2X3 + . . . + C 3 2x3X s = 2x3 y, 
 
 Ci2xix s + c 2 ^x 2 x a + . . . + c s 2x a 2 = 2 x s y, 
 
 from which 
 
 2*i y 
 2*2 y 
 
 2 x 3 y 
 
 2xiX2 
 
 2* 2 2 
 
 2*2*3 
 
 2*1*3 
 
 2*2*3 
 
 2x 3 2 
 
 2*1*8 
 
 2*2*8 
 
 2*3*8 
 
 S x a y 
 
 2X2* 8 
 
 2*3*8 
 
 2 
 
 2xixi 
 
 2*1*2 
 
 2*1*3 
 
 2*1*8 
 
 2xix 2 
 
 2*2*2 
 
 2*2*3 
 
 2*2*8 
 
 2*1*3 
 
 2*2*3 
 
 2*3*3 
 
 2*3*, 
 
 2*i*, 
 
 2*2*8 
 
 2*3*, 
 
 2*,*8 
 
190 
 
 CORRELATION COEFFICIENT 
 
 Remembering that 2xiX,- = no H ■ o Xj ■ r XjXj . the quantity ci becomes 
 
 Cl 
 
 n<T Xl (7y 
 
 r x\v 
 
 no Xl cr X2 T Xl x 2 
 
 
 no Xl o x r ZiXa 
 
 ]/ 
 
 r HV 
 
 no z 2 r HZ% 
 
 
 no X2 o Xa r XiXii 
 
 Tiff xj£T^ 
 
 r *3l/ 
 
 n<7 X 2°’l 3 r x 2 I, 
 
 
 no Xl o x r X3xs 
 
 na I> a i 
 
 fx a y 
 
 no X2 <j x r xlls 
 
 
 n<Jx ? r Ie x s 
 
 no x 2 • 
 
 1 
 
 no Zl o l2 r XlZi 
 
 
 no Zl (T x r XlXa 
 
 Tiff X] ff zja-2 
 
 no^ ■ 1 
 
 
 no X2 o Za r XiXa 
 
 Tiff X) ff xJT 
 
 no l2 c x 3 ■ i'z 2 x l 
 
 
 no X3 o Xa r XjXa 
 
 n<Xx l <Tx , r ' x l x , 
 
 nox 2 o Xe i' X2 i a 
 
 
 no x 2 -1 
 
 
 r i 1 y 
 
 r *\ x % 
 
 T x \ x z 
 
 Tx \ x , 
 
 
 r x 2 y 
 
 r x 2 x 2 
 
 rx 2 x ) 
 
 r x 2 i t 
 
 a v 
 
 r x 3 V 
 
 r HH 
 
 r V3 
 
 rx 3 x t 
 
 
 T x,v 
 
 r *i*» 
 
 Tx 3 x » 
 
 Tx , x , 
 
 
 1 
 
 r x 1 x 2 
 
 Vx l x 3 
 
 Tx l x a 
 
 
 T X\ x 2 
 
 1 
 
 r H x l 
 
 rx 2 z s 
 
 ° x i 
 
 r x 2 x, 
 
 r x 2 x i 
 
 1 
 
 r x 3 x a 
 
 
 r *l x t 
 
 r *t x s 
 
 r H x » 
 
 1 
 
 ov D\ 
 o Xl ■ D 0 ’ 
 
 where D i is equal to the determinant in the numerator and Do is 
 the determinant in the denominator. 
 
 In a similar way it can be shown that 
 
 OyDj 
 o H ■ Do 
 
 Hence the predicting equation becomes 
 
 (10.17) 
 
 or 
 
 <7 y \ ^ D iAJ x ^ (Ty \ ^ Di 
 
 Do <7 Xi Do <7 Xi 
 
 V = My 
 
MULTIPLE CORRELATION—OVER THREE VARIABLES 191 
 
 The standard error of estimate is 
 
 (10.18) a. 
 
 1 Di 
 
 —ft / y 
 
 na y D 0 <r x . f—' 
 
 (10.19) 
 
 j = i. 
 
 which is the multiple correlation coefficient between y and the 
 other variables. 
 
 Example. 
 
 Given that y = 
 
 ■■ CiXi 
 
 + c 2 x 2 + c 3 x 3 , and 
 
 
 My 
 
 = 36.6, 
 
 <Ty = 
 
 1.9, 
 
 T yi x 
 
 — -52, t X iX3 
 
 = 0.43, 
 
 M Xx 
 
 = 13.2, 
 
 O’xj = 
 
 0.8, 
 
 r VX 2 
 
 — .74, ^x 2 x 2 
 
 = 0.51. 
 
 M Xi 
 
 = 20.2, 
 
 &X2 = 
 
 1.2, 
 
 T *V3 
 
 = 0.80, 
 
 
 M x 3 
 
 = 29.4, 
 
 <7*3 = 
 
 1.7, 
 
 r x l x 2 
 
 = 0.93; 
 
 
 Find the predicting equation and the standard error of prediction. 
 
 Ci = 
 
 c 2 = 
 
 1.7 
 
 
 0.52 
 
 0.93 
 
 0.43 
 
 
 (1.9) 
 
 0.74 
 
 1 
 
 0.51 
 
 
 
 0.80 
 
 0.51 
 
 1 
 
 (1.9) (-0.11) 
 
 
 1 
 
 0.93 
 
 0.43 
 
 (0.8) (0.10) 
 
 (0.8) 
 
 0.93 
 
 1 
 
 0.51 
 
 
 
 0.43 
 
 0.51 
 
 1 
 
 
 
 1 
 
 0.52 
 
 0.43 
 
 
 (1.9) 
 
 0.93 
 
 0.74 
 
 0.51 
 
 
 
 0.43 
 
 0.80 
 
 1 
 
 (1.9) (+0.15) 
 
 1.2 
 
 
 0.10 
 
 
 (1.2) (0.10) 
 
 
 1 
 
 0.93 
 
 0.52 
 
 
 (1.9) 
 
 0.93 
 
 1 
 
 0.74 
 
 
 
 0.43 
 
 0.51 
 
 0.80 
 
 (1.9) (+0.05) 
 
 = -2.63, 
 
 (1.7) (0.10) 
 
 = +2.38, 
 
 = 0.56. 
 
 0.10 
 
192 
 
 CORRELATION COEFFICIENT 
 
 The predicting equation is 
 
 y = —2.63 xi + 2.38 x 2 + 0.56 x 3 . 
 
 <r e = (1.91 - [(-0.11) (0.52) + (0.15) (0.74) + (0.05) (0.80)] 
 
 = 0.48. 
 
 The multiple correlation coefficient is 
 
 [(-0.11) (0.52) + (0.15) (0.74) + (0.05) (0.80)] 
 
 = 0.97. 
 
 PROBLEMS 
 
 1. Using the data on pages 151, 157, and 161 concerning weights, 
 heights, chest measurements, and right thigh measurements, find the 
 multiple correlation, /? wt . ht . ch . th . by means of the correlation coefficients. 
 Write Do in terms of the r ’s and find its numerical value. 
 
 2. Derive a formula for c 5 as was done for C\. 
 
 3. Assume that weight is connected linearly -with height, waist 
 measurement, hip measurement, shoulder measurement, and chest 
 measurement. Find the multiple correlation between weights and heights, 
 waist measurements, shoulder measurements, hip measurements, chest 
 measurements, for the data on page 193 if y = weights, Xi = heights, 
 x 2 = shoulder measurements, x 3 = chest measurements, x 4 = waist 
 measurements, x 3 = hip measurements. Let different members of the 
 class do certain parts of this problem. 
 
 4. Assume that y is connected linearly with the 3 variables x h x 2 , and 
 x 3 , by the relation y = c 0 + CiXi + c 2 x 2 + c 3 x 3 , and that the following 
 information is given concerning the various means, standard deviations, 
 and correlation coefficients: 
 
 My 
 
 = 140.2, 
 
 <j v = 
 
 18.1, 
 
 fyxi 
 
 = 0.71, 
 
 r xx x 3 = 0.47, 
 
 M Xl 
 
 = 67.6, 
 
 
 2.5, 
 
 Tyx 2 
 
 = 0.80, 
 
 r x 2 i 3 = 0.76, 
 
 M X1 
 
 = 36.8, 
 
 = 
 
 1.9, 
 
 r vz 3 
 
 = 0.92, 
 
 
 M x 3 
 
 = 20.1, 
 
 = 
 
 1.3, 
 
 t x\x 2 
 
 = 0.54. 
 
 
 Find the D’s, the predicting equation, and the standard error of predic¬ 
 tion. Find the multiple correlation between y and the other variables. 
 
PROBLEMS 
 
 193 
 
 Data* for Problem 3, Page 192 
 
 Weight 
 
 V 
 
 Height 
 
 X 
 
 Shoul¬ 
 
 der 
 
 z 
 
 Chest 
 
 w 
 
 Waist 
 
 u 
 
 Hip 
 
 s 
 
 Weight 
 
 y 
 
 Height 
 
 X 
 
 Shoul¬ 
 
 der 
 
 z 
 
 Chest 
 
 w 
 
 Waist 
 
 u 
 
 Hip 
 
 s 
 
 142 
 
 67 
 
 5 
 
 16 
 
 9 
 
 35 
 
 3 
 
 30 
 
 2 
 
 36 
 
 5 
 
 144 
 
 68 
 
 5 
 
 17 
 
 5 
 
 38 
 
 0 
 
 28 
 
 5 
 
 36 
 
 5 
 
 151 
 
 73 
 
 0 
 
 16 
 
 5 
 
 36 
 
 2 
 
 29 
 
 0 
 
 37 
 
 3 
 
 124 
 
 68 
 
 2 
 
 16 
 
 2 
 
 33 
 
 3 
 
 25 
 
 8 
 
 35 
 
 0 
 
 130 
 
 67 
 
 5 
 
 17 
 
 2 
 
 35 
 
 0 
 
 25 
 
 5 
 
 35 
 
 0 
 
 143 
 
 69 
 
 7 
 
 16 
 
 4 
 
 33 
 
 7 
 
 28 
 
 2 
 
 35 
 
 3 
 
 127 
 
 67 
 
 0 
 
 16 
 
 5 
 
 36 
 
 5 
 
 28 
 
 i 
 
 34 
 
 3 
 
 122 
 
 64 
 
 2 
 
 16 
 
 2 
 
 34 
 
 7 
 
 26 
 
 4 
 
 33 
 
 5 
 
 143 
 
 65 
 
 9 
 
 17 
 
 2 
 
 37 
 
 5 
 
 29 
 
 5 
 
 36 
 
 0 
 
 140 
 
 67 
 
 5 
 
 16 
 
 4 
 
 36 
 
 6 
 
 28 
 
 5 
 
 34 
 
 8 
 
 132 
 
 66 
 
 8 
 
 16 
 
 0 
 
 34 
 
 2 
 
 28 
 
 5 
 
 35 
 
 0 
 
 150 
 
 66 
 
 0 
 
 16 
 
 0 
 
 36 
 
 2 
 
 30 
 
 i 
 
 37 
 
 2 
 
 155 
 
 69 
 
 5 
 
 16 
 
 9 
 
 35 
 
 0 
 
 28 
 
 5 
 
 37 
 
 3 
 
 141 
 
 67 
 
 0 
 
 16 
 
 8 
 
 36 
 
 0 
 
 28 
 
 0 
 
 36 
 
 0 
 
 113 
 
 66 
 
 1 
 
 15 
 
 5 
 
 31 
 
 8 
 
 25 
 
 0 
 
 33 
 
 0 
 
 102 
 
 67 
 
 6 
 
 15 
 
 4 
 
 32 
 
 7 
 
 26 
 
 2 
 
 32 
 
 2 
 
 161 
 
 67 
 
 7 
 
 17 
 
 0 
 
 37 
 
 6 
 
 30 
 
 6 
 
 38 
 
 6 
 
 134 
 
 67 
 
 1 
 
 16 
 
 4 
 
 34 
 
 5 
 
 27 
 
 7 
 
 34 
 
 2 
 
 125 
 
 68 
 
 5 
 
 16 
 
 1 
 
 32 
 
 3 
 
 27 
 
 2 
 
 34 
 
 0 
 
 136 
 
 66 
 
 7 
 
 15 
 
 9 
 
 34 
 
 2 
 
 28 
 
 2 
 
 35 
 
 8 
 
 136 
 
 68 
 
 0 
 
 16 
 
 0 
 
 34 
 
 3 
 
 27 
 
 7 
 
 34 
 
 2 
 
 143 
 
 67 
 
 6 
 
 17 
 
 8 
 
 36 
 
 3 
 
 27 
 
 2 
 
 36 
 
 2 
 
 136 
 
 68 
 
 0 
 
 17 
 
 0 
 
 34 
 
 2 
 
 28 
 
 7 
 
 35 
 
 0 
 
 148 
 
 68 
 
 0 
 
 16 
 
 0 
 
 35 
 
 0 
 
 28 
 
 2 
 
 36 
 
 2 
 
 130 
 
 64 
 
 2 
 
 17 
 
 5 
 
 34 
 
 5 
 
 26 
 
 5 
 
 35 
 
 5 
 
 131 
 
 69 
 
 7 
 
 15 
 
 9 
 
 32 
 
 7 
 
 26 
 
 2 
 
 34 
 
 1 
 
 132 
 
 63 
 
 3 
 
 16 
 
 1 
 
 35 
 
 7 
 
 28 
 
 0 
 
 34 
 
 2 
 
 127 
 
 67 
 
 5 
 
 14 
 
 7 
 
 32 
 
 7 
 
 25 
 
 2 
 
 35 
 
 3 
 
 122 
 
 65 
 
 0 
 
 17 
 
 3 
 
 34 
 
 0 
 
 27 
 
 6 
 
 34 
 
 5 
 
 138 
 
 66 
 
 7 
 
 16 
 
 4 
 
 35 
 
 9 
 
 28 
 
 2 
 
 35 
 
 5 
 
 142 
 
 70 
 
 5 
 
 16 
 
 5 
 
 34 
 
 3 
 
 28 
 
 5 
 
 36 
 
 5 
 
 140 
 
 69 
 
 4 
 
 15 
 
 4 
 
 33 
 
 6 
 
 25 
 
 7 
 
 35 
 
 2 
 
 138 
 
 70 
 
 0 
 
 17 
 
 0 
 
 38 
 
 0 
 
 27 
 
 0 
 
 34 
 
 0 
 
 117 
 
 63 
 
 5 
 
 15 
 
 4 
 
 34 
 
 8 
 
 27 
 
 1 
 
 34 
 
 5 
 
 129 
 
 67 
 
 0 
 
 17 
 
 0 
 
 36 
 
 5 
 
 27 
 
 1 
 
 34 
 
 0 
 
 114 
 
 67 
 
 9 
 
 14 
 
 5 
 
 31 
 
 8 
 
 24 
 
 2 
 
 32 
 
 2 
 
 132 
 
 67 
 
 4 
 
 16 
 
 5 
 
 35 
 
 8 
 
 28 
 
 6 
 
 34 
 
 7 
 
 142 
 
 67 
 
 6 
 
 16 
 
 2 
 
 35 
 
 4 
 
 27 
 
 2 
 
 36 
 
 2 
 
 148 
 
 72 
 
 7 
 
 16 
 
 6 
 
 34 
 
 0 
 
 27 
 
 2 
 
 37 
 
 3 
 
 128 
 
 69 
 
 7 
 
 15 
 
 7 
 
 34 
 
 4 
 
 27 
 
 6 
 
 33 
 
 3 
 
 139 
 
 70 
 
 2 
 
 16 
 
 4 
 
 35 
 
 2 
 
 27 
 
 8 
 
 35 
 
 0 
 
 150 
 
 67 
 
 6 
 
 16 
 
 8 
 
 36 
 
 0 
 
 29 
 
 2 
 
 36 
 
 2 
 
 108 
 
 62 
 
 4 
 
 16 
 
 5 
 
 30 
 
 5 
 
 26 
 
 0 
 
 32 
 
 5 
 
 132 
 
 64 
 
 2 
 
 15 
 
 0 
 
 34 
 
 7 
 
 29 
 
 0 
 
 34 
 
 9 
 
 142 
 
 67 
 
 0 
 
 16 
 
 7 
 
 37 
 
 6 
 
 29 
 
 3 
 
 35 
 
 7 
 
 150 
 
 70 
 
 8 
 
 17 
 
 2 
 
 35 
 
 8 
 
 27 
 
 3 
 
 36 
 
 0 
 
 135 
 
 69 
 
 3 
 
 17 
 
 0 
 
 34 
 
 0 
 
 27 
 
 8 
 
 35 
 
 1 
 
 117 
 
 68 
 
 8 
 
 15 
 
 7 
 
 32 
 
 8 
 
 25 
 
 4 
 
 33 
 
 2 
 
 145 
 
 67 
 
 3 
 
 16 
 
 7 
 
 35 
 
 1 
 
 30 
 
 0 
 
 35 
 
 3 
 
 138 
 
 67 
 
 2 
 
 17 
 
 4 
 
 36 
 
 5 
 
 29 
 
 9 
 
 35 
 
 9 
 
 152 
 
 71 
 
 0 
 
 17 
 
 3 
 
 37 
 
 3 
 
 29 
 
 5 
 
 36 
 
 6 
 
 138 
 
 64 
 
 4 
 
 17 
 
 4 
 
 35 
 
 3 
 
 29 
 
 5 
 
 35 
 
 3 
 
 132 
 
 65 
 
 9 
 
 16 
 
 5 
 
 36 
 
 7 
 
 28 
 
 0 
 
 34 
 
 6 
 
 139 
 
 67 
 
 8 
 
 16 
 
 4 
 
 35 
 
 8 
 
 28 
 
 2 
 
 34 
 
 2 
 
 172 
 
 65 
 
 1 
 
 16 
 
 4 
 
 38 
 
 3 
 
 35 
 
 0 
 
 39 
 
 2 
 
 122 
 
 65 
 
 7 
 
 15 
 
 0 
 
 34 
 
 7 
 
 28 
 
 2 
 
 33 
 
 0 
 
 133 
 
 67 
 
 5 
 
 15 
 
 5 
 
 36 
 
 0 
 
 28 
 
 5 
 
 35 
 
 5 
 
 144 
 
 66 
 
 4 
 
 17 
 
 2 
 
 36 
 
 2 
 
 28 
 
 4 
 
 35 
 
 7 
 
 144 
 
 70 
 
 3 
 
 16 
 
 7 
 
 34 
 
 5 
 
 26 
 
 5 
 
 36 
 
 0 
 
 155 
 
 71 
 
 4 
 
 17 
 
 2 
 
 36 
 
 8 
 
 27 
 
 7 
 
 37 
 
 3 
 
 155 
 
 68 
 
 3 
 
 16 
 
 9 
 
 36 
 
 0 
 
 27 
 
 7 
 
 36 
 
 4 
 
 125 
 
 61 
 
 2 
 
 14 
 
 7 
 
 32 
 
 0 
 
 30 
 
 0 
 
 34 
 
 3 
 
 124 
 
 69 
 
 6 
 
 15 
 
 3 
 
 33 
 
 5 
 
 25 
 
 2 
 
 34 
 
 7 
 
 138 
 
 70 
 
 0 
 
 16 
 
 0 
 
 34 
 
 8 
 
 28 
 
 0 
 
 34 
 
 9 
 
 141 
 
 69 
 
 2 
 
 15 
 
 7 
 
 33 
 
 0 
 
 25 
 
 8 
 
 33 
 
 2 
 
 113 
 
 66 
 
 5 
 
 15 
 
 4 
 
 33 
 
 8 
 
 24 
 
 0 
 
 33 
 
 3 
 
 129 
 
 63 
 
 0 
 
 16 
 
 5 
 
 35 
 
 2 
 
 28 
 
 3 
 
 34 
 
 5 
 
 144 
 
 67 
 
 1 
 
 16 
 
 0 
 
 35 
 
 8 
 
 28 
 
 8 
 
 36 
 
 3 
 
 124 
 
 64 
 
 1 
 
 17 
 
 2 
 
 36 
 
 2 
 
 28 
 
 0 
 
 36 
 
 8 
 
 137 
 
 68 
 
 2 
 
 15 
 
 8 
 
 34 
 
 8 
 
 28 
 
 6 
 
 34 
 
 0 
 
 115 
 
 66 
 
 3 
 
 15 
 
 3 
 
 32 
 
 1 
 
 25 
 
 2 
 
 32 
 
 3 
 
 140 
 
 67 
 
 8 
 
 16 
 
 2 
 
 34 
 
 5 
 
 27 
 
 2 
 
 36 
 
 2 
 
 142 
 
 68 
 
 6 
 
 16 
 
 2 
 
 35 
 
 0 
 
 27 
 
 6 
 
 36 
 
 0 
 
 143 
 
 67 
 
 8 
 
 16 
 
 4 
 
 34 
 
 8 
 
 28 
 
 6 
 
 35 
 
 2 
 
 137 
 
 64 
 
 4 
 
 16 
 
 5 
 
 35 
 
 3 
 
 26 
 
 5 
 
 32 
 
 0 
 
 128 
 
 68 
 
 3 
 
 16 
 
 4 
 
 34 
 
 3 
 
 27 
 
 3 
 
 33 
 
 7 
 
 144 
 
 71 
 
 0 
 
 16 
 
 4 
 
 34 
 
 0 
 
 29 
 
 3 
 
 36 
 
 4 
 
 136 
 
 68 
 
 1 
 
 16 
 
 2 
 
 34 
 
 6 
 
 26 
 
 2 
 
 35 
 
 0 
 
 146 
 
 67 
 
 0 
 
 17 
 
 6 
 
 34 
 
 6 
 
 28 
 
 0 
 
 37 
 
 0 
 
 152 
 
 67 
 
 4 
 
 16 
 
 6 
 
 37 
 
 0 
 
 28 
 
 0 
 
 37 
 
 0 
 
 167 
 
 67 
 
 7 
 
 17 
 
 2 
 
 39 
 
 3 
 
 30 
 
 5 
 
 39 
 
 5 
 
 122 
 
 66 
 
 6 
 
 15 
 
 3 
 
 32 
 
 7 
 
 25 
 
 1 
 
 33 
 
 2 
 
 140 
 
 66 
 
 5 
 
 17 
 
 4 
 
 38 
 
 9 
 
 28 
 
 3 
 
 35 
 
 7 
 
 130 
 
 65 
 
 6 
 
 14 
 
 9 
 
 33 
 
 9 
 
 28 
 
 3 
 
 36 
 
 1 
 
 146 
 
 69 
 
 5 
 
 17 
 
 3 
 
 35 
 
 5 
 
 29 
 
 1 
 
 36 
 
 5 
 
 109 
 
 62 
 
 5 
 
 15 
 
 2 
 
 31 
 
 4 
 
 23 
 
 3 
 
 33 
 
 6 
 
 125 
 
 69 
 
 6 
 
 16 
 
 0 
 
 32 
 
 8 
 
 27 
 
 0 
 
 33 
 
 7 
 
 147 
 
 68 
 
 6 
 
 16 
 
 6 
 
 37 
 
 0 
 
 30 
 
 4 
 
 35 
 
 3 
 
 129 
 
 68 
 
 5 
 
 16 
 
 2 
 
 32 
 
 7 
 
 27 
 
 3 
 
 35 
 
 8 
 
 150 
 
 71 
 
 7 
 
 15 
 
 9 
 
 34 
 
 0 
 
 28 
 
 0 
 
 36 
 
 3 
 
 157 
 
 71 
 
 0 
 
 16 
 
 3 
 
 35 
 
 8 
 
 30 
 
 2 
 
 38 
 
 0 
 
 159 
 
 69 
 
 3 
 
 16 
 
 4 
 
 37 
 
 0 
 
 28 
 
 8 
 
 36 
 
 8 
 
 116 
 
 63 
 
 0 
 
 16 
 
 3 
 
 32 
 
 1 
 
 26 
 
 5 
 
 32 
 
 5 
 
 136 
 
 68 
 
 7 
 
 17 
 
 0 
 
 36 
 
 5 
 
 27 
 
 8 
 
 36 
 
 _2 
 
 152 
 
 70 
 
 2 
 
 16 
 
 8 
 
 37 
 
 2 
 
 29 
 
 9 
 
 35 
 
 7 
 
 136 
 
 68 
 
 3 
 
 15 
 
 2 
 
 34 
 
 2 
 
 28 
 
 0 
 
 35 
 
 0 
 
 146 
 
 68 
 
 2 
 
 1 5 
 
 8 
 
 36 
 
 2 
 
 30 
 
 0 
 
 36 
 
 8 
 
 172 
 
 69 
 
 1 
 
 17 
 
 2 
 
 36 
 
 8 
 
 28 
 
 1 
 
 36 
 
 4 
 
 132 
 
 67 
 
 0 
 
 16 
 
 1 
 
 36 
 
 1 
 
 29 
 
 2 
 
 35 
 
 0 
 
 143 
 
 70 
 
 7 
 
 16 
 
 9 
 
 35 
 
 8 
 
 28 
 
 3 
 
 36 
 
 3 
 
 148 
 
 62 
 
 4 
 
 16 
 
 0 
 
 37 
 
 2 
 
 31 
 
 8 
 
 37 
 
 5 
 
 121 
 
 60 
 
 2 
 
 15 
 
 9 
 
 33 
 
 9 
 
 28 
 
 2 
 
 34 
 
 2 
 
 130 
 
 65 
 
 3 
 
 15 
 
 6 
 
 34 
 
 2 
 
 28 
 
 2 
 
 34 
 
 2 
 
 159 
 
 71 
 
 1 
 
 16 
 
 3 
 
 36 
 
 8 
 
 28 
 
 8 
 
 35 
 
 8 
 
 159 
 
 71 
 
 8 
 
 16 
 
 8 
 
 37 
 
 8 
 
 29 
 
 8 
 
 37 
 
 8 
 
 170 
 
 72 
 
 0 
 
 17 
 
 5 
 
 39 
 
 0 
 
 30 
 
 3 
 
 37 
 
 2 
 
 145 
 
 67 
 
 0 
 
 15 
 
 4 
 
 34 
 
 6 
 
 28 
 
 2 
 
 35 
 
 2 
 
 133 
 
 63 
 
 1 
 
 16 
 
 2 
 
 36 
 
 1 
 
 28 
 
 2 
 
 34 
 
 2 
 
 131 
 
 69 
 
 2 
 
 15 
 
 7 
 
 34 
 
 8 
 
 26 
 
 5 
 
 33 
 
 8 
 
 144 
 
 70 
 
 1 
 
 17 
 
 7 
 
 36 
 
 0 
 
 27 
 
 0 
 
 35 
 
 0 
 
 140 
 
 66 
 
 6 
 
 16 
 
 2 
 
 35 
 
 3 
 
 28 
 
 1 
 
 36 
 
 1 
 
 138 
 
 69 
 
 4 
 
 17 
 
 1 
 
 35 
 
 6 
 
 26 
 
 2 
 
 34 
 
 7 
 
 153 
 
 70 
 
 1 
 
 16 
 
 0 
 
 35 
 
 7 
 
 29 
 
 7 
 
 35 
 
 7 
 
 163 
 
 70 
 
 7 
 
 18 
 
 4 
 
 38 
 
 6 
 
 30 
 
 0 
 
 37 
 
 7 
 
 119 
 
 61 
 
 9 
 
 15 
 
 9 
 
 34 
 
 8 
 
 29 
 
 2 
 
 33 
 
 8 
 
 169 
 
 72 
 
 2 
 
 17 
 
 7 
 
 38 
 
 2 
 
 30 
 
 0 
 
 38 
 
 3 
 
 155 
 
 68 
 
 0 
 
 16 
 
 3 
 
 37 
 
 0 
 
 29 
 
 2 
 
 38 
 
 2 
 
 121 
 
 69 
 
 0 
 
 15 
 
 7 
 
 34 
 
 5 
 
 27 
 
 5 
 
 33 
 
 7 
 
 150 
 
 70 
 
 0 
 
 16 
 
 5 
 
 35 
 
 3 
 
 29 
 
 2 
 
 36 
 
 8 
 
 140 
 
 66 
 
 4 
 
 16 
 
 2 
 
 35 
 
 8 
 
 35 
 
 3 
 
 20 
 
 2 
 
 * These data were taken from "Synopsis of Elementary Mathematical Statistics," by 
 II. C. Carver. 
 
194 
 
 CORRELATION COEFFICIENT 
 
 PARTIAL CORRELATION COEFFICIENT 
 
 When x and y are connected linearly by the relation 
 
 y = a + bx, 
 
 values of y are predicted from values of x. By making the sum of 
 the squares of the residual errors a minimum, the value of b is 
 found to be b = Zxy/Zx 2 . When x and y are connected linearly 
 the x values are predicted from values of y by the relation 
 
 x = c + dy. 
 
 By making the sum of the squares of the residual errors a minimum, 
 the value of d is found to be d = Zxy/Zy 2 . 
 
 The correlation coefficient between x and y is 
 
 T vx 
 
 2xy 
 
 VZx 2 Zy 2 
 
 This same quantity can be found by taking the square root of the 
 product of b and d, for 
 
 r BI = V bd 
 
 U?xy ) 2 
 
 V Zx 2 Zy 2 
 
 Zxy 
 
 2 V Vzx 2 z f 
 
 This shows that the correlation coefficient between x and y is the 
 square root of the product of the coefficient of x in the predicting 
 equation for y and the coefficient of y in the predicting equation 
 for x, or r yx = \/db. 
 
 Consider the three variables x, y, and z connected by the linear 
 relation 
 
 y = a + bx + cz, or y = bx + cz. 
 
 It is also assumed that any 2 of these 3 variables are connected 
 linearly. 
 
 It is assumed that there are n “corresponding” values of x, y, and z 
 given from observations. From the observational equations the 
 following normal equations are obtained: 
 
 bZx 2 + cZzx = 2 xy 
 
 bZxz + cZz 2 = 2 zy, 
 
 * r vx is the geometric mean of b and d. 
 
PARTIAL CORRELATION COEFFICIENT 
 
 195 
 
 from which 
 
 2z 2 2xy — 2xz2zy 
 = 2x 2 2z 2 - ( 2 i 2 ) 2 ' 
 
 Now consider that x is predicted from y and z by the equation 
 
 x = dy + gz. 
 
 The normal equations are 
 
 from which 
 
 d2y 2 + g^zy = 2 xy, 
 ddLyz + g'Zz 2 — 2 xz, 
 
 2z 2 2xy — 2zi/2xz 
 2y 2 2z 2 - (2 yz) 2 * 
 
 The square root of the product of bd is 
 
 (2z 2 ) 2 (2xy) 2 - 22z 2 2x£2zy2xz + (2xz) 2 (2zy) 
 [2x 2 2z 2 - (2xz) 2 ][2z 2 2y 2 - (2 yz) 2 ] 
 
 r 2 __ Or* r r _I— r 2y» 2 
 xy xy' xz' yz I ' xz ' zy 
 
 (1 - r„ 2 )(l - r zy 2 ) 
 
 2 
 
 The partial correlation coefficient is defined to be 
 
 ( 10 . 20 ) 
 
 ' yx*z 
 
 = Vbd = 
 
 — r zz r. 
 
 V(1 - r„ 2 )(l - r zy 2 ) 
 
 The above correlation coefficient was obtained as in finding 
 r yz , that is, by finding y/bd. The values of c and g were not used. 
 This amounts to holding z constant. This correlation is called 
 partial because it is the correlation obtained when the third vari¬ 
 able is held constant. Suppose height is predicted from weight and 
 chest measurements, then the partial correlation coefficient 
 
 ^height weight'Chest measurement 
 
 is the correlation between height and weight when chest measure¬ 
 ment is held constant, that is, when people are considered who 
 have the same chest measurement. The Pearson linear correlation 
 coefficient between heights and weights for people with 37-inch 
 chest measurement is what is meant by holding chest measurement 
 constant. The correlation should be about the same if people 
 
196 
 
 CORRELATION COEFFICIENT 
 
 were considered with chest 38 inches or 40 inches. Instead of sep¬ 
 arating all the people in the distribution considered with a certain 
 chest measurement and then finding the ordinary correlation coeffi¬ 
 cient between heights and weights, it is only necessary to find the 
 partial correlation coefficient, for it does the same thing. 
 
 To be able to use the partial correlation coefficient it must be 
 ascertained whether or not the variables are linearly connected in 
 all possible pairs, and in every other way. This should be done, 
 but it is very seldom done. The average person uses the partial 
 correlation coefficient without determining whether or not the 
 variables are connected linearly in all possible pairs. 
 
 Example. Consider that the four-year average of a university 
 student is a measure of his general ability. For a group of students at a 
 certain university the following correlations were obtained, where m = 
 freshman mathematics average, s = science average, and a = four-year 
 average: r mB = 0.54, r ma = 0.80, r 8a = 0.57. 
 
 The correlation between freshman mathematics and science with 
 general ability eliminated is 
 
 ?ms Tma Tas n 17 
 
 Tma.a = V =0.17, 
 
 V (1 - Tas 2 ) (1 - r ma 2 ) 
 
 which seems to indicate that there is not very much correlation between 
 freshman mathematics average and science average. 
 
 PROBLEMS 
 
 1. Find the correlation coefficient between freshman mathematics 
 averages and education averages with general ability eliminated, if 
 m = mathematics average, e = education average, a = four-year average, 
 or general ability, and r me = 0.34, r ma = 0.66, and r ea = 0.54. Discuss 
 your result for poor students, for average students, and for the best 
 students. 
 
 2. Find the correlation between weights and thigh measurements with 
 heights eliminated for the data on pages 151 and 161. 
 
 3. Find the correlation coefficient between freshman mathematics 
 averages and freshman English averages with general ability eliminated 
 for data on page 188. 
 
 4. Discuss formula (10.20) when r 2 yx . z = 1. 
 
 5. Discuss this same formula when r xl 2 = 1. 
 
 6. Discuss this when r xy 2 = 1. 
 
 7. Find the partial correlation coefficient between weights and waist 
 
TETRACHORIC CORRELATION 
 
 197 
 
 measurements, with shoulder measurements held constant, for the data 
 on page 193. 
 
 8. Use data on page 193 for finding the correlation between weights 
 and waist measurements for those whose shoulder measurements were 
 between 15.5 inches and 16.0 inches inclusive. How does this compare 
 with the result of problem 7? 
 
 9. Use data on page 193 for finding the correlation coefficient between 
 weights and waist measurements for those whose shoulder measurements 
 were between 16.5 inches and 17.0 inches inclusive. Compare this with 
 results of problems 7 and 8. 
 
 10. By using formula (10.16) and formula (10.20) prove that 
 
 (10.2D i - = Lpirrf. 
 
 l - v 
 
 11. Use this formula to find the partial correlation coefficient between 
 mathematics grades and English grades with general ability eliminated for 
 data on page 188 and the result of problem 5 on page 188. 
 
 12. If y = bx + cl, prove that this equation can be written in the form 
 
 [V(l — rM)]y = — (Vl - r yz 2 )r vx . z -x + — (Vl - r xv 2 )r vz . x -z • 
 
 (T x G z 
 
 13. If r vz = ± 0.80, plot relation (10.21), after reducing to the simplest 
 form. Discuss the results. 
 
 14. If the multiple correlation coefficient between y and ( x and z) is 
 equal to 1, what is the partial correlation coefficient between y and x with 
 z held constant? 
 
 15. If r vz . z = +0.80, plot relation (10.21). Discuss. 
 
 16. If y = bx + cl + dw, find an expression for the partial correlation 
 coefficient between y and x with z and w held constant, or find an expression 
 for r, ,x.zw in terms of the fundamental summations and also in terms of r’s. 
 
 17. Prove that 
 
 /in nrn 1'yx.z I'yw.z^xwz 'I'zy.yj Tyz.w'Xxz.w 
 
 ( 10 . 22 ). Tyx-ZW — — - - — 
 
 V (1 - r vw J) (1 - r xw J) V [1 - r yz . w 2 ] [1 - r xl . w 2 } 
 
 18. Use formula (10.22) and the data on page 193 to find the partial 
 correlation between weight and hip measurement with height and shoulder 
 measurement held constant. 
 
 TETRACHORIC CORRELATION 
 
 This section presents a correlation coefficient between variables 
 which cannot be measured or which are not quantitative; for exam¬ 
 ple, blindness, deafness, dispositions, mealiness of apples, desirable¬ 
 ness of potatoes, etc. Karl Pearson derived a formula for finding 
 
198 
 
 CORRELATION COEFFICIENT 
 
 a measure of the correlation between two qualitative variables.* 
 Consider the following example concerning inheritance of eye-color 
 in man. 
 
 Grandmothers 
 
 Tint 
 
 Gray or 
 Lighter 
 
 Dark Gray 
 or Darker 
 
 Totals 
 
 Gray or 
 
 a 
 
 b 
 
 cl -f- b 
 
 Lighter 
 
 254 
 
 136 
 
 390 
 
 Dark Gray 
 
 c 
 
 d 
 
 c + d 
 
 or Darker 
 
 156 
 
 193 
 
 349 
 
 
 a + c 
 
 b + d 
 
 N 
 
 Totals 
 
 410 
 
 329 
 
 739 
 
 The above table gives the number of granddaughters with gray 
 or lighter eyes whose grandmothers had eyes which were gray 
 or lighter and those which had eyes that were dark gray or darker, 
 and also the number of granddaughters with dark gray or darker 
 eyes whose grandmothers had eyes which were gray or lighter and 
 those with eyes which were dark gray or darker. Pearson’s for¬ 
 mulas enable one to find the correlation coefficient between eye- 
 color of granddaughter and eye-color of grandmother. These 
 formulas are as follows: 
 
 c "I - d 
 
 (10.23) —-— = the area under the normal curve between k and oo , 
 
 (10.24) 
 
 N 
 b + d 
 N 
 
 = the area under the normal curve between h and oo , 
 
 1 
 
 *2 
 
 (10.25) H = —-= e 2 , or the height of the normal curve at h, 
 \/27r 
 
 1 
 
 *2 
 
 (10.26) K = — 7 = e 2 , or the height of the normal curve at k, 
 
 \'2lT 
 
 0 2 (ad — be) 
 
 (10.27) hkr 2 + 2r « ^ ^ , 
 
 * Karl Pearson, “Mathematical Contributions to the Theory of Evolu¬ 
 tion—VII. On the Correlation of Characters Not Quantitatively Measur¬ 
 able,” Phil. Trans., 195-A, pages 1-47. 
 
PROBLEMS 
 
 199 
 
 where h and k can be found from the area table for the normal 
 curve, and H and K can be found from the ordinate table of the 
 normal curve. The quantity r is found by solving the quadratic 
 equation (10.27), and is the coefficient of correlation sought. For 
 the above table pertaining to eye-color we have 
 
 c + d 
 ' N 
 
 b + d 
 N 
 
 H 
 
 K 
 
 349 
 
 739 
 
 329 
 
 739 
 
 0.4723 = the area under the normal curve between 
 0.0696 and oo ; from Table I 
 (k = 0.06961) 
 
 0.4452 = the area under the normal curve between 
 0.1381 and oo ; from Table I 
 (h = 0.1381) 
 
 **(0.1381)2 _ 
 
 0.3952, from Table II; 
 
 __L g— **(o.o696) 2 _ o.398, from Table II. 
 V2t r 
 
 The quadratic equation (10.27) becomes 
 
 from which 
 
 0.009612r 2 + 2r = 
 
 55,612 
 
 85,899,154 
 
 r = 0.3225 + 
 
 0.6475, 
 
 which is the correlation coefficient between eye-color of grand¬ 
 mother and granddaughter. 
 
 PROBLEMS 
 
 1. Find the tetrachoric correlation coefficient between blindness and 
 deafness for the following data taken from the 1930 census of the United 
 States. 
 
 Blindness 
 
 
 Present 
 
 Absent 
 
 Totals 
 
 Present 
 
 1,942 
 
 55,142 
 
 57,084 
 
 Not 
 
 Present 
 
 61,747 
 
 122,656,215 
 
 122,717,962 
 
 Totals 
 
 63,689 
 
 122,711,357 
 
 122,775,046 
 
200 
 
 CORRELATION COEFFICIENT 
 
 2. Find the tetrachoric correlation coefficient between vaccination 
 and recoveries from the following data: 
 
 
 Deaths 
 
 Recoveries 
 
 Vaccinated 
 
 546 
 
 2,602 
 
 Unvaccinated 
 
 130 
 
 100 
 
 3. Find the tetrachoric correlation coefficient between toughness of 
 skin and mealiness of potatoes of samples taken from stores at random. 
 
 Skin 
 
 
 Tough 
 
 Not tough 
 
 Mealy 
 
 12 
 
 93 
 
 Not Mealy 
 
 30 
 
 14 
 
CHAPTER 11 
 
 SAMPLING 
 
 AVERAGE OF SAMPLE AVERAGES 
 
 The most important part of statistics is sampling from an 
 unknown parent population. Often all the information concern¬ 
 ing the parent population is obtained from a rather small sample 
 or from a small number of samples. In physics, many times, 10 
 or 12 experiments are made to discover the nature or law of a 
 certain phenomenon. The botanist picks at random a sample of 
 a certain plant, investigates the characteristics of this sample, and 
 assigns to this species the information obtained from the sample. 
 Business forecasting depends on information obtained from sam¬ 
 ples; in fact, almost every empirical formula is obtained from sam¬ 
 pling data. 
 
 Since sampling is so important, a few of the basic ideas con¬ 
 cerning sampling will be presented in this chapter. These ideas 
 will be presented first from the study of sampling from a known 
 parent population and later from an unknown parent. One of the 
 first things to investigate is the nature of the distribution made up 
 of averages of the samples drawn from a parent. 
 
 An example will be given to introduce this phase of sampling. 
 
 Consider the following parent distribution: 
 
 v 
 
 /(*>) 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 1 
 
 2 
 
 9 
 
 28 
 
 66 
 
 121 
 
 175 
 
 197 
 
 175 
 
 121 
 
 66 
 
 28 
 
 9 
 
 2 
 
 1 
 
 The mean, standard deviation and skew¬ 
 ness are as follows: 
 
 M v = 7.0000, 
 
 cr„ = 2.002, 
 a 3 -.v = 0.0000. 
 
 1,001 
 
 201 
 
202 
 
 SAMPLING 
 
 From this parent distribution 400 samples, each of 200 variates, were 
 drawn at random. Cards with the variates written thereon were put in 
 a bag, from which drawings were made. The average of each sample 
 of 200 was obtained, giving 400 values. These 400 values formed the 
 frequency distribution given below, which is the distribution of the means 
 of these 400 samples. 
 
 Classes 
 
 z 
 
 M 
 
 
 6.625-6.674 
 
 0 
 
 
 6.675-6.724 
 
 2 
 
 
 6.725-6.774 
 
 12 
 
 
 6.775-6.824 
 
 20 
 
 
 6.825-6.874 
 
 22 
 
 The mean, standard deviation, 
 
 6.875-6.924 
 
 46 
 
 and skewness of the distribution of 
 
 6.925-6.974 
 
 57 
 
 the means are as follows: 
 
 6.975-7.024 
 
 66 
 
 
 7.025-7.074 
 
 55 
 
 M z = 7.0055, 
 
 7.075-7.124 
 
 50 
 
 = 0.121, 
 
 7.125-7.174 
 
 40 
 
 a 3 : 2 = 0.0001. 
 
 7.175-7.224 
 
 15 
 
 
 7.225-7.274 
 
 11 
 
 
 7.275-7.324 
 
 3 
 
 
 7.325-7.374 
 
 1 
 
 
 
 400 
 
 
 The mean of the parent is 7.0000, and the mean of the distribution of 
 sample means is 7.0055. These values are nearly the same. The 
 standard deviation of the parent distribution is 2.002; the standard devia¬ 
 tion of the distribution of sample averages is 0.121. (These distributions 
 were taken from “An Application of Thiele’s Semi-Invariants to the 
 Sampling Problem,” by C. C. Craig, Metron, Yol. 7, No. 4, 1928, pages 
 3-75.) The smallness of the standard deviation of the distribution of 
 the sample averages shows that all the sample averages are very near their 
 mean and the mean of the parent. 
 
 The range of the variates in the parent is 15, while the range of 
 the distribution of sample means is only 0.75, which again shows 
 how near the variates in the distribution of sample averages lie 
 with reference to the mean of the sample averages and to the 
 mean of the parent. Compare the <r’s of the two distributions. 
 In this example all the sample averages were within a distance of 
 0.375 from the mean of the parent population. From the theory 
 
AVERAGE OF SAMPLE AVERAGES 
 
 203 
 
 of probability it is almost impossible to get a sample average, 
 where the sample consists of 200 variates, to differ from the mean 
 of the parent by as much as 0.5, hence this example shows that as 
 far as the mean is concerned a mean of a good size sample is about 
 as good as the mean of the parent. When the parent is unknown, 
 the mean of a sample, or of a few samples, is about as good as we 
 can get without going to a great deal of trouble to secure other 
 samples or a larger sample. The two distributions on pages 201 
 and 202 are plotted in Fig. 11.1 in percentages. Notice the ranges 
 of the distributions and the locations of the larger frequencies. 
 
 Distribution of Sample Averages 
 Fig. 11.1 
 
 Let us now consider the general case. Out of a parent popula¬ 
 tion consisting of s variates, iq, vo, v$, ... , v„ are taken at random 
 samples of r variates. Each sample average is obtained, and then 
 the arithmetic mean of all possible sample averages is found. The 
 parent population may consist of the heights of soldiers in one par¬ 
 ticular camp, or the lung capacities of students in a certain college, 
 or the numbers of bushels of wheat grown per year in a certain 
 state for the last 50 years, or the numbers of petals on a certain 
 kind of flower, etc. Let the sample averages be represented by z, 
 where z\ is the first sample average, z-i the second, etc. Let the 
 mean, M v , the standard deviation, a V) and the skewness, <23 be 
 known of the parent. The sample averages are: 
 
204 
 
 SAMPLING 
 
 Vl + V 2 + ... V r 
 
 Z 1 = 
 
 r 
 
 2 2 = 
 
 Vl + V2 + . . . + V r —i + V r+ \ 
 r 
 
 Vs- r-fl + Vz-r+2 + . . . + Vs-\ + V» 
 
 r 
 
 Adding 
 
 where M z is the mean of all possible sample means; no sample is 
 taken twice. 
 
 The above summation was obtained by adding all the variates 
 in every row. There are s C r rows or possible samples of r variates 
 which can be formed from the s variates in the parent. Hence the 
 sum of all the variates in the s C r samples is r- s C T as there are r 
 variates in each of the s C r rows. Any variate, say vz, appears in 
 this sum as often as any other variate. Since there are s variates 
 in the parent and since each variate appears as often as any other, 
 we divide the above sum by s and let the summation run from 1 to 
 s. Dividing by s gives the number of times each variate appears 
 in the sum. After reducing the above average of the 2 ’s we find 
 that the mean of all possible sample averages is equal to the mean 
 of the parent population, or that 
 
 ( 11 . 1 ) 
 
 Parent Distribution 
 
 Z 
 
 Distribution of the sample means 
 Fig. 11.2 
 
PROBLEMS 
 
 205 
 
 Let the frequency curves in Fig. (11.2) represent the parent 
 distribution and the distribution made up of all possible sample 
 averages. 
 
 The lower distribution represents frequencies of the sample 
 averages which are plotted on a z axis about the mean of the z’s 
 which is equal to the mean of the parent. The range of the distri¬ 
 bution made up of all possible sample averages is much less than 
 the range of the parent. 
 
 PROBLEMS 
 
 1. Let the parent population consist of the variates V\, v 2 , . . ., v l0 . 
 Write all possible sample averages of 3 in the sample where no sample is 
 repeated. Find the average of these sample averages. Does this check 
 with (11.1)? 
 
 2. How many possible samples of 6 variates to the sample can be taken 
 from a parent of 10 variates, where repetitions of samples are not allowed? 
 With 4 variates in the samples? Five in the sample? 
 
 3. If there are 1,000 students whose average weight is 140 pounds 
 and if 500 students are taken at random from this 1,000, what would you 
 expect the average weight of these 500 to be? What would you expect 
 if 100 were drawn at random? Would the standard deviation of the dis¬ 
 tribution of the sample means help you to answer this question? Explain. 
 
 4. The following data give the right thigh measurements in inches of 
 
 20 
 
 22 
 
 20 
 
 17 
 
 17 
 
 19 
 
 20 
 
 21 
 
 21 
 
 22 
 
 19 
 
 20 
 
 21 
 
 18 
 
 20 
 
 21 
 
 19 
 
 22 
 
 19 
 
 20 
 
 23 
 
 20 
 
 20 
 
 17 
 
 22 
 
 20 
 
 19 
 
 19 
 
 23 
 
 18 
 
 20 
 
 24 
 
 23 
 
 18 
 
 21 
 
 21 
 
 19 
 
 19 
 
 26 
 
 20 
 
 20 
 
 21 
 
 21 
 
 20 
 
 20 
 
 19 
 
 22 
 
 19 
 
 18 
 
 20 
 
 22 
 
 21 
 
 20 
 
 19 
 
 21 
 
 19 
 
 20 
 
 22 
 
 22 
 
 19 
 
 18 
 
 20 
 
 21 
 
 17 
 
 23 
 
 20 
 
 21 
 
 20 
 
 25 
 
 24 
 
 21 
 
 19 
 
 18 
 
 18 
 
 17 
 
 20 
 
 19 
 
 24 
 
 21 
 
 18 
 
 18 
 
 18 
 
 19 
 
 19 
 
 20 
 
 19 
 
 23 
 
 19 
 
 23 
 
 18 
 
 19 
 
 20 
 
 19 
 
 21 
 
 21 
 
 20 
 
 19 
 
 19 
 
 20 
 
 20 
 
 21 
 
 22 
 
 21 
 
 19 
 
 19 
 
 18 
 
 19 
 
 23 
 
 24 
 
 23 
 
 20 
 
 19 
 
 21 
 
 20 
 
 18 
 
 17 
 
 22 
 
 18 
 
 20 
 
 20 
 
 17 
 
 19 
 
 20 
 
 21 
 
 20 
 
 21 
 
 23 
 
 18 
 
 20 
 
 19 
 
 23 
 
 22 
 
 24 
 
 22 
 
 23 
 
 18 
 
 20 
 
 19 
 
 23 
 
 20 
 
 20 
 
 20 
 
 24 
 
 19 
 
 20 
 
 19 
 
 20 
 
 22 
 
 21 
 
 21 
 
 19 
 
 19 
 
 19 
 
 20 
 
 23 
 
 18 
 
 21 
 
 19 
 
 20 
 
 21 
 
 23 
 
 22 
 
 19 
 
 19 
 
 20 
 
 18 
 
 23 
 
 21 
 
 26 
 
 19 
 
 22 
 
 19 
 
 23 
 
 20 
 
 19 
 
 18 
 
 21 
 
 19 
 
 21 
 
 19 
 
 18 
 
 26 
 
 20 
 
 18 
 
 23 
 
 21 
 
 22 
 
 18 
 
 20 
 
 21 
 
 19 
 
 21 
 
 21 
 
 22 
 
 20 
 
 21 
 
 21 
 
 20 
 
 19 
 
 20 
 
206 
 
 SAMPLING 
 
 men of 19 years of age. Have each member of the class pick 50 measure¬ 
 ments and find the average of this sample of 50 measurements. Examine 
 these sample averages with reference to the average of the parent, which 
 is M v = 20.28. Find the average of all sample averages obtained by 
 members of the class, and compare this with the mean of the parent. 
 Find some scheme where each member will secure a sample which is 
 different from the others. For example, let one member take every other 
 one until 50 are obtained, another take every fifth one until 50 are ob¬ 
 tained, etc. 
 
 STANDARD DEVIATION OF THE DISTRIBUTION OF 
 SAMPLE AVERAGES 
 
 As before, let = Vi — M v , which is the deviation of variate 
 Vi from the mean of the v’s or the mean of the parent. Let the 
 deviation of the first sample average from the mean of all sample 
 averages be 
 
 (11.2) zi = Z\ — M z = 21 — M v = - M v 
 
 r 
 
 (rq — M v ) + (v 2 — M v ) -f- (v r — M v ) 
 
 r 
 
 V\ -\- V2 + ... -Mr 
 r 
 
 which expresses the deviation of the first sample average from the 
 mean of all possible sample averages in terms of the deviations of 
 the variates in the parent from the mean of the parent. The 
 standard deviation of the distribution of all possible sample aver¬ 
 ages is the square root of the average of the z v s. These deviations 
 squared are: 
 
 
 t’l + V2 + • • • + Vr 
 
 2 2Si 2 + 2 'Zv&j . 
 
 = -- -, where i ^ j. 
 
 Vi + V 2 + . . . + Vr- 
 
 Vr-l + Vr + l j 
 
 y Vi 2 + 22 ViVj 
 
 i 9^ j, i, j 7* r 
 
 9 
 
STANDARD DEVIATION OF SAMPLE AVERAGES 207 
 
 Vs-r + l + • • • + w s _i + Vs 2 2ii i 2 + 225*5,• . . 
 
 - =-;-J i,j>s — r 
 
 r 
 
 where i ^ j. 
 
 The sum of the square terms was found by adding the squares 
 of the variates in each row. Since there are r squares in each row 
 and a C r rows, there are r- S C T variates squared in all the rows. 
 
 The average of these square terms is 
 
 S 
 
 where the summation runs from 1 to s. Since each variate squared 
 appears as often as any other and since there are s variates, then 
 putting an s in the denominator gives the number of times each 
 variate squared appears. 
 
 The sum of the product terms is found in a similar manner. 
 There are £2 product terms in each row and s C r rows, hence there 
 are r C 2 - s C r product terms in all rows. Each product term like 
 V 3 V 7 appears as often as any other product term. As there are S C 2 
 possible product terms the average of the product terms is equal to 
 
 2- T C2- a Cr'2ViV j 
 
 Dividing by a C 2 gives the number of times each product term is 
 repeated. 
 
 S 
 
 The summation, 25,-5,-, i 7 ^ j; can be expressed in terms of 2iq 2 . 
 The sum of the deviations of the variates from the mean of the 
 variates is zero as was proven in an earlier chapter, hence 
 
 0 = (h + v 2 + ... + v a ) 2 = 'Lv? + 225,1), • , i 9 * j, 
 
 or 
 
 ( 11 . 3 ) 
 
 22 vj)j = — 25 2 . 
 
208 
 
 SAMPLING 
 
 Substituting this in the expression for n 2 -.z, gives 
 
 Therefore 
 
 (11.4) 
 
 M2:z 
 
 1 r — 1 
 
 ~ M2 : v r M2 : v 
 
 r L s — 1 
 
 Gz 
 
 
 s — r 
 r(s - 1) 
 
 ov 
 
 This formula shows that the standard deviation of all possible 
 sample averages is equal to v(s — r)/r(s — 1) times the standard 
 deviation of the parent. This gives the standard deviation of the 
 distribution of sample averages in terms of s, the number of variates 
 in the parent; r, the number of variates in each sample; and a v , the 
 standard deviation of the parent. When the mean and standard 
 deviation of the parent are known, the mean and standard devia¬ 
 tion of the distribution of sample averages can be obtained. 
 
 PROBLEMS 
 
 1. Given a parent population of 5 variates, v h v 2 , v 3 , v t , v 3 . Set up all 
 possible sample averages of 3 variates, find the z’s and the standard 
 deviation of all possible sample averages, where repetitions are not 
 allowed. 
 
 2. Given a parent population consisting of the heights of 1,000 men, 
 
 with mean 67.6 inches, and standard deviation equal to 2.5 inches. Find 
 the standard deviation of all possible sample averages if each sample 
 contains 200 men. If each sample contains 500 men. If each sample 
 contains 800 men. _ 
 
 3. Plot <r 2 = y / (s — r)/r(s — l)-a„, where s = 1,000, a v = 2.5. Dis¬ 
 cuss the plot. How large must r be so that a v is less than one-half of cr„? 
 Less than one-tenth as large? 
 
 4. If all the sample averages are within 3.2 standard deviations of the 
 mean of all possible sample averages, what are the ranges for the sample 
 averages in the parts of problem 2? 
 
 SKEWNESS OF THE DISTRIBUTION OF THE MEANS 
 
 It can be shown that the skewness of the distribution made up 
 of all possible sample means is 
 
 (11.5) 
 
 «3 :* 
 
 s — 2r / s — 1 
 •c — 2 ' r(s — r) 
 
 «3:( 
 
SKEWNESS OF DISTRIBUTION OF MEANS 
 
 209 
 
 If s approaches infinity, formulas (11.4) and (11.5) become 
 (11.6) = a-*/Vr; 
 
 (H.7) <* 3 : 2 = ot Z: jVr, 
 
 as can be seen by dividing both numerator and denominator in the 
 formulas for a finite parent by s and then allowing s to become 
 infinitely large. 
 
 Formula (11.7) indicates that when r is large the skewness of 
 the distribution of the means is nearly equal to zero; this means 
 that the distribution of the sample means is approximately sym¬ 
 metrical for large values of r. 
 
 Example. There are 4,000 students on a college campus. The 
 average weight of these students is 145 pounds; the standard deviation is 
 20 pounds; the skewness of the distribution made up of the weights of the 
 students is 0.3. If a sample of 1,000 students is taken at random from 
 the student body, what is the probability that the average weight shall be 
 greater than 148 pounds? 
 
 „ _ *1 + Xi + . . . + *1,000 
 
 1 ~~ F000 
 
 M z = M x — 145 pounds, which is the mean of all possible averages 
 of 1,000 weights. The standard deviation and skewness are: 
 
 a M = 20 
 
 s — r 
 
 *r(s — 1) 
 
 -20 J 3 ' 0M 
 
 \ 1,000(3,999) 
 
 = 0.55 lb. 
 
 a 3 = 0.006; call it zero. 
 
 148 - 145 
 0.55 
 
 5.5 standard units, 
 
 which means that 148 pounds, the average of 1 sample taken at random, 
 deviates from the average of all possible sample averages by 5.5 standard 
 units. On examining normal probability tables it is seen that the prob¬ 
 ability that a sample average as large as or larger than 148 is very small 
 indeed. Very few tables record any area beyond 5.5 standard units. 
 The difference between 145 and 148 is only 3 pounds, yet it is almost 
 impossible to take at random a sample of 1,000 students from the original 
 4,000 and secure an average which differs by as little as 3 pounds from 
 the mean of the entire group. The work so far shows that the mean of a 
 representative sample differs by a very small amount from the mean of the 
 parent, when the sample is large. The mean is usually written as 
 Mz ± cta ii- 
 
210 
 
 SAMPLING 
 
 PROBLEMS 
 
 1. The mean, standard deviation, and skewness of a parent population 
 with 1,000 variates are respectively 36.8 inches, 1.2 inches, and 0.6. 
 Find the mean, standard deviation, and skewness of the distribution made 
 up of all possible sample means of 400 to the sample. When the sample 
 contains 500 variates. When the sample contains 800 variates. 
 
 2. Consider a parent with an infinite number of variates, where 
 M x = 40.3 gallons, <j x = 1.7 gallons, and a 3 -. x = 0.4. Find the mean, 
 standard deviation, and skewness of the distribution made up of all 
 possible sample averages, when the samples contain 1,000 variates. 
 When the samples contain 100 variates; 10,000 variates. 
 
 3. There are 1,200 first-year men on the campus. The following 
 is known concerning the distribution of their heights: M v = 68.1 inches, 
 a v = 2.3 inches, 0:3 : p = 0.2 Two hundred first-year students are picked 
 at random; find the probability that the average of these heights will be: 
 (a) greater than 68.7 inches; (6) less than 67.8 inches; (c) greater than 
 67.6 inches and less than 68.7 inches if measurements are made to the 
 nearest 0.1 inch. 
 
 4. Out of 400 head of cattle 200 are picked. It is found that the 
 average number of pounds of milk for the first group is 39, for the second 
 group is 36; also that the standard deviations are respectively 4.5 and 
 5.4 pounds. Is the difference due to random sampling? ( Hint. Find the 
 mean and standard deviation of the pounds of milk from the 400 cows.) 
 
 6. Solve problem 4 when the first group contains 100 cows and the 
 second contains 300. 
 
 6. If the standard deviation of all possible samples of 1,000 from a 
 parent population consisting of s items is 0.55 and the standard devia¬ 
 tion of the parent is 19.8, find s the number of items in the parent. 
 
 7. Find the standard deviation of all sample averages obtained by the 
 class when solving problem 4 on page 205. Compare this value with the 
 standard deviation of the parent and with the value obtained from 
 formula (11.4) (although this formula is for all possible samples). 
 Compare this value with the standard deviation of the parent divided by 
 s/n, where n is the number in the sample. 
 
 8. Find the skewness of the distribution made up of the sample 
 averages which the class obtained when solving problem 4 on page 205. 
 Compare this with that obtained from formula (11.5). 
 
 9. Given a parent population of the variates v h v 2 , v 3 , and v 4 . Set up 
 all possible sample averages of 2 variates, and find the skewness of the 
 averages of all possible samples. 
 
 10. How many possible samples of 5 variates can be drawn from a parent 
 consisting of 100 variates? How many when there are 25 variates in the 
 sample? 
 
SAMPLING FROM AN UNKNOWN PARENT 
 
 211 
 
 11 . A parent contains 10,001 variates, with mean equal to 69.7 feet and 
 standard deviation equal to 7.4 feet. How large must the sample be so 
 that the standard deviation of the mean of the sample averages will be 
 less than 2 feet? Less than 1 foot? Less than f foot? 
 
 12 . A parent population contains an infinite number of variates with 
 mean equal to 1,927 gallons and standard deviation equal to 160 gallons. 
 How large wall a sample have to be so that the standard deviation of the 
 distribution made up of all possible sample means will be less than 80 
 gallons? Less than 40 gallons? Less than 20 gallons? Less than 
 10 gallons? Less than 5 gallons? Less than 1 gallon? 
 
 13 . If the skewness of the parent population in problem 12 is 1.1, how 
 large will a sample have to be so that the skewness of the distribution of 
 all possible sample averages will be less than 0.1? 
 
 14 . What is the mean of all possible sample averages of 10,000 variates 
 taken at random from the parent in problem 12? Of 100,000 variates? 
 Of 10 variates? 
 
 16 . A parent population contains an infinite number of variates with 
 standard deviation equal to 25 cubic feet. Plot and discuss the curve 
 
 <7„ _ 25 
 
 VT VT 
 
 16 . Assume that the skewness of the parent in problem 15 is 0.8. Plot 
 and discuss the curve 
 
 oiz-v _ 0.8 
 
 ClZ ' Z ~Vr~Vr 
 
 SAMPLING FROM AN UNKNOWN PARENT 
 
 It was shown, when the parent population -was known, that 
 the average of all possible sample averages was the same as the 
 average of the parent. It was also pointed out that, when the 
 sample was large, its mean did not differ very much from the 
 mean of the parent. In sampling, one never takes all possible 
 samples, but one or two good-sized samples, using the average 
 of these to represent the mean of the parent. This average will 
 not differ very much from the mean of the parent if the sample 
 is fairly large. 
 
 When the parent is unknown the average of a large sample is 
 also considered to be near the mean of the parent, since the range 
 for the distribution of all possible sample averages is very small. 
 Hence the mean of a good-sized sample or the average of the means 
 of several good-sized samples will give a good approximation of 
 
212 
 
 SAMPLING 
 
 the mean of the parent. This is usually accepted as a good esti¬ 
 mate of the mean of the parent. 
 
 It is now necessary to find the standard deviation of all possible 
 sample means, or the standard deviation of the distribution made 
 up of all possible sample averages when the parent is unknown. 
 This standard deviation when found will indicate how accurate the 
 mean we have obtained from one or several samples is. When the 
 parent was known the standard deviation of the sample averages 
 was found on page 208. There the standard deviation of the 
 parent was known and was used to find the standard deviation of 
 the distribution of the means of the samples. When the parent is 
 unknown its standard deviation is unknown, hence this formula 
 cannot be used. It is now necessary to find an estimate for this 
 standard deviation. 
 
 The standard deviation of the sample means shall be considered 
 to be the square root of the average of all possible second moments 
 of the samples about their respective means. Consider for the 
 present that the parent of s variates is known and that all possible 
 samples can be obtained. Take samples of r variates from the 
 parent and find the second moment about the mean of this sample. 
 Take another sample and find the second moment about the mean 
 of this sample. Do this for all possible samples. Find the average 
 of all these second moments. The second moment about the 
 mean of the first sample is 
 
 iv? ( 2pA 2 
 
 ^ = ~ "V77 ' 
 
 The following gives the second moments about the means of the 
 respective samples, and the average of all of them 
 
 M2:2j = Xivi-Mjyr = 
 
 r* L 
 
 7,1 
 
 (r - 1) v ? ~ 
 
 2 
 
 * 
 
 3 
 
 M2; 22 = 
 
 7,2 
 
 ( r “ x ) 2 V ' 2 ~ 
 
 ) 
 
SAMPLING FROM AN UNKNOWN PARENT 
 
 213 
 
 Me :z 
 
 aCr 
 
 72 [( r - !) X) v < 
 
 sCr 
 
 E 1 "' , 
 
 <= 1 1 
 
 aC r 
 
 (r - l)r(,C r ) 2 «« 2 %C 2 ){,C r ) v *>i 
 
 i = 1 
 
 s( 8 C r ) 
 
 GC 2 )GC r ) 
 
 ,*^ i 
 
 [mG :, - tY T (*V 2 i : • “ s mG :.)] J (2^) 2 = S 2 M V 2 
 
 r— 1 
 
 s(s-l) 
 s(mG :* — M ,2 1 :«) 
 
 r L s — 1 
 
 s(r — 1) 
 r(s — 1) 
 
 M2 
 
 Thus the average of all possible second moments about the means 
 of the respective sample means is 
 
 , s(r — 1) 
 
 (11-8) Mh2:z = - — M2 :v) 
 
 r(s — 1) 
 
 which is not the same as the second moment about the mean of 
 the parent. The quantity Mm 2 : z is the average of all second 
 moments of the samples about the respective means of the samples. 
 
 Take at random a sample of r variates from a parent population 
 and find the second moment of this sample about its mean; this 
 may be a value which is not far from Mm 2 ■. z, and again it may not 
 be. Take several samples, and find the average of their second 
 moments about their respective means. This average will be a 
 better estimate of Mu 2 : * than the second moment about the mean 
 of just one sample. The quantity Mm 2 -. z will be considered to be 
 the best value for the second moment about the mean of any par¬ 
 ticular sample or the average of the second moments about the 
 means of several samples. Since Mm 2 : z is never known, an esti¬ 
 mate of it will be considered to be the second moment about the 
 mean of a good-sized sample. We shall call the second moment 
 about the mean of a sample the square root of Mm 2 :2 , or 
 
 ^sample ^ ^ M2 : z ^parent 
 
 
 s(r - 1) 
 r(s - 1) 
 
 As a rule, the parent population is not known, and hence <r piirent 
 and s are not known. All the information that is known is that 
 
 * The r, 1 above 2 means the first sample with r variates in the sample; 
 r, 2, the second, etc. 
 
214 
 
 SAMPLING 
 
 obtained from one sample or that obtained from several samples. 
 From this information it is desired to obtain certain characteristics 
 of the parent population. The exact values of these characteris¬ 
 tics, of course, cannot be found. Yet results can be obtained which 
 appear to be good approximations of the characteristics of the 
 parent. 
 
 The quantity 
 
 s(r — 1) . 
 
 is less than 1; hence the average of the 
 
 r{s — 1) 
 
 respective second moments about the means of all possible sample 
 averages is less than the second moment about the mean of the 
 parent. It is necessary to adjust the value of the second moment 
 about the mean of any sample taken at random to make it a better 
 approximation for the second moment about the mean of the 
 parent. 
 
 It was proved before that the standard deviation of all possible 
 sample means is 
 
 (11.4) 
 
 <Tz = 
 
 (Ty 
 
 
 s — r 
 r(s — 1) 
 
 
 when the parent was known. Formula (11.4) gives the standard 
 deviation of all possible sample averages in terms of the standard 
 deviation, a v , of the parent. If this, a v , were known, the standard 
 deviation of the distribution of the sample means would be known. 
 It is the object of this section to find a similar formula for the 
 standard deviation of the sample means when the parent is not 
 known. Since a v , the standard deviation of the parent, is not 
 known the best that can be done is to replace it by something that 
 seems to approximate it. From what has been said above, the 
 best value of the standard deviation of the sample is 
 
 therefore 
 
 ^sample ^parent 
 
 ^parent ^sample 
 
 
 s(r - 1) 
 r(s — 1) ’ 
 
 r(s - 1) 
 s(r - 1) 
 
 Substituting this value of o- parent in (11.4) for the standard deviation 
 of the parent population gives 
 
 &z ^sample 
 
 
 s — r 
 s(r - 1) 
 
 s — r 
 s(r - 1) ’ 
 
 (11.9) 
 
SAMPLING FROM AN UNKNOWN PARENT 
 
 215 
 
 where a' v is the standard deviation of the particular sample taken 
 at random. Formula (11.9) gives the best estimate of the stand¬ 
 ard deviation of the distribution made up of sample averages when 
 the parent is unknowm. 
 
 When s —> oo formula (11.9) reduces to 
 
 ( 11 . 10 ) 
 
 or 
 
 Oz = 
 
 S V 
 
 V~r’ 
 
 which is the standard deviation of the sample means when the 
 parent population is not known, and is considered to be the most 
 
 plausible value. The numerator s. 
 
 I 2v 2 . 
 
 - = V-7,1 
 
 v r — 1 
 
 in the right-hand 
 
 member of (11.10) is considered to be the best estimate of the stand¬ 
 ard deviation of the parent population from which the sample came. 
 
 / / 2D 2 
 
 vis a v = yj - 
 
 the best 
 
 The standard deviation of a set of r items 
 estimate of the standard deviation of the parent from which a 
 
 It is not the exact value 
 
 sample of r items came is s, 
 
 'v = J- 
 
 * 1 
 
 hv 2 
 
 1 
 
 and may not be the best that can be obtained, yet it is used 
 as an estimate of the true value. Formula (11.9) is not an 
 exact formula because we do not know how near the standard 
 deviation, <r' v , of 1 sample, although large, is to the square root of 
 the average of all possible second moments about the various means 
 of the samples. We only assume that it is near it. After all, this 
 is a plausible assumption for large samples. 
 
 Formula (11.10) gives the standard deviation of the mean of a 
 sample consisting of r variates, when the parent contains an infinite 
 number of items. Formula (11.10) should be used when the 
 parent is unknown. 
 
 Formulas (11.6) and (11.10) are about the same for large values 
 of r. 
 
 Example. Twenty-seven stalks of wheat were measured with the 
 following results: M v = 5.76 cm., 2SF 2 = 80.4 sq. cm. Find the standard 
 
216 
 
 SAMPLING 
 
 deviation of the set of measurements, the best estimate of the standard 
 deviation of the parent from which this sample was taken, and the best 
 estimate of the standard deviation of the mean. These values are 
 respectively 
 
 r', - AlJ - 
 \ 27 
 
 5 .-= 
 V 26 
 
 <s m 
 
 26 
 1.823 
 V?7 
 
 1.789 cm., 
 1.823 cm., 
 
 .351 cm. 
 
 M v ± an = 5.76 ± .351 cm. 
 
 PROBLEMS 
 
 1. The length of a table is measured 10 times by A and 10 times by B 
 with the following results: 
 
 A B 
 
 Mean = 274.1 in., Mean = 274.1 in., 
 
 a = 0.15 in., cr = 0.66 in. 
 
 By means of a diagram, show that the results are not consistent. 
 
 2. If a random sample of 36 variates is taken from an infinite parent 
 whose standard deviation is known, how many variates should be taken 
 in another sample to make the standard deviation of the mean of the 
 second sample 6/10 of the standard deviation of the mean of the first 
 sample? 
 
 3. The mean and standard deviation of lengths of 26 snakes are 
 respectively: M v = 24.8 in., a v = 2.3 in. Find the standard deviation 
 of this mean, or the distribution of all possible means. 
 
 4. If the sum of the squares of the deviations of 37 items from the 
 mean of these 37 items is 144 square feet, find the best estimate of the 
 standard deviation of the parent from which the sample came. Find the 
 standard deviation of the mean. Find the standard deviation of the set 
 of 37 items. 
 
 THE STANDARD DEVIATION OF A SUM AND DIFFERENCE 
 
 Consider the line ABC. Let n measurements be made of AB 
 and m be made of BC. Let the error of an individual measurement 
 of AB from the mean of all measurements of AB be represented by 
 v and that of BC be represented by w. 
 
 A 
 
 B 
 
 C 
 
STANDARD DEVIATION OF A SUM AND DIFFERENCE 217 
 
 After one measurement has been made of AB, then any one of 
 the m measurements can be made of BC) hence the first error of 
 AB can be added to each of the m errors of BC in finding the error 
 of the sum ABC. The second error of AB can also be added to 
 the m errors of BC in finding the error of ABC , and so on for each 
 of the n errors of AB. The errors v, w, and v + w are fisted below: 
 
 Error 
 
 For AB For BC Error for ABC 
 
 v w v + w 
 
 VI WI V\ + WI 
 
 V2 m V2 + Wl 
 
 V3 W3 V3 + W l 
 
 V\ + W2 
 V2 + W2 
 V3 + U)2 
 
 vi+m... 
 V 2 + W3 . . . 
 V 3 + m . . . 
 
 V\ + W m 
 V 2 + W m 
 V3 + W m 
 
 V m Vrn W\ 
 
 V m + W 2 
 
 V m +W 3 ... 
 
 V m + Wm 
 
 V n V n + Wl 
 
 V n + W 2 
 
 Vn + W3 • ■ • 
 
 V n ”1“ Wm 
 
 Square each error for ABC and add by adding the columns sep¬ 
 arately; this gives 
 
 n n n 
 
 ^ (Vi + wi) 2 = "y ^ Vi 2 + 2wi y Vi + nwi 2 , for the 1st column, 
 
 i =1 «=i i=i 
 
 n n n 
 
 y ^ ( Vi -\-w 2 ) 2 = y ] Vi 2 + 2u>2 y ^ Vi + nw-A, for the 2d column, 
 
 <=i 
 
 i=i 
 
 n n n 
 
 (v { +w 3 ) 2 = y>2 2 + 2m y^ 
 <=1 <=1 (=1 
 
 Vi + nw 3 2 , for the 3d column, 
 
 n n n 
 
 y ^ ( Vi -f- w m ) 2 = y ' Vi 2 -f 2 w m y ^ Vi + nu; m 2 , for the last column 
 
 ^=.l t= i (-i 
 
 Now sum all columns; this gives the sum of the squares of all 
 errors. The sum of the squares of all possible errors of measure¬ 
 ments of ABC is 
 
 n vi m n n rn 
 
 '“’i 2 + di = m y^ v ?+ n yi , 
 
 i~ i ^-i j-i (-i <-i j-i 
 
218 
 
 SAMPLING 
 
 since the Zv- = Sic = 0. The standard deviation of the sum is 
 
 ( 11 . 11 ) &AB + BC ~ 
 
 winch is equal to 
 
 ImZvi 2 nSwj 2 /— -— 
 
 O'sum = V -1-= v a v 2 + a w 2 , 
 
 y mn mn 
 
 <?ABC 
 
 = <tab~ + 
 
 <TBC 2 - 
 
 If the signs in the errors for the sums are changed to minus 
 when finding the errors for differences, formula (" 11 . 11 ) will give 
 the standard deviation of the difference, hence 
 
 (11.12) ^difference = Vo-* 2 + U w 2 . 
 
 ANOTHER DERIVATION OF THE STANDARD DEVIATION 
 
 OF A SUM 
 
 Let v and w be 2 independently measured quantities with stand¬ 
 ard deviations respectively, a v and <r w . Let the standard deviation 
 of the sum v + w be <r v+w , the errors from the mean of the measure¬ 
 ments made of v be v\, i> 2 , V 3 , ■ ■ ■ , v n , the errors from the mean of 
 measurements of w be w\, ijbo, w 3 , . . . , w n , and the errors for the 
 sum be di + w,-. Then 
 
 ]2vi 2 i 2 Wj 2 | 2'EviWj 
 
 <?v+w = \ I— ' d-, 
 
 v n n n 
 
 = V <J 2 + <T W 2 j 
 
 since it is assumed that ViWj will be positive as often as it will be 
 negative and that the sum will be zero or very near to zero. This 
 assumption is plausible when there are a large number of measure¬ 
 ments. 
 
 In general, 
 
 ^sum = <7v 2 + <Tw 2 + <r 8 2 + . . . + a a 2 . 
 
 PROBLEMS 
 
 1. The segments AB, BC, CD, DE, EF of the line AF were measured 
 with the following results for the means: 
 
 AB = 7.3 ft. ± 0.01,* 
 
 BC = 9.7 ft. ± 0 . 02 , j -j- j -—- 1 
 
 CD = 4.9 ft. ± 0.02, A B C DE F 
 
 DE = 1.8 ft. ±0.03, 
 
 EF = 14.1 ft. ± 0.01. 
 
 * Plus or minus standard deviation of the mean. 
 
SIGNIFICANCE OF DIFFERENCE BETWEEN TWO MEANS 219 
 
 Find the length of AF and its standard error. * If 16 measurements were 
 made of DE, find the standard deviation of the individual measurements. 
 (The standard deviation of a set of items and the standard deviation of 
 each item in the set are the same.) 
 
 2. To measure a certain distance, a tape is laid down 529 times, each 
 with a standard error of 0.11 inch. What is the standard error of the 
 whole distance measured? 
 
 3. The segments AC and CB, where B is on line AC, were measured 
 with the following results for their means: 
 
 AC = 39.33 in. ± 0.05, 
 
 CB = 27.89 in. ± 0.03. 
 
 Find the length of AB and its standard error. 
 
 SIGNIFICANCE OF THE DIFFERENCE BETWEEN TWO MEANS 
 
 Consider a frequency distribution made up of the weights of 
 men, with mean 140 pounds. The probability that a person, taken 
 at random from this group of men, will weigh more than the mean, 
 140 pounds, by more than 3 standard deviations is very small. 
 The same is true for picking a person whose weight is less than the 
 mean by 3 standard deviations of the distribution. 
 
 Assume that one person took at random 50 men and found the 
 mean of the weights of these 50 men to be 141 pounds, and that 
 another person took a sample of 50 and found the mean weight to 
 be 145 pounds. Are these results significant, or are they due to 
 fluctuations in random sampling? Would one expect a difference 
 as large as this difference in the means to happen often or very 
 seldom? The answer to this question is found in the theory of 
 probability. Set up the distribution of the differences of the means 
 for all possible samples. This distribution will have a mean and 
 a standard deviation as all distributions. Since the distribution 
 of all possible sample averages is normal (when the number in the 
 sample is largef), the distribution made up of the differences of the 
 averages will also be normal. About 68 per cent of this distribu¬ 
 tion will lie within 1 standard deviation of the mean of this distri¬ 
 bution. Since we know from the first of this chapter that the 
 means of the samples lie close to the mean of the parent, the differ¬ 
 ences of any two sample means will be small, and hence most of 
 these differences will be close to zero. The mean of all possible 
 
 * The standard error is the standard deviation of the mean, 
 t Proved in more advanced works in statistics. 
 
220 
 
 SAMPLING 
 
 differences between the sample means is zero.* Since the differ¬ 
 ences of the means of the samples are small and close to zero the 
 standard deviation of the distribution made up of the differences 
 of the means will naturally be small. If the difference of 2 sample 
 means were as large as several standard deviations of this distri¬ 
 bution made up of the differences of sample averages, then the 
 probability of getting such a large difference by random sampling 
 would be very small. The answer to this question is obtained by 
 finding the size of this probability. We know that the probability 
 of a variate (that is, one chosen at random) being more than 3 
 standard deviations from the mean is very small. Hence if the 
 difference between the means is more than 3 standard deviations 
 of the distribution, made up of the differences of all possible means, 
 then the probability is small and the results cannot be due to ran¬ 
 dom sampling. Since the mean of the distribution made up of all 
 possible differences of sample means is zero, the test for significance 
 of the difference between 2 means is found by obtaining a t value 
 as was done on page 88; this test for the significance between 2 
 means is 
 
 (11.13) 5 = 
 
 Actual difference of the means — zero 
 
 Significance = 
 
 Standard error of the difference of the means 
 
 If this is greater than or equal to 2.6 the difference between the 
 mean is significant and we can say that such a large difference was 
 not due to chance, or did not arise from random sampling from 
 the same parent population. The following problem will bring 
 out the meaning of formula (11.13). 
 
 Out of 400 head of cattle, 200 are picked. It is found that the average 
 number of pounds of milk for the first group is 39, for the second group 
 it is 36; also that the best estimates of the standard deviations of the 
 parents from which the first and second groups came are respectively 4.5 
 and 5.4. Is the difference between the means significant? 
 
 The standard error of the first mean is 4.5/ \/ 200. 
 
 The standard error of the second mean is 5.4/\/200. 
 
 The standard error of the difference of the means, by formula (11.12), is 
 
 This is proved in more advanced works in statistics. 
 
PROBLEMS 
 
 221 
 
 Significance = S = 3/.49 = 6+ standard errors. Since the difference 
 of the 2 means is over 6 standard errors of the difference of the means 
 from zero, the mean of all possible sample means, the 200 cows were not 
 picked by random sampling. The difference is significant. 
 
 PROBLEMS 
 
 1. An epidemic resulted fatally in 12.2 per cent of the cases, this being 
 an average of the records from 82 cities, with a standard deviation of 
 1.3 per cent for the individual records. Upon a recurrence of the epidemic 
 a serum was used and the average this time was 11.0 per cent, and the 
 standard deviation of the individual records was 1.2 per cent. Does the 
 evidence support the use of the serum? 
 
 2. At a certain experimental station 50 sheep were fed spelt with an 
 average gain of 25 pounds per animal and standard deviation per animal 
 9.44 pounds. Another 50 were fed barley with an average gain of 
 33.9 pounds per animal and standard deviation per animal 2.28 pounds. 
 Is the evidence in favor of the barley conclusive? 
 
 3. Two orchards A and B have the same number of trees, n, and the 
 standard deviation of the yield of fruit for the individual trees is 0.51 
 bushel in both orchards. If orchard A yields on the average 0.15 bushel 
 per tree more than orchard B, how large must n be in order that it may be 
 asserted safely that A is the better orchard? 
 
 4. A marksman who is due south of a post has hit the post 41 times in 
 75 shots. The post subtends at the eye an angle of 2 minutes. Due east 
 of the post stands a cow such that the distance between the cow and the 
 center of the post subtends an angle of 8 minutes at the eyes of the 
 marksman. Is the cow in danger of being shot? 
 
 6. The following table gives the frequency distribution of the number 
 of rays per inflorescence of the Golden Alexander ( Zizia aurea), taken 
 from two different localities. Find the means, the standard deviations 
 of these means, and the significance of the means. Discuss the results. 
 
 of Rays per 
 
 A 
 
 B 
 
 Cluster 
 
 Frequency 
 
 Frequency 
 
 8- 9 
 
 0 
 
 28 
 
 10-11 
 
 21 
 
 51 
 
 12-13 
 
 39 
 
 72 
 
 14-15 
 
 98 
 
 103 
 
 16-17 
 
 127 
 
 109 
 
 18-19 
 
 111 
 
 71 
 
 20-21 
 
 66 
 
 46 
 
 22-23 
 
 22 
 
 12 
 
 24-25 
 
 6 
 
 8 
 
 25-26 
 
 11 
 
 0 
 
222 
 
 SAMPLING 
 
 6. Counts of bacteria in 1 cc. of milk: 
 
 Best Estimate of <j of 
 Average the Parents 
 
 First lot of 25 samples... . 150,000 18,000 
 
 Second lot of 36 samples.. 120,000 12,000 
 
 Does the second lot show improvement not attributable to chance? 
 
CHAPTER 12 
 
 NON-LINEAR REGRESSION 
 NON-LINEAR CORRELATION 
 
 In Chapters 9 and 10 relations between the variables were 
 assumed to be linear. This is not always the case. In physics it 
 is shown that the volume of a gas and the pressure applied to the 
 gas are related according to a law similar to 
 
 (12.1) V = C/P, 
 
 where E represents the volume, P the pressure applied, and C some 
 constant. Suppose a physicist has made n observations of the 
 volume of a certain gas after different amounts of pressure have 
 been applied. He has n observed values for E and n correspond¬ 
 ing values for P. Substituting these values in (12.1) gives the fol¬ 
 lowing n observational equations: 
 
 Fi = C/P i, E 2 = C/P 2 , E 3 = C/P;, . . . , V n = C/P n - 
 
 To find the “best value” of C it is necessary to find the value 
 which will make the sum of the squares of the residual errors a 
 minimum. The squares of the residual errors are as follows: 
 
 ei* = (Ei - C/P,) 2 = Ex 2 - 2C- Ei/Pi + C 2 /P i 2 
 
 e 2 2 = (E 2 - C/P 2 ) 2 = E 2 2 - 2C- E 2 /P 2 + C 2 /P 2 2 
 
 e 3 2 = (E 3 - C/P; Y = E 3 2 - 2C-E 3 /P 3 + C 2 /P 3 2 
 
 e„ 2 = (E„ 2 - C/P„) 2 = E„ 2 - 2C- Vn/Pn + C 2 /P„ 2 
 2e 2 = 2E 2 - 2C2E/P + C 2 2(l/P 2 ) 
 
 = ( s ^) c2 + 2 ( _s ?) c + sy2 - 
 
 The sum of the squares of the residual errors is a quadratic 
 
 223 
 
224 
 
 NON-LINEAR REGRESSION 
 
 expression in C; by the lemma on page 147 the quadratic ex¬ 
 pression is a minimum when 
 
 2(1 /P 2 )C - 2F/P = 0, or when C = 
 
 S(F/P) 
 2 (1/P 2 )' 
 
 This value for C is the value which makes the sum of the squares 
 of the residual errors a minimum; hence from the definition this 
 value is the “best value” for the quantity C. 
 
 The standard error of prediction found as in Chapters 9 and 
 10 is 
 
 
 C2(F/P) 
 
 n 
 
 4 
 
 2T 2 - (2F/P) 2 /2(1/P 2 ) 
 
 n 
 
 If the quantity 2F 2 + (2 V) 2 /n is substituted for 2F 2 , the 
 standard error of prediction reduces to 
 
 (2 VIP) 2 (2F) 2 
 n 
 
 G e — Gu 
 
 1 - 
 
 ;(i/p 2 ) 
 
 vt;2 
 
 21 
 
 = <r,Vl ~ r 2 , 
 
 where 
 
 (2 F/P) 2 (2F) 2 
 
 r = 
 
 2(1/P 2 ) 
 
 n 
 
 2T 2 
 
 The quantity r is the non-linear correlation coefficient for the par¬ 
 ticular relation (12.1). In many books it is called the correlation 
 ratio. The relation between a e , and r may be written as follows: 
 
 2 _ 
 
 = a 2 (1 — r 2 ) or 
 
 r 2 = 
 
 = 1 - 
 
 Let y and x be connected by the relation 
 (12.2) y = a + bx + cx 2 , 
 
 and let the n observations for x and the n corresponding observed 
 values for y be substituted in this equation. This gives rise to n 
 observational equations which, according to the least squares 
 method, lead to the following normal equations: 
 
 na + bZx + c2x 2 = 2 y 
 
 2 x-a + bZx 2 + cZx 3 = Zxy 
 
 (12.3) 
 
 2x 2 -a + bZx 3 + c2x 4 = 2 x 2 y 
 
PROBLEMS 
 
 225 
 
 from which values of a, b, and c can be found. If relation (12.2) is 
 written in terms of the deviations from the means the normal 
 equations will be easier to solve. 
 
 The standard error of prediction is 
 
 (12.4) 
 
 — aly — blxy — clx 2 y 
 n 
 
 If 2 y 2 + nM y 2 is substituted in (12.4) for 2 y 2 , the standard error 
 of prediction becomes 
 
 (12.5) 
 
 ally + blxy + cllx 2 y — nM y 2 
 
 ~ W 2 
 
 r 2 , 
 
 where r is the non-linear correlation coefficient between x and y. 
 
 If the predicting equation had been written in terms of the 
 deviations from the means the standard error of prediction could 
 be written as 
 
 ( 12 . 6 ) 
 
 where 
 
 a e = o-y-yj] 
 
 blxy + clxry 
 
 v " "■ Vr ^ 5 
 
 r =4 
 
 blxy + clx 2 y 
 
 2 y 
 
 .•72 
 
 as before we have 
 
 r = 
 
 The quantity r satisfies the inequalities — 1 ^ r ^ 1. 
 
 PROBLEMS 
 
 1. Write out the value of b in (12.3) in terms of the fundamental 
 summations. 
 
 2. Set up the normal equations for finding the constants a and b if 
 the predicting equation is y = a + b x 2 . 
 
 3. Assume that x and y are connected by the relation y — a + bx 4- 
 cx 2 ; find the values of a, b, and c which will make the sum of the squares 
 of the residual error a minimum for the following data: 
 
 x: 1.3, 2.0, 2.5, 3.0, 3.4, 3.5, 4.0, 5.0, 5.5, 6.0, 7.0, 7.5, 8.0, 
 y: 2.6, 5.1, 7.3, 9.9, 12.5, 13.3, 16.8, 26.2, 31.3, 36.9, 50.4, 56.6, 73.4, 
 x: 8.2, 9.0, 9.3, 9.8, 10. 
 y : 65.4,81.4,87.7, 97.3, 101. 
 
226 
 
 NON-LINEAR REGRESSION 
 
 Plot the predicting line and the lines parallel to it at a distance of 1 stand¬ 
 ard error of prediction from it. What percentage of the observed values 
 for y falls in this band? Find the value of y when x is equal to 5.2, also 
 when x = 8.7. Find the value of the non-linear correlation coefficient. 
 
 4. By means of the lemma on page 147 prove that the normal equations 
 (12.3) arise when the relation between y and x is relation (12.2). 
 
 5. The following data give the relation between thread angle in worm 
 gear and efficiency. Plot the data, and find the equation of the curve 
 which seems to fit the data best. Find this equation. Let y represent 
 efficiency. 
 
 Angle Thread 
 
 Efficiency 
 
 Angle Thread 
 
 Efficiency 
 
 1° 
 
 14% 
 
 25° 
 
 82% 
 
 5 
 
 52 
 
 30 
 
 84 
 
 7 
 
 60 
 
 35 
 
 86 
 
 10 
 
 78 
 
 40 
 
 87 
 
 15 
 
 75 
 
 45 
 
 88 
 
 20 
 
 79 
 
 50 
 
 87 
 
 6. Find the value of r which arises when the equation in problem 2 
 is used for data in problem 5. 
 
 CORRELATION RATIO 
 
 The non-linear correlation coefficient, as defined above, is 
 obtained after a predicting equation has been assumed; that is, it 
 is a function of the predicting equation. Different predicting 
 equations lead to different values of r. When a non-linear corre¬ 
 lation coefficient is desired without making any assumption con¬ 
 cerning the type of curve which appears to fit the data, it is cus¬ 
 tomary to obtain a coefficient -which arises from the average of the 
 squares of the deviations of the y’s from the respective means of 
 the y arrays. The quantity oy 2 , as was shown above, is equal to 
 the average of the squares of the deviations of the y’s from the 
 regression line. Let s y 2 represent the average of the squares of the 
 deviations of the y values from the respective means of the y 
 arrays. From each value of y in the first y array subtract the 
 mean of the first y array; from each y value in the second y array 
 subtract the mean of the second y array; etc. The average of the 
 squares of these deviations from the respective means of the y 
 arrays is equal to s 2 . The quantity s v is a standard error of pre- 
 
CORRELATION RATIO 
 
 227 
 
 diction when the predicted values for y are the respective means 
 of the y arrays.* According to this, s y isf 
 
 (12.7) Sy 
 
 where Si is the sum of the y’s in the ith y array; Si/f Vi is the mean 
 of the ith y array; f Vi is the number or frequency of the y’s in the 
 zth y array; k is the number of y arrays in the distribution; and n 
 is the number in all arrays. This can be written as follows: 
 
 ( 12 . 8 ) 
 
 Add and subtract M y 2 under the radical; this gives 
 
 Sy 
 
 Sy 
 
 or 
 
 (12.9) s„ 
 where 
 
 ( 12 . 10 ) r, vx 
 
 The quantity rj yz is called the correlation ratio and is used as a 
 * The y array on x is composed of the y values for certain x or a certain 
 x class. 
 
 t Si =/yi. 
 
228 
 
 NON-LINEAR REGRESSION 
 
 measure of non-linear correlation. The advantage of r} yx over r is 
 that it does not depend upon any predicting equation. On exam¬ 
 ining the quantity rj yx> it is found to be the standard deviation of 
 the means of the y arrays divided by the standard deviation of the 
 y’s for the entire distribution; it can be found from a correlation 
 surface table similar to that on page 178. 
 
 In a similar way it can be shown that 
 
 ( 12 . 11 ) 
 
 X 
 
 where T > is the sum of the x values in the zth x array on y, Ti/f Zi 
 is the mean of the zth x array; M x is the mean of the x’s for the 
 entire distribution. 
 
 The following example will show the difference between r, r, 
 Vyij and rjiy. 
 
 Table 12.1 gives data pertaining to observed tree radii at breast 
 height and areas of cross sections at this height. The radius of a tree 
 at breast height is the average of the maximum and minimum radii. 
 Areas of cross sections are obtained by a planimeter. Values in row S 
 in the above table are found by summing the y' values, which are multi¬ 
 plied by the particular x' in finding the sum of the products, Sx'z/'. The 
 first value in row S, —16, is found by multiplying y’ = —4 by its fre¬ 
 quency 4. The third number, —41, in this row is found by summing the 
 values of y' which are multiplied by x' = — 2 in finding the sum of the 
 products ~2x'y'. There are 3 values of x’ = —2 and y' = —4, 9 values 
 of x’ = — 2 and y' — — 3, and 1 value of a;' = — 2 and y' = — 2; hence the 
 sum of these y' values which are to be multiplied by x = — 2 is 3( —4) + 
 9(— 3) + 1 ( — 2) = —41; and so on for the other values in row S. The 
 sum of the quantities in the row designated by S-x r is the sum of the 
 product terms, or 'Zx'y'. 
 
 Values in column T are obtained in a similar way. The second value, 
 27, in this column is found by finding the sum of the x' values which are 
 multiplied by y' = 2 in finding the sum of the product terms. There are 3 
 values of y' = 2 and x = 4 and 5 values of y' = 2 and x' — 3; hence 
 the sum of these x' values which are multiplied by y = 2 in finding the 
 sum of the product terms is 3(4) + 5(3) = 27. The sum of the values 
 in the column represented by Ty' is the sum of the product terms or 
 Z.t y. The sum of the Ty ' column should equal the sum of the Sx' row. 
 
 The rows S 2 and S 2 /f are used in finding y yx . The quantities rj yx and 
 t] xu are not always equal. The sum of the values in the row designated 
 
Correlation Surface Table for the Radii of Trees and the Areas of the Cross Sections at Breast Height 
 
 x radius 
 
 CORRELATION RATIO 
 
 229 
 
 ci 
 
 Si 
 
 128.000 
 
 91.125 
 
 146.260 
 
 49.846 
 
 14.205 
 
 8.036 
 
 76.409 
 
 93.091 
 
 606.972 
 
 
 
 
 
 
 
 
 
 
 
 C-4 
 
 Si 
 
 1,024 
 
 729 
 
 3,364 
 
 1,296 
 
 625 
 
 225 
 
 1,681 
 
 1,024 
 
 
 Si 
 
 96 
 
 54 
 
 58 
 
 o 
 
 -25 
 
 30 
 
 CO 
 
 CM 
 
 128 
 
 464 
 
 &■« 
 
 32 
 
 27 
 
 58 
 
 36 
 
 25 
 
 iO 
 
 T 
 
 i—i 
 
 l 
 
 -32 
 
 
 d 
 
 72 
 
 32 
 
 23 
 
 o 
 
 44 
 
 112 
 
 198 
 
 176 
 
 657 
 
 
 24 
 
 CD 
 
 l-H 
 
 23 
 
 o 
 
 1 
 
 -56 
 
 99- 
 
 -44 
 
 — 
 
 7 
 
 *?> 
 
 CO 
 
 
 
 o 
 
 7 
 
 1 
 
 CO 
 
 1 
 
 Tf< 
 
 1 
 
 
 
 00 
 
 00 
 
 23 
 
 26 
 
 44 
 
 28 
 
 22 
 
 - 
 
 170 
 
 
 90 
 
 688 
 
 
 
 
 594.681 
 
 -52 
 
 816 
 
 6,820 
 
 5.25 — 
 
 00 
 
 CO 
 
 
 
 
 
 
 
 Ol 
 
 
 48 
 
 192 
 
 CO 
 
 <N 
 
 961 
 
 80.083 
 
 496 
 
 768 
 
 3,072 
 
 4.95- 
 
 
 
 1-0 
 
 o 
 
 r —1 
 
 
 
 
 
 
 r- 
 
 CO 
 
 LO 
 
 CO 
 
 LO 
 
 20 
 
 09 
 
 400 
 
 23.529 
 
 180 
 
 459 
 
 1,377 
 
 4.65- 
 
 
 
 
 
 'M 
 
 CO 
 
 Tt< 
 
 
 
 
 22 
 
 CM 
 
 44 
 
 oo 
 
 00 
 
 X 
 
 CD 
 
 64 
 
 2.909 
 
 32 
 
 176 
 
 352 
 
 4.35 — 
 
 
 
 
 
 
 
 18 
 
 17 
 
 <N 
 
 
 
 37 
 
 - 
 
 37 
 
 37 
 
 Ol 
 
 1 
 
 Ol 
 
 1 
 
 
 GIG 11 
 
 -21 
 
 37 
 
 37 
 
 4.05- 
 
 
 
 
 
 
 
 
 23 
 
 O 
 
 
 
 35 
 
 O 
 
 o 
 
 o 
 
 -49 
 
 O 
 
 o 
 
 ■>* 
 
 <N 
 
 68.600 
 
 O 
 
 o 
 
 o 
 
 3.75- 
 
 
 
 
 
 
 
 
 
 1C 
 
 lO 
 
 - 
 
 Dl 
 
 r— < 
 
 1 
 
 01 
 
 1 
 
 21 
 
 -49 
 
 49 
 
 2,401 
 
 114.333 
 
 -49 
 
 CM 
 
 1 
 
 
 3.45- 
 
 
 
 
 
 
 
 
 
 - 
 
 C5 
 
 CO 
 
 CO 
 
 Ol 
 
 1 
 
 -26 
 
 52 
 
 1 
 
 82 
 
 1,681 
 
 129.308 
 
 -164 
 
 o 
 
 7 
 
 208 
 
 3.15- 
 
 
 
 
 
 
 
 
 
 
 CD 
 
 CO 
 
 C5 
 
 CO 
 
 1 
 
 -27 
 
 X 
 
 -30 
 
 90 
 
 006 
 
 100 
 
 -270 
 
 -243 
 
 729 
 
 2.85- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 CD 
 
 7 
 
 64 
 
 CD 
 
 1 
 
 CD 
 
 256 
 
 64 
 
 -256 
 
 -256 
 
 1,024 
 
 / 
 
 94.5 — 104.5 
 
 84.5 - 94.5 
 
 74.5 - 84.5 
 
 64.5 — 74.5 
 
 54.5 - 64.5 
 
 44.5 - 54.5 
 
 34.5 - 44.5 
 
 24.5 - 34.5 
 
 < 
 
 V H 
 
 "h 
 
 M 
 
 'h 
 
 CO 
 
 "h 
 
 CO 
 
 Cl 
 
 CO 
 
 CO 
 
 N 
 
 n 
 
 
 (101)039 SSOJO JO 0311! = H 
 
230 
 
 NON-LINEAR REGRESSION 
 
 by Sr/f x is part of the first term in formula (12.10). The sum of column 
 T- f y gives a part of the first term in formula (12.11). According to these 
 formulas* 
 
 V\x 
 
 and 
 
 594.681 
 
 170 
 
 (0.8647) 2 
 
 /3.117 = 0.8824; Vyx = 0.9394, t 
 
 V~iy — 
 
 606.972 
 
 170 
 
 0.2803 
 
 /3.767 = 0.8734; rj xy = 0.9346-f 
 
 By using data on page 229 and the predicting equation y = ax 2 , 
 where a is obtained by the least squares method, the non-linear 
 correlation coefficient r is 
 
 ja~Zx 2 y — nMy 2 
 T ~ \ y.,2 _ v 
 
 ZyMy ’ 
 
 where according to the least squares method 
 
 a 
 
 2x 4 
 
 3.1556. 
 
 The quantity 2x 2 y is found from the following equation: 
 
 (12.12) 2 x 2 y — 2 (wx' + h) 2 ( qy ' + k) = 'Zw 2 qx' 2 y' 
 
 + 2wqh'Ex'y' + h 2 qZy' + kw 2 Zx' 2 
 + 2hkw’Zx' + nh 2 k, 
 
 where w is the width of the class interval for x, q the width of the 
 class interval for y, h the provisional mean which w T as subtracted 
 from the mid-points of the x classes, and k the provisional mean 
 which was subtracted from the mid-points of the y classes. From 
 the above table this sum is 
 
 2 x 2 y = 0.09(10) (-52) + 2(0.3) (10) (4.2) (464) + 10(17.64) (-147) 
 
 + 0.09(69.5) (688) + 2(4.2) (0.3) (69.5) (90) 
 + 170(17.64)(69.5) 
 
 = 214,197.84. 
 
 The quantity 2x 4 is found by the formula 
 
 (12.13) 2x 4 = 2 (rex' + h ) 4 = w 4 2x' 4 + 4/m> 3 2x' 3 + 6/i 2 u> 2 2x' 2 
 
 + 4/i 3 ie2x' + nh 4 . 
 
 * In Table 12.1 frequencies of y-arrays are f z - 
 
 f Formulas (12.10) and (12.11) hold for deviations from provisional means, 
 as can be seen by examining formula (12.7). 
 
CORRELATION RATIO 
 
 231 
 
 From the sums of the last two rows in the table and the other 
 computed values this sum according to (12.13) is 
 
 2x 4 = 67,879.39; 
 
 hence a = 3.1556. The non-linear correlation coefficient r is 
 
 r = V8,757 = 0.9358. 
 
 The values of r and y yx are nearly the same because the means of 
 the y arrays lie close to the predicting line y = 3.1556a; 2 . 
 
 The above example shows that it is much easier to obtain y yx 
 than r. 
 
 If the predicting equation is a straight line the linear correla¬ 
 tion coefficient is 
 
 r xy = 0.9295+. 
 
 Linear regression should not be used here for the data fit a curved 
 line much better than a straight fine. The example shows the 
 meaning of r yx , r yx , rj yx , and rj xy . 
 
 The correlation ratio should be used for non-linear correlation, 
 for it gives a measure of dependence based upon the means of the 
 y arrays as the predicted values. 
 
 1. For the following data find the linear correlation coefficient and 
 the two correlation ratios. 
 
 No. of bracts per inflorescence of Daucus carota (wild carrot) for first 
 branch terminal inflorescence 
 
 <D 
 
 O 
 
 C2 
 
 o 
 
 o 
 
 cn 
 
 <D 
 
 (-> 
 
 O 
 
 t-i 
 
 o 
 
 a, 
 
 o 
 
 c3 
 
 S-H 
 
 X5 
 
 O 
 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 
 13 
 
 
 
 
 
 
 
 1 
 
 
 12 
 
 
 
 
 
 
 2 
 
 
 
 11 
 
 
 
 
 10 
 
 4 
 
 
 
 
 10 
 
 0 
 
 1 
 
 10 
 
 13 
 
 7 
 
 
 
 
 9 
 
 1 
 
 12 
 
 23 
 
 14 
 
 6 
 
 
 
 
 8 
 
 3 
 
 9 
 
 11 
 
 
 
 
 
 
 7 
 
 2 
 
 
 
 
 
 
 
 
232 
 
 NON-LINEAR REGRESSION 
 
 2. Rats were fed on a diet containing fixed amounts of vitamin G 
 with the following data pertaining to gains in grams for a certain period. 
 
 Amount of 
 
 Vitamin G Gain 
 
 .5. +2 
 
 .5. - 3 
 
 .5. +8 
 
 .5. +15 
 
 .5. +3 
 
 .5. 5 
 
 .5. 10 
 
 .5. 6 
 
 Amount of 
 
 Vitamin G Gain 
 
 1.5. 20 
 
 1.5. 28 
 
 1.5. 33 
 
 1.5. +30 
 
 1.5. 25 
 
 1.5. 30 
 
 Amount of 
 
 Vitamin G Gain 
 
 1. 14 
 
 1. 20 
 
 1. 8 
 
 1 . 17 
 
 1. 26 
 
 1. 18 
 
 1. 16 
 
 Amount of 
 
 Vitamin G Gain 
 
 2 . 31 
 
 2. 35 
 
 2. 36 
 
 2. 28 
 
 2. 29 
 
 2. 38 
 
 2. 36 
 
 Find the correlation ratio between amount of vitamin G and gain in 
 weight. 
 
Wholesale Prices per Pound of Butter 
 
 CHAPTER 13 
 
 THE ANALYSIS OF TIME SERIES 
 SECULAR TREND 
 
 Time series arise when one considers prices, yields, rates, etc., 
 for a period of time intervals. During each time interval there is 
 a price of a certain commodity, a yield of a certain crop, a rate of 
 exchange of money for a certain country, etc. These values, 
 respectively, form time series. The object of this chapter is to 
 point out some of the ways of analyzing times series. Three char¬ 
 acteristics of such series, namely, secular trend, seasonal varia¬ 
 tions, and cycles, will be introduced together with methods for 
 obtaining them. 
 
 Fia. 13.1. — Average monthly wholesale prices of butter and the straight line 
 
 trend for yearly averages. 
 
 Consider prices of butter for each month from 1928 to 1934 as 
 given in Table 13.1. These prices, plotted in Fig. 13.1, show a 
 downward trend for this period of 7 years including depression 
 years. This trend appears to be linear for most of the data; hence 
 
 233 
 
234 
 
 THE ANALYSIS OF TIME SERIES 
 
 a straight line was fitted by the method of least squares to the 
 yearly averages given in Table 13.1. This straight line is 
 
 (13.1) y = 8,818.28 - 4.55x, 
 
 where x represents years and y represents expected yearly averages. 
 This fine, shown in Fig. 13.1, inchoates the general downward trend 
 of these prices of butter. Equation (13.1) will be used as the trend 
 fine for this time series. 
 
 A second-degree curve might have been used instead of a 
 straight line, for there is an upward trend in prices of butter from 
 1932 on. A smooth curve may be drawn in freehand which with 
 care will give a very good trend curve. 
 
 The slope of the trend line (13.1) is —4.55, meaning that during 
 1 year the price of butter dropped 4.55 cents. The drop per month 
 is 1/12 of this, or 0.38 cent. 
 
 TABLE 13.1 
 
 Average Wholesale Prices per Pound of Butter, 
 92-Score Creamery at Chicago* 
 
 Year 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 Apr. 
 
 May 
 
 Jun. 
 
 July 
 
 Aug. 
 
 Sep. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
 Ave. 
 
 1928 
 
 48.76 
 
 46.62 
 
 49.44 
 
 45.49 
 
 44.93 
 
 44.13 
 
 44.93 
 
 46.94 
 
 48.75 
 
 47.79 
 
 50.57 
 
 50.46 
 
 47.40 
 
 1929 
 
 47.94 
 
 49.89 
 
 48.45 
 
 45.35 
 
 43.54 
 
 43.54 
 
 42.42 
 
 43.45 
 
 46.22 
 
 45.56 
 
 42.70 
 
 41.1C 
 
 45.01 
 
 1930 
 
 36.63 
 
 35.70 
 
 37.27 
 
 38.53 
 
 34.85 
 
 32.93 
 
 35.31 
 
 38.92 
 
 39.77 
 
 39.98 
 
 36.09 
 
 32.18 
 
 36.51 
 
 1931 
 
 28.50 
 
 28.40 
 
 28.88 
 
 26.10 
 
 23.70 
 
 23.33 
 
 24.95 
 
 28.12 
 
 32.50 
 
 33.76 
 
 30.93 
 
 30.55 
 
 28.31 
 
 1932 
 
 23.59 
 
 22.46 
 
 22.61 
 
 20.08 
 
 18.84 
 
 16.99 
 
 18.18 
 
 20.31 
 
 20.76 
 
 20.72 
 
 23.30 
 
 24.11 
 
 21.00 
 
 1933 
 
 18.85 
 
 18.65 
 
 18.17 
 
 20.66 
 
 22.54 
 
 22.84 
 
 24.53 
 
 21.31 
 
 23.60 
 
 24.04 
 
 23.60 
 
 20.08 
 
 21.66 
 
 1934 
 
 19.84 
 
 25.35 
 
 25.35 
 
 23.66 
 
 24.49 
 
 24.8S 
 
 24.49 
 
 27.3S 
 
 25.78 
 
 26.93 
 
 29.36 
 
 30.95 
 
 25.71 
 
 Ave. 
 
 32.16 
 
 32.44 
 
 32.88 
 
 31.41 
 
 30.41 
 
 29.81 
 
 30.69 
 
 32.35 
 
 33.91 
 
 34.11 
 
 33.79 
 
 32.78 
 
 
 * Center at middle of month. 
 
 SEASONAL VARIATIONS 
 
 The average of the January prices is 32.16; the average of the 
 February prices is 32.44. These averages, listed in Table 13.2, 
 contain the effect of the trend. To find the seasonal variations of 
 prices it is necessary to remove the effect of trend from these 
 averages for the monthly prices. To remove trend from the prices 
 it is necessary to add 0.38 cent to the average February prices 
 (since the trend is downward) and 0.76 cent to the average March 
 prices, etc. These averages with the effect of trend removed are 
 given in the second column of Table 13.2. These values are now 
 written as percentages of the average of the values in column 3; 
 
SEASONAL VARIATIONS 
 
 235 
 
 TABLE 13.2 
 
 Construction of Indexes of Seasonal Variations From 
 Arithmetic Averages of Actual Prices 
 
 Month 
 
 Average Monthly 
 Prices per Pound 
 
 Average Corrected 
 for Secular Trend 
 
 Indexes of Seasonal 
 Variations 
 
 Jan. 
 
 32.16 
 
 32.16 
 
 93.71 
 
 Feb. 
 
 32.44 
 
 32.82 
 
 95.63 
 
 Mar. 
 
 32.88 
 
 33.64 
 
 98.02 
 
 Apr. 
 
 31.41 
 
 32.55 
 
 94.84 
 
 May 
 
 30.41 
 
 31.93 
 
 93.04 
 
 June 
 
 29.81 
 
 31.71 
 
 92.40 
 
 Jul. 
 
 30.69 
 
 32.97 
 
 96.07 
 
 Aug. 
 
 32.35 
 
 35.01 
 
 102.10 
 
 Sep. 
 
 33.91 
 
 36.95 
 
 107.66 
 
 Oct. 
 
 34.11 
 
 37.53 
 
 109.35 
 
 Nov. 
 
 33.79 
 
 35.59 
 
 109.53 
 
 Dec. 
 
 32.78 
 
 36.96 
 
 107.69 
 
 Ave. 
 
 
 34.32 
 
 100.00 
 
 these are called the indexes of seasonal variation and indicate how 
 the seasons of the years affect prices, or how prices fluctuate for 
 
 Fid. 13.2. — Seasonal indexes pertaining to prices of butter based on averages 
 of actual prices and chain relatives. 
 
 the different periods of the year. These indexes show a decrease 
 from March to June, an increase from June to November, and a 
 decrease from November to January. These indexes of seasonal 
 variation are plotted in Fig. 13.2. 
 
236 
 
 THE ANALYSIS OF TIME SERIES 
 
 PROBLEMS 
 
 1. The following table contains the average price in dollars per 100 
 pounds of milk received by producers in the United States 1930-1934. 
 
 Year Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. 
 
 1930 2.53 2.44 2.38 2.35 2.28 2.22 2.15 2.18 2.25 2.30 2.31 2.20 
 
 1931 2.04 1.96 1.92 1.85 1.73 1.66 1.62 1.64 1.70 1.72 1.73 1.67 
 
 1932 1.56 1.49 1.43 1.39 1.29 1.17 1.20 1.21 1.25 1.28 1.26 1.26 
 
 1933 1.25 1.16 1.10 1.08 1.14 1.21 1.33 1.39 1.47 1.51 1.51 1.49 
 
 1934 1.44 1.48 1.50 1.46 1.45 1.47 1.50 1.52 1.57 1.60 1.65 1.69 
 
 Find the linear trend line for the yearly averages. Plot this with the data. 
 
 2. Find the seasonal variations and plot them. 
 
 LINK RELATIVE METHOD OF FINDING SEASONALS 
 
 Another method of finding indexes of seasonal variations is by 
 using link relatives. A link relative of prices is the ratio of the 
 price for 1 month to the price of the preceding month. The price 
 of each month must be expressed as a percentage of the price of 
 the preceding month; that is, the January price is expressed as a 
 percentage of the December price, the February price as a percent¬ 
 age of the January price, etc. Table 13.3 contains the link rela¬ 
 tives of prices given in Table 13.1. The medians of the link rela¬ 
 tives of prices for the various months are recorded in the first col¬ 
 umn in Table 13.5. These median link relatives measure monthly 
 fluctuations in terms of a shifting base, each monthly price being 
 the base for the next monthly price. 
 
 Chain relatives are constructed for the purpose of giving one 
 base monthly price. The median for the link relatives of prices 
 for Januaries will be considered 100. The February median is 
 97.46 and the February chain relative is also 97.46; the March 
 chain relative is 100.67 X 97.46, or the median for the March link 
 relatives times the February chain relative. The April chain rela¬ 
 tive is equal to the April median times the [March chain relative, 
 or 93.33 X 98.11 = 91.57.* These chain relatives are listed in 
 column 2 of Table 13.5. The December chain relative times the 
 January median should give 100 if no trend were present. The 
 December chain relative times the January median gives 
 
 103.29 X 89.12*= 92.05 instead of 100, 
 
 * Divided by 100. 
 
TABLE 13.3 —Link Relatives of Prices of Butter Given in Table 13.1 
 
 LINK RELATIVE METHOD OF FINDING SEASONALS 237 
 
 Dec. 
 
 + 
 
 00 iO l> t>- C30 00 Ol 
 ninhn-^o^ 
 
 03 CO 03 00 CO LO 1C 
 03 03 00 C3 o 00 o 
 
 96.85 
 
 98.77 
 
 
 + 
 
 -+- 
 
 
 03 1> Ol iD I'- 03 
 
 id r- 
 
 > 
 
 co 03 co tjh t-h o 
 
 r-T T-H 
 
 A 
 
 \D CO O — ' 03 00 03 
 
 o cd 
 
 O 03 03 03 I— 1 1 03 O 
 
 O 03 
 
 
 t-H t-H T-H 
 
 1—1 
 
 
 CO N CO 00 ^ o o 
 
 03 CO 
 
 
 O lO »o 00 00 GO-t 
 
 O iO 
 
 O 
 
 00 00 O CO C5 —< H 
 
 — o 
 
 o 
 
 03 03 O O 03 o o 
 
 o o 
 
 
 1—H T— ( hH i—I 
 
 T-H T-H 
 
 
 1 
 
 CO 00 00 00 co io 00 
 
 00 CO 
 
 d 
 
 00 CO t-h iQ Cl I> io 
 
 o co 
 
 02 
 
 wdNiONO^ 
 
 co CO 
 
 OOO—O—O 
 
 o o 
 
 
 1—1 T-H 1—1 1—1 1—1 T-H 
 
 T "H ’" H 
 
 
 N CO Cl >h Ol N O 
 
 1 
 
 iO 03 
 
 bb 
 
 ^ Tji 03 I> 00 GO 
 
 t"- 03 
 
 <! 
 
 ^CIOCIHCH 
 
 id o 
 
 OOhhhCCh 
 
 O -H 
 
 
 hH t— 1 t-H t-H t-H t-H 
 
 r-H t-H 
 
 
 r-H CO CO ^ o o co 
 
 ^ rt< 
 
 >> 
 
 CO ^ Cl 03 O CO't 
 
 L- 03 
 
 
 
 
 o 
 
 '—' N N o' N N CO 
 
 CO CO 
 
 *""3 
 
 O 03 o O O O 03 
 
 o o 
 
 
 t-H t-H t-H t-H t-H 
 
 T-H T-H 
 
 03 
 
 fl 
 
 -h O 03 Tf oo CO 03 
 
 1 
 
 IO Tf 
 
 C^O^^rHCOlO 
 
 
 O 
 
 COO^COO ^ 
 
 d cd 
 
 
 03 o 03 03 03 © © 
 
 03 O 
 
 
 1 
 
 NHIOOWO^ 
 
 03 t-h 
 
 £ 
 
 O 00 00 T-H IQ 
 
 ^ o 
 
 § 
 
 00 co O O CO 03 CO 
 
 L'- d 
 
 03 03 03 03 03 <0 <© 
 
 03 03 
 
 
 HO00NHOM 
 
 r-4 CO 
 
 
 o co co oo co 
 
 io CO 
 
 a 
 
 03 CO CO O CO CO CO 
 
 co co 
 
 <3 
 
 03 03 O 03 00 T—1 03 
 
 03 03 
 
 
 1 
 
 IOH003NCOO 
 
 OrH^ocOrf o 
 
 io D- 
 
 c3 
 
 O CO 
 
 HH 
 
 CON^HQNO 
 o © o o o 03 o 
 
 H o 
 
 
 o o 
 
 
 i-H t-H t-H t-H t-H 
 
 
 
 1 
 
 H N CO lO H 1C N 
 
 CO CO 
 
 d 
 
 O O ^ co C) 03 N 
 
 03 Tt< 
 
 
 io d 03 io 03 d 
 
 -H td 
 
 03 O 03 03 03 03 Cl 
 
 O 03 
 
 
 T-H T-H 
 
 •—' 
 
 
 CO H M co Cl CO O 
 
 O 03 
 
 
 HOHICCICOOO 
 
 CO T"H 
 
 cj 
 
 co io co oo 03 oo 
 
 d 03 
 
 03 CO 
 
 
 O 03 03 00 00 03 
 
 t- 
 
 CO 03 O 'H ci co ^ 
 
 Ol Ol CO CO CO CO CO 
 
 a 
 
 aj 
 
 g’S 
 
 <iS 
 
 
 03 03 03 03 03 03 03 
 
 
 r-H t-H t-H t-H t-H hH i-H 
 
 rJl 
 
 a 
 
 > 
 
 a 
 
 a 
 
 a 
 
 g 
 
 a 
 
 o 
 
 a 
 
 a 
 
 « 
 
 < 
 
 E-i 
 
 >- 
 
 o 
 
 a 
 
 a 
 
 a 
 
 a 
 
 a 
 
 a 
 
 CO 
 
 W 
 
 Dec. 
 
 ; ; ; ;hcici ;03 \ ; 
 
 Nov. 
 
 ! * * t-h 03 t-h | co * * \ 
 
 Oct. 
 
 * ; • • • id <n * • • * 
 
 <D 
 
 02 
 
 : :—* : : : : : 
 
 fcf 
 
 < 
 
 • • • r*4 03 ■ 't-h • • 
 
 July 
 
 ’ ■ * • * * * 
 
 June 
 
 . co <n 03 ; ; ; 
 
 May 
 
 ; ; ;hhcico * [ * 
 
 Apr. 
 
 ; t-H | t-H | Tf H j | 
 
 s 
 
 ; ; ; ; rH (M ; * ; ; 
 
 Feb. 
 
 H ! ! ! ;ncoh ; ; ; 
 
 Jan. 
 
 ; ! I ^ 03 ; <m t-h t-h 
 
 Classes 
 
 125-129 
 
 120-124 
 
 115-119 
 
 110-114 
 
 105-109 
 
 100-104 
 
 95-99 
 
 90-94 
 
 85-89 
 
 80-84 
 
 75-79 
 
238 
 
 THE ANALYSIS OF TIME SERIES 
 
 TABLE 13.5 
 
 Indexes of Seasonal Variations by the Method of Link 
 
 Relatives 
 
 Month 
 
 Median 
 
 Link 
 
 Relatives 
 
 Chain 
 
 Relatives 
 
 Corrected 
 
 Chain 
 
 Relatives 
 
 Indexes of 
 Seasonal 
 Variations 
 
 Jan. 
 
 89.12 
 
 100 
 
 100 
 
 97.85 
 
 Feb. 
 
 97.46 
 
 97.46 
 
 97.46 
 
 95.66 
 
 Mar. 
 
 100.67 
 
 98.11 
 
 99.48 
 
 97.34 
 
 Apr. 
 
 93.33 
 
 91.57 
 
 93.51 
 
 91.50 
 
 May 
 
 96.01 
 
 87.92 
 
 90.41 
 
 88.46 
 
 June 
 
 98.44 
 
 86.55“ 
 
 89.61 
 
 87.68 
 
 July 
 
 106.94 
 
 92.56 
 
 96.51 
 
 94.43 
 
 Aug. 
 
 110.22 
 
 102.02 
 
 107.11 
 
 104.80 
 
 Sep. 
 
 103.86 
 
 105.96 
 
 112.03 
 
 109.62 
 
 Oct. 
 
 100.53 
 
 106.52 
 
 113.41 
 
 110.97 
 
 Nov. 
 
 98.17 
 
 104.58 
 
 112.10 
 
 109.69 
 
 Dec. 
 
 98.77 
 
 103.29 
 
 114.48 
 
 112.02 
 
 Jan. 
 
 
 92.05 
 
 Ave. 102.20 
 
 Ave. 100.00 
 
 which shows that trend perhaps has entered into the data and 
 caused this product to differ from 100. During this 12-month 
 period 100 has been reduced to 92.05. It is usually assumed that 
 the rate of increase or decrease is constant for each month. Hence 
 
 or 
 
 from which 
 
 92.05 = 100(1 + r) 12 , 
 
 log .9205 
 12 
 
 = log (1 + r), 
 
 r = —0.0069. 
 
 To adjust the chain relatives because of the effect of trend, the 
 February chain relative is divided by 1 + r = 0.9931, the March 
 chain relative by (1 + r) 2 , the April chain relative by (1 + r) 3 , 
 etc. These adjusted chain relatives are given in column 4 of 
 Table 13.5. Column 5 of the above table contains the corrected 
 
MOVING AVERAGE METHOD OF FINDING SEASONALS 239 
 
 chain relatives with base equal to the average of the chain relatives 
 in column 4; these are the indexes of seasonal variations and indi¬ 
 cate the relative fluctuations in prices for the seasons. These are 
 shown in Fig. 13.2 
 
 THE MOVING AVERAGE METHOD OF FINDING SEASONALS 
 
 If prices of butter for 12 months, December through Novem¬ 
 ber, are averaged, a value is obtained which is centered at June 1; 
 if prices for 12 months, January through December, are averaged, 
 a value is obtained, centered at July 1. Successive 12-month 
 averages are obtained by dropping the prices for the first month, 
 adding a new month and then finding the average. These aver¬ 
 ages, called moving averages, give a good idea of the trend of prices 
 for the period under consideration. 
 
 Table 13.6 gives the sums of prices for the different 12-month 
 .periods. Table 13.7 records moving averages centered at the first 
 of the months. Table 13.8 contains the moving averages centered 
 at the fifteenth of the months; these were obtained by averaging 
 two consecutive moving averages in Table 13.7. 
 
 TABLE 13.6 
 
 Sums of Prices of Butter for Various 12-Month Periods 
 
 Month 
 
 1928 
 
 1929 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 Dec. 
 
 570.22 
 
 
 
 
 
 
 
 
 
 
 Jan.* 
 
 568.81 
 
 540 
 
 16 
 
 439.64 
 
 341 
 
 20 
 
 253 
 
 43 
 
 261 
 
 35 
 
 Feb. 
 
 567.99 
 
 528 
 
 85 
 
 431.51 
 
 336 
 
 29 
 
 249 
 
 69 
 
 261 
 
 34 
 
 Mar. 
 
 571.26 
 
 514 
 
 66 
 
 424.21 
 
 330 
 
 35 
 
 245 
 
 88 
 
 268 
 
 04 
 
 Apr. 
 
 570.27 
 
 507 
 
 84 
 
 415.82 
 
 324 
 
 08 
 
 241 
 
 44 
 
 275 
 
 22 
 
 May 
 
 570.13 
 
 499 
 
 15 
 
 403.39 
 
 318 
 
 06 
 
 242 
 
 02 
 
 278 
 
 22 
 
 June 
 
 568.74 
 
 490 
 
 46 
 
 392.24 
 
 313 
 
 20 
 
 245 
 
 72 
 
 280 
 
 17 
 
 July 
 
 568.15 
 
 479 
 
 84 
 
 382.64 
 
 306 
 
 86 
 
 251 
 
 57 
 
 282 
 
 21 
 
 Aug. 
 
 565.64 
 
 472 
 
 73 
 
 372.28 
 
 300 
 
 09 
 
 257 
 
 92 
 
 282 
 
 17 
 
 Sep. 
 
 562.15 
 
 468 
 
 20 
 
 361.48 
 
 292 
 
 28 
 
 258 
 
 92 
 
 288 
 
 24 
 
 Oct. 
 
 559.62 
 
 461 
 
 75 
 
 354.21 
 
 280 
 
 54 
 
 261 
 
 76 
 
 290 
 
 42 
 
 Nov. 
 
 557.39 
 
 456 
 
 17 
 
 347.99 
 
 267 
 
 50 
 
 265 
 
 08 
 
 293 
 
 31 
 
 Dec. 
 
 549.52 
 
 449 
 
 56 
 
 342.83 
 
 359 
 
 87 
 
 265 
 
 38 
 
 299 
 
 07 
 
 * Sum for 1928, from January fco December, inclusive. Price for December, 1927, is 51.87. 
 
240 
 
 THE ANALYSIS OF TIME SERIES 
 
 The first figure in column 2 of Table 13.6 is the sum of the prices 
 from December, 1927. through November, 1928; the next figure in 
 
 TABLE 13.7 
 
 Moving 12-Month Averages of Prices, Centered at 
 First of Month 
 
 Month 
 
 1928 
 
 1929 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Jan. 
 
 
 47.35 
 
 39.99 
 
 31.89 
 
 25.57 
 
 20.96 
 
 23.52 
 
 Feb. 
 
 
 47.14 
 
 39.39 
 
 31.02 
 
 25.01 
 
 21.49 
 
 23.51 
 
 Mar. 
 
 
 46.85 
 
 39.02 
 
 30.12 
 
 24.36 
 
 21.58 
 
 24.02 
 
 Apr. 
 
 
 46.64 
 
 38.48 
 
 29.52 
 
 23.38 
 
 21.81 
 
 24.20 
 
 May 
 
 
 46.45 
 
 38.01 
 
 29.00 
 
 22.29 
 
 22.09 
 
 24.44 
 
 June 
 
 47.52 
 
 45.79 
 
 37.46 
 
 28.57 
 
 21.66 
 
 22.12 
 
 24.92 
 
 July 
 
 47.40 
 
 45.01 
 
 36.64 
 
 28.43 
 
 21.12 
 
 21.78 
 
 
 Aug. 
 
 47.33 
 
 44.07 
 
 35.96 
 
 28.02 
 
 20.81 
 
 21.78 
 
 
 Sept. 
 
 47.61 
 
 42.89 
 
 35.35 
 
 27.52 
 
 20.49 
 
 22.34 
 
 
 Oct. 
 
 47.52 
 
 42.32 
 
 34.65 
 
 27.01 
 
 20.12 
 
 22.94 
 
 
 Nov. 
 
 47.51 
 
 41.60 
 
 33.62 
 
 26.51 
 
 20.17 
 
 23.19 
 
 
 Dec. 
 
 47.40 
 
 40.87 
 
 32.69 
 
 26.10 
 
 20.48 
 
 23.35 
 
 
 TABLE 13.8 
 
 Moving 12-Month Averages of Prices, Centered at 
 Fifteenth of Month 
 
 Month 
 
 1928 
 
 1929 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Jan. 
 
 
 47 
 
 245 
 
 39 
 
 69 
 
 31.455 
 
 25.29 
 
 21 
 
 225 
 
 23 
 
 515 
 
 Feb. 
 
 
 46 
 
 995 
 
 39 
 
 205 
 
 30.57 
 
 24.685 
 
 21 
 
 54 
 
 23 
 
 815 
 
 Alar. 
 
 
 46 
 
 795 
 
 38 
 
 75 
 
 29.82 
 
 23.87 
 
 21 
 
 685 
 
 24 
 
 11 
 
 Apr. 
 
 
 46 
 
 545 
 
 38 
 
 245 
 
 29.26 
 
 22.835 
 
 21 
 
 95 
 
 24 
 
 32 
 
 May 
 
 
 46 
 
 12 
 
 37 
 
 735 
 
 28.785 
 
 21.975 
 
 22 
 
 105 
 
 24 
 
 73 
 
 June 
 
 47.46 
 
 45 
 
 40 
 
 37 
 
 50 
 
 28.50 
 
 21.39 
 
 21 
 
 95 
 
 
 
 July 
 
 47.365 
 
 44 
 
 54 
 
 36 
 
 30 
 
 28.225 
 
 20.965 
 
 21 
 
 78 
 
 
 
 Aug. 
 
 47.47 
 
 43 
 
 98 
 
 35 
 
 655 
 
 27.77 
 
 20.65 
 
 22 
 
 06 
 
 
 
 Sept. 
 
 47.565 
 
 42 
 
 605 
 
 35 
 
 00 
 
 27.265 
 
 20.305 
 
 22 
 
 69 
 
 
 
 Oct. 
 
 47.515 
 
 41 
 
 96 
 
 34 
 
 135 
 
 26.76 
 
 20.145 
 
 23 
 
 065 
 
 
 
 Nov. 
 
 47.455 
 
 41 
 
 235 
 
 33 
 
 155 
 
 26.305 
 
 20.325 
 
 23 
 
 27 
 
 
 
 Dec. 
 
 47.375 
 
 40 
 
 43 
 
 32 
 
 29 
 
 25.835 
 
 20.72 
 
 23 
 
 435 
 
 
 
MOVING AVERAGE METHOD OF FINDING SEASONALS 241 
 
 this column is the sum of the prices from January, 1928, through 
 December, 1928, etc. 
 
 The moving averages in Table 13.8 indicate the trend of prices 
 from 1928 to 1934. This trend is downward from 1928 to the last 
 of 1932; from here to 1934 there is an upward trend of prices. 
 These moving averages, plotted in Fig. 13.3, give a better idea of 
 trend than the straight line used in the first of the chapter. 
 
 60 
 55 
 50 
 45 
 40 
 
 u 35 
 
 O 
 
 30 
 25 
 20 
 15 
 10 
 
 Fig. 13.3.—Twelve-month moving average trend line plotted 
 with the original prices of butter. 
 
 Table 13.9 contains ratios of the actual monthly prices to the 
 moving averages given in Table 13.8. These values represent 
 seasonal variations with moving-average-trend removed. 
 
 The average of the January percentages or ratios in Table 13.9 
 is 92.59; the average of the February percentages is 95.69, etc. 
 These averages of monthly percentages are listed in Table 13.10, 
 together with their medians.* Indexes of seasonal variations are 
 given in columns 4 and 5 of this table on the basis of the mean and 
 median respectively. There is an increase from January to March, 
 a decrease from March to June, an increase from June to October, 
 and a decrease from October to December. 
 
 * The median is the arithmetic average of the middle two. 
 
242 
 
 THE ANALYSIS OF TIME SERIES 
 
 TABLE 13.9 
 
 Ratio of Actual Prices to Moving Averages 
 
 Month 
 
 1928 
 
 1929 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Jan. 
 
 
 101.47 
 
 92.29 
 
 90.61 
 
 93.28 
 
 93.52 
 
 84.37 
 
 Feb. 
 
 
 106.16 
 
 91.06 
 
 92.90 
 
 90.99 
 
 86.60 
 
 106.43 
 
 Mar. 
 
 
 103.54 
 
 96.18 
 
 96.85- 
 
 94.72 
 
 83.79 
 
 105.14 
 
 Apr. 
 
 
 97.43 
 
 100.75- 
 
 89.20 
 
 87.94 
 
 94.12 
 
 97.29 
 
 May 
 
 
 94.40 
 
 92.35+ 
 
 82.33 
 
 85.73 
 
 101.97 
 
 99.03 
 
 June 
 
 92.98 
 
 95.90 
 
 87.81 
 
 81.86 
 
 79.43 
 
 104.05+ 
 
 
 July 
 
 94.86 
 
 95.24 
 
 97.27 
 
 88.40 
 
 86.72 
 
 94.26 
 
 
 Aug. 
 
 98.88 
 
 98.79 
 
 109.16 
 
 101.26 
 
 98.35+ 
 
 96.60 
 
 
 Sept. 
 
 102.49 
 
 108.48 
 
 113.29 
 
 119.20 
 
 102.24 
 
 104.01 
 
 
 Oct. 
 
 100.58 
 
 108.59 
 
 117.12 
 
 126.16 
 
 102.85+ 
 
 104.23 
 
 
 Nov. 
 
 106.56 
 
 103.55— 
 
 108.85+ 
 
 117.58 
 
 114.64 
 
 101.42 
 
 
 Dec. 
 
 106.51 
 
 101.66 
 
 99.66 
 
 118.25- 
 
 116.36 
 
 85.68 
 
 
 TABLE 13.10 
 
 Indexes of Seasonal Variations by Method of Moving Averages 
 on Basis of Mean and Median; Moving-average-trend Removed 
 
 Month 
 
 Average 
 
 * Median 
 
 Corrected 
 
 Average 
 
 Corrected 
 
 Median 
 
 Jan. 
 
 92.59 
 
 92.78 
 
 93.57 
 
 94.72 
 
 Feb. 
 
 95.69 
 
 91.98 
 
 96.70 
 
 93.90 
 
 Mar. 
 
 96.70 
 
 91.52 
 
 97.73 
 
 93.43 
 
 Apr. 
 
 94.46 
 
 95.71 
 
 95.46 
 
 97.71 
 
 May 
 
 92.64 
 
 98.38 
 
 93.62 
 
 100.44 
 
 June 
 
 90.34 
 
 86.86 
 
 91.30 
 
 88.68 
 
 July 
 
 72.79 
 
 99.56 
 
 93.77 
 
 101.64 
 
 Aug. 
 
 100.51 
 
 98.84 
 
 101.58 
 
 100.91 
 
 Sept. 
 
 108.29 
 
 101.50 
 
 109.44 
 
 103.62 
 
 Oct. 
 
 109.92 
 
 106.41 
 
 111.09 
 
 108.63 
 
 Nov. 
 
 108.77 
 
 107.72 
 
 109.92 
 
 109.97 
 
 Dec. 
 
 104.69 
 
 104.08 
 
 105.80 
 
 106.26 
 
 Ave. 
 
 98.95 
 
 97.95 
 
 100.00 
 
 99.99 
 
 *The median is the arithmetic average of the middle two. 
 
INDEXES OF SEASONAL VARIATIONS FROM TREND LINE 243 
 
 PROBLEM 
 
 1. Find the moving 12-month averages for data on page 236. Set up a 
 table similar to Table 13.10 for these same data. Plot the moving averages. 
 
 INDEXES OF SEASONAL VARIATIONS FROM TREND LINE 
 
 Table 13.11 contains the values of prices of butter obtained 
 from the trend line y = 8,818.28 — 4.55a:. This trend line was 
 found from the yearly averages which were centered at July 1, the 
 middle of the year. Hence the predicted values are also centered 
 at this date. The predicted value for 1928 is 45.60. From this the 
 predicted value for January 15 is found to be 45.60 + 0.379 X 5.5 
 = 47.684, since the monthly slope of the line is minus 0.379 and 
 there are 5.5 months between July 1 and January 15. The pre¬ 
 dicted value for February, 1928, is found by subtracting 0.379 from 
 the January predicted price; this is 47.305. The other values in 
 this table are found in a similar way. These values lie on the trend 
 line. 
 
 TABLE 13.11 
 
 Prices Obtained from the Trend Line ij = 8,818.28 — 4.55x 
 
 Month 
 
 1928 
 
 1929 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Jan. 
 
 47.684 
 
 43.136 
 
 38 
 
 588 
 
 34 
 
 040 
 
 29 
 
 492 
 
 24 
 
 944 
 
 20.396 
 
 Feb. 
 
 47.305 
 
 42.757 
 
 38 
 
 209 
 
 33 
 
 661 
 
 29 
 
 113 
 
 24 
 
 565 
 
 20.017 
 
 Mar. 
 
 46.926 
 
 42.378 
 
 37 
 
 830 
 
 33 
 
 282 
 
 28 
 
 734 
 
 24 
 
 185 
 
 19.638 
 
 Apr. 
 
 46.547 
 
 41.999 
 
 37 
 
 451 
 
 32 
 
 903 
 
 28 
 
 355 
 
 23 
 
 807 
 
 19.259 
 
 May 
 
 46.168 
 
 41.620 
 
 37 
 
 072 
 
 32 
 
 524 
 
 27 
 
 876 
 
 23 
 
 428 
 
 18.880 
 
 June 
 
 45.789 
 
 41.241 
 
 36 
 
 693 
 
 32 
 
 145 
 
 27 
 
 597 
 
 23 
 
 049 
 
 18.501 
 
 July 
 
 45.410 
 
 40.862 
 
 36 
 
 314 
 
 31 
 
 766 
 
 27 
 
 218 
 
 22 
 
 670 
 
 18.122 
 
 Aug. 
 
 45.031 
 
 40.483 
 
 35 
 
 935 
 
 31 
 
 387 
 
 26 
 
 839 
 
 22 
 
 291 
 
 17.743 
 
 Sept. 
 
 44.652 
 
 40.104 
 
 35 
 
 556 
 
 30 
 
 008 
 
 26 
 
 460 
 
 21 
 
 912 
 
 17.364 
 
 Oct. 
 
 44.273 
 
 39.725 
 
 35 
 
 177 
 
 30 
 
 629 
 
 26 
 
 081 
 
 21 
 
 533 
 
 16.985 
 
 Nov. 
 
 43.894 
 
 39.345 
 
 34 
 
 798 
 
 30 
 
 250 
 
 25 
 
 702 
 
 21 
 
 154 
 
 16.606 
 
 Dec. 
 
 43.515 
 
 38.967 
 
 34 
 
 419 
 
 29 
 
 871 
 
 25 
 
 323 
 
 20 
 
 775 
 
 16.227 
 
 Table 13.12 contains ratios of the actual monthly prices to the 
 monthly prices obtained from the trend line. These values are 
 
244 
 
 THE ANALYSIS OF TIME SERIES 
 
 monthly prices with trend removed, for every price is expressed as 
 a percentage of the trend line value. The ninth column contains 
 the medians for the different months; the last column contains the 
 adjusted medians. These adjusted values are obtained by divid¬ 
 ing each median by the average of column 9. The adjusted 
 medians are the seasonal variations, and they indicate how prices 
 fluctuated during the seasons of the year. Averages could be used 
 as well as medians for this purpose. Effect of trend has now been 
 removed from the prices of butter. 
 
 TABLE 13.12 
 
 Actual Values Divided by Values Obtained from Trend Line 
 
 Month 
 
 1928 
 
 1929 
 
 1930 
 
 1931 
 
 1932 
 
 1933 
 
 1934 
 
 Median 
 
 Ad¬ 
 
 justed 
 
 median 
 
 Jan... 
 
 102.27 
 
 111.13 
 
 94.92 
 
 83.73 
 
 79.99 
 
 79.59 
 
 97.25 
 
 94. 92 
 
 94.06 
 
 Feb.. . 
 
 98.54 
 
 116.67 
 
 93.43 
 
 84.37 
 
 77. 16 
 
 75.91 
 
 126.62 
 
 93.40 
 
 92.58 
 
 Mar. . 
 
 105.35 
 
 114.30 
 
 98.52 
 
 86.78 
 
 78.70 
 
 75.11 
 
 129.07 
 
 98.52 
 
 97.63 
 
 Apr.. . 
 
 97.72 
 
 107.98 
 
 102.88 
 
 79.33 
 
 70.80 
 
 86.77 
 
 122.85 
 
 97.72 
 
 96.84 
 
 May. . 
 
 97.31 
 
 104.61 
 
 94.01 
 
 72.88 
 
 67.36 
 
 96.20 
 
 129.71 
 
 96.20 
 
 95.33 
 
 June. . 
 
 96.37 
 
 105.58 
 
 89. 75 
 
 72.57 
 
 61.56 
 
 99.09 
 
 134.49 
 
 96.37 
 
 95.50 
 
 July. . 
 
 98.72 
 
 103.82 
 
 97.25 
 
 78.53 
 
 66.79 
 
 108.20 
 
 135.15 
 
 98.72 
 
 97.83 
 
 Aug.. . 
 
 104.24 
 
 107.34 
 
 108.29 
 
 89.58 
 
 75. 67 
 
 95.60 
 
 154.34 
 
 104.24 
 
 103.30 
 
 Sept. . 
 
 109.18 
 
 115.26 
 
 111.84 
 
 10S.30 
 
 78.46 
 
 107.71 
 
 148.50 
 
 109.18 
 
 108.19 
 
 Oct. .. 
 
 107.95 
 
 114.67 
 
 100. 63 
 
 110.87 
 
 79.45 
 
 111.66 
 
 158.51 
 
 110.89 
 
 109.87 
 
 Nov. . 
 
 115.22 
 
 108.51 
 
 103.71 
 
 102.25 
 
 90.66 
 
 111.58 
 
 176.76 
 
 108.51 
 
 107.53 
 
 Dec.. . 
 
 115.95 
 
 105.47 
 
 93.49 
 
 102.28 
 
 95.22 
 
 96.63 
 
 190.70 
 
 102.28 
 
 101.35 
 
 It is now possible to remove seasonal variations from the data 
 by subtracting the seasonal variations from the data after trend 
 has been removed. This is done in Table 13.13. Here the sea¬ 
 sonal variations in the third column are those in the last column in 
 Table 13.12. Prices of butter with trend and seasonals removed 
 are found in column headed a — b; these are expressed as differ¬ 
 ences of percentages. Notice that the seasonals are repeated every 
 12 months as the fourth column shows. 
 
INDEXES OF SEASONAL VARIATIONS FROM TREND LINE 245 
 
 TABLE 13.13 
 
 Cycle Fluctuations after Trend and Seasonal Variations 
 Have Been Removed from Prices of Butter 
 
 Year 
 
 Month 
 
 Ratio of 
 Actual 
 Prices to 
 Prices from 
 Trend Line, 
 a 
 
 Seasonal 
 
 Variations, 
 
 b 
 
 Deviations, 
 a — b 
 
 Deviations 
 
 Squared, 
 
 (a — b) 2 
 
 Deviations 
 in Terms of 
 Standard 
 Deviations 
 
 1928 
 
 Jan. 
 
 102.27 
 
 94.06 
 
 8.21 
 
 67.4041 
 
 0.37 
 
 
 Feb. 
 
 98.54 
 
 92.58 
 
 5.96 
 
 35.5216 
 
 0.28 
 
 
 Mar. 
 
 105.35 
 
 97.63 
 
 7.72 
 
 59.5984 
 
 0.35 
 
 
 Apr. 
 
 97.92 
 
 96.84 
 
 0.88 
 
 0.7744 
 
 0.04 
 
 
 May 
 
 97.31 
 
 95.33 
 
 1.98 
 
 3.9204 
 
 0.09 
 
 
 June 
 
 96.37 
 
 95.50 
 
 0.87 
 
 0.7569 
 
 0.04 
 
 
 July 
 
 98.72 
 
 97.83 
 
 0.89 
 
 0.7921 
 
 0.04 
 
 
 Aug. 
 
 104.24 
 
 103.30 
 
 0.94 
 
 0.8836 
 
 0.04 
 
 
 Sept. 
 
 109.18 
 
 108.19 
 
 0.99 
 
 0.9801 
 
 0.04 
 
 
 Oct. 
 
 107.95 
 
 109.87 
 
 -1.92 
 
 3.6864 
 
 -0.09 
 
 
 Nov. 
 
 115.22 
 
 107. 53 
 
 7.69 
 
 59.1361 
 
 0.34 
 
 
 Dec. 
 
 115.95 
 
 101.35 
 
 4.60 
 
 21.1600 
 
 0.21 
 
 1929 
 
 Jan. 
 
 111.13 
 
 94.06 
 
 17.07 
 
 291 . 3849 
 
 0.77 
 
 
 Feb. 
 
 116.67 
 
 92.58 
 
 24.09 
 
 580.3281 
 
 1.08 
 
 
 Mar. 
 
 114.30 
 
 97.63 
 
 16.67 
 
 277.8889 
 
 0.75 
 
 
 Apr. 
 
 107.98 
 
 96.84 
 
 11.14 
 
 124.0996 
 
 0.50 
 
 
 May 
 
 104.61 
 
 95.33 
 
 9.28 
 
 86.1184 
 
 0.42 
 
 
 June 
 
 105.58 
 
 95.50 
 
 10.08 
 
 101.6064 
 
 0.45 
 
 
 July 
 
 103.82 
 
 97.83 
 
 5.99 
 
 35.8801 
 
 0.27 
 
 
 Aug. 
 
 107.34 
 
 103.30 
 
 4.04 
 
 16.3216 
 
 0.18 
 
 
 Sept. 
 
 115.26 
 
 108.19 
 
 7.07 
 
 49.9849 
 
 0.32 
 
 
 Oct. 
 
 114.67 
 
 109.87 
 
 4.80 
 
 23.0400 
 
 0.22 
 
 
 Nov. 
 
 108.50 
 
 107.53 
 
 0.98 
 
 0.9604 
 
 0.04 
 
 
 Dec. 
 
 105.47 
 
 101.35 
 
 4.12 
 
 16.9744 
 
 0.18 
 
 1930 
 
 Jan. 
 
 94.92 
 
 94.06 
 
 0.86 
 
 0.7396 
 
 0.22 
 
 
 Feb. 
 
 93.43 
 
 92.58 
 
 0.85 
 
 0.7225 
 
 0.04 
 
 
 Mar. 
 
 98.52 
 
 97.63 
 
 0.89 
 
 0.7921 
 
 0.04 
 
 
 Apr. 
 
 102.88 
 
 96.84 
 
 6.04 
 
 36.4816 
 
 0.27 
 
 
 May 
 
 94.01 
 
 95.33 
 
 -1.32 
 
 1.7424 
 
 -0.06 
 
 
 June 
 
 89.75 
 
 95.50 
 
 -5.75 
 
 33.0625 
 
 -0.26 
 
 
 July 
 
 97.25 
 
 97.83 
 
 -0.58 
 
 0.3364 
 
 -0.03 
 
 
 Aug. 
 
 108.29 
 
 103.30 
 
 4.99 
 
 24.9001 
 
 0.22 
 
 
 Sept. 
 
 111.84 
 
 108.19 
 
 3.65 
 
 13.3225 
 
 0.16 
 
 
 Oct. 
 
 100.63 
 
 109.87 
 
 -9.24 
 
 85.5625 
 
 -0.41 
 
 
 Nov. 
 
 103.71 
 
 107.53 
 
 -3.82 
 
 14.5924 
 
 -0.17 
 
 
 Dec. 
 
 93.49 
 
 101.35 
 
 -7.86 
 
 61.7796 
 
 -0.35 
 
 1931 
 
 Jan. 
 
 83.73 
 
 94.06 
 
 -10.33 
 
 106.7089 
 
 -0.46 
 
 
 Feb. 
 
 84.37 
 
 92.58 
 
 -3.26 
 
 10.6276 
 
 -0.15 
 
 
 Mar. 
 
 86.78 
 
 97.63 
 
 -10.06 
 
 101.2036 
 
 -0.45 
 
 
 Apr. 
 
 79.33 
 
 96.84 
 
 17.51 
 
 306.6001 
 
 -0.78 
 
 
 May 
 
 72.88 
 
 95.33 
 
 -22.45 
 
 504.0025 
 
 -1.01 
 
 
 June 
 
 72.57 
 
 95.50 
 
 -22.93 
 
 525.7849 
 
 -1.03 
 
246 
 
 THE ANALYSIS OF TIME SERIES 
 
 TABLE 13.13—( Continued) 
 
 Cycle Fluctuations after Trend and Seasonal Variations 
 
 Have Been Removed from Prices of Butter 
 
 Year 
 
 Month 
 
 Ratio of 
 Active 
 Prices to 
 Prices from 
 Trend Line, 
 a 
 
 Seasonal 
 
 Variations, 
 
 b 
 
 Deviations, 
 a — b 
 
 Deviations 
 
 Squared, 
 
 (a - by 
 
 Deviations 
 in Terms of 
 Standard 
 Deviations 
 
 1931 
 
 July 
 
 78.53 
 
 97.83 
 
 -19.30 
 
 372.4900 
 
 -0.86 
 
 
 Aug. 
 
 89.58 
 
 103.30 
 
 -13.72 
 
 188.2384 
 
 -0.61 
 
 
 Sept. 
 
 108.30 
 
 108.19 
 
 Oil 
 
 0.0121 
 
 0 00 
 
 
 Oct. 
 
 110.87 
 
 109.87 
 
 1.00 
 
 1.0000 
 
 0.04 
 
 
 Nov. 
 
 102.25 
 
 107.53 
 
 -5.28 
 
 27.8784 
 
 -0.24 
 
 
 Dec. 
 
 102.28 
 
 101.35 
 
 0.93 
 
 0.8649 
 
 0.04 
 
 1932 
 
 Jan. 
 
 79.99 
 
 94.06 
 
 -14.07 
 
 197.9649 
 
 -0.63 
 
 
 Feb. 
 
 77.16 
 
 92.58 
 
 -15.42 
 
 237.7764 
 
 -0.69 
 
 
 Mar. 
 
 78.70 
 
 97.63 
 
 -18.93 
 
 358.3449 
 
 -0.85 
 
 
 Apr. 
 
 70.80 
 
 96.84 
 
 -26.04 
 
 678.0816 
 
 -1.17 
 
 
 May 
 
 67.36 
 
 95.33 
 
 -27.97 
 
 782.3209 
 
 — 1.25 
 
 
 June 
 
 61.56 
 
 95.50 
 
 -33.94 
 
 1,151.9236 
 
 -1.52 
 
 
 July 
 
 66.79 
 
 97.83 
 
 -31.04 
 
 963.4816 
 
 -1.39 
 
 
 Aug. 
 
 75 67 
 
 103.30 
 
 -27.63 
 
 763.4169 
 
 -1.24 
 
 
 Sept. 
 
 78.46 
 
 108.19 
 
 -29.73 
 
 883.8729 
 
 -1.33 
 
 
 Oct. 
 
 79.45 
 
 109.87 
 
 -30.42 
 
 925.3764 
 
 -1.36 
 
 
 Nov. 
 
 90.66 
 
 107.53 
 
 -16.87 
 
 284.5969 
 
 -0.76 
 
 
 Dec. 
 
 95.22 
 
 101.35 
 
 -6.13 
 
 37.5769 
 
 -0.27 
 
 1933 
 
 Jan. 
 
 79.59 
 
 94.06 
 
 -14.47 
 
 209.3809 
 
 -0.65 
 
 
 Feb. 
 
 75.91 
 
 92.58 
 
 -16.67 
 
 277.8889 
 
 -0.75 
 
 
 Mar. 
 
 75.11 
 
 97.63 
 
 -22.52 
 
 507.1504 
 
 -1.01 
 
 
 Apr. 
 
 86.77 
 
 96.84 
 
 -10 07 
 
 101.4049 
 
 -0.45 
 
 
 May 
 
 96.20 
 
 95.33 
 
 0.87 
 
 0.7569 
 
 0.04 
 
 
 June 
 
 99.09 
 
 95.50 
 
 3 59 
 
 12.8881 
 
 0.16 
 
 
 July 
 
 108.20 
 
 97.83 
 
 10.37 
 
 107.5369 
 
 0 46 
 
 
 Aug. 
 
 95.60 
 
 103.30 
 
 -7.70 
 
 59.2900 
 
 -0 34 
 
 
 Sept. 
 
 107.71 
 
 108.19 
 
 -0.48 
 
 0 2304 
 
 -0 02 
 
 
 Oct. 
 
 111.66 
 
 109.87 
 
 1.79 
 
 3.2041 
 
 0.08 
 
 
 Nov. 
 
 111.58 
 
 107.53 
 
 4.05 
 
 16.4025 
 
 0.18 
 
 
 Dec. 
 
 96.63 
 
 101.35 
 
 -4.72 
 
 22.3729 
 
 -0.21 
 
 1934 
 
 Jan. 
 
 97.25 
 
 94.06 
 
 3.19 
 
 10.1761 
 
 0.14 
 
 
 Feb. 
 
 126.62 
 
 92.58 
 
 34.04 
 
 1,158 7216 
 
 1.50 
 
 
 Mar. 
 
 129.07 
 
 97.63 
 
 31.44 
 
 988.4736 
 
 1.38 
 
 
 Apr. 
 
 122.85 
 
 96.84 
 
 26.01 
 
 676.5201 
 
 1.14 
 
 
 May 
 
 129.71 
 
 95.33 
 
 34.38 
 
 1,178.5464 
 
 1.51 
 
 
 June 
 
 134.49 
 
 95.50 
 
 38.99 
 
 1,520.2201 
 
 1.72 
 
 
 July 
 
 135.15 
 
 97.83 
 
 37.32 
 
 1,392.7824 
 
 1.64 
 
 
 Aug. 
 
 154.34 
 
 103.30 
 
 51.04 
 
 2,005.0816 
 
 2.25 
 
 
 Sept. 
 
 148.50 
 
 108.19 
 
 40.31 
 
 1,624.8961 
 
 1.77 
 
 
 Oct. 
 
 158.51 
 
 109.87 
 
 57.64 
 
 3,322.3696 
 
 2.54 
 
 
 Nov. 
 
 176.76 
 
 107.53 
 
 69.23 
 
 4,792.7929 
 
 3.05 
 
 
 Dec. 
 
 190.70 
 
 101 35 
 
 89 35 
 
 9,672.7225 
 
 3.93 
 
 a = I 41,805 - 191 . 2 = V497.6808 = 22.31, 
 
 \ 84 
 
 found from the total of column 6. 
 
CYCLE FLUCTUATIONS 
 
 247 
 
 CYCLE FLUCTUATIONS 
 
 Time series are usually influenced by trend, seasonal variations, 
 cycle fluctuations, and residuals which cannot always be measured. 
 After the effects of trend and seasonal variations have been 
 removed from the original data there remain cyclical fluctuations 
 and residuals which should be located and analyzed if possible. 
 One way of determining cycle variation is to eliminate the effect 
 of seasonal variations from trend line values by correcting the 
 ratios of actual prices to the ordinates of the trend line for sea- 
 sonals. The necessary computations are presented in Table 13.13. 
 Column 5 contains the data with trend and seasonals removed. 
 These values are plotted in Fig. 13.4. 
 
 1928 1929 1930 1931 1932 1933 1934 
 
 Fiq. 13.4. — Wholesale prices of butter and eggs with trends and seasonal 
 variations removed, leaving cyclical and residual variations. 
 
248 
 
 THE ANALYSIS OF TIME SERIES 
 
 The last column in Table 13.13 gives the size of the deviations 
 from zero in terms of the standard deviations. Some business 
 firms like to plot these values instead of the values a — b. There 
 do not appear to be definite cycles for this period. Cycles are diffi¬ 
 cult to detect in time series when the time under consideration 
 extends over a short period. A cycle chart pertaining to one com¬ 
 modity can be compared with cycle charts of other commodities 
 by plotting on the same axes or by placing them over a glass under 
 which is a bright light. 
 
 Figure 13.4 shows the cyclical variations and residual errors 
 pertaining to prices of eggs after trend and seasonals have been 
 removed. Cycles pertaining to butter prices nearly coincide with 
 those pertaining to prices of eggs. The broken lines in this figure 
 follow about the same general course during this period of seven 
 years. 
 
 PROBLEMS 
 
 Analyze the following time series, securing secular trend, seasonal 
 variations, and cyclic variations. 
 
 1. Average prices per 100 pounds paid producers by condensaries for 
 milk testing 3.5 butterfat, f.o.b. factory. Part of Table 416 in Agricultural 
 Statistics, 1937. 
 
 Year Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. 
 
 1932 1.12 0.99 0.95 0.93 0.86 0.81 0.77 0.80 0.85 0.86 0.86 0.92 
 
 1933 0.95 0.84 0.82 0.81 0.93 1.00 1.07 1.10 1.09 1.10 1.08 1.00 
 
 1934 0.97 1.10 1.11 1.02 1.06 1.09 1.09 1.21 1.17 1.20 1.32 1.35 
 
 1935 1.46 1.57 1.42 1.46 1.23 1.13 1.13 1.18 1.22 1.31 1.49 1.57 
 
 1936 1.58 1.62 1.47 1.40 1.29 1.39 1.63 1.74 1.74 1.65 1.64 1.62 
 
 2. Butterfat: Average price per pound received by farmers, U.S.A. 
 Part of Table 420 in Agricultural Statistics. 
 
 Year Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec. 
 
 1932 22.8 19.8 19.5 17.8 16.3 14.6 14.4 17.5 17.6 17.8 18.4 21.1 
 
 1933 18.9 15.8 15.1 16.5 20.2 19.7 23.0 18.4 19.6 20.1 20.4 18.0 
 
 1934 16.1 21.6 23.5 21.0 21.5 22.2 22.1 24.3 24.0 24.3 27.2 28.2 
 
 1935 30.5 35.9 31.2 33.8 27.5 23.7 22.3 22.9 24.9 25.9 29.9 33.0 
 
 1936 33.5 34.9 31.7 31.2 27.1 27.7 32.6 35.7 35.5 33.5 33.1 33.6 
 
CHAPTER 14 
 
 ANALYSIS OF VARIANCE 
 SUMS OF SQUARES 
 
 In Chapter 11 a method was given for finding the significance 
 of 2 arithmetic means. The present chapter presents methods of 
 testing the significance of 2 or more means by the analysis of vari¬ 
 ances. Consider the following example of wheat yields of 3 
 varieties, A, B, and C, each of which was planted on 5 plots. 
 Table 14.1 contains the yields for the plots. 
 
 TABLE 14.1 
 Yields of Wheat 
 
 A 
 
 B 
 
 C 
 
 36 
 
 34 
 
 26 
 
 33 
 
 24 
 
 27 
 
 40 
 
 39 
 
 29 
 
 31 
 
 26 
 
 24 
 
 35 
 
 27 
 
 34 
 
 Total 175 
 
 150 
 
 140 465 
 
 Mean 35 
 
 30 
 
 28 31 
 
 If there are no differences between the varieties the means of 
 the yields should be about the same; that is, the differences should 
 be small and due only to fluctuations arising in random sampling. 
 To be able to test whether or not there are significant differences 
 between means it is necessary to find standard errors and standard 
 errors for differences of means. These standard errors are found 
 from the analysis of variances, as will be explained. 
 
 249 
 
250 
 
 ANALYSIS OF VARIANCE 
 
 The sum of the squares of the deviations of the plot yields from 
 the mean of all the yields will be designated by the phrase total 
 sum of squares. This total sum of squares can be broken up into 
 two sums of squares, namely, the sum of squares of the deviations 
 of the variety means from the mean of all yields, and the sum of 
 the squares of the deviations of the plot yields from their corre¬ 
 sponding variety means. The total sum of squares can be broken 
 up into the sum of squares between variety means and the sum of 
 squares within varieties. 
 
 Let Xij represent the yield from the jth plot of the zth variety, 
 x the mean of all plot yields, xi* the mean of the yields of variety 
 A, X 2 the mean of the yields of variety B, £3 the mean of yields of 
 variety C, or x p (p = 1 , 2 , 3) represents the means of varieties. 
 The relation between these sums of squares can be expressed as 
 
 (14.1) 2 ( Xi,- — x ) 2 = 2 (x p - x ) 2 + 2 ( x^ — Xp) 2 . 
 
 The quantity 2(xi,- — x ) 2 is the total sum of squares, the quantity 
 2 (ip — x ) 2 is the sum of squares between variety means, and the 
 quantity 2(£,-,• — x p ) 2 is the sum of squares within varieties. For 
 the data in Table 14.1 these sums of squares are as follows: 
 
 2(X(j - x) 2 = (36 - 31) 2 + (33 - 31) 2 + • • • + (24 - 31) 2 + (34 - 31) 2 
 
 = 392. 
 
 2(fp - x) 2 = 5(35 - 31) 2 + 5(30 - 31) 2 + 5(28 - 31) 2 = 130. 
 2(x l7 - Xp) 2 = (36 - 35) 2 + (33 - 35) 2 + • • ■ + (35 - 35) 2 + (34 - 30) 2 
 + (24 - 30) 2 H-b (27 - 30) 2 + (26 - 28) 2 
 
 + (27 - 28) 2 H-b (34 - 28) 2 = 262. 
 
 In the first sum in the right member of equation (14.1) the mean 
 Xp = xi for items in the first column, £2 for items in the second 
 column, and X 3 for items in the third column; thus each of the 
 quantities (xi — x) 2 , (X 2 — x) 2 , and (£3 — x) 2 is repeated 5 times. 
 In the last sum of squares in equation (14.1), x p = xi, when x i; - 
 takes values in the first column; x p = X 2 , when x i; - takes values in 
 
 * The notation will be changed for this chapter because almost all writers 
 in their work in the analysis of variances use these notations. 
 
SUMS OF SQUARES 
 
 251 
 
 the second column; and x p = X 3 , when x i3 takes values in the third 
 column. Equation (14.1) can be written for the general case as 
 follows: 
 
 (14.2) 2(x i} - — x) 2 = /c2(x p — x) 2 + 2(xij — x p ) 2 , 
 
 where k is the number of values from which x p comes. 
 
 Equation (14.1) may be proved to be true from the identity 
 
 (14.3) (Xij — X) = (X p — x) + (;Xij — x p ). 
 
 Square both members of this identity and sum for all values of x I; ; 
 this gives 
 
 (14.4) 2 (xj/ — x) 2 = 2(x p — x) 2 + 22(x p — x) (x i3 — x p ) 
 
 + 2 (xa - x p ) 2 . 
 
 The middle term in the right member of (14.4) is equal to zero; 
 for consider the simple table with three varieties and two replica¬ 
 tions. 
 
 ABC 
 
 X\\ X21 Xzi 
 
 X\n X 22 %32 
 
 Consider the middle term in the right member of (14.4) for the 
 above table; this term expanded is 
 
 22(x p — x)(xij — x) = 2(xi — x)(xn — xi) + 2(xi — x)(xi 2 — xi) 
 + 2(x 2 — x)(x 2 l — x 2 ) + 2(x 2 - X) (x 2 2 — X 2 ) 
 + 2(x3 — x)(X31 — X'i) + 2(xs — x) (x 32 — x) 
 
 = 2(xi - x) (0) + 2(x 2 - x) (0) + 2(X3 — x)(0) 
 = 0, 
 
 since the sum of the items in any column is equal to the mean of 
 the items times the number of items. Equation (14.4) reduces to 
 equation (14.2) when k = 2. 
 
 The sum of squares within varieties is usually found by sub¬ 
 tracting the sum of squares between means from the total sum of 
 squares. Table 14.2 gives the analysis of variances for the wheat 
 yields. 
 
252 
 
 ANALYSIS OF VARIANCE 
 
 TABLE 14.2 
 
 Analysis of Variances for Wheat Yields 
 
 Source of 
 
 Sums of 
 
 Degrees of 
 
 Variance 
 
 Standard 
 
 Variation 
 
 Squares 
 
 Freedom 
 
 Deviation 
 
 Total. 
 
 392 
 
 14 
 
 28.000 
 
 5.292 
 
 Between means 
 
 
 
 
 
 of varieties... 
 
 130 
 
 2 
 
 65 
 
 8.062 
 
 Within varieties 
 
 262 
 
 12 
 
 21.833 
 
 4.673 
 
 The variances in column 4 are found by dividing the sums of 
 squares by the corresponding degrees of freedom. The degrees of 
 freedom for the total variance is 1 less than the number of plot 
 yields, or 15 — 1 = 14; the degrees of freedom for the between 
 means variance is 1 less than the number of varieties, or 3 — 1 = 2; 
 the degrees of freedom for the within varieties variance is found by 
 subtracting the last number of degrees of freedom from the former, 
 or by the product h(k — 1 ), where k is the number of rows 
 in each column and h is the number of columns. Each of the 3 
 variances, provided the data are homogeneous, are independent 
 estimates of the variance of the assumed normal parent population 
 from which this sample was taken. The within-varieties variance 
 is usually taken as the estimate of the variance of the population 
 from which the yields were taken. It is found from the sum of 
 squares which is obtained by subtracting from the total sum of 
 squares the sum of the squares between means', that is, it is obtained 
 after subtracting out the variations due to different means. The 
 standard deviation, or an estimate of the standard deviation of the 
 parent from which the sample came, is considered to be the square 
 root of the variance within varieties', for the wheat yields it is 4.673. 
 The standard deviation of any of the means is equal to 
 
 Estimated a of the parent 
 
 ^mean /T7 
 
 V N 
 
 a for within varieties 4.67 
 
 = - 7 =- = - 7 = = 2.09, 
 
 VN n /5 
 
RANDOMIZED BLOCKS 
 
 253 
 
 where N is the number from which the mean was found. Means 
 can now be tested as in Chapter 11. Snedecor’s table enables one 
 to test for significance between the means by finding a value F 
 where 
 
 „ Larger variance 65 
 F =--- =- = 2.98. 
 
 Smaller variance 21.833 
 
 Is this ratio too large to be attributable to fluctuations due to 
 chance? Snedecor’s table, Table III in this book, shows that 
 F should be greater than 3.88 to be significant. Thus there are no 
 significant differences, between the means of the varieties, and as 
 far as this experiment goes one variety is as good as each of the 
 others. The number 3.88 is found in Table III by going 
 down the second column (2 degrees of freedom) to the twelfth 
 row (12 degrees of freedom) for the variance in the denomi¬ 
 nator of F. 
 
 In the analysis of variance all the data are pooled in finding 
 the variance within varieties ; this gives a much better estimate 
 of the variance of the parent population than using only 5 yields. 
 The variance within varieties is used as experimental error because 
 the variations due to different means have been subtracted from 
 the total sum of squares. The value of F shows whether there are 
 significant differences between the means, and in the above example 
 its size showed that there were no significant differences between 
 any 2 of the means. If the value of F suggests no significant differ¬ 
 ences between any 2 means it is useless to test further; F tests 
 significance between variances and suggests significance or no sig¬ 
 nificance between means. 
 
 RANDOMIZED BLOCKS 
 
 Consider the case where 5 varieties of beans, A, B, C, D, and E 
 are grown on 4 blocks of land, each of which is divided into 5 equal 
 plots. Varieties are assigned to the plots in each block at random; 
 this might be done by drawing tickets from a box, where each 
 variety is represented by a ticket. The first variety drawn is 
 planted in the first plot in the block under consideration, the next 
 variety drawn is planted in the second plot of this block, etc. 
 Each block contains 1 plot of each variety; thus each variety is 
 replicated 4 times in this example. The following chart gives an 
 arrangement of the plots. 
 
254 
 
 ANALYSIS OF VARIANCE 
 
 Blocks 
 
 1 A 
 
 2 B 
 
 3 A 
 
 4 E 
 
 Plots 
 D E 
 
 C A 
 
 E D 
 
 B C 
 
 C B 
 
 E D 
 
 B C 
 
 D A 
 
 The following table gives yields in 
 arranged for computations. 
 
 grams per plot for beans 
 
 TABLE 14.3 
 
 Yields of 5 Varieties of Beans 
 
 
 Varieties 
 
 
 Blocks 
 
 
 
 
 
 
 Totals 
 
 
 A 
 
 B 
 
 C 
 
 D 
 
 E 
 
 
 1 
 
 42 
 
 36 
 
 34 
 
 46 
 
 32 
 
 190 
 
 2 
 
 38 
 
 34 
 
 30 
 
 39 
 
 37 
 
 178 
 
 3 
 
 47 
 
 42 
 
 37 
 
 42 
 
 36 
 
 204 
 
 4 
 
 41 
 
 40 
 
 39 
 
 37 
 
 35 
 
 192 
 
 Totals 
 
 168 
 
 152 
 
 140 
 
 164 
 
 140 
 
 764 
 
 Means 
 
 42 
 
 38 
 
 35 
 
 41 
 
 35 
 
 38.2 
 
 The total sum of squares of the deviations of the yields from 
 the general mean can be broken up into three sums of squares, 
 namely, the sum of squares of the deviations of the means of the 
 varieties from the general mean, the sum of squares of the devia¬ 
 tions of the means of the blocks from the general mean, and the 
 sum of squares due to error. The equation which expresses this 
 relation is 
 
 (14.5) 2 (Xu — x) 2 = 42(x„ — x) 2 + 52(x 6 — x) 2 
 
 + 2 (xn — x„ — x b + x) 2 
 
 where x v represents the variety means and Xb represents the block 
 means. The quantity xt is equal to the mean of the third block 
 when Xu takes values, in the summations, in the third block, etc. 
 
RANDOMIZED BLOCKS 
 
 255 
 
 The last summation in the above equation is usually found by 
 subtracting the first two from the total sum of squares. Formula 
 (14.5) can be proved as Formula (14.1). Equation (14.5) may be 
 written as 
 
 Total sum of squares = Sum of square between variety means 
 
 + Sum of squares between block means 
 + Sum of squares due to error. 
 
 The sum of squares 42(x„ — x) 2 can be written as follows: 
 
 5 5 / 5 
 
 4^(f„— x) 2 = 4y^(x„ — 2x v x+x 2 ) = 4 ( y^ y x v 2 — 2x- E^»+5x 2 
 
 c=l v=l ' 1 
 
 2x 
 
 = 4(2x„ 2 —2x-5 — I +5x 2 )=4(2x i , 2 -10-x 2 + 5x 2 ) 
 
 5 
 
 = 4(2x„ 2 — 5-x 2 ) 
 
 = 4(xi 2 + X2 2 + X3 2 + X4 2 + X5 2 — 5x 2 ) 
 
 - 20x 2 
 
 = Tl+Tl+Tl+Tl+Tj _ 
 
 4 V 20 / 
 
 E T ? ( 2x ) 2 
 
 A _ _ 
 
 _ 4 20 ’ 
 
 (i = A, B, C, D, E ) 
 
 where T B is the total of the yields for variety B, etc. The term 
 
 n 
 
 (2x) 2 /20 is called the correcting factor or term. The sum of 
 squares between block means may be written in a similar manner as 
 
 5 E < A tb ~ 2 = 
 
 b= 1 
 
 2 
 
 20 ’ 
 
 * The values above the summation signs indicate that these sums take in 
 variety A, B, etc. 
 
256 
 
 ANALYSIS OF VARIANCE 
 
 <2 represents the totals of the second block. The correcting factor 
 is the same for all the sums of squares. 
 
 The total sum of squares is equal to 
 
 (764) 2 
 
 2(x - x) 2 = 42 2 + 38 2 + . . . + 36 2 + 35 2 - p- = 359.2, 
 the sum of squares between means of blocks is 
 
 Zt? (2x) 2 190 2 + 178 2 + 204 2 + 192 2 
 
 52(z 6 - x) 2 = 
 
 20 
 
 (764) 2 
 20 
 
 = 68 . 0 , 
 
 the sum of squares between variety means is 
 
 27\ 2 (2a-) 2 168 2 +152 2 + 140 2 + 164 2 + 140 2 
 
 42(x„— x) 2 — 
 
 20 
 
 (764) 2 
 20 
 
 = 171.2. 
 
 The sum of squares due to error is found by subtracting the sum 
 of the last 2 sums of squares from the total sum of squares. 
 
 Table 14.4 gives the analysis of variances for the yields of beans. 
 
 TABLE 14.4 
 Analysis of Variance 
 
 Source of Variation 
 
 Sum of 
 Squares 
 
 Degrees of 
 Freedom 
 
 Variance 
 
 Total. 
 
 359.2 
 
 19 
 
 18.91 
 
 Between means of varieties. 
 
 171.2 
 
 4 
 
 42.80 
 
 Between means of blocks. 
 
 68.0 
 
 3 
 
 22.67 
 
 Error. 
 
 120.0 
 
 12 
 
 10.00 
 
 The error variance is commonly used as the estimate of the 
 variance of the parent population from which the sample of bean 
 
RANDOMIZED BLOCKS 
 
 257 
 
 yields was taken at random. By comparing the error variance 
 with the variance between means of blocks it is seen that F is 
 
 22.67 
 
 10.00 
 
 2.27. 
 
 From Table III, F should be as large or larger than 3.49 to be sig¬ 
 nificant; that is, there are no significant differences between the 
 means of the blocks. This shows that the soil was rather homo¬ 
 geneous. Comparing the error variance with the variance between 
 means of varieties, F is 
 
 42.80 
 
 10.00 
 
 4.80, 
 
 which is too large to be the result of random sampling from the 
 same parent: hence, there is a significance between at least two of 
 the variety means. Let us test for the difference between the 
 means of variety A and variety C. The standard error of any of 
 the means of the varieties is 
 
 -4 
 
 Error variance 
 
 10.00 
 
 = 1.58 
 
 4 4 
 
 and the standard error of the difference of the means is 
 
 ^dlflerence of means = V(1.58) 2 + (1.58) 2 = 1.58 V2 = 2.23. 
 Mean of A — Mean of C 
 
 t = 
 
 2.23 
 
 = 3.14. 
 
 The values of t at the 5% level and the 1% level are given in 
 Table III. Variety A is better than variety C or E. Other means 
 can be tested in this way. A difference between two means as 
 great as 4.8 grams is significant. Variety D is significantly better 
 than varieties C and E. The 4.8 is found by solving the equation. 
 
 Difference of means 
 
 The number 2.179 is found in the t column in Table III at the 5% 
 level for 12 degrees of freedom for error. 
 
 The sums of squares can be found by using a provisional mean. 
 This enables one to reduce the sizes of the items and hence to reduce 
 
258 
 
 ANALYSIS OF VARIANCE 
 
 the amount of computing. Consider Table 14.3. Subtract 40 
 from each value. These new data are recorded in Table 14.6. 
 
 TABLE 14.6 
 
 Yields of 5 Varieties of Beans Less 40 
 
 Blocks 
 
 Varieties 
 
 Totals 
 
 A 
 
 B 
 
 C 
 
 D 
 
 E 
 
 1 
 
 +2 
 
 -4 
 
 -6 
 
 6 
 
 -8 
 
 -10 
 
 2 
 
 _2 
 
 -6 
 
 -10 
 
 -1 
 
 -3 
 
 -22 
 
 3 
 
 7 
 
 2 
 
 -3 
 
 2 
 
 -4 
 
 4 
 
 4 
 
 1 
 
 0 
 
 -1 
 
 -3 
 
 -5 
 
 -8 
 
 Totals 
 
 8 
 
 -8 
 
 -20 
 
 4 
 
 -20 
 
 -36 
 
 The sum of squares 
 
 Z(x - x) 2 = 2 2 + (-2) 2 + • • • + (-4) 2 + (-5)2 - (-36)2/20 
 
 = 359.2, as found before. 
 
 The sum of squares 
 
 8 2+(- 8) 2 + (- 20 ) 2 + 4 2 +(- 20)2 
 42(x„-x) 2 = ----- 
 
 (-36) 2 /20 
 
 = 171.2 as found before. 
 
 The sum of square 52(xt — x) 2 
 
 (-lO) 2 + (-22)2 + 4 2 + (_ 8 )2 
 5 
 
 (-36)2 
 
 20 
 
 68 .0, as found before. 
 
 The sum of squares for error is found as before. 
 
 THE LATIN SQUARE 
 
 Consider a field experiment carried out on a piece of land 
 divided into plots which are arranged in rows and columns, where 
 there are the same number of rows as columns and where each 
 
THE LATIN SQUARE 
 
 259 
 
 variety or treatment occurs once in each row and once in each 
 column. The varieties or treatments are assigned to the plots at 
 random. An example of a 4 X 4 Latin square is shown in the 
 following diagram, where A, B, C, and D represent treatments. 
 
 A 
 
 C 
 
 B 
 
 D 
 
 C 
 
 B 
 
 D 
 
 A 
 
 D 
 
 A 
 
 C 
 
 B 
 
 B 
 
 D 
 
 A 
 
 C 
 
 The following example pertaining to the heights of oats grown 
 in four treatments, will help to explain how the Latin square is 
 used to test significance between means of varieties or treatments. 
 
 Heights of Oats in cm. from Plots Arranged in a Latin 
 
 Square 
 
 
 Columns 
 
 Total 
 
 Rows 
 
 Total 
 
 Treatments 
 
 Treatment 
 
 Means 
 
 
 A 
 
 C 
 
 B 
 
 D 
 
 
 
 
 
 18 
 
 19 
 
 11 
 
 15 
 
 63 
 
 A = 60 
 
 A 15 
 
 
 C 
 
 B 
 
 D 
 
 A 
 
 
 
 
 GO 
 
 £ 
 
 20 
 
 9 
 
 16 
 
 16 
 
 61 
 
 B = 47 
 
 B 11.75 
 
 o 
 
 D 
 
 A 
 
 C 
 
 B 
 
 
 
 
 
 17 
 
 14 
 
 17 
 
 13 
 
 61 
 
 II 
 
 O 
 
 C 17.75 
 
 
 B 
 
 D 
 
 A 
 
 C 
 
 
 
 
 
 14 
 
 13 
 
 12 
 
 15 
 
 54 
 
 D = 61 
 
 D 15.25 
 
 
 
 
 
 
 
 
 Grand 
 
 Total 
 
 
 
 
 
 Grand total 
 
 mean 
 
 Column 
 
 G9 
 
 55 
 
 56 
 
 59 
 
 
 239 
 
 14.94 
 
260 
 
 ANALYSIS OF VARIANCE 
 
 The total sum of squares is broken up into 4 sums of squares as 
 follows: 
 
 Total sum of squares = Sum of squares between means of columns 
 + Sum of squares between means of rows 
 + Sum of squares between means of treatments 
 + Sum of squares for error. 
 
 This is written as follows: 
 
 4 4 
 
 (14.6) S(x ,7 — x) 2 = 4 (x c — x) 2 + 4 (x r — x) 2 
 
 C=1 T -1 
 
 4 
 
 + 4 ^2 (Xt - x) 2 
 
 t= i 
 4 
 
 + ^ (Xij ~ Xc ~ X r ~ X t + 2x) 2 , 
 
 1.7 = 1 
 
 where x c represents column means, x r represents row means, and x t 
 represents treatment means. Formula (14.6) can be derived as 
 Formulas (14.1) and (14.5). The estimate of the parent popula¬ 
 tion variance is usually taken to be that of the error variance; that 
 is, the variance after the variations due to rows, columns, and 
 treatments have been subtracted from the total sum of squares. 
 The following table gives the analysis of variance for the data con¬ 
 cerning the growth of oats. 
 
 TABLE 14.7 
 Analysis of Variance 
 
 Source of Variation 
 
 Sum of 
 Squares 
 
 Degrees of 
 Freedom 
 
 Variance 
 
 Total. 
 
 130.94 
 
 15 
 
 8.73 
 
 Between means of columns... 
 
 30.69 
 
 3 
 
 10.23 
 
 Between means of rows. 
 
 11.69 
 
 3 
 
 3.90 
 
 Between means of treatments 
 
 72.69 
 
 3 
 
 24.23 
 
 Error. 
 
 15.87 
 
 6 
 
 2.65 
 
PROBLEMS 
 
 261 
 
 The sum of squares between treatment means is equal to 
 2 T 2 (2x) 2 60 2 + 47 2 + 71 2 + 61 2 
 
 42(x, - x) 2 = 
 
 16 
 
 (239) 2 
 
 16 
 
 72.69. 
 
 There is a significant difference between treatment means, for 
 the value of F is 
 
 F = 
 
 Treatment variance 
 Error variance 
 
 24.23 
 
 2.65 
 
 = 9.14, 
 
 which according to Table III is too large to be due to random 
 sampling. The experimental error is the square root of the vari¬ 
 ance for error or is \Z2.65 = 1.63. The standard deviation of 
 any treatment mean is 
 
 
 Error variance 
 4 
 
 1.63 
 
 2 
 
 0.815. 
 
 The difference between 2 treatment means, to be significant, is 
 found, as before, to be 2.0 cm. Hence treatment C is significantly 
 better for growth than the other treatments. Treatments A and C 
 are better than B. 
 
 The analysis of variance is better for testing the significance of 
 the difference between 2 means than the test on page 220. 
 
 PROBLEMS 
 
 1. Analyze the data in the following Latin square by the analysis of 
 variance; where the letters represent different fertilizer treatments and 
 the numbers give yields of a certain variety of corn in pounds. 
 
 N 
 
 P 
 
 K 
 
 C 
 
 M 
 
 72 
 
 64 
 
 54 
 
 60 
 
 78 
 
 C 
 
 K 
 
 M 
 
 P 
 
 N 
 
 56 
 
 52 
 
 80 
 
 61 
 
 71 
 
 M 
 
 N 
 
 C 
 
 K 
 
 P 
 
 75 
 
 67 
 
 53 
 
 58 
 
 60 
 
 P 
 
 C 
 
 N 
 
 M 
 
 K 
 
 57 
 
 58 
 
 70 
 
 83 
 
 62 
 
 K 
 
 M 
 
 P 
 
 N 
 
 C 
 
 55 
 
 73 
 
 54 
 
 66 
 
 59 
 
262 
 
 ANALYSIS OF VARIANCE 
 
 where N represents a nitrogen fertilizer, P a phosphorus fertilizer, K a 
 potassium fertilizer, C the control and M manure. 
 
 2 . The following data are grades of potatoes from three states judged 
 as to desirability: 
 
 State A 
 
 State B 
 
 State C 
 
 State A 
 
 State B 
 
 State C 
 
 4.2 
 
 3.8 
 
 4.0 
 
 4.6 
 
 4.0 
 
 3.9 
 
 3.4 
 
 3.5 
 
 4.1 
 
 2.7 
 
 3.2 
 
 4.8 
 
 4.9 
 
 4.0 
 
 3.7 
 
 3.4 
 
 3.6 
 
 4.3 
 
 3.0 
 
 4.1 
 
 3.1 
 
 3.9 
 
 4.0 
 
 4.5 
 
 3.6 
 
 2.9 
 
 3.6 
 
 3.0 
 
 3.1 
 
 3.7 
 
 4.6 
 
 3.9 
 
 3.3 
 
 3.7 
 
 2.8 
 
 3.9 
 
 4.4 
 
 3.7 
 
 4.0 
 
 3.5 
 
 4.0 
 
 4.2 
 
 3.3 
 
 2.8 
 
 3.9 
 
 2.9 
 
 2.8 
 
 3.8 
 
 3.7 
 
 3.0 
 
 3.9 
 
 3.9 
 
 3.0 
 
 3.8 
 
 3.5 
 
 3.4 
 
 3.8 
 
 3.4 
 
 3.6 
 
 4.3 
 
 By the analysis of variance, test whether or not there are significant 
 differences between the means. 
 
 3 . The following data pertain to fertilizers and varieties of a certain 
 crop. Test for significant differences between means of varieties and 
 means of yields on different fertilizers. 
 
 Yields in Grams 
 
 
 Varieties 
 
 A 
 
 B 
 
 C 
 
 D 
 
 Fertilizer I. 
 
 36 
 
 42 
 
 34 
 
 34 
 
 Fertilizer II. 
 
 38 
 
 39 
 
 29 
 
 39 
 
 Fertilizer III. 
 
 30 
 
 44 
 
 28 
 
 32 
 
CHAPTER 15 
 
 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 STANDARD ERROR OF THE MEAN 
 
 In Chapter 11 the standard deviation of the distribution made 
 up of all sample averages was found to be 
 
 (15.1) 
 
 S(i; - M V Y 
 
 n — 1 s* 
 
 where s v is the best estimate of the standard deviation of the 
 parent from which the sample came and n is number of items in 
 the sample. 
 
 Example. The mean and value of s v for the weights of 25 steers in a 
 feeding experiment are 542.3 pounds and 22.1 pounds respectively. 
 According to (15.1), the standard deviation of the distribution made up 
 of all possible averages of the weights of 25 steers similar to the above is 
 
 G’mean 
 
 22.1 
 
 V25 
 
 4.42 lb.; 
 
 this means that the probability of the true mean lying between (542.3 — 
 4.42 = 537.88) and (542.3 + 4.42 = 546.72) is about 0.682. 
 
 The standard deviation of the distribution made up of all sam¬ 
 ple averages is called the standard error of the mean, and shows 
 how accurate the mean of the sample under consideration is. The 
 following example will illustrate the use of the standard error of 
 the mean. 
 
 The following information is known concerning the average 
 weight of the chicks of the first and third brood of a certain hen. 
 
 2G3 
 
264 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 Mean Weight 
 at 9 Weeks 
 
 First brood. 
 Third brood 
 Difference.. 
 
 549.2 grams 
 421.7 “ 
 
 127.5 “ 
 
 5.1 grams 
 4.6 “ 
 
 Are the two averages significantly different ? 
 
 The test for significance, as given on page 220, is 
 
 Significance = 
 
 Mi - Mo 
 
 Standard deviation of the difference of the means 
 
 549.2 - 421.7 127.5 
 
 > 2 . 6 , 
 
 V(5.1) 2 + (4.6) 2 V47.17 
 
 which shows that there is a significant difference between the 
 average weight of the first and third brood at 9 weeks of age. 
 
 STANDARD ERROR OF A PERCENTAGE 
 
 In Chapter 7 it was shown that the standard deviation of a 
 Bernoulli distribution is 
 
 cr = V npq, 
 
 (15.2) 
 
 where n is the number of independent trials, p the probability of a 
 success in each trial, and q the probability of a failure. This will 
 be used to derive the standard error of a percentage. 
 
 Let N items be divided into classes with frequencies, /i", fa", 
 / 3 ", . . . , f k " , and let n items be drawn at random from the popu¬ 
 lation of N items. The probability of getting an item from the 
 class with frequency//' is f/'/N — p, and the probability of get¬ 
 ting no item from this class is 1 — fi'/N = q. The probabilities 
 of getting the various numbers from 0 to n from this class are given 
 by terms of (q + p) n . The standard deviation of this distribution 
 according to (15.2) is 
 
 (15.3) 
 
STANDARD ERROR OF A PERCENTAGE 
 
 265 
 
 expect to come from the class with frequency //' in n random 
 drawings. Substituting this in (15.3) gives 
 
 
 In practice N and //' are never known and hence// is not known. 
 If a drawing of n is made from the population the number of items, 
 fi , coming from the 1th class will usually lie within /, ± 3<x; hence 
 if a is small // may be replaced by fi , the observed frequency for 
 the class under consideration. Formula (15.4) becomes 
 
 (15.5) 
 
 Of, — 
 
 
 which is considered to be the standard error of the frequency/,-. It 
 is necessary to prove a theorem before the standard error of the 
 percentage fi/n can be obtained. 
 
 Theorem 15.1. Let the variates of a given distribution be 
 multiplied by a constant, thus forming a new distribution. The 
 mean and the standard deviation of the new distribution are equal 
 respectively to the mean and standard deviation of the given dis¬ 
 tribution multiplied by the constant. 
 
 Proof: Let the given distribution be given by the first and third 
 columns in the following table and the new distribution be repre¬ 
 sented by the second and third columns, where k is the constant. 
 
 Variates 
 
 Variates 
 
 Frequencies 
 
 V 
 
 kv 
 
 f 
 
 V \ 
 
 kv i 
 
 fx 
 
 Vi 
 
 kv 2 
 
 fi 
 
 Vl 
 
 kv 3 
 
 /a 
 
 V n 
 
 kv n 
 
 fn 
 
 The mean of the new distribution is 
 
 ,, kvtfi + kv 2 fo + • • • + kv„f n Zvf 
 
 M -• = /,+/, + ... +/. = vr “ 
 
 The standard deviation of the new distribution is 
 
 O lev 
 
 1 2 {lev - kM v ) 2 Ik 2 2(v - M v ) 2 , 
 
 = V " = V- Tf -** 
 
266 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 The standard deviation of the frequency of a class was found 
 to be (15.5). By employing the results of the above theorem and 
 by allowing k = 1 /n, the standard deviation of the percentage 
 fi/n = p is 
 
 (15.6) 
 
 The following example will illustrate the use of formula (15.6). Com¬ 
 munity A, of population 1,500, contains 30 blind people, and community 
 B, of population 2,000, contains 45 blind; is there a significant difference 
 between the percentages of blind in the two communities? 
 
 According to the above formula the standard error of the first per¬ 
 centage, pi = 30/1,500 = 1/50, is 
 
 c Pl 
 
 4 
 
 1 49 1 
 
 50 50 1500 
 
 0.0036- 
 
 The standard error of the second percentage p 2 = 45/2,000 = 9/400 is 
 0.0016. The test for significance is 
 
 Significance = 
 
 Pi ~ V 2 
 
 ^(pi-pz) 
 
 9/400 - 1/50 
 V0.0036 2 + 0.0016 2 
 
 1 
 
 1.56 
 
 < 2 . 6 - 
 
 Therefore there is no significance between the percentages of blind people 
 in the communities. 
 
 THE STANDARD ERROR OF A PRODUCT 
 
 When the quantity A ± <r A is multiplied by the quantity 
 B ± cr B , the question arises as to what is the error of the product. 
 Let the zth measurement of the quantity a be i,- = A + x,-, where 
 X; is the difference between the measurement and the mean of the 
 measurements; let the jth measurement of the quantity b be 
 Bj = B + y, where y, is the difference between the jth measure¬ 
 ment and the mean of all measurements of b. The quantities x» 
 and ijj are considered to be errors. Let there be n measurements 
 of a and m measurements of b. The first error of the product 
 
 A\B\ = AB + (Bx i + Ayi + xiyi) 
 
 is the sum of the quantities in parentheses. The errors of all 
 possible products are listed below. 
 
THE STANDARD ERROR OF A QUOTIENT 
 
 267 
 
 Bx i + Ayi + xiyi, Bx 2 + Ayi + x 2 y 1 , . . . Bx n + Ay i + x n yi 
 Bx i + Ay 2 + xiy 2 , Bx 2 + Ay 2 +.x 2 y 2 , . . . Bx n + Ay 2 + x n y 2 
 
 Bxi + Ay m + x\y m , Bx 2 + Ay m + x 2 y m ,. . . Bx n + Ay m + x n y m 
 
 The square root of the average of the squares of the errors of the 
 product is 
 
 lB 2 Zx 2 A 2 Xi] 2 2 AB-Zx-^y 2BXx 2 • Zy 2 AZy 2 -2x 2z 2 • S y 2 
 
 c ab = \ l -1-1-1-1-1- 
 
 \ n m n-m n-m n-m n-m 
 
 The third, fourth, and fifth terms vanish because = 'Ey = 0, 
 which leaves for the standard error of the product AB 
 
 (15.7) <sab = "\/(B<sa) 2 + ( Acts ) 2 + (<sa<sb) 2 - 
 
 If the quantities <s A and as are small the product (o- a ctb) 2 will be 
 negligible; hence the standard error of the product is 
 
 (15.8) ctab = V ( Bct a ) 2 + (Aa B ) 2 - 
 
 Example. The width of a piece of land is 76 ± 0.2 feet; the length is 
 137 ± 0.4 feet. According to (15.8), the area of the land is 10,412 ± 
 41 square feet. 
 
 THE STANDARD ERROR OF A QUOTIENT 
 
 Let the quantity a be measured n times with measurements 
 A i, A 2 , . . . , A n and the quantity b be measured m times with 
 measurements B\, B 2 , . . . , B m ; and let A and B be the means of 
 
 the measurements of a and b respectively. Let Xi be the deviation 
 
 of Ai from A , and y, the deviation of Bj from B. The ratio 
 
 A\ A -\- x i A + xi B — yi AB — Ayi + Bx i — xiyi 
 
 Bi B + yi B + yx B - yi B 2 - y{ 2 
 
 _ A Bx i - Ayi + (A/B)yi 2 - xiyi 
 
 ~ B + B 2 - yi 2 
 
 The last fraction is the deviation of the quotient A\/B\ of the 
 measurements A i and By from the quotient A/B of the means. 
 The standard error of the quotient A/B is the square root of the 
 
26S 
 
 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 average of the squares of all possible errors. The square of the 
 numerator of the first error is 
 
 B 2 x i 2 + A 2 y x 2 + xi 2 yi 2 + A - yS - 2ABx 1 y 1 + 2Ax 1 y l 2 
 - 2Bxryi - 2~ yi 3 + 2Ax x iji 2 — 2 A - xiyi 3 . 
 
 £> L> 
 
 If the error y\ is small, y\ 2 is much smaller. Let us neglect the 
 quantity y\ 2 in the denominator. The average of the squares of all 
 possible errors, if y? is neglected in the denominators, is 
 
 v , (Ba A ) 2 + (Aa B ) 2 + o 2 -a 2 + (A 2 /B 2 ) ^ - 2{A 2 /B) ^ 
 Zje* mn mn 
 
 mn B 4 
 
 If the last 3 terms in the numerator are negligible, the standard 
 error of the quotient A/B is 
 
 (15.8) 
 
 &A/B 
 
 V(Ba A ) 2 + (Aa B ) 2 
 B 2 
 
 where it is understood that B is not zero. 
 
 Example. The number of calories produced by a given food varies 
 directly as the weight of the food taken. The average of the readings 
 showed the average number of calories to be 63 ± 0.004. The average of 
 the weight readings was 9 ± 0.002 grams. Find the value of 
 
 0.63 ± 0.004 
 
 - and its standard error. 
 
 0.9 ± 0.002 
 
 According to (15.8) the quotient is 
 
 y/ (0.63 X 0.002) 2 + (0.9 X 0.004) 2 
 °' 7 ± 0.9 2 
 
 = 0.7 ± 0.0047 calorie-gram. 
 
 STANDARD ERRORS OF OTHER STATISTICS 
 
 Derivations for many of the standard errors of statistics are 
 too complicated to be given in an elementary textbook; the follow¬ 
 ing standard errors are given without derivation. They are based 
 upon a normal parent population and also on a rather large number 
 in the sample. 
 
PROBLEMS 
 
 269 
 
 TABLE 15.1 
 
 Standard Errors of Certain Statistics 
 
 Statistics 
 
 Standard Error 
 
 Statistics 
 
 Standard Error 
 
 Median 
 
 1.2533 a/VN 
 
 Rank correlation 
 coefficient 
 
 v - P [l+0.086p 2 + 
 
 0.013p 4 + 0.002p 6 ] 
 
 Standard 
 
 a/V2N 
 
 Mulitple correla- 
 
 1 - (R,- 23 • • -n) 2 * 
 
 deviation 
 
 tion coefficient 
 
 VN — n 
 
 Mean 
 
 0.6028 cr 
 
 Partial correlation 
 
 1 — (r 12-34 ■ • -n) 2 
 
 deviation 
 
 VN 
 
 coefficient 
 
 VN -2 
 
 Coefficient of 
 
 v \!i,o( v v 
 
 Semi-interquartile 
 
 range 
 
 0.7867 <r/V~N 
 
 variation f 
 
 V2N ' 1+ “(l00/ 
 
 Correlation 
 
 1 - r 2 
 
 Regression _ <j v 
 
 1 
 
 h 
 
 coefficient 
 
 VN - 2 
 
 coefficient r n 
 
 o X 
 
 <7 x y N 
 
 Either 
 
 quartile 
 
 1.3626 -£= 
 
 Correlation ratio 
 
 1 - v 
 
 
 
 VN - mt 
 
 * Where n is the number of constants in the regression equations. 
 
 t V = a/m. 
 
 t m is the number of arrays. 
 
 When any statistic is obtained from a set of data its standard 
 error or probable error should always be computed, for this indi¬ 
 cates the accuracy of the statistic under consideration. The fol¬ 
 lowing problems will enable one to use these standard errors. 
 
 PROBLEMS 
 
 1. Find the area of the circle and the standard error of the area if the 
 average of 25 measurements of the diameter is 59 ± 0.2 ft. 
 
 2. Find the circumference and its standard error of the circle in 
 Problem 1. 
 
 3. A cylindrical hole was measured with the following results: 
 
 Mean of depth measurements == 53 ± 0.3 ft., 
 
 Mean of diameter measurements = 7 ± 0.2 ft. 
 
 Find the volume and its standard error. 
 
270 
 
 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 4. The following pertain to measurements made on the length and 
 width of a certain rectangular field. 
 
 Mean of length measurements = 172 ± 0.3 ft., 
 
 Mean of width measurements = 81 ± 0.1 ft. 
 
 Find the area in acres and its standard error in terms of acres. 
 
 5. The following data relate to the pressure and volume of a certain 
 gas held at a constant temperature. 
 
 Mean of readings of the pressure = 77 ± 0.1 cm., 
 
 Mean of readings of the volume = 2 ± 0.2 cu. cm. 
 
 Find the product of the pressure and volume and its standard error. 
 
 6. The following pertain to measurements made on a dam: 
 
 Mean of length measurements = 374 ± 0.4 ft., 
 
 Mean of width measurements = 23 ± 0.2 ft., 
 
 Mean of height measurements = 16 ± 0.1 ft. 
 
 Find the volume and its standard error. 
 
 7. A machine put out 16 imperfect articles in a sample of 500. After 
 the machine was overhauled it put out 3 imperfect articles in a sample of 
 100. Has the machine been improved? 
 
 8. Find the standard errors of the standard deviations found in prob¬ 
 lem 5 on page 221. Are these standard deviations significantly different? 
 
 9. Find the coefficients of variation of the distributions in problem 5 
 on page 81. Are these significantly different? 
 
 10. Find the standard errors of the means of the data pertaining to 
 length of feet given on page 54. Use the f-values in Table III to deter¬ 
 mine whether or not these means are significantly different. 
 
 11. Find the standard error of the correlation coefficient given on 
 page 181. 
 
 12. Find the standard error of the rank correlation coefficient given 
 on page 175. 
 
 13. The multiple correlation between the number of petals on the 
 second branch terminal flower and the number of petals on the stem 
 terminal flower and the first branch terminal flower is 0.82; this infor¬ 
 mation was obtained from 300 plants on plot .4. The multiple correlation 
 coefficient between the number of petals on similar flowers on 200 plants 
 on plot B is 0.74. Are these multiple correlation coefficients significantly 
 different? 
 
SIGNIFICANCE BETWEEN MEANS OF SMALL SAMPLES 271 
 
 14. Partial coefficients of correlation between weight of grain and 
 length of head with length of stalk held constant for barley growing on 
 two study plots are as follows: 
 
 r , oh . 3 = 0.67 for 400 plants, 
 r w h .s — 0.60 for 300 plants, 
 
 where w represents weight, h head length, and s length of stalk below 
 the head. Are the partial correlation coefficients significantly different? 
 
 15. The correlation ratios between the amount of yellow and red in 
 the color of potatoes from two states are: 
 
 State A: rj yT = 0.89 from 50 potatoes, 
 
 State B: i) yT — 0.68 from 73 potatoes. 
 
 Are the correlation ratios significantly different? 
 
 16. Find the standard error of the regression coefficient which gives 
 the slope of the linear relation between the variables in the example on 
 page 181. 
 
 17. Find the standard error of the median of the example on page 41. 
 
 18. The average height of one hundred 10-year-old girls is 53 ± 0.3 
 inches; the average height of one hundred 18-year-old girls is 65 ± 0.4 
 inches. Are the coefficients of variation significantly different? 
 
 19. A zoologist puts each day the same number of white and brown 
 mice in a cage with owls. During the month of September the following 
 was observed. 
 
 No. of white mice eaten = 76, 
 
 No. of brown mice eaten = 59. 
 
 Can one say that the color of the mice determined what the owls ate? 
 
 TESTS OF SIGNIFICANCE BETWEEN MEANS OF SMALL SAMPLES 
 
 The formulas used for testing significance between means 
 obtained from large samples should not be used when testing sig¬ 
 nificance between means of small samples. “Student” and R. A. 
 Fisher obtained distributions whereby one can make accurate tests 
 between means regardless of the size of the sample. Consider the 
 following example. 
 
 Counts were made of the number of pedicels per inflorescence 
 of wild carrots (Daucus carota ) from two localities A and B. 
 
'272 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 Locality A 
 No. of Pedicels 
 per Stem Cluster 
 x 
 50 
 55 
 60 
 
 47 
 
 52 
 58 
 46 
 50 
 
 53 
 
 54 
 50 
 49 
 44 
 42 
 
 48 
 42 
 
 Total 800 
 
 M x = 50 pedicels. 
 2(x - M x y - 412. 
 
 Locality B 
 No. of Pedicels 
 per Stem Cluster 
 
 y 
 
 47 
 
 43 
 
 44 
 
 48 
 40 
 48 
 39 
 46 
 
 45 
 43 
 45 
 52 
 
 Total 540 
 
 M y = 45 pedicels. 
 2(i/ - M V Y = 142. 
 
 Let us combine the sums of the squares of the items from their 
 respective means and divide by the number of degrees of freedom, 
 or the number of items in both sets less 2. The square root of this 
 quantity is called the best estimate of the standard deviation of the 
 parent from which these 2 samples came, or 
 
 (15.9) 
 
 -V 1 
 
 (x - Mx) 2 + 2(y - M y ) 2 
 
 n\ + ri2 — 2 
 
 -4 
 
 412 + 142 
 26 
 
 = 4.6 pedicels. 
 
 The quantity s is obtained by pooling the sums of the squares of 
 
SIGNIFICANCE BETWEEN MEANS OF SMALL SAMPLES 273 
 
 the items from their respective means. The standard deviations 
 of the means of the sets are respectively 
 
 
 s 
 
 \/l6 
 
 1.15 pedicels, cfm v 
 
 s 
 
 Vl2 
 
 1.38. 
 
 The standard deviation of the difference of the means is 
 
 ^Difference of means 
 
 
 The value of t is 
 
 (M x - M y ) - 0 
 
 ^Difference of means 
 
 (50 - 45) - 0 
 1.75 
 
 2 . 86 . 
 
 Look in Table III for Lvalues at 26 degrees of freedom. The 
 value of t at the 5 per cent point is 2.056, and at the 1 per cent 
 point is 2.779. The value of t obtained above is larger than these 
 two values obtained from the table, hence there is a significant 
 difference between the means. This means that the difference 
 between the means, {M x — M y = 5), differs significantly from zero 
 and that the probability that the two samples were taken at ran¬ 
 dom from the same population is very small; from the Ltable this 
 probability is less than 0.01. 
 
 In general, the value of t is 
 
 (15.10) 
 
 t = 
 
 {M x — M v ) — 0 \ n\-n 2 
 
 n\ + n 2 ’ 
 
 where there are n\ items in the first set and n 2 items in the second, 
 and s is the value given in (15.9). The number of degrees of free¬ 
 dom at which the Ltable is entered is always equal to the denomi¬ 
 nator in s, that is, n\ -f- n 2 — 2. 
 
 Consider the following data pertaining to percentages of butterfat in 
 milk at 2 different temperatures: 
 
274 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 
 Low 
 
 High 
 
 
 
 Temperatures 
 
 Temperatures 
 
 
 
 60° - 68° F. 
 
 85° - 100° F. 
 
 Difference 
 
 Patron 
 
 X 
 
 y 
 
 x - y 
 
 1. 
 
 . 4.03 
 
 3.95 
 
 0.08 
 
 2. 
 
 . 3.54 
 
 3.49 
 
 0.05 
 
 3. 
 
 . 3.35 
 
 3.39 
 
 -0.04 
 
 4. 
 
 . 3.81 
 
 3.86 
 
 -0.05 
 
 5. 
 
 . 3.69 
 
 3.70 
 
 -0.01 
 
 6. 
 
 . 3.15 
 
 3.20 
 
 -0.05 
 
 7. 
 
 . 4.11 
 
 4.05 
 
 0.06 
 
 8. 
 
 . 3.51 
 
 3.64 
 
 -0.13 
 
 9. 
 
 . 4.09 
 
 4.11 
 
 -0.02 
 
 10. 
 
 . 3.31 
 
 3.40 
 
 -0.09 
 
 11. 
 
 . 3.99 
 
 3.94 
 
 0.05 
 
 12. 
 
 . 3.70 
 
 3.74 
 
 -0.04 
 
 13. 
 
 . 3.11 
 
 3.14 
 
 -0.03 
 
 14. 
 
 . 4.90 
 
 4.91 
 
 -0.01 
 
 15. 
 
 . 3.60 
 
 3.65 
 
 -0.05 
 
 16. 
 
 . 3.33 
 
 3.44 
 
 -0.11 
 
 17. 
 
 . 4.33 
 
 4.38 
 
 -0.05 
 
 18. 
 
 . 3.90 
 
 3.96 
 
 -0.06 
 
 19. 
 
 . 3.51 
 
 3.51 
 
 0.00 
 
 20. 
 
 . 4.30 
 
 4.30 
 
 0.00 
 
 21. 
 
 . 3.98 
 
 3.98 
 
 0.00 
 
 22. 
 
 . 3.70 
 
 3.76 
 
 -0.06 
 
 23. 
 
 . 3.48 
 
 3.48 
 
 0.00 
 
 24. 
 
 . 3.51 
 
 3.58 
 
 -0.07 
 
 25 . 
 
 . 3.94 
 
 3.94 
 
 0.00 
 
 Total.. .. 
 
 . 93.87 
 
 94.50 
 
 -0.63 
 
 Mean. .. 
 
 . 3.7548 
 
 3.78 
 
 -0.0252 
 
 Let us apply the above test. 
 
 _ 14.0776 + 3788 _ 
 
 \ 48 
 
 t = ° :C ^ 2 (3.5355) = 0.221. 
 
 0.4035 
 
 Entering the f-table at 48 degrees of freedom we see that this test shows 
 no significance between the means, or that the difference between the 
 means does not differ significantly from zero. 
 
SIGNIFICANCE BETWEEN MEANS OF SMALL SAMPLES 275 
 
 If there is no difference between the tests for butterfat at the 2 
 temperatures the average of the differences of the corresponding 
 readings should be close to zero. The column x — y gives the 
 differences of corresponding readings. The mean of the column of 
 differences is —0.0252. The best estimate of the standard devia¬ 
 tion of the parent from which these differences came is 
 
 si 
 
 si 
 
 n — 1 
 
 J 2[(z - y) - (M x -y)] 2 
 
 4 
 
 1 0.063024 
 ~2A 
 
 = 0.0512. 
 
 ) 
 
 The standard deviation of the mean of the differences is 
 
 
 si _ 0.0512 
 y/n 5 
 
 0.01025, 
 
 where n is the number of differences. 
 The lvalue which “Student” gave is 
 
 (15.11) 
 
 M x - V -0.0252 
 ^ " 0.01025 
 
 2.461. 
 
 Entering the Ltable at 49 degrees of freedom it is seen that the 
 average of the differences does differ significantly from zero for 
 the 5 per cent point. This contradicts the first test which was 
 used. 
 
 The first test cannot be applied when the measurements are 
 correlated. In the case above, the milk for the 2 determinations 
 of butterfat for patron 1 at the 2 temperatures was taken from the 
 same milk can. There is very high correlation between the deter¬ 
 minations at low temperatures and high temperatures. The first 
 test cannot be applied here because of this high correlation. 
 Another test for determining significance is 
 
 (M x — My) - 0 / n-n 
 
 s 2 'n + n’ 
 
 Z(x-M I ) 2 +2(y-My) 2 -22(x-M I )(y-My) 
 n + n — 2 
 
 (15.12) 
 where 
 
 (15.13) 
 
276 STANDARD ERRORS OF CERTAIN STATISTICS 
 The quantity S 2 is like s, except the product term, 
 
 22(;r - M x )(y - M v ), 
 
 has been subtracted from the numerator of s. This takes into 
 consideration the correlation without actually finding the correla¬ 
 tion coefficient. Applying this test, the value of S 2 is 
 
 S2 
 
 =V— 
 
 776 + 3.7380 - 7.7526 
 
 48 
 
 = 0.0360 
 
 and 
 
 t = 
 
 0.0252 - 0 
 
 0.360 > 50 
 
 / 25-25 
 
 2.461, 
 
 which is the same as was obtained in the second test.* The degrees 
 of freedom at which to enter the /-table equal n — 1, since this test 
 is the same as that where si is used. 
 
 If the correlation coefficient is negative the quantity 
 
 -22(z - M x )(y - M y ) 
 
 becomes a positive quantity, and the numerator in the value of S 2 
 is increased, and hence the value of t is decreased. The last two 
 tests do not apply if the values of x and y are not corresponding 
 values, that is, the x for patron 2 cannot be placed opposite the y 
 value of patron 6. 
 
 When there is independence between the sets of variates or 
 measurements the first test is the one to use. The first test can 
 be used to test significance between several sets of independent 
 measurements. The following example concerning length of heads 
 of wheat grown on three different soils, A, B, and C, will illustrate 
 this. 
 
 * This last test is the same as the following test: Let 
 
 Si — X ffj. "" f" G y~ 2r jyO'jO'y j 
 
 $4 
 
 the standard deviation of the difference between the means M z and M v is — - 
 
 M x - M v _ v 71 
 
 where n is the number of pairs. The 1-value is t = --- V n - 
 
 04 
 
SIGNIFICANCE BETWEEN MEANS OF SMALL SAMPLES 277 
 
 Soil A 
 
 Length of Head 
 cm. 
 
 7.1 
 
 7.3 
 7.0 
 
 6.7 
 
 6.8 
 
 7.1 
 
 6.8 
 7.0 
 
 6.9 
 
 6.9 
 
 7.4 
 7.0 
 
 7.2 
 
 7.1 
 6.8 
 
 6.9 
 
 Soil B 
 
 Length of Head 
 
 CM. 
 
 6.6 
 
 6.7 
 
 6.4 
 
 6.1 
 
 6.2 
 
 6.7 
 6.0 
 
 5.8 
 
 5.9 
 
 6.3 
 6.0 
 6.7 
 
 6.2 
 6.6 
 
 Soil C 
 
 Length of Head 
 cm. 
 
 7.2 
 
 7.2 
 
 6.9 
 
 7.4 
 7.1 
 
 7.6 
 
 7.1 
 
 7.5 
 
 6.7 
 
 7.3 
 
 7.2 
 
 -4 
 
 Let S 3 , similar to the s in (15.9), be 
 
 S3 
 
 2(x - M x ) 2 + 2 (y - M y ) 2 + 2 (z - M z ) 2 
 
 ni + ri2 + «3 — 3 
 
 0.56 + 1.32 + 0.1056 
 
 38 
 
 = 0.229. 
 
 The standard deviations of the means are 
 
 °m x 
 
 S.3 
 
 Vl6 
 
 0.057; <tm v 
 
 S3 
 
 Vl4 
 
 0.061 
 
 M z 
 
 S3 
 
 VTl 
 
 0.069. 
 
 The values of t for testing the differences between these means are 
 
 H 
 
 1 
 
 t: 
 
 / n r n 2 
 
 S3 
 
 ^ n\ + n 2 
 
 M x - M z 
 
 1 ny-m 
 
 S3 
 
 ^ ny + n 3 
 
 My - M z 
 
 / n 2 -n3 
 
 S 3 
 
 ^ ri2 + nq 
 
 8.35, 
 
 2.23, 
 
 9.75, 
 
278 
 
 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 The above values of / for 38 degrees of freedom show that there 
 are significant differences between the means, or that soil type has 
 much to do with the length of wheat heads. 
 
 PROBLEMS 
 
 1. The following data pertain to growth of wheat under 2 treatments 
 A and B. 
 
 Treatment A 
 No. of stalks = 34 
 Mean height = 15.6 cm., 
 
 2 (x — M z y = 167.52. 
 
 Test for significance between the means. 
 
 2. The following data pertain to corn planted on 3 different dates. 
 
 Treatment B 
 26 
 
 12.1 cm., 
 
 2 {y - M v y = 145.94. 
 
 April 22 May 1 
 
 No. of stalks. 36 49 
 
 Mean yield, pounds... 1.7 0.8 
 
 2 (x — M ,) 2 . 72.3 78.4 
 
 May 15 
 64 
 0.5 
 89.6 
 
 By the use of S 3 find whether or not there are significant differences 
 between the means. 
 
 3. Given the following measurements concerning gain in weights of 
 rats on diets A and B: 
 
 Diet A, 
 
 Diet B, 
 
 Diet A, 
 
 Diet B, 
 
 Gains, Grams 
 
 Gains, Grams 
 
 Gains, Grams 
 
 Gains, Grams 
 
 12 
 
 21 
 
 6 
 
 24 
 
 16 
 
 17 
 
 5 
 
 27 
 
 17 
 
 25 
 
 16 
 
 11 
 
 8 
 
 30 
 
 17 
 
 18 
 
 29 
 
 14 
 
 12 
 
 23 
 
 20 
 
 20 
 
 15 
 
 26 
 
 16 
 
 16 
 
 
 18 
 
 Is it safe to say that one diet is better than the other? 
 
 4. Given the following information concerning two sets of yield 
 measurements made on 16 pairs of parallel plots, A and B : 
 
 A B 
 
 Mean 60 grams. 63 grams. tab = 0.4. 
 
 Standard deviation 3 grams. 4 grams. 
 
SIGNIFICANCE OF A CORRELATION COEFFICIENT 279 
 
 Use the test given in the footnote on page 276 for testing the significance 
 between the means. 
 
 5. Suppose there were 9 parallel plots in problem 4 and that the value 
 of r — —0.4, and the means and standard deviations are as in problem 4. 
 Use the same test for testing significance between means. 
 
 6. The following data pertain to heights of first and second sons at 
 21 years of age and the heights of their fathers at the same age. 
 
 Father, 
 
 First Son, 
 
 Second Son, 
 
 Inches 
 
 Inches 
 
 Inches 
 
 66 
 
 66.3 
 
 66.1 
 
 67 
 
 66.8 
 
 66.9 
 
 68 
 
 68.5 
 
 68.3 
 
 69 
 
 69.2 
 
 69.0 
 
 70 
 
 70.6 
 
 70.4 
 
 71 
 
 71.8 
 
 70.9 
 
 72 
 
 72.2 
 
 71.9 
 
 72 
 
 71.8 
 
 71.5 
 
 73 
 
 74.0 
 
 73.6 
 
 74 
 
 73.9 
 
 73.8 
 
 75 
 
 76.2 
 
 75.7 
 
 Test for significance between the means of the heights of the sons by 
 employing Si and s 2 . 
 
 SIGNIFICANCE OF A CORRELATION COEFFICIENT 
 
 The value of t for testing whether or not a linear correlation 
 coefficient is significant, that is, whether or not it is significantly 
 different from zero, is 
 
 (15.14) t = —/ r — • Vn - 2, 
 
 Vl - r 2 
 
 where r is the linear correlation coefficient and n is the number of 
 pairs of measurements. The number n — 2 is the number of 
 degrees of freedom to use for entering the 2-table. 
 
 Consider the following example. 
 
 The correlation coefficient from 27 pairs of measurements is 0.8. The 
 value of t is 
 
280 STANDARD ERRORS OF CERTAIN STATISTICS 
 
 for 25 degrees of freedom it is seen that r = 0.8 is significantly different 
 from zero. This means that the probability that the 27 pairs of measure¬ 
 ments came at random from a non-correlated parent is very small, in 
 fact less than 0.01. 
 
 PROBLEM 
 
 1. Test for significance the correlation coefficient r = 0.4, which was 
 determined from 18 pairs of observations. 
 
 SIGNIFICANCE BETWEEN STANDARD DEVIATIONS 
 
 Table III can be used for testing significance between two 
 standard deviations or two variances for 
 
 Larger mean square Larger Variance 
 
 Smaller mean square Smaller Variance 
 
 Larger (standard deviation) 2 
 
 Smaller (standard deviation) 2 
 
 This will be illustrated by measurements of cherry stems from two 
 branches. 
 
 Branch without 
 bud sports 
 
 Branch with 
 bud sports 
 
 N = 31, 
 
 N = 25, 
 
 2(x - M x ) 2 = 0.6750. 
 
 2(y - My) 2 = 1.500, 
 
 Entering Table III in the column headed 24 and the row headed 
 30 we see that the value of F to be significant must be as large 
 or larger than 1.89 at the 5% point or 2.47 at the 1% point. 
 
PROBLEMS 
 
 281 
 
 Hence the two standard deviations are significantly different. 
 Cherries from bud sports have more variability in stem lengths 
 than ordinary cherries. 
 
 PROBLEM 
 
 1. Use the above method for testing significance between standard 
 deviations found in problem 8 on page 270. 
 
282 
 
 TABLES 
 
 TABLE I 
 
 Area Under the Normal Curve From — oo to Values of t* 
 
 t 
 
 A = Area 
 
 t 
 
 A 
 
 t 
 
 A 
 
 t 
 
 A 
 
 -4.00 
 
 -3.99 
 
 .00003 
 
 .00003 
 
 -3.49 
 
 .00024 
 
 -2.99 
 
 .00140 
 
 -2.49 
 
 .00639 
 
 -3.98 
 
 .00003 
 
 -3.48 
 
 .00025 + 
 
 -2.98 
 
 .00144 
 
 -2.48 
 
 .00657 
 
 -3.97 
 
 .00004 
 
 -3.47 
 
 .00026 
 
 -2.97 
 
 .00149 
 
 -2.47 
 
 .00676 
 
 -3.96 
 
 .00004 
 
 -3.46 
 
 .00027 
 
 -2.96 
 
 .00154 
 
 -2.46 
 
 .00695 - 
 
 -3.95 
 
 .00004 
 
 -3.45 
 
 .00028 
 
 -2.95 
 
 .00159 
 
 -2.45 
 
 .00714 
 
 -3.94 
 
 .00004 
 
 -3.44 
 
 .00029 
 
 -2.94 
 
 .00164 
 
 -2.44 
 
 .00734 
 
 -3.93 
 
 .00004 
 
 -3.43 
 
 .00030 
 
 -2.93 
 
 .00170 
 
 -2.43 
 
 .00755- 
 
 -3.92 
 
 .00004 
 
 -3.42 
 
 .00031 
 
 -2.92 
 
 .00175 + 
 
 -2.42 
 
 .00776 
 
 -3.91 
 
 .00005 - 
 
 -3.41 
 
 .00033 
 
 -2.91 
 
 .00181 
 
 -2.41 
 
 .00798 
 
 -3.90 
 
 .00005 - 
 
 -3.40 
 
 .00034 
 
 -2.90 
 
 .00187 
 
 -2.40 
 
 .00820 
 
 -3.89 
 
 .00005 + 
 
 -3.39 
 
 .00035- 
 
 -2.89 
 
 .00193 
 
 -2.39 
 
 .00842 
 
 -3.88 
 
 .00005 + 
 
 -3.38 
 
 .00036 
 
 -2.88 
 
 .00199 
 
 -2.38 
 
 .00866 
 
 -3.87 
 
 .00005 + 
 
 -3.37 
 
 .00038 
 
 -2.87 
 
 .00205 + 
 
 -2.37 
 
 .00889 
 
 -3.86 
 
 .00006 
 
 -3.36 
 
 .00039 
 
 -2.86 
 
 .00212 
 
 -2.36 
 
 .00914 
 
 -3.85 
 
 .00006 
 
 -3.35 
 
 .00040 
 
 -2.85 
 
 .00219 
 
 -2.35 
 
 .00939 
 
 -3.84 
 
 .00006 
 
 -3.34 
 
 .00042 
 
 -2.84 
 
 .00226 
 
 -2.34 
 
 .00964 
 
 -3.83 
 
 .00006 
 
 -3.33 
 
 .00043 
 
 -2.83 
 
 .00233 
 
 -2.33 
 
 .00990 
 
 -3.82 
 
 .00007 
 
 -3.32 
 
 .00045 + 
 
 -2.82 
 
 .00240 
 
 -2.32 
 
 .01017 
 
 -3.81 
 
 .00007 
 
 -3.31 
 
 .00047 
 
 -2.81 
 
 .00248 
 
 -2.31 
 
 .01044 
 
 -3.80 
 
 .00007 
 
 -3.30 
 
 .00048 
 
 -2.80 
 
 .00256 
 
 -2.30 
 
 .01072 
 
 -3.79 
 
 .00008 
 
 -3.29 
 
 .00050 
 
 -2.79 
 
 .00264 
 
 -2.29 
 
 .01101 
 
 -3.78 
 
 .00008 
 
 -3.28 
 
 .00052 
 
 -2.78 
 
 .00272 
 
 -2.28 
 
 .01130 
 
 -3.77 
 
 .00008 
 
 -3.27 
 
 .00054 
 
 -2.77 
 
 .00280 
 
 -2.27 
 
 .01160 
 
 -3.76 
 
 .00009 
 
 -3.26 
 
 .00056 
 
 -2.76 
 
 .00289 
 
 -2.26 
 
 .01191 
 
 -3.75 
 
 .00009 
 
 -3.25 
 
 . 00058 
 
 -2.75 
 
 .00298 
 
 -2.25 
 
 .01222 
 
 -3.74 
 
 .00009 
 
 -3.24 
 
 .00060 
 
 -2.74 
 
 .00307 
 
 -2.24 
 
 .01255- 
 
 -3.73 
 
 .00010 
 
 -3.23 
 
 . 00062 
 
 -2.73 
 
 .00317 
 
 -2.23 
 
 .01287 
 
 -3.72 
 
 .00010 
 
 -3.22 
 
 .00064 
 
 -2.72 
 
 .00326 
 
 -2.22 
 
 .01321 
 
 -3.71 
 
 .00010 
 
 -3.21 
 
 . 00066 
 
 -2.71 
 
 .00336 
 
 -2.21 
 
 .01355 + 
 
 -3.70 
 
 .00011 
 
 -3.20 
 
 .00069 
 
 -2.70 
 
 .00347 
 
 -2.20 
 
 .01390 
 
 -3.69 
 
 .00011 
 
 -3.19 
 
 .00071 
 
 -2.69 
 
 .00357 
 
 -2.19 
 
 .01426 
 
 -3.68 
 
 .00012 
 
 -3.18 
 
 .00074 
 
 -2.68 
 
 . 00368 
 
 -2.18 
 
 .01463 
 
 -3.67 
 
 .00012 
 
 -3.17 
 
 .00076 
 
 -2.67 
 
 .00379 
 
 —2.17 
 
 .01500 
 
 -3.66 
 
 .00013 
 
 -3.16 
 
 .00079 
 
 -2.66 
 
 .00391 
 
 -2.16 
 
 .01539 
 
 -3.65 
 
 .00013 
 
 -3.15 
 
 . 00082 
 
 -2.65 
 
 .00403 
 
 -2.15 
 
 .01578 
 
 -3.64 
 
 .00014 
 
 -3.14 
 
 .00085- 
 
 -2.64 
 
 .00415- 
 
 -2.14 
 
 .01618 
 
 -3.63 
 
 .00014 
 
 -3.13 
 
 .00087 
 
 -2.63 
 
 .00427 
 
 -2.13 
 
 .01659 
 
 -3.62 
 
 .00015- 
 
 -3.12 
 
 . 00090 
 
 -2.62 
 
 .00440 
 
 -2.12 
 
 .01700 
 
 -3.61 
 
 .00015 + 
 
 -3.11 
 
 .00094 
 
 -2.61 
 
 .00453 
 
 -2.11 
 
 .01743 
 
 -3.60 
 
 .00016 
 
 -3.10 
 
 .00097 
 
 -2.60 
 
 .00466 
 
 -2.10 
 
 .01786 
 
 -3.59 
 
 .00017 
 
 -3.09 
 
 .00100 
 
 -2.59 
 
 .00480 
 
 -2.09 
 
 .01831 
 
 -3.58 
 
 .00017 
 
 -3.08 
 
 .00104 
 
 -2.58 
 
 .00494 
 
 -2.08 
 
 .01876 
 
 — 3.57 
 
 .00018 
 
 -3.07 
 
 .00107 
 
 -2.57 
 
 .00509 
 
 -2.07 
 
 .01923 
 
 -3.56 
 
 .00019 
 
 -3.06 
 
 .00111 
 
 -2.56 
 
 .00523 
 
 -2.06 
 
 .01970 
 
 -3.55 
 
 .00019 
 
 -3.05 
 
 .00114 
 
 -2.55 
 
 .00539 
 
 -2.05 
 
 .02018 
 
 -3.54 
 
 .00020 
 
 -3.04 
 
 .00118 
 
 -2.54 
 
 .00554 
 
 -2.04 
 
 .02068 
 
 -3.53 
 
 .00021 
 
 -3.03 
 
 .00122 
 
 -2.53 
 
 .00570 
 
 -2.03 
 
 .02118 
 
 -3.52 
 
 .00022 
 
 -3.02 
 
 .00126 
 
 -2.52 
 
 .00587 
 
 -2.02 
 
 .02169 
 
 -3.51 
 
 .00022 
 
 -3.01 
 
 .00131 
 
 -2.51 
 
 .00604 
 
 -2.01 
 
 .02222 
 
 -3.50 
 
 .00023 
 
 -3.00 
 
 .00135 + 
 
 -2.50 
 
 .00621 
 
 -2.00 
 
 .02275+ 
 
 * Tables I and II were taken from L. R. Salvosa’s “Tables,” Annals of Mathematical 
 Statistics, May, 1930. The fifth figure is rounded off from his table with 6 figures. 
 
TABLES 
 
 283 
 
 TABLE I —(Continued) 
 
 Area Under the Normal Curve From — °o to Values of t 
 
 t 
 
 A 
 
 t 
 
 A 
 
 t 
 
 A 
 
 t 
 
 A 
 
 -1.99 
 
 .02330 
 
 -1.49 
 
 .06811 
 
 - .99 
 
 .16109 
 
 - .49 
 
 .31207 
 
 -1.98 
 
 .02385 + 
 
 -1.48 
 
 .06944 
 
 - .98 
 
 .16354 
 
 - .48 
 
 .31561 
 
 -1.97 
 
 .02442 
 
 -1.47 
 
 .07078 
 
 - .97 
 
 .16602 
 
 - .47 
 
 .31918 
 
 -1.96 
 
 .02500 
 
 -1.46 
 
 .07215- 
 
 - .96 
 
 .16853 
 
 - .46 
 
 .32276 
 
 -1.95 
 
 .02559 
 
 -1.45 
 
 .07353 
 
 - .95 
 
 .17106 
 
 — .45 
 
 .32636 
 
 -1.94 
 
 .02619 
 
 -1.44 
 
 .07493 
 
 - .94 
 
 .17361 
 
 - .44 
 
 . 32997 
 
 -1.93 
 
 .02680 
 
 -1.43 
 
 .07636 
 
 - .93 
 
 .17619 
 
 - .43 
 
 .33360 
 
 -1.92 
 
 .02743 
 
 -1.42 
 
 .07780 
 
 - .92 
 
 . 17879 
 
 - .42 
 
 .33724 
 
 -1.91 
 
 .02807 
 
 -1.41 
 
 .07927 
 
 - .91 
 
 .18141 
 
 - .41 
 
 .34090 
 
 -1.90 
 
 .02872 
 
 -1.40 
 
 .08076 
 
 - .90 
 
 .18406 
 
 - .40 
 
 .34458 
 
 -1.89 
 
 .02938 
 
 -1.39 
 
 .08226 
 
 - .89 
 
 .18673 
 
 - .39 
 
 .34827 
 
 -1.88 
 
 .03005 + 
 
 -1.38 
 
 .08379 
 
 - .88 
 
 .18943 
 
 - .38 
 
 .35197 
 
 -1.87 
 
 .03074 
 
 -1.37 
 
 .08534 
 
 - .87 
 
 .19215 + 
 
 - .37 
 
 . 35569 
 
 -1.86 
 
 .03144 
 
 -1.36 
 
 .08692 
 
 - .86 
 
 .19490 
 
 - .36 
 
 .35942 
 
 -1.85 
 
 .03216 
 
 -1.35 
 
 .08851 
 
 - .85 
 
 .19766 
 
 - .35 
 
 .36317 
 
 -1.84 
 
 .03288 
 
 -1.34 
 
 .09012 
 
 - .84 
 
 .20045 + 
 
 - .34 
 
 .36693 
 
 -1.83 
 
 .03363 
 
 -1.33 
 
 .09176 
 
 - .83 
 
 .20327 
 
 - .33 
 
 .37070 
 
 -1.82 
 
 .03438 
 
 -1.32 
 
 .09342 
 
 - .82 
 
 .20611 
 
 - .32 
 
 .37448 
 
 -1.81 
 
 .03515- 
 
 -1.31 
 
 .09510 
 
 - .81 
 
 .20897 
 
 - .31 
 
 .37828 
 
 -1.80 
 
 .03593 
 
 -1.30 
 
 .09680 
 
 - .80 
 
 .21186 
 
 - .30 
 
 .38209 
 
 -1.79 
 
 .03673 
 
 -1.29 
 
 .09853 
 
 - .79 
 
 .21476 
 
 - .29 
 
 .38591 
 
 -1.78 
 
 .03754 
 
 -1.28 
 
 .10027 
 
 - .78 
 
 .21770 
 
 - .28 
 
 .38974 
 
 -1.77 
 
 .03836 
 
 -1.27 
 
 .10204 
 
 - .77 
 
 .22065 + 
 
 - .27 
 
 .39358 
 
 -1.76 
 
 .03920 
 
 -1.26 
 
 .10384 
 
 - .76 
 
 .22363 
 
 - .26 
 
 .39743 
 
 -1.75 
 
 .04006 
 
 -1.25 
 
 .10565 + 
 
 - .75 
 
 .22663 
 
 - .25 
 
 .40129 
 
 -1.74 
 
 .04093 
 
 -1.24 
 
 .10749 
 
 - .74 
 
 .22965 + 
 
 - .24 
 
 .40517 
 
 -1.73 
 
 .04182 
 
 -1.23 
 
 .10935- 
 
 - .73 
 
 .23270 
 
 - .23 
 
 .40905 - 
 
 -1.72 
 
 .04272 
 
 -1.22 
 
 .11123 
 
 - .72 
 
 .23576 
 
 - .22 
 
 .41294 
 
 -1.71 
 
 .04363 
 
 -1.21 
 
 .11314 
 
 - .71 
 
 .23885 + 
 
 - .21 
 
 .41683 
 
 -1.70 
 
 .04457 
 
 -1.20 
 
 .11507 
 
 - .70 
 
 .24196 
 
 - .20 
 
 .42074 
 
 -1.69 
 
 .04551 
 
 -1.19 
 
 .11702 
 
 - .69 
 
 .24510 
 
 - .19 
 
 .42466 
 
 -1.68 
 
 .04648 
 
 -1.18 
 
 .11900 
 
 - .68 
 
 .24825 + 
 
 - .18 
 
 .42858 
 
 -1.67 
 
 . 04746 
 
 -1.17 
 
 .12100 
 
 - .67 
 
 .25143 
 
 - .17 
 
 .43251 
 
 -1.66 
 
 .04846 
 
 -1.16 
 
 .12302 
 
 - .66 
 
 .25463 
 
 - .16 
 
 .43644 
 
 -1.65 
 
 .04947 
 
 -1.15 
 
 .12507 
 
 - .65 
 
 .25785- 
 
 - .15 
 
 . 44038 
 
 -1.64 
 
 .05050 
 
 -1.14 
 
 .12714 
 
 - .64 
 
 .26109 
 
 - .14 
 
 .44433 
 
 -1.63 
 
 .05155 + 
 
 -1.13 
 
 .12924 
 
 - .63 
 
 .26435- 
 
 - .13 
 
 .44828 
 
 -1.62 
 
 .05262 
 
 -1. 12 
 
 .13136 
 
 - .62 
 
 .26763 
 
 - .12 
 
 .45224 
 
 -1.61 
 
 . 05370 
 
 -1.11 
 
 .13350 
 
 - .61 
 
 .27093 
 
 - .11 
 
 .45621 
 
 -1.60 
 
 .05480 
 
 -1.10 
 
 . 13567 
 
 - .60 
 
 .27425 + 
 
 - .10 
 
 .46017 
 
 -1.59 
 
 .05592 
 
 -1.09 
 
 . 13786 
 
 - .59 
 
 .27760 
 
 - .09 
 
 .46414 
 
 -1.58 
 
 .05705 + 
 
 —1.08 
 
 . 14007 
 
 - .58 
 
 .28096 
 
 - .08 
 
 .46812 
 
 -1.57 
 
 .05821 
 
 -1.07 
 
 .14231 
 
 - .57 
 
 .28434 
 
 - .07 
 
 .47210 
 
 -1.56 
 
 .05938 
 
 -1.06 
 
 .14457 
 
 - .56 
 
 .28774 
 
 - .06 
 
 . 47608 
 
 -1.55 
 
 .06057 
 
 -1.05 
 
 .14686 
 
 - .55 
 
 .29116 
 
 - .05 
 
 .48006 
 
 -1.54 
 
 .06178 
 
 -1.04 
 
 .14917 
 
 - .54 
 
 .29460 
 
 - .04 
 
 . 48405 - 
 
 -1.53 
 
 .06301 
 
 -1.03 
 
 .15151 
 
 - .53 
 
 .29806 
 
 - .03 
 
 .48803 
 
 -1.52 
 
 . 06426 
 
 -1.02 
 
 . 15386 
 
 - .52 
 
 .30153 
 
 - .02 
 
 . 49202 
 
 -1.51 
 
 . 06522 
 
 -1.01 
 
 .15625- 
 
 - .51 
 
 .30503 
 
 - .01 
 
 .49601 
 
 -1.50 
 
 .06681 
 
 -1.00 
 
 . 15866 
 
 - .50 
 
 .30854 
 
 - .00 
 
 .50000 
 
284 
 
 TABLES 
 
 TABLE I —( Continued) 
 
 Area Lender the Normal Curve From — oo to Values of t 
 
 t 
 
 A 
 
 t 
 
 A 
 
 t 
 
 A 
 
 t 
 
 A 
 
 .00 
 
 . 50000 
 
 .50 
 
 .69146 
 
 1.00 
 
 .84135- 
 
 1.50 
 
 .93319 
 
 .01 
 
 .50399 
 
 .51 
 
 .69497 
 
 1.01 
 
 .84375 + 
 
 1.51 
 
 .93448 
 
 .02 
 
 .50798 
 
 .52 
 
 .69847 
 
 1.02 
 
 .84614 
 
 1.52 
 
 .93575- 
 
 .03 
 
 .51197 
 
 .53 
 
 .70194 
 
 1.03 
 
 .84850 
 
 1.53 
 
 .93699 
 
 .04 
 
 .51595 + 
 
 .54 
 
 .70540 
 
 1.04 
 
 . 850S3 
 
 1.54 
 
 .93822 
 
 .05 
 
 .51994 
 
 . 55 
 
 .70884 
 
 1.05 
 
 .85314 
 
 1.55 
 
 .93943 
 
 .06 
 
 .52392 
 
 .56 
 
 .71226 
 
 1.06 
 
 .85543 
 
 1.56 
 
 .94062 
 
 .07 
 
 .52790 
 
 .57 
 
 .71566 
 
 1.07 
 
 .85769 
 
 1.57 
 
 .94179 
 
 .08 
 
 .53188 
 
 .58 
 
 .71904 
 
 1.08 
 
 .85993 
 
 1.58 
 
 .94295 - 
 
 .09 
 
 .53586 
 
 .59 
 
 .72241 
 
 1.09 
 
 .86214 
 
 1.59 
 
 .94408 
 
 .10 
 
 .53983 
 
 .60 
 
 .72575- 
 
 1.10 
 
 .86433 
 
 1.60 
 
 .94520 
 
 . 11 
 
 .543S0 
 
 .61 
 
 .72907 
 
 1.11 
 
 .86650 
 
 1.61 
 
 .94630 
 
 .12 
 
 .54776 
 
 .62 
 
 . 73237 
 
 1.12 
 
 .86864 
 
 1.62 
 
 .94738 
 
 .13 
 
 .55172 
 
 .63 
 
 .73565 + 
 
 1.13 
 
 .87076 
 
 1.63 
 
 .94845 - 
 
 . 14 
 
 .55567 
 
 .64 
 
 .73891 
 
 1.14 
 
 .87286 
 
 1.64 
 
 .94950 
 
 . 15 
 
 .55962 
 
 .65 
 
 .74215 + 
 
 1.15 
 
 .87493 
 
 1.65 
 
 .95053 
 
 .16 
 
 .56356 
 
 .66 
 
 .74537 ’ 
 
 1.16 
 
 .87698 
 
 1.66 
 
 .95154 
 
 .17 
 
 .56750 
 
 .67 
 
 . 74857 
 
 1.17 
 
 .87900 
 
 1.67 
 
 .95254 
 
 .18 
 
 .57142 
 
 .68 
 
 . 75175 — 
 
 1.18 
 
 .8S100 
 
 1.68 
 
 .95352 
 
 .19 
 
 .57535- 
 
 .69 
 
 .75490 
 
 1.19 
 
 .88298 
 
 1.69 
 
 .95449 
 
 .20 
 
 .57926 
 
 .70 
 
 .75804 
 
 1.20 
 
 .88493 
 
 1.70 
 
 .95544 
 
 .21 
 
 .58317 
 
 .71 
 
 .76115- 
 
 1.21 
 
 .88686 
 
 1.71 
 
 .95637 
 
 .22 
 
 .58706 
 
 .72 
 
 .76424 
 
 1.22 
 
 .88877 
 
 1.72 
 
 .95728 
 
 .23 
 
 .59095 + 
 
 .73 
 
 .76731 
 
 1.23 
 
 .89065 + 
 
 1.73 
 
 .95819 
 
 .24 
 
 .59484 
 
 .74 
 
 .77035 + 
 
 1.24 
 
 .89251 
 
 1.74 
 
 .95907 
 
 .25 
 
 .59871 
 
 .75 
 
 .77337 
 
 1.25 
 
 .89435 + 
 
 1.75 
 
 .95994 
 
 .26 
 
 .60257 
 
 .76 
 
 .77637 
 
 1.26 
 
 .89617 
 
 1.76 
 
 .960S0 
 
 .27 
 
 .60642 
 
 .77 
 
 .77935 + 
 
 1.27 
 
 .89796 
 
 1.77 
 
 .96164 
 
 .28 
 
 .61026 
 
 .78 
 
 .78231 
 
 1.28 
 
 .89973 
 
 1.78 
 
 .96246 
 
 .29 
 
 .61409 
 
 .79 
 
 .78524 
 
 1.29 
 
 . 90148 
 
 1.79 
 
 .96327 
 
 .30 
 
 .61791 
 
 .80 
 
 .78815- 
 
 1.30 
 
 .90320 
 
 1.80 
 
 .96407 
 
 .31 
 
 .62172 
 
 .81 
 
 .79103 
 
 1.31 
 
 .90490 
 
 1.81 
 
 .96485 + 
 
 .32 
 
 . 62552 
 
 .82 
 
 .79389 
 
 1.32 
 
 .90658 
 
 1.82 
 
 .96562 
 
 .33 
 
 .62930 
 
 .83 
 
 .79673 
 
 1.33 
 
 .90824 
 
 1.83 
 
 .96638 
 
 .34 
 
 . 63307 
 
 .84 
 
 .79955- 
 
 1.34 
 
 .90988 
 
 1.84 
 
 .96712 
 
 .35 
 
 .63683 
 
 .85 
 
 .80234 
 
 1.35 
 
 .91149 
 
 1.85 
 
 .96784 
 
 .36 
 
 .64058 
 
 .86 
 
 .80511 
 
 1.36 
 
 .91309 
 
 1.86 
 
 .96856 
 
 .37 
 
 .64431 
 
 .87 
 
 .80785 + 
 
 1.37 
 
 .91466 
 
 1.87 
 
 .96926 
 
 .38 
 
 .64803 
 
 .88 
 
 .81057 
 
 1.38 
 
 .91621 
 
 1.88 
 
 .96995- 
 
 .39 
 
 .65173 
 
 .89 
 
 .81327 
 
 1.39 
 
 .91774 
 
 1.89 
 
 .97062 
 
 .40 
 
 .65542 
 
 .90 
 
 .81594 
 
 1.40 
 
 .91924 
 
 1.90 
 
 .97128 
 
 .41 
 
 .65910 
 
 .91 
 
 .81859 
 
 1.41 
 
 .92073 
 
 1.91 
 
 .97193 
 
 .42 
 
 .66276 
 
 .92 
 
 .82121 
 
 1.42 
 
 .92220 
 
 1.92 
 
 .97257 
 
 .43 
 
 .66640 
 
 .93 
 
 . 82381 
 
 1.43 
 
 .92364 
 
 1.93 
 
 .97320 
 
 .44 
 
 .67003 
 
 .94 
 
 .82639 
 
 1.44 
 
 .92507 
 
 1.94 
 
 .97381 
 
 .45 
 
 .67365- 
 
 .95 
 
 .82894 
 
 1.45 
 
 .92647 
 
 1.95 
 
 .97441 
 
 .46 
 
 .67724 
 
 .96 
 
 .83147 
 
 1.46 
 
 .92786 
 
 1.96 
 
 .97500 
 
 .47 
 
 .68082 
 
 .97 
 
 .83398 
 
 1.47 
 
 .92922 
 
 1.97 
 
 .97558 
 
 .48 
 
 .68439 
 
 .98 
 
 .83646 
 
 1.48 
 
 .93056 
 
 1.98 
 
 .97615- 
 
 .49 
 
 .68793 
 
 .99 
 
 .83891 
 
 1.49 
 
 .93189 
 
 1.99 
 
 .97671 
 
TABLES 
 
 285 
 
 TABLE I—( Continued) 
 
 Area Under the Normal Curve From — oo to Values of t 
 
 t 
 
 A 
 
 t 
 
 A 
 
 t 
 
 A 
 
 t 
 
 A 
 
 2.00 
 
 .97725 + 
 
 2.50 
 
 .99379 
 
 3.00 
 
 .99865 + 
 
 3.50 
 
 .99977 
 
 2.01 
 
 .97778 
 
 2.51 
 
 .99396 
 
 3.01 
 
 .99869 
 
 3.51 
 
 .99978 
 
 2.02 
 
 .97831 
 
 2.52 
 
 .99413 
 
 3.02 
 
 .99874 
 
 3.52 
 
 .99978 
 
 2.03 
 
 .97882 
 
 2.53 
 
 .99430 
 
 3.03 
 
 .99878 
 
 3.53 
 
 .99979 
 
 2.04 
 
 .97933 
 
 2.54 
 
 .99446 
 
 3.04 
 
 .99882 
 
 3.54 
 
 .99980 
 
 2.05 
 
 .97982 
 
 2.55 
 
 .99461 
 
 3.05 
 
 .99886 
 
 3.55 
 
 .99981 
 
 2.06 
 
 .98030 
 
 2.56 
 
 .99477 
 
 3.06 
 
 .99889 
 
 3.56 
 
 .99982 
 
 2.07 
 
 .98077 
 
 2.57 
 
 .99492 
 
 3.07 
 
 .99893 
 
 3.57 
 
 .99982 
 
 2.08 
 
 .98124 
 
 2.58 
 
 .99506 
 
 3.08 
 
 .99897 
 
 3.58 
 
 . 99983 
 
 2.09 
 
 .98169 
 
 2.59 
 
 .99520 
 
 3.09 
 
 .99900 
 
 3.59 
 
 .99984 
 
 2.10 
 
 .98214 
 
 2.60 
 
 .99534 
 
 3.10 
 
 .99903 
 
 3.60 
 
 .99984 
 
 2.11 
 
 .98257 
 
 2.61 
 
 .99547 
 
 3.11 
 
 .99907 
 
 3.61 
 
 .99985 - 
 
 2.12 
 
 .98300 
 
 2.62 
 
 .99560 
 
 3.12 
 
 .99910 
 
 3.62 
 
 .99985 + 
 
 2.13 
 
 .98341 
 
 2.63 
 
 .99573 
 
 3.13 
 
 .99913 
 
 3.63 
 
 .99986 
 
 2.14 
 
 .98382 
 
 2.64 
 
 .99586 
 
 3.14 
 
 .99916 
 
 3.64 
 
 .99986 
 
 2.15 
 
 .98422 
 
 2.65 
 
 .99598 
 
 3.15 
 
 .99918 
 
 3.65 
 
 .99987 
 
 2.16 
 
 .98461 
 
 2.66 
 
 .99609 
 
 3.16 
 
 .99921 
 
 3.66 
 
 .99987 
 
 2.17 
 
 . 98500 
 
 2.67 
 
 .99621 
 
 3.17 
 
 .99924 
 
 3.67 
 
 .99988 
 
 2.18 
 
 .98537 
 
 2.68 
 
 . 99632 
 
 3.18 
 
 .99926 
 
 3.68 
 
 .99988 
 
 2.19 
 
 .98574 
 
 2.69 
 
 .99643 
 
 3.19 
 
 .99929 
 
 3.69 
 
 . 99989 
 
 2.20 
 
 .98610 
 
 2.70 
 
 .99653 
 
 3.20 
 
 .99931 
 
 3.70 
 
 .99989 
 
 2.21 
 
 .98645- 
 
 2.71 
 
 .99664 
 
 3.21 
 
 .99934 
 
 3.71 
 
 .99990 
 
 2.22 
 
 .98679 
 
 2.72 
 
 .99674 
 
 3.22 
 
 .99936 
 
 3.72 
 
 .99990 
 
 2.23 
 
 .98713 
 
 2.73 
 
 .99683 
 
 3.23 
 
 .99938 
 
 3.73 
 
 .99990 
 
 2.24 
 
 .98746 
 
 2.74 
 
 .99693 
 
 3.24 
 
 . 99940 
 
 3.74 
 
 .99991 
 
 2.25 
 
 .98778 
 
 2.75 
 
 .99702 
 
 3.25 
 
 .99942 
 
 3.75 
 
 .99991 
 
 2.26 
 
 .98809 
 
 2.76 
 
 .99711 
 
 3.26 
 
 .99944 
 
 3.76 
 
 .99992 
 
 2.27 
 
 .98840 
 
 2.77 
 
 .99720 
 
 3.27 
 
 .99946 
 
 3.77 
 
 .99992 
 
 2.28 
 
 .98870 
 
 2.78 
 
 .99728 
 
 3.28 
 
 .99948 
 
 3.78 
 
 .99992 
 
 2.29 
 
 .98899 
 
 2.79 
 
 .99737 
 
 3.29 
 
 .99950 
 
 3.79 
 
 .99993 
 
 2.30 
 
 .98928 
 
 2.80 
 
 .99745- 
 
 3.30 
 
 .99952 
 
 3.80 
 
 .99993 
 
 2.31 
 
 .98956 
 
 2.81 
 
 .99752 
 
 3.31 
 
 .99953 
 
 3.81 
 
 .99993 
 
 2.32 
 
 .98983 
 
 2.82 
 
 .99760 
 
 3.32 
 
 .99955 + 
 
 3.82 
 
 .99993 
 
 2.33 
 
 .99010 
 
 2.83 
 
 .99767 
 
 3.33 
 
 .99957 
 
 3.83 
 
 .99994 
 
 2.34 
 
 .99036 
 
 2.84 
 
 .99774 
 
 3.34 
 
 .99958 
 
 3.84 
 
 .99994 
 
 2.35 
 
 .99061 
 
 2.85 
 
 .99781 
 
 3.35 
 
 .99960 
 
 3.85 
 
 .99994 
 
 2.36 
 
 .99086 
 
 2.86 
 
 .99788 
 
 3.36 
 
 .99961 
 
 3.86 
 
 .99994 
 
 2.37 
 
 .99111 
 
 2.87 
 
 .99795 - 
 
 3.37 
 
 .99962 
 
 3.87 
 
 .99995- 
 
 2.38 
 
 .99134 
 
 2.SS 
 
 .99801 
 
 3.38 
 
 .99964 
 
 3.88 
 
 .99995- 
 
 2.39 
 
 .99158 
 
 2.89 
 
 .99807 
 
 3.39 
 
 .99965 + 
 
 3.89 
 
 .99995 + 
 
 2.40 
 
 .99180 
 
 2.90 
 
 .99813 
 
 3.40 
 
 .99966 
 
 3.90 
 
 .99995 + 
 
 2.41 
 
 . 99202 
 
 2.91 
 
 .99819 
 
 3.41 
 
 .99967 
 
 3.91 
 
 .99995 + 
 
 2.42 
 
 .99224 
 
 2.92 
 
 .99825 + 
 
 3.42 
 
 .99969 
 
 3.92 
 
 .99996 
 
 2.43 
 
 .99245 + 
 
 2.93 
 
 .99831 
 
 3.43 
 
 .99970 
 
 3.93 
 
 .99996 
 
 2.44 
 
 .99266 
 
 2.94 
 
 .99836 
 
 3.44 
 
 .99971 
 
 3.94 
 
 .99996 
 
 2.45 
 
 . 99286 
 
 2.95 
 
 .99841 
 
 3.45 
 
 .99972 
 
 3.95 
 
 .99996 
 
 2.46 
 
 .99305 + 
 
 2.96 
 
 .99846 
 
 3.46 
 
 . 99973 
 
 3.96 
 
 .99996 
 
 2.47 
 
 .99324 
 
 2.97 
 
 .99851 
 
 3.47 
 
 .99974 
 
 3.97 
 
 .99996 
 
 2.48 
 
 . 99343 
 
 2.98 
 
 .99856 
 
 3.48 
 
 .99975- 
 
 3.98 
 
 .99997 
 
 2.49 
 
 .99361 
 
 2.99 
 
 .99861 
 
 3.49 
 
 .99976 
 
 3.99 
 
 4.00 
 
 4.50 
 
 .99997 
 
 .99997 
 
 .999997 
 
286 
 
 TABLES 
 
 TABLE II 
 
 Ordinates of the Normal Curve for Values of t 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 -4.00 
 
 .00013 
 
 
 
 
 
 
 
 -3.99 
 
 .00014 
 
 -3.49 
 
 .00090 
 
 -2.99 
 
 .00457 
 
 -2.49 
 
 .01797 
 
 -3.98 
 
 .00014 
 
 -3.48 
 
 .00094 
 
 -2.98 
 
 .00471 
 
 -2.48 
 
 .01823 
 
 -3.97 
 
 .00015 + 
 
 -3.47 
 
 .00097 
 
 -2.97 
 
 .00485- 
 
 -2.47 
 
 .01889 
 
 -3.96 
 
 .00016 
 
 -3.46 
 
 .00100 
 
 -2.96 
 
 .00499 
 
 -2.46 
 
 .01936 
 
 -3.95 
 
 .00016 
 
 -3.45 
 
 .00104 
 
 -2.95 
 
 .00514 
 
 -2.45 
 
 .01984 
 
 -3.94 
 
 .00017 
 
 -3.44 
 
 .00108 
 
 -2.94 
 
 .00530 
 
 -2.44 
 
 .02033 
 
 -3.93 
 
 .00018 
 
 -3.43 
 
 .00111 
 
 -2.93 
 
 .00545 + 
 
 -2.43 
 
 .02083 
 
 -3.92 
 
 .00018 
 
 -3.42 
 
 .00115 + 
 
 -2.92 
 
 .00562 
 
 -2.42 
 
 .02134 
 
 -3.91 
 
 .00019 
 
 -3.41 
 
 .00119 
 
 -2.91 
 
 .00578 + 
 
 -2.41 
 
 .02186 
 
 -3.90 
 
 .00020 
 
 -3.40 
 
 .00123 
 
 -2.90 
 
 .00595 + 
 
 -2.40 
 
 .02240 
 
 -3.89 
 
 .00021 
 
 -3.39 
 
 .00128 
 
 -2.89 
 
 .00613 
 
 -2.39 
 
 .02294 
 
 -3.88 
 
 .00022 
 
 -3.38 
 
 .00132 
 
 -2.88 
 
 .00631 
 
 -2.38 
 
 .02349 
 
 -3.87 
 
 .00022 
 
 -3.37 
 
 .00136 
 
 -2.87 
 
 .00649 
 
 -2.37 
 
 .02406 
 
 -3.86 
 
 .00023 
 
 -3.36 
 
 .00141 
 
 -2.86 
 
 .00668 
 
 -2.36 
 
 .02463 
 
 -3.85 
 
 .00024 
 
 -3.35 
 
 .00146 
 
 -2.85 
 
 .00687 
 
 -2.35 
 
 .02522 
 
 -3.84 
 
 .00025 + 
 
 -3.34 
 
 .00151 
 
 -2.84 
 
 .00707 
 
 -2.34 
 
 .02582 
 
 -3.83 
 
 .00026 
 
 -3.33 
 
 .00156 
 
 -2.83 
 
 .00727 
 
 -2.33 
 
 .02643 
 
 -3.82 
 
 .00027 
 
 -3.32 
 
 .00161 
 
 -2.82 
 
 .00748 
 
 -2.32 
 
 .02705- 
 
 -3.81 
 
 .00028 
 
 -3.31 
 
 .00167 
 
 -2.81 
 
 .00770 
 
 -2.31 
 
 .02768 
 
 -3.80 
 
 .00029 
 
 -3.30 
 
 .00172 
 
 -2.80 
 
 .00792 
 
 -2.30 
 
 .02833 
 
 -3.79 
 
 .00030 
 
 -3.29 
 
 .00178 
 
 -2.79 
 
 .00S14 
 
 -2.29 
 
 .02899 
 
 -3.78 
 
 .00032 
 
 -3.28 
 
 .00184 
 
 -2.78 
 
 .00837 
 
 -2.28 
 
 .02966 
 
 -3.77 
 
 .00033 
 
 -3.27 
 
 .00190 
 
 -2.77 
 
 .00861 
 
 -2.27 
 
 .03034 
 
 -3.76 
 
 .00034 
 
 -3.26 
 
 .00196 
 
 -2.76 
 
 .00885- 
 
 -2.26 
 
 .03103 
 
 -3.75 
 
 .00035 + 
 
 -3.25 
 
 .00203 
 
 -2.75 
 
 .00909 
 
 -2.25 
 
 .03174 
 
 -3.74 
 
 .00037 
 
 -3.24 
 
 .00210 
 
 -2.74 
 
 .00935- 
 
 -2.24 
 
 .03246 
 
 -3.73 
 
 .00038 
 
 -3.23 
 
 .00217 
 
 -2.73 
 
 .00961 
 
 -2.23 
 
 .03319 
 
 -3.72 
 
 .00039 
 
 -3.22 
 
 .00224 
 
 -2.72 
 
 .00987 
 
 -2.22 
 
 .03394 
 
 -3.71 
 
 .00041 
 
 -3.21 
 
 .00231 
 
 -2.71 
 
 .01014 
 
 -2.21 
 
 .03470 
 
 -3.70 
 
 .00043 
 
 -3.20 
 
 .00238 
 
 -2.70 
 
 .01042 
 
 -2.20 
 
 .03548 
 
 -3.69 
 
 .00044 
 
 -3.19 
 
 .00246 
 
 -2.69 
 
 .01071 
 
 -2.19 
 
 .03626 
 
 -3.68 
 
 .00046 
 
 -3.18 
 
 .00254 
 
 -2.68 
 
 .01100 
 
 -2.18 
 
 .03706 
 
 -3.67 
 
 .00047 
 
 — 3 17 
 
 .00262 
 
 -2.67 
 
 .01130 
 
 -2.17 
 
 .03789 
 
 -3 66 
 
 .00049 
 
 -3.16 
 
 .00271 
 
 -2.66 
 
 .01160 
 
 -2 16 
 
 .03871 
 
 -3.65 
 
 .00051 
 
 -3.15 
 
 .00279 
 
 -2 65 
 
 .01191 
 
 -2.15 
 
 .03955 + 
 
 -3.64 
 
 .00053 
 
 -3.14 
 
 .00288 
 
 -2.64 
 
 01223 
 
 -2.14 
 
 .04041 
 
 -3.63 
 
 .00055- 
 
 -3.13 
 
 .00298 
 
 -2.63 
 
 .01256 
 
 -2.13 
 
 .04128 
 
 -3.62 
 
 .00057 
 
 -3.12 
 
 .00307 
 
 -2.62 
 
 .01289 
 
 -2.12 
 
 .04217 
 
 -3.61 
 
 .00059 
 
 -3.11 
 
 .00317 
 
 -2.61 
 
 .01323 
 
 -2 11 
 
 .04307 
 
 -3.60 
 
 .00061 
 
 -3.10 
 
 .00327 
 
 -2.60 
 
 .01358 
 
 -2.10 
 
 .04398 
 
 -3 59 
 
 .00063 
 
 -3.09 
 
 .00337 
 
 -2.59 
 
 .01394 
 
 -2 09 
 
 .04492 
 
 -3.58 
 
 .00066 
 
 -3.08 
 
 .00348 
 
 -2.58 
 
 .01431 
 
 -2.08 
 
 .04586 
 
 -3.57 
 
 .00068 
 
 -3 07 
 
 .00358 
 
 -2.57 
 
 .01468 
 
 -2.07 
 
 .04682 
 
 -3.56 
 
 .00071 
 
 -3.06 
 
 .00370 
 
 -2.56 
 
 .01506 
 
 -2.06 
 
 .04780 
 
 -3.55 
 
 .00073 
 
 -3.05 
 
 .00381 
 
 -2.55 
 
 .01545- 
 
 -2.05 
 
 .04879 
 
 -3.54 
 
 .00076 
 
 -3.04 
 
 .00393 
 
 -2.54 
 
 .01585 — 
 
 -2.04 
 
 .04980 
 
 -3.53 
 
 .00079 
 
 -3.03 
 
 .00405- 
 
 -2.53 
 
 .01625 + 
 
 -2.03 
 
 .05082 
 
 -3.52 
 
 .000S1 
 
 -3.02 
 
 .00417 
 
 -2.52 
 
 .01667 
 
 -2 02 
 
 .05186 
 
 -3.51 
 
 .00084 
 
 -3.01 
 
 .00430 
 
 -2.51 
 
 .01710 
 
 -2.01 
 
 .05292 
 
 -3.50 
 
 .00087 
 
 -3.00 
 
 .00443 
 
 -2.50 
 
 .01753 
 
 -2.00 
 
 .05399 
 
TABLES 
 
 287 
 
 TABLE II—( Continued) 
 
 Ordinates of the Normal Curve for Values of t 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 -1.99 
 
 .05508 
 
 -1.49 
 
 .13147 
 
 — 
 
 .99 
 
 .24439 
 
 — 
 
 .49 
 
 .35381 
 
 -1.98 
 
 .05618 
 
 -1.48 
 
 .13344 
 
 — 
 
 .98 
 
 .24681 
 
 — 
 
 .48 
 
 .35553 
 
 -1.97 
 
 .05730 
 
 -1.47 
 
 .13542 
 
 — 
 
 .97 
 
 .24923 
 
 — 
 
 .47 
 
 .35723 
 
 -1.96 
 
 .05844 
 
 -1.46 
 
 .13742 
 
 — 
 
 .96 
 
 .25164 
 
 — 
 
 .46 
 
 .35889 
 
 -1.95 
 
 .05960 
 
 -1.45 
 
 .13943 
 
 — 
 
 .95 
 
 .25406 
 
 — 
 
 .45 
 
 .36053 
 
 -1.94 
 
 .06077 
 
 -1.44 
 
 .14146 
 
 — 
 
 .94 
 
 .25647 
 
 — 
 
 .44 
 
 .36124 
 
 -1.93 
 
 .06195 + 
 
 -1.43 
 
 .14351 
 
 — 
 
 .93 
 
 .25888 
 
 — 
 
 .43 
 
 .36371 
 
 -1.92 
 
 .06316 
 
 -1.42 
 
 .14556 
 
 — 
 
 .92 
 
 .26129 
 
 — 
 
 .42 
 
 .36526 
 
 -1.91 
 
 .06439 
 
 -1.41 
 
 .14764 
 
 — 
 
 .91 
 
 .26369 
 
 — 
 
 .41 
 
 .36678 
 
 -1.90 
 
 .06562 
 
 -1.40 
 
 .14973 
 
 — 
 
 .90 
 
 .26609 
 
 
 .40 
 
 .36827 
 
 -1.89 
 
 .06687 
 
 -1.39 
 
 .15183 
 
 — 
 
 .89 
 
 .26848 
 
 — 
 
 .39 
 
 .36973 
 
 -1.88 
 
 .06814 
 
 -1.38 
 
 .15395- 
 
 — 
 
 .88 
 
 .27086 
 
 — 
 
 .38 
 
 .37115 + 
 
 -1.87 
 
 .06943 
 
 -1.37 
 
 .15608 
 
 — 
 
 .87 
 
 .27324 
 
 — 
 
 .37 
 
 .37255- 
 
 -1.86 
 
 .07074 
 
 -1.36 
 
 .15823 
 
 — 
 
 .86 
 
 .27562 
 
 — 
 
 .36 
 
 .37391 
 
 -1.85 
 
 .07206 
 
 -1.35 
 
 .16038 
 
 — 
 
 .85 
 
 .27799 
 
 — 
 
 .35 
 
 . 37524 
 
 -1.84 
 
 .07341 
 
 -1.34 
 
 .16256 
 
 — 
 
 .84 
 
 .28034 
 
 — 
 
 .34 
 
 .37654 
 
 -1.83 
 
 .07477 
 
 -1.33 
 
 .16474 
 
 — 
 
 .83 
 
 .28269 
 
 — 
 
 .33 
 
 .37780 
 
 -1.82 
 
 .07614 
 
 -1.32 
 
 .16694 
 
 — 
 
 .82 
 
 .28504 
 
 — 
 
 .32 
 
 .37903 
 
 -1.81 
 
 .07754 
 
 -1.31 
 
 .16915- 
 
 — 
 
 .81 
 
 .28737 
 
 — 
 
 .31 
 
 .38023 
 
 -1.80 
 
 .07895 + 
 
 -1.30 
 
 .17137 
 
 — 
 
 .80 
 
 .28969 
 
 — 
 
 .30 
 
 .38139 
 
 -1.79 
 
 .08038 
 
 -1.29 
 
 .17360 
 
 — 
 
 .79 
 
 .29200 
 
 — 
 
 .29 
 
 .38252 
 
 -1.78 
 
 .08183 
 
 -1.28 
 
 .17585- 
 
 — 
 
 .78 
 
 .29431 
 
 — 
 
 .28 
 
 .38361 
 
 -1.77 
 
 .08329 
 
 -1.27 
 
 .17810 
 
 — 
 
 .77 
 
 .29660 
 
 — 
 
 .27 
 
 .38466 
 
 -1.76 
 
 .08478 
 
 -1.26 
 
 .18037 
 
 — 
 
 .76 
 
 .29887 
 
 — 
 
 .26 
 
 .38568 
 
 — 1.75 
 
 .08628 
 
 -1.25 
 
 .18265- 
 
 
 .75 
 
 .30114 
 
 — 
 
 .25 
 
 . 38667 
 
 -1.74 
 
 .08780 
 
 -1.24 
 
 . 18494 
 
 — 
 
 .74 
 
 .30339 
 
 — 
 
 .24 
 
 .38762 
 
 -1.73 
 
 .08933 
 
 -1.23 
 
 .18724 
 
 — 
 
 .73 
 
 .30563 
 
 — 
 
 .23 
 
 .38853 
 
 -1.72 
 
 .09089 
 
 -1.22 
 
 . 18954 
 
 — 
 
 .72 
 
 .30785 + 
 
 — 
 
 .22 
 
 .38940 
 
 -1.71 
 
 .09246 
 
 -1.21 
 
 .19186 
 
 — 
 
 .71 
 
 .31006 
 
 — 
 
 .21 
 
 .39024 
 
 -1.70 
 
 .09405- 
 
 -1.20 
 
 .19419 
 
 — 
 
 .70 
 
 .31225 + 
 
 — 
 
 .20 
 
 .39104 
 
 -1.69 
 
 .09566 
 
 -1.19 
 
 19652 
 
 — 
 
 .69 
 
 .31443 
 
 _ 
 
 .19 
 
 .39181 
 
 -1.68 
 
 .09728 
 
 -1.18 
 
 .19886 
 
 — 
 
 .68 
 
 .31659 
 
 — 
 
 .18 
 
 .39253 
 
 -1.67 
 
 .09893 
 
 -1.17 
 
 .20121 
 
 — 
 
 .67 
 
 .31874 
 
 — 
 
 .17 
 
 .39322 
 
 -1.66 
 
 .10059 
 
 -1.16 
 
 .20357 
 
 — 
 
 .66 
 
 .32086 
 
 — 
 
 .16 
 
 .39387 
 
 -1.65 
 
 .10227 
 
 -1.15 
 
 .20594 
 
 — 
 
 .65 
 
 .32297 
 
 — 
 
 .15 
 
 .39448 
 
 -1.64 
 
 .10396 
 
 — 1.14 
 
 .20831 
 
 — 
 
 .64 
 
 .32506 
 
 — 
 
 .14 
 
 .39505 + 
 
 -1.63 
 
 . 10568 
 
 -1.13 
 
 .21069 
 
 — 
 
 .63 
 
 .32713 
 
 — 
 
 .13 
 
 .39559 
 
 -1.62 
 
 .10741 
 
 — 1.12 
 
 .21307 
 
 — 
 
 .62 
 
 .32918 
 
 — 
 
 .12 
 
 .39608 
 
 -1.61 
 
 .10916 
 
 — 1.11 
 
 .21546 
 
 — 
 
 .61 
 
 .33122 
 
 — 
 
 .11 
 
 .39654 
 
 -1.60 
 
 .11092 
 
 -1.10 
 
 .21785 + 
 
 — 
 
 .60 
 
 .33323 
 
 — 
 
 .10 
 
 .39695 + 
 
 -1.59 
 
 .11270 
 
 -1.09 
 
 .22025 + 
 
 — 
 
 .59 
 
 .33521 
 
 — 
 
 .09 
 
 .39733 
 
 -1.58 
 
 .11451 
 
 -1.08 
 
 .22265 + 
 
 — 
 
 .58 
 
 .33718 
 
 — 
 
 .08 
 
 .39767 
 
 -1.57 
 
 .11632 
 
 -1.07 
 
 . 22506 
 
 — 
 
 .57 
 
 .33912 
 
 — 
 
 .07 
 
 .39797 
 
 -1.56 
 
 .11816 
 
 -1.06 
 
 .22747 
 
 — 
 
 .56 
 
 .34105- 
 
 — 
 
 .06 
 
 .39823 
 
 -1.55 
 
 .12001 
 
 -1.05 
 
 .22988 
 
 — 
 
 .55 
 
 .34294 
 
 — 
 
 .05 
 
 .39844 
 
 -1.54 
 
 .12188 
 
 -1.04 
 
 .23230 
 
 — 
 
 .54 
 
 .34482 
 
 — 
 
 .04 
 
 .39862 
 
 - 1.53 
 
 .12376 
 
 -1.03 
 
 .23471 
 
 — 
 
 .53 
 
 .34667 
 
 — 
 
 .03 
 
 .39876 
 
 -1.52 
 
 .12567 
 
 - 1.02 
 
 .23713 
 
 — 
 
 .52 
 
 .34849 
 
 — 
 
 .02 
 
 .39886 
 
 -1.51 
 
 .12758 
 
 -1.01 
 
 .23955 + 
 
 — 
 
 .51 
 
 .35029 
 
 — 
 
 .01 
 
 .39892 
 
 -1.50 
 
 .12952 
 
 -1.00 
 
 .24197 
 
 
 .50 
 
 .35207 
 
 
 .00 
 
 .39894 
 
288 
 
 TABLES 
 
 TABLE II—( Continued ) 
 
 Ordinates of the Normal Curve for Values of t 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 .00 
 
 .39894 
 
 .50 
 
 .35207 
 
 1 00 
 
 .24197 
 
 1.50 
 
 . 12952 
 
 .01 
 
 .39S92 
 
 .51 
 
 . 35029 
 
 1.01 
 
 .23955 + 
 
 1.51 
 
 .12758 
 
 .02 
 
 39886 
 
 .52 
 
 .34849 
 
 1 02 
 
 .23713 
 
 1.52 
 
 .12567 
 
 .03 
 
 .39876 
 
 .53 
 
 .34667 
 
 1.03 
 
 .23471 
 
 1.53 
 
 . 12376 
 
 .04 
 
 .39862 
 
 .54 
 
 .344S2 
 
 1.04 
 
 .23230 
 
 1.54 
 
 .12188 
 
 .05 
 
 .39844 
 
 .55 
 
 .34294 
 
 1.05 
 
 .22988 
 
 1.55 
 
 .12001 
 
 .06 
 
 .39823 
 
 .56 
 
 .34105- 
 
 1.06 
 
 .22747 
 
 1.56 
 
 .11816 
 
 .07 
 
 .39797 
 
 57 
 
 .33912 
 
 1.07 
 
 .22506 
 
 1.57 
 
 .11632 
 
 .08 
 
 .39767 
 
 .58 
 
 .33718 
 
 1.08 
 
 .22265 + 
 
 1.58 
 
 .11451 
 
 .09 
 
 .39733 
 
 .59 
 
 .33521 
 
 1 09 
 
 .22025 + 
 
 1.59 
 
 .11270 
 
 .10 
 
 .39695 + 
 
 .60 
 
 .33323 
 
 1.10 
 
 .21785 + 
 
 1.60 
 
 .11092 
 
 .11 
 
 .39654 
 
 .61 
 
 .33122 
 
 111 
 
 .21546 
 
 1.61 
 
 .10916 
 
 .12 
 
 .39608 
 
 .62 
 
 .32918 
 
 1.12 
 
 .21307 
 
 1 62 
 
 .10741 
 
 .13 
 
 .39559 
 
 .63 
 
 .32713 
 
 1 13 
 
 .21069 
 
 1.63 
 
 .10568 
 
 .14 
 
 . 39o05 
 
 .64 
 
 .32506 
 
 1.14 
 
 .20830 
 
 1.64 
 
 .10396 
 
 . 15 
 
 .39448 
 
 .65 
 
 .32297 
 
 1.15 
 
 .20594 
 
 1.65 
 
 .10227 
 
 .16 
 
 .39387 
 
 .66 
 
 .32086 
 
 1.16 
 
 .20357 
 
 1 66 
 
 .10059 
 
 .17 
 
 .39322 
 
 .67 
 
 .31874 
 
 1.17 
 
 .20121 
 
 1.67 
 
 .09893 
 
 .18 
 
 .39253 
 
 .68 
 
 .31659 
 
 1.18 
 
 .19886 
 
 1.68 
 
 .09728 
 
 .19 
 
 .39181 
 
 .69 
 
 .31443 
 
 1.19 
 
 .19652 
 
 1.69 
 
 .09566 
 
 .20 
 
 .39104 
 
 .70 
 
 .31225 + 
 
 1 20 
 
 .19419 
 
 1.70 
 
 .09405- 
 
 .21 
 
 .39024 
 
 .71 
 
 .31006 
 
 1.21 
 
 .19186 
 
 1.71 
 
 .09246 
 
 .22 
 
 .38940 
 
 .72 
 
 .30785 + 
 
 1.22 
 
 .18954 
 
 1.72 
 
 .09089 
 
 .23 
 
 .38853 
 
 .73 
 
 .30563 
 
 1.23 
 
 . 18724 
 
 1.73 
 
 .08933 
 
 .24 
 
 .38762 
 
 .74 
 
 .30339 
 
 1.24 
 
 .18494 
 
 1.74 
 
 .08780 
 
 .25 
 
 .38667 
 
 .75 
 
 .30114 
 
 1.25 
 
 .18265- 
 
 1.75 
 
 .08628 
 
 .26 
 
 .38568 
 
 .76 
 
 .29887 
 
 1.26 
 
 .18037 
 
 1.76 
 
 .08478 
 
 .27 
 
 .38466 
 
 . i 7 
 
 .29060 
 
 1.27 
 
 .17S10 
 
 1.77 
 
 .08329 
 
 .28 
 
 .38361 
 
 .78 
 
 .29431 
 
 1.28 
 
 .17485- 
 
 1.78 
 
 .08183 
 
 .29 
 
 .38252 
 
 .79 
 
 .29200 
 
 1.29 
 
 .17360 
 
 1.79 
 
 .08038 
 
 .30 
 
 .38139 
 
 SO 
 
 .28969 
 
 1.30 
 
 .17137 
 
 ICO 
 
 .07S95 + 
 
 .31 
 
 .38023 
 
 .81 
 
 .28737 
 
 1.31 
 
 .16915- 
 
 1 Cl 
 
 .07754 
 
 .32 
 
 .37903 
 
 .82 
 
 .28504 
 
 1.32 
 
 .16694 
 
 1.82 
 
 .07614 
 
 .33 
 
 .37780 
 
 .83 
 
 .28269 
 
 1 33 
 
 .16474 
 
 1.83 
 
 .07477 
 
 .34 
 
 .37654 
 
 .84 
 
 .28034 
 
 1.34 
 
 .16256 
 
 1.84 
 
 .07341 
 
 .35 
 
 .37524 
 
 .85 
 
 .27799 
 
 1.35 
 
 .16038 
 
 1.85 
 
 .07207 
 
 .36 
 
 .37391 
 
 .86 
 
 .27562 
 
 1.36 
 
 .15823 
 
 1.C6 
 
 .07074 
 
 .37 
 
 .37255- 
 
 .87 
 
 .27324 
 
 1.37 
 
 .15608 
 
 1.87 
 
 .06943 
 
 .38 
 
 .37115 + 
 
 .88 
 
 .27086 
 
 1.38 
 
 .15395- 
 
 1.88 
 
 .06814 
 
 .39 
 
 .36973 
 
 .89 
 
 .26848 
 
 1.39 
 
 .15183 
 
 1.89 
 
 .06687 
 
 .40 
 
 .36827 
 
 .90 
 
 .26609 
 
 1.40 
 
 .14973 
 
 1.90 
 
 .06562 
 
 .41 
 
 .36678 
 
 .91 
 
 .26369 
 
 1.41 
 
 .14764 
 
 1.91 
 
 .06438 
 
 .42 
 
 .36526 
 
 .92 
 
 .26129 
 
 1.42 
 
 .14556 
 
 1.92 
 
 .06316 
 
 .43 
 
 .36371 
 
 .93 
 
 .258S8 
 
 1.43 
 
 .14351 
 
 1.93 
 
 .06195 + 
 
 .44 
 
 .36214 
 
 .94 
 
 .25647 
 
 1.44 
 
 .14146 
 
 1 94 
 
 .06077 
 
 .45 
 
 .36053 
 
 .95 
 
 .25406 
 
 1.45 
 
 .13943 
 
 1.95 
 
 .05960 
 
 .46 
 
 .35889 
 
 .96 
 
 .25164 
 
 1.46 
 
 .13742 
 
 1.96 
 
 .05844 
 
 .47 
 
 35723 
 
 .97 
 
 .24923 
 
 1.47 
 
 .13542 
 
 1.97 
 
 05730 
 
 .48 
 
 .35553 
 
 .98 
 
 .24681 
 
 1.48 
 
 .13344 
 
 1.98 
 
 .05618 
 
 .49 
 
 .35381 
 
 .99 
 
 .24439 
 
 1.49 
 
 .13147 
 
 1.99 
 
 .05508 
 
TABLES 
 
 289 
 
 TABLE II — ( Continued) 
 
 Ordinates of the Normal Curve for Values of t 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 t 
 
 Ordinate 
 
 2.00 
 
 .05399 
 
 2.50 
 
 .01753 
 
 3.00 
 
 .00443 
 
 3.50 
 
 .00087 
 
 2.01 
 
 .05292 
 
 2.51 
 
 .01710 
 
 3.01 
 
 .00430 
 
 3.51 
 
 .00084 
 
 2.02 
 
 .05186 
 
 2.52 
 
 .01667 
 
 3.02 
 
 .00417 
 
 3.52 
 
 .00081 
 
 2.03 
 
 .050S2 
 
 2.53 
 
 .01625 + 
 
 3.03 
 
 .00405 - 
 
 3.53 
 
 .00079 
 
 2.04 
 
 .04980 
 
 2.54 
 
 .01585- 
 
 3.04 
 
 .00393 
 
 3.54 
 
 .00076 
 
 2.05 
 
 .04879 
 
 2.55 
 
 .01545- 
 
 3.05 
 
 .00381 
 
 3.55 
 
 .00073 
 
 2.06 
 
 .04780 
 
 2.56 
 
 .01506 
 
 3.06 
 
 .00370 
 
 3.56 
 
 .00071 
 
 2.07 
 
 .04682 
 
 2.57 
 
 .01468 
 
 3.07 
 
 .00358 
 
 3.57 
 
 .00068 
 
 2.08 
 
 .04586 
 
 2.58 
 
 .01431 
 
 3.08 
 
 .00348 
 
 3.58 
 
 .00066 
 
 2.09 
 
 .04492 
 
 2.59 
 
 .01394 
 
 3.09 
 
 .00337 
 
 3.59 
 
 .00063 
 
 2.10 
 
 .04398 
 
 2.60 
 
 .01358 
 
 3.10 
 
 .00327 
 
 3.60 
 
 .00061 
 
 2.11 
 
 .04307 
 
 2.61 
 
 .01323 
 
 3.11 
 
 .00317 
 
 3.61 
 
 .00059 
 
 2.12 
 
 .04217 
 
 2.62 
 
 .01289 
 
 3.12 
 
 .00307 
 
 3.62 
 
 .00057 
 
 2.13 
 
 .04128 
 
 2.63 
 
 .01256 
 
 3.13 
 
 .00298 
 
 3.63 
 
 .00055- 
 
 2.14 
 
 .04041 
 
 2.64 
 
 .01223 
 
 3.14 
 
 .00288 
 
 3.64 
 
 .00053 
 
 2.15 
 
 .03955 + 
 
 2.65 
 
 .01191 
 
 3.15 
 
 .00279 
 
 3.65 
 
 .00051 
 
 2.16 
 
 .03871 
 
 2.66 
 
 .01160 
 
 3.16 
 
 .00271 
 
 3.66 
 
 .00049 
 
 2.17 
 
 .03788 
 
 2.67 
 
 .01130 
 
 3.17 
 
 .00262 
 
 3.67 
 
 .00047 
 
 2.18 
 
 .03706 
 
 2.68 
 
 .01100 
 
 3.18 
 
 .00254 
 
 3.68 
 
 .00046 
 
 2.19 
 
 .03626 
 
 2.69 
 
 .01071 
 
 3.19 
 
 .00246 
 
 3.69 
 
 .00044 
 
 2.20 
 
 .03548 
 
 2.70 
 
 .01042 
 
 3.20 
 
 .00238 
 
 3.70 
 
 .00043 
 
 2.21 
 
 .03470 
 
 2.71 
 
 .01014 
 
 3.21 
 
 .00231 
 
 3.71 
 
 .00041 
 
 2.22 
 
 .03394 
 
 2.72 
 
 .00987 
 
 3.22 
 
 .00224 
 
 3.72 
 
 .00039 
 
 2.23 
 
 .03319 
 
 2.73 
 
 .00961 
 
 3.23 
 
 .00217 
 
 3.73 
 
 .00038 
 
 2.24 
 
 .03246 
 
 2.74 
 
 .00935- 
 
 3.24 
 
 .00210 
 
 3.74 
 
 .00037 
 
 2.25 
 
 .03174 
 
 2.75 
 
 .00909 
 
 3.25 
 
 .00203 
 
 3.75 
 
 .00035+ 
 
 2.26 
 
 .03103 
 
 2.76 
 
 .00S85 — 
 
 3.26 
 
 .00196 
 
 3.76 
 
 .00034 
 
 2.27 
 
 .03034 
 
 2.77 
 
 .00861 
 
 3.27 
 
 .00190 
 
 3.77 
 
 .00033 
 
 2.28 
 
 .02966 
 
 2.78 
 
 .00837 
 
 3.23 
 
 .00184 
 
 3.78 
 
 .00032 
 
 2.29 
 
 .02899 
 
 2.79 
 
 .00814 
 
 3.2D 
 
 .00178 
 
 3.79 
 
 .00030 
 
 2.30 
 
 .02833 
 
 2.80 
 
 .00792 
 
 3.30 
 
 .00172 
 
 3.SO 
 
 .00029 
 
 2.31 
 
 .02768 
 
 2.81 
 
 .00770 
 
 3.31 
 
 .00167 
 
 3.81 
 
 .00028 
 
 2.32 
 
 .02705- 
 
 2.82 
 
 .00748 
 
 3.32 
 
 .00161 
 
 3.82 
 
 .00027 
 
 2.33 
 
 .02643 
 
 2.83 
 
 .00727 
 
 3.33 
 
 .00156 
 
 3.83 
 
 .00026 
 
 2.34 
 
 .02582 
 
 2.84 
 
 .00707 
 
 3.34 
 
 .00151 
 
 3.84 
 
 .00025 + 
 
 2.35 
 
 .02522 
 
 2.85 
 
 .00687 
 
 3 35 
 
 .00146 
 
 3 85 
 
 .00024 
 
 2.30 
 
 .02463 
 
 2.86 
 
 .00668 
 
 3.36 
 
 .00141 
 
 3.86 
 
 .00023 
 
 2.37 
 
 .02406 
 
 2.87 
 
 .00649 
 
 3.37 
 
 .00136 
 
 3.87 
 
 .00022 
 
 2.38 
 
 .02349 
 
 2.88 
 
 .00631 
 
 3 38 
 
 .00132 
 
 3.88 
 
 .00022 
 
 2.39 
 
 .02294 
 
 2.89 
 
 .00613 
 
 3.39 
 
 .00128 
 
 3.89 
 
 .00021 
 
 2.40 
 
 .02240 
 
 2.90 
 
 .00595 + 
 
 3.40 
 
 .00123 
 
 3.90 
 
 .00020 
 
 2.41 
 
 .02186 
 
 2.91 
 
 .00578 
 
 3.41 
 
 .00119 
 
 3.91 
 
 .00019 
 
 2.42 
 
 .02134 
 
 2.92 
 
 .00562 
 
 3.42 
 
 .00115 + 
 
 92 
 
 .00018 
 
 2.43 
 
 .02083 
 
 2.93 
 
 .00545 + 
 
 3.43 
 
 .00111 
 
 3.93 
 
 .00018 
 
 2.44 
 
 .02033 
 
 2.94 
 
 .00530 
 
 3.44 
 
 .00108 
 
 3 94 
 
 .00017 
 
 2.45 
 
 .01984 
 
 2 95 
 
 .00514 
 
 3.45 
 
 .00104 
 
 3.95 
 
 .00016 
 
 2.46 
 
 .01936 
 
 2.96 
 
 .00499 
 
 3.46 
 
 .00100 
 
 3 96 
 
 .00016 
 
 2.47 
 
 .01889 
 
 2.97 
 
 .00485- 
 
 3.47 
 
 .00097 
 
 3 '.17 
 
 .00015 + 
 
 2.48 
 
 .01842 
 
 2.98 
 
 .00471 
 
 3.48 
 
 .00094 
 
 on 
 
 .00014 
 
 2.49 
 
 .01797 
 
 2.99 
 
 .00457 
 
 3.49 
 
 .00090 
 
 > 
 
 3.99 
 
 4.00 
 
 4.50 
 
 .00014 
 
 .00013 
 
 .00002 
 
Degrees of Freedom for Greater Mean Square 
 
 290 
 
 TABLES 
 
 o 
 
 12.706 
 
 63 667 
 
 4.303 
 
 9.926 
 
 3.182 
 
 6.841 
 
 2.776 
 
 4 604 
 
 2.571 
 
 4 032 
 
 2.447 
 
 3.707 
 
 2.365 
 
 3 499 
 
 2.306 
 
 3 356 
 
 2.262 
 
 3 250 
 
 2.228 
 
 3 169 
 
 2.201 
 
 3 106 
 
 8 
 
 ci oo 
 
 CO t* 
 
 © © 
 1-0 lO 
 
 CO CM 
 
 © iH 
 
 CO © 
 
 © ^ 
 
 © CM 
 
 CO © 
 
 OO 
 © co 
 
 CO © 
 CM © 
 
 X © 
 © CO 
 
 •— 1 iH 
 
 X 
 
 Tt- tH 
 © © 
 
 © © 
 
 Tf © 
 
 CD 
 © CD 
 Cl CO 
 CD 
 
 C5 05 
 r- C5 
 
 X © 
 CM 
 
 © CO 
 
 i-H 
 
 ^ © 
 
 CO © 
 
 co © 
 
 CM t* 
 
 CM 
 
 CM X 
 
 CM X 
 
 
 t* CD 
 © 
 
 LO CD 
 
 © 
 
 © © 
 
 D- CO 
 t- © 
 
 co t> 
 
 IO T* 
 
 ^ iH 
 
 x co 
 
 ■— t— 
 
 rr © 
 
 CM CO 
 CM 
 
 O X 
 © t- 
 
 ^ X 
 
 r- x 
 
 — CM 
 © © 
 
 CM 
 
 © ^ 
 CO 
 CM CM 
 CD 
 
 05 05 
 i-h 05 
 
 X © 
 CM 
 
 © CO 
 
 iH 
 
 rf © 
 
 CO c- 
 
 CO © 
 
 X © 
 
 CM t* 
 
 CM Ttt 
 
 CM t* 
 
 Cl 
 
 — CO 
 05 00 
 
 »-H CM 
 vt< 
 
 Tf lO 
 
 r- © 
 
 T-H t- 
 
 © co 
 
 X © 
 © OO 
 
 © CM 
 
 © c- 
 
 © ^ 
 
 X t— 
 
 CM © 
 
 th 
 
 O iH 
 
 ^ H 
 
 © c— 
 
 ©© 
 
 T* 
 
 
 CO © 
 
 -r o 
 
 Cl tH 
 
 © 
 
 05 05 
 i-H 05 
 
 x t- 
 
 CM 
 
 © 
 
 H 
 
 Tf © 
 
 t- 
 
 CO © 
 
 X © 
 
 X © 
 
 CM t* 
 
 CM Til 
 
 CO 
 
 05 ^ 
 X CO 
 
 t - CD 
 CO CO 
 
 a 
 
 x 
 
 Tf © 
 © 00 
 
 01 t- 
 X CM 
 
 LO © 
 
 ■— i—l 
 
 X ^ 
 t- CO 
 
 rt< X 
 
 Tj< O 
 
 X tr- 
 CM t}< 
 
 D- © 
 © © 
 
 © Til 
 
 © t- 
 
 X -rH 
 
 CO 00 
 Cl 05 
 © 
 
 05 05 
 — 05 
 
 X t- 
 CM 
 
 © ^ 
 
 H 
 
 © 
 
 iH 
 
 Tf co 
 
 CO © 
 
 X © 
 
 X © 
 
 X © 
 
 CM Til 
 
 o 
 
 233.97 
 
 6859 39 
 
 19.33 
 
 99 33 
 
 8.94 
 
 27 91 
 
 6.16 
 
 16.21 
 
 LO tr— 
 © © 
 
 r?- O 
 
 tH 
 
 4.28 
 
 8 47 
 
 3.87 
 
 7.19 
 
 3.58 
 
 6 37 
 
 3.37 
 
 5 80 
 
 3.22 
 
 6 39 
 
 3.09 
 
 6 07 
 
 © 
 
 I - 00 
 
 — o 
 
 © © 
 CO CO 
 
 — T* 
 
 © CM 
 
 © CM 
 
 CM © 
 
 LO C— 
 © © 
 
 © © 
 CO c— 
 
 L— © 
 © t* 
 
 © X 
 © © 
 
 X © 
 
 TT o 
 
 X 
 
 X © 
 
 © CM 
 
 CM X 
 
 O t* 
 CO CD 
 Cl tr¬ 
 io 
 
 © 05 
 r- 05 
 
 © © 
 CM 
 
 © © 
 H 
 
 LO o 
 
 H 
 
 rf CO 
 
 X t- 
 
 X © 
 
 X © 
 
 X © 
 
 X © 
 
 
 224.57 
 
 5625.14 
 
 19.25 
 
 99 25 
 
 9.12 
 
 28 71 
 
 6.39 
 
 15 98 
 
 5.19 
 
 11 39 
 
 4.53 
 
 9 15 
 
 4.12 
 
 7 85 
 
 3.84 
 
 7.01 
 
 3.63 
 
 6 42 
 
 3.48 
 
 5.99 
 
 3.36 
 
 6 67 
 
 CO 
 
 Cl 05 
 
 r- ^ 
 
 © t— 
 
 r- y—i 
 
 X © 
 CM t* 
 
 © © 
 © © 
 
 — © 
 -r o 
 
 © OO 
 t- c— 
 
 © © 
 X ^ 
 
 © 
 o © 
 
 © © 
 
 X © 
 
 lO 
 
 © 
 
 ©CM 
 © CM 
 
 © CO 
 
 — o 
 
 Cl ^ 
 
 © 
 
 © 05 
 
 — <j> 
 
 © © 
 CM 
 
 © © 
 
 iH 
 
 LO CM 
 
 tH 
 
 TT © 
 
 Tt oo 
 
 ^ c- 
 
 X © 
 
 X © 
 
 X © 
 
 CM 
 
 O CO 
 © © 
 
 © rH 
 © © 
 
 © 
 
 © 00 
 
 -r © 
 © © 
 
 © t'- 
 t- CM 
 
 CM 
 
 — © 
 
 ■rf © 
 © 
 
 © © 
 -r © 
 
 © CM 
 
 CM O 
 
 © © 
 lO 
 
 X © 
 © CM 
 
 05 05 
 C5 05 
 — 05 
 T* 
 
 © 05 
 i-h 05 
 
 © © 
 CO 
 
 © oo 
 
 H 
 
 LO CO 
 
 H 
 
 LO © 
 
 H 
 
 Tf © 
 
 co 
 
 tt co 
 
 t- 
 
 X t- 
 
 - 
 
 161.45 
 
 4052.10 
 
 18.51 
 
 98 49 
 
 10.13 
 
 34 12 
 
 7.71 
 
 21 20 
 
 6.61 
 
 16 26 
 
 5.99 
 
 13 74 
 
 5.59 
 
 12 25 
 
 5.32 
 
 11.26 
 
 5 12 
 
 10.66 
 
 4.96 
 
 10 04 
 
 4.84 
 
 9.65 
 
 
 - 
 
 CM 
 
 CO 
 
 Tt< 
 
 LO 
 
 
 
 X 
 
 © 
 
 © 
 
 T-* 
 
 
 
 
 aj-snbg uB3j\[ .laqeuig joj 
 
 uiopaajj jo saajSaQ 
 
 
 
TABLES 
 
 291 
 
 05 lO 
 lq 
 
 t—i o 
 
 O 05 
 
 CD rH 
 rH O 
 
 lO t- 
 TP C*> 
 
 1-H <J5 
 
 rH fr- 
 
 co ^ 
 
 rH 05 
 
 O rH 
 05 05 
 
 rH 05 
 
 o oo 
 
 rH 05 
 r-H 00 
 
 rH 00 
 
 o c-» 
 
 T-H CO 
 
 CO rH 
 
 C5 CD 
 
 o oo 
 
 CD LO 
 
 00 
 
 o co 
 
 OH 
 
 CO CO 
 O CO 
 
 05 
 1^ rH 
 
 O 00 
 
 C5 C- 
 'D O 
 
 O 00 
 
 05 CO 
 
 05 CO 
 
 05 05 
 
 05 05 
 
 05 05 
 
 05 05 
 
 05 05 
 
 05 05 
 
 05 05 
 
 Ol C5 
 
 05 05 
 
 05 05 
 
 O CD 
 
 CO CO 
 
 rH CD 
 
 05 rH 
 
 co o 
 
 T- O 
 
 b- ts- 
 O CO 
 
 rH LO 
 
 O 0- 
 
 CD LO 
 
 05 CD 
 
 05 
 
 C5 LO 
 
 OO 05 
 CO tp 
 
 ^P 05 
 
 co tp 
 
 rH CD 
 
 OC CO 
 
 CO o 
 
 CO 
 
 CD O 
 
 05 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 05 
 
 05 05 
 
 r-H 05 
 
 rH 05 
 
 r-H 05 
 
 rH 05 
 
 rH C5 
 
 rH 05 
 
 rH 05 
 
 O 00 
 lO t- 
 
 05 05 
 rp iq 
 
 LO CO 
 
 co 
 
 05 05 
 05 05 
 
 rP 00 
 05 rH 
 
 05 00 
 r-H O 
 
 LO O 
 h rH 
 
 rH 05 
 r-H 05 
 
 CO CD 
 
 o oo 
 
 LQ O 
 
 o oo 
 
 CO LO 
 O C~ 
 
 O o 
 
 O Cr 
 
 05 CO 
 
 C5 CO 
 
 Ol CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 05 
 
 05 05 
 
 Ol 05 
 
 05 05 
 
 05 05 
 
 05 CD 
 
 CO rH 
 
 O CD 
 
 CO 05 
 
 co o 
 
 lO CO 
 
 oo o- 
 
 Tf CD 
 
 05 lO 
 rp LO 
 
 CO LO 
 
 CO 
 
 ^P tr 
 co CO 
 
 H O 
 CO CO 
 
 co co 
 
 05 05 
 
 LO t- 
 Ol rH 
 
 CO 05 
 
 05 tH 
 
 O t- 
 Ol O 
 
 05 tP 
 
 05 CO 
 
 <M CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 iO O 
 oo lq 
 
 t^O 
 
 CO 
 
 o ^P 
 
 rH 
 
 ^ o 
 
 CO o 
 
 05 05 
 LO 00 
 
 LO C5 
 
 LO t- 
 
 *— 1 rH 
 LO Ct 
 
 oo CO 
 -r cd 
 
 LO CD 
 rp LO 
 
 Ol rH 
 TP lO 
 
 o LO 
 
 00 rH 
 
 CO TP 
 
 05 TP 
 
 05 
 
 05 tP 
 
 05 Hp 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 Ol CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 05 CO 
 
 O 05 
 O CO 
 
 oi oq 
 
 05 CD 
 
 ID CD 
 
 CO tP 
 
 05 05 
 t- CO 
 
 ^ o 
 
 05 
 
 o o 
 
 l- rH 
 
 CD rH 
 
 CD O 
 
 CO tP 
 CD 05 
 
 O Ct 
 CD 00 
 
 1^ rH 
 
 LO co 
 
 LQ LO 
 LO Cr 
 
 CO rH 
 
 LO t> 
 
 CO tP 
 
 Ol tP 
 
 05 tP 
 
 05 -<p 
 
 05 tP 
 
 05 
 
 05 tP 
 
 Ol CO 
 
 05 CO 
 
 Ol CO 
 
 05 CO 
 
 05 CO 
 
 t-h CD 
 
 rH O 
 
 05 CD 
 O CO 
 
 CO 05 
 
 05 CD 
 
 O CD 
 05 LQ 
 
 LO Tp 
 CO TP 
 
 r-H Tp 
 
 GO CO 
 
 t^. lO 
 t> 05 
 
 Hf t— 
 b- rH 
 
 rH O 
 t'r rH 
 
 CO tP 
 
 CD O 
 
 CD 05 
 CD 05 
 
 rp tP 
 
 CD 05 
 
 CO lO 
 
 CO hP 
 
 05 ^P 
 
 05 tP 
 
 05 tP 
 
 05 tP 
 
 05 tP 
 
 05 "<P 
 
 Ol tP 
 
 05 tP 
 
 05 CO 
 
 05 CO 
 
 CO rH 
 
 oi ^ 
 
 CO o 
 
 T-H 05 
 
 t-h co 
 
 r-H O 
 
 CO 05 
 O 00 
 
 rH t* 
 
 O 
 
 CD t> 
 
 C5 CD 
 
 CO 00 
 C5 LO 
 
 o o 
 
 C5 LO 
 
 CO 
 
 00 ^ 
 
 rP 
 
 00 CO 
 
 05 rH 
 
 CO co 
 
 O CD 
 
 00 05 
 
 CO LO 
 
 CO lO 
 
 CO io 
 
 CO -<p 
 
 co tp 
 
 05 tP 
 
 05 tP 
 
 05 tP 
 
 Ol Tp 
 
 05 rjl 
 
 05 Til 
 
 05 tP 
 
 05 LQ 
 TP 05 
 
 rH tP 
 
 ^P O- 
 
 rp CD 
 
 CO LO 
 
 05 05 
 Ol -<p 
 
 rP 05 
 Ol C5 
 
 o oo 
 
 Ol lH 
 
 CD 05 
 rn O 
 
 CO rH 
 
 rH O 
 
 O 
 
 rH O 
 
 O 00 
 
 LO 05 
 
 o co 
 
 CO CD 
 
 O tr 
 
 CO LO 
 
 CO lO 
 
 CO io 
 
 CO LO 
 
 CO LO 
 
 CO LO 
 
 CO LO 
 
 CO LO 
 
 CO Tp 
 
 CO Tjl 
 
 CO 
 
 CO tP 
 
 co co 
 
 00 05 
 
 o o 
 co o- 
 
 ^P rH 
 
 t'- to 
 
 CO CD 
 CO CO 
 
 CO CO 
 CO 05 
 
 05 rH 
 lO rH 
 
 LO rH 
 LO O 
 
 05 CO 
 LO 05 
 
 05 LO 
 Tp 00 
 
 00 
 
 Cr 
 
 Tp 05 
 TP 
 
 05 CD 
 
 tP CD 
 
 CO CD 
 
 CO CD 
 
 CO CD 
 
 CO CD 
 
 CO CD 
 
 CO CD 
 
 CO CD 
 
 CO LO 
 
 CO LO 
 
 CO LO 
 
 CO LO 
 
 CO LO 
 
 LO CO 
 CO 
 
 b- t- 
 
 CO o 
 
 O CD 
 
 CO CO 
 
 tP 00 
 >C CD 
 
 C5 CO 
 Tp LO 
 
 LO O 
 Tp Tp 
 
 r-H 00 
 
 tP 05 
 
 CO oo 
 
 CO rH 
 
 LO O 
 
 CO rH 
 
 Ol 05 
 CO o 
 
 O TP 
 CO 05 
 
 CO oo 
 
 05 00 
 
 05 
 
 tP 05 
 
 rt« 00 
 
 rp oo 
 
 Tp 00 
 
 Tp 00 
 
 rf 00 
 
 TP 00 
 
 ^ 00 
 
 oo 
 
 rp 
 
 rp t> 
 
 05 
 
 rH 
 
 CO 
 
 H 
 
 TP 
 »—1 
 
 »o 
 
 CD 
 
 
 2 
 
 05 
 
 o 
 
 Ol 
 
 Ol 
 
 22 
 
 23 
 
 
 
 Oil] 
 
 nby 
 
 
 iajI-BUIg .IOJ Ui0p39J w 
 
 I jo soajSaQ 
 
 
 
 This table was taken from “Calculation and Interpretation of Analysis of Variance and Covariance,” by G. W. Snedecor, 1934, by permission of 
 the author and the publisher, Collegiate Press, Inc., Ames, Iowa. 
 
TABLE III —( Continued ) 
 Degrees of Freedom for Greater Mean Square 
 
 292 
 
 TABLES 
 
 r Jl 
 
 M 4 b- 
 
 O b- 
 
 CD Cl 
 
 Cl rH 
 
 00 CO 
 
 to CD 
 
 CM O 
 
 O Hi 
 
 rH H* 
 
 CM O 
 
 H 1 O 
 
 co oo 
 
 
 CD Cl 
 
 CD 00 
 
 tO b- 
 
 tO b- 
 
 H CD 
 
 H tO 
 
 H tO 
 
 CO CM 
 
 rr Cl 
 
 O b- 
 
 
 © b- 
 
 O b- 
 
 o b- 
 
 O b- 
 
 O b- 
 
 O b- 
 
 O b- 
 
 O b- 
 
 O b- 
 
 o CD 
 
 O CD 
 
 > 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 
 CO rH 
 
 rH b- 
 
 © CO 
 
 b- O 
 
 to CD 
 
 H CO 
 
 CM rH 
 
 b~ O 
 
 CM CM 
 
 00 to 
 
 H 4 00 
 
 8 
 
 r ^ cm 
 
 rH 
 
 CD rH 
 
 CD rH 
 
 CD O 
 
 CD O 
 
 CD O 
 
 to Cl 
 
 to oo 
 
 H< b- 
 
 H 4 CD 
 
 
 —• CM 
 
 rH CM 
 
 rH CM 
 
 h CM 
 
 rH CM 
 
 rH CM 
 
 rH CM 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 
 CO CD 
 
 CD CM 
 
 to 00 
 
 CO to 
 
 rH CM 
 
 O ci 
 
 Cl b- 
 
 CO b- 
 
 Cl Cl 
 
 CD CO 
 
 H 4 00 
 
 
 © cd 
 
 © CD 
 
 © to 
 
 © to 
 
 © to 
 
 Cl Hi 
 
 00 Hi 
 
 cc co 
 
 cm 
 
 1> CM 
 
 b- rH 
 
 CM 
 
 rH CM 
 
 rH CM 
 
 h CM 
 
 rH CM 
 
 rH CM 
 
 rH CM 
 
 rH CM 
 
 rH CM 
 
 rH CM 
 
 h CM 
 
 rH CM 
 
 
 00 CO 
 
 CD Cl 
 
 to CD 
 
 CO co 
 
 Cl O 
 
 O b- 
 
 Cl HI 
 
 Hi H< 
 
 O CD 
 
 ^ rH 
 
 tO CD 
 
 CM 
 
 rH O 
 
 H Cl 
 
 rr Cl 
 
 rH O 
 
 rH Cl 
 
 r 00 
 
 o oo 
 
 o b- 
 
 O CD 
 
 Cl CD 
 
 Cl to 
 
 
 CM CO 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 rH CM 
 
 rH CM 
 
 
 o cd 
 
 cm 
 
 ci ci 
 
 O CD 
 
 ci co 
 
 00 o 
 
 b- 
 
 CM b- 
 
 00 Cl 
 
 to Hi 
 
 CO Ol 
 
 00 
 
 CO CO 
 
 co co 
 
 CO CM 
 
 CO CM 
 
 Cl CM 
 
 Cl CM 
 
 CM rH 
 
 Cl O 
 
 T-H Cl 
 
 r <J1 
 
 rn 00 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CM 
 
 CM CM 
 
 CM CM 
 
 
 rH b* 
 
 o co 
 
 I- ci 
 
 CD CD 
 
 H CO 
 
 CO o 
 
 CM b- 
 
 b~ b- 
 
 H< Cl 
 
 rH CO 
 
 Cl cn 
 
 
 to cd 
 
 TT 4 CD 
 
 -r to 
 
 rr io 
 
 H to 
 
 H* tO 
 
 H H 
 
 CO CO 
 
 CO CM 
 
 CO CM 
 
 CM rH 
 
 cd 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 Cl CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 
 01 o 
 
 C CD 
 
 COM 
 
 r- oo 
 
 CD to 
 
 H CO 
 
 CO O 
 
 co ci 
 
 tO rH 
 
 CM tO 
 
 O rH 
 
 
 cd © 
 
 CD 00 
 
 to oo 
 
 to b- 
 
 tO b- 
 
 to b- 
 
 to b- 
 
 H 1 to 
 
 rr to 
 
 H< H< 
 
 rf 4 ^ 
 
 tO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 ci co 
 
 CM CO 
 
 CM CO 
 
 ci co 
 
 CM CO 
 
 CM CO 
 
 
 CO CM 
 
 CD 00 
 
 Hi 
 
 CO rH 
 
 rH b- 
 
 O HI 
 
 Cl CM 
 
 H^ rH 
 
 rH eo 
 
 CO b- 
 
 CD CM 
 
 
 I- CM 
 
 I- rH 
 
 b« rH 
 
 l- rH 
 
 b- o 
 
 I- o 
 
 «o o 
 
 CD Cl 
 
 CD CO 
 
 to b- 
 
 tO b- 
 
 
 CM t* 
 
 CM 
 
 CM Til 
 
 Cl HI 
 
 CM Hi 
 
 CM H 
 
 CM Hi 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 CM CO 
 
 
 rH CM 
 
 © 00 
 
 CO Hi 
 
 CD O 
 
 tO b- 
 
 CO Hi 
 
 01 rH 
 
 br O 
 
 H 1 rH 
 
 r- IO 
 
 o o 
 
 CO 
 
 O b- 
 
 © CD 
 
 © cd 
 
 © CD 
 
 Cl to 
 
 Cl to 
 
 Cl to 
 
 CO H< 
 
 CO co 
 
 CC CM 
 
 b- CM 
 
 CO t* 
 
 CM rt< 
 
 CM Hi 
 
 Cl Hi 
 
 CM H 
 
 CM Hi 
 
 CM Hi 
 
 CM H< 
 
 CM H 1 
 
 CM H< 
 
 CM Til 
 
 
 O rH 
 
 00 b- 
 
 CO 
 
 to Cl 
 
 H tO 
 
 CO CM 
 
 CM Cl 
 
 br b- 
 
 CO 00 
 
 rH r-l 
 
 00 CD 
 
 CM 
 
 cd 
 
 CO to 
 
 CO to 
 
 CO HI 
 
 CO HI 
 
 CO H 
 
 CO CO 
 
 Cl CM 
 
 CM rH 
 
 CM rH 
 
 rn O 
 
 CO tO 
 
 CO to 
 
 CO to 
 
 CO to 
 
 CO to 
 
 co to 
 
 CO to 
 
 co to 
 
 CO to 
 
 CO to 
 
 CO to 
 
 
 CD CM 
 
 T^ b- 
 
 CM CM 
 
 rH 00 
 
 O H* 
 
 00 o 
 
 CD 
 
 Cl CM 
 
 CO rH 
 
 CD CO 
 
 CO b- 
 
 
 ci co 
 
 Cl b- 
 
 Cl b- 
 
 Cl CD 
 
 CM CD 
 
 rr CD 
 
 rH IO 
 
 rH Tj 4 
 
 c co 
 
 O CM 
 
 O rH 
 
 H 
 
 Ttf b- 
 
 rf 4 b- 
 
 Hi b- 
 
 H b- 
 
 H b- 
 
 H b- 
 
 Hi b- 
 
 H* b- 
 
 H 1 b- 
 
 H 4 b- 
 
 Tt 4 b- 
 
 
 
 to 
 
 CO 
 
 
 CO 
 
 Cl 
 
 O 
 
 to 
 
 O 
 
 to 
 
 o 
 
 
 CM 
 
 CM 
 
 CM 
 
 CM 
 
 CM 
 
 CM 
 
 CO 
 
 CO 
 
 H* 
 
 
 to 
 
 
 
 a.renbg ueoj^ joqeuig joj uiopaajj jo sooiSoQ 
 
 
 
TABLES 
 
 293 
 
 o o 
 
 o CO 
 o CO 
 
 tP 00 
 05 tP 
 05 CO 
 
 o oo 
 
 CD CO 
 05 CO 
 
 in 03 
 CO CO 
 
 O co 
 
 TP CO 
 CO 03 
 CD CO 
 
 05 CO 
 t>- tH 
 C5 CO 
 
 CD 05 
 
 tH o 
 
 CD CO 
 
 03 rH 
 
 tn O 
 C5 CO 
 
 00 03 
 CD 05 
 
 05 m 
 
 CD oo 
 CD 00 
 
 cd m 
 
 m co 
 
 CD 00 
 
 cd m 
 
 03 rH 
 
 CD 00 
 
 05 m 
 
 O CD 
 
 CD 0- 
 
 cd m 
 
 03 03 
 
 T-H 03 
 
 rH 03 
 
 t-H 03 
 
 rH 03 
 
 t-h 03 
 
 t-h 03 
 
 rH 03 
 
 H 03 
 
 t-h 03 
 
 tH 03 
 
 tH 03 
 
 rH 03 
 
 o o 
 
 CO CO 
 
 ID CO 
 
 CO ^ 
 
 03 05 
 CO TP 
 
 O 
 
 CO Ttf 
 
 co co 
 
 03 tP 
 
 ID Eh 
 03 CO 
 
 03 CO 
 03 CO 
 
 CD CO 
 rH 03 
 
 in 03 
 
 rH 03 
 
 CO 05 
 i—i rH 
 
 rH CO 
 
 tH tH 
 
 CO rH 
 O rH 
 
 o o 
 o o 
 
 H rH 
 
 T —1 rH 
 
 T—1 rH 
 
 rH tH 
 
 t-H tH 
 
 t-h rH 
 
 H iH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 tH rH 
 
 rH rH 
 
 O 03 
 N rH 
 
 in o- 
 
 CO O 
 
 ID CO 
 
 o o 
 
 tP O 
 CD O 
 
 CO 00 
 CD 05 
 
 O TP 
 CD 05 
 
 05 03 
 IO 05 
 
 1> C30 
 
 in co 
 
 in m 
 m co 
 
 tP tP 
 
 m oo 
 
 tP CO 
 
 m oo 
 
 CO rH 
 
 m oo 
 
 03 00 
 
 m t>. 
 
 T—1 N 
 
 T-H 03 
 
 T-H 03 
 
 rH 03 
 
 T-H tH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 rH rH 
 
 03 O 
 
 05 IO 
 
 CD iO 
 00 tP 
 
 CO 03 
 CO tP 
 
 CD 05 
 CO CO 
 
 in O- 
 
 oo co 
 
 co co 
 co co 
 
 03 rH 
 
 oo co 
 
 o co 
 
 00 03 
 
 CD rp 
 tn 03 
 
 00 co 
 
 tH 03 
 
 tn 03 
 Ih 03 
 
 CD O 
 
 tn 03 
 
 m oo 
 
 tn rH 
 
 rH 03 
 
 rH 03 
 
 t-h 03 
 
 t-h 03 
 
 TH 03 
 
 tH 03 
 
 TH 03 
 
 tH 03 
 
 rH 03 
 
 rH 03 
 
 tH 03 
 
 t-h 03 
 
 tH 03 
 
 O 03 
 rH 00 
 
 r- oo 
 o o- 
 
 CD rp 
 
 o o- 
 
 tP 03 
 
 o o- 
 
 CO 05 
 o CO 
 
 rH CO 
 O CO 
 
 o co 
 
 o CO 
 
 oo o 
 
 C5 CO 
 
 tH |H 
 
 cd m 
 
 CD CO 
 
 cd m 
 
 cd m 
 cd m 
 
 m co 
 
 cd m 
 
 rp rH 
 
 05 m 
 
 03 Cl 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 TH 03 
 
 TH 03 
 
 TH 03 
 
 rH 03 
 
 rH 03 
 
 TH 03 
 
 ID 03 
 
 03 rH 
 
 CO 0- 
 03 O 
 
 t-H tP 
 
 Ol O 
 
 O tH 
 
 03 O 
 
 CD 05 
 TH 05 
 
 i> m 
 
 tH 05 
 
 CD 03 
 
 rH 05 
 
 TP 05 
 rH OO 
 
 CO co 
 
 rH 00 
 
 03 m 
 
 TH 00 
 
 rH ^ 
 
 TH 00 
 
 O 03 
 
 TH C0 
 
 05 O 
 
 G 00 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 03 03 
 
 tn-^P 
 CO CO 
 
 ID 05 
 CO 03 
 
 CO co 
 
 CO 03 
 
 03 CO 
 CO 03 
 
 O rH 
 
 CO 03 
 
 05 tH 
 03 tH 
 
 in tp 
 
 03 rH 
 
 CD tH 
 
 Ol rH 
 
 in co 
 
 Ol o 
 
 rp CO 
 
 Ol o 
 
 co m 
 Ol o 
 
 03 tP 
 Ol O 
 
 rH 03 
 
 Ol o 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 ID 
 iO CO 
 
 o o 
 
 ID CO 
 
 CD CO 
 tp m 
 
 CO 
 TP IO 
 
 CD rH 
 
 rp m 
 
 TP t> 
 TP tP 
 
 CO ID 
 
 TP tP 
 
 03 rH 
 
 TP tP 
 
 TH 00 
 
 rp CO 
 
 O tH 
 rp CO 
 
 05 CO 
 CO co 
 
 CO tP 
 CO CO 
 
 tn 03 
 
 CO CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 Ol CO 
 
 03 CO 
 
 oi co 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 CO CO 
 
 Ih rH 
 
 tP 0- 
 tn O 
 
 03 tP 
 
 t> o 
 
 rH t-H 
 
 t- O 
 
 o 00 
 
 t - 05 
 
 CO tP 
 
 CD 05 
 
 CD rH 
 
 CD 05 
 
 in oo 
 CD oo 
 
 rp in 
 
 CD 00 
 
 CO co 
 CD 00 
 
 03 03 
 
 CD 00 
 
 TH o 
 CD 00 
 
 o oo 
 
 CD IT- 
 
 03 tP 
 
 03 tP 
 
 03 tP 
 
 03 tP 
 
 03 CO 
 
 03 CO 
 
 03 CO 
 
 Ol co 
 
 03 CO 
 
 03 CO 
 
 Ol CO 
 
 Ol CO 
 
 03 CO 
 
 ID 00 
 r-i CD 
 
 CO 03 
 
 t~h 05 
 
 rH 00 
 
 H 00 
 
 o iO 
 
 I-H 00 
 
 CD 03 
 
 o oo 
 
 In 00 
 
 O !> 
 
 CD ID 
 
 O C- 
 
 rp rH 
 
 O t- 
 
 CO oo 
 
 O CO 
 
 03 CO 
 
 O co 
 
 rH in 
 
 o co 
 
 O CO 
 o CD 
 
 o o 
 
 O CD 
 
 CO tP 
 
 CO tP 
 
 CO rp 
 
 CO tH 
 
 CO Tp 
 
 co tp 
 
 CO rp 
 
 CO TP 
 
 CO 03 
 
 CO TP 
 
 CO tP 
 
 CO rP 
 
 CO rp 
 
 o oo 
 o o 
 
 CO rH 
 
 CD O 
 
 CD CO 
 
 C5 05 
 
 IO 03 
 
 CD 05 
 
 r* O 
 
 CD 05 
 
 03 tP 
 
 CD 00 
 
 O rH 
 
 CD 00 
 
 CD CO 
 
 00 tn 
 
 tH C>q 
 
 00 tH 
 
 CD O 
 
 CO |H 
 
 CD 05 
 
 CO co 
 
 IO CD 
 
 CO CD 
 
 tP tP 
 
 00 CD 
 
 TPt> 
 
 CO t* 
 
 CO CO 
 
 CO CO 
 
 CO CO 
 
 CO co 
 
 CO CO 
 
 CO CO 
 
 CO co 
 
 CO co 
 
 CO co 
 
 CO CD 
 
 CO CD 
 
 09 
 
 70 
 
 o 
 
 00 
 
 06 
 
 o 
 
 o 
 
 rH 
 
 125 
 
 o 
 
 »o 
 
 rH 
 
 o 
 
 o 
 
 03 
 
 300 
 
 400 
 
 500 
 
 O 
 
 o 
 
 CD 
 
 8 
 
 
 
 
 OJTjnby urcaj^ ja[p3uiy joj mopaajy[ jo saajSoQ 
 
 rH 
 
 
294 
 
 TABLES 
 
 TABLE IV 
 
 Squares, Square Roots, and Reciprocals to 1000* 
 
 No. 
 
 Square 
 
 Square Root 
 
 Reciprocal 
 
 X 100 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Reciprocal 
 X 100 
 
 1 
 
 1 
 
 1.0000000 
 
 100.0000000 
 
 
 51 
 
 26 01 
 
 7.1414284 
 
 1.9607843 
 
 2 
 
 4 
 
 1.4142136 
 
 50.0000000 
 
 
 52 
 
 27 04 
 
 7.2111026 
 
 1.9230769 
 
 3 
 
 9 
 
 1.7320508 
 
 33.3333333 
 
 
 53 
 
 28 09 
 
 7.2801099 
 
 1.8867925 
 
 4 
 
 16 
 
 2.0000000 
 
 25.0000000 
 
 
 54 
 
 29 16 
 
 7.3484692 
 
 1.8518519 
 
 5 
 
 25 
 
 2.2360680 
 
 20.0000000 
 
 
 55 
 
 30 25 
 
 7.4161985 
 
 1.8181818 
 
 6 
 
 36 
 
 2.4494897 
 
 16.6666667 
 
 
 56 
 
 31 36 
 
 7.4833148 
 
 1.7857143 
 
 7 
 
 49 
 
 2.0457513 
 
 14.2S57143 
 
 
 57 
 
 32 49 
 
 7.5498344 
 
 1.7543860 
 
 8 
 
 64 
 
 2.8284271 
 
 12.5000000 
 
 
 58 
 
 33 64 
 
 7.6157731 
 
 1.7241379 
 
 9 
 
 81 
 
 3.0000000 
 
 11.1111111 
 
 
 59 
 
 34 81 
 
 7.6811457 
 
 1.6949153 
 
 10 
 
 1 00 
 
 3.1622777 
 
 10.0000000 
 
 
 60 
 
 36 00 
 
 7.7459667 
 
 1.6666667 
 
 11 
 
 1 21 
 
 3 3166248 
 
 9.0909091 
 
 
 61 
 
 37 21 
 
 7.8102497 
 
 1.6393443 
 
 12 
 
 1 44 
 
 3.4641016 
 
 8.3333333 
 
 
 62 
 
 38 44 
 
 7.8740079 
 
 1.6129032 
 
 13 
 
 1 69 
 
 3.6055513 
 
 7.6923077 
 
 
 63 
 
 39 69 
 
 7.9372539 
 
 1.5873016 
 
 14 
 
 1 96 
 
 3 7416574 
 
 7.142S571 
 
 
 64 
 
 40 96 
 
 8.0000000 
 
 1.5625000 
 
 15 
 
 2 25 
 
 3.8729833 
 
 6.6666667 
 
 
 65 
 
 42 25 
 
 8.0622577 
 
 1.5384615 
 
 16 
 
 2 56 
 
 4 0000000 
 
 6.2500000 
 
 
 66 
 
 43 56 
 
 8.1240384 
 
 1.5151515 
 
 17 
 
 2 89 
 
 4 1231056 
 
 5.8823529 
 
 
 67 
 
 44 89 
 
 8.1853528 
 
 1.4925373 
 
 18 
 
 3 24 
 
 4.2426407 
 
 5.5555556 
 
 
 68 
 
 46 24 
 
 8.2462113 
 
 1.4705882 
 
 19 
 
 3 61 
 
 4.3588989 
 
 5.2631579 
 
 
 69 
 
 47 61 
 
 8.3066239 
 
 1.4492754 
 
 20 
 
 4 00 
 
 4.4721360 
 
 5.0000000 
 
 
 70 
 
 49 00 
 
 8.3666003 
 
 1.4285714 
 
 21 
 
 4 41 
 
 4.5825757 
 
 4.761904S 
 
 
 71 
 
 50 41 
 
 8.4261498 
 
 1.4084507 
 
 22 
 
 4 84 
 
 4.6904158 
 
 4.5454545 
 
 
 72 
 
 51 84 
 
 8.4852814 
 
 1.3888889 
 
 23 
 
 5 29 
 
 4.7958315 
 
 4.3478261 
 
 
 73 
 
 53 29 
 
 8.5440037 
 
 1.3698630 
 
 24 
 
 5 76 
 
 4.8989795 
 
 4.1666667 
 
 
 74 
 
 54 76 
 
 8.6023253 
 
 1.3513514 
 
 25 
 
 6 25 
 
 5.0000000 
 
 4.0000000 
 
 
 <0 
 
 56 25 
 
 8.6602540 
 
 1.3333333 
 
 26 
 
 6 76 
 
 5.0990195 
 
 3.8461538 
 
 
 76 
 
 57 76 
 
 8.7177979 
 
 1.3157895 
 
 27 
 
 7 29 
 
 5.1961524 
 
 3.7037037 
 
 
 77 
 
 59 29 
 
 8.7749644 
 
 1.2987013 
 
 28 
 
 7 84 
 
 5.2915026 
 
 3.5714286 
 
 
 78 
 
 60 84 
 
 8.8317609 
 
 1.2820513 
 
 29 
 
 8 41 
 
 5.3851648 
 
 3.4482759 
 
 
 79 
 
 62 41 
 
 8.8881944 
 
 1.2658228 
 
 30 
 
 9 00 
 
 5.4772256 
 
 3.3333333 
 
 
 80 
 
 64 00 
 
 8.9442719 
 
 1.2500000 
 
 31 
 
 9 61 
 
 5.5677644 
 
 3.2258065 
 
 
 81 
 
 65 61 
 
 9.0000000 
 
 1.2345679 
 
 32 
 
 10 24 
 
 5.6568542 
 
 3.1250000 
 
 
 82 
 
 67 24 
 
 9.0553851 
 
 1 2195122 
 
 33 
 
 10 89 
 
 5.7445626 
 
 3.0303030 
 
 
 S3 
 
 68 89 
 
 9.1104336 
 
 1.2048193 
 
 34 
 
 11 56 
 
 5.8309519 
 
 2.9411765 
 
 
 84 
 
 70 56 
 
 9.1651514 
 
 1.1904762 
 
 35 
 
 12 25 
 
 5.9160798 
 
 2.8571429 
 
 
 85 
 
 72 25 
 
 9.2195445 
 
 1.1764706 
 
 36 
 
 12 96 
 
 6.0000000 
 
 2.7777778 
 
 
 86 
 
 73 96 
 
 9.2736185 
 
 1.1627907 
 
 37 
 
 13 69 
 
 6 0S27625 
 
 2.7027027 
 
 
 87 
 
 75 69 
 
 9.3273791 
 
 1 1494253 
 
 38 
 
 14 44 
 
 6.1644140 
 
 2.6315789 
 
 
 S8 
 
 77 44 
 
 9.3S08315 
 
 1 1363636 
 
 39 
 
 15 21 
 
 6 2449980 
 
 2.5641026 
 
 
 89 
 
 79 21 
 
 9.4339811 
 
 1 1235955 
 
 40 
 
 16 00 
 
 0.3245553 
 
 2.5000000 
 
 
 90 
 
 81 00 
 
 9.4868330 
 
 1.1111111 
 
 41 
 
 16 81 
 
 6.4031242 
 
 2.4390244 
 
 
 91 
 
 82 81 
 
 9.5393920 
 
 1.0989011 
 
 42 
 
 17 64 
 
 6.4807407 
 
 2.3809524 
 
 
 92 
 
 84 64 
 
 9.5916630 
 
 1.0869565 
 
 43 
 
 18 49 
 
 6.55743S5 
 
 2.3255814 
 
 
 93 
 
 86 49 
 
 9.6436508 
 
 1.0752688 
 
 44 
 
 19 36 
 
 6 6332496 
 
 2.2727273 
 
 
 94 
 
 88 36 
 
 9.6953597 
 
 1.0638298 
 
 45 
 
 20 25 
 
 6.7082039 
 
 2.2222222 
 
 
 95 
 
 90 25 
 
 9.7467943 
 
 1.0526316 
 
 46 
 
 21 16 
 
 6 7823300 
 
 2 1739130 
 
 
 96 
 
 92 16 
 
 9.7979590 
 
 1.0416667 
 
 47 
 
 22 09 
 
 6 8556546 
 
 2.1276596 
 
 
 97 
 
 94 09 
 
 9.8488578 
 
 1 0309278 
 
 48 
 
 23 04 
 
 6.9282032 
 
 2.0833333 
 
 
 98 
 
 96 04 
 
 9.8994949 
 
 1.0204082 
 
 49 
 
 24 01 
 
 7 0000000 
 
 2.0408163 
 
 
 99 
 
 98 01 
 
 9.9498744 
 
 1.0101010 
 
 50 
 
 25 00 
 
 7 0710678 
 
 2 0000000 
 
 
 100 
 
 1 00 00 
 
 10.0000000 
 
 1 0000000 
 
 * This table was taken from "Business Statistics,” by G. R. Davies and D. Yoder by 
 permission of the authors and John Wiley & Sons, Inc., the publisher. 
 
TABLES 
 
 295 
 
 TABLE IV —Continued 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10’ 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10° 
 
 101 
 
 1 02 01 
 
 10 0498756 
 
 9900990 
 
 
 151 
 
 2 28 01 
 
 12.2882057 
 
 6622517 
 
 102 
 
 1 04 04 
 
 10.0995049 
 
 9803922 
 
 
 152 
 
 2 31 04 
 
 12.3288280 
 
 6578947 
 
 103 
 
 1 06 09 
 
 10.1488916 
 
 9708738 
 
 
 153 
 
 2 34 09 
 
 12.3693169 
 
 6535948 
 
 104 
 
 1 08 16 
 
 10.1980390 
 
 9615385 
 
 
 154 
 
 2 37 16 
 
 12.4096736 
 
 6493506 
 
 105 
 
 1 10 25 
 
 10.2469508 
 
 9523810 
 
 
 155 
 
 2 40 25 
 
 12.4498996 
 
 6451613 
 
 10G 
 
 1 12 36 
 
 10.2950301 
 
 9433962 
 
 
 156 
 
 2 43 36 
 
 12.4899960 
 
 6410256 
 
 107 
 
 1 14 49 
 
 10.3440804 
 
 9345794 
 
 
 157 
 
 2 46 49 
 
 12.5299641 
 
 6369427 
 
 108 
 
 1 16 64 
 
 10.3923048 
 
 9259259 
 
 
 158 
 
 2 49 64 
 
 12.5698051 
 
 6329114 
 
 109 
 
 1 18 81 
 
 10.4403065 
 
 9174312 
 
 
 159 
 
 2 52 81 
 
 12.6095202 
 
 62S9308 
 
 110 
 
 1 21 00 
 
 10.4880885 
 
 9090909 
 
 
 160 
 
 2 56 00 
 
 12.6491106 
 
 6250000 
 
 111 
 
 1 23 21 
 
 10.5356538 
 
 9009009 
 
 
 161 
 
 2 59 21 
 
 12.6885775 
 
 6211180 
 
 112 
 
 1 25 44 
 
 10.5830052 
 
 8928571 
 
 
 162 
 
 2 62 44 
 
 12.7279221 
 
 6172840 
 
 113 
 
 1 27 69 
 
 10.6301458 
 
 8849558 
 
 
 103 
 
 2 65 69 
 
 12.7671453 
 
 6134969 
 
 114 
 
 1 29 96 
 
 10.6770783 
 
 8771930 
 
 
 164 
 
 2 68 96 
 
 12.8062485 
 
 6097561 
 
 115 
 
 1 32 25 
 
 10.7238053 
 
 8695652 
 
 
 165 
 
 2 72 25 
 
 12.8452326 
 
 6060606 
 
 116 
 
 1 34 56 
 
 10.7703296 
 
 8620690 
 
 
 166 
 
 2 75 56 
 
 12.8840987 
 
 6024096 
 
 117 
 
 1 36 89 
 
 10.8166538 
 
 8547009 
 
 
 167 
 
 2 78 89 
 
 12.9228480 
 
 5988024 
 
 118 
 
 1 39 24 
 
 10.8627805 
 
 8474576 
 
 
 168 
 
 2 82 24 
 
 12.9614814 
 
 5952381 
 
 119 
 
 1 41 61 
 
 10.9087121 
 
 8403361 
 
 
 169 
 
 2 85 61 
 
 13.0000000 
 
 5917160 
 
 120 
 
 1 44 00 
 
 10.9544512 
 
 8333333 
 
 
 170 
 
 2 89 00 
 
 13.0384048 
 
 5882353 
 
 121 
 
 1 46 41 
 
 11.0000000 
 
 8264463 
 
 
 171 
 
 2 92 41 
 
 13.0766968 
 
 5847953 
 
 122 
 
 1 48 84 
 
 11.0453610 
 
 8196721 
 
 
 172 
 
 2 95 84 
 
 13.1148770 
 
 5813953 
 
 123 
 
 1 51 29 
 
 11.0905365 
 
 8130081 
 
 
 173 
 
 2 99 29 
 
 13.1529464 
 
 57S0347 
 
 124 
 
 1 53 76 
 
 11.1355287 
 
 8064516 
 
 
 174 
 
 3 02 76 
 
 13.1909060 
 
 5747126 
 
 125 
 
 1 56 25 
 
 11.1803399 
 
 8000000 
 
 
 175 
 
 3 06 25 
 
 13.2287566 
 
 5714286 
 
 126 
 
 1 58 76 
 
 11.2249722 
 
 7936508 
 
 
 176 
 
 3 09 76 
 
 13 2664992 
 
 5681818 
 
 127 
 
 1 61 29 
 
 11.2694277 
 
 7874016 
 
 
 177 
 
 3 13 29 
 
 13.3041347 
 
 5649718 
 
 128 
 
 1 63 84 
 
 11 3137085 
 
 7812500 
 
 
 178 
 
 3 16 84 
 
 13 3416641 
 
 5617978 
 
 129 
 
 1 66 41 
 
 11 3578167 
 
 7751938 
 
 
 179 
 
 3 20 41 
 
 13 3790882 
 
 5586592 
 
 130 
 
 1 69 00 
 
 11.4017543 
 
 7692308 
 
 
 180 
 
 3 24 00 
 
 13.4164079 
 
 5555556 
 
 131 
 
 1 71 61 
 
 11 4455231 
 
 7633588 
 
 
 181 
 
 3 27 61 
 
 13.4536240 
 
 5524862 
 
 132 
 
 1 74 24 
 
 11.4891253 
 
 7575758 
 
 
 182 
 
 3 31 24 
 
 13.4907376 
 
 5494505 
 
 133 
 
 1 76 89 
 
 11.5325626 
 
 7518797 
 
 
 183 
 
 3 34 89 
 
 13.5277493 
 
 5464481 
 
 134 
 
 1 79 56 
 
 11 5758369 
 
 7462687 
 
 
 184 
 
 3 38 56 
 
 13 5646000 
 
 5434783 
 
 135 
 
 1 82 25 
 
 11.6189500 
 
 7407407 
 
 
 185 
 
 3 42 25 
 
 13.6014705 
 
 5405405 
 
 136 
 
 1 84 96 
 
 11 6019038 
 
 7352941 
 
 
 186 
 
 3 45 96 
 
 13.6381817 
 
 5376344 
 
 137 
 
 1 87 69 
 
 11 7046999 
 
 7299270 
 
 
 187 
 
 3 49 69 
 
 13 6747943 
 
 5347594 
 
 138 
 
 1 90 44 
 
 11 7473401 
 
 7246377 
 
 
 188 
 
 3 53 44 
 
 13 7113092 
 
 5319149 
 
 139 
 
 1 93 21 
 
 11 7898261 
 
 7194245 
 
 
 189 
 
 3 57 21 
 
 13.7477271 
 
 5291005 
 
 140 
 
 1 96 00 
 
 11 8321596 
 
 7142857 
 
 
 190 
 
 3 61 00 
 
 13.7840488 
 
 5263158 
 
 141 
 
 1 98 81 
 
 11 8743421 
 
 7092199 
 
 
 191 
 
 3 64 81 
 
 13 8202750 
 
 5235602 
 
 142 
 
 2 01 64 
 
 11 9163753 
 
 7042254 
 
 
 192 
 
 3 68 64 
 
 13 8564065 
 
 5208333 
 
 143 
 
 2 04 49 
 
 11 9582607 
 
 6993007 
 
 
 193 
 
 3 72 49 
 
 13 8924440 
 
 5181347 
 
 144 
 
 2 07 30 
 
 12.0000000 
 
 6944444 
 
 
 194 
 
 3 76 36 
 
 13 9283883 
 
 5154639 
 
 145 
 
 2 10 25 
 
 12 0415946 
 
 6896552 
 
 
 195 
 
 3 80 25 
 
 13.9642400 
 
 5128205 
 
 146 
 
 2 13 16 
 
 12 0830160 
 
 6849315 
 
 
 196 
 
 3 84 16 
 
 14.0000000 
 
 5102041 
 
 147 
 
 2 16 09 
 
 12 1243557 
 
 6802721 
 
 
 197 
 
 3 88 09 
 
 14.0356688 
 
 5076142 
 
 148 
 
 2 19 04 
 
 12.1655251 
 
 0756757 
 
 
 198 
 
 3 92 04 
 
 14 0712473 
 
 5050505 
 
 149 
 
 2 22 01 
 
 12 2065556 
 
 6711409 
 
 
 199 
 
 3 96 01 
 
 14.1067360 
 
 5025126 
 
 150 
 
 2 25 00 
 
 12.2474487 
 
 6666667 
 
 
 200 
 
 4 00 00 
 
 14.1421356 
 
 5000000 
 
296 
 
 TABLES 
 
 TABLE IV—( Continued ) 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Reciprocal 
 X 10* 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Reciprocal 
 X 10 9 
 
 201 
 
 4 04 01 
 
 14.1774469 
 
 4975124 
 
 
 251 
 
 6 30 01 
 
 15.8429795 
 
 3984064 
 
 202 
 
 4 OS 04 
 
 14.2126704 
 
 4950495 
 
 
 252 
 
 6 35 04 
 
 15 8745079 
 
 3968254 
 
 203 
 
 4 12 09 
 
 14.2478068 
 
 4926108 
 
 
 253 
 
 6 40 09 
 
 15.9059737 
 
 3952569 
 
 204 
 
 4 16 16 
 
 14.2828569 
 
 4901961 
 
 
 254 
 
 6 45 16 
 
 15.9373775 
 
 3937008 
 
 205 
 
 4 20 25 
 
 14 3178211 
 
 4S7S049 
 
 
 255 
 
 6 50 25 
 
 15.9687194 
 
 3921569 
 
 200 
 
 4 24 36 
 
 14 3527001 
 
 4S54369 
 
 
 256 
 
 6 55 36 
 
 16.0000000 
 
 3906250 
 
 207 
 
 4 28 49 
 
 14 3S74946 
 
 4S30918 
 
 
 257 
 
 6 60 49 
 
 16.0312195 
 
 3891051 
 
 20S 
 
 4 32 64 
 
 14.4222051 
 
 4S07692 
 
 
 25S 
 
 6 65 64 
 
 16.0023784 
 
 3875969 
 
 209 
 
 4 36 81 
 
 14.4568323 
 
 4784689 
 
 
 259 
 
 6 70 81 
 
 16.0934769 
 
 3861004 
 
 210 
 
 4 41 00 
 
 14.4913767 
 
 4761905 
 
 
 260 
 
 6 76 00 
 
 16.1245155 
 
 3846154 
 
 211 
 
 4 45 21 
 
 14 5258390 
 
 4739336 
 
 
 261 
 
 6 81 21 
 
 16.1554944 
 
 3S31418 
 
 212 
 
 4 49 44 
 
 14.5602198 
 
 4710981 
 
 
 262 
 
 6 86 44 
 
 16.1864141 
 
 3816794 
 
 213 
 
 4 53 69 
 
 14.5945195 
 
 4694836 
 
 
 263 
 
 6 91 69 
 
 16.2172747 
 
 3802281 
 
 214 
 
 4 57 96 
 
 14 6287388 
 
 4672897 
 
 
 264 
 
 6 96 96 
 
 10.2480768 
 
 3787879 
 
 215 
 
 4 62 25 
 
 14.6628783 
 
 4651163 
 
 
 265 
 
 7 02 25 
 
 16.2788206 
 
 3773585 
 
 216 
 
 4 66 56 
 
 14.6969385 
 
 4629630 
 
 
 266 
 
 7 07 56 
 
 16.3095064 
 
 375939S 
 
 217 
 
 4 70 89 
 
 14.7309199 
 
 4608295 
 
 
 267 
 
 7 12 89 
 
 16.3401346 
 
 3745318 
 
 218 
 
 4 75 24 
 
 14.764S231 
 
 4587156 
 
 
 208 
 
 7 18 24 
 
 16.3707055 
 
 3731343 
 
 219 
 
 4 79 61 
 
 14.79S64S6 
 
 4566210 
 
 
 269 
 
 7 23 61 
 
 16.4012195 
 
 3717472 
 
 220 
 
 4 84 00 
 
 14.8323970 
 
 4545455 
 
 
 270 
 
 7 29 00 
 
 16.4316767 
 
 3703704 
 
 221 
 
 4 S8 41 
 
 14.8660687 
 
 4524887 
 
 
 271 
 
 7 34 41 
 
 16.4620776 
 
 3690037 
 
 222 
 
 4 92 84 
 
 14.8996644 
 
 4504505 
 
 
 272 
 
 7 39 84 
 
 16.4924225 
 
 3676471 
 
 223 
 
 4 97 29 
 
 14.9331845 
 
 4484305 
 
 
 273 
 
 7 45 29 
 
 16.5227116 
 
 3663004 
 
 224 
 
 5 01 76 
 
 14.9666295 
 
 4464286 
 
 
 274 
 
 7 50 76 
 
 16.5529454 
 
 3649635 
 
 225 
 
 5 06 25 
 
 15.0000000 
 
 4444444 
 
 
 275 
 
 7 56 25 
 
 16.5S31240 
 
 3636364 
 
 226 
 
 5 10 76 
 
 15.0332964 
 
 4424779 
 
 
 276 
 
 7 61 76 
 
 16.0132477 
 
 362318S 
 
 227 
 
 5 15 29 
 
 15.0665192 
 
 44052S6 
 
 
 277 
 
 7 67 29 
 
 10.6433170 
 
 3610108 
 
 228 
 
 5 19 84 
 
 15.09966S9 
 
 4385965 
 
 
 278 
 
 7 72 84 
 
 16.6733320 
 
 3597122 
 
 229 
 
 5 24 41 
 
 15.1327460 
 
 4366S12 
 
 
 279 
 
 7 78 41 
 
 16.7032931 
 
 3584229 
 
 230 
 
 5 29 00 
 
 15.1657509 
 
 4347826 
 
 
 280 
 
 7 84 00 
 
 16.7332005 
 
 3571429 
 
 231 
 
 5 33 61 
 
 15.19S6842 
 
 4329004 
 
 
 2S1 
 
 7 89 61 
 
 16.7630546 
 
 3558719 
 
 232 
 
 5 38 24 
 
 15.2315462 
 
 4310345 
 
 
 282 
 
 7 95 24 
 
 16.7928556 
 
 3546099 
 
 233 
 
 5 42 89 
 
 15.2643375 
 
 4291845 
 
 
 283 
 
 8 00 89 
 
 16.8226038 
 
 3533569 
 
 234 
 
 5 47 56 
 
 15.2970585 
 
 4273504 
 
 
 2S4 
 
 8 06 56 
 
 16.8522995 
 
 3521127 
 
 235 
 
 5 52 25 
 
 15.3297097 
 
 4255319 
 
 
 285 
 
 8 12 25 
 
 16.8819430 
 
 3508772 
 
 236 
 
 5 56 96 
 
 15.3622915 
 
 42372S8 
 
 
 286 
 
 8 17 96 
 
 16 9115345 
 
 3496503 
 
 237 
 
 5 61 69 
 
 15.3948043 
 
 4219409 
 
 
 287 
 
 8 23 69 
 
 16.9410743 
 
 3484321 
 
 238 
 
 5 66 44 
 
 15.42724S6 
 
 4201681 
 
 
 288 
 
 8 29 44 
 
 16.9705627 
 
 3472222 
 
 239 
 
 5 71 21 
 
 15.4596248 
 
 4184100 
 
 
 289 
 
 8 35 21 
 
 17.0000000 
 
 3460208 
 
 240 
 
 5 76 00 
 
 15.4919334 
 
 4166667 
 
 
 290 
 
 8 41 00 
 
 17.0293864 
 
 3448276 
 
 241 
 
 5 80 81 
 
 15.5241747 
 
 4149378 
 
 
 291 
 
 8 46 81 
 
 17.0587221 
 
 3436426 
 
 242 
 
 5 S5 64 
 
 15.5563492 
 
 4132231 
 
 
 292 
 
 8 52 64 
 
 17.0880075 
 
 3424658 
 
 243 
 
 5 90 49 
 
 15.5S84573 
 
 4115226 
 
 
 293 
 
 8 58 49 
 
 17.1172428 
 
 3412969 
 
 244 
 
 5 95 36 
 
 15.6204994 
 
 4098361 
 
 
 294 
 
 8 64 36 
 
 17.1464282 
 
 3401361 
 
 245 
 
 6 00 25 
 
 15.6524758 
 
 4081633 
 
 
 295 
 
 8 70 25 
 
 17.1755640 
 
 3389831 
 
 246 
 
 6 05 16 
 
 15.6843871 
 
 4065041 
 
 
 296 
 
 8 76 16 
 
 17.2046505 
 
 3378378 
 
 247 
 
 6 10 09 
 
 15.7162336 
 
 404S583 
 
 
 297 
 
 8 82 09 
 
 17.2336879 
 
 3367003 
 
 248 
 
 6 15 04 
 
 15.7480157 
 
 4032258 
 
 
 298 
 
 8 88 04 
 
 17.2626765 
 
 3355705 
 
 249 
 
 6 20 01 
 
 15.7797338 
 
 4016064 
 
 
 299 
 
 8 94 01 
 
 17 2916165 
 
 3344482 
 
 250 
 
 6 25 00 
 
 15.8113883 
 
 4000000 
 
 
 300 
 
 9 00 00 
 
 17.32050S1 
 
 3333333 
 
TABLES 
 
 297 
 
 TABLE IV—( Continued ) 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 301 
 
 9 06 01 
 
 17.3493516 
 
 3322259 
 
 
 351 
 
 12 32 01 
 
 18.7349940 
 
 2849003 
 
 302 
 
 9 12 04 
 
 17.3781472 
 
 3311258 
 
 
 352 
 
 12 39 04 
 
 18.7616630 
 
 2840909 
 
 303 
 
 9 18 09 
 
 17.4068952 
 
 3300330 
 
 
 353 
 
 12 46 09 
 
 18.7882942 
 
 2832861 
 
 304 
 
 9 24 16 
 
 17.4355958 
 
 3289474 
 
 
 354 
 
 12 53 16 
 
 18.8148877 
 
 2824859 
 
 305 
 
 9 30 25 
 
 17.4642492 
 
 3278689 
 
 
 355 
 
 12 60 25 
 
 18.8414437 
 
 2816901 
 
 306 
 
 9 36 36 
 
 17.4928557 
 
 3267974 
 
 
 356 
 
 12 67 36 
 
 18.8679623 
 
 2808989 
 
 307 
 
 9 42 49 
 
 17.5214155 
 
 3257329 
 
 
 357 
 
 12 74 49 
 
 18.8944436 
 
 2801120 
 
 308 
 
 9 48 64 
 
 17.5499288 
 
 3246753 
 
 
 358 
 
 12 81 64 
 
 18.9208879 
 
 2793296 
 
 309 
 
 9 54 81 
 
 17.5783958 
 
 3236246 
 
 
 359 
 
 12 88 81 
 
 18.9472953 
 
 2785515 
 
 310 
 
 9 61 00 
 
 17.6068169 
 
 3225806 
 
 
 360 
 
 12 96 00 
 
 18.9736660 
 
 2777778 
 
 311 
 
 9 67 21 
 
 17.6351921 
 
 3215434 
 
 
 361 
 
 13 03 21 
 
 19 0000000 
 
 2770083 
 
 312 
 
 9 73 44 
 
 17.6635217 
 
 3205128 
 
 
 362 
 
 13 10 44 
 
 19.0262976 
 
 2762431 
 
 313 
 
 9 79 69 
 
 17.6918060 
 
 3194888 
 
 
 363 
 
 13 17 69 
 
 19.0525589 
 
 2754821 
 
 314 
 
 9 85 96 
 
 17.7200451 
 
 3184713 
 
 
 364 
 
 13 24 96 
 
 19 0787840 
 
 2747253 
 
 315 
 
 9 92 25 
 
 17.7482393 
 
 3174603 
 
 
 365 
 
 13 32 25 
 
 19.1049732 
 
 2739726 
 
 316 
 
 9 98 56 
 
 17.7763888 
 
 3164557 
 
 
 366 
 
 13 39 56 
 
 19.1311265 
 
 2732240 
 
 317 
 
 10 04 89 
 
 17.8044938 
 
 3154574 
 
 
 367 
 
 13 46 89 
 
 19.1572441 
 
 2724796 
 
 318 
 
 10 11 24 
 
 17.8325545 
 
 3144654 
 
 
 368 
 
 13 54 24 
 
 19.1833261 
 
 2717391 
 
 319 
 
 10 17 61 
 
 17.8605711 
 
 3134796 
 
 
 369 
 
 13 61 61 
 
 19.2093727 
 
 2710027 
 
 320 
 
 10 24 00 
 
 17.8885438 
 
 3125000 
 
 
 370 
 
 13 69 00 
 
 19.2353841 
 
 2702703 
 
 321 
 
 10 30 41 
 
 17.9164729 
 
 3115265 
 
 
 371 
 
 13 76 41 
 
 19.2613603 
 
 2695418 
 
 322 
 
 10 36 84 
 
 17.9443584 
 
 3105590 
 
 
 372 
 
 13 83 84 
 
 19.2873015 
 
 268S172 
 
 323 
 
 10 43 29 
 
 17.9722008 
 
 3095975 
 
 
 373 
 
 13 91 29 
 
 19.3132079 
 
 2680965 
 
 324 
 
 10 49 76 
 
 18.0000000 
 
 3086420 
 
 
 374 
 
 13 98 76 
 
 19.3390796 
 
 2673797 
 
 325 
 
 10 56 25 
 
 18.0277564 
 
 3076923 
 
 
 375 
 
 14 06 25 
 
 19.3649167 
 
 2666667 
 
 326 
 
 10 62 76 
 
 18.0554701 
 
 3067485 
 
 
 376 
 
 14 13 76 
 
 19.3907194 
 
 2659574 
 
 327 
 
 10 69 29 
 
 18.0831413 
 
 3058104 
 
 
 377 
 
 14 21 29 
 
 19.4164878 
 
 2652520 
 
 328 
 
 10 75 84 
 
 18.1107703 
 
 30487S0 
 
 
 378 
 
 14 28 84 
 
 19.4422221 
 
 2645503 
 
 329 
 
 10 82 41 
 
 18.1383571 
 
 3039514 
 
 
 379 
 
 14 36 41 
 
 19.4679223 
 
 2638522 
 
 330 
 
 10 89 00 
 
 18.1659021 
 
 3030303 
 
 
 380 
 
 14 44 00 
 
 19.4935887 
 
 2631579 
 
 331 
 
 10 95 61 
 
 18.1934054 
 
 3021148 
 
 
 381 
 
 14 51 61 
 
 19.5192213 
 
 2624672 
 
 332 
 
 11 02 24 
 
 18.2208672 
 
 3012048 
 
 
 382 
 
 14 59 24 
 
 19.5448203 
 
 2617801 
 
 333 
 
 11 08 89 
 
 18.2482S76 
 
 3003003 
 
 
 383 
 
 14 66 89 
 
 19.5703858 
 
 2610966 
 
 334 
 
 11 15 56 
 
 18.2756669 
 
 2994012 
 
 
 384 
 
 14 74 56 
 
 19 5959179 
 
 2604167 
 
 335 
 
 11 22 25 
 
 18.3030052 
 
 2985075 
 
 
 385 
 
 14 82 25 
 
 19.6214169 
 
 2597403 
 
 33G 
 
 11 28 96 
 
 18.3303028 
 
 2976190 
 
 
 386 
 
 14 89 96 
 
 19.6468827 
 
 2590674 
 
 337 
 
 11 35 69 
 
 18.3575598 
 
 2967359 
 
 
 387 
 
 14 97 69 
 
 19.6723156 
 
 2583979 
 
 338 
 
 11 42 44 
 
 18.3847763 
 
 2958580 
 
 
 388 
 
 15 05 44 
 
 19.6977156 
 
 2577320 
 
 339 
 
 11 49 21 
 
 18 4119526 
 
 2949853 
 
 
 3S9 
 
 15 13 21 
 
 19.7230829 
 
 2570694 
 
 340 
 
 11 56 00 
 
 18.4390889 
 
 2941176 
 
 
 390 
 
 15 21 00 
 
 19.7484177 
 
 2564103 
 
 341 
 
 11 62 81 
 
 18.4661853 
 
 2932551 
 
 
 391 
 
 15 28 81 
 
 19.7737199 
 
 2557545 
 
 342 
 
 11 69 64 
 
 18.4932420 
 
 2923977 
 
 
 392 
 
 15 36 64 
 
 19.7989899 
 
 2551020 
 
 343 
 
 11 76 49 
 
 18.5202592 
 
 2915452 
 
 
 393 
 
 15 44 49 
 
 19.8242276 
 
 2.544529 
 
 344 
 
 11 83 36 
 
 18.5472370 
 
 2906977 
 
 
 394 
 
 15 52 36 
 
 19 8494332 
 
 2538071 
 
 345 
 
 11 90 25 
 
 18.5741756 
 
 2898551 
 
 
 395 
 
 15 60 25 
 
 19.8746069 
 
 2531646 
 
 346 
 
 11 97 16 
 
 18.6010752 
 
 2890173 
 
 
 396 
 
 15 68 16 
 
 19.8997487 
 
 2525253 
 
 347 
 
 12 04 09 
 
 18.6279360 
 
 2881844 
 
 
 397 
 
 15 76 09 
 
 19.9248588 
 
 2518892 
 
 348 
 
 12 11 04 
 
 18.6547581 
 
 2873563 
 
 
 398 
 
 15 84 04 
 
 19 9499373 
 
 2512563 
 
 349 
 
 12 18 01 
 
 18 6815417 
 
 2865330 
 
 
 399 
 
 15 92 01 
 
 19.9749844 
 
 2506266 
 
 350 
 
 12 25 00 
 
 18.7082869 
 
 2857143 
 
 
 400 
 
 16 00 00 
 
 20.0000000 
 
 2500000 
 
298 
 
 TABLES 
 
 TABLE IV—( Continued ) 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10* 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 401 
 
 16 08 01 
 
 20.0249844 
 
 2493766 
 
 
 451 
 
 20 34 01 
 
 21.2367606 
 
 2217295 
 
 402 
 
 16 16 04 
 
 20.0499377 
 
 2487562 
 
 
 452 
 
 20 43 04 
 
 21.2602916 
 
 2212389 
 
 403 
 
 16 24 09 
 
 20 0748599 
 
 24S1390 
 
 
 453 
 
 20 52 09 
 
 21.2837967 
 
 2207506 
 
 404 
 
 16 32 16 
 
 20 0997512 
 
 2475248 
 
 
 454 
 
 20 61 16 
 
 21.3072758 
 
 2202643 
 
 405 
 
 16 40 25 
 
 20.1246118 
 
 2469136 
 
 
 455 
 
 20 70 25 
 
 21.3307290 
 
 2197802 
 
 406 
 
 16 48 36 
 
 20.1494417 
 
 2463054 
 
 
 456 
 
 20 79 36 
 
 21.3541565 
 
 2192982 
 
 407 
 
 16 56 49 
 
 20.1742410 
 
 2457002 
 
 
 457 
 
 20 88 49 
 
 21.3775583 
 
 2188184 
 
 408 
 
 16 64 64 
 
 20.1990099 
 
 2450980 
 
 
 458 
 
 20 97 64 
 
 21.4009346 
 
 2183406 
 
 409 
 
 16 72 81 
 
 20.2237484 
 
 2444988 
 
 
 459 
 
 21 06 81 
 
 21.4242853 
 
 2178649 
 
 410 
 
 16 81 00 
 
 20.2484567 
 
 2439024 
 
 
 460 
 
 21 16 00 
 
 21.4476106 
 
 2173913 
 
 411 
 
 16 89 21 
 
 20.2731349 
 
 2433090 
 
 
 461 
 
 21 25 21 
 
 21.4709106 
 
 2169197 
 
 412 
 
 16 97 44 
 
 20 2977831 
 
 2427184 
 
 
 462 
 
 21 34 44 
 
 21.4941853 
 
 2164502 
 
 413 
 
 17 05 69 
 
 20.3224014 
 
 242130S 
 
 
 463 
 
 21 43 69 
 
 21.5174348 
 
 2159827 
 
 414 
 
 17 13 96 
 
 20.3469899 
 
 2415459 
 
 
 464 
 
 21 52 96 
 
 21.5406592 
 
 2155172 
 
 415 
 
 17 22 25 
 
 20.3715488 
 
 2409639 
 
 
 465 
 
 21 62 25 
 
 21.5638587 
 
 2150538 
 
 416 
 
 17 30 56 
 
 20.3960781 
 
 2403S46 
 
 
 466 
 
 21 71 56 
 
 21.5870331 
 
 2145923 
 
 417 
 
 17 38 89 
 
 20.4205779 
 
 2398082 
 
 
 467 
 
 21 80 89 
 
 21 610182S 
 
 2141328 
 
 418 
 
 17 47 24 
 
 20.4450483 
 
 2392344 
 
 
 468 
 
 21 90 24 
 
 21.6333077 
 
 2136752 
 
 419 
 
 17 55 61 
 
 20.4694895 
 
 2386635 
 
 
 469 
 
 21 99 61 
 
 21.6564078 
 
 2132196 
 
 420 
 
 17 64 00 
 
 20.4939015 
 
 2380952 
 
 
 470 
 
 22 09 00 
 
 21.6794834 
 
 2127660 
 
 421 
 
 17 72 41 
 
 20.5182845 
 
 2375297 
 
 
 471 
 
 22 18 41 
 
 21.7025344 
 
 2123142 
 
 422 
 
 17 80 84 
 
 20.5426386 
 
 2369668 
 
 
 472 
 
 22 27 84 
 
 21.7255610 
 
 2118644 
 
 423 
 
 17 89 29 
 
 20.566963S 
 
 2364066 
 
 
 473 
 
 22 37 29 
 
 21.7485632 
 
 2114165 
 
 424 
 
 17 97 76 
 
 20.5912603 
 
 2358491 
 
 
 474 
 
 22 46 76 
 
 21.7715411 
 
 2109705 
 
 425 
 
 18 06 25 
 
 20.6155281 
 
 2352941 
 
 
 475 
 
 22 56 25 
 
 21.7944947 
 
 2105263 
 
 426 
 
 18 14 76 
 
 20.6397674 
 
 2347418 
 
 
 476 
 
 22 65 76 
 
 21.8174242 
 
 2100840 
 
 427 
 
 18 23 29 
 
 20.6639783 
 
 2341920 
 
 
 477 
 
 22 75 29 
 
 21.8403297 
 
 2096436 
 
 428 
 
 18 31 84 
 
 20.6881609 
 
 2336449 
 
 
 478 
 
 22 84 84 
 
 21.8632111 
 
 2092050 
 
 429 
 
 18 40 41 
 
 20.7123152 
 
 2331002 
 
 
 479 
 
 22 94 41 
 
 21.8860686 
 
 2087683 
 
 430 
 
 18 49 00 
 
 20.7364414 
 
 2325581 
 
 
 4S0 
 
 23 04 00 
 
 21.9089023 
 
 2083333 
 
 431 
 
 18 57 61 
 
 20.7605395 
 
 2320186 
 
 
 4S1 
 
 23 13 61 
 
 21.9317122 
 
 2079002 
 
 432 
 
 IS 66 24 
 
 20.7846097 
 
 2314815 
 
 
 482 
 
 23 23 24 
 
 21.9544984 
 
 2074689 
 
 433 
 
 18 74 89 
 
 20.8086520 
 
 2309469 
 
 
 4S3 
 
 23 32 89 
 
 21.9772610 
 
 2070393 
 
 434 
 
 18 83 56 
 
 20.8326667 
 
 2304147 
 
 
 484 
 
 23 42 56 
 
 22.0000000 
 
 2066116 
 
 435 
 
 18 92 25 
 
 20.8566536 
 
 2298851 
 
 
 485 
 
 23 52 25 
 
 22.0227155 
 
 2061856 
 
 436 
 
 19 00 96 
 
 20.8806130 
 
 2293578 
 
 
 4S6 
 
 23 61 96 
 
 22.0454077 
 
 2057613 
 
 437 
 
 19 09 69 
 
 20.9045450 
 
 22SS330 
 
 
 487 
 
 23 71 69 
 
 22.0080765 
 
 2053388 
 
 438 
 
 19 18 44 
 
 20.9284495 
 
 22S3105 
 
 
 488 
 
 23 81 44 
 
 22.0907220 
 
 2049180 
 
 439 
 
 19 27 21 
 
 20.952326S 
 
 2277904 
 
 
 489 
 
 23 91 21 
 
 22.1133444 
 
 2044990 
 
 440 
 
 19 36 00 
 
 20.9761770 
 
 2272727 
 
 
 490 
 
 24 01 00 
 
 22.1359436 
 
 2040S16 
 
 441 
 
 19 44 81 
 
 21.0000000 
 
 2267574 
 
 
 491 
 
 24 10 81 
 
 22.1585198 
 
 2036660 
 
 442 
 
 19 53 64 
 
 21.0237960 
 
 2262443 
 
 
 492 
 
 24 20 64 
 
 22.1S10730 
 
 2032520 
 
 443 
 
 19 62 49 
 
 21.0475652 
 
 2257336 
 
 
 493 
 
 24 30 49 
 
 22.2030033 
 
 2028398 
 
 444 
 
 19 71 36 
 
 21.0713075 
 
 2252252 
 
 
 494 
 
 24 40 36 
 
 22.2261108 
 
 2024291 
 
 445 
 
 19 80 25 
 
 21.0950231 
 
 2247191 
 
 
 495 
 
 24 50 25 
 
 22.24S5955 
 
 2020202 
 
 446 
 
 19 89 16 
 
 21.1187121 
 
 2242152 
 
 
 496 
 
 24 60 16 
 
 22 2710575 
 
 2016129 
 
 447 
 
 19 98 09 
 
 21 1423745 
 
 2237136 
 
 
 497 
 
 24 70 09 
 
 22.2934968 
 
 2012072 
 
 448 
 
 20 07 04 
 
 21.1660105 
 
 2232143 
 
 
 498 
 
 24 80 04 
 
 22.3159136 
 
 2008032 
 
 449 
 
 20 16 01 
 
 21.1896201 
 
 2227171 
 
 
 499 
 
 24 90 01 
 
 22 3383079 
 
 2004008 
 
 450 
 
 20 25 00 
 
 21 2132034 
 
 2222222 
 
 
 500 
 
 25 00 00 
 
 22.3006798 
 
 2000000 
 
TABLES 
 
 299 
 
 TABLE IV—( Continued ) 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 501 
 
 25 10 01 
 
 22.3830293 
 
 1996008 
 
 
 551 
 
 30 36 01 
 
 23.4733892 
 
 1814882 
 
 502 
 
 25 20 04 
 
 22.4053565 
 
 1992032 
 
 
 552 
 
 30 47 04 
 
 23.4946802 
 
 1811594 
 
 503 
 
 25 30 09 
 
 22.4276615 
 
 1988072 
 
 
 553 
 
 30 58 09 
 
 23.5159520 
 
 1808318 
 
 504 
 
 25 40 16 
 
 22.4499443 
 
 1984127 
 
 
 554 
 
 30 69 16 
 
 23.5372046 
 
 1805054 
 
 505 
 
 25 50 25 
 
 22.4722051 
 
 1980198 
 
 
 555 
 
 30 80 25 
 
 23.5584380 
 
 1801802 
 
 506 
 
 25 60 36 
 
 22.4944438 
 
 1976285 
 
 
 556 
 
 30 91 36 
 
 23.5796522 
 
 1798561 
 
 507 
 
 25 70 49 
 
 22.5166605 
 
 1972387 
 
 
 557 
 
 31 02 49 
 
 23.6008474 
 
 1795332 
 
 508 
 
 25 80 64 
 
 22.5388553 
 
 1968504 
 
 
 558 
 
 31 13 64 
 
 23.6220236 
 
 1792115 
 
 509 
 
 25 90 81 
 
 22.5610283 
 
 1964637 
 
 
 559 
 
 31 24 81 
 
 23.6431808 
 
 1788909 
 
 510 
 
 26 01 00 
 
 22.5831796 
 
 1960784 
 
 
 560 
 
 31 36 00 
 
 23.6643191 
 
 1785714 
 
 511 
 
 26 11 21 
 
 22.6053091 
 
 1956947 
 
 
 561 
 
 31 47 21 
 
 23.6854386 
 
 1782531 
 
 512 
 
 26 21 44 
 
 22.6274170 
 
 1953125 
 
 
 562 
 
 31 58 44 
 
 23.7065392 
 
 1779359 
 
 513 
 
 26 31 69 
 
 22.6495033 
 
 1949318 
 
 
 563 
 
 31 69 69 
 
 23.7276210 
 
 1776199 
 
 514 
 
 26 41 96 
 
 22.6715681 
 
 1945525 
 
 
 564 
 
 31 80 96 
 
 23.7486842 
 
 1773050 
 
 515 
 
 26 52 25 
 
 22.6936114 
 
 1941748 
 
 
 565 
 
 31 92 25 
 
 23.7697286 
 
 1769912 
 
 516 
 
 26 62 56 
 
 22.7156334 
 
 1937984 
 
 
 566 
 
 32 03 56 
 
 23.7907545 
 
 1766784 
 
 517 
 
 26 72 89 
 
 22.7376340 
 
 1934236 
 
 
 567 
 
 32 14 89 
 
 23.8117618 
 
 1763668 
 
 518 
 
 26 83 24 
 
 22.7596134 
 
 1930502 
 
 
 568 
 
 32 26 24 
 
 23.8327506 
 
 1760563 
 
 519 
 
 26 93 61 
 
 22.7815715 
 
 1926782 
 
 
 569 
 
 32 37 61 
 
 23.8537209 
 
 1757469 
 
 520 
 
 27 04 00 
 
 22.8035085 
 
 1923077 
 
 
 570 
 
 32 49 00 
 
 23.8746728 
 
 1754386 
 
 521 
 
 27 14 41 
 
 22.8254244 
 
 1919386 
 
 
 571 
 
 32 60 41 
 
 23.8956063 
 
 1751313 
 
 522 
 
 27 24 84 
 
 22.8473193 
 
 1915709 
 
 
 572 
 
 32 71 84 
 
 23.9165215 
 
 1748252 
 
 523 
 
 27 35 29 
 
 22.8691933 
 
 1912046 
 
 
 573 
 
 32 83 29 
 
 23.9374184 
 
 1745201 
 
 524 
 
 27 45 76 
 
 22.8910463 
 
 1908397 
 
 
 574 
 
 32 94 76 
 
 23.9582971 
 
 1742160 
 
 525 
 
 27 56 25 
 
 22.9128785 
 
 1904762 
 
 
 575 
 
 33 06 25 
 
 23.9791576 
 
 1739130 
 
 526 
 
 27 66 76 
 
 22.9346899 
 
 1901141 
 
 
 576 
 
 33 17 76 
 
 24.0000000 
 
 1736111 
 
 527 
 
 27 77 29 
 
 22.9564806 
 
 1897533 
 
 
 577 
 
 33 29 29 
 
 24.020S243 
 
 1733102 
 
 528 
 
 27 87 84 
 
 22.9782506 
 
 1893939 
 
 
 578 
 
 33 40 84 
 
 24.0416306 
 
 1730104 
 
 529 
 
 27 98 41 
 
 23.0000000 
 
 1890359 
 
 
 579 
 
 33 52 41 
 
 24.0624188 
 
 1727116 
 
 530 
 
 28 09 00 
 
 23.0217289 
 
 1886792 
 
 
 5S0 
 
 33 64 00 
 
 24.0831892 
 
 1724138 
 
 531 
 
 28 19 61 
 
 23.0434372 
 
 1883239 
 
 
 581 
 
 33 75 61 
 
 24.1039416 
 
 1721170 
 
 532 
 
 28 30 24 
 
 23.0651252 
 
 1879699 
 
 
 582 
 
 33 87 24 
 
 24.1246762 
 
 1718213 
 
 533 
 
 28 40 89 
 
 23.0867928 
 
 1876173 
 
 
 583 
 
 33 98 89 
 
 24.1453929 
 
 1715266 
 
 534 
 
 28 51 56 
 
 23.1084400 
 
 1872659 
 
 
 584 
 
 34 10 56 
 
 24.1660919 
 
 1712329 
 
 535 
 
 28 62 25 
 
 23.1300670 
 
 1869159 
 
 
 585 
 
 34 22 25 
 
 24.1867732 
 
 1709402 
 
 536 
 
 28 72 96 
 
 23.1516738 
 
 1865072 
 
 
 586 
 
 34 33 96 
 
 24.2074369 
 
 1706485 
 
 537 
 
 28 83 69 
 
 23.1732605 
 
 1862197 
 
 
 587 
 
 34 45 69 
 
 24 2280829 
 
 1703578 
 
 538 
 
 28 94 44 
 
 23.1948270 
 
 1858730 
 
 
 588 
 
 34 57 44 
 
 24.2487113 
 
 1700680 
 
 539 
 
 29 05 21 
 
 23.2163735 
 
 1855288 
 
 
 5S9 
 
 34 09 21 
 
 24.2693222 
 
 1697793 
 
 540 
 
 29 16 00 
 
 23.2379001 
 
 1S51852 
 
 
 590 
 
 34 81 00 
 
 24.2S99156 
 
 1694915 
 
 541 
 
 29 26 81 
 
 23.2594007 
 
 1848429 
 
 
 591 
 
 34 92 81 
 
 24.3104916 
 
 1692047 
 
 542 
 
 29 37 04 
 
 23.2808935 
 
 1845018 
 
 
 592 
 
 35 04 64 
 
 24.3310501 
 
 1689189 
 
 543 
 
 29 48 49 
 
 23.3023604 
 
 1841621 
 
 
 593 
 
 35 10 49 
 
 24.3515913 
 
 1686341 
 
 544 
 
 29 59 36 
 
 23.323S070 
 
 1838235 
 
 
 594 
 
 35 28 30 
 
 24.3721152 
 
 1683502 
 
 545 
 
 29 70 25 
 
 23.3452351 
 
 1834802 
 
 
 595 
 
 35 40 25 
 
 24.39262IS 
 
 1680672 
 
 546 
 
 29 81 16 
 
 23.3600429 
 
 1831502 
 
 
 596 
 
 35 52 16 
 
 24.4131112 
 
 1677852 
 
 547 
 
 29 92 09 
 
 23.3880311 
 
 1828154 
 
 
 597 
 
 35 64 09 
 
 24.4335834 
 
 1675042 
 
 548 
 
 30 03 04 
 
 23.4093998 
 
 1824818 
 
 
 598 
 
 35 76 04 
 
 24.4540385 
 
 1672241 
 
 549 
 
 30 14 01 
 
 23.4307490 
 
 1821494 
 
 
 599 
 
 35 8S 01 
 
 24.4744765 
 
 1669449 
 
 550 
 
 30 25 00 
 
 23.4520788 
 
 1818182 
 
 
 000 
 
 36 00 00 
 
 24.4948974 
 
 1066667 
 
300 
 
 TABLES 
 
 TABLE IV—( Continued) 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 s 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 601 
 
 36 12 01 
 
 24 5153013 
 
 1663894 
 
 
 651 
 
 42 3S 01 
 
 25.5147016 
 
 1536098 
 
 602 
 
 36 24 04 
 
 24.53568S3 
 
 1661130 
 
 
 652 
 
 42 51 04 
 
 25.5342907 
 
 1533742 
 
 603 
 
 36 36 09 
 
 24.5560583 
 
 1658375 
 
 
 653 
 
 42 64 09 
 
 25.5538647 
 
 1531394 
 
 604 
 
 36 48 16 
 
 24.5764115 
 
 1655629 
 
 
 654 
 
 42 77 16 
 
 25.5734237 
 
 1529052 
 
 605 
 
 36 60 25 
 
 24.5967478 
 
 1652S93 
 
 
 655 
 
 42 90 25 
 
 25.5929678 
 
 1526718 
 
 606 
 
 36 72 36 
 
 24.6170673 
 
 1650165 
 
 
 656 
 
 43 03 30 
 
 25.6124969 
 
 1524390 
 
 607 
 
 36 84 49 
 
 24.6373700 
 
 1647446 
 
 
 657 
 
 43 16 49 
 
 25.6320112 
 
 1522070 
 
 608 
 
 36 96 64 
 
 24.6576560 
 
 1644737 
 
 
 658 
 
 43 29 64 
 
 25.6515107 
 
 1519757 
 
 609 
 
 37 08 81 
 
 24.6779254 
 
 1642036 
 
 
 659 
 
 43 42 81 
 
 25.6709953 
 
 1517451 
 
 610 
 
 37 21 00 
 
 24.69S1781 
 
 1639344 
 
 
 660 
 
 43 50 00 
 
 25.0904652 
 
 1515152 
 
 611 
 
 37 33 21 
 
 24.7184142 
 
 1636661 
 
 
 661 
 
 43 69 21 
 
 25.7099203 
 
 1512S59 
 
 612 
 
 37 45 44 
 
 24.73S6338 
 
 1633987 
 
 
 662 
 
 43 82 44 
 
 25.7293607 
 
 1510574 
 
 613 
 
 37 57 69 
 
 24.758S368 
 
 1631321 
 
 
 663 
 
 43 95 09 
 
 25.7487864 
 
 1508296 
 
 614 
 
 37 69 96 
 
 24.7790234 
 
 162S664 
 
 
 664 
 
 44 OS 96 
 
 25.7681975 
 
 1506024 
 
 615 
 
 37 82 25 
 
 24.7991935 
 
 1626016 
 
 
 665 
 
 44 22 25 
 
 25.7875939 
 
 1503759 
 
 616 
 
 37 94 56 
 
 24.8193473 
 
 1623377 
 
 
 666 
 
 44 35 56 
 
 25.8069758 
 
 1501502 
 
 617 
 
 3S 06 89 
 
 24.8394S47 
 
 1620746 
 
 
 667 
 
 44 48 89 
 
 25.8263431 
 
 1499250 
 
 618 
 
 38 19 24 
 
 24.8596058 
 
 1618123 
 
 
 668 
 
 44 62 24 
 
 25.8456960 
 
 1497006 
 
 619 
 
 38 31 61 
 
 24.8797106 
 
 1615509 
 
 
 669 
 
 44 75 61 
 
 25.8650343 
 
 1494768 
 
 620 
 
 38 44 00 
 
 24.8997992 
 
 1612903 
 
 
 670 
 
 44 89 00 
 
 25.8843582 
 
 1492537 
 
 621 
 
 38 56 41 
 
 24.9198716 
 
 1610306 
 
 
 671 
 
 45 02 41 
 
 25.9036677 
 
 1490313 
 
 622 
 
 38 68 84 
 
 24.9399278 
 
 1607717 
 
 
 672 
 
 45 15 84 
 
 25.9229628 
 
 148S095 
 
 623 
 
 38 81 29 
 
 24 9599679 
 
 1605136 
 
 
 673 
 
 45 29 29 
 
 25.9422435 
 
 1485884 
 
 624 
 
 38 93 76 
 
 24 9799920 
 
 1602564 
 
 
 674 
 
 45 42 76 
 
 25.9615100 
 
 14S3680 
 
 625 
 
 39 06 25 
 
 25.0000000 
 
 1600000 
 
 
 675 
 
 45 56 25 
 
 25.9S07621 
 
 1481481 
 
 626 
 
 39 18 76 
 
 25.0199920 
 
 1597444 
 
 
 676 
 
 45 69 76 
 
 26.0000000 
 
 1479290 
 
 627 
 
 39 31 29 
 
 25.0399681 
 
 1594896 
 
 
 677 
 
 45 83 29 
 
 26.0192237 
 
 1477105 
 
 628 
 
 39 43 84 
 
 25.0599282 
 
 1592357 
 
 
 678 
 
 45 96 84 
 
 26.0384331 
 
 1474926 
 
 629 
 
 39 56 41 
 
 25.0798724 
 
 1589S25 
 
 
 679 
 
 46 10 41 
 
 26.0576284 
 
 1472754 
 
 630 
 
 39 69 00 
 
 25.099800S 
 
 15S7302 
 
 
 680 
 
 46 24 00 
 
 26.0768096 
 
 14705S8 
 
 631 
 
 39 81 61 
 
 25.1197134 
 
 15S47S6 
 
 
 681 
 
 40 37 61 
 
 26.0959767 
 
 1468429 
 
 632 
 
 39 94 24 
 
 25.1396102 
 
 158227S 
 
 
 682 
 
 46 51 24 
 
 26.1151297 
 
 1466276 
 
 633 
 
 40 06 89 
 
 25.1594913 
 
 1579779 
 
 
 683 
 
 46 64 89 
 
 26.1342687 
 
 1464129 
 
 634 
 
 40 19 56 
 
 25.1793566 
 
 15772S7 
 
 
 684 
 
 46 78 56 
 
 26.1533937 
 
 1461988 
 
 635 
 
 40 32 25 
 
 25.1992063 
 
 1574S03 
 
 
 685 
 
 46 92 25 
 
 26.1725047 
 
 1459S54 
 
 636 
 
 40 44 96 
 
 25.2190404 
 
 1572327 
 
 
 686 
 
 47 05 96 
 
 26.1916017 
 
 1457726 
 
 637 
 
 40 57 69 
 
 25.238S5S9 
 
 1569859 
 
 
 687 
 
 47 19 69 
 
 26.2106848 
 
 1455604 
 
 638 
 
 40 70 44 
 
 25 2586619 
 
 1567398 
 
 
 6S8 
 
 47 33 44 
 
 26.2297541 
 
 1453488 
 
 639 
 
 40 S3 21 
 
 25 2784493 
 
 1564945 
 
 
 6S9 
 
 47 47 21 
 
 26.24SS095 
 
 1451379 
 
 640 
 
 40 90 00 
 
 25.2982213 
 
 1562500 
 
 
 690 
 
 47 61 00 
 
 26.267S511 
 
 1449275 
 
 641 
 
 41 OS 81 
 
 25.3179778 
 
 1560062 
 
 
 691 
 
 47 74 81 
 
 26.2S687S9 
 
 1447178 
 
 642 
 
 41 21 64 
 
 25 33771S9 
 
 1557632 
 
 
 692 
 
 47 88 64 
 
 26.3058929 
 
 1445087 
 
 643 
 
 41 34 49 
 
 25.3574447 
 
 1555210 
 
 
 693 
 
 4S 02 49 
 
 26.3248932 
 
 1443001 
 
 644 
 
 41 47 36 
 
 25.3771551 
 
 1552795 
 
 
 694 
 
 48 16 36 
 
 26.343S797 
 
 1440922 
 
 645 
 
 41 60 25 
 
 25.3968502 
 
 1550388 
 
 
 695 
 
 48 30 25 
 
 26.3628527 
 
 143SS49 
 
 640 
 
 41 73 16 
 
 25.4165301 
 
 1547988 
 
 
 696 
 
 48 44 16 
 
 26 3818119 
 
 1436782 
 
 647 
 
 41 86 09 
 
 25.4361947 
 
 1545595 
 
 
 697 
 
 4S 58 09 
 
 26.4007576 
 
 1434720 
 
 648 
 
 41 99 04 
 
 25.455S441 
 
 1543210 
 
 
 698 
 
 48 72 04 
 
 26.4196896 
 
 1432665 
 
 649 
 
 42 12 01 
 
 25.4754784 
 
 1540832 
 
 
 699 
 
 48 86 01 
 
 26 43S60S1 
 
 1430615 
 
 650 
 
 42 25 00 
 
 25.4950976 
 
 153S462 
 
 
 700 
 
 49 00 00 
 
 26.4575131 
 
 142S571 
 
TABLES 
 
 301 
 
 TABLE IV—( Continued ) 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 701 
 
 49 14 01 
 
 26.4764046 
 
 1426534 
 
 
 751 
 
 56 40 01 
 
 27.4043792 
 
 1331558 
 
 702 
 
 49 28 04 
 
 26.4952826 
 
 1424501 
 
 
 752 
 
 56 55 04 
 
 27.4226184 
 
 1329787 
 
 703 
 
 49 42 09 
 
 26.5141472 
 
 1422475 
 
 
 753 
 
 56 70 09 
 
 27.4408455 
 
 1328021 
 
 704 
 
 49 56 16 
 
 26.5329983 
 
 1420455 
 
 
 754 
 
 56 85 16 
 
 27.4590604 
 
 1326260 
 
 705 
 
 49 70 25 
 
 26.5518361 
 
 1418440 
 
 
 755 
 
 57 00 25 
 
 27.4772633 
 
 1324503 
 
 706 
 
 49 84 36 
 
 26.5706605 
 
 1416431 
 
 
 756 
 
 57 15 36 
 
 27.4954542 
 
 1322751 
 
 707 
 
 49 98 49 
 
 26.5894716 
 
 1414427 
 
 
 757 
 
 57 30 49 
 
 27.5136330 
 
 1321004 
 
 708 
 
 50 12 64 
 
 26.6082694 
 
 1412429 
 
 
 758 
 
 57 45 64 
 
 27.5317998 
 
 1319261 
 
 709 
 
 50 26 81 
 
 26.6270539 
 
 1410437 
 
 
 759 
 
 57 60 SI 
 
 27.5499546 
 
 1317523 
 
 710 
 
 50 41 00 
 
 26.6458252 
 
 1408451 
 
 
 760 
 
 57 76 00 
 
 27.5680975 
 
 1315789 
 
 711 
 
 50 55 21 
 
 26.6645833 
 
 1406470 
 
 
 761 
 
 57 91 21 
 
 27.5862284 
 
 1314060 
 
 712 
 
 50 69 44 
 
 26.6833281 
 
 1404494 
 
 
 762 
 
 58 06 44 
 
 27.6043475 
 
 1312336 
 
 713 
 
 50 83 69 
 
 26.7020598 
 
 1402525 
 
 
 763 
 
 58 21 69 
 
 27.6224546 
 
 1310616 
 
 714 
 
 50 97 96 
 
 26.7207784 
 
 1400560 
 
 
 764 
 
 58 36 96 
 
 27.6405499 
 
 1308901 
 
 715 
 
 51 12 25 
 
 26.7394839 
 
 1398601 
 
 
 765 
 
 58 52 25 
 
 27.6586334 
 
 1307190 
 
 716 
 
 51 26 56 
 
 26.7581763 
 
 1396648 
 
 
 766 
 
 58 67 56 
 
 27.6767050 
 
 1305483 
 
 717 
 
 51 40 89 
 
 26.7768557 
 
 1394700 
 
 
 767 
 
 58 82 89 
 
 27.6947648 
 
 1303781 
 
 718 
 
 51 55 24 
 
 26.7955220 
 
 1392758 
 
 
 768 
 
 58 98 24 
 
 27.7128129 
 
 1302083 
 
 719 
 
 51 69 61 
 
 26.8141754 
 
 1390821 
 
 
 769 
 
 59 13 61 
 
 27.7308492 
 
 1300390 
 
 720 
 
 51 84 00 
 
 26.8328157 
 
 1388889 
 
 
 770 
 
 59 29 00 
 
 27.7488739 
 
 1298701 
 
 721 
 
 51 98 41 
 
 26.8514432 
 
 1386963 
 
 
 771 
 
 59 44 41 
 
 27.7668868 
 
 1297017 
 
 722 
 
 52 12 84 
 
 26.8700577 
 
 1385042 
 
 
 772 
 
 59 59 84 
 
 27.7848880 
 
 1295337 
 
 723 
 
 52 27 29 
 
 26.8886593 
 
 1383126 
 
 
 773 
 
 59 75 29 
 
 27.8028775 
 
 1293661 
 
 724 
 
 52 41 76 
 
 26.9072481 
 
 1381215 
 
 
 774 
 
 59 90 70 
 
 27.8208555 
 
 1291990 
 
 725 
 
 52 56 25 
 
 26.9258240 
 
 1379310 
 
 
 775 
 
 60 06 25 
 
 27.8388218 
 
 1290323 
 
 726 
 
 52 70 76 
 
 26.9443872 
 
 1377410 
 
 
 776 
 
 60 21 76 
 
 27.8567766 
 
 1288660 
 
 727 
 
 52 85 29 
 
 26.9629375 
 
 1375516 
 
 
 777 
 
 60 37 29 
 
 27.8747197 
 
 1287001 
 
 728 
 
 52 99 84 
 
 26.9814751 
 
 1373626 
 
 
 778 
 
 60 52 84 
 
 27.8926514 
 
 1285347 
 
 729 
 
 53 14 41 
 
 27.0000000 
 
 1371742 
 
 
 779 
 
 60 68 41 
 
 27.9105715 
 
 1283697 
 
 730 
 
 53 29 00 
 
 27.0185122 
 
 1369863 
 
 
 780 
 
 60 84 00 
 
 27.9284801 
 
 1282051 
 
 731 
 
 53 43 61 
 
 27.0370117 
 
 1367989 
 
 
 781 
 
 60 99 61 
 
 27.9463772 
 
 1280410 
 
 732 
 
 53 58 24 
 
 27.0554985 
 
 1366120 
 
 
 782 
 
 61 15 24 
 
 27.9642629 
 
 1278772 
 
 733 
 
 53 72 89 
 
 27.0739727 
 
 1364256 
 
 
 783 
 
 61 30 89 
 
 27.9821372 
 
 1277139 
 
 734 
 
 53 87 56 
 
 27.0924344 
 
 136239S 
 
 
 7S4 
 
 61 46 50 
 
 28.0000000 
 
 1275510 
 
 735 
 
 54 02 25 
 
 27.1108834 
 
 1360544 
 
 
 785 
 
 61 62 25 
 
 28.0178515 
 
 1273885 
 
 736 
 
 54 16 96 
 
 27.1293199 
 
 1358696 
 
 
 780 
 
 61 77 96 
 
 28.0350915 
 
 1272265 
 
 737 
 
 54 31 69 
 
 27.1477439 
 
 1356852 
 
 
 787 
 
 61 93 69 
 
 28.0535203 
 
 1270648 
 
 738 
 
 54 46 44 
 
 27.1661554 
 
 1355014 
 
 
 788 
 
 62 09 44 
 
 28.0713377 
 
 1269036 
 
 739 
 
 54 61 21 
 
 27.1845544 
 
 1353180 
 
 
 789 
 
 62 25 21 
 
 28.0891438 
 
 1267427 
 
 740 
 
 54 76 00 
 
 27.2029410 
 
 1351351 
 
 
 790 
 
 62 41 00 
 
 28.1069386 
 
 1265823 
 
 741 
 
 54 90 81 
 
 27.2213152 
 
 1349528 
 
 
 791 
 
 62 56 81 
 
 28.1247222 
 
 1264223 
 
 742 
 
 55 05 64 
 
 27.2396709 
 
 1347709 
 
 
 792 
 
 02 72 64 
 
 28.1424946 
 
 1262626 
 
 743 
 
 55 20 49 
 
 27.2580263 
 
 1345895 
 
 
 793 
 
 62 88 49 
 
 28.1602557 
 
 1261034 
 
 744 
 
 55 35 30 
 
 27.2763034 
 
 1344086 
 
 
 794 
 
 63 04 36 
 
 28.1780056 
 
 1259446 
 
 745 
 
 55 50 25 
 
 27.2946881 
 
 1342282 
 
 
 795 
 
 63 20 25 
 
 28.1957444 
 
 1257862 
 
 746 
 
 55 65 16 
 
 27.3130006 
 
 1340483 
 
 
 796 
 
 63 36 16 
 
 28.2134720 
 
 1256281 
 
 747 
 
 55 80 09 
 
 27.3313007 
 
 1338088 
 
 
 797 
 
 63 52 09 
 
 28.2311884 
 
 1254705 
 
 748 
 
 55 95 04 
 
 27.3495887 
 
 1336898 
 
 
 798 
 
 63 68 04 
 
 28.2488938 
 
 1253133 
 
 749 
 
 56 10 01 
 
 27.3678644 
 
 1335113 
 
 
 799 
 
 63 84 01 
 
 28.2665881 
 
 1251564 
 
 750 
 
 56 25 00 
 
 27.3861279 
 
 1333333 
 
 
 800 
 
 64 00 00 
 
 28.2842712 
 
 1250000 
 
302 
 
 TABLES 
 
 TABLE IV—( Continued ) 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 801 
 
 64 16 01 
 
 28.3019434 
 
 1248439 
 
 
 851 
 
 72 42 01 
 
 29.1719043 
 
 1175088 
 
 802 
 
 64 32 04 
 
 28.3196045 
 
 12468S3 
 
 
 852 
 
 72 59 04 
 
 29.1890390 
 
 1173709 
 
 803 
 
 64 4S 09 
 
 28.3372546 
 
 1245330 
 
 
 853 
 
 72 76 09 
 
 29 2061637 
 
 1172333 
 
 804 
 
 64 64 16 
 
 28.3548938 
 
 12437S1 
 
 
 854 
 
 72 93 16 
 
 29.2232784 
 
 1170960 
 
 805 
 
 64 80 25 
 
 2S.3725219 
 
 1242236 
 
 
 855 
 
 73 10 25 
 
 29.2403830 
 
 1169591 
 
 806 
 
 64 96 36 
 
 28.3901391 
 
 1240695 
 
 
 856 
 
 73 27 36 
 
 29.2574777 
 
 116S224 
 
 807 
 
 65 12 49 
 
 28.4077454 
 
 1239157 
 
 
 857 
 
 73 44 49 
 
 29.2745623 
 
 1166861 
 
 SOS 
 
 65 28 64 
 
 28.4253408 
 
 1237624 
 
 
 858 
 
 73 61 64 
 
 29.2916370 
 
 1165501 
 
 809 
 
 65 44 81 
 
 28.4429253 
 
 1236094 
 
 
 859 
 
 73 78 81 
 
 29.30S7018 
 
 1164144 
 
 810 
 
 65 61 00 
 
 28.46049S9 
 
 1234568 
 
 
 860 
 
 73 96 00 
 
 29.3257566 
 
 1162791 
 
 811 
 
 65 77 21 
 
 28.4780617 
 
 1233046 
 
 
 861 
 
 74 13 21 
 
 29.3428015 
 
 1161440 
 
 S12 
 
 65 93 44 
 
 28.4956137 
 
 1231527 
 
 
 862 
 
 74 30 44 
 
 20.3598365 
 
 1160093 
 
 813 
 
 66 09 69 
 
 28 5131549 
 
 1230012 
 
 
 863 
 
 74 47 69 
 
 29.3768616 
 
 1158749 
 
 814 
 
 66 25 96 
 
 28.5306852 
 
 1228501 
 
 
 864 
 
 74 64 96 
 
 29.3938769 
 
 1157407 
 
 815 
 
 66 42 25 
 
 28.54S2048 
 
 1226994 
 
 
 S65 
 
 74 82 25 
 
 29.4108823 
 
 1156069 
 
 816 
 
 66 58 56 
 
 28 5657137 
 
 1225490 
 
 
 866 
 
 74 99 56 
 
 29.4278779 
 
 1154734 
 
 817 
 
 66 74 89 
 
 28.5S32119 
 
 1223990 
 
 
 867 
 
 75 16 89 
 
 29.4448637 
 
 1153403 
 
 818 
 
 66 91 24 
 
 28 6006993 
 
 1222494 
 
 
 868 
 
 75 34 24 
 
 29.4618397 
 
 1152074 
 
 819 
 
 67 07 61 
 
 28 6181760 
 
 1221001 
 
 
 S69 
 
 75 51 61 
 
 29.4788059 
 
 1150748 
 
 820 
 
 67 24 00 
 
 28.6356421 
 
 1219512 
 
 
 870 
 
 75 69 00 
 
 29.4957624 
 
 114942.' 
 
 821 
 
 67 40 41 
 
 28 6530976 
 
 1218027 
 
 
 871 
 
 75 86 41 
 
 29.5127091 
 
 1148106 
 
 822 
 
 67 56 84 
 
 28 6705424 
 
 1216545 
 
 
 872 
 
 76 03 84 
 
 29.5296461 
 
 1146789 
 
 823 
 
 67 73 29 
 
 28.6879766 
 
 1215067 
 
 
 873 
 
 76 21 29 
 
 29.5465734 
 
 1145475 
 
 824 
 
 67 89 76 
 
 28.7054002 
 
 1213592 
 
 
 874 
 
 76 38 76 
 
 29 5634910 
 
 1144165 
 
 825 
 
 68 06 25 
 
 28.7228132 
 
 1212121 
 
 
 875 
 
 76 56 25 
 
 29.5803989 
 
 1142857 
 
 826 
 
 68 22 76 
 
 28 7402157 
 
 1210654 
 
 
 876 
 
 76 73 76 
 
 29 5972972 
 
 1141553 
 
 827 
 
 68 39 29 
 
 28.7576077 
 
 1209190 
 
 
 877 
 
 76 91 29 
 
 20.6141858 
 
 1140251 
 
 828 
 
 6S 55 84 
 
 28.7749S91 
 
 1207729 
 
 
 878 
 
 77 08 84 
 
 29.6310648 
 
 1138952 
 
 829 
 
 68 72 41 
 
 28 792360! 
 
 1206273 
 
 
 879 
 
 77 26 41 
 
 29 6479342 
 
 1137656 
 
 830 
 
 68 89 00 
 
 28.8097206 
 
 1204819 
 
 
 880 
 
 77 44 00 
 
 29.6647939 
 
 1136364 
 
 831 
 
 69 05 61 
 
 28 8270706 
 
 1203369 
 
 
 881 
 
 77 61 61 
 
 29 6816442 
 
 1135074 
 
 832 
 
 69 22 24 
 
 28 8444102 
 
 1201923 
 
 
 882 
 
 77 79 24 
 
 29.6984848 
 
 1133787 
 
 833 
 
 69 38 89 
 
 28.8617394 
 
 1200480 
 
 
 883 
 
 77 96 89 
 
 29.7153159 
 
 1132503 
 
 834 
 
 69 55 56 
 
 28.8790582 
 
 1199041 
 
 
 884 
 
 78 14 56 
 
 29.7321375 
 
 1131222 
 
 835 
 
 69 72 25 
 
 28.8963666 
 
 1197605 
 
 
 885 
 
 78 32 25 
 
 29.7489496 
 
 1129944 
 
 836 
 
 69 88 96 
 
 28 9136646 
 
 1196172 
 
 
 886 
 
 78 49 96 
 
 29 7657521 
 
 1128668 
 
 837 
 
 70 05 69 
 
 28.9309523 
 
 1194743 
 
 
 887 
 
 7S 67 69 
 
 29.7825452 
 
 1127396 
 
 838 
 
 70 22 44 
 
 28 9482297 
 
 1193317 
 
 
 888 
 
 78 85 44 
 
 29.7993289 
 
 1126126 
 
 839 
 
 70 39 21 
 
 28 9654967 
 
 1191S95 
 
 
 889 
 
 79 03 21 
 
 29.8161030 
 
 1124859 
 
 840 
 
 70 56 00 
 
 28 9S27535 
 
 1190476 
 
 
 890 
 
 79 21 00 
 
 29.8328678 
 
 1123596 
 
 841 
 
 70 72 81 
 
 29.0000000 
 
 1189061 
 
 
 891 
 
 79 38 81 
 
 29 8496231 
 
 1122334 
 
 842 
 
 70 89 64 
 
 29.0172363 
 
 1187648 
 
 
 892 
 
 79 56 64 
 
 29.8663690 
 
 1121076 
 
 843 
 
 71 06 49 
 
 29 0344623 
 
 1186240 
 
 
 893 
 
 79 74 49 
 
 29 8831056 
 
 1119821 
 
 844 
 
 71 23 36 
 
 29 0516781 
 
 U84834 
 
 
 894 
 
 79 92 36 
 
 29.8998328 
 
 1118568 
 
 845 
 
 71 40 25 
 
 29.06S8S37 
 
 1183432 
 
 
 895 
 
 80 10 25 
 
 29.9165506 
 
 1117318 
 
 846 
 
 71 57 16 
 
 29 0860791 
 
 1182033 
 
 
 896 
 
 80 28 16 
 
 29 9332591 
 
 1116071 
 
 847 
 
 71 74 09 
 
 29.1032644 
 
 1180638 
 
 
 897 
 
 80 46 09 
 
 29.9499583 
 
 1114827 
 
 848 
 
 71 91 04 
 
 29.1204396 
 
 1179245 
 
 
 89S 
 
 80 64 04 
 
 29 9666481 
 
 1113586 
 
 849 
 
 72 08 01 
 
 29 1376046 
 
 1177856 
 
 
 899 
 
 SO 82 01 
 
 29 98332S7 
 
 1112347 
 
 850 
 
 72 25 00 
 
 29.1547595 
 
 1176471 
 
 
 900 
 
 81 00 00 
 
 30 0000000 
 
 1111111 
 
TABLES 
 
 303 
 
 TABLE IV—( Continued ) 
 
 Squares, Square Roots, and Reciprocals to 1000 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 3 
 
 
 No. 
 
 Square 
 
 Square Root 
 
 Recipro¬ 
 cal X 10 9 
 
 901 
 
 81 18 01 
 
 30.0166620 
 
 1109878 
 
 
 951 
 
 90 44 01 
 
 30.8382879 
 
 1051525 
 
 902 
 
 81 36 04 
 
 30.0333148 
 
 1108647 
 
 
 952 
 
 90 63 04 
 
 30.8544972 
 
 1050420 
 
 903 
 
 81 54 09 
 
 30.0499584 
 
 1107420 
 
 
 953 
 
 90 82 09 
 
 30.8706981 
 
 1049318 
 
 904 
 
 81 72 16 
 
 30.0665928 
 
 1106195 
 
 
 954 
 
 91 01 16 
 
 30.8868904 
 
 1048218 
 
 905 
 
 81 90 25 
 
 30.0832179 
 
 1104972 
 
 
 955 
 
 91 20 25 
 
 30.9030743 
 
 1047120 
 
 906 
 
 82 08 36 
 
 30.0998339 
 
 1103753 
 
 
 956 
 
 91 39 36 
 
 30.9192497 
 
 1046025 
 
 907 
 
 82 26 49 
 
 30.1164407 
 
 1102536 
 
 
 957 
 
 91 58 49 
 
 30.9354166 
 
 1044932 
 
 908 
 
 82 44 64 
 
 30.1330383 
 
 1101322 
 
 
 958 
 
 91 77 64 
 
 30.9515751 
 
 1043841 
 
 909 
 
 82 62 81 
 
 30.1496269 
 
 1100110 
 
 
 959 
 
 91 96 81 
 
 30.9677251 
 
 1042753 
 
 910 
 
 82 81 00 
 
 30.1662063 
 
 1098901 
 
 
 960 
 
 92 16 00 
 
 30.9838668 
 
 1041667 
 
 911 
 
 82 99 21 
 
 30.1827765 
 
 1097695 
 
 
 961 
 
 92 35 21 
 
 31.0000000 
 
 1040583 
 
 912 
 
 83 17 44 
 
 30.1993377 
 
 1096491 
 
 
 962 
 
 92 54 44 
 
 31 0161248 
 
 1039501 
 
 913 
 
 83 35 69 
 
 30.2158899 
 
 1095290 
 
 
 963 
 
 92 73 69 
 
 31.0322413 
 
 1038422 
 
 914 
 
 83 53 96 
 
 30.2324329 
 
 1094092 
 
 
 964 
 
 92 92 96 
 
 31.0483494 
 
 1037344 
 
 915 
 
 83 72 25 
 
 30.2489609 
 
 1092896 
 
 
 965 
 
 93 12 25 
 
 31.0644491 
 
 1036269 
 
 916 
 
 83 90 56 
 
 30.2654919 
 
 1091703 
 
 
 966 
 
 93 31 56 
 
 31.0805405 
 
 1035197 
 
 917 
 
 84 08 89 
 
 30.2820079 
 
 1090513 
 
 
 967 
 
 93 50 89 
 
 31.0966236 
 
 1034126 
 
 918 
 
 84 27 24 
 
 30.2985148 
 
 1089325 
 
 
 968 
 
 93 70 24 
 
 31.1126984 
 
 1033058 
 
 919 
 
 84 45 61 
 
 30.3150128 
 
 1088139 
 
 
 969 
 
 93 89 61 
 
 31.1287648 
 
 1031992 
 
 920 
 
 84 64 00 
 
 30.3315018 
 
 1086957 
 
 
 970 
 
 94 09 00 
 
 31.1448230 
 
 1030928 
 
 921 
 
 84 82 41 
 
 30.3479818 
 
 1085776 
 
 
 971 
 
 94 28 41 
 
 31.1608729 
 
 1029866 
 
 922 
 
 85 00 84 
 
 30.3644529 
 
 1084599 
 
 
 972 
 
 94 47 84 
 
 31.1769145 
 
 1028807 
 
 923 
 
 85 19 29 
 
 30.3809151 
 
 1083424 
 
 
 973 
 
 94 67 29 
 
 31.1929479 
 
 1027749 
 
 924 
 
 85 37 76 
 
 30.3973683 
 
 1082251 
 
 
 974 
 
 94 86 70 
 
 31.2089731 
 
 1026694 
 
 925 
 
 85 56 25 
 
 30.4138127 
 
 1081081 
 
 
 975 
 
 95 06 25 
 
 31.2249900 
 
 1025641 
 
 926 
 
 85 74 76 
 
 30.4302481 
 
 1079914 
 
 
 976 
 
 95 25 76 
 
 31.2409987 
 
 1024590 
 
 927 
 
 85 93 29 
 
 30.4466747 
 
 1078749 
 
 
 977 
 
 95 45 29 
 
 31.2569992 
 
 1023541 
 
 928 
 
 86 11 84 
 
 30.4030924 
 
 1077586 
 
 
 978 
 
 95 64 84 
 
 31.2729915 
 
 1022495 
 
 929 
 
 86 30 41 
 
 30.4795013 
 
 1076426 
 
 
 979 
 
 95 84 41 
 
 31.2889757 
 
 1021450 
 
 930 
 
 86 49 00 
 
 30.4959014 
 
 1075269 
 
 
 980 
 
 96 04 00 
 
 31.3049517 
 
 1020408 
 
 931 
 
 86 67 61 
 
 30.5122926 
 
 1074114 
 
 
 981 
 
 96 23 61 
 
 31.3209195 
 
 1019368 
 
 932 
 
 86 86 24 
 
 30.5286750 
 
 1072961 
 
 
 982 
 
 90 43 24 
 
 31.3368792 
 
 1018330 
 
 933 
 
 87 04 89 
 
 30.5450487 
 
 1071811 
 
 
 983 
 
 96 62 89 
 
 31.3528308 
 
 1017294 
 
 934 
 
 87 23 56 
 
 30.5614136 
 
 1070664 
 
 
 984 
 
 96 82 50 
 
 31.3687743 
 
 1016200 
 
 935 
 
 87 42 25 
 
 30.5777097 
 
 1069519 
 
 
 985 
 
 97 02 25 
 
 31.3847097 
 
 1015228 
 
 936 
 
 87 60 96 
 
 30.5941171 
 
 1068376 
 
 
 980 
 
 97 21 96 
 
 31.4006369 
 
 1014199 
 
 937 
 
 87 79 69 
 
 30.6104557 
 
 1067236 
 
 
 987 
 
 97 41 69 
 
 31.4165561 
 
 1013171 
 
 938 
 
 87 98 44 
 
 30.6267857 
 
 1066098 
 
 
 988 
 
 97 01 44 
 
 31.4324673 
 
 1012146 
 
 939 
 
 88 17 21 
 
 30.6431069 
 
 1064963 
 
 
 989 
 
 97 81 21 
 
 31.4483704 
 
 1011122 
 
 940 
 
 88 36 00 
 
 30.0594194 
 
 1003830 
 
 
 990 
 
 98 01 00 
 
 31.4642054 
 
 1010101 
 
 941 
 
 88 54 81 
 
 30.6757233 
 
 1062699 
 
 
 991 
 
 98 20 81 
 
 31.4801525 
 
 1009082 
 
 942 
 
 88 73 64 
 
 30.6920185 
 
 1061571 
 
 
 992 
 
 98 40 64 
 
 31.4960315 
 
 1008065 
 
 943 
 
 88 92 49 
 
 30.7083051 
 
 1060445 
 
 
 993 
 
 98 60 49 
 
 31.5119025 
 
 1007049 
 
 944 
 
 89 11 36 
 
 30.7245830 
 
 1059322 
 
 
 994 
 
 98 80 30 
 
 31.5277655 
 
 1006036 
 
 945 
 
 89 30 25 
 
 30.7408523 
 
 1058201 
 
 
 995 
 
 99 00 25 
 
 31.5436206 
 
 1005025 
 
 946 
 
 89 49 16 
 
 30.7571130 
 
 1057082 
 
 
 996 
 
 99 20 16 
 
 31.5594677 
 
 1004016 
 
 947 
 
 89 68 09 
 
 30.7733651 
 
 1055966 
 
 
 997 
 
 99 40 09 
 
 31.5753068 
 
 1003009 
 
 948 
 
 89 87 04 
 
 30 7896086 
 
 1054852 
 
 
 998 
 
 99 60 04 
 
 31.5911380 
 
 1002004 
 
 949 
 
 90 06 01 
 
 30.8058436 
 
 1053741 
 
 
 999 
 
 99 80 01 
 
 31.6069613 
 
 1001001 
 
 950 
 
 90 25 00 
 
 30.8220700 
 
 1052632 
 
 
 1000 
 
 1 00 00 00 
 
 31.6227766 
 
 1000000 
 
304' 
 
 TABLES 
 
 TABLE V* 
 
 Five-place Logarithms: 100-150* 
 
 N 
 
 O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
 100 
 
 00 000 
 
 043 
 
 087 
 
 130 
 
 173 
 
 217 
 
 260 
 
 303 
 
 346 
 
 389 
 
 
 
 01 
 
 432 
 
 475 
 
 518 
 
 561 
 
 604 
 
 647 
 
 689 
 
 732 
 
 775 
 
 817 
 
 
 
 02 
 
 00 860 
 
 903 
 
 945 
 
 988 
 
 *030 
 
 *072 
 
 *115 
 
 *157 
 
 *199 
 
 *242 
 
 
 
 05 
 
 01 284 
 
 326 
 
 368 
 
 410 
 
 452 
 
 494 
 
 536 
 
 578 
 
 620 
 
 662 
 
 1 
 
 2 
 
 4.4 4.3 4.2 
 8.8 8.6 8.4 
 
 04 
 
 01 703 
 
 745 
 
 787 
 
 828 
 
 870 
 
 912 
 
 953 
 
 995 
 
 *036 
 
 *078 
 
 3 
 
 13.2 12.9 12.6 
 
 05 
 
 02 119 
 
 160 
 
 202 
 
 243 
 
 284 
 
 325 
 
 366 
 
 407 
 
 449 
 
 490 
 
 5 
 
 22 o 21 z 9 i n 
 
 06 
 
 531 
 
 572 
 
 612 
 
 655 
 
 694 
 
 735 
 
 776 
 
 816 
 
 857 
 
 898 
 
 G 
 
 26.4 25.8 25.2 
 
 07 
 
 02 938 
 
 979 
 
 *019 
 
 *060 
 
 *100 
 
 *141 
 
 *181 
 
 *222 
 
 *262 
 
 *302 
 
 7 
 
 8 
 
 30.8 30.1 29.4 
 35.2 34.4 33.6 
 
 08 
 
 03 342 
 
 383 
 
 423 
 
 463 
 
 503 
 
 543 
 
 583 
 
 623 
 
 663 
 
 703 
 
 9 
 
 39.6 38.7 37.8 
 
 09 
 
 03 743 
 
 782 
 
 822 
 
 862 
 
 902 
 
 941 
 
 981 
 
 *021 
 
 *060 
 
 *100 
 
 
 
 110 
 
 04 159 
 
 179 
 
 218 
 
 258 
 
 297 
 
 336 
 
 376 
 
 415 
 
 454 
 
 493 
 
 
 
 11 
 
 532 
 
 571 
 
 610 
 
 650 
 
 689 
 
 727 
 
 766 
 
 805 
 
 844 
 
 883 
 
 
 
 12 
 
 04 922 
 
 961 
 
 999 
 
 *058 
 
 *077 
 
 *115 
 
 *154 
 
 *192 
 
 *231 
 
 *269 
 
 
 41 40 39 
 
 15 
 
 05 308 
 
 346 
 
 385 
 
 423 
 
 461 
 
 500 
 
 538 
 
 576 
 
 614 
 
 652 
 
 1 
 
 4.1 4 3.9 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 8.2 8 7.8 
 
 14 
 
 05 690 
 
 729 
 
 767 
 
 805 
 
 843 
 
 881 
 
 918 
 
 956 
 
 994 
 
 *032 
 
 3 
 
 12.3 12 11.7 
 
 15 
 
 06 070 
 
 108 
 
 145 
 
 183 
 
 221 
 
 258 
 
 296 
 
 333 
 
 371 
 
 408 
 
 4 
 
 16.4 16 15.6 
 
 16 
 
 446 
 
 483 
 
 521 
 
 658 
 
 595 
 
 633 
 
 670 
 
 707 
 
 744 
 
 781 
 
 5 
 
 20.5 20 19.5 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 24.6 24 23.4 
 
 17 
 
 06 819 
 
 856 
 
 893 
 
 930 
 
 967 
 
 *004 
 
 *041 
 
 *078 
 
 *115 
 
 *151 
 
 7 
 
 28.7 28 27.3 
 
 18 
 
 07 188 
 
 225 
 
 262 
 
 298 
 
 335 
 
 372 
 
 408 
 
 445 
 
 482 
 
 518 
 
 9 
 
 32.8 32 31.2 
 
 19 
 
 555 
 
 591 
 
 628 
 
 664 
 
 700 
 
 737 
 
 773 
 
 809 
 
 846 
 
 882 
 
 
 
 120 
 
 07 918 
 
 954 
 
 990 
 
 *027 
 
 *063 
 
 *099 
 
 *135 
 
 *171 
 
 *207 
 
 *243 
 
 
 
 21 
 
 08 279 
 
 314 
 
 350 
 
 386 
 
 422 
 
 458 
 
 493 
 
 529 
 
 565 
 
 600 
 
 
 
 22 
 
 636 
 
 672 
 
 707 
 
 743 
 
 778 
 
 814 
 
 849 
 
 884 
 
 920 
 
 955 
 
 
 38 37 36 
 
 25 
 
 08 991 
 
 *026 
 
 *061 
 
 *096 
 
 *132 
 
 *167 
 
 *202 
 
 *237 
 
 *272 
 
 *307 
 
 i 
 
 3.8 3.7 3.6 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 7.6 7.4 7.2 
 
 24 
 
 09 342 
 
 377 
 
 412 
 
 447 
 
 482 
 
 517 
 
 552 
 
 587 
 
 621 
 
 656 
 
 3 
 
 11.4 11.1 10.8 
 
 25 
 
 09 691 
 
 726 
 
 760 
 
 795 
 
 830 
 
 864 
 
 899 
 
 934 
 
 968 
 
 *003 
 
 4 
 
 15.2 14.8 14.4 
 
 26 
 
 10 037 
 
 072 
 
 106 
 
 140 
 
 175 
 
 209 
 
 243 
 
 278 
 
 312 
 
 346 
 
 6 
 
 22.8 22.2 21.6 
 
 27 
 
 380 
 
 415 
 
 449 
 
 483 
 
 517 
 
 551 
 
 585 
 
 619 
 
 653 
 
 687 
 
 7 
 
 8 
 
 26.6 25.9 25.2 
 50 4 29 6 28 ft 
 
 28 
 
 10 721 
 
 755 
 
 789 
 
 823 
 
 857 
 
 890 
 
 924 
 
 958 
 
 992 
 
 *025 
 
 9 
 
 34.2 33.3 32.4 
 
 29 
 
 11 059 
 
 093 
 
 126 
 
 160 
 
 193 
 
 227 
 
 261 
 
 294 
 
 327 
 
 361 
 
 
 
 130 
 
 394 
 
 428 
 
 461 
 
 494 
 
 528 
 
 561 
 
 594 
 
 628 
 
 661 
 
 694 
 
 
 
 31 
 
 11 727 
 
 760 
 
 793 
 
 826 
 
 860 
 
 893 
 
 926 
 
 959 
 
 992 
 
 *024 
 
 
 as aj. aa 
 
 52 
 
 12 057 
 
 090 
 
 123 
 
 156 
 
 189 
 
 222 
 
 254 
 
 287 
 
 520 
 
 352 
 
 
 
 35 
 
 385 
 
 418 
 
 450 
 
 483 
 
 516 
 
 548 
 
 581 
 
 613 
 
 646 
 
 678 
 
 2 
 
 3.5 3.4 3.3 
 7.0 6.8 6.6 
 
 34 
 
 12 710 
 
 743 
 
 775 
 
 808 
 
 840 
 
 872 
 
 905 
 
 937 
 
 969 
 
 *001 
 
 3 
 
 4 
 
 10.5 10.2 9.9 
 
 14 0 15 6 15 2 
 
 35 
 
 13 033 
 
 066 
 
 098 
 
 130 
 
 162 
 
 194 
 
 226 
 
 258 
 
 290 
 
 322 
 
 5 
 
 17.5 17.0 16.5 
 
 36 
 
 354 
 
 386 
 
 418 
 
 450 
 
 481 
 
 513 
 
 545 
 
 577 
 
 609 
 
 640 
 
 6 
 
 21.0 20.4 19.8 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 24.5 23.8 23.1 
 
 37 
 
 672 
 
 704 
 
 735 
 
 767 
 
 799 
 
 830 
 
 862 
 
 893 
 
 925 
 
 956 
 
 8 
 
 28.0 27.2 26.4 
 
 38 
 
 13 988 
 
 *019 
 
 *051 
 
 *082 
 
 *114 
 
 *145 
 
 *176 
 
 *208 
 
 *239 
 
 *270 
 
 9 
 
 31.5 30.6 29.7 
 
 39 
 
 14 301 
 
 333 
 
 364 
 
 395 
 
 426 
 
 457 
 
 489 
 
 520 
 
 551 
 
 582 
 
 
 
 140 
 
 613 
 
 644 
 
 675 
 
 706 
 
 737 
 
 768 
 
 799 
 
 829 
 
 860 
 
 891 
 
 
 
 41 
 
 14 922 
 
 953 
 
 983 
 
 *014 
 
 *045 
 
 *076 
 
 *106 
 
 *137 
 
 *168 
 
 *198 
 
 
 
 42 
 
 15 229 
 
 259 
 
 290 
 
 320 
 
 351 
 
 381 
 
 412 
 
 442 
 
 473 
 
 503 
 
 
 
 43 
 
 534 
 
 564 
 
 594 
 
 625 
 
 655 
 
 685 
 
 715 
 
 746 
 
 776 
 
 806 
 
 2 
 
 3.2 3.1 3 
 
 6.4 6.2 6 
 
 44 
 
 15 836 
 
 866 
 
 897 
 
 927 
 
 957 
 
 987 
 
 *017 
 
 *047 
 
 *077 
 
 *107 
 
 3 
 
 4 
 
 9.6 9.3 9 
 
 12 8 12 4 12 
 
 45 
 
 16 137 
 
 167 
 
 197 
 
 227 
 
 256 
 
 286 
 
 316 
 
 346 
 
 376 
 
 406 
 
 5 
 
 16.0 15.5 15 
 
 46 
 
 435 
 
 465 
 
 495 
 
 524 
 
 554 
 
 584 
 
 613 
 
 643 
 
 673 
 
 702 
 
 6 
 
 19.2 18.6 18 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 22.4 21.7 21 
 
 47 
 
 16 732 
 
 761 
 
 791 
 
 820 
 
 850 
 
 879 
 
 909 
 
 938 
 
 967 
 
 997 
 
 8 
 
 25.6 24.8 24 
 
 48 
 
 17 026 
 
 056 
 
 085 
 
 114 
 
 143 
 
 173 
 
 202 
 
 231 
 
 260 
 
 289 
 
 9 
 
 28.8 27.9 27 
 
 49 
 
 319 
 
 348 
 
 377 
 
 406 
 
 435 
 
 464 
 
 493 
 
 522 
 
 551 
 
 580 
 
 
 
 150 
 
 17 609 
 
 638 
 
 667 
 
 696 
 
 725 
 
 754 
 
 782 
 
 811 
 
 840 
 
 869 
 
 
 
 N 
 
 O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
 ♦This table was taken from "Plane Trigonometry with Five-place Tables” by H. A. 1 
 Simmons and G. D. Gore by permission of the authors and the publisher, John Wiley 
 & Sons, Ine. 
 
TABLES 
 
 305 
 
 TABLE V—( Continued ) 
 
 Five-place Logarithms: 150-200 
 
 Prop 
 
 Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 
 150 
 
 17 609 
 
 638 
 
 667 
 
 696 
 
 725 
 
 754 
 
 782 
 
 811 
 
 840 
 
 869 
 
 
 
 
 51 
 
 17 898 
 
 926 
 
 955 
 
 984 
 
 *013 
 
 *041 
 
 *070 
 
 *099 
 
 *127 
 
 *156 
 
 
 29 28 
 
 52 
 
 18 184 
 
 213 
 
 241 
 
 270 
 
 298 
 
 327 
 
 355 
 
 384 
 
 412 
 
 441 
 
 I 
 
 2.9 2.8 
 
 53 
 
 469 
 
 498 
 
 526 
 
 554 
 
 583 
 
 611 
 
 639 
 
 667 
 
 696 
 
 724 
 
 2 
 
 5.8 5.6 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 8.7 8.4 
 
 54 
 
 18 752 
 
 780 
 
 808 
 
 837 
 
 865 
 
 893 
 
 921 
 
 949 
 
 977 
 
 *005 
 
 4 
 
 11.6 11.2 
 
 55 
 
 19 033 
 
 061 
 
 089 
 
 117 
 
 145 
 
 173 
 
 201 
 
 229 
 
 257 
 
 285 
 
 6 
 
 17.4 16.8 
 
 56 
 
 312 
 
 340 
 
 368 
 
 396 
 
 424 
 
 451 
 
 479 
 
 507 
 
 535 
 
 562 
 
 7 
 
 20.3 19.6 
 
 57 
 
 590 
 
 618 
 
 645 
 
 673 
 
 700 
 
 728 
 
 756 
 
 783 
 
 811 
 
 838 
 
 9 
 
 26.1 25.2 
 
 58 
 
 19 866 
 
 893 
 
 921 
 
 948 
 
 976 
 
 *003 
 
 *030 
 
 *058 
 
 *085 
 
 *112 
 
 
 
 
 59 
 
 20 140 
 
 167 
 
 194 
 
 222 
 
 249 
 
 276 
 
 303 
 
 330 
 
 358 
 
 385 
 
 
 
 
 160 
 
 412 
 
 439 
 
 466 
 
 493 
 
 520 
 
 548 
 
 575 
 
 602 
 
 629 
 
 656 
 
 
 
 
 61 
 
 683 
 
 710 
 
 737 
 
 763 
 
 790 
 
 817 
 
 844 
 
 871 
 
 898 
 
 925 
 
 
 27 21! 
 
 62 
 
 20 952 
 
 978 
 
 *005 
 
 *032 
 
 *059 
 
 *085 
 
 *112 
 
 *139 
 
 *165 
 
 *192 
 
 i 
 
 2.7 2.6 
 
 63 
 
 21 219 
 
 245 
 
 272 
 
 299 
 
 325 
 
 352 
 
 378 
 
 405 
 
 431 
 
 458 
 
 2 
 
 5.4 5.2 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 8.1 7.8 
 
 64 
 
 484 
 
 511 
 
 537 
 
 564 
 
 590 
 
 617 
 
 643 
 
 669 
 
 696 
 
 722 
 
 4 
 
 10.8 10.4 
 
 65 
 
 21 748 
 
 775 
 
 801 
 
 827 
 
 854 
 
 880 
 
 906 
 
 932 
 
 958 
 
 985 
 
 5 
 
 G 
 
 13.6 13.0 
 
 16.2 15.6 
 
 66 
 
 22 011 
 
 037 
 
 063 
 
 089 
 
 115 
 
 141 
 
 167 
 
 194 
 
 220 
 
 246 
 
 7 
 
 18.9 18.2 
 
 67 
 
 272 
 
 298 
 
 324 
 
 350 
 
 376 
 
 401 
 
 427 
 
 453 
 
 479 
 
 505 
 
 9 
 
 24.3 23 4 
 
 68 
 
 531 
 
 557 
 
 583 
 
 608 
 
 634 
 
 660 
 
 686 
 
 712 
 
 737 
 
 763 
 
 
 
 
 69 
 
 22 789 
 
 814 
 
 840 
 
 866 
 
 891 
 
 917 
 
 943 
 
 968 
 
 994 
 
 *019 
 
 
 
 
 170 
 
 23 045 
 
 070 
 
 096 
 
 121 
 
 147 
 
 172 
 
 198 
 
 223 
 
 249 
 
 274 
 
 
 
 
 71 
 
 300 
 
 325 
 
 350 
 
 376 
 
 401 
 
 426 
 
 452 
 
 477 
 
 502 
 
 528 
 
 
 
 25 
 
 72 
 
 553 
 
 578 
 
 603 
 
 629 
 
 654 
 
 679 
 
 704 
 
 729 
 
 754 
 
 779 
 
 
 i 
 
 2.5 
 
 73 
 
 23 805 
 
 830 
 
 855 
 
 880 
 
 905 
 
 930 
 
 955 
 
 980 
 
 *005 
 
 *030 
 
 
 2 
 
 5.0 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 7.5 
 
 74 
 
 24 055 
 
 080 
 
 105 
 
 130 
 
 155 
 
 180 
 
 204 
 
 229 
 
 254 
 
 279 
 
 
 4 
 
 10.0 
 
 75 
 
 304 
 
 329 
 
 353 
 
 378 
 
 403 
 
 428 
 
 452 
 
 477 
 
 502 
 
 527 
 
 
 5 
 
 6 
 
 12.5 
 
 15.0 
 
 76 
 
 551 
 
 576 
 
 601 
 
 625 
 
 650 
 
 674 
 
 699 
 
 724 
 
 748 
 
 773 
 
 
 7 
 
 17.5 
 
 77 
 
 24 797 
 
 822 
 
 846 
 
 871 
 
 895 
 
 920 
 
 944 
 
 969 
 
 993 
 
 *018 
 
 
 9 
 
 22.5 
 
 78 
 
 25 042 
 
 066 
 
 091 
 
 115 
 
 139 
 
 164 
 
 188 
 
 212 
 
 237 
 
 261 
 
 
 
 
 79 
 
 285 
 
 310 
 
 334 
 
 358 
 
 382 
 
 406 
 
 431 
 
 455 
 
 479 
 
 503 
 
 
 
 
 180 
 
 527 
 
 551 
 
 575 
 
 600 
 
 624 
 
 648 
 
 672 
 
 696 
 
 720 
 
 744 
 
 
 
 
 81 
 
 25 768 
 
 792 
 
 816 
 
 840 
 
 864 
 
 888 
 
 912 
 
 935 
 
 959 
 
 983 
 
 
 24 23 
 
 82 
 
 26 007 
 
 031 
 
 055 
 
 079 
 
 102 
 
 126 
 
 150 
 
 174 
 
 198 
 
 221 
 
 i 
 
 2.4 2.3 
 
 83 
 
 245 
 
 269 
 
 293 
 
 516 
 
 340 
 
 364 
 
 387 
 
 411 
 
 435 
 
 458 
 
 2 
 
 4.8 4.6 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 7.2 6.9 
 
 84 
 
 482 
 
 505 
 
 529 
 
 553 
 
 576 
 
 600 
 
 623 
 
 647 
 
 670 
 
 694 
 
 4 
 
 9.6 9.2 
 
 85 
 
 717 
 
 741 
 
 764 
 
 788 
 
 811 
 
 834 
 
 858 
 
 881 
 
 905 
 
 928 
 
 5 
 
 G 
 
 12.0 11.5 
 
 14.4 13.8 
 
 86 
 
 26 951 
 
 975 
 
 998 
 
 *021 
 
 *045 
 
 *068 
 
 *091 
 
 *114 
 
 *138 
 
 *161 
 
 7 
 
 16.8 16.1 
 
 87 
 
 27 184 
 
 207 
 
 231 
 
 254 
 
 277 
 
 300 
 
 323 
 
 346 
 
 370 
 
 393 
 
 9 
 
 21 6 20 7 
 
 88 
 
 416 
 
 439 
 
 462 
 
 485 
 
 508 
 
 531 
 
 554 
 
 577 
 
 600 
 
 623 
 
 
 
 
 89 
 
 646 
 
 669 
 
 692 
 
 715 
 
 738 
 
 761 
 
 784 
 
 807 
 
 830 
 
 852 
 
 
 
 
 190 
 
 27 875 
 
 898 
 
 921 
 
 944 
 
 967 
 
 989 
 
 *012 
 
 *035 
 
 *058 
 
 *081 
 
 
 
 
 91 
 
 28 103 
 
 126 
 
 149 
 
 171 
 
 194 
 
 217 
 
 240 
 
 262 
 
 285 
 
 507 
 
 
 22 21 
 
 92 
 
 330 
 
 353 
 
 375 
 
 398 
 
 421 
 
 443 
 
 466 
 
 488 
 
 511 
 
 533 
 
 i 
 
 2.2 2.1 
 
 93 
 
 556 
 
 578 
 
 601 
 
 623 
 
 646 
 
 668 
 
 691 
 
 713 
 
 755 
 
 758 
 
 2 
 
 4.4 4.2 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 6.6 6.3 
 
 94 
 
 28 780 
 
 805 
 
 825 
 
 847 
 
 870 
 
 892 
 
 914 
 
 937 
 
 959 
 
 981 
 
 4 
 
 8.8 8.4 
 
 95 
 
 29 003 
 
 026 
 
 048 
 
 070 
 
 092 
 
 115 
 
 137 
 
 159 
 
 181 
 
 203 
 
 G 
 
 11.0 10.5 
 13.2 12.6 
 
 96 
 
 226 
 
 248 
 
 270 
 
 292 
 
 314 
 
 336 
 
 358 
 
 380 
 
 403 
 
 425 
 
 7 
 
 8 
 
 15.4 14.7 
 
 17 5 1 ft R 
 
 97 
 
 447 
 
 469 
 
 491 
 
 513 
 
 535 
 
 557 
 
 579 
 
 601 
 
 623 
 
 645 
 
 9 
 
 19 8 IK 9 
 
 98 
 
 667 
 
 688 
 
 710 
 
 732 
 
 754 
 
 776 
 
 798 
 
 820 
 
 842 
 
 863 
 
 
 
 
 99 
 
 29 885 
 
 907 
 
 929 
 
 951 
 
 973 
 
 994 
 
 *016 
 
 *038 
 
 *060 
 
 *081 
 
 
 
 
 200 
 
 30 103 
 
 125 
 
 146 
 
 168 
 
 190 
 
 211 
 
 233 
 
 255 
 
 276 
 
 298 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 G 
 
 7 
 
 8 
 
 9 
 
306 
 
 TABLES 
 
 TABLE V—( Continued ) 
 Five-place Logarithms: 200-250 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
 200 
 
 30 103 
 
 125 
 
 146 
 
 168 
 
 190 
 
 211 
 
 233 
 
 255 
 
 276 
 
 298 
 
 
 
 
 01 
 
 320 
 
 341 
 
 363 
 
 384 
 
 406 
 
 428 
 
 449 
 
 471 
 
 492 
 
 514 
 
 
 
 
 02 
 
 535 
 
 557 
 
 578 
 
 600 
 
 621 
 
 643 
 
 664 
 
 685 
 
 707 
 
 728 
 
 
 22 21 
 
 03 
 
 750 
 
 771 
 
 792 
 
 814 
 
 835 
 
 856 
 
 878 
 
 899 
 
 920 
 
 942 
 
 1 
 
 2.2 2,1 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 4.4 4.2 
 
 04 
 
 30 963 
 
 984 
 
 *006 
 
 *027 
 
 *048 
 
 *069 
 
 *091 
 
 *112 
 
 *133 
 
 *154 
 
 3 
 
 6.6 6.3 
 
 05 
 
 31 175 
 
 197 
 
 218 
 
 259 
 
 260 
 
 281 
 
 502 
 
 323 
 
 345 
 
 566 
 
 4 
 
 8.8 8.4 
 
 06 
 
 387 
 
 408 
 
 429 
 
 450 
 
 471 
 
 492 
 
 513 
 
 534 
 
 555 
 
 576 
 
 5 
 
 11.0 10.5 
 
 
 
 
 
 
 
 
 
 
 
 
 0 
 
 13.2 12.6 
 
 07 
 
 597 
 
 618 
 
 639 
 
 660 
 
 681 
 
 702 
 
 723 
 
 744 
 
 765 
 
 785 
 
 7 
 
 15.4 14.7 
 
 08 
 
 31 806 
 
 827 
 
 848 
 
 869 
 
 890 
 
 911 
 
 931 
 
 952 
 
 973 
 
 994 
 
 8 
 
 17.6 16.8 
 
 09 
 
 32 015 
 
 035 
 
 056 
 
 077 
 
 098 
 
 118 
 
 139 
 
 160 
 
 181 
 
 201 
 
 
 
 210 
 
 222 
 
 243 
 
 263 
 
 284 
 
 305 
 
 325 
 
 346 
 
 366 
 
 387 
 
 408 
 
 
 
 
 11 
 
 428 
 
 449 
 
 469 
 
 490 
 
 510 
 
 531 
 
 552 
 
 572 
 
 593 
 
 613 
 
 
 
 
 12 
 
 634 
 
 654 
 
 675 
 
 695 
 
 715 
 
 756 
 
 756 
 
 777 
 
 797 
 
 818 
 
 
 
 20 
 
 13 
 
 32 838 
 
 858 
 
 879 
 
 899 
 
 919 
 
 940 
 
 960 
 
 980 
 
 *001 
 
 *021 
 
 i 
 
 
 2 
 
 14 
 
 33 041 
 
 062 
 
 082 
 
 102 
 
 122 
 
 143 
 
 163 
 
 183 
 
 203 
 
 224 
 
 3 
 
 
 6 
 
 15 
 
 244 
 
 264 
 
 284 
 
 304 
 
 325 
 
 345 
 
 365 
 
 585 
 
 405 
 
 425 
 
 4 
 
 
 8 
 
 16 
 
 445 
 
 465 
 
 486 
 
 506 
 
 526 
 
 546 
 
 566 
 
 586 
 
 606 
 
 626 
 
 5 
 
 6 
 
 
 10 
 
 12 
 
 17 
 
 646 
 
 666 
 
 686 
 
 706 
 
 726 
 
 746 
 
 766 
 
 786 
 
 806 
 
 826 
 
 7 
 
 
 14 
 
 18 
 
 33 846 
 
 866 
 
 885 
 
 905 
 
 925 
 
 945 
 
 965 
 
 985 
 
 *005 
 
 *025 
 
 8 
 
 
 16 
 
 19 
 
 34 044 
 
 064 
 
 084 
 
 104 
 
 124 
 
 143 
 
 163 
 
 183 
 
 203 
 
 223 
 
 
 
 
 220 
 
 242 
 
 262 
 
 282 
 
 301 
 
 321 
 
 541 
 
 361 
 
 380 
 
 400 
 
 420 
 
 
 
 
 21 
 
 439 
 
 459 
 
 479 
 
 498 
 
 518 
 
 537 
 
 557 
 
 577 
 
 596 
 
 616 
 
 
 
 
 22 
 
 635 
 
 655 
 
 674 
 
 694 
 
 713 
 
 733 
 
 753 
 
 772 
 
 792 
 
 811 
 
 
 
 19 
 
 23 
 
 34 830 
 
 850 
 
 869 
 
 889 
 
 908 
 
 928 
 
 947 
 
 967 
 
 986 
 
 *005 
 
 1 
 
 
 1.9 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 3.8 
 
 24 
 
 35 025 
 
 044 
 
 064 
 
 083 
 
 102 
 
 1 22 
 
 141 
 
 160 
 
 180 
 
 199 
 
 3 
 
 
 5.7 
 
 25 
 
 218 
 
 258 
 
 257 
 
 276 
 
 295 
 
 315 
 
 334 
 
 353 
 
 372 
 
 392 
 
 4 
 
 
 7.6 
 
 26 
 
 411 
 
 450 
 
 449 
 
 468 
 
 4S8 
 
 507 
 
 526 
 
 545 
 
 564 
 
 583 
 
 5 
 
 
 9.5 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 
 11.4 
 
 27 
 
 603 
 
 622 
 
 641 
 
 660 
 
 679 
 
 698 
 
 717 
 
 736 
 
 755 
 
 774 
 
 7 
 
 
 13.3 
 
 28 
 
 793 
 
 813 
 
 832 
 
 851 
 
 870 
 
 889 
 
 908 
 
 927 
 
 946 
 
 965 
 
 8 
 
 
 15.2 
 
 29 
 
 35 984 
 
 *003 
 
 *021 
 
 *040 
 
 *059 
 
 *078 
 
 *097 
 
 *116 
 
 *135 
 
 *154 
 
 
 
 
 230 
 
 36 173 
 
 192 
 
 211 
 
 229 
 
 248 
 
 267 
 
 286 
 
 305 
 
 324 
 
 342 
 
 
 
 
 51 
 
 561 
 
 380 
 
 399 
 
 418 
 
 456 
 
 455 
 
 474 
 
 493 
 
 511 
 
 530 
 
 
 
 
 32 
 
 549 
 
 568 
 
 586 
 
 605 
 
 624 
 
 642 
 
 661 
 
 6S0 
 
 698 
 
 717 
 
 
 
 18 
 
 33 
 
 756 
 
 754 
 
 773 
 
 791 
 
 810 
 
 829 
 
 847 
 
 866 
 
 884 
 
 903 
 
 i 
 
 
 1.8 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 3.6 
 
 34 
 
 36 922 
 
 940 
 
 959 
 
 977 
 
 996 
 
 *014 
 
 *033 
 
 *051 
 
 *070 
 
 *088 
 
 3 
 
 
 5.4 
 
 35 
 
 37 107 
 
 125 
 
 144 
 
 162 
 
 181 
 
 199 
 
 218 
 
 236 
 
 254 
 
 273 
 
 4 
 
 
 7.2 
 
 56 
 
 291 
 
 310 
 
 328 
 
 346 
 
 565 
 
 383 
 
 401 
 
 420 
 
 438 
 
 457 
 
 5 
 
 G 
 
 
 9.0 
 
 10.8 
 
 37 
 
 475 
 
 493 
 
 511 
 
 530 
 
 548 
 
 566 
 
 585 
 
 603 
 
 621 
 
 639 
 
 7 
 
 
 12.6 
 
 38 
 
 658 
 
 676 
 
 694 
 
 712 
 
 751 
 
 749 
 
 767 
 
 785 
 
 803 
 
 822 
 
 
 
 16 2 
 
 39 
 
 37 840 
 
 858 
 
 876 
 
 894 
 
 912 
 
 951 
 
 949 
 
 967 
 
 985 
 
 *003 
 
 
 
 
 240 
 
 38 021 
 
 059 
 
 057 
 
 075 
 
 093 
 
 112 
 
 130 
 
 148 
 
 166 
 
 184 
 
 
 
 
 41 
 
 202 
 
 220 
 
 238 
 
 256 
 
 274 
 
 292 
 
 310 
 
 328 
 
 546 
 
 364 
 
 
 
 
 42 
 
 382 
 
 399 
 
 417 
 
 435 
 
 453 
 
 471 
 
 489 
 
 507 
 
 525 
 
 543 
 
 
 
 17 
 
 43 
 
 561 
 
 578 
 
 596 
 
 614 
 
 652 
 
 650 
 
 668 
 
 686 
 
 705 
 
 721 
 
 1 
 
 
 1.7 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 
 3.4 
 
 44 
 
 739 
 
 757 
 
 775 
 
 792 
 
 810 
 
 828 
 
 846 
 
 863 
 
 881 
 
 899 
 
 3 
 
 
 5.1 
 
 45 
 
 38 917 
 
 934 
 
 952 
 
 970 
 
 987 
 
 *005 
 
 *023 
 
 *041 
 
 *058 
 
 *076 
 
 4 
 
 
 6.8 
 
 46 
 
 39 094 
 
 111 
 
 129 
 
 146 
 
 164 
 
 182 
 
 199 
 
 217 
 
 235 
 
 252 
 
 5 
 
 6 
 
 
 8.5 
 
 10.2 
 
 47 
 
 270 
 
 287 
 
 305 
 
 322 
 
 540 
 
 358 
 
 375 
 
 393 
 
 410 
 
 428 
 
 7 
 
 11.9 
 
 48 
 
 445 
 
 463 
 
 480 
 
 498 
 
 515 
 
 
 550 
 
 568 
 
 585 
 
 602 
 
 9 
 
 1 j.U 
 
 IS 3 
 
 49 
 
 620 
 
 657 
 
 6 55 
 
 672 
 
 690 
 
 707 
 
 724 
 
 742 
 
 759 
 
 777 
 
 
 
 
 250 
 
 39 794 
 
 811 
 
 829 
 
 846 
 
 863 
 
 881 
 
 898 
 
 915 
 
 933 
 
 950 
 
 
 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
TABLES 
 
 307 
 
 TABLE V —( Continued ) 
 Five-place Logarithms: 250-300 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 250 
 
 39 794 
 
 811 
 
 829 
 
 846 
 
 863 
 
 881 
 
 898 
 
 915 
 
 933 
 
 950 
 
 
 
 51 
 
 39 967 
 
 985 
 
 *002 
 
 *019 
 
 *037 
 
 *054 
 
 *071 
 
 *088 
 
 *106 
 
 *123 
 
 
 18 
 
 52 
 
 40 140 
 
 157 
 
 175 
 
 192 
 
 209 
 
 226 
 
 243 
 
 261 
 
 278 
 
 295 
 
 1 
 
 1.8 
 
 53 
 
 312 
 
 329 
 
 346 
 
 364 
 
 381 
 
 398 
 
 415 
 
 432 
 
 449 
 
 466 
 
 2 
 
 3.6 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 5.4 
 
 54 
 
 483 
 
 500 
 
 518 
 
 535 
 
 552 
 
 569 
 
 586 
 
 603 
 
 620 
 
 637 
 
 i 
 
 7.2 
 
 55 
 
 654 
 
 671 
 
 688 
 
 705 
 
 722 
 
 739 
 
 756 
 
 773 
 
 790 
 
 807 
 
 5 
 
 9.0 
 
 56 
 
 824 
 
 841 
 
 858 
 
 875 
 
 892 
 
 909 
 
 926 
 
 943 
 
 960 
 
 976 
 
 C 
 
 10.8 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 12.6 
 
 57 
 
 40 993 
 
 *010 
 
 *027 
 
 *044 
 
 *061 
 
 *078 
 
 *095 
 
 *111 
 
 *128 
 
 *145 
 
 8 
 
 14.4 
 
 58 
 
 41 162 
 
 179 
 
 196 
 
 212 
 
 229 
 
 246 
 
 263 
 
 280 
 
 296 
 
 513 
 
 
 
 59 
 
 330 
 
 347 
 
 363 
 
 380 
 
 397 
 
 414 
 
 430 
 
 447 
 
 464 
 
 481 
 
 
 
 260 
 
 497 
 
 514 
 
 531 
 
 547 
 
 564 
 
 581 
 
 597 
 
 614 
 
 631 
 
 647 
 
 
 
 61 
 
 664 
 
 681 
 
 697 
 
 714 
 
 731 
 
 747 
 
 764 
 
 780 
 
 797 
 
 814 
 
 
 17 
 
 62 
 
 830 
 
 847 
 
 863 
 
 880 
 
 896 
 
 913 
 
 929 
 
 946 
 
 963 
 
 979 
 
 i 
 
 1.7 
 
 63 
 
 41 996 
 
 *012 
 
 *029 
 
 *045 
 
 *062 
 
 *078 
 
 *095 
 
 *111 
 
 *127 
 
 *144 
 
 2 
 
 3.4 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 5.1 
 
 64 
 
 42 160 
 
 177 
 
 193 
 
 210 
 
 226 
 
 243 
 
 259 
 
 275 
 
 292 
 
 308 
 
 4 
 
 6.8 
 
 65 
 
 325 
 
 341 
 
 357 
 
 374 
 
 390 
 
 406 
 
 423 
 
 439 
 
 455 
 
 472 
 
 5 
 
 8.5 
 
 66 
 
 488 
 
 504 
 
 521 
 
 537 
 
 553 
 
 570 
 
 586 
 
 602 
 
 619 
 
 635 
 
 6 
 
 10.2 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 11.9 
 
 67 
 
 651 
 
 667 
 
 684 
 
 700 
 
 716 
 
 732 
 
 749 
 
 765 
 
 781 
 
 797 
 
 8 
 
 13.6 
 
 68 
 
 813 
 
 830 
 
 846 
 
 862 
 
 878 
 
 894 
 
 911 
 
 927 
 
 943 
 
 959 
 
 
 
 69 
 
 42 975 
 
 991 
 
 *008 
 
 *024 
 
 *040 
 
 *056 
 
 *072 
 
 *088 
 
 *104 
 
 *120 
 
 
 
 270 
 
 43 136 
 
 152 
 
 169 
 
 185 
 
 201 
 
 217 
 
 233 
 
 249 
 
 265 
 
 281 
 
 
 
 71 
 
 297 
 
 313 
 
 329 
 
 345 
 
 361 
 
 377 
 
 393 
 
 409 
 
 425 
 
 441 
 
 
 16 
 
 72 
 
 457 
 
 473 
 
 489 
 
 505 
 
 521 
 
 537 
 
 553 
 
 569 
 
 584 
 
 600 
 
 i 
 
 1.6 
 
 73 
 
 616 
 
 632 
 
 648 
 
 664 
 
 680 
 
 696 
 
 712 
 
 727 
 
 743 
 
 759 
 
 2 
 
 3.2 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 4.8 
 
 74 
 
 775 
 
 791 
 
 807 
 
 823 
 
 838 
 
 854 
 
 870 
 
 886 
 
 902 
 
 917 
 
 4 
 
 6.4 
 
 75 
 
 43 933 
 
 949 
 
 965 
 
 981 
 
 996 
 
 *012 
 
 *028 
 
 *044 
 
 *059 
 
 *075 
 
 5 
 
 6 
 
 8.0 
 
 9.6 
 
 76 
 
 44 091 
 
 107 
 
 122 
 
 138 
 
 154 
 
 170 
 
 185 
 
 201 
 
 217 
 
 232 
 
 7 
 
 11.2 
 
 77 
 
 248 
 
 264 
 
 279 
 
 295 
 
 311 
 
 326 
 
 342 
 
 358 
 
 373 
 
 389 
 
 J) 
 
 14 4 
 
 78 
 
 404 
 
 420 
 
 436 
 
 451 
 
 467 
 
 483 
 
 498 
 
 514 
 
 529 
 
 545 
 
 
 
 79 
 
 560 
 
 576 
 
 592 
 
 607 
 
 623 
 
 638 
 
 654 
 
 669 
 
 685 
 
 700 
 
 
 
 280 
 
 716 
 
 731 
 
 747 
 
 762 
 
 778 
 
 793 
 
 809 
 
 824 
 
 840 
 
 855 
 
 
 
 81 
 
 44 871 
 
 886 
 
 902 
 
 917 
 
 932 
 
 948 
 
 963 
 
 979 
 
 994 
 
 *010 
 
 
 15 
 
 82 
 
 45 025 
 
 040 
 
 056 
 
 071 
 
 086 
 
 102 
 
 117 
 
 133 
 
 148 
 
 163 
 
 1 
 
 1.5 
 
 83 
 
 179 
 
 194 
 
 209 
 
 225 
 
 240 
 
 255 
 
 271 
 
 286 
 
 301 
 
 317 
 
 2 
 
 3.0 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 4.5 
 
 84 
 
 332 
 
 347 
 
 362 
 
 378 
 
 393 
 
 408 
 
 423 
 
 439 
 
 454 
 
 469 
 
 4 
 
 6.0 
 
 85 
 
 484 
 
 500 
 
 515 
 
 530 
 
 545 
 
 561 
 
 576 
 
 591 
 
 606 
 
 621 
 
 5 
 
 6 
 
 7.5 
 
 9.0 
 
 86 
 
 637 
 
 652 
 
 667 
 
 682 
 
 697 
 
 712 
 
 728 
 
 743 
 
 758 
 
 773 
 
 7 
 
 10.5 
 
 87 
 
 788 
 
 803 
 
 818 
 
 834 
 
 849 
 
 864 
 
 879 
 
 894 
 
 909 
 
 924 
 
 
 13 5 
 
 88 
 
 45 939 
 
 954 
 
 969 
 
 984 
 
 *000 
 
 *015 
 
 *030 
 
 *045 
 
 *060 
 
 *075 
 
 
 
 89 
 
 46 090 
 
 105 
 
 120 
 
 135 
 
 150 
 
 165 
 
 180 
 
 195 
 
 21C 
 
 225 
 
 
 
 290 
 
 240 
 
 255 
 
 270 
 
 285 
 
 300 
 
 315 
 
 330 
 
 345 
 
 359 
 
 374 
 
 
 
 91 
 
 389 
 
 404 
 
 419 
 
 434 
 
 449 
 
 464 
 
 479 
 
 494 
 
 509 
 
 523 
 
 
 14 
 
 92 
 
 538 
 
 553 
 
 568 
 
 583 
 
 598 
 
 613 
 
 627 
 
 642 
 
 657 
 
 672 
 
 1 
 
 1.4 
 
 93 
 
 687 
 
 702 
 
 716 
 
 731 
 
 746 
 
 761 
 
 776 
 
 790 
 
 805 
 
 820 
 
 2 
 
 2.8 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 4.2 
 
 94 
 
 835 
 
 850 
 
 864 
 
 879 
 
 894 
 
 909 
 
 923 
 
 938 
 
 953 
 
 967 
 
 4 
 
 6.6 
 
 95 
 
 46 982 
 
 997 
 
 *012 
 
 *026 
 
 *041 
 
 *056 
 
 *070 
 
 *085 
 
 *100 
 
 *1 14 
 
 5 
 
 6 
 
 7.0 
 
 8.4 
 
 96 
 
 47 129 
 
 144 
 
 159 
 
 173 
 
 188 
 
 202 
 
 217 
 
 232 
 
 246 
 
 261 
 
 7 
 
 9.8 
 
 97 
 
 276 
 
 290 
 
 305 
 
 319 
 
 334 
 
 349 
 
 363 
 
 378 
 
 392 
 
 407 
 
 0 
 
 12.6 
 
 98 
 
 422 
 
 436 
 
 451 
 
 465 
 
 480 
 
 494 
 
 509 
 
 524 
 
 538 
 
 553 
 
 
 
 99 
 
 567 
 
 582 
 
 596 
 
 611 
 
 625 
 
 640 
 
 654 
 
 669 
 
 683 
 
 698 
 
 
 
 300 
 
 47 712 
 
 727 
 
 741 
 
 756 
 
 770 
 
 784 
 
 799 
 
 813 
 
 828 
 
 842 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 •> 
 
 3 
 
 4 
 
 5 
 
 (> 
 
 7 
 
 8 
 
 9 
 
308 
 
 TABLES 
 
 TABLE V—( Continued) 
 Five-place Logarithms: 300-350 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
 300 
 
 47 712 
 
 727 
 
 741 
 
 756 
 
 770 
 
 784 
 
 799 
 
 813 
 
 828 
 
 842 
 
 
 
 01 
 
 47 857 
 
 871 
 
 885 
 
 900 
 
 914 
 
 929 
 
 943 
 
 958 
 
 972 
 
 986 
 
 
 
 02 
 
 48 001 
 
 015 
 
 029 
 
 044 
 
 058 
 
 075 
 
 087 
 
 101 
 
 116 
 
 130 
 
 
 
 03 
 
 144 
 
 159 
 
 173 
 
 187 
 
 202 
 
 216 
 
 230 
 
 244 
 
 259 
 
 273 
 
 
 15 
 
 04 
 
 287 
 
 302 
 
 316 
 
 330 
 
 344 
 
 359 
 
 373 
 
 387 
 
 401 
 
 416 
 
 1 
 
 1.5 
 
 05 
 
 430 
 
 444 
 
 458 
 
 473 
 
 487 
 
 501 
 
 515 
 
 530 
 
 544 
 
 558 
 
 2 
 
 3.0 
 
 4.5 
 
 6.0 
 
 06 
 
 572 
 
 586 
 
 601 
 
 615 
 
 629 
 
 643 
 
 657 
 
 671 
 
 686 
 
 700 
 
 4 
 
 07 
 
 714 
 
 728 
 
 742 
 
 756 
 
 770 
 
 785 
 
 799 
 
 813 
 
 827 
 
 841 
 
 5 
 
 6 
 
 7.5 
 
 Q 0 
 
 08 
 
 855 
 
 869 
 
 883 
 
 897 
 
 911 
 
 926 
 
 940 
 
 954 
 
 968 
 
 982 
 
 
 10.5 
 
 12.0 
 
 09 
 
 48 996 
 
 *010 
 
 *024 
 
 *058 
 
 *052 
 
 *066 
 
 *080 
 
 *094 
 
 *108 
 
 *122 
 
 8 
 
 310 
 
 49 156 
 
 150 
 
 164 
 
 178 
 
 192 
 
 206 
 
 220 
 
 234 
 
 248 
 
 262 
 
 9 
 
 13.5 
 
 11 
 
 276 
 
 290 
 
 304 
 
 318 
 
 332 
 
 346 
 
 360 
 
 374 
 
 588 
 
 402 
 
 
 
 12 
 
 415 
 
 429 
 
 443 
 
 457 
 
 471 
 
 485 
 
 499 
 
 515 
 
 527 
 
 541 
 
 
 
 13 
 
 554 
 
 568 
 
 582 
 
 596 
 
 610 
 
 624 
 
 658 
 
 651 
 
 665 
 
 679 
 
 
 
 14 
 
 693 
 
 707 
 
 721 
 
 734 
 
 748 
 
 762 
 
 776 
 
 790 
 
 803 
 
 817 
 
 
 
 15 
 
 831 
 
 845 
 
 859 
 
 872 
 
 886 
 
 900 
 
 914 
 
 927 
 
 941 
 
 955 
 
 
 
 16 
 
 49 969 
 
 982 
 
 996 
 
 *010 
 
 *024 
 
 *057 
 
 *051 
 
 *065 
 
 *079 
 
 *092 
 
 
 14 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 1.4 
 
 17 
 
 50 106 
 
 120 
 
 133 
 
 147 
 
 161 
 
 174 
 
 188 
 
 202 
 
 215 
 
 229 
 
 2 
 
 2.8 
 
 18 
 
 243 
 
 256 
 
 270 
 
 284 
 
 297 
 
 311 
 
 325 
 
 338 
 
 352 
 
 365 
 
 3 
 
 4.2 
 
 19 
 
 379 
 
 393 
 
 406 
 
 420 
 
 433 
 
 447 
 
 461 
 
 474 
 
 488 
 
 501 
 
 4 
 
 5 6 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 7.0 
 
 8.4 
 
 320 
 
 515 
 
 529 
 
 542 
 
 556 
 
 569 
 
 583 
 
 596 
 
 610 
 
 623 
 
 637 
 
 6 
 
 7 
 
 9.8 
 
 
 
 
 
 
 
 
 
 
 
 
 21 
 
 651 
 
 6b4 
 
 678 
 
 691 
 
 705 
 
 718 
 
 732 
 
 745 
 
 759 
 
 772 
 
 8 
 
 11.2 
 
 22 
 
 786 
 
 799 
 
 813 
 
 826 
 
 840 
 
 853 
 
 866 
 
 880 
 
 893 
 
 907 
 
 9 
 
 12.6 
 
 23 
 
 50 920 
 
 934 
 
 947 
 
 961 
 
 974 
 
 987 
 
 *001 
 
 *014 
 
 *028 
 
 *041 
 
 
 
 24 
 
 51 055 
 
 068 
 
 081 
 
 095 
 
 108 
 
 121 
 
 135 
 
 148 
 
 162 
 
 175 
 
 
 
 25 
 
 188 
 
 202 
 
 215 
 
 228 
 
 242 
 
 255 
 
 268 
 
 282 
 
 295 
 
 308 
 
 
 
 26 
 
 322 
 
 335 
 
 348 
 
 362 
 
 575 
 
 388 
 
 402 
 
 415 
 
 428 
 
 441 
 
 
 
 27 
 
 455 
 
 468 
 
 481 
 
 495 
 
 508 
 
 521 
 
 534 
 
 548 
 
 561 
 
 574 
 
 
 
 28 
 
 587 
 
 601 
 
 614 
 
 627 
 
 640 
 
 654 
 
 667 
 
 680 
 
 693 
 
 706 
 
 
 13 
 
 29 
 
 720 
 
 733 
 
 746 
 
 759 
 
 772 
 
 786 
 
 799 
 
 812 
 
 825 
 
 858 
 
 1 
 
 1.3 
 
 330 
 
 851 
 
 865 
 
 878 
 
 891 
 
 904 
 
 917 
 
 930 
 
 943 
 
 957 
 
 970 
 
 2 
 
 3 
 
 2.6 
 
 3.9 
 
 31 
 
 51 983 
 
 996 
 
 *009 
 
 *022 
 
 *035 
 
 *048 
 
 *061 
 
 *075 
 
 *088 
 
 *101 
 
 4 
 
 5.2 
 
 6 5 
 
 32 
 
 52 114 
 
 127 
 
 140 
 
 153 
 
 166 
 
 179 
 
 192 
 
 205 
 
 218 
 
 251 
 
 o 
 
 53 
 
 244 
 
 257 
 
 270 
 
 284 
 
 297 
 
 310 
 
 323 
 
 356 
 
 349 
 
 362 
 
 7 
 
 9.1 
 
 34 
 
 375 
 
 388 
 
 401 
 
 414 
 
 427 
 
 440 
 
 453 
 
 466 
 
 479 
 
 492 
 
 8 
 
 9 
 
 10.4 
 
 11.7 
 
 55 
 
 504 
 
 517 
 
 550 
 
 543 
 
 556 
 
 569 
 
 582 
 
 595 
 
 608 
 
 621 
 
 
 36 
 
 654 
 
 647 
 
 660 
 
 675 
 
 6S6 
 
 699 
 
 711 
 
 724 
 
 737 
 
 750 
 
 
 
 37 
 
 763 
 
 776 
 
 789 
 
 802 
 
 815 
 
 827 
 
 840 
 
 853 
 
 866 
 
 879 
 
 
 
 38 
 
 52 892 
 
 905 
 
 917 
 
 950 
 
 943 
 
 956 
 
 969 
 
 9S2 
 
 994 
 
 *007 
 
 
 
 39 
 
 53 020 
 
 035 
 
 046 
 
 058 
 
 071 
 
 084 
 
 097 
 
 110 
 
 122 
 
 135 
 
 
 
 340 
 
 148 
 
 161 
 
 173 
 
 186 
 
 199 
 
 212 
 
 224 
 
 237 
 
 250 
 
 263 
 
 
 12 
 
 1.2 
 
 41 
 
 275 
 
 288 
 
 301 
 
 314 
 
 326 
 
 339 
 
 352 
 
 364 
 
 377 
 
 390 
 
 i 
 
 42 
 
 403 
 
 415 
 
 428 
 
 441 
 
 453 
 
 466 
 
 479 
 
 491 
 
 504 
 
 517 
 
 2 
 
 2.4 
 
 45 
 
 529 
 
 542 
 
 ODD 
 
 567 
 
 580 
 
 593 
 
 605 
 
 618 
 
 651 
 
 643 
 
 3 
 
 3.6 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 4.8 
 
 44 
 
 656 
 
 668 
 
 681 
 
 694 
 
 706 
 
 719 
 
 732 
 
 744 
 
 757 
 
 769 
 
 5 
 
 6.0 
 
 45 
 
 782 
 
 794 
 
 807 
 
 820 
 
 832 
 
 845 
 
 857 
 
 870 
 
 882 
 
 895 
 
 6 
 
 7.2 
 
 46 
 
 53 908 
 
 920 
 
 933 
 
 945 
 
 958 
 
 970 
 
 983 
 
 995 
 
 *008 
 
 *020 
 
 7 
 
 8.4 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 9.6 
 
 47 
 
 54 033 
 
 045 
 
 058 
 
 070 
 
 083 
 
 095 
 
 108 
 
 120 
 
 133 
 
 145 
 
 9 
 
 10.8 
 
 48 
 
 158 
 
 170 
 
 183 
 
 195 
 
 208 
 
 220 
 
 233 
 
 245 
 
 258 
 
 270 
 
 
 
 49 
 
 283 
 
 295 
 
 307 
 
 320 
 
 332 
 
 345 
 
 357 
 
 370 
 
 382 
 
 394 
 
 
 
 350 
 
 54 407 
 
 419 
 
 432 
 
 444 
 
 456 
 
 469 
 
 481 
 
 494 
 
 506 
 
 518 
 
 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
TABLES 
 
 309 
 
 TABLE V—( Continued ) 
 Five-place Logarithms: 350-400 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 350 
 
 54 407 
 
 419 
 
 432 
 
 444 
 
 456 
 
 469 
 
 481 
 
 494 
 
 506 
 
 518 
 
 
 
 51 
 
 531 
 
 543 
 
 555 
 
 568 
 
 580 
 
 593 
 
 605 
 
 617 
 
 630 
 
 642 
 
 
 
 52 
 
 654 
 
 667 
 
 679 
 
 691 
 
 704 
 
 716 
 
 728 
 
 741 
 
 753 
 
 765 
 
 
 13 
 
 53 
 
 777 
 
 790 
 
 802 
 
 814 
 
 827 
 
 839 
 
 851 
 
 864 
 
 876 
 
 888 
 
 1 
 
 1.3 
 
 54 
 
 54 900 
 
 913 
 
 925 
 
 937 
 
 949 
 
 962 
 
 974 
 
 986 
 
 998 
 
 *011 
 
 2 
 
 2.6 
 
 55 
 
 55 023 
 
 035 
 
 047 
 
 060 
 
 072 
 
 084 
 
 096 
 
 108 
 
 121 
 
 153 
 
 3 
 
 
 56 
 
 145 
 
 157 
 
 169 
 
 182 
 
 194 
 
 206 
 
 218 
 
 230 
 
 242 
 
 255 
 
 4 
 
 5.2 
 
 
 
 
 
 
 
 
 
 
 5 
 
 a 
 
 6.5 
 
 7 8 
 
 57 
 
 267 
 
 279 
 
 291 
 
 303 
 
 315 
 
 328 
 
 340 
 
 352 
 
 364 
 
 376 
 
 
 9.1 
 
 10.4 
 
 58 
 
 388 
 
 400 
 
 413 
 
 425 
 
 457 
 
 449 
 
 461 
 
 473 
 
 485 
 
 497 
 
 8 
 
 59 
 
 509 
 
 522 
 
 534 
 
 546 
 
 558 
 
 570 
 
 582 
 
 594 
 
 606 
 
 618 
 
 9 
 
 11.7 
 
 360 
 
 630 
 
 642 
 
 654 
 
 666 
 
 678 
 
 691 
 
 703 
 
 715 
 
 727 
 
 739 
 
 
 
 61 
 
 751 
 
 763 
 
 775 
 
 787 
 
 799 
 
 811 
 
 823 
 
 835 
 
 847 
 
 859 
 
 
 
 62 
 
 871 
 
 883 
 
 895 
 
 907 
 
 919 
 
 931 
 
 943 
 
 955 
 
 967 
 
 979 
 
 
 
 63 
 
 55 991 
 
 *003 
 
 *015 
 
 *027 
 
 *038 
 
 *050 
 
 *062 
 
 *074 
 
 *086 
 
 *098 
 
 
 
 64 
 
 56 110 
 
 122 
 
 134 
 
 146 
 
 158 
 
 170 
 
 182 
 
 194 
 
 205 
 
 217 
 
 
 
 65 
 
 229 
 
 241 
 
 253 
 
 265 
 
 277 
 
 289 
 
 301 
 
 312 
 
 324 
 
 336 
 
 
 12 
 
 66 
 
 348 
 
 360 
 
 372 
 
 384 
 
 396 
 
 407 
 
 419 
 
 431 
 
 443 
 
 455 
 
 1 
 
 2 
 
 1.2 
 
 2.4 
 
 67 
 
 467 
 
 478 
 
 490 
 
 502 
 
 514 
 
 526 
 
 538 
 
 549 
 
 561 
 
 573 
 
 3 
 
 3.6 
 
 68 
 
 585 
 
 597 
 
 608 
 
 620 
 
 632 
 
 644 
 
 656 
 
 667 
 
 679 
 
 691 
 
 4 
 
 5 
 
 6 
 
 4.8 
 
 6.0 
 
 7.2 
 
 8.4 
 
 69 
 
 703 
 
 714 
 
 726 
 
 758 
 
 750 
 
 761 
 
 773 
 
 785 
 
 797 
 
 808 
 
 370 
 
 820 
 
 832 
 
 844 
 
 855 
 
 867 
 
 879 
 
 891 
 
 902 
 
 914 
 
 926 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 9.6 
 
 71 
 
 56 937 
 
 949 
 
 961 
 
 972 
 
 984 
 
 996 
 
 *008 
 
 *019 
 
 *031 
 
 *043 
 
 9 
 
 10.8 
 
 72 
 
 57 054 
 
 066 
 
 078 
 
 089 
 
 101 
 
 115 
 
 124 
 
 136 
 
 148 
 
 159 
 
 
 
 73 
 
 171 
 
 183 
 
 194 
 
 206 
 
 217 
 
 229 
 
 241 
 
 252 
 
 264 
 
 276 
 
 
 
 74 
 
 287 
 
 299 
 
 310 
 
 322 
 
 334 
 
 345 
 
 357 
 
 368 
 
 380 
 
 392 
 
 
 
 75 
 
 403 
 
 415 
 
 426 
 
 438 
 
 449 
 
 461 
 
 473 
 
 484 
 
 496 
 
 507 
 
 
 
 76 
 
 519 
 
 530 
 
 542 
 
 553 
 
 565 
 
 576 
 
 588 
 
 600 
 
 611 
 
 623 
 
 
 
 77 
 
 634 
 
 646 
 
 657 
 
 669 
 
 680 
 
 692 
 
 703 
 
 715 
 
 726 
 
 738 
 
 
 11 
 
 78 
 
 749 
 
 761 
 
 772 
 
 784 
 
 795 
 
 807 
 
 818 
 
 850 
 
 841 
 
 852 
 
 
 79 
 
 864 
 
 875 
 
 887 
 
 898 
 
 910 
 
 921 
 
 933 
 
 944 
 
 955 
 
 967 
 
 1 
 
 1.1 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 3 
 
 2.2 
 
 3.3 
 
 380 
 
 57 978 
 
 990 
 
 *001 
 
 *013 
 
 *024 
 
 *035 
 
 *047 
 
 *058 
 
 *070 
 
 *081 
 
 4 
 
 4.4 
 
 81 
 
 58 092 
 
 104 
 
 115 
 
 127 
 
 138 
 
 149 
 
 161 
 
 172 
 
 184 
 
 195 
 
 5 
 
 5.5 
 
 6.6 
 
 82 
 
 206 
 
 218 
 
 229 
 
 240 
 
 252 
 
 263 
 
 274 
 
 286 
 
 297 
 
 309 
 
 
 83 
 
 320 
 
 331 
 
 343 
 
 354 
 
 365 
 
 377 
 
 388 
 
 399 
 
 410 
 
 422 
 
 7 
 
 7.7 
 
 
 
 
 
 
 
 
 8 
 
 9 
 
 8.8 
 
 9.9 
 
 84 
 
 433 
 
 444 
 
 456 
 
 467 
 
 478 
 
 490 
 
 501 
 
 512 
 
 524 
 
 535 
 
 
 85 
 
 546 
 
 557 
 
 569 
 
 580 
 
 591 
 
 602 
 
 614 
 
 625 
 
 636 
 
 647 
 
 
 
 86 
 
 659 
 
 670 
 
 681 
 
 692 
 
 704 
 
 715 
 
 726 
 
 757 
 
 749 
 
 760 
 
 
 
 87 
 
 771 
 
 782 
 
 794 
 
 805 
 
 816 
 
 827 
 
 838 
 
 850 
 
 861 
 
 872 
 
 
 
 88 
 
 883 
 
 894 
 
 906 
 
 917 
 
 928 
 
 959 
 
 950 
 
 961 
 
 973 
 
 984 
 
 
 
 89 
 
 58 995 
 
 *006 
 
 *017 
 
 *028 
 
 *040 
 
 *051 
 
 *062 
 
 *073 
 
 *084 
 
 *095 
 
 
 10 
 
 1 0 
 
 390 
 
 59 106 
 
 118 
 
 129 
 
 140 
 
 151 
 
 162 
 
 173 
 
 184 
 
 195 
 
 207 
 
 1 
 
 91 
 
 218 
 
 229 
 
 240 
 
 251 
 
 262 
 
 273 
 
 284 
 
 295 
 
 306 
 
 318 
 
 2 
 
 2.0 
 
 92 
 
 329 
 
 340 
 
 351 
 
 362 
 
 373 
 
 384 
 
 395 
 
 406 
 
 417 
 
 428 
 
 3 
 
 3.0 
 
 93 
 
 439 
 
 450 
 
 461 
 
 472 
 
 483 
 
 494 
 
 506 
 
 517 
 
 528 
 
 539 
 
 4 
 
 4.0 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 6.0 
 
 94 
 
 550 
 
 561 
 
 572 
 
 583 
 
 594 
 
 605 
 
 616 
 
 627 
 
 638 
 
 649 
 
 6 
 
 6.0 
 
 95 
 
 660 
 
 671 
 
 682 
 
 693 
 
 704 
 
 715 
 
 726 
 
 757 
 
 748 
 
 759 
 
 7 
 
 7.0 
 
 96 
 
 770 
 
 780 
 
 791 
 
 802 
 
 813 
 
 824 
 
 835 
 
 846 
 
 857 
 
 868 
 
 8 
 
 8.0 
 
 
 
 
 
 
 
 
 
 
 
 
 9 
 
 9.0 
 
 97 
 
 879 
 
 890 
 
 901 
 
 912 
 
 923 
 
 934 
 
 945 
 
 956 
 
 966 
 
 977 
 
 
 
 98 
 
 59 988 
 
 999 
 
 *010 
 
 *021 
 
 *032 
 
 *043 
 
 *054 
 
 *065 
 
 *076 
 
 *086 
 
 
 
 99 
 
 60 097 
 
 108 
 
 119 
 
 130 
 
 141 
 
 152 
 
 163 
 
 173 
 
 184 
 
 195 
 
 
 
 400 
 
 60 206 
 
 217 
 
 228 
 
 239 
 
 249 
 
 260 
 
 271 
 
 282 
 
 293 
 
 304 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 <> 
 
 7 
 
 8 
 
 9 
 
310 
 
 TABLES 
 
 TABLE V—( Continued ) 
 Five-place Logarithms: 400-450 
 
 N 
 
 0 
 
 1 
 
 o 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop . Parts 
 
 400 
 
 60 206 
 
 217 
 
 228 
 
 239 
 
 249 
 
 260 
 
 271 
 
 282 
 
 293 
 
 304 
 
 
 
 01 
 
 314 
 
 325 
 
 336 
 
 547 
 
 358 
 
 369 
 
 379 
 
 390 
 
 401 
 
 412 
 
 
 
 02 
 
 423 
 
 433 
 
 444 
 
 455 
 
 466 
 
 477 
 
 487 
 
 498 
 
 509 
 
 520 
 
 
 
 03 
 
 531 
 
 541 
 
 552 
 
 563 
 
 574 
 
 584 
 
 595 
 
 606 
 
 617 
 
 627 
 
 
 
 04 
 
 658 
 
 649 
 
 660 
 
 670 
 
 681 
 
 692 
 
 703 
 
 713 
 
 724 
 
 735 
 
 
 
 05 
 
 746 
 
 756 
 
 767 
 
 778 
 
 788 
 
 799 
 
 810 
 
 821 
 
 831 
 
 842 
 
 
 
 06 
 
 853 
 
 863 
 
 874 
 
 885 
 
 895 
 
 906 
 
 917 
 
 927 
 
 938 
 
 949 
 
 
 11 
 
 1.1 
 
 07 
 
 60 959 
 
 970 
 
 981 
 
 991 
 
 *002 
 
 *013 
 
 *023 
 
 *034 
 
 *045 
 
 *055 
 
 1 
 
 08 
 
 61 066 
 
 077 
 
 087 
 
 098 
 
 109 
 
 119 
 
 130 
 
 140 
 
 151 
 
 162 
 
 2 
 
 2.2 
 
 09 
 
 172 
 
 183 
 
 194 
 
 204 
 
 215 
 
 225 
 
 236 
 
 247 
 
 257 
 
 268 
 
 3 
 
 3.3 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 4.4 
 
 5.5 
 
 410 
 
 278 
 
 289 
 
 300 
 
 310 
 
 321 
 
 331 
 
 342 
 
 352 
 
 363 
 
 374 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 6.6 
 
 11 
 
 384 
 
 595 
 
 405 
 
 416 
 
 426 
 
 437 
 
 448 
 
 458 
 
 469 
 
 479 
 
 7 
 
 7.7 
 
 12 
 
 490 
 
 500 
 
 511 
 
 521 
 
 532 
 
 542 
 
 553 
 
 563 
 
 574 
 
 584 
 
 8 
 
 8.8 
 
 13 
 
 595 
 
 606 
 
 616 
 
 627 
 
 637 
 
 648 
 
 658 
 
 669 
 
 679 
 
 690 
 
 9 
 
 9.9 
 
 14 
 
 700 
 
 711 
 
 721 
 
 731 
 
 742 
 
 752 
 
 763 
 
 773 
 
 784 
 
 794 
 
 
 
 15 
 
 805 
 
 815 
 
 826 
 
 836 
 
 847 
 
 857 
 
 868 
 
 878 
 
 888 
 
 899 
 
 
 
 16 
 
 61 909 
 
 920 
 
 930 
 
 941 
 
 951 
 
 962 
 
 972 
 
 982 
 
 993 
 
 *003 
 
 
 
 17 
 
 62 014 
 
 024 
 
 034 
 
 045 
 
 055 
 
 066 
 
 076 
 
 086 
 
 097 
 
 107 
 
 
 
 18 
 
 118 
 
 128 
 
 138 
 
 149 
 
 159 
 
 170 
 
 ISO 
 
 190 
 
 201 
 
 211 
 
 
 
 19 
 
 221 
 
 232 
 
 242 
 
 252 
 
 26J 
 
 273 
 
 284 
 
 294 
 
 304 
 
 315 
 
 
 
 420 
 
 325 
 
 335 
 
 346 
 
 356 
 
 366 
 
 377 
 
 387 
 
 397 
 
 408 
 
 418 
 
 
 
 21 
 
 428 
 
 439 
 
 449 
 
 459 
 
 469 
 
 4S0 
 
 490 
 
 500 
 
 511 
 
 521 
 
 
 
 22 
 
 531 
 
 542 
 
 552 
 
 562 
 
 572 
 
 583 
 
 593 
 
 603 
 
 613 
 
 624 
 
 
 10 
 
 23 
 
 634 
 
 644 
 
 655 
 
 665 
 
 675 
 
 685 
 
 696 
 
 706 
 
 716 
 
 726 
 
 1 
 
 1.0 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 2.0 
 
 24 
 
 737 
 
 747 
 
 757 
 
 767 
 
 778 
 
 788 
 
 798 
 
 808 
 
 818 
 
 829 
 
 3 
 
 3.0 
 
 25 
 
 839 
 
 849 
 
 859 
 
 870 
 
 880 
 
 890 
 
 900 
 
 910 
 
 921 
 
 931 
 
 4 
 
 4.0 
 
 26 
 
 62 941 
 
 951 
 
 961 
 
 972 
 
 982 
 
 992 
 
 *002 
 
 *012 
 
 *022 
 
 *055 
 
 5 
 
 6 
 
 5.0 
 
 6.0 
 
 27 
 
 63 043 
 
 053 
 
 063 
 
 073 
 
 083 
 
 094 
 
 104 
 
 114 
 
 124 
 
 134 
 
 7 
 
 8 
 
 9 
 
 7.0 
 
 8.0 
 
 9.0 
 
 28 
 
 144 
 
 155 
 
 165 
 
 175 
 
 185 
 
 195 
 
 205 
 
 215 
 
 225 
 
 236 
 
 29 
 
 246 
 
 256 
 
 266 
 
 276 
 
 286 
 
 296 
 
 306 
 
 317 
 
 327 
 
 337 
 
 
 430 
 
 347 
 
 357 
 
 367 
 
 377 
 
 387 
 
 397 
 
 407 
 
 417 
 
 428 
 
 438 
 
 
 
 31 
 
 448 
 
 458 
 
 468 
 
 478 
 
 488 
 
 498 
 
 50S 
 
 518 
 
 528 
 
 538 
 
 
 
 32 
 
 548 
 
 558 
 
 568 
 
 579 
 
 589 
 
 599 
 
 609 
 
 619 
 
 629 
 
 659 
 
 
 
 33 
 
 649 
 
 659 
 
 669 
 
 679 
 
 689 
 
 699 
 
 709 
 
 719 
 
 729 
 
 739 
 
 
 
 34 
 
 749 
 
 759 
 
 769 
 
 779 
 
 789 
 
 799 
 
 809 
 
 819 
 
 829 
 
 839 
 
 
 
 55 
 
 849 
 
 859 
 
 869 
 
 879 
 
 889 
 
 899 
 
 909 
 
 919 
 
 929 
 
 959 
 
 
 
 56 
 
 63 949 
 
 959 
 
 969 
 
 979 
 
 988 
 
 998 
 
 *008 
 
 *018 
 
 *028 
 
 *058 
 
 
 
 37 
 
 64 048 
 
 058 
 
 068 
 
 078 
 
 088 
 
 098 
 
 108 
 
 118 
 
 128 
 
 137 
 
 
 
 38 
 
 147 
 
 157 
 
 167 
 
 177 
 
 187 
 
 197 
 
 207 
 
 217 
 
 227 
 
 257 
 
 
 
 59 
 
 246 
 
 256 
 
 266 
 
 276 
 
 286 
 
 296 
 
 306 
 
 516 
 
 326 
 
 335 
 
 1 
 
 2 
 
 0.9 
 
 1.8 
 
 440 
 
 345 
 
 555 
 
 565 
 
 575 
 
 385 
 
 395 
 
 404 
 
 414 
 
 424 
 
 434 
 
 3 
 
 4 
 
 2.7 
 
 3.6 
 
 41 
 
 444 
 
 454 
 
 464 
 
 473 
 
 483 
 
 493 
 
 503 
 
 513 
 
 523 
 
 552 
 
 5 
 
 6 
 
 4.5 
 
 5.4 
 
 42 
 
 542 
 
 552 
 
 562 
 
 572 
 
 582 
 
 591 
 
 601 
 
 611 
 
 621 
 
 651 
 
 43 
 
 640 
 
 650 
 
 660 
 
 670 
 
 680 
 
 689 
 
 699 
 
 709 
 
 719 
 
 729 
 
 7 
 
 8 
 
 6.3 
 
 7.2 
 
 44 
 
 738 
 
 748 
 
 758 
 
 768 
 
 777 
 
 787 
 
 797 
 
 807 
 
 816 
 
 826 
 
 9 
 
 8.1 
 
 45 
 
 856 
 
 846 
 
 856 
 
 865 
 
 875 
 
 885 
 
 895 
 
 904 
 
 914 
 
 924 
 
 
 
 46 
 
 64 933 
 
 943 
 
 953 
 
 963 
 
 972 
 
 982 
 
 992 
 
 *002 
 
 *011 
 
 *021 
 
 
 
 47 
 
 65 031 
 
 040 
 
 050 
 
 060 
 
 070 
 
 079 
 
 089 
 
 099 
 
 108 
 
 118 
 
 
 
 48 
 
 128 
 
 137 
 
 147 
 
 157 
 
 167 
 
 176 
 
 186 
 
 196 
 
 205 
 
 215 
 
 
 
 49 
 
 225 
 
 234 
 
 244 
 
 254 
 
 263 
 
 273 
 
 283 
 
 292 
 
 302 
 
 312 
 
 
 
 450 
 
 65 321 
 
 331 
 
 341 
 
 350 
 
 360 
 
 369 
 
 379 
 
 389 
 
 398 
 
 408 
 
 
 
 N 
 
 0 
 
 1 
 
 2 j 
 
 3 
 
 4 
 
 5 1 6 
 
 7 
 
 8 
 
 9 
 
 Prop . Parts 
 
TABLES 
 
 311 
 
 TABLE V — ( Continued ) 
 Five-place Logarithms: 450-500 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 450 
 
 65 321 
 
 331 
 
 341 
 
 350 
 
 360 
 
 369 
 
 379 
 
 389 
 
 398 
 
 408 
 
 
 
 51 
 
 418 
 
 427 
 
 437 
 
 447 
 
 456 
 
 466 
 
 475 
 
 485 
 
 495 
 
 504 
 
 
 
 52 
 
 514 
 
 523 
 
 533 
 
 543 
 
 552 
 
 562 
 
 571 
 
 581 
 
 591 
 
 600 
 
 
 
 53 
 
 610 
 
 619 
 
 629 
 
 639 
 
 648 
 
 658 
 
 667 
 
 677 
 
 686 
 
 696 
 
 
 
 54 
 
 706 
 
 715 
 
 725 
 
 734 
 
 744 
 
 753 
 
 763 
 
 772 
 
 782 
 
 792 
 
 
 
 55 
 
 801 
 
 811 
 
 820 
 
 830 
 
 839 
 
 849 
 
 858 
 
 868 
 
 877 
 
 887 
 
 
 in 
 
 56 
 
 896 
 
 906 
 
 916 
 
 925 
 
 935 
 
 944 
 
 954 
 
 963 
 
 973 
 
 982 
 
 1 
 
 1.0 
 
 57 
 
 65 992 
 
 *001 
 
 *011 
 
 *020 
 
 *030 
 
 *039 
 
 *049 
 
 *058 
 
 *068 
 
 *077 
 
 2 
 
 2.0 
 
 58 
 
 66 087 
 
 096 
 
 106 
 
 115 
 
 124 
 
 134 
 
 143 
 
 153 
 
 162 
 
 172 
 
 3 
 
 3.0 
 
 59 
 
 181 
 
 191 
 
 200 
 
 210 
 
 219 
 
 229 
 
 238 
 
 247 
 
 257 
 
 266 
 
 4 
 
 4 0 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 5.0 
 
 460 
 
 276 
 
 285 
 
 295 
 
 304 
 
 314 
 
 323 
 
 332 
 
 342 
 
 351 
 
 361 
 
 6 
 
 6.0 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 7.0 
 
 61 
 
 370 
 
 380 
 
 389 
 
 398 
 
 408 
 
 417 
 
 427 
 
 436 
 
 445 
 
 455 
 
 8 
 
 8.0 
 
 62 
 
 464 
 
 474 
 
 483 
 
 492 
 
 502 
 
 511 
 
 521 
 
 530 
 
 539 
 
 549 
 
 9 
 
 9.0 
 
 63 
 
 558 
 
 567 
 
 577 
 
 586 
 
 596 
 
 605 
 
 614 
 
 624 
 
 633 
 
 642 
 
 
 
 64 
 
 652 
 
 661 
 
 671 
 
 680 
 
 689 
 
 699 
 
 708 
 
 717 
 
 727 
 
 736 
 
 
 
 65 
 
 745 
 
 755 
 
 764 
 
 773 
 
 783 
 
 792 
 
 801 
 
 811 
 
 820 
 
 829 
 
 
 
 66 
 
 839 
 
 848 
 
 857 
 
 867 
 
 876 
 
 885 
 
 894 
 
 904 
 
 913 
 
 922 
 
 
 
 67 
 
 66 932 
 
 941 
 
 950 
 
 960 
 
 969 
 
 978 
 
 987 
 
 997 
 
 *006 
 
 *015 
 
 
 
 68 
 
 67 025 
 
 034 
 
 043 
 
 052 
 
 062 
 
 071 
 
 080 
 
 089 
 
 099 
 
 108 
 
 
 
 69 
 
 117 
 
 127 
 
 136 
 
 145 
 
 154 
 
 164 
 
 173 
 
 182 
 
 191 
 
 201 
 
 
 
 470 
 
 210 
 
 219 
 
 228 
 
 237 
 
 247 
 
 256 
 
 265 
 
 274 
 
 284 
 
 293 
 
 
 
 71 
 
 302 
 
 311 
 
 321 
 
 330 
 
 339 
 
 348 
 
 357 
 
 367 
 
 376 
 
 385 
 
 
 9 
 
 72 
 
 394 
 
 403 
 
 413 
 
 422 
 
 431 
 
 440 
 
 449 
 
 459 
 
 468 
 
 477 
 
 i 
 
 0.9 
 
 73 
 
 486 
 
 495 
 
 504 
 
 514 
 
 523 
 
 532 
 
 541 
 
 550 
 
 560 
 
 569 
 
 2 
 
 1.8 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 2.7 
 
 74 
 
 578 
 
 587 
 
 596 
 
 605 
 
 614 
 
 624 
 
 633 
 
 642 
 
 651 
 
 660 
 
 4 
 
 3.6 
 
 75 
 
 669 
 
 679 
 
 688 
 
 697 
 
 706 
 
 715 
 
 724 
 
 733 
 
 742 
 
 752 
 
 5 
 
 6 
 
 4.5 
 
 5.4 
 
 76 
 
 761 
 
 770 
 
 779 
 
 788 
 
 797 
 
 806 
 
 815 
 
 825 
 
 834 
 
 843 
 
 7 
 
 6.3 
 
 77 
 
 852 
 
 861 
 
 870 
 
 879 
 
 888 
 
 897 
 
 906 
 
 916 
 
 925 
 
 934 
 
 9 
 
 ft 1 
 
 78 
 
 67 943 
 
 952 
 
 961 
 
 970 
 
 979 
 
 988 
 
 997 
 
 *006 
 
 *015 
 
 *024 
 
 
 
 79 
 
 68 034 
 
 043 
 
 052 
 
 061 
 
 070 
 
 079 
 
 088 
 
 097 
 
 106 
 
 115 
 
 
 
 480 
 
 124 
 
 133 
 
 142 
 
 151 
 
 160 
 
 169 
 
 178 
 
 187 
 
 196 
 
 205 
 
 
 
 81 
 
 215 
 
 224 
 
 233 
 
 242 
 
 251 
 
 260 
 
 269 
 
 278 
 
 287 
 
 296 
 
 
 
 82 
 
 305 
 
 314 
 
 323 
 
 332 
 
 341 
 
 350 
 
 359 
 
 368 
 
 377 
 
 386 
 
 
 
 83 
 
 395 
 
 404 
 
 413 
 
 422 
 
 431 
 
 440 
 
 449 
 
 458 
 
 467 
 
 476 
 
 
 
 84 
 
 485 
 
 494 
 
 502 
 
 511 
 
 520 
 
 529 
 
 538 
 
 547 
 
 556 
 
 565 
 
 
 
 85 
 
 574 
 
 583 
 
 592 
 
 601 
 
 610 
 
 619 
 
 628 
 
 637 
 
 646 
 
 655 
 
 
 
 86 
 
 664 
 
 673 
 
 681 
 
 690 
 
 699 
 
 708 
 
 717 
 
 726 
 
 735 
 
 744 
 
 
 
 87 
 
 753 
 
 762 
 
 771 
 
 780 
 
 789 
 
 797 
 
 806 
 
 815 
 
 824 
 
 833 
 
 
 
 88 
 
 842 
 
 851 
 
 860 
 
 869 
 
 878 
 
 886 
 
 895 
 
 904 
 
 913 
 
 922 
 
 1 
 
 3 
 
 0.8 
 
 1.6 
 
 89 
 
 68 931 
 
 940 
 
 949 
 
 958 
 
 966 
 
 975 
 
 984 
 
 993 
 
 *002 
 
 *011 
 
 3 
 
 2.4 
 
 400 
 
 69 020 
 
 028 
 
 037 
 
 046 
 
 055 
 
 064 
 
 073 
 
 082 
 
 090 
 
 099 
 
 4 
 
 3.2 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 4.0 
 
 91 
 
 108 
 
 117 
 
 126 
 
 135 
 
 144 
 
 152 
 
 161 
 
 170 
 
 179 
 
 188 
 
 
 
 92 
 
 197 
 
 205 
 
 214 
 
 223 
 
 232 
 
 241 
 
 249 
 
 258 
 
 267 
 
 276 
 
 7 
 
 8 
 
 5.6 
 
 6.4 
 
 93 
 
 285 
 
 294 
 
 302 
 
 311 
 
 320 
 
 329 
 
 338 
 
 346 
 
 355 
 
 564 
 
 9 
 
 7.2 
 
 94 
 
 373 
 
 381 
 
 390 
 
 399 
 
 408 
 
 417 
 
 425 
 
 434 
 
 443 
 
 452 
 
 
 
 95 
 
 461 
 
 469 
 
 478 
 
 487 
 
 496 
 
 504 
 
 513 
 
 522 
 
 531 
 
 539 
 
 
 
 96 
 
 548 
 
 557 
 
 566 
 
 574 
 
 583 
 
 592 
 
 601 
 
 609 
 
 618 
 
 627 
 
 
 
 97 
 
 636 
 
 644 
 
 653 
 
 662 
 
 671 
 
 679 
 
 688 
 
 697 
 
 705 
 
 714 
 
 
 
 98 
 
 723 
 
 732 
 
 740 
 
 749 
 
 758 
 
 767 
 
 775 
 
 784 
 
 793 
 
 801 
 
 
 
 99 
 
 810 
 
 819 
 
 827 
 
 836 
 
 845 
 
 854 
 
 862 
 
 871 
 
 880 
 
 888 
 
 
 
 500 
 
 69 897 
 
 906 
 
 914 
 
 923 
 
 932 
 
 940 
 
 949 
 
 958 
 
 966 
 
 975 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
312 
 
 TABLES 
 
 TABLE V—( Continued ) 
 Five-place Logarithms: 500-550 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop 
 
 Parts 
 
 500 
 
 69 897 
 
 906 
 
 914 
 
 923 
 
 932 
 
 940 
 
 949 
 
 958 
 
 966 
 
 975 
 
 
 
 01 
 
 69 984 
 
 992 
 
 *001 
 
 *010 
 
 *018 
 
 *027 
 
 *036 
 
 *044 
 
 *053 
 
 *062 
 
 
 
 02 
 
 70 070 
 
 079 
 
 088 
 
 096 
 
 105 
 
 114 
 
 122 
 
 131 
 
 140 
 
 148 
 
 
 
 03 
 
 157 
 
 165 
 
 174 
 
 183 
 
 191 
 
 200 
 
 209 
 
 217 
 
 226 
 
 234 
 
 
 
 04 
 
 243 
 
 252 
 
 260 
 
 269 
 
 278 
 
 286 
 
 295 
 
 303 
 
 312 
 
 321 
 
 
 
 05 
 
 329 
 
 338 
 
 546 
 
 355 
 
 364 
 
 372 
 
 381 
 
 389 
 
 398 
 
 406 
 
 
 
 06 
 
 415 
 
 424 
 
 432 
 
 441 
 
 449 
 
 458 
 
 467 
 
 475 
 
 484 
 
 492 
 
 
 
 07 
 
 501 
 
 509 
 
 518 
 
 526 
 
 535 
 
 544 
 
 552 
 
 561 
 
 569 
 
 578 
 
 1 
 
 0.9 
 
 08 
 
 586 
 
 595 
 
 605 
 
 612 
 
 621 
 
 629 
 
 638 
 
 646 
 
 655 
 
 665 
 
 2 
 
 1.8 
 
 09 
 
 672 
 
 680 
 
 689 
 
 697 
 
 706 
 
 714 
 
 7 23 
 
 731 
 
 740 
 
 749 
 
 3 
 
 2.7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 510 
 
 757 
 
 766 
 
 774 
 
 783 
 
 791 
 
 800 
 
 808 
 
 817 
 
 825 
 
 834 
 
 o 
 
 4.5 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 5 4 
 
 11 
 
 842 
 
 851 
 
 859 
 
 868 
 
 876 
 
 885 
 
 893 
 
 902 
 
 910 
 
 919 
 
 7 
 
 6.3 
 
 12 
 
 70 927 
 
 035 
 
 944 
 
 952 
 
 961 
 
 969 
 
 978 
 
 986 
 
 995 
 
 *003 
 
 8 
 
 7.2 
 
 13 
 
 71 012 
 
 020 
 
 029 
 
 037 
 
 046 
 
 054 
 
 063 
 
 071 
 
 079 
 
 088 
 
 9 
 
 8.1 
 
 14 
 
 096 
 
 105 
 
 113 
 
 122 
 
 150 
 
 139 
 
 147 
 
 155 
 
 164 
 
 172 
 
 
 
 15 
 
 181 
 
 189 
 
 198 
 
 206 
 
 214 
 
 223 
 
 231 
 
 240 
 
 248 
 
 257 
 
 
 
 16 
 
 265 
 
 273 
 
 282 
 
 290 
 
 299 
 
 307 
 
 315 
 
 324 
 
 332 
 
 341 
 
 
 
 17 
 
 349 
 
 357 
 
 566 
 
 374 
 
 383 
 
 391 
 
 399 
 
 408 
 
 416 
 
 425 
 
 
 
 18 
 
 433 
 
 441 
 
 450 
 
 458 
 
 466 
 
 475 
 
 483 
 
 492 
 
 500 
 
 508 
 
 
 
 19 
 
 517 
 
 525 
 
 533 
 
 542 
 
 550 
 
 559 
 
 567 
 
 575 
 
 584 
 
 592 
 
 
 
 520 
 
 600 
 
 609 
 
 617 
 
 625 
 
 634 
 
 642 
 
 650 
 
 659 
 
 667 
 
 675 
 
 
 
 21 
 
 684 
 
 692 
 
 700 
 
 709 
 
 717 
 
 725 
 
 734 
 
 742 
 
 750 
 
 759 
 
 
 
 22 
 
 767 
 
 775 
 
 784 
 
 792 
 
 800 
 
 809 
 
 817 
 
 825 
 
 834 
 
 842 
 
 
 8 
 
 23 
 
 850 
 
 858 
 
 867 
 
 875 
 
 883 
 
 892 
 
 900 
 
 908 
 
 917 
 
 925 
 
 1 
 
 0.8 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 1.6 
 
 24 
 
 71 933 
 
 941 
 
 950 
 
 958 
 
 966 
 
 975 
 
 983 
 
 991 
 
 999 
 
 *008 
 
 3 
 
 2.4 
 
 25 
 
 72 016 
 
 024 
 
 032 
 
 041 
 
 049 
 
 057 
 
 066 
 
 074 
 
 082 
 
 090 
 
 4 
 
 3.2 
 
 26 
 
 099 
 
 107 
 
 115 
 
 123 
 
 132 
 
 140 
 
 148 
 
 156 
 
 165 
 
 173 
 
 5 
 
 6 
 
 4.0 
 
 4.8 
 
 27 
 
 181 
 
 189 
 
 198 
 
 206 
 
 214 
 
 222 
 
 230 
 
 239 
 
 247 
 
 255 
 
 7 
 
 5.6 
 
 28 
 
 263 
 
 272 
 
 280 
 
 288 
 
 296 
 
 504 
 
 313 
 
 321 
 
 329 
 
 337 
 
 9 
 
 7.2 
 
 29 
 
 346 
 
 354 
 
 362 
 
 570 
 
 378 
 
 587 
 
 395 
 
 403 
 
 411 
 
 419 
 
 
 530 
 
 428 
 
 436 
 
 444 
 
 452 
 
 460 
 
 469 
 
 477 
 
 485 
 
 493 
 
 501 
 
 
 
 31 
 
 509 
 
 518 
 
 526 
 
 534 
 
 542 
 
 550 
 
 558 
 
 567 
 
 575 
 
 583 
 
 
 
 32 
 
 591 
 
 599 
 
 607 
 
 616 
 
 624 
 
 632 
 
 640 
 
 648 
 
 656 
 
 665 
 
 
 
 33 
 
 673 
 
 681 
 
 689 
 
 697 
 
 705 
 
 713 
 
 722 
 
 730 
 
 738 
 
 746 
 
 
 
 34 
 
 754 
 
 762 
 
 770 
 
 779 
 
 787 
 
 795 
 
 803 
 
 811 
 
 819 
 
 827 
 
 
 
 55 
 
 835 
 
 843 
 
 852 
 
 860 
 
 868 
 
 876 
 
 884 
 
 892 
 
 900 
 
 908 
 
 
 
 56 
 
 916 
 
 925 
 
 933 
 
 941 
 
 949 
 
 957 
 
 965 
 
 973 
 
 981 
 
 989 
 
 
 
 37 
 
 72 997 
 
 *006 
 
 *014 
 
 *022 
 
 *030 
 
 *058 
 
 *046 
 
 *054 
 
 *062 
 
 *070 
 
 
 
 38 
 
 73 078 
 
 086 
 
 094 
 
 102 
 
 111 
 
 119 
 
 127 
 
 135 
 
 143 
 
 151 
 
 
 
 39 
 
 159 
 
 167 
 
 175 
 
 183 
 
 191 
 
 199 
 
 207 
 
 215 
 
 223 
 
 231 
 
 1 
 
 2 
 
 0.7 
 
 1.4 
 
 540 
 
 259 
 
 247 
 
 255 
 
 263 
 
 272 
 
 280 
 
 288 
 
 296 
 
 504 
 
 312 
 
 3 
 
 2.1 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 2.8 
 
 41 
 
 320 
 
 328 
 
 336 
 
 344 
 
 352 
 
 360 
 
 368 
 
 376 
 
 384 
 
 392 
 
 5 
 
 3.5 
 
 42 
 
 400 
 
 408 
 
 416 
 
 424 
 
 452 
 
 440 
 
 448 
 
 456 
 
 464 
 
 472 
 
 
 
 43 
 
 480 
 
 488 
 
 496 
 
 504 
 
 512 
 
 520 
 
 528 
 
 536 
 
 544 
 
 552 
 
 7 
 
 8 
 
 4.9 
 
 5.6 
 
 44 
 
 560 
 
 568 
 
 576 
 
 584 
 
 592 
 
 600 
 
 608 
 
 616 
 
 624 
 
 632 
 
 9 
 
 6.3 
 
 45 
 
 640 
 
 648 
 
 656 
 
 664 
 
 672 
 
 679 
 
 687 
 
 695 
 
 703 
 
 711 
 
 
 
 46 
 
 719 
 
 727 
 
 735 
 
 743 
 
 751 
 
 759 
 
 767 
 
 775 
 
 785 
 
 791 
 
 
 
 47 
 
 799 
 
 807 
 
 815 
 
 823 
 
 830 
 
 838 
 
 846 
 
 854 
 
 862 
 
 870 
 
 
 
 48 
 
 878 
 
 886 
 
 894 
 
 902 
 
 910 
 
 918 
 
 926 
 
 953 
 
 941 
 
 949 
 
 
 
 49 
 
 73 957 
 
 965 
 
 975 
 
 981 
 
 989 
 
 997 
 
 *005 
 
 *013 
 
 *020 
 
 *028 
 
 
 
 550 
 
 74 036 
 
 044 
 
 052 
 
 060 
 
 068 
 
 076 
 
 084 
 
 092 
 
 099 
 
 107 
 
 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
TABLES 
 
 313 
 
 TABLE V—( Continued ) 
 Five-place Logarithms: 550-600 
 
 Prop 
 
 . Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 550 
 
 74 036 
 
 044 
 
 052 
 
 060 
 
 068 
 
 076 
 
 084 
 
 092 
 
 099 
 
 107 
 
 
 
 51 
 
 115 
 
 123 
 
 131 
 
 139 
 
 147 
 
 155 
 
 162 
 
 170 
 
 178 
 
 186 
 
 
 
 52 
 
 194 
 
 202 
 
 210 
 
 218 
 
 225 
 
 233 
 
 241 
 
 249 
 
 257 
 
 265 
 
 
 
 53 
 
 273 
 
 280 
 
 288 
 
 296 
 
 304 
 
 312 
 
 320 
 
 327 
 
 335 
 
 343 
 
 
 
 54 
 
 351 
 
 359 
 
 367 
 
 374 
 
 382 
 
 390 
 
 398 
 
 406 
 
 414 
 
 421 
 
 
 
 55 
 
 429 
 
 437 
 
 445 
 
 453 
 
 461 
 
 468 
 
 476 
 
 484 
 
 492 
 
 500 
 
 
 
 56 
 
 507 
 
 515 
 
 523 
 
 531 
 
 539 
 
 547 
 
 554 
 
 562 
 
 570 
 
 578 
 
 
 
 57 
 
 586 
 
 593 
 
 601 
 
 609 
 
 617 
 
 624 
 
 632 
 
 640 
 
 648 
 
 656 
 
 
 
 58 
 
 663 
 
 671 
 
 679 
 
 687 
 
 695 
 
 702 
 
 710 
 
 718 
 
 726 
 
 733 
 
 
 
 59 
 
 741 
 
 749 
 
 757 
 
 764 
 
 772 
 
 780 
 
 788 
 
 796 
 
 803 
 
 811 
 
 
 
 560 
 
 819 
 
 827 
 
 834 
 
 842 
 
 850 
 
 858 
 
 865 
 
 873 
 
 881 
 
 889 
 
 
 
 61 
 
 896 
 
 904 
 
 912 
 
 920 
 
 927 
 
 935 
 
 943 
 
 950 
 
 958 
 
 966 
 
 
 8 
 
 62 
 
 74 974 
 
 981 
 
 989 
 
 997 
 
 *005 
 
 *012 
 
 *020 
 
 *028 
 
 *035 
 
 *043 
 
 1 
 
 0.8 
 
 63 
 
 75 051 
 
 059 
 
 066 
 
 074 
 
 082 
 
 089 
 
 097 
 
 105 
 
 113 
 
 120 
 
 2 
 
 1.6 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 2.4 
 
 64 
 
 128 
 
 136 
 
 143 
 
 151 
 
 159 
 
 166 
 
 174 
 
 182 
 
 189 
 
 197 
 
 4 
 
 3.2 
 
 65 
 
 205 
 
 213 
 
 220 
 
 228 
 
 236 
 
 243 
 
 251 
 
 259 
 
 266 
 
 274 
 
 5 
 
 4.0 
 
 66 
 
 282 
 
 289 
 
 297 
 
 305 
 
 312 
 
 320 
 
 328 
 
 335 
 
 343 
 
 351 
 
 6 
 
 4.8 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 5.6 
 
 67 
 
 358 
 
 366 
 
 374 
 
 381 
 
 389 
 
 397 
 
 404 
 
 412 
 
 420 
 
 427 
 
 8 
 
 6.4 
 
 68 
 
 435 
 
 442 
 
 450 
 
 458 
 
 465 
 
 473 
 
 481 
 
 488 
 
 496 
 
 504 
 
 9 
 
 7.2 
 
 69 
 
 511 
 
 519 
 
 526 
 
 534 
 
 542 
 
 549 
 
 557 
 
 565 
 
 572 
 
 580 
 
 
 
 570 
 
 587 
 
 595 
 
 603 
 
 610 
 
 618 
 
 626 
 
 633 
 
 641 
 
 648 
 
 656 
 
 
 
 71 
 
 664 
 
 671 
 
 679 
 
 686 
 
 694 
 
 702 
 
 709 
 
 717 
 
 724 
 
 732 
 
 
 
 72 
 
 740 
 
 747 
 
 755 
 
 762 
 
 770 
 
 778 
 
 785 
 
 793 
 
 800 
 
 808 
 
 
 
 73 
 
 815 
 
 823 
 
 831 
 
 838 
 
 846 
 
 853 
 
 861 
 
 868 
 
 876 
 
 884 
 
 
 
 74 
 
 891 
 
 899 
 
 906 
 
 914 
 
 921 
 
 929 
 
 937 
 
 944 
 
 952 
 
 959 
 
 
 
 75 
 
 75 967 
 
 974 
 
 982 
 
 989 
 
 997 
 
 *005 
 
 *012 
 
 *020 
 
 *027 
 
 *035 
 
 
 
 76 
 
 76 042 
 
 050 
 
 057 
 
 065 
 
 072 
 
 080 
 
 087 
 
 095 
 
 103 
 
 110 
 
 
 
 77 
 
 118 
 
 125 
 
 133 
 
 140 
 
 148 
 
 155 
 
 163 
 
 170 
 
 178 
 
 185 
 
 
 
 78 
 
 193 
 
 200 
 
 208 
 
 215 
 
 223 
 
 230 
 
 238 
 
 245 
 
 253 
 
 260 
 
 
 
 79 
 
 268 
 
 275 
 
 283 
 
 290 
 
 298 
 
 305 
 
 313 
 
 320 
 
 328 
 
 335 
 
 
 
 580 
 
 343 
 
 350 
 
 358 
 
 365 
 
 373 
 
 380 
 
 388 
 
 395 
 
 403 
 
 410 
 
 
 7 
 
 81 
 
 418 
 
 425 
 
 433 
 
 440 
 
 448 
 
 455 
 
 462 
 
 470 
 
 477 
 
 485 
 
 1 
 
 0.7 
 
 82 
 
 492 
 
 500 
 
 507 
 
 515 
 
 522 
 
 530 
 
 537 
 
 545 
 
 552 
 
 559 
 
 2 
 
 1.4 
 
 83 
 
 567 
 
 574 
 
 582 
 
 589 
 
 597 
 
 604 
 
 612 
 
 619 
 
 626 
 
 634 
 
 3 
 
 2.1 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 2.8 
 
 84 
 
 641 
 
 649 
 
 656 
 
 664 
 
 671 
 
 678 
 
 686 
 
 693 
 
 701 
 
 708 
 
 5 
 
 3.5 
 
 85 
 
 716 
 
 723 
 
 730 
 
 738 
 
 745 
 
 753 
 
 760 
 
 768 
 
 775 
 
 782 
 
 V 
 
 4.2 
 
 86 
 
 790 
 
 797 
 
 805 
 
 812 
 
 819 
 
 827 
 
 834 
 
 842 
 
 849 
 
 856 
 
 7 
 
 4.9 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 5.6 
 
 87 
 
 864 
 
 871 
 
 879 
 
 886 
 
 893 
 
 901 
 
 908 
 
 916 
 
 923 
 
 930 
 
 9 
 
 6.3 
 
 88 
 
 76 938 
 
 945 
 
 953 
 
 960 
 
 967 
 
 975 
 
 982 
 
 989 
 
 997 
 
 *004 
 
 
 
 89 
 
 77 012 
 
 019 
 
 026 
 
 034 
 
 041 
 
 048 
 
 056 
 
 063 
 
 070 
 
 078 
 
 
 
 590 
 
 085 
 
 093 
 
 100 
 
 107 
 
 115 
 
 122 
 
 129 
 
 137 
 
 144 
 
 151 
 
 
 
 91 
 
 159 
 
 166 
 
 173 
 
 181 
 
 188 
 
 195 
 
 203 
 
 210 
 
 217 
 
 225 
 
 
 
 92 
 
 232 
 
 240 
 
 247 
 
 254 
 
 262 
 
 269 
 
 276 
 
 283 
 
 291 
 
 298 
 
 
 
 93 
 
 305 
 
 313 
 
 320 
 
 327 
 
 335 
 
 342 
 
 349 
 
 357 
 
 364 
 
 371 
 
 
 
 94 
 
 379 
 
 386 
 
 393 
 
 401 
 
 408 
 
 415 
 
 422 
 
 430 
 
 437 
 
 444 
 
 
 
 95 
 
 452 
 
 459 
 
 466 
 
 474 
 
 481 
 
 488 
 
 495 
 
 503 
 
 510 
 
 517 
 
 
 
 96 
 
 525 
 
 532 
 
 539 
 
 546 
 
 554 
 
 561 
 
 568 
 
 576 
 
 583 
 
 590 
 
 
 
 97 
 
 597 
 
 605 
 
 612 
 
 619 
 
 627 
 
 634 
 
 641 
 
 648 
 
 656 
 
 663 
 
 
 
 98 
 
 670 
 
 677 
 
 685 
 
 692 
 
 699 
 
 706 
 
 714 
 
 721 
 
 728 
 
 735 
 
 
 
 99 
 
 743 
 
 750 
 
 757 
 
 764 
 
 772 
 
 779 
 
 786 
 
 793 
 
 801 
 
 808 
 
 
 
 600 
 
 77 815 
 
 822 
 
 830 
 
 837 
 
 844 
 
 851 
 
 859 
 
 866 
 
 873 
 
 880 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 :t 
 
 4 
 
 5 
 
 « 
 
 7 
 
 8 
 
 9 
 
314 
 
 TABLES 
 
 TABLE V—( Continued) 
 Five-place Logarithms: 600-650 
 
 N 
 
 0 
 
 1 
 
 o 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
 GOO 
 
 77 815 
 
 822 
 
 830 
 
 837 
 
 844 
 
 851 
 
 859 
 
 866 
 
 873 
 
 880 
 
 
 
 01 
 
 887 
 
 895 
 
 902 
 
 909 
 
 916 
 
 924 
 
 931 
 
 938 
 
 945 
 
 952 
 
 
 
 02 
 
 77 960 
 
 967 
 
 974 
 
 981 
 
 988 
 
 996 
 
 *003 
 
 *010 
 
 *017 
 
 *025 
 
 
 
 03 
 
 78 032 
 
 039 
 
 046 
 
 05 o 
 
 061 
 
 068 
 
 075 
 
 082 
 
 089 
 
 097 
 
 
 
 04 
 
 104 
 
 111 
 
 118 
 
 125 
 
 132 
 
 140 
 
 147 
 
 154 
 
 161 
 
 168 
 
 
 
 05 
 
 176 
 
 183 
 
 190 
 
 197 
 
 204 
 
 211 
 
 219 
 
 226 
 
 233 
 
 240 
 
 
 
 06 
 
 247 
 
 254 
 
 262 
 
 269 
 
 276 
 
 283 
 
 290 
 
 297 
 
 305 
 
 312 
 
 
 
 07 
 
 319 
 
 326 
 
 333 
 
 340 
 
 347 
 
 355 
 
 362 
 
 369 
 
 376 
 
 383 
 
 1 
 
 0.8 
 
 08 
 
 390 
 
 398 
 
 405 
 
 412 
 
 419 
 
 426 
 
 433 
 
 440 
 
 447 
 
 455 
 
 2 
 
 1.6 
 
 09 
 
 462 
 
 469 
 
 476 
 
 483 
 
 490 
 
 497 
 
 504 
 
 512 
 
 519 
 
 526 
 
 3 
 
 2.4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 G 10 
 
 533 
 
 540 
 
 547 
 
 554 
 
 561 
 
 569 
 
 576 
 
 583 
 
 590 
 
 597 
 
 5 
 
 4.0 
 
 
 
 
 
 
 
 
 
 
 
 
 0 
 
 
 11 
 
 604 
 
 611 
 
 618 
 
 625 
 
 653 
 
 640 
 
 647 
 
 654 
 
 661 
 
 668 
 
 7 
 
 5.6 
 
 12 
 
 675 
 
 682 
 
 689 
 
 696 
 
 704 
 
 711 
 
 718 
 
 725 
 
 732 
 
 739 
 
 8 
 
 6.4 
 
 13 
 
 746 
 
 753 
 
 760 
 
 767 
 
 774 
 
 781 
 
 789 
 
 796 
 
 805 
 
 810 
 
 9 
 
 7.2 
 
 14 
 
 817 
 
 824 
 
 831 
 
 838 
 
 845 
 
 852 
 
 859 
 
 866 
 
 873 
 
 880 
 
 
 
 15 
 
 888 
 
 895 
 
 902 
 
 909 
 
 916 
 
 923 
 
 950 
 
 937 
 
 944 
 
 951 
 
 
 
 16 
 
 78 958 
 
 965 
 
 972 
 
 979 
 
 986 
 
 993 
 
 *000 
 
 *007 
 
 *014 
 
 *021 
 
 
 
 17 
 
 79 029 
 
 036 
 
 043 
 
 050 
 
 057 
 
 064 
 
 071 
 
 078 
 
 085 
 
 092 
 
 
 
 18 
 
 099 
 
 106 
 
 113 
 
 120 
 
 127 
 
 134 
 
 141 
 
 148 
 
 155 
 
 162 
 
 
 
 19 
 
 169 
 
 176 
 
 183 
 
 190 
 
 197 
 
 204 
 
 211 
 
 218 
 
 225 
 
 232 
 
 
 
 620 
 
 239 
 
 246 
 
 253 
 
 260 
 
 267 
 
 274 
 
 281 
 
 288 
 
 295 
 
 302 
 
 
 
 21 
 
 309 
 
 316 
 
 323 
 
 330 
 
 337 
 
 344 
 
 351 
 
 358 
 
 365 
 
 372 
 
 
 
 22 
 
 379 
 
 386 
 
 393 
 
 400 
 
 407 
 
 414 
 
 421 
 
 428 
 
 435 
 
 442 
 
 
 7 
 
 23 
 
 449 
 
 456 
 
 463 
 
 470 
 
 477 
 
 484 
 
 491 
 
 498 
 
 505 
 
 511 
 
 1 
 
 0.7 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 1.4 
 
 24 
 
 518 
 
 525 
 
 532 
 
 539 
 
 546 
 
 553 
 
 560 
 
 567 
 
 574 
 
 581 
 
 3 
 
 2.1 
 
 25 
 
 588 
 
 595 
 
 602 
 
 609 
 
 616 
 
 623 
 
 630 
 
 637 
 
 644 
 
 650 
 
 4 
 
 2.8 
 
 26 
 
 657 
 
 664 
 
 671 
 
 678 
 
 685 
 
 692 
 
 699 
 
 706 
 
 713 
 
 720 
 
 5 
 
 3.5 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 4.2 
 
 27 
 
 727 
 
 734 
 
 741 
 
 748 
 
 754 
 
 761 
 
 768 
 
 775 
 
 782 
 
 789 
 
 7 
 
 4.9 
 
 28 
 
 796 
 
 803 
 
 810 
 
 817 
 
 824 
 
 831 
 
 857 
 
 844 
 
 851 
 
 858 
 
 8 
 
 5.6 
 
 29 
 
 865 
 
 872 
 
 879 
 
 886 
 
 893 
 
 900 
 
 906 
 
 913 
 
 920 
 
 927 
 
 9 
 
 6.3 
 
 630 
 
 79 934 
 
 941 
 
 948 
 
 955 
 
 962 
 
 969 
 
 975 
 
 982 
 
 989 
 
 996 
 
 
 
 31 
 
 80 003 
 
 010 
 
 017 
 
 024 
 
 050 
 
 037 
 
 044 
 
 051 
 
 058 
 
 065 
 
 
 
 32 
 
 072 
 
 079 
 
 085 
 
 092 
 
 099 
 
 106 
 
 113 
 
 120 
 
 127 
 
 134 
 
 
 
 33 
 
 140 
 
 147 
 
 154 
 
 161 
 
 168 
 
 175 
 
 182 
 
 188 
 
 195 
 
 202 
 
 
 
 34 
 
 209 
 
 216 
 
 223 
 
 229 
 
 236 
 
 243 
 
 250 
 
 257 
 
 264 
 
 271 
 
 
 
 JO 
 
 277 
 
 284 
 
 291 
 
 298 
 
 305 
 
 312 
 
 318 
 
 325 
 
 332 
 
 339 
 
 
 
 36 
 
 346 
 
 353 
 
 559 
 
 366 
 
 573 
 
 380 
 
 387 
 
 393 
 
 400 
 
 407 
 
 
 
 37 
 
 414 
 
 421 
 
 428 
 
 454 
 
 441 
 
 448 
 
 455 
 
 462 
 
 468 
 
 475 
 
 
 
 38 
 
 482 
 
 489 
 
 496 
 
 502 
 
 509 
 
 516 
 
 525 
 
 550 
 
 536 
 
 543 
 
 
 6 
 
 39 
 
 550 
 
 557 
 
 564 
 
 570 
 
 577 
 
 584 
 
 591 
 
 598 
 
 604 
 
 611 
 
 i 
 
 0.6 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 1.2 
 
 640 
 
 618 
 
 625 
 
 632 
 
 658 
 
 645 
 
 652 
 
 659 
 
 665 
 
 672 
 
 679 
 
 3 
 
 1.8 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 ? 4 
 
 41 
 
 686 
 
 693 
 
 699 
 
 706 
 
 713 
 
 720 
 
 726 
 
 733 
 
 740 
 
 747 
 
 5 
 
 3.0 
 
 42 
 
 754 
 
 760 
 
 767 
 
 774 
 
 781 
 
 7 S 7 
 
 794 
 
 801 
 
 808 
 
 814 
 
 6 
 
 3.6 
 
 43 
 
 821 
 
 828 
 
 835 
 
 841 
 
 848 
 
 855 
 
 862 
 
 868 
 
 875 
 
 882 
 
 7 
 
 4.2 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 4.8 
 
 44 
 
 889 
 
 895 
 
 902 
 
 909 
 
 916 
 
 922 
 
 929 
 
 936 
 
 943 
 
 949 
 
 9 
 
 5.4 
 
 45 
 
 80 956 
 
 963 
 
 969 
 
 976 
 
 983 
 
 990 
 
 996 
 
 *003 
 
 *010 
 
 *017 
 
 
 
 46 
 
 81 023 
 
 030 
 
 037 
 
 043 
 
 050 
 
 057 
 
 064 
 
 070 
 
 077 
 
 084 
 
 
 
 47 
 
 090 
 
 097 
 
 104 
 
 111 
 
 117 
 
 124 
 
 131 
 
 137 
 
 144 
 
 151 
 
 
 
 48 
 
 158 
 
 164 
 
 171 
 
 178 
 
 184 
 
 191 
 
 198 
 
 204 
 
 211 
 
 218 
 
 
 
 49 
 
 224 
 
 231 
 
 238 
 
 245 
 
 251 
 
 258 
 
 265 
 
 271 
 
 278 
 
 285 
 
 
 
 650 
 
 81 291 
 
 298 
 
 305 
 
 311 
 
 318 
 
 325 
 
 331 
 
 338 
 
 345 
 
 351 
 
 
 
 N 
 
 O 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
TABLES 
 
 315 
 
 TABLE V —( Continued ) 
 Five-place Logarithms: 650-700 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 650 
 
 81 291 
 
 298 
 
 305 
 
 311 
 
 318 
 
 325 
 
 331 
 
 338 
 
 345 
 
 351 
 
 
 
 51 
 
 358 
 
 365 
 
 371 
 
 378 
 
 385 
 
 391 
 
 398 
 
 405 
 
 411 
 
 418 
 
 
 
 52 
 
 425 
 
 451 
 
 438 
 
 445 
 
 451 
 
 458 
 
 465 
 
 471 
 
 478 
 
 485 
 
 
 
 53 
 
 491 
 
 498 
 
 505 
 
 511 
 
 518 
 
 525 
 
 531 
 
 538 
 
 544 
 
 551 
 
 
 
 54 
 
 558 
 
 564 
 
 571 
 
 578 
 
 584 
 
 591 
 
 598 
 
 604 
 
 611 
 
 617 
 
 
 
 55 
 
 624 
 
 631 
 
 637 
 
 644 
 
 651 
 
 657 
 
 664 
 
 671 
 
 677 
 
 684 
 
 
 
 56 
 
 690 
 
 697 
 
 704 
 
 710 
 
 717 
 
 723 
 
 730 
 
 737 
 
 743 
 
 750 
 
 
 
 57 
 
 757 
 
 763 
 
 770 
 
 776 
 
 783 
 
 790 
 
 796 
 
 803 
 
 809 
 
 816 
 
 
 
 58 
 
 823 
 
 829 
 
 856 
 
 842 
 
 849 
 
 856 
 
 862 
 
 869 
 
 875 
 
 882 
 
 
 
 59 
 
 889 
 
 895 
 
 902 
 
 908 
 
 915 
 
 921 
 
 928 
 
 935 
 
 941 
 
 948 
 
 
 
 660 
 
 81 954 
 
 961 
 
 968 
 
 974 
 
 981 
 
 987 
 
 994 
 
 *000 
 
 *007 
 
 *014 
 
 
 
 61 
 
 82 020 
 
 027 
 
 033 
 
 040 
 
 046 
 
 053 
 
 060 
 
 066 
 
 073 
 
 079 
 
 
 7 
 
 62 
 
 086 
 
 092 
 
 099 
 
 105 
 
 112 
 
 119 
 
 125 
 
 132 
 
 138 
 
 145 
 
 1 
 
 0.7 
 
 63 
 
 151 
 
 158 
 
 164 
 
 171 
 
 178 
 
 184 
 
 191 
 
 197 
 
 204 
 
 210 
 
 2 
 
 3 
 
 1.4 
 
 2.1 
 
 64 
 
 217 
 
 223 
 
 230 
 
 236 
 
 243 
 
 249 
 
 256 
 
 263 
 
 269 
 
 276 
 
 4 
 
 2.8 
 
 65 
 
 282 
 
 289 
 
 295 
 
 302 
 
 308 
 
 315 
 
 321 
 
 328 
 
 334 
 
 341 
 
 5 
 
 3.5 
 
 66 
 
 347 
 
 354 
 
 360 
 
 367 
 
 373 
 
 380 
 
 387 
 
 393 
 
 400 
 
 406 
 
 6 
 
 4.2 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 4.9 
 
 67 
 
 413 
 
 419 
 
 426 
 
 432 
 
 439 
 
 445 
 
 452 
 
 458 
 
 465 
 
 471 
 
 8 
 
 5.6 
 
 68 
 
 478 
 
 484 
 
 491 
 
 497 
 
 504 
 
 510 
 
 517 
 
 523 
 
 530 
 
 536 
 
 9 
 
 6.3 
 
 69 
 
 543 
 
 549 
 
 556 
 
 562 
 
 569 
 
 575 
 
 582 
 
 588 
 
 595 
 
 601 
 
 
 
 670 
 
 607 
 
 614 
 
 620 
 
 627 
 
 633 
 
 640 
 
 646 
 
 653 
 
 659 
 
 666 
 
 
 
 71 
 
 672 
 
 679 
 
 685 
 
 692 
 
 698 
 
 705 
 
 711 
 
 718 
 
 724 
 
 730 
 
 
 
 72 
 
 737 
 
 743 
 
 750 
 
 756 
 
 763 
 
 769 
 
 776 
 
 782 
 
 789 
 
 795 
 
 
 
 73 
 
 802 
 
 808 
 
 814 
 
 821 
 
 827 
 
 834 
 
 840 
 
 847 
 
 853 
 
 860 
 
 
 
 74 
 
 866 
 
 872 
 
 879 
 
 885 
 
 892 
 
 898 
 
 905 
 
 911 
 
 918 
 
 924 
 
 
 
 75 
 
 930 
 
 937 
 
 943 
 
 950 
 
 956 
 
 963 
 
 969 
 
 975 
 
 982 
 
 988 
 
 
 
 76 
 
 82 995 
 
 *001 
 
 *008 
 
 *014 
 
 *020 
 
 *027 
 
 *033 
 
 *040 
 
 *046 
 
 *052 
 
 
 
 77 
 
 83 059 
 
 065 
 
 072 
 
 078 
 
 085 
 
 091 
 
 097 
 
 104 
 
 110 
 
 117 
 
 
 
 78 
 
 123 
 
 129 
 
 136 
 
 142 
 
 149 
 
 155 
 
 161 
 
 168 
 
 174 
 
 181 
 
 
 
 79 
 
 187 
 
 193 
 
 200 
 
 206 
 
 213 
 
 219 
 
 225 
 
 232 
 
 238 
 
 245 
 
 
 
 680 
 
 251 
 
 257 
 
 264 
 
 270 
 
 276 
 
 283 
 
 289 
 
 296 
 
 302 
 
 308 
 
 
 6 
 
 81 
 
 315 
 
 321 
 
 327 
 
 334 
 
 340 
 
 347 
 
 353 
 
 359 
 
 366 
 
 372 
 
 1 
 
 0 6 
 
 82 
 
 378 
 
 385 
 
 391 
 
 398 
 
 404 
 
 410 
 
 417 
 
 423 
 
 429 
 
 436 
 
 2 
 
 1.2 
 
 83 
 
 442 
 
 448 
 
 455 
 
 461 
 
 467 
 
 474 
 
 480 
 
 487 
 
 493 
 
 499 
 
 3 
 
 1.8 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 2.4 
 
 84 
 
 506 
 
 512 
 
 518 
 
 525 
 
 531 
 
 537 
 
 544 
 
 550 
 
 556 
 
 563 
 
 5 
 
 3.0 
 
 85 
 
 569 
 
 575 
 
 582 
 
 588 
 
 594 
 
 601 
 
 607 
 
 613 
 
 620 
 
 626 
 
 (i 
 
 3.6 
 
 86 
 
 632 
 
 639 
 
 645 
 
 651 
 
 658 
 
 664 
 
 670 
 
 677 
 
 683 
 
 689 
 
 7 
 
 4.2 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 4.8 
 
 87 
 
 696 
 
 702 
 
 70 S 
 
 715 
 
 721 
 
 727 
 
 734 
 
 740 
 
 746 
 
 753 
 
 9 
 
 5.4 
 
 88 
 
 759 
 
 765 
 
 771 
 
 778 
 
 784 
 
 790 
 
 797 
 
 803 
 
 809 
 
 816 
 
 
 
 89 
 
 822 
 
 828 
 
 835 
 
 841 
 
 847 
 
 853 
 
 860 
 
 866 
 
 872 
 
 879 
 
 
 
 600 
 
 885 
 
 891 
 
 897 
 
 904 
 
 910 
 
 916 
 
 923 
 
 929 
 
 935 
 
 942 
 
 
 
 91 
 
 83 948 
 
 954 
 
 960 
 
 967 
 
 973 
 
 979 
 
 985 
 
 992 
 
 998 
 
 *004 
 
 
 
 92 
 
 84 011 
 
 017 
 
 023 
 
 029 
 
 036 
 
 042 
 
 048 
 
 055 
 
 061 
 
 067 
 
 
 
 93 
 
 073 
 
 080 
 
 086 
 
 092 
 
 098 
 
 105 
 
 111 
 
 117 
 
 123 
 
 130 
 
 
 
 94 
 
 136 
 
 142 
 
 148 
 
 155 
 
 161 
 
 167 
 
 173 
 
 180 
 
 186 
 
 192 
 
 
 
 95 
 
 198 
 
 205 
 
 211 
 
 217 
 
 223 
 
 230 
 
 236 
 
 242 
 
 248 
 
 255 
 
 
 
 96 
 
 261 
 
 267 
 
 273 
 
 280 
 
 286 
 
 292 
 
 298 
 
 305 
 
 311 
 
 317 
 
 
 
 97 
 
 323 
 
 330 
 
 336 
 
 342 
 
 348 
 
 354 
 
 361 
 
 367 
 
 373 
 
 379 
 
 
 
 98 
 
 386 
 
 392 
 
 398 
 
 404 
 
 410 
 
 417 
 
 423 
 
 429 
 
 435 
 
 442 
 
 
 
 99 
 
 448 
 
 454 
 
 460 
 
 466 
 
 473 
 
 479 
 
 485 
 
 491 
 
 497 
 
 504 
 
 
 
 700 
 
 84 510 
 
 516 
 
 522 
 
 528 
 
 535 
 
 541 
 
 547 
 
 553 
 
 559 
 
 566 
 
 Prop. Parts 
 --- 
 
 N 
 
 0 
 
 1 
 
 *> 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
316 
 
 TABLES 
 
 TABLE V—( Continued ) 
 
 Five-place Logarithms: 700-750 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop 
 
 Parts 
 
 700 
 
 84 510 
 
 516 
 
 522 
 
 528 
 
 555 
 
 541 
 
 547 
 
 553 
 
 559 
 
 566 
 
 
 
 01 
 
 572 
 
 578 
 
 584 
 
 590 
 
 597 
 
 603 
 
 609 
 
 615 
 
 621 
 
 628 
 
 
 
 02 
 
 634 
 
 640 
 
 646 
 
 652 
 
 658 
 
 665 
 
 671 
 
 677 
 
 683 
 
 689 
 
 
 
 03 
 
 696 
 
 702 
 
 708 
 
 714 
 
 720 
 
 726 
 
 73 3 
 
 739 
 
 745 
 
 751 
 
 
 
 04 
 
 757 
 
 763 
 
 770 
 
 776 
 
 782 
 
 788 
 
 794 
 
 800 
 
 807 
 
 813 
 
 
 
 05 
 
 819 
 
 825 
 
 831 
 
 837 
 
 844 
 
 850 
 
 856 
 
 862 
 
 868 
 
 874 
 
 
 
 06 
 
 880 
 
 887 
 
 893 
 
 899 
 
 905 
 
 911 
 
 917 
 
 924 
 
 930 
 
 936 
 
 
 7 
 
 07 
 
 84 942 
 
 948 
 
 954 
 
 960 
 
 967 
 
 973 
 
 979 
 
 985 
 
 991 
 
 997 
 
 1 
 
 0.7 
 
 08 
 
 85 003 
 
 009 
 
 016 
 
 022 
 
 028 
 
 034 
 
 040 
 
 046 
 
 052 
 
 058 
 
 2 
 
 3 
 
 1.4 
 
 2.1 
 
 09 
 
 065 
 
 071 
 
 077 
 
 085 
 
 089 
 
 095 
 
 101 
 
 107 
 
 114 
 
 120 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 2.8 
 
 710 
 
 126 
 
 132 
 
 138 
 
 144 
 
 150 
 
 156 
 
 163 
 
 169 
 
 175 
 
 181 
 
 5 
 
 6 
 
 3.5 
 
 4.2 
 
 11 
 
 187 
 
 193 
 
 199 
 
 205 
 
 211 
 
 217 
 
 224 
 
 230 
 
 236 
 
 242 
 
 7 
 
 4.9 
 
 12 
 
 248 
 
 254 
 
 260 
 
 266 
 
 272 
 
 278 
 
 285 
 
 291 
 
 297 
 
 303 
 
 8 
 
 9 
 
 5.6 
 
 6.3 
 
 13 
 
 309 
 
 515 
 
 321 
 
 327 
 
 333 
 
 339 
 
 345 
 
 352 
 
 358 
 
 364 
 
 14 
 
 370 
 
 376 
 
 382 
 
 388 
 
 394 
 
 400 
 
 406 
 
 412 
 
 418 
 
 425 
 
 
 
 15 
 
 431 
 
 437 
 
 443 
 
 449 
 
 455 
 
 461 
 
 467 
 
 473 
 
 479 
 
 485 
 
 
 
 16 
 
 491 
 
 497 
 
 505 
 
 509 
 
 516 
 
 522 
 
 528 
 
 534 
 
 540 
 
 546 
 
 
 
 17 
 
 552 
 
 558 
 
 564 
 
 570 
 
 576 
 
 582 
 
 588 
 
 594 
 
 600 
 
 606 
 
 
 
 18 
 
 612 
 
 618 
 
 625 
 
 631 
 
 637 
 
 643 
 
 649 
 
 655 
 
 661 
 
 667 
 
 
 
 19 
 
 673 
 
 679 
 
 685 
 
 691 
 
 697 
 
 703 
 
 709 
 
 715 
 
 721 
 
 727 
 
 
 
 720 
 
 735 
 
 739 
 
 745 
 
 751 
 
 757 
 
 763 
 
 769 
 
 775 
 
 781 
 
 788 
 
 
 
 21 
 
 794 
 
 800 
 
 806 
 
 812 
 
 818 
 
 824 
 
 830 
 
 836 
 
 842 
 
 848 
 
 
 6 
 
 22 
 
 854 
 
 860 
 
 866 
 
 872 
 
 878 
 
 884 
 
 890 
 
 896 
 
 902 
 
 908 
 
 
 23 
 
 914 
 
 920 
 
 926 
 
 932 
 
 938 
 
 944 
 
 950 
 
 956 
 
 962 
 
 968 
 
 i 
 
 0.6 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 1.2 
 
 24 
 
 85 974 
 
 980 
 
 986 
 
 992 
 
 998 
 
 *004 
 
 *010 
 
 *016 
 
 *022 
 
 *028 
 
 3 
 
 1.8 
 
 25 
 
 86 034 
 
 040 
 
 046 
 
 052 
 
 058 
 
 064 
 
 070 
 
 076 
 
 082 
 
 088 
 
 4 
 
 5 
 
 6 
 
 2.4 
 
 26 
 
 094 
 
 100 
 
 106 
 
 112 
 
 118 
 
 124 
 
 130 
 
 136 
 
 141 
 
 147 
 
 3.6 
 
 27 
 
 153 
 
 159 
 
 165 
 
 171 
 
 177 
 
 185 
 
 189 
 
 195 
 
 201 
 
 207 
 
 7 
 
 8 
 9 
 
 4.2 
 
 4.8 
 
 5.4 
 
 28 
 
 215 
 
 219 
 
 225 
 
 231 
 
 237 
 
 243 
 
 249 
 
 255 
 
 261 
 
 267 
 
 29 
 
 273 
 
 279 
 
 285 
 
 291 
 
 297 
 
 303 
 
 308 
 
 514 
 
 320 
 
 326 
 
 730 
 
 332 
 
 338 
 
 544 
 
 350 
 
 356 
 
 362 
 
 568 
 
 374 
 
 380 
 
 386 
 
 
 
 31 
 
 392 
 
 398 
 
 404 
 
 410 
 
 415 
 
 421 
 
 427 
 
 433 
 
 439 
 
 445 
 
 
 
 32 
 
 451 
 
 457 
 
 465 
 
 469 
 
 475 
 
 481 
 
 487 
 
 493 
 
 499 
 
 504 
 
 
 
 33 
 
 510 
 
 516 
 
 522 
 
 528 
 
 534 
 
 540 
 
 546 
 
 552 
 
 558 
 
 564 
 
 
 
 34 
 
 570 
 
 576 
 
 581 
 
 587 
 
 593 
 
 599 
 
 605 
 
 611 
 
 617 
 
 623 
 
 
 
 35 
 
 629 
 
 635 
 
 641 
 
 646 
 
 652 
 
 658 
 
 664 
 
 670 
 
 676 
 
 682 
 
 
 
 36 
 
 688 
 
 694 
 
 700 
 
 705 
 
 711 
 
 717 
 
 723 
 
 729 
 
 735 
 
 741 
 
 
 
 57 
 
 747 
 
 753 
 
 759 
 
 764 
 
 770 
 
 776 
 
 782 
 
 788 
 
 794 
 
 800 
 
 
 5 
 
 58 
 
 806 
 
 812 
 
 817 
 
 825 
 
 829 
 
 855 
 
 841 
 
 847 
 
 853 
 
 859 
 
 
 59 
 
 864 
 
 870 
 
 876 
 
 882 
 
 888 
 
 894 
 
 900 
 
 906 
 
 911 
 
 917 
 
 1 
 
 2 
 
 0.5 
 
 1.0 
 
 740 
 
 923 
 
 929 
 
 935 
 
 941 
 
 947 
 
 953 
 
 958 
 
 964 
 
 970 
 
 976 
 
 3 
 
 4 
 
 1.5 
 
 2.0 
 
 41 
 
 86 982 
 
 988 
 
 994 
 
 999 
 
 *005 
 
 *011 
 
 *017 
 
 *023 
 
 *029 
 
 *035 
 
 5 
 
 2.5 
 
 3.0 
 
 42 
 
 87 040 
 
 046 
 
 052 
 
 058 
 
 064 
 
 070 
 
 075 
 
 081 
 
 087 
 
 093 
 
 
 43 
 
 099 
 
 105 
 
 111 
 
 116 
 
 122 
 
 128 
 
 134 
 
 140 
 
 146 
 
 151 
 
 7 
 
 8 
 
 3.5 
 
 4.0 
 
 44 
 
 157 
 
 163 
 
 169 
 
 175 
 
 181 
 
 186 
 
 192 
 
 198 
 
 204 
 
 210 
 
 9 
 
 4.5 
 
 45 
 
 216 
 
 221 
 
 227 
 
 233 
 
 259 
 
 245 
 
 251 
 
 256 
 
 262 
 
 268 
 
 
 
 46 
 
 274 
 
 280 
 
 286 
 
 291 
 
 297 
 
 303 
 
 309 
 
 315 
 
 320 
 
 326 
 
 
 
 47 
 
 332 
 
 338 
 
 344 
 
 349 
 
 355 
 
 361 
 
 367 
 
 373 
 
 379 
 
 384 
 
 
 
 48 
 
 590 
 
 396 
 
 402 
 
 408 
 
 413 
 
 419 
 
 425 
 
 431 
 
 437 
 
 442 
 
 
 
 49 
 
 448 
 
 454 
 
 460 
 
 466 
 
 471 
 
 477 
 
 483 
 
 489 
 
 495 
 
 500 
 
 
 
 750 
 
 87 506 
 
 512 
 
 518 
 
 523 
 
 529 
 
 535 
 
 541 
 
 547 
 
 552 
 
 558 
 
 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop. Parts 
 
TABLES 
 
 317 
 
 TABLE V —( Continued ) 
 Five-place Logarithms: 750-800 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 750 
 
 87 506 
 
 512 
 
 518 
 
 523 
 
 529 
 
 535 
 
 541 
 
 547 
 
 552 
 
 558 
 
 
 
 51 
 
 564 
 
 570 
 
 576 
 
 581 
 
 587 
 
 593 
 
 599 
 
 604 
 
 610 
 
 616 
 
 
 
 52 
 
 622 
 
 628 
 
 633 
 
 639 
 
 645 
 
 651 
 
 656 
 
 662 
 
 668 
 
 674 
 
 
 
 53 
 
 679 
 
 685 
 
 691 
 
 697 
 
 703 
 
 708 
 
 714 
 
 720 
 
 726 
 
 731 
 
 
 
 54 
 
 737 
 
 743 
 
 749 
 
 754 
 
 760 
 
 766 
 
 772 
 
 777 
 
 783 
 
 789 
 
 
 
 55 
 
 795 
 
 800 
 
 806 
 
 812 
 
 818 
 
 823 
 
 829 
 
 835 
 
 841 
 
 846 
 
 
 
 56 
 
 852 
 
 858 
 
 864 
 
 869 
 
 875 
 
 881 
 
 887 
 
 892 
 
 898 
 
 904 
 
 
 
 57 
 
 910 
 
 915 
 
 921 
 
 927 
 
 933 
 
 938 
 
 944 
 
 950 
 
 955 
 
 961 
 
 
 
 58 
 
 87 967 
 
 973 
 
 978 
 
 984 
 
 990 
 
 996 
 
 *001 
 
 *007 
 
 *013 
 
 *018 
 
 
 
 59 
 
 88 024 
 
 030 
 
 036 
 
 041 
 
 047 
 
 053 
 
 058 
 
 064 
 
 070 
 
 076 
 
 
 
 760 
 
 081 
 
 087 
 
 093 
 
 098 
 
 104 
 
 110 
 
 116 
 
 121 
 
 127 
 
 133 
 
 
 
 61 
 
 138 
 
 144 
 
 150 
 
 156 
 
 161 
 
 167 
 
 173 
 
 178 
 
 184 
 
 190 
 
 
 6 
 
 62 
 
 195 
 
 201 
 
 207 
 
 213 
 
 218 
 
 224 
 
 230 
 
 235 
 
 241 
 
 247 
 
 1 
 
 0.6 
 
 63 
 
 252 
 
 258 
 
 264 
 
 270 
 
 275 
 
 281 
 
 287 
 
 292 
 
 298 
 
 304 
 
 2 
 
 3 
 
 1.2 
 
 1.8 
 
 64 
 
 309 
 
 315 
 
 321 
 
 326 
 
 332 
 
 338 
 
 343 
 
 349 
 
 355 
 
 360 
 
 4 
 
 2.4 
 
 65 
 
 366 
 
 372 
 
 377 
 
 383 
 
 389 
 
 395 
 
 400 
 
 406 
 
 412 
 
 417 
 
 5 
 
 3.0 
 
 66 
 
 423 
 
 429 
 
 434 
 
 440 
 
 446 
 
 451 
 
 457 
 
 463 
 
 468 
 
 474 
 
 6 
 
 3.6 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 4.2 
 
 67 
 
 480 
 
 485 
 
 491 
 
 497 
 
 502 
 
 508 
 
 513 
 
 519 
 
 525 
 
 530 
 
 8 
 
 4.8 
 
 68 
 
 536 
 
 542 
 
 547 
 
 553 
 
 559 
 
 564 
 
 570 
 
 5/6 
 
 581 
 
 587 
 
 9 
 
 5.4 
 
 69 
 
 593 
 
 598 
 
 604 
 
 610 
 
 615 
 
 621 
 
 627 
 
 632 
 
 638 
 
 643 
 
 
 
 770 
 
 649 
 
 655 
 
 660 
 
 666 
 
 672 
 
 677 
 
 683 
 
 689 
 
 694 
 
 700 
 
 
 
 71 
 
 705 
 
 711 
 
 717 
 
 722 
 
 728 
 
 734 
 
 739 
 
 745 
 
 750 
 
 756 
 
 
 
 72 
 
 762 
 
 767 
 
 773 
 
 779 
 
 784 
 
 790 
 
 795 
 
 801 
 
 807 
 
 812 
 
 
 
 73 
 
 818 
 
 824 
 
 829 
 
 835 
 
 840 
 
 846 
 
 852 
 
 857 
 
 863 
 
 868 
 
 
 
 74 
 
 874 
 
 880 
 
 885 
 
 891 
 
 897 
 
 902 
 
 908 
 
 913 
 
 919 
 
 925 
 
 
 
 75 
 
 930 
 
 936 
 
 941 
 
 947 
 
 953 
 
 958 
 
 964 
 
 969 
 
 975 
 
 981 
 
 
 
 76 
 
 88 986 
 
 992 
 
 997 
 
 *003 
 
 *009 
 
 *014 
 
 *020 
 
 *025 
 
 *031 
 
 *037 
 
 
 
 77 
 
 89 042 
 
 048 
 
 053 
 
 059 
 
 064 
 
 070 
 
 076 
 
 081 
 
 087 
 
 092 
 
 
 
 78 
 
 098 
 
 104 
 
 109 
 
 115 
 
 120 
 
 126 
 
 131 
 
 137 
 
 143 
 
 148 
 
 
 
 79 
 
 154 
 
 159 
 
 165 
 
 170 
 
 176 
 
 182 
 
 187 
 
 193 
 
 198 
 
 204 
 
 
 
 780 
 
 209 
 
 215 
 
 221 
 
 226 
 
 232 
 
 237 
 
 243 
 
 248 
 
 254 
 
 260 
 
 
 5 
 
 81 
 
 265 
 
 271 
 
 276 
 
 282 
 
 287 
 
 293 
 
 298 
 
 304 
 
 310 
 
 315 
 
 1 
 
 0 5 
 
 82 
 
 321 
 
 326 
 
 332 
 
 337 
 
 343 
 
 348 
 
 354 
 
 360 
 
 365 
 
 371 
 
 2 
 
 1.0 
 
 83 
 
 376 
 
 382 
 
 387 
 
 393 
 
 398 
 
 404 
 
 409 
 
 415 
 
 421 
 
 426 
 
 3 
 
 1.5 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 2.0 
 
 84 
 
 432 
 
 437 
 
 443 
 
 448 
 
 454 
 
 459 
 
 465 
 
 470 
 
 476 
 
 481 
 
 5 
 
 2.5 
 
 85 
 
 487 
 
 492 
 
 498 
 
 504 
 
 509 
 
 515 
 
 520 
 
 526 
 
 531 
 
 537 
 
 G 
 
 3.0 
 
 86 
 
 542 
 
 548 
 
 553 
 
 559 
 
 564 
 
 570 
 
 575 
 
 581 
 
 586 
 
 592 
 
 7 
 
 3.5 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 4.0 
 
 87 
 
 597 
 
 603 
 
 609 
 
 614 
 
 620 
 
 625 
 
 631 
 
 636 
 
 642 
 
 647 
 
 9 
 
 4.5 
 
 88 
 
 653 
 
 658 
 
 664 
 
 669 
 
 675 
 
 680 
 
 686 
 
 691 
 
 697 
 
 702 
 
 
 
 89 
 
 708 
 
 713 
 
 719 
 
 724 
 
 730 
 
 735 
 
 741 
 
 746 
 
 752 
 
 757 
 
 
 
 790 
 
 763 
 
 768 
 
 774 
 
 779 
 
 785 
 
 790 
 
 796 
 
 801 
 
 807 
 
 812 
 
 
 
 91 
 
 818 
 
 823 
 
 829 
 
 834 
 
 840 
 
 845 
 
 851 
 
 856 
 
 862 
 
 867 
 
 
 
 92 
 
 873 
 
 878 
 
 883 
 
 889 
 
 894 
 
 900 
 
 905 
 
 911 
 
 916 
 
 922 
 
 
 
 93 
 
 927 
 
 933 
 
 938 
 
 944 
 
 949 
 
 955 
 
 960 
 
 966 
 
 971 
 
 977 
 
 
 
 94 
 
 89 982 
 
 988 
 
 993 
 
 998 
 
 *004 
 
 *009 
 
 *015 
 
 *020 
 
 *026 
 
 *031 
 
 
 
 95 
 
 90 037 
 
 042 
 
 048 
 
 053 
 
 059 
 
 064 
 
 069 
 
 075 
 
 080 
 
 086 
 
 
 
 96 
 
 091 
 
 097 
 
 102 
 
 108 
 
 113 
 
 119 
 
 124 
 
 129 
 
 155 
 
 140 
 
 
 
 97 
 
 146 
 
 151 
 
 157 
 
 162 
 
 168 
 
 173 
 
 179 
 
 184 
 
 189 
 
 195 
 
 
 
 98 
 
 200 
 
 206 
 
 211 
 
 217 
 
 222 
 
 227 
 
 255 
 
 238 
 
 244 
 
 249 
 
 
 
 99 
 
 255 
 
 260 
 
 266 
 
 271 
 
 276 
 
 282 
 
 287 
 
 293 
 
 298 
 
 304 
 
 
 
 800 
 
 90 309 
 
 314 
 
 320 
 
 325 
 
 331 
 
 336 
 
 342 
 
 347 
 
 352 
 
 358 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 0 
 
 7 
 
 8 
 
 9 
 
318 
 
 TABLES 
 
 TABLE V—( Continued) 
 Five-place Logarithms: 800-850 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop . 
 
 Parts 
 
 800 
 
 90 309 
 
 314 
 
 320 
 
 325 
 
 331 
 
 336 
 
 342 
 
 347 
 
 352 
 
 358 
 
 
 
 01 
 
 363 
 
 369 
 
 374 
 
 380 
 
 385 
 
 390 
 
 396 
 
 401 
 
 407 
 
 412 
 
 
 
 02 
 
 417 
 
 423 
 
 428 
 
 434 
 
 439 
 
 445 
 
 450 
 
 455 
 
 461 
 
 466 
 
 
 
 03 
 
 472 
 
 477 
 
 482 
 
 488 
 
 493 
 
 499 
 
 504 
 
 509 
 
 515 
 
 520 
 
 
 
 04 
 
 526 
 
 531 
 
 536 
 
 542 
 
 547 
 
 553 
 
 558 
 
 563 
 
 569 
 
 574 
 
 
 
 05 
 
 580 
 
 585 
 
 590 
 
 596 
 
 601 
 
 607 
 
 612 
 
 617 
 
 623 
 
 628 
 
 
 
 06 
 
 634 
 
 639 
 
 644 
 
 650 
 
 655 
 
 660 
 
 666 
 
 671 
 
 677 
 
 682 
 
 
 
 07 
 
 68 7 
 
 693 
 
 698 
 
 703 
 
 709 
 
 714 
 
 720 
 
 725 
 
 730 
 
 736 
 
 
 
 08 
 
 741 
 
 747 
 
 752 
 
 757 
 
 763 
 
 768 
 
 773 
 
 779 
 
 784 
 
 789 
 
 
 
 09 
 
 795 
 
 800 
 
 806 
 
 811 
 
 816 
 
 822 
 
 827 
 
 832 
 
 838 
 
 843 
 
 
 
 810 
 
 849 
 
 854 
 
 859 
 
 865 
 
 870 
 
 875 
 
 881 
 
 886 
 
 891 
 
 897 
 
 
 
 11 
 
 902 
 
 907 
 
 913 
 
 918 
 
 924 
 
 929 
 
 934 
 
 940 
 
 945 
 
 950 
 
 
 
 12 
 
 90 956 
 
 961 
 
 966 
 
 972 
 
 977 
 
 982 
 
 988 
 
 993 
 
 998 
 
 *004 
 
 
 6 
 
 13 
 
 91 009 
 
 014 
 
 020 
 
 025 
 
 030 
 
 036 
 
 041 
 
 046 
 
 052 
 
 057 
 
 1 
 
 0.6 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 1.2 
 
 14 
 
 062 
 
 068 
 
 073 
 
 078 
 
 084 
 
 089 
 
 094 
 
 100 
 
 105 
 
 110 
 
 3 
 
 1.8 
 
 15 
 
 116 
 
 121 
 
 126 
 
 132 
 
 157 
 
 142 
 
 148 
 
 153 
 
 158 
 
 164 
 
 4 
 
 2 4 
 
 16 
 
 169 
 
 174 
 
 180 
 
 185 
 
 190 
 
 196 
 
 201 
 
 206 
 
 212 
 
 217 
 
 5 
 
 3.0 
 
 
 
 
 
 
 
 
 
 
 
 
 G 
 
 3.6 
 
 17 
 
 222 
 
 228 
 
 233 
 
 238 
 
 243 
 
 249 
 
 254 
 
 259 
 
 265 
 
 270 
 
 7 
 
 4.2 
 
 18 
 
 275 
 
 281 
 
 286 
 
 291 
 
 297 
 
 502 
 
 307 
 
 312 
 
 318 
 
 323 
 
 8 
 
 4.8 
 
 19 
 
 328 
 
 334 
 
 339 
 
 344 
 
 350 
 
 355 
 
 360 
 
 365 
 
 371 
 
 376 
 
 9 
 
 5.4 
 
 820 
 
 381 
 
 387 
 
 392 
 
 397 
 
 403 
 
 408 
 
 413 
 
 418 
 
 424 
 
 429 
 
 
 
 21 
 
 434 
 
 440 
 
 445 
 
 450 
 
 455 
 
 461 
 
 466 
 
 471 
 
 477 
 
 482 
 
 
 
 22 
 
 487 
 
 492 
 
 498 
 
 503 
 
 508 
 
 514 
 
 519 
 
 524 
 
 529 
 
 535 
 
 
 
 23 
 
 540 
 
 545 
 
 551 
 
 556 
 
 561 
 
 566 
 
 572 
 
 577 
 
 582 
 
 587 
 
 
 
 24 
 
 593 
 
 598 
 
 603 
 
 609 
 
 614 
 
 619 
 
 624 
 
 630 
 
 635 
 
 640 
 
 
 
 25 
 
 645 
 
 651 
 
 656 
 
 661 
 
 666 
 
 672 
 
 677 
 
 682 
 
 687 
 
 693 
 
 
 
 26 
 
 698 
 
 703 
 
 709 
 
 714 
 
 719 
 
 724 
 
 730 
 
 735 
 
 740 
 
 745 
 
 
 
 27 
 
 751 
 
 756 
 
 761 
 
 766 
 
 772 
 
 777 
 
 782 
 
 787 
 
 793 
 
 798 
 
 
 
 28 
 
 803 
 
 808 
 
 814 
 
 819 
 
 824 
 
 829 
 
 834 
 
 840 
 
 845 
 
 850 
 
 
 
 29 
 
 855 
 
 861 
 
 866 
 
 871 
 
 876 
 
 882 
 
 887 
 
 892 
 
 897 
 
 903 
 
 
 
 830 
 
 908 
 
 913 
 
 918 
 
 924 
 
 929 
 
 934 
 
 939 
 
 944 
 
 950 
 
 955 
 
 
 
 31 
 
 91 960 
 
 965 
 
 971 
 
 976 
 
 981 
 
 986 
 
 991 
 
 997 
 
 *002 
 
 *007 
 
 
 5 
 
 32 
 
 92 012 
 
 018 
 
 023 
 
 028 
 
 033 
 
 038 
 
 044 
 
 049 
 
 054 
 
 059 
 
 1 
 
 0 5 
 
 33 
 
 065 
 
 070 
 
 075 
 
 080 
 
 085 
 
 091 
 
 096 
 
 101 
 
 106 
 
 111 
 
 2 
 
 1.0 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 1.5 
 
 34 
 
 117 
 
 122 
 
 127 
 
 132 
 
 137 
 
 143 
 
 148 
 
 153 
 
 158 
 
 163 
 
 4 
 
 2.0 
 
 35 
 
 169 
 
 174 
 
 179 
 
 184 
 
 189 
 
 195 
 
 200 
 
 205 
 
 210 
 
 215 
 
 5 
 
 2.0 
 
 36 
 
 221 
 
 226 
 
 231 
 
 236 
 
 241 
 
 247 
 
 252 
 
 257 
 
 262 
 
 267 
 
 6 
 
 3.0 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 3.5 
 
 37 
 
 273 
 
 278 
 
 283 
 
 288 
 
 293 
 
 298 
 
 304 
 
 309 
 
 314 
 
 319 
 
 8 
 
 4.0 
 
 38 
 
 324 
 
 330 
 
 535 
 
 340 
 
 345 
 
 350 
 
 355 
 
 361 
 
 366 
 
 371 
 
 9 
 
 4.5 
 
 39 
 
 376 
 
 381 
 
 387 
 
 392 
 
 397 
 
 402 
 
 407 
 
 412 
 
 418 
 
 423 
 
 
 
 840 
 
 428 
 
 433 
 
 438 
 
 443 
 
 449 
 
 454 
 
 459 
 
 464 
 
 469 
 
 474 
 
 
 
 41 
 
 480 
 
 485 
 
 490 
 
 495 
 
 500 
 
 505 
 
 511 
 
 516 
 
 521 
 
 526 
 
 
 
 42 
 
 531 
 
 536 
 
 542 
 
 547 
 
 552 
 
 557 
 
 562 
 
 567 
 
 572 
 
 578 
 
 
 
 43 
 
 583 
 
 588 
 
 593 
 
 598 
 
 603 
 
 609 
 
 614 
 
 619 
 
 624 
 
 629 
 
 
 
 44 
 
 634 
 
 639 
 
 645 
 
 650 
 
 655 
 
 660 
 
 665 
 
 670 
 
 675 
 
 681 
 
 
 
 45 
 
 686 
 
 691 
 
 696 
 
 701 
 
 706 
 
 711 
 
 716 
 
 722 
 
 727 
 
 732 
 
 
 
 46 
 
 737 
 
 742 
 
 747 
 
 752 
 
 758 
 
 763 
 
 768 
 
 773 
 
 778 
 
 783 
 
 
 
 47 
 
 788 
 
 793 
 
 799 
 
 804 
 
 809 
 
 814 
 
 819 
 
 824 
 
 829 
 
 834 
 
 
 
 48 
 
 840 
 
 845 
 
 850 
 
 855 
 
 860 
 
 865 
 
 870 
 
 875 
 
 881 
 
 886 
 
 
 
 49 
 
 891 
 
 896 
 
 901 
 
 906 
 
 911 
 
 916 
 
 921 
 
 927 
 
 932 
 
 937 
 
 
 
 850 
 
 92 942 
 
 947 
 
 952 
 
 957 
 
 962 
 
 967 
 
 973 
 
 978 
 
 983 
 
 988 
 
 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop . Parts 
 
TABLES 
 
 319 
 
 TABLE V — ( Continued) 
 Five-place Logarithms: 850-900 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 850 
 
 92 942 
 
 947 
 
 952 
 
 957 
 
 962 
 
 967 
 
 973 
 
 978 
 
 983 
 
 988 
 
 
 
 51 
 
 92 993 
 
 998 
 
 *003 
 
 *008 
 
 *013 
 
 *018 
 
 *024 
 
 *029 
 
 *034 
 
 *039 
 
 
 
 52 
 
 93 044 
 
 049 
 
 054 
 
 059 
 
 064 
 
 069 
 
 075 
 
 080 
 
 085 
 
 090 
 
 
 
 53 
 
 095 
 
 100 
 
 105 
 
 no 
 
 115 
 
 120 
 
 125 
 
 131 
 
 136 
 
 141 
 
 
 
 54 
 
 146 
 
 151 
 
 156 
 
 161 
 
 166 
 
 171 
 
 176 
 
 181 
 
 186 
 
 192 
 
 
 
 55 
 
 197 
 
 202 
 
 207 
 
 212 
 
 217 
 
 222 
 
 227 
 
 232 
 
 237 
 
 242 
 
 
 6 
 
 56 
 
 247 
 
 252 
 
 258 
 
 263 
 
 268 
 
 273 
 
 278 
 
 283 
 
 288 
 
 293 
 
 1 
 
 0.6 
 
 57 
 
 298 
 
 303 
 
 308 
 
 313 
 
 318 
 
 323 
 
 328 
 
 334 
 
 339 
 
 344 
 
 2 
 
 1.2 
 
 58 
 
 349 
 
 354 
 
 359 
 
 364 
 
 369 
 
 374 
 
 379 
 
 384 
 
 389 
 
 394 
 
 3 
 
 1.8 
 
 59 
 
 399 
 
 404 
 
 409 
 
 414 
 
 420 
 
 425 
 
 430 
 
 435 
 
 440 
 
 445 
 
 4 
 
 2.4 
 
 
 
 
 
 
 
 
 
 
 5 
 
 6 
 
 3.0 
 
 3.6 
 
 860 
 
 450 
 
 455 
 
 460 
 
 465 
 
 470 
 
 475 
 
 480 
 
 485 
 
 490 
 
 495 
 
 7 
 
 4.2 
 
 61 
 
 500 
 
 505 
 
 510 
 
 515 
 
 520 
 
 526 
 
 531 
 
 536 
 
 541 
 
 546 
 
 8 
 
 4.8 
 
 62 
 
 551 
 
 556 
 
 561 
 
 566 
 
 571 
 
 576 
 
 581 
 
 586 
 
 591 
 
 596 
 
 9 
 
 5.4 
 
 63 
 
 601 
 
 606 
 
 611 
 
 616 
 
 621 
 
 626 
 
 631 
 
 636 
 
 641 
 
 646 
 
 
 
 64 
 
 651 
 
 656 
 
 661 
 
 666 
 
 671 
 
 676 
 
 682 
 
 687 
 
 692 
 
 697 
 
 
 
 65 
 
 702 
 
 707 
 
 712 
 
 717 
 
 722 
 
 727 
 
 732 
 
 737 
 
 742 
 
 747 
 
 
 
 66 
 
 752 
 
 757 
 
 762 
 
 767 
 
 772 
 
 777 
 
 782 
 
 787 
 
 792 
 
 797 
 
 
 
 67 
 
 802 
 
 807 
 
 812 
 
 817 
 
 822 
 
 827 
 
 832 
 
 837 
 
 842 
 
 847 
 
 
 
 68 
 
 852 
 
 857 
 
 862 
 
 867 
 
 872 
 
 877 
 
 882 
 
 887 
 
 892 
 
 897 
 
 
 
 69 
 
 902 
 
 907 
 
 912 
 
 917 
 
 922 
 
 927 
 
 932 
 
 937 
 
 942 
 
 947 
 
 
 
 870 
 
 93 952 
 
 957 
 
 962 
 
 967 
 
 972 
 
 977 
 
 982 
 
 987 
 
 992 
 
 997 
 
 
 
 71 
 
 94 002 
 
 007 
 
 012 
 
 017 
 
 022 
 
 027 
 
 032 
 
 037 
 
 042 
 
 047 
 
 
 5 
 
 72 
 
 052 
 
 057 
 
 062 
 
 067 
 
 072 
 
 077 
 
 082 
 
 086 
 
 091 
 
 096 
 
 i 
 
 0.5 
 
 73 
 
 101 
 
 106 
 
 111 
 
 116 
 
 121 
 
 126 
 
 131 
 
 136 
 
 141 
 
 146 
 
 2 
 
 1.0 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 1.5 
 
 74 
 
 151 
 
 156 
 
 161 
 
 166 
 
 171 
 
 176 
 
 181 
 
 186 
 
 191 
 
 196 
 
 4 
 
 2.0 
 
 75 
 
 201 
 
 206 
 
 211 
 
 216 
 
 221 
 
 226 
 
 231 
 
 236 
 
 240 
 
 245 
 
 5 
 
 6 
 
 2.5 
 
 3.0 
 
 76 
 
 250 
 
 255 
 
 260 
 
 265 
 
 270 
 
 275 
 
 280 
 
 285 
 
 290 
 
 295 
 
 7 
 
 8 
 
 9 
 
 3.5 
 
 4.0 
 
 4.5 
 
 77 
 
 300 
 
 305 
 
 310 
 
 315 
 
 320 
 
 325 
 
 330 
 
 335 
 
 340 
 
 345 
 
 78 
 
 349 
 
 354 
 
 359 
 
 364 
 
 369 
 
 374 
 
 379 
 
 384 
 
 389 
 
 394 
 
 
 79 
 
 399 
 
 404 
 
 409 
 
 414 
 
 419 
 
 424 
 
 429 
 
 433 
 
 438 
 
 443 
 
 
 
 880 
 
 448 
 
 453 
 
 458 
 
 463 
 
 468 
 
 473 
 
 478 
 
 483 
 
 488 
 
 493 
 
 
 
 81 
 
 498 
 
 503 
 
 507 
 
 512 
 
 517 
 
 522 
 
 527 
 
 532 
 
 537 
 
 542 
 
 
 
 82 
 
 547 
 
 552 
 
 557 
 
 562 
 
 567 
 
 571 
 
 576 
 
 581 
 
 586 
 
 591 
 
 
 
 83 
 
 596 
 
 601 
 
 606 
 
 611 
 
 616 
 
 621 
 
 626 
 
 630 
 
 635 
 
 640 
 
 
 
 84 
 
 645 
 
 650 
 
 655 
 
 660 
 
 665 
 
 670 
 
 675 
 
 680 
 
 685 
 
 689 
 
 
 
 85 
 
 694 
 
 699 
 
 704 
 
 709 
 
 714 
 
 719 
 
 724 
 
 729 
 
 734 
 
 738 
 
 
 
 86 
 
 743 
 
 748 
 
 753 
 
 758 
 
 763 
 
 768 
 
 773 
 
 778 
 
 783 
 
 787 
 
 
 
 87 
 
 792 
 
 797 
 
 802 
 
 807 
 
 812 
 
 817 
 
 822 
 
 827 
 
 832 
 
 836 
 
 
 
 88 
 
 841 
 
 846 
 
 851 
 
 856 
 
 861 
 
 866 
 
 871 
 
 876 
 
 880 
 
 885 
 
 1 
 
 2 
 
 0.4 
 
 0.8 
 
 89 
 
 890 
 
 895 
 
 900 
 
 905 
 
 910 
 
 915 
 
 919 
 
 924 
 
 929 
 
 934 
 
 3 
 
 1.2 
 
 890 
 
 939 
 
 944 
 
 949 
 
 954 
 
 959 
 
 963 
 
 968 
 
 973 
 
 978 
 
 983 
 
 
 
 
 
 
 
 
 
 
 
 
 5 
 
 2.0 
 
 2.4 
 
 91 
 
 94 988 
 
 993 
 
 998 
 
 *002 
 
 *007 
 
 *012 
 
 *017 
 
 *022 
 
 *027 
 
 *032 
 
 
 92 
 
 95 036 
 
 041 
 
 046 
 
 05 1 
 
 056 
 
 061 
 
 066 
 
 071 
 
 075 
 
 080 
 
 7 
 
 8 
 
 2.8 
 
 3.2 
 
 93 
 
 085 
 
 090 
 
 095 
 
 100 
 
 105 
 
 109 
 
 114 
 
 119 
 
 124 
 
 129 
 
 9 
 
 3.6 
 
 94 
 
 134 
 
 139 
 
 143 
 
 148 
 
 153 
 
 158 
 
 163 
 
 168 
 
 173 
 
 177 
 
 
 
 95 
 
 182 
 
 187 
 
 192 
 
 197 
 
 202 
 
 207 
 
 21 1 
 
 216 
 
 221 
 
 226 
 
 
 
 96 
 
 231 
 
 236 
 
 240 
 
 245 
 
 250 
 
 255 
 
 260 
 
 265 
 
 270 
 
 274 
 
 
 
 97 
 
 279 
 
 284 
 
 289 
 
 294 
 
 299 
 
 303 
 
 308 
 
 313 
 
 318 
 
 323 
 
 
 
 98 
 
 328 
 
 332 
 
 337 
 
 342 
 
 347 
 
 352 
 
 357 
 
 361 
 
 366 
 
 371 
 
 
 
 99 
 
 376 
 
 381 
 
 386 
 
 390 
 
 395 
 
 400 
 
 405 
 
 410 
 
 415 
 
 419 
 
 
 
 900 
 
 95 424 
 
 429 
 
 434 
 
 439 
 
 444 
 
 448 
 
 453 
 
 458 
 
 463 
 
 468 
 
 Prop. Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 « 
 
 7 
 
 8 
 
 9 
 
320 
 
 TABLES 
 
 TABLE V— {Continued) 
 Five-place Logarithms: 900-950 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop . Parts 
 
 900 
 
 95 424 
 
 429 
 
 434 
 
 439 
 
 444 
 
 448 
 
 453 
 
 458 
 
 463 
 
 468 
 
 
 
 01 
 
 472 
 
 477 
 
 482 
 
 487 
 
 492 
 
 497 
 
 501 
 
 506 
 
 511 
 
 516 
 
 
 
 02 
 
 521 
 
 525 
 
 550 
 
 535 
 
 540 
 
 545 
 
 550 
 
 554 
 
 559 
 
 564 
 
 
 
 03 
 
 569 
 
 574 
 
 578 
 
 583 
 
 588 
 
 593 
 
 598 
 
 602 
 
 607 
 
 612 
 
 
 
 04 
 
 617 
 
 622 
 
 626 
 
 631 
 
 636 
 
 641 
 
 646 
 
 650 
 
 655 
 
 660 
 
 
 
 05 
 
 665 
 
 670 
 
 674 
 
 679 
 
 684 
 
 689 
 
 694 
 
 698 
 
 703 
 
 708 
 
 
 
 06 
 
 713 
 
 718 
 
 722 
 
 727 
 
 732 
 
 737 
 
 742 
 
 746 
 
 751 
 
 756 
 
 
 
 07 
 
 761 
 
 766 
 
 770 
 
 775 
 
 780 
 
 785 
 
 789 
 
 794 
 
 799 
 
 804 
 
 
 
 OS 
 
 809 
 
 813 
 
 818 
 
 823 
 
 828 
 
 832 
 
 837 
 
 842 
 
 847 
 
 852 
 
 
 
 09 
 
 856 
 
 861 
 
 866 
 
 871 
 
 875 
 
 880 
 
 885 
 
 890 
 
 895 
 
 899 
 
 
 
 910 
 
 904 
 
 909 
 
 914 
 
 918 
 
 923 
 
 928 
 
 933 
 
 938 
 
 942 
 
 947 
 
 
 
 11 
 
 952 
 
 957 
 
 961 
 
 966 
 
 971 
 
 976 
 
 980 
 
 985 
 
 990 
 
 995 
 
 
 
 12 
 
 95 999 
 
 *004 
 
 *009 
 
 *014 
 
 *019 
 
 *023 
 
 *028 
 
 *033 
 
 *038 
 
 *042 
 
 
 5 
 
 13 
 
 96 047 
 
 052 
 
 057 
 
 061 
 
 066 
 
 071 
 
 076 
 
 080 
 
 085 
 
 090 
 
 1 
 
 0.5 
 
 14 
 
 095 
 
 099 
 
 104 
 
 109 
 
 114 
 
 118 
 
 123 
 
 128 
 
 133 
 
 137 
 
 2 
 
 3 
 
 1.0 
 
 1.5 
 
 15 
 
 142 
 
 147 
 
 152 
 
 156 
 
 161 
 
 166 
 
 171 
 
 175 
 
 180 
 
 185 
 
 4 
 
 2 0 
 
 16 
 
 190 
 
 194 
 
 199 
 
 204 
 
 209 
 
 213 
 
 218 
 
 223 
 
 227 
 
 232 
 
 a 
 
 2.5 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 3.0 
 
 17 
 
 237 
 
 242 
 
 246 
 
 251 
 
 256 
 
 261 
 
 265 
 
 270 
 
 275 
 
 280 
 
 7 
 
 3.5 
 
 18 
 
 284 
 
 289 
 
 294 
 
 298 
 
 503 
 
 508 
 
 313 
 
 317 
 
 322 
 
 327 
 
 8 
 
 4.0 
 
 19 
 
 332 
 
 336 
 
 541 
 
 346 
 
 550 
 
 355 
 
 360 
 
 365 
 
 369 
 
 374 
 
 9 
 
 4.5 
 
 920 
 
 379 
 
 384 
 
 388 
 
 393 
 
 398 
 
 402 
 
 407 
 
 412 
 
 417 
 
 421 
 
 
 
 21 
 
 426 
 
 431 
 
 435 
 
 440 
 
 445 
 
 450 
 
 454 
 
 459 
 
 464 
 
 468 
 
 
 
 22 
 
 473 
 
 478 
 
 483 
 
 487 
 
 492 
 
 497 
 
 501 
 
 506 
 
 511 
 
 515 
 
 
 
 23 
 
 520 
 
 525 
 
 530 
 
 534 
 
 539 
 
 544 
 
 548 
 
 553 
 
 558 
 
 562 
 
 
 
 24 
 
 567 
 
 572 
 
 577 
 
 581 
 
 586 
 
 591 
 
 595 
 
 600 
 
 605 
 
 609 
 
 
 
 25 
 
 614 
 
 619 
 
 624 
 
 628 
 
 633 
 
 638 
 
 642 
 
 647 
 
 652 
 
 656 
 
 
 
 26 
 
 661 
 
 666 
 
 670 
 
 675 
 
 680 
 
 685 
 
 689 
 
 694 
 
 699 
 
 703 
 
 
 
 27 
 
 708 
 
 713 
 
 717 
 
 722 
 
 727 
 
 731 
 
 736 
 
 741 
 
 745 
 
 750 
 
 
 
 28 
 
 755 
 
 759 
 
 764 
 
 769 
 
 774 
 
 778 
 
 783 
 
 788 
 
 792 
 
 797 
 
 
 
 29 
 
 802 
 
 806 
 
 811 
 
 816 
 
 820 
 
 825 
 
 830 
 
 834 
 
 839 
 
 844 
 
 
 
 9 SO 
 
 848 
 
 853 
 
 858 
 
 862 
 
 867 
 
 872 
 
 876 
 
 881 
 
 886 
 
 890 
 
 
 
 31 
 
 895 
 
 900 
 
 904 
 
 909 
 
 914 
 
 918 
 
 923 
 
 928 
 
 932 
 
 937 
 
 
 4 
 
 32 
 
 942 
 
 946 
 
 951 
 
 956 
 
 960 
 
 965 
 
 970 
 
 974 
 
 979 
 
 984 
 
 1 
 
 n 4 
 
 33 
 
 96 988 
 
 993 
 
 997 
 
 *002 
 
 *007 
 
 *011 
 
 *016 
 
 *021 
 
 *025 
 
 *030 
 
 2 
 
 0.8 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 1.2 
 
 34 
 
 97 035 
 
 039 
 
 044 
 
 049 
 
 053 
 
 058 
 
 063 
 
 067 
 
 072 
 
 077 
 
 4 
 
 1.6 
 
 35 
 
 081 
 
 086 
 
 090 
 
 095 
 
 100 
 
 104 
 
 109 
 
 114 
 
 118 
 
 123 
 
 5 
 
 2.0 
 
 56 
 
 128 
 
 132 
 
 157 
 
 142 
 
 146 
 
 151 
 
 155 
 
 160 
 
 165 
 
 169 
 
 6 
 
 2.4 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 2.8 
 
 37 
 
 174 
 
 179 
 
 183 
 
 188 
 
 192 
 
 197 
 
 202 
 
 206 
 
 211 
 
 216 
 
 8 
 
 3.2 
 
 38 
 
 220 
 
 225 
 
 230 
 
 254 
 
 239 
 
 243 
 
 248 
 
 253 
 
 257 
 
 262 
 
 9 
 
 5.6 
 
 59 
 
 267 
 
 271 
 
 276 
 
 280 
 
 285 
 
 290 
 
 294 
 
 299 
 
 304 
 
 308 
 
 
 
 940 
 
 313 
 
 317 
 
 322 
 
 327 
 
 331 
 
 336 
 
 340 
 
 345 
 
 350 
 
 354 
 
 
 
 41 
 
 359 
 
 364 
 
 368 
 
 373 
 
 377 
 
 382 
 
 387 
 
 391 
 
 396 
 
 400 
 
 
 
 42 
 
 405 
 
 410 
 
 414 
 
 419 
 
 424 
 
 428 
 
 433 
 
 437 
 
 442 
 
 447 
 
 
 
 43 
 
 451 
 
 456 
 
 460 
 
 465 
 
 470 
 
 474 
 
 479 
 
 483 
 
 488 
 
 493 
 
 
 
 44 
 
 497 
 
 502 
 
 506 
 
 511 
 
 516 
 
 520 
 
 525 
 
 529 
 
 534 
 
 539 
 
 
 
 45 
 
 543 
 
 548 
 
 552 
 
 557 
 
 562 
 
 566 
 
 571 
 
 575 
 
 580 
 
 585 
 
 
 
 46 
 
 589 
 
 594 
 
 598 
 
 603 
 
 607 
 
 612 
 
 617 
 
 621 
 
 626 
 
 630 
 
 
 
 47 
 
 635 
 
 640 
 
 644 
 
 649 
 
 655 
 
 658 
 
 663 
 
 667 
 
 672 
 
 676 
 
 
 
 48 
 
 681 
 
 685 
 
 690 
 
 695 
 
 699 
 
 704 
 
 708 
 
 713 
 
 717 
 
 722 
 
 
 
 49 
 
 727 
 
 731 
 
 756 
 
 740 
 
 745 
 
 749 
 
 754 
 
 759 
 
 763 
 
 768 
 
 
 
 950 
 
 97 772 
 
 777 
 
 782 
 
 786 
 
 791 
 
 795 
 
 800 
 
 804 
 
 809 
 
 813 
 
 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 Prop . Parts 
 
TABLES 
 
 321 
 
 TABLE V — {Continued) 
 Five-place Logarithms: 950-1000 
 
 Prop . Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 950 
 
 97 772 
 
 777 
 
 782 
 
 786 
 
 791 
 
 795 
 
 800 
 
 804 
 
 809 
 
 813 
 
 
 
 51 
 
 818 
 
 823 
 
 827 
 
 832 
 
 836 
 
 841 
 
 845 
 
 850 
 
 855 
 
 859 
 
 
 
 52 
 
 864 
 
 868 
 
 873 
 
 877 
 
 882 
 
 886 
 
 891 
 
 896 
 
 900 
 
 905 
 
 
 
 53 
 
 909 
 
 914 
 
 918 
 
 923 
 
 928 
 
 932 
 
 937 
 
 941 
 
 946 
 
 950 
 
 
 
 54 
 
 97 955 
 
 959 
 
 964 
 
 968 
 
 973 
 
 978 
 
 982 
 
 987 
 
 991 
 
 996 
 
 
 
 55 
 
 98 000 
 
 005 
 
 009 
 
 014 
 
 019 
 
 023 
 
 028 
 
 032 
 
 037 
 
 041 
 
 
 
 56 
 
 046 
 
 050 
 
 055 
 
 059 
 
 064 
 
 068 
 
 073 
 
 078 
 
 082 
 
 087 
 
 
 
 57 
 
 091 
 
 096 
 
 100 
 
 105 
 
 109 
 
 114 
 
 118 
 
 123 
 
 127 
 
 132 
 
 
 
 58 
 
 137 
 
 141 
 
 146 
 
 150 
 
 155 
 
 159 
 
 164 
 
 168 
 
 173 
 
 177 
 
 
 
 59 
 
 182 
 
 186 
 
 191 
 
 195 
 
 200 
 
 204 
 
 209 
 
 214 
 
 218 
 
 223 
 
 
 
 960 
 
 227 
 
 232 
 
 236 
 
 241 
 
 245 
 
 250 
 
 254 
 
 259 
 
 263 
 
 268 
 
 
 
 61 
 
 272 
 
 277 
 
 281 
 
 286 
 
 290 
 
 295 
 
 299 
 
 304 
 
 308 
 
 313 
 
 
 5 
 
 62 
 
 318 
 
 322 
 
 327 
 
 331 
 
 336 
 
 340 
 
 345 
 
 349 
 
 354 
 
 358 
 
 1 
 
 0.5 
 
 63 
 
 363 
 
 367 
 
 372 
 
 376 
 
 381 
 
 385 
 
 390 
 
 394 
 
 399 
 
 403 
 
 2 
 
 1.0 
 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 1.5 
 
 64 
 
 408 
 
 412 
 
 417 
 
 421 
 
 426 
 
 430 
 
 435 
 
 439 
 
 444 
 
 448 
 
 4 
 
 2.0 
 
 65 
 
 453 
 
 457 
 
 462 
 
 466 
 
 471 
 
 475 
 
 480 
 
 484 
 
 489 
 
 493 
 
 5 
 
 2.5 
 
 66 
 
 498 
 
 502 
 
 507 
 
 511 
 
 516 
 
 520 
 
 525 
 
 529 
 
 534 
 
 538 
 
 6 
 
 3.0 
 
 
 
 
 
 
 
 
 
 
 
 
 7 
 
 3.5 
 
 67 
 
 543 
 
 547 
 
 552 
 
 556 
 
 561 
 
 565 
 
 570 
 
 574 
 
 579 
 
 583 
 
 8 
 
 4.0 
 
 68 
 
 588 
 
 592 
 
 597 
 
 601 
 
 605 
 
 610 
 
 614 
 
 619 
 
 623 
 
 628 
 
 9 
 
 4.5 
 
 69 
 
 632 
 
 637 
 
 641 
 
 646 
 
 650 
 
 655 
 
 659 
 
 664 
 
 668 
 
 673 
 
 
 
 970 
 
 677 
 
 682 
 
 686 
 
 691 
 
 695 
 
 700 
 
 704 
 
 709 
 
 713 
 
 717 
 
 
 
 71 
 
 722 
 
 726 
 
 731 
 
 735 
 
 740 
 
 744 
 
 749 
 
 753 
 
 758 
 
 762 
 
 
 
 72 
 
 767 
 
 771 
 
 776 
 
 780 
 
 784 
 
 789 
 
 793 
 
 798 
 
 802 
 
 S07 
 
 
 
 73 
 
 811 
 
 816 
 
 820 
 
 825 
 
 829 
 
 834 
 
 838 
 
 843 
 
 847 
 
 851 
 
 
 
 74 
 
 856 
 
 860 
 
 865 
 
 869 
 
 874 
 
 878 
 
 883 
 
 887 
 
 892 
 
 896 
 
 
 
 75 
 
 900 
 
 905 
 
 909 
 
 914 
 
 918 
 
 923 
 
 927 
 
 932 
 
 936 
 
 941 
 
 
 
 76 
 
 945 
 
 949 
 
 954 
 
 958 
 
 963 
 
 967 
 
 972 
 
 976 
 
 981 
 
 985 
 
 
 
 77 
 
 98 989 
 
 994 
 
 998 
 
 *003 
 
 *007 
 
 *012 
 
 *016 
 
 *021 
 
 *025 
 
 *029 
 
 
 
 78 
 
 99 034 
 
 038 
 
 043 
 
 047 
 
 052 
 
 056 
 
 061 
 
 065 
 
 069 
 
 074 
 
 
 
 79 
 
 078 
 
 083 
 
 087 
 
 092 
 
 096 
 
 100 
 
 105 
 
 109 
 
 114 
 
 118 
 
 
 
 980 
 
 123 
 
 127 
 
 131 
 
 136 
 
 140 
 
 145 
 
 149 
 
 154 
 
 158 
 
 162 
 
 
 4 
 
 81 
 
 167 
 
 171 
 
 176 
 
 180 
 
 185 
 
 189 
 
 193 
 
 198 
 
 202 
 
 207 
 
 1 
 
 0.4 
 
 82 
 
 211 
 
 216 
 
 220 
 
 224 
 
 229 
 
 233 
 
 238 
 
 242 
 
 247 
 
 251 
 
 2 
 
 0.8 
 
 83 
 
 255 
 
 260 
 
 264 
 
 269 
 
 273 
 
 277 
 
 282 
 
 286 
 
 291 
 
 295 
 
 3 
 
 1.2 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 1.6 
 
 84 
 
 300 
 
 304 
 
 308 
 
 313 
 
 317 
 
 322 
 
 326 
 
 330 
 
 335 
 
 339 
 
 5 
 
 2.0 
 
 85 
 
 344 
 
 348 
 
 352 
 
 35 7 
 
 361 
 
 366 
 
 370 
 
 374 
 
 379 
 
 383 
 
 6 
 
 2.4 
 
 86 
 
 388 
 
 392 
 
 396 
 
 401 
 
 405 
 
 410 
 
 414 
 
 419 
 
 423 
 
 427 
 
 7 
 
 2.8 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 3.2 
 
 87 
 
 432 
 
 436 
 
 441 
 
 445 
 
 449 
 
 454 
 
 458 
 
 463 
 
 467 
 
 471 
 
 9 
 
 3.6 
 
 88 
 
 476 
 
 480 
 
 484 
 
 489 
 
 493 
 
 498 
 
 502 
 
 506 
 
 511 
 
 515 
 
 
 
 89 
 
 520 
 
 524 
 
 528 
 
 533 
 
 537 
 
 542 
 
 546 
 
 550 
 
 555 
 
 559 
 
 
 
 990 
 
 564 
 
 568 
 
 572 
 
 577 
 
 581 
 
 585 
 
 590 
 
 594 
 
 599 
 
 603 
 
 
 
 91 
 
 607 
 
 612 
 
 616 
 
 621 
 
 625 
 
 629 
 
 634 
 
 638 
 
 642 
 
 647 
 
 
 
 92 
 
 651 
 
 656 
 
 660 
 
 664 
 
 669 
 
 673 
 
 677 
 
 682 
 
 686 
 
 691 
 
 
 
 93 
 
 695 
 
 699 
 
 704 
 
 708 
 
 712 
 
 717 
 
 721 
 
 726 
 
 730 
 
 734 
 
 
 
 94 
 
 739 
 
 743 
 
 747 
 
 752 
 
 756 
 
 760 
 
 765 
 
 769 
 
 774 
 
 778 
 
 
 
 95 
 
 782 
 
 787 
 
 791 
 
 795 
 
 800 
 
 804 
 
 808 
 
 813 
 
 817 
 
 822 
 
 
 
 96 
 
 826 
 
 830 
 
 835 
 
 839 
 
 845 
 
 848 
 
 852 
 
 856 
 
 861 
 
 865 
 
 
 
 97 
 
 870 
 
 874 
 
 878 
 
 883 
 
 887 
 
 891 
 
 896 
 
 900 
 
 904 
 
 909 
 
 
 
 98 
 
 915 
 
 917 
 
 922 
 
 926 
 
 930 
 
 935 
 
 939 
 
 944 
 
 948 
 
 952 
 
 
 
 99 
 
 99 957 
 
 961 
 
 965 
 
 970 
 
 974 
 
 978 
 
 983 
 
 987 
 
 991 
 
 996 
 
 
 
 1000 
 
 00 000 
 
 004 
 
 009 
 
 013 
 
 017 
 
 022 
 
 026 
 
 030 
 
 035 
 
 039 
 
 Prop . Parts 
 
 N 
 
 0 
 
 1 
 
 2 
 
 S 
 
 4 
 
 1 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
322 
 
 TABLES 
 
 TABLE VI* 
 
 Natural Logarithms of Numbers 
 
 Note. ■ 
 
 Base e 
 loge 10 N 
 
 ^ N 
 
 loge 
 
 10 
 loge 10 
 Examples: loge 35 
 
 loge .35 = 
 
 2.71828... 
 
 loge N + loge 10 
 
 loge N — loge 10 
 2.30259 
 
 loge 3.5 + loge 10 
 
 1.25276 + 2.30259 = 3.55535 
 
 loge 3.5 — loge 10 
 
 1.25276 — 2.30259 = 8.95017 — 10 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1.0 
 
 0.0 0000 
 
 0995 
 
 1980 
 
 2956 
 
 3922 
 
 4879 
 
 5827 
 
 6766 
 
 7696 
 
 8618 
 
 1.1 
 
 9531 
 
 *0436 
 
 *1333 
 
 *2222 
 
 *3103 
 
 *3976 
 
 *4842 
 
 *5700 
 
 *6551 
 
 *7595 
 
 1.2 
 
 0.1 8232 
 
 9062 
 
 9885 
 
 *0701 
 
 *1511 
 
 *2314 
 
 *5111 
 
 *3902 
 
 *4686 
 
 *5464 
 
 1.3 
 
 0.2 6236 
 
 7003 
 
 7763 
 
 8518 
 
 9267 
 
 *0010 
 
 *0748 
 
 *1481 
 
 *2208 
 
 *2950 
 
 1.4 
 
 0.3 3647 
 
 4359 
 
 5066 
 
 5767 
 
 6464 
 
 7156 
 
 7844 
 
 8526 
 
 9204 
 
 9878 
 
 1.5 
 
 0.4 0547 
 
 1211 
 
 1871 
 
 2527 
 
 3178 
 
 3825 
 
 4469 
 
 5108 
 
 5742 
 
 6373 
 
 1.6 
 
 7000 
 
 7623 
 
 8243 
 
 8858 
 
 9470 
 
 *0078 
 
 *0682 
 
 *1282 
 
 *1879 
 
 *2473 
 
 1.7 
 
 0.5 3063 
 
 3649 
 
 4232 
 
 4812 
 
 5389 
 
 5962 
 
 6531 
 
 7098 
 
 7661 
 
 8222 
 
 1.8 
 
 8779 
 
 9333 
 
 9884 
 
 *0432 
 
 *0977 
 
 *1519 
 
 *2058 
 
 *2594 
 
 *3127 
 
 *5658 
 
 1.9 
 
 0.6 4185 
 
 4710 
 
 5253 
 
 5752 
 
 6269 
 
 6783 
 
 7294 
 
 7803 
 
 8310 
 
 8813 
 
 2.0 
 
 9315 
 
 9813 
 
 *0510 
 
 *0804 
 
 *1295 
 
 *1784 
 
 *2271 
 
 *2755 
 
 *3237 
 
 *5716 
 
 2.1 
 
 0.7 4194 
 
 4669 
 
 5142 
 
 5612 
 
 6081 
 
 6547 
 
 7011 
 
 7473 
 
 7932 
 
 8390 
 
 2.2 
 
 8846 
 
 9299 
 
 9751 
 
 *0200 
 
 *0648 
 
 *1093 
 
 *1536 
 
 *1978 
 
 *2418 
 
 *2855 
 
 2.3 
 
 0.8 3291 
 
 3725 
 
 4157 
 
 4587 
 
 5015 
 
 5442 
 
 5866 
 
 6289 
 
 6710 
 
 7129 
 
 2.4 
 
 7547 
 
 7963 
 
 8377 
 
 8789 
 
 9200 
 
 9609 
 
 *0016 
 
 *0422 
 
 *0826 
 
 *1228 
 
 2.5 
 
 0.9 1629 
 
 2028 
 
 2426 
 
 2822 
 
 3216 
 
 3609 
 
 4001 
 
 4591 
 
 4779 
 
 5166 
 
 2.6 
 
 5551 
 
 5935 
 
 6317 
 
 6698 
 
 7078 
 
 7456 
 
 7833 
 
 8208 
 
 8582 
 
 8954 
 
 2.7 
 
 9325 
 
 9695 
 
 *0063 
 
 *0430 
 
 *0796 
 
 *1160 
 
 *1523 
 
 *1885 
 
 *2245 
 
 *2604 
 
 2.8 
 
 1.0 2962 
 
 3318 
 
 3674 
 
 4028 
 
 4380 
 
 4732 
 
 5082 
 
 5431 
 
 5779 
 
 6126 
 
 2.9 
 
 6471 
 
 6815 
 
 7158 
 
 7500 
 
 7841 
 
 8181 
 
 8519 
 
 8856 
 
 9192 
 
 9527 
 
 3.0 
 
 9861 
 
 *0194 
 
 *0526 
 
 *0856 
 
 *1186 
 
 *1514 
 
 *1841 
 
 *2168 
 
 *2493 
 
 *2817 
 
 3.1 
 
 1.1 3140 
 
 3462 
 
 3783 
 
 4103 
 
 4422 
 
 4740 
 
 5057 
 
 5373 
 
 5688 
 
 6002 
 
 3.2 
 
 6315 
 
 6627 
 
 6938 
 
 7248 
 
 7557 
 
 7865 
 
 8173 
 
 8479 
 
 8784 
 
 9089 
 
 3.3 
 
 9392 
 
 9695 
 
 9996 
 
 *0297 
 
 *0597 
 
 *0896 
 
 *1194 
 
 *1491 
 
 *1788 
 
 *2083 
 
 3.4 
 
 1.2 2378 
 
 2671 
 
 2964 
 
 3256 
 
 3547 
 
 3837 
 
 4127 
 
 4415 
 
 4703 
 
 4990 
 
 3.5 
 
 5276 
 
 5562 
 
 5846 
 
 6130 
 
 6413 
 
 6695 
 
 6976 
 
 7257 
 
 7556 
 
 7815 
 
 3.6 
 
 8093 
 
 8371 
 
 8647 
 
 8923 
 
 9198 
 
 9473 
 
 9746 
 
 *0019 
 
 *0291 
 
 *0563 
 
 3.7 
 
 1.3 0833 
 
 1103 
 
 1372 
 
 1641 
 
 1909 
 
 2176 
 
 2442 
 
 2708 
 
 2972 
 
 3237 
 
 3.8 
 
 3500 
 
 3763 
 
 4025 
 
 4286 
 
 4547 
 
 4807 
 
 5067 
 
 5325 
 
 5584 
 
 5841 
 
 3.9 
 
 6098 
 
 6354 
 
 6609 
 
 6864 
 
 7118 
 
 7572 
 
 7624 
 
 7877 
 
 8128 
 
 8379 
 
 4.0 
 
 8629 
 
 8879 
 
 9128 
 
 9377 
 
 9624 
 
 9872 
 
 *0118 
 
 *0564 
 
 *0610 
 
 *0854 
 
 4.1 
 
 1.4 1099 
 
 1342 
 
 1585 
 
 1828 
 
 2070 
 
 2311 
 
 2552 
 
 2792 
 
 3031 
 
 3270 
 
 4.2 
 
 3508 
 
 3746 
 
 3984 
 
 4220 
 
 4456 
 
 4692 
 
 4927 
 
 5161 
 
 5395 
 
 5629 
 
 4.3 
 
 5862 
 
 6094 
 
 6326 
 
 6557 
 
 6787 
 
 7018 
 
 7247 
 
 7476 
 
 7705 
 
 7933 
 
 4.4 
 
 8160 
 
 8387 
 
 8614 
 
 8840 
 
 9065 
 
 9290 
 
 9515 
 
 9739 
 
 9962 
 
 *0185 
 
 4.5 
 
 1.5 0408 
 
 0650 
 
 0851 
 
 1072 
 
 1293 
 
 1513 
 
 1732 
 
 1951 
 
 2170 
 
 2388 
 
 4.6 
 
 2606 
 
 2823 
 
 3039 
 
 3256 
 
 3471 
 
 3687 
 
 3902 
 
 4116 
 
 4330 
 
 4543 
 
 4.7 
 
 4756 
 
 4969 
 
 5181 
 
 5393 
 
 5604 
 
 5814 
 
 6025 
 
 6235 
 
 6444 
 
 6653 
 
 4.8 
 
 6862 
 
 7070 
 
 7277 
 
 7485 
 
 7691 
 
 7898 
 
 8104 
 
 8309 
 
 8515 
 
 8719 
 
 4.9 
 
 8924 
 
 9127 
 
 9331 
 
 9534 
 
 9737 
 
 9939 
 
 *0141 
 
 *0342 
 
 *0543 
 
 *0744 
 
 5.0 
 
 1.6 0944 
 
 1144 
 
 1343 
 
 1542 
 
 1741 
 
 1959 
 
 2137 
 
 2334 
 
 2531 
 
 2728 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 * This table was taken from "Plane Trigonometry with Five-place Tables" by H. A. 
 Simmons and G. D. Gore by permission of the authors and the publisher, John Wiley 
 & Sons, Inc. 
 
TABLES 
 
 323 
 
 TABLE VI—( Continued ) 
 Five-place Natural Logarithms 
 
 N 
 
 0 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 5.0 
 
 1.6 0944 
 
 1144 
 
 1343 
 
 1542 
 
 1741 
 
 1939 
 
 2137 
 
 2334 
 
 2531 
 
 2728 
 
 5.1 
 
 2924 
 
 3120 
 
 3315 
 
 3511 
 
 3705 
 
 3900 
 
 4094 
 
 4287 
 
 4481 
 
 4673 
 
 5.2 
 
 4866 
 
 5058 
 
 5250 
 
 5441 
 
 5632 
 
 5823 
 
 6013 
 
 6203 
 
 6393 
 
 6582 
 
 5.3 
 
 6771 
 
 6959 
 
 7147 
 
 7335 
 
 7523 
 
 7710 
 
 7896 
 
 8083 
 
 8269 
 
 8455 
 
 5.4 
 
 8640 
 
 8825 
 
 9010 
 
 9194 
 
 9378 
 
 9562 
 
 9745 
 
 9928 
 
 *0111 
 
 *0293 
 
 5.5 
 
 1.7 0475 
 
 0656 
 
 0838 
 
 1019 
 
 1199 
 
 1580 
 
 1560 
 
 1740 
 
 1919 
 
 2098 
 
 5.6 
 
 2277 
 
 2455 
 
 2633 
 
 2811 
 
 2988 
 
 3166 
 
 3342 
 
 3519 
 
 3695 
 
 3871 
 
 5.7 
 
 4047 
 
 4222 
 
 4397 
 
 4572 
 
 4746 
 
 4920 
 
 5094 
 
 5267 
 
 5440 
 
 5613 
 
 5.8 
 
 5786 
 
 5958 
 
 6130 
 
 6502 
 
 6473 
 
 6644 
 
 6815 
 
 6985 
 
 7156 
 
 7326 
 
 5.9 
 
 7495 
 
 7665 
 
 7834 
 
 8002 
 
 8171 
 
 8339 
 
 8507 
 
 8675 
 
 8842 
 
 9009 
 
 6.0 
 
 9176 
 
 9342 
 
 9509 
 
 9675 
 
 9840 
 
 *0006 
 
 *0171 
 
 *0336 
 
 *0500 
 
 *0665 
 
 6.1 
 
 1.8 0829 
 
 0993 
 
 1156 
 
 1319 
 
 1482 
 
 1645 
 
 1808 
 
 1970 
 
 2132 
 
 2294 
 
 6.2 
 
 2455 
 
 2616 
 
 2777 
 
 2938 
 
 3098 
 
 3258 
 
 3418 
 
 3578 
 
 3737 
 
 3896 
 
 6.3 
 
 4055 
 
 4214 
 
 4372 
 
 4530 
 
 4688 
 
 4845 
 
 5003 
 
 5160 
 
 5317 
 
 5473 
 
 6.4 
 
 5630 
 
 5786 
 
 5942 
 
 6097 
 
 6253 
 
 6408 
 
 6563 
 
 6718 
 
 6872 
 
 7026 
 
 6.5 
 
 7180 
 
 7334 
 
 7487 
 
 7641 
 
 7794 
 
 7947 
 
 8099 
 
 8251 
 
 8403 
 
 8555 
 
 6.6 
 
 8707 
 
 8858 
 
 9010 
 
 9160 
 
 9311 
 
 9462 
 
 9612 
 
 9762 
 
 9912 
 
 *0061 
 
 6.7 
 
 1.9 0211 
 
 0360 
 
 0509 
 
 0658 
 
 0806 
 
 0954 
 
 1102 
 
 1250 
 
 1398 
 
 1545 
 
 6.8 
 
 1692 
 
 1839 
 
 1986 
 
 2132 
 
 2279 
 
 2425 
 
 2571 
 
 2716 
 
 2862 
 
 3007 
 
 6.9 
 
 3152 
 
 3297 
 
 3442 
 
 3586 
 
 3730 
 
 3874 
 
 4018 
 
 4162 
 
 4305 
 
 4448 
 
 7.0 
 
 4591 
 
 4734 
 
 4876 
 
 5019 
 
 5161 
 
 5303 
 
 5445 
 
 5586 
 
 5727 
 
 5869 
 
 7.1 
 
 6009 
 
 6150 
 
 6291 
 
 6431 
 
 6571 
 
 6711 
 
 6851 
 
 6991 
 
 7130 
 
 7269 
 
 7.2 
 
 7408 
 
 7547 
 
 7685 
 
 7824 
 
 7962 
 
 8100 
 
 8238 
 
 8376 
 
 8513 
 
 8650 
 
 7.3 
 
 8787 
 
 8924 
 
 9061 
 
 9198 
 
 9334 
 
 9470 
 
 9606 
 
 9742 
 
 9877 
 
 *0013 
 
 7.4 
 
 2.0 0148 
 
 0283 
 
 0418 
 
 0553 
 
 0687 
 
 0821 
 
 0956 
 
 1089 
 
 1223 
 
 1357 
 
 7.5 
 
 1490 
 
 1624 
 
 1757 
 
 1890 
 
 2022 
 
 2155 
 
 2287 
 
 2419 
 
 2551 
 
 2683 
 
 7.6 
 
 2815 
 
 2946 
 
 3078 
 
 3209 
 
 3340 
 
 3471 
 
 5601 
 
 3732 
 
 3862 
 
 3992 
 
 7.7 
 
 4122 
 
 4252 
 
 4381 
 
 4511 
 
 4640 
 
 4769 
 
 4898 
 
 5027 
 
 5156 
 
 5284 
 
 7.8 
 
 5412 
 
 5540 
 
 5668 
 
 5796 
 
 5924 
 
 6051 
 
 6179 
 
 6506 
 
 6433 
 
 6560 
 
 7.9 
 
 6686 
 
 6813 
 
 6939 
 
 7065 
 
 7191 
 
 7317 
 
 7443 
 
 7568 
 
 7694 
 
 7819 
 
 8.0 
 
 7944 
 
 8069 
 
 8194 
 
 8318 
 
 8443 
 
 8567 
 
 8691 
 
 8815 
 
 8939 
 
 9063 
 
 8.1 
 
 9186 
 
 9310 
 
 9433 
 
 9556 
 
 9679 
 
 9802 
 
 9924 
 
 *0047 
 
 *0169 
 
 *0291 
 
 8.2 
 
 2.1 0413 
 
 0535 
 
 0657 
 
 0779 
 
 0900 
 
 1021 
 
 1142 
 
 1263 
 
 1384 
 
 1505 
 
 8.3 
 
 1626 
 
 1746 
 
 1866 
 
 1986 
 
 2106 
 
 2226 
 
 2346 
 
 2465 
 
 2585 
 
 2704 
 
 8.4 
 
 2823 
 
 2942 
 
 3061 
 
 3180 
 
 3298 
 
 3417 
 
 3535 
 
 3653 
 
 3771 
 
 3889 
 
 8.5 
 
 4007 
 
 4124 
 
 4242 
 
 4359 
 
 4476 
 
 4593 
 
 4710 
 
 4827 
 
 4943 
 
 5060 
 
 8.6 
 
 5176 
 
 5292 
 
 5409 
 
 5524 
 
 5640 
 
 5756 
 
 5871 
 
 5987 
 
 6102 
 
 6217 
 
 8.7 
 
 6332 
 
 6447 
 
 6562 
 
 6677 
 
 6791 
 
 6905 
 
 7020 
 
 7134 
 
 7248 
 
 7361 
 
 8.8 
 
 7475 
 
 7589 
 
 7702 
 
 7816 
 
 7929 
 
 8042 
 
 8155 
 
 8267 
 
 8380 
 
 8493 
 
 8.9 
 
 8605 
 
 8717 
 
 8830 
 
 8942 
 
 9054 
 
 9165 
 
 9277 
 
 9389 
 
 9500 
 
 9611 
 
 9.0 
 
 9722 
 
 9834 
 
 9944 
 
 *0055 
 
 *0166 
 
 *0276 
 
 *0387 
 
 *0497 
 
 *0607 
 
 *0717 
 
 9.1 
 
 2.2 0827 
 
 0937 
 
 1047 
 
 1157 
 
 1266 
 
 1375 
 
 1485 
 
 1594 
 
 1703 
 
 1812 
 
 9.2 
 
 1920 
 
 2029 
 
 2138 
 
 2246 
 
 2354 
 
 2462 
 
 2570 
 
 2678 
 
 2786 
 
 2894 
 
 9.3 
 
 3001 
 
 3109 
 
 3216 
 
 3324 
 
 3431 
 
 3538 
 
 3645 
 
 3751 
 
 3858 
 
 3965 
 
 9.4 
 
 4071 
 
 4177 
 
 4284 
 
 4390 
 
 4496 
 
 4601 
 
 4707 
 
 4813 
 
 4918 
 
 5024 
 
 9.5 
 
 5129 
 
 5234 
 
 5339 
 
 5444 
 
 5549 
 
 5654 
 
 5759 
 
 5863 
 
 5968 
 
 6072 
 
 9.6 
 
 6176 
 
 6280 
 
 6384 
 
 6488 
 
 6592 
 
 669<> 
 
 6799 
 
 6903 
 
 7006 
 
 7109 
 
 9.7 
 
 7213 
 
 7316 
 
 7419 
 
 7521 
 
 7624 
 
 7727 
 
 7829 
 
 7932 
 
 8034 
 
 8136 
 
 9.8 
 
 8238 
 
 8340 
 
 8442 
 
 8544 
 
 8646 
 
 8747 
 
 8849 
 
 8950 
 
 9051 
 
 9152 
 
 9.9 
 
 9253 
 
 9354 
 
 9455 
 
 9556 
 
 9657 
 
 9757 
 
 9858 
 
 9958 
 
 *0058 
 
 *0158 
 
 10.0 
 
 2.3 0259 
 
 0358 
 
 0458 
 
 0558 
 
 0658 
 
 0757 
 
 0857 
 
 0956 
 
 1055 
 
 1154 
 
 N 
 
 0 
 
 1 
 
 
 3 
 
 4 
 
 5 
 
 0 
 
 7 
 
 8 
 
 9 
 
ANSWERS 
 
 Pages 4 and 5 
 
 2 56- 
 
 13. n(n + 1) (4 n - l)/6. 14. 2n 2 (n + 1). 15. n(n + 1) (n + 2) (n + 3)/4. 
 
 x = 15 
 
 _ V- 
 
 1=10 y 1 
 
 16. 2n 2 (n 2 - 1). 17. 2,485. 18. X x(x + 3). 
 
 x = 7 
 
 12 18 ^ 30 
 
 20. 2 v\v + 1). 21. X —. 22. Vif(vi). 
 
 4 
 11 
 
 23. x; XC3X 2 + 5x — 4a:). 
 
 « = 6 
 
 Pages 22 and 23 
 
 1. 11 seeds. 2. (a) 2.16 in. (6) 2.17 in. 
 
 4. (a) 204.35 eggs. (6) 5.11 eggs. 
 
 42 
 
 19. X */(»)■ 
 
 15 
 
 3. 461.2 gal. 
 
 Pages 24 and 25 
 
 1. 17.32 ligulate flowers. 2. 200.27 loaves. 
 
 3. (a) 73.62. ( b ) Per cent d.’s = 0.93, per cent of C’s = 47.56. 
 
 Page 27 
 
 1. 146 students. 
 
 Pages 28 and 29 
 
 1. 9.312 petals. 2. 88.45 strokes. 3. 24,755.7 lb. 4. 25 pupils. 
 6. 40.075 qt. 6. 12.76 rays. 
 
 Pages 29 and 30 
 
 1. 75.7. 2. 139.51 lb. 3. (a) 4,804 gal. per day. ( b ) 148,924. 6. $199.70. 
 
 Pages 35 and 36 
 
 1. 79.33. 2. 17.088 rays. 3. 171.82 cu. in. 4. 120.02 lb. 
 
 6. (a) 53.51. ( b) 53.26. 6. $5.75. 
 
 Page 38 
 
 4. 0.119. 
 
 Page 40 
 
 4. (a) 53. (b) 20. 6. (a) 215. 
 
 Page 41 
 
 1. 79 grade pt. 2. 22.5 years. 
 
 325 
 
326 
 
 ANSWERS 
 
 Pages 44 and 45 
 
 1. 74.08. Area to right of median is 497. 2. 25.96 years of age. 3. $4.80. 
 
 Page 46 
 
 1. 11.06. 2. 17.14 ft. 3. 2.92 mice. 
 
 Pages 48 and 49 
 
 1. 0.0047. 2. (a) rate = 0.01425. ( 6 ) r = 0.00509. 
 
 3. (a) 0.1767. ( b ) 0.1042. 4. 0.1946. 5. (a) 0.0291. ( 6 ) 0.2419. 6 . 1.710. 
 
 7. 10. 
 
 Pages 51 and 52 
 
 1. 121.78 mi. per hr. 2. (a) 12 cents per lb. ( b ) 85 . 
 
 3. (a) 9.92. ( b ) 49.68. 4. $1.75. 5. (a) 0.65. ( b ) about 2 men. 
 
 7. 4. 8 . 0.04. 
 
 Pages 58 and 59 
 
 1. (a) 20.6 tires. ( b ) 8 including 18. 2. 1.96 ker. 3. (a) 1.77 Al. Part. 
 ( 6 ) 47 per cent, (c) 88 per cent. 
 
 Pages 65 and 66 
 
 1. (a) M = 10.108 sheets, ( 6 ) S. D. = 5.804 sheets. 2. 69, 98, 100. 
 3. 573 or 69.4 per cent, 947 or 98.24 per cent, 964 or 100 per cent. 4. 101.26. 
 6 . 4.94. 7. 635,104. 8 . 5.07. 9. n = 500. 
 
 Pages 72 and 73 
 
 1. M = 17.52, S. D. = 2.65, 63 per cent, 98.7 per cent, 99.6 per cent. 
 2. M = 74.2, S. D. = 11.067. 3. M = 74.05, S. D. = 11.074. 
 
 4. M = 170.121, S. D. = 29.74, 67.2 per cent., 94.7 per cent, 99.8 per cent. 
 ( b ) Prob. = 0.116. 
 
 Pages 76 and 77 
 
 1. M = 66.53, S. D. = 2.96. 2. M = 66.67, S. D. = 3.02. 
 
 4. M =72.1, S. D. = 4.6. 5. M = 138.92, S. D. = 16.8. 
 
 6. M = 17.24, S. D. = 1.17. 
 
 Pages 80 and 81 
 
 1. 178. 2. S. D. = 12. 3. M = 132.86, S. D. = 14.17 lb., V = 10.7 
 
 (6) M = 228.66 lb., S. D. = 16.88 lb., V = 7.4. 4. Q = 8.05; 395. 
 
 5. V = 15.00, V = 14.97. 
 
 7. P 83 = Cg3 + 
 
 (rffo n - w 83 ) 
 
 f &3 
 
 W 
 
 83- 
 
 9. Z) 7 = Ci + 
 
 (to n - m) 
 
 h 
 
 Wj. 
 
 Pages 89 and 90 
 
 
 1. Exp. freq. 
 
 3, 8, 
 
 16, 
 
 29, 45, 59, 65, 61, 
 
 48, 33, 19, 9, 4, 1; 
 
 sum 
 
 = 400. 
 
 2 . 
 
 A’s 
 
 = 11, 
 
 B’s 
 
 = D’s = 
 
 71, 
 
 C’s = 
 
 136, E’s = 11; 
 
 sum 
 
 = 300. 
 
 3. 
 
 A’s 
 
 = E’s = 
 
 1.77, 
 
 
 B’s = 
 
 = D’s = 
 
 22.41, 
 
 C’s = 51.60; sum = 
 
 99.96. 
 
 4. 
 
 A+ 
 
 = E_ = 
 
 0.33, 
 
 A 
 
 = E = 
 
 = 0.92, 
 
 A_ = 
 
 E + = 2.20, B+ = 
 
 D_ = 
 
 = 4.48, 
 
 B 
 
 = D 
 
 = 7.79. 
 
 B. 
 
 _ = 
 
 D+ = 
 
 = 11.56, 
 
 c 4 
 
 . = C_ = 14.65, 
 
 C = 
 
 15.85, 
 
 sum = 
 
 99.71. 
 
 5. Exp. freq. 4, 
 
 11, 27, 
 
 52, 84 
 
 , 109, 123, 112, 85, 53, 27, 
 
 12, 4; 
 
 sum = 
 
 703; 2 
 
 without 
 
 this 
 
 range. 
 
 7. V 
 
 = 98.28. 9. 216.56 
 
 gal. 
 
 = M. 
 
 1C 
 
 1. 4.44 cc. 
 
 
 
 
 
 
 
 
 
ANSWERS 
 
 327 
 
 Page 95 
 
 1. Exp. freq. 1, 3, 7, 13, 18, 20, 18, 12, 6, 2, 1; sum = 101. 
 
 2. t\ = 0.80, ti = 1.32; area between = 0.118437, N = 2,930. 3. N = 5,561. 
 
 4. N = 3,897. 5. h = 101. 6. 7’s = ll’s = 48, l\ = 10j = 404, 
 
 8’s = 10’s = 1,846, 8| = 9| = 4,590, 9’s = 6,217. 
 
 Pages 102 and 103 
 
 1. 1,010.6 = A, B = 2,642.6, C = 2,886.60, D = 719.4, E = 225.0. 
 
 2. S. D. = 2 cm. 3. 1,721.6. 4. N = 500. 5. 53.67 rays. S. D. = 
 
 2.131 rays. Skew. = —.079. 6. M = 24.87 yr., S. D. = 7.8 yr. Skew. = 1.8. 
 
 „ 1 | n\[a x Z <xz:x-\-3M z -< t x 2 -\- M x 2 }-\-riQ.\ay^ ■az :y -\-SMy-a y ' 2 -\-M y 3 \ 
 
 <• c*3:i+y — ~ q 1 
 
 <J x+y l 7l\ + 712 
 
 SMx+y&x-H? Mx+i ' 
 
 8 . M x+y = 139.03 lb., S. D. x+y = 17.89 lb., Skew. I41/ = 0.92. 
 
 Page 107 
 
 1. Skew. = —0.077, K = 1.10. 2. M = 12.804 stamen, S. D. = 
 
 1.4407 stamen, Skew. = .644, K = 0.464. 3. a 4 = 0.88. 
 
 Pages 108 and 109 
 
 1. 15. 2. 216. 3. 24. 4. 25. 6 . (a) 9, ( b) 9. 6 . 49. 7. 60. 8 . 90. 9. 5 5 . 
 
 Page 110 
 
 1. 504. 2. 648. 3. 60,480. 4. 10! 6 . 120. 6 . 3,628,800. 7. 4,320. 9. 4. 
 
 Page 113 
 
 1. 55. 2. 10. 3. 700. 4. 360. 5. (a) 210, ( b ) 214. 6 . (a) 60, ( b ) 21, (c) 
 480, (cl) 1, (e) 630. 7.3,150. 8.210. 9. 52 C 13 . 10.132. 11.63. 12.52. 
 
 Pages 114 and 115 
 
 1- (a) fa, ( b ) fa, (c) fa, (<*) fa- 
 
 2- (a) |y, (&) It- ( c ) 0- (d) -fi> ( e ) tt- 
 
 3- fa, fa, fa, fa- 4. (a) 0.0032, ( b ) 0.0368, (c) 0.8832, (d) 1 - (H) 5 , 
 (e) (fa) b - 5. 1 — (f-f-) 5 , ( b ) (fa) & , (c) (y-jr) 5 , ( d ) 1 — (y^-) 5 . 6. (a) 0.374 ( 
 
 (b) 0.047, (c) 0.2299, (d) 0.08199. 7. 0.001. 
 
 Pages 118 and 119 
 
 ToVff- (&) -fafa, (c) ToVff- (d) yog-f, (e) fa fa 
 
 2 - («) if 1 ST’ (&) 2TTT- ( c ) fafa’ (d) fa fa, (e ) t nVf- (/) axW 
 
 3- (a) T 6 §T- (5) yV^T- ( c ) t 6 "§T> (d) fafa 
 
 4. (a) 6(0.98 ) b (0.02), (6)(0.98) 6 +6(0.98) 5 (0.02)+15(0.98) 4 (0.02) 2 , (c) (0.98) 6 . 
 6 . (a) (0.92) 3 , ( 6 ) 0.203136, (c) 0.067712, (cl) 0.000512. 6 . (a) (5) 
 
 (c) (cl) £££. 7. (a) 0.0446, (6) 0.3050. (c) 0.78642, (cl) 0.0307, (e) 0.02951, 
 (/) 0.1095- 8. w = 20.23. 9. (a) 0.6828, (5) 0.0214. 10. (a) 0.50, ( b ) 0.25. 
 
328 
 
 ANSWERS 
 
 Pages 128 and 129 
 
 1. V = 0.02076. 2. (a) §f§ ( b) 3. (a) 0.051551, ( b ) 0.09680, 
 
 (c) 0.851649, (d) 0.026594, (e) 1200 C 300 • (|) 300 (f) 900 . 4. (a) 0.351973, 
 
 (b) 0.344578, (c) 0.03128. 5. Af new = H M 0 id, S.D. new = H( S.D. oW ), 
 
 Skew. new = Skew.aid. 6. K = (1 — Qpq)/npq. (a) Zero. 
 
 Pages 142 and 143 
 6. Ideal index for 1934 = 0.734. 
 
 Page 144 
 
 1. ( b ) 1926, 1927, 1928, 1929, 1930. 
 
 51, 52, 53, 63, 100. 
 
 2. (a) 1930, 1931, 1932, 1933, 1934, 1935, 1936, 1937. 
 
 112, 101, 85, 90, 100, 107, 110, 117. 
 
 Page 147 
 
 1. (6) min. = -Ilf. 2. x = 4.2, y = 5.2, 2e 2 = 0.05. 
 
 Pages 149 and 150 
 
 1. x = 3.71, y = 7.72, 2e 2 = 0.0014. 2. x = 4.8, y = 2.2, 2 = 7.3, 
 
 ei = 0.1, e 3 = — 0.2. 3. x = 1, y = 2. 4. 10. 6. AB = 7.3 mi., 
 
 BC = 11.3 mi. 
 
 Pages 153 and 154 
 
 1. y = —1.695 + + 1.646 x, a e = 7.30 in. 2. Between 11.5 and 15.7, may 
 be between 10 and 16. 3. Zero. 
 
 Page 155 
 
 1. y = 11.122 + 0.732 x, <r e = 0.287. 2. n = 153. 
 
 Page 159 
 
 1. y = —137, 017.82 + 72.408 x, a e = 115.9 pupils. 
 
 Pages 162 and 163 
 
 1. y = 3.089 + 0.976 x, a e = 0.456 cm. 2. y = 38.6 + 1.57 x, c e = 3.52, 
 (b) averages. 
 
 Page 165 
 
 2. y = -272.7 + 2.633 x + 6.653 2 , = 8.452 lb. 3. y = -106.654 + 
 
 3.109 x + 7.144 2 , = 5.631b. 
 
 Page 167 
 
 1. y = -197.21 + 1-388 x + 3.247 2 + 6.119 w, c e = 4.56 lb. 
 
 8. na + + Zx 2 z = Zy, 
 
 2xa + 1.x 2 b + 1x z c = Ixy, 
 
 1x 2 a + 1x 3 b + 1 x 4 c = 1x?y. 
 
 9 . 1(a/x ) = 1y. 
 
 Pages 171 and 172 
 
 1. r = 0.568. 2. r = 0.807. 4. r = 0.707. 5. When the predicting line 
 
 is parallel to horizontal axis and all points fall on the line. 
 
ANSWERS 
 
 329 
 
 Pages 175 and 176 
 1. R = .681. 2 . R = 0.751. 
 
 Page 177 
 
 1. x = 56.014 + 0.0814 y, 8 = 29° 35', R = 30° 50'. 
 
 Pages 182 and 183 
 1. r = 0.622. 2. r = 0.788. 
 
 Pages 185 and 186 
 
 1. y = 66.88 + 0.0121 z. 2. y = 15.59. 3. y = -217.7 + 11.99 z, y = 
 152.8 when z = 30.9. 4. a y = 25 lb. 
 
 Pages 188 and 189 
 
 1. r = ± Vo99. 2. (a) = 0. 4. R„. xz = 0.885- 5. R v . xz = 0.849. 
 
 s. r = v^. 9 . g - 3 r ~ z + 3 h; ~ i- 
 
 L 1 ^XZ _J L 1 fxz 
 
 10. Ry.xz = 0.9992. 
 
 Pages 192 and 193 
 
 1. D 0 = 0.42, Ry.xzw = 0.97._ 4. D 0 =_0.3039, D L = 0.1001, D 3 = 0.2043, 
 Z ) 2 = 0.0326. y = 2.382 z x + 1.024 z 2 + 3.627 z 3 , R u .x 1X2 x 3 = 0.977. 
 
 Pages 196 and 197 
 
 1. r me . a = —0.0247. 2. = 0.566, r wt = 0.919, = 0.448, r wt .h = 0.437. 
 
 3. fine = 0.53, t m 4 yj.^ = 0.74 ,4 yr, = 0. / 3, r mfi . 4 = 0.022. 14. Zero. 
 
 Pages 199 and 200 
 
 1. r = 0.23. 2. r = -.493. 3. r = -.869. 
 
 Page 205 
 
 2. (a) 210, ( b ) 210, (c) 252. 
 
 Page 208 
 
 2. (a) = 0.159, ( b ) 0.08, (c) 0.04. 
 
 Pages 210 and 211 
 
 1. M m = 36.8 in., S.D. = 0.046, Skew. = 0.008. 2. S.D. = 0.054, 
 
 Skew. = 0.013. 3. (a) 0.000032, ( b ) 0.022216, (c) = 1 - . 4. t = 5, not due 
 
 to random sampling. 6 . s > 4,375. 10. iooCs, 100 ^ 25 . 11. (a) r > 14, 
 ( 6 ) r > 54, (c) r > 214. 12. (a) r = 4, ( 6 ) r = 16, (c) r = 64, (d) r =256, 
 (e) r = 25600. 13. r = 121. 
 
 Page 216 
 
 2. n = 100. 3. 0.56. 4. (a) 2, ( b ) 2/V37, (c) V 144/37. 
 
 Pages 218 and 219 
 
 1. (a) AF = 37.8 ± 0.44, ( b ) S.D. = 0.12. 2. 2.53. 3. AB = 1.44±04. 
 
330 
 
 ANSWERS 
 
 Pages 221 and 222 
 
 1. t > 6, sign. 2. t > 2.6, sign. 3. n > 78 trees. 4. No danger. 
 6. t = 1.718/0.218 > 2.6; sign. 6. t > 2.6; sign. 
 
 Pages 225 and 226 
 
 3. y = -22.595 + 10.2944 x + 0.1342 x 2 . <r e = 4.8. r = 0.989. 
 
 Pages 231 and 232 
 
 1. r yx = 0.707, v vx = 0.792. Vxy = 0.784. 2. 0.857. 
 
 Page 236 
 
 1. y = 398.976 - .201 x. 
 
 Pages 261 and 263 
 
 1. Analysis of Variances 
 
 Source of 
 Variation 
 
 Degrees of 
 Freedom 
 
 Sum of 
 Squares 
 
 Variances 
 
 Exp. 
 
 Error 
 
 Total 
 
 24 
 
 1,977.84 
 
 
 
 Col. 
 
 4 
 
 61.04 
 
 15.26 
 
 
 Rows 
 
 4 
 
 76.24 
 
 19.06 
 
 
 Treatments 
 
 4 
 
 1,737.84 
 
 434.46 
 
 
 Error 
 
 12 
 
 102.72 
 
 9.56 
 
 3.09 
 
 Treatments M and N significantly better than the others. 
 
 2. Analyses of Variances 
 
 Source of 
 Variation 
 
 Degrees of 
 Freedom 
 
 Sum of 
 Squares 
 
 Variances 
 
 Exp. 
 
 Error 
 
 Total 
 
 59 
 
 16.27 
 
 
 
 Between 
 
 
 
 
 
 means 
 
 2 
 
 2.21 
 
 111 
 
 
 Within 
 
 
 
 
 
 states 
 
 57 
 
 14.06 
 
 0.25 
 
 0.5 
 
 State C better than State B. 
 
 3. Variety B is better than the others. 
 
 Page 243 
 
 July 1, 1930, July 1 , 1931, July 1, 1932, July 1, 1933, July 1, 1934, 
 2.30 1.77 1.32 1.30 1.53 
 
ANSWERS 
 
 331 
 
 Pages 269, 270 and 271 
 
 1. Area = 1,211.74 ± 5.8 sq. ft. 2. Cir. = 185.354 ± .628. 3. Vol. = 
 8,158.74 ± 33.96. 4 . 29.7 sq. ft. 6 . 15.4. 6 . 137,632 ± 1,481.2 cu. ft. 
 7. t = 0.002/0.019; not sign. 9. V = 14.97, V = 14.95, not sign. 10. t = 
 .21/.18; not sign. 11. <r r = 0.0198. 13. t = 0.08/0.03; sign. 16. b = 12.79±.051. 
 18. <r Cl = 0.00028, = 0.00033; t = 0.0051/0.0004; sign. 19. Yes. 
 
 Pages 278 and 279 
 
 1. t = 1.83; not sign. 2. April 22 better than other dates. 3. Diet B is 
 the better. 
 
 Page 280 
 
 1. Not significantly different from zero. 
 
INDEX 
 
 (Numbers refer to pages) 
 
 Aggregative index numbers, 131 
 Analysis of time series, 233 
 cycles, 245 
 link relatives, 236 
 moving averages, 239 
 residuals, 245 
 seasonal variations, 235 
 trend, 233 
 
 Analysis of variances, 249 
 degrees of freedom, 252 
 
 fundamental identity for sums of squares, 250 
 Latin square, 258 
 randomized blocks, 253 
 significance between means, 257 
 Answers to problems, 325 
 Areas under the normal curve, 282 
 Arithmetic mean, 20 
 definition of, 21 
 for combination of sets, 29 
 indirect method of computing, 25 
 of grouped data, 31 
 of relatives, 133 
 of sample means, 203, 204 
 Array, 227 
 Asymmetry, 96 
 Average, see arithmetic mean 
 Average deviation, see mean deviation 
 
 Bar diagrams, 6, 7 
 Belt charts, 9, 11 
 Bernoulli distribution, 120 
 moments of, 123 
 Binomial expansion, 117 
 
 Carver, H. C., 69, 106 
 Charts, 5 
 Class limits, 31 
 Class marks, 31 
 
 333 
 
334 
 
 INDEX 
 
 Coefficient of correlation, see correlation coefficient, 168 
 Coefficient of determination, 185 
 Coefficient of variation, 79 
 standard error of, 269 
 Combinations, 110 
 
 Constants in regression equations, 154, 163, 189, 190 
 Continuous variates, 71 
 Correlation 
 linear, 168 
 non-linear, 223 
 rank, 172 
 ratio, 226 
 
 Correlation coefficient 
 
 for grouped data, 178, 180 
 geometric meaning of, 176 
 linear, 168 
 multiple, 186 
 non-linear, 223 
 other forms, 171 
 partial, 194 
 ratio, 226 
 tetrachoric, 197 
 value of in predicting, 184 
 Correlation ratio, 226 
 Craig, C. C., 202 
 Cycles, 245 
 
 Cyclical variations, 245, 247 
 
 Davies, G. R., 294 
 Deciles, 82 
 
 Degrees of freedom, 252 
 Difference between two means, 219 
 Discrete variates, 67 
 Dispersion, see measures of 
 
 Errors of prediction, 152 
 Estimating, 157, 161 
 
 Factor reversal test, 137 
 Fisher, I., 136 
 Fisher, R. A., 271 
 
 Fisher’s “Ideal” index number, 136 
 Fractional frequencies, 120 
 Frequency polygon, 36, 37 
 
 Geographic charts, 10, 12 
 Geometric mean, 47 
 as a median, 138 
 of relatives, 134 
 
INDEX 
 
 335 
 
 Geographic representation of frequency distributions, 36 
 Gore, G. D., 322 
 Graduation by use of 
 
 areas under the normal curve, 86 
 ordinates of the normal curve, 91 
 Grouped data, 30 
 
 Harmonic mean, 49 
 Histogram, 36, 38 
 
 “Ideal” index number, 136 
 Index numbers, 130 
 aggregative, 131 
 as simple relatives, 130 
 average of relatives, 133 
 Factor reversal test, 139 
 Fisher’s “Ideal”, 136 
 geometric mean of relatives, 134 
 median of relatives, 133 
 splicing, 143 
 time reversal test, 137 
 weighted aggregative, 136, 137 
 Index of seasonal variations, 243 
 
 Kurtosis, 103 
 
 Latin square, 258 
 
 Least squares method, 147, 148 
 
 Line charts, 5, 6 
 
 Linear correlation coefficient, 168 
 Link relatives, 236 
 Logarithmic charts, 
 
 double logarithmic paper, 17, 18 
 semi-logarithmic paper, 16, 17 
 Logarithms of numbers, 304, 322 
 
 Mean, 
 
 arithmetic, 20 
 geometric, 47 
 harmonic, 49 
 significance of, 219, 220 
 standard error of, 208, 209, 263 
 Mean deviation, 53 
 indirect method of computing, 57 
 Measures of dispersion, 53 
 mean deviation, 53 
 quartiles, 77 
 
 semi-interquartile range, 78 
 standard deviation, 59 
 
336 
 
 INDEX 
 
 Median 
 
 for grouped data, 41 
 for ungrouped data, 40 
 Mode, 45 
 
 Moments of a distribution, 64 
 of Bernoulli distribution, 123 
 Moving average, 162, 239 
 Multiple correlation coefficient, 186 
 for more than three variables, 189 
 standard error of, 269 
 
 Non-linear correlation, 224 
 Non-linear regression, 223 
 Normal curve, 83 
 area table of, 282 
 ordinate table of, 286 
 use in graduating, 86, 91 
 Normal equations, 151, 154 
 
 Observational equations, 145, 150, 151 
 Ogive, 38 
 
 Ordinates of the normal curve, 286 
 
 Parent population, 201 
 Partial correlation coefficient, 194 
 standard error of, 269 
 Pearson, K., 168, 197 
 Percentiles, 82 
 Permutations, 109 
 Pie charts, 7, 8 
 
 Predicting equations, 151, 163, 183 
 Price index numbers, 130 
 Probability, 113 
 in repeated trials, 115 
 Provisional mean, 26, 159 
 
 Quantity index numbers, 139 
 Quartiles, 77 
 
 Randomized blocks, 253 
 Rank correlation coefficient, 172 
 standard error of, 269 
 Ratio charts, 16, 17, 18 
 Reciprocals of numbers, 294 
 Regression coefficients, 154- 189, 190 * 
 Relatives, 236 
 
INDEX 
 
 337 
 
 Salvosa, L. R,., 282 
 Sampling, 201 
 
 average of sample means, 203, 204 
 from infinite parent, 209 
 from unknown parent, 211 
 skewness of the means, 208 
 standard deviation of the means, 206 
 Semi-interquartile range, 78 
 Sheppard’s corrections, 71 
 Significance 
 
 between correlation coefficients, 279 
 between means, 219, 220, 271 
 between standard deviations, 280 
 between variances, 280 
 Simmons, H. A., 304, 322 
 Skewness, 96 
 
 Slope of prediction line, 177 
 Snedecor, G. W., 291 
 Solution of simultaneous equations, 166 
 Splicing, 143 
 Squares of numbers, 294 
 Square roots of numbers, 294 
 Standard deviation, 59 
 of a difference, 218 
 of a sum, 216 
 of grouped data, 67, 71 
 of the combination of sets, 73 
 standard error of, 269 
 Standard error of 
 coefficient of variation, 269 
 correlation coefficient, 269 
 correlation ratio, 269 
 frequency, 265 
 mean, 263 
 median, 269 
 mean deviation, 269 
 multiple correlation coefficient, 269 
 partial correlation coefficient, 269 
 percentage, 264, 266 
 product, 266 
 quartiles, 269 
 quotient, 267 
 
 rank correlation coefficient, 269 
 regression coefficient, 269 
 semi-interquartile range, 269 
 standard deviation, 269 
 “Student”, 271 
 Summation symbols, 1, 2 
 
338 
 
 INDEX 
 
 Tables 
 
 area under the normal curve, 282 
 logarithms of numbers, 304 
 natural logarithms, 322 
 ordinates of the normal curve, 286 
 reciprocals of numbers, 294 
 squares of numbers, 294 
 square roots of numbers, 294 
 stable for testing significance, 290 
 
 values of F for testing significance between variances, 290 
 Tests for index numbers, 137, 139 
 
 Tests of significance between means of small samples, 271 
 Tests of significance of correlation coefficients, 279 
 Tests of significance between means for correlated data, 274 
 Tetrachoric correlation coefficient, 197 
 Three dimensional charts, 7, 9 
 Ties in ranks, 174 
 
 Time reversal test for index numbers, 137 
 
 Time series, 233 
 
 Trend 
 
 from moving averages, 239 
 secular from straight line, 233 
 
 Variance, 252 
 Variates 
 continuous, 71 
 discrete, 67 
 
 Weighted aggregative index numbers, 137 
 Yoder, D., 294 
 
 Zee chart, 9, 10 
 

 

7 * 
 
 ; 
 
 
 311.2 
 
 B328E 
 
 33695