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TABLES FOR STATISTICIANS

AND BIOMETRICIANS

EDITED BY

KARL PEARSON, F.R.S.

GALTON PROFESSOR, UNIVERSITY OF LONDON

ISSUED WITH ASSISTANCE FROM THE GRANT MADE BY

THE WORSHIPFUL COMPANY OF DRAPERS TO THE

BIOMETRIC LABORATORY

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CAMBRIDGE UNIVERSITY PRESS

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THE LIFE, LETTERS, AND LABOURS OF FRANCIS

GALTON. Vol. I (1822â€”1854). By Karl Pearson.

This volume contains more than sixty plates of illustration and five pedigree plates of Galton

and Darwin ancestry. It deals with the life of Sir Francis Galton from birth to marriage.

BIOMETRIKA. A Journal for the Statistical Study of Biological

Problems. Founded by W. F. R. Weldon, Karl Pearson and Francis Galton.

Biometrika appears about four times .a year. A volume containing about 500 pages, with

plates and tables, is issued annually. The subscription price, payable in advance, is 30s. net ($7.50)

per volume (post free) ; single parts 10s. net ($2.50) each. The current volume is Volume X.

Volumes I â€” IX (1902 â€” 1913) complete, 30s. net per volume; bound in buckram, 34s. %d.

net per volume. Till further notice new subscribers to Biometrika may obtain Vols. I â€” IX

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TABLES FOR STATISTICIANS

AND BIOMETRICIANS

EDITED BY

KARL PEARSON, F.R.S.

GALTON PROFESSOR, UNIVERSITY OF LONDON

ISSUED WITH ASSISTANCE FROM THE GRANT MADE BY

THE WORSHIPFUL COMPANY OF DRAPERS TO THE

BIOMETRIC LABORATORY

UNIVERSITY COLLEGE

LONDON

Cambridge :

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PREFACE

|~ AM very conscious of the delay which has intervened between the announce-

ment of the publication of these Tables and their appearance. This delay has

been chiefly due to two causes." First the great labour necessary, which largely

fell on those otherwise occupied, and secondly the great expense involved (a) in

the calculation of the Tables, and (b) in their publication. This matter of expense

is one which my somewhat urgent correspondents, I venture to think, have entirely

overlooked. It is perfectly true that only one single Table in this volume has

been directly paid for, but a very large part of the labour of calculation has been

done by the Staff of the Biometric Laboratory, whose very existence depends on

the generous grant made to that laboratory by the Worshipful Company of

Drapers. Our staff is not a large one and it has many duties, so that the progress

of calculation has of necessity been slow. Even now I am omitting projected

Tables, which I can only hope may be incorporated in a later edition of this

work, e.g. Tables of the Incomplete B- and T-functions, and the Table needed to

complete Everitt's work on High Values of Tetrachoric r when r lies between

â€” - 80 and â€” TOO. It would only satisfy my ideal of what these Tables should be,

had I been able to throw into one volume with the present special tables,

extensive tables of squares, of square roots, of reciprocals and of the natural

trigonometric functions tabled to decimals of a degree. Logarithmic tables are

relatively little used by the statistician to-day, which is the age of mechanical

calculators, and he is perfectly ready to throw aside the fiction that there is any

gain in the cumbersome notation of minutes and seconds of angle â€” a system

which would have disappeared long ago, but for the appalling 'scrapping' of

astronomical apparatus it would involve. But the ideal of one handy book for

the statistician cannot be realised until we have a body of scientific statisticians

far more numerous than at present. Statisticians must for the time being carry

about with them not only this volume but a copy of Barlow's Tables, and a

set of Tables of the Trigonometrical Functions.

vi Tables for Statisticians and Biometricians

Beside the cost of calculating these Tables, to which I have referred, must be

added the cost of printing them. I had to do this slowly as opportunity offered

in my Journal Biometrika, and the Tables as printed were moulded, in order

that stereos might be taken for reproduction. Even as it is, there are a number

of Tables in this volume, either printed here for the first time (e.g. Tables of the

Logarithm of the Factorial and of the Fourth Moment), or published here for the

first time (e.g. Tables of the G(r, v) Integrals), the setting up of which has

naturally been very expensive.

From the beginning of this work in 1901* when the first of these Tables was

published and moulded, I have had one end in view, the publication, as funds

would permit, of as full a series of Tables as possible. It is needless to say that

no anticipation of profit was ever made, the contributors worked for the sake

of science, and the aim was to provide what was possible at the lowest rate we

could. The issue may appear to many as even now costly ; let me assure those

inclined to cavil, that to pay its way with our existing public double or treble

the present price would not have availed; we are able to publish because of the

direct aid provided by initial publication in Biometrika and by direct assistance

from the Drapers' Company Grant. Yet a few years ago when a reprint of these

Tables in America was only stopped by the threat to prevent the circulation of

the book in which they were to appear entering any country with which we had

a reasonable copyright law, I was vigorously charged with checking the progress

of science and acting solely from commercial ends ! Meanwhile without any leave,

large portions of these tables have been reprinted, sometimes without even citing

the originals, in American psychological text-books. Two Russian subjects have

reissued many of these Tables in Russian and Polish versions, and copies of their

works in contravention of copyright are carried into other European countries.

It does not seem to have occurred to these men of science that there was any-

thing blameworthy in depriving Biometrika of such increased circulation as it

obtained from being the sole locus of these Tables, nor did they see in their

actions any injury to science as a whole resulting from lessening my power to

publish other work of a similar character. It is a singular phase of modern science

that it steals with a plagiaristic right hand while it stabs with a critical left.

The Introduction gives a brief description of each individual table ; it is by no

means intended to replace actual instruction in the use of the tables such as

* When issuing their prospectus in the spring of 1901 the Editors of Biometrika promised to

provide " numerical tables tending to reduce the labour of statistical arithmetic"

Preface vii

is given in a statistical laboratory, nor does it profess to provide an account

of the innumerable uses to which they may be put, or to warn the reader of the

many difficulties which may arise from inept handling of them. Additional aid

may be found in the text which usually accompanies the original publication of

the tables.

In conclusion here I wish to thank the loyal friends and colleagues â€” Dr W. F.

Sheppard, Mr W. Palin Elderton, Dr Alice Lee, Mr P. F. Everitt, Miss Julia Bell,

Miss Winifred Gibson, Mr A. Rhind, Mr H. E. Soper and others â€” whose un-

remitting exertions have enabled so much to be accomplished, if that much is

indeed not the whole we need. I have further to acknowledge the courtesy

of the Council of the British Association, who have permitted the republication

of the Tables of the G (r, v) Integrals, originally published in their Transactions.

To the Syndics of the Cambridge Press I owe a deep debt of gratitude for

allowing me the services of their staff in the preparation of this work. Pages and

pages of these Tables' were originally set up for Biometrika, or were set up afresh

here, without the appearance of a single error. To those who have had experience

of numerical tables prepared elsewhere, the excellence of the Cambridge first proof

of columns of figures is a joy, which deserves the fullest acknowledgement.

Should this work ever reach a second edition I will promise two things,

rendered possible by the stereotyping of the tables : it shall not only appear

at a much reduced price, but it shall be largely increased in extent.

KARL PEARSON.

Biometric Laboratory,

February 7, 1914.

Errata

The reader is requested to make before using these Tables the following corrections on

pp. 82, 83, 84 and 85 :

For 177 VF2i and 177 sfFs 2 at the top of the Tables read 1-177 ViVSi and 1-177 V#2 a .

When you can measure what you are speaking about and express it in

numbers, you know something about it, but when you cannot measure it, when

you cannot express it in numbers, your knowledge is of a meagre and unsatis-

factory kind.

Lord Kelvin.

La theorie des probabilites n'est au fond que le bon sens reduit au calcul ;

elle fait apprÂ£cier avec exactitude ce que les esprits justes sentent par une sorte

d'instinct, sans qu'ils puissent souvent s'en rendre compte.

Laplace.

ERRATA, ANTE USUM DILIGENTER CORRIGENDA.

Introduction.

p. xiii. Equation (i) cancel the + sign which follows A 3 w > or replace by - sign,

p. xiv. For Equation (vii) bis

ff i i(u -u 1 + u_ l + u i ) + 6%(5u 1 -5u -u_ l -u i )+u a -u (6) = 0,

read 2 i (-2Â« -Mi + Â«-i+Â»2) + <5i( 5 "i- 3M o- U-i-u 2 )+u -u (d) = O,

and add : " This equation is most effectively dealt with by finding the value of

%-Â«o(d)

<?o=

and then calculating :

kiZuo+U-i + Uz-bua)

p. xxxiv. Formula (xxxi), For A 7 (ab â€” cdf read N(adâ€”bcf-

p. xxxv. Table (3) was taken from Biometrika, Vol. IX. p. 292. Unfortunately it was not

there noted that Mr Yule's unit was 1000 houses : see his Theory of Statistics, p. 61. He

has drawn the Editor's attention to this regrettable omission. The table for the statistical

constants at the centre of the page should read :

(3) Houses xÂ»- 1439-2998, P=8-730/10 312 , while # 2 , <j> and C 2 remain unchanged.

On p. xxxvi. Lines 3 â€” 9 while correct for the illustration actually given as table (3)

on p. xxv, are of course incorrect for the true unit of 1000 houses. The statement in

Lines 19 â€” 23 with regard to the houses building or built is incorrect ; there is very marked

positive association. We must now include the house-data, and Lines 26 â€” 27 should

read : " If we regard these four tables the order of ascending association judged by either

<j> or <7 2 is (3), (4), (5), (2) as against Mr Yule's (2), (3), (4), (5)."

p. xlvii. Line 6. For 4th -071,162 read 4th -073,116.

p. xlviii. Table column (i), 22nd Line of figures. For 45 read 5-0, and for S(x) at foot

read S(y).

p. xlix. Line 1. For &i = 10e read 6 1 = 10c l .

p. lv. Formula (xlii).

For log CJft/ ) = -0399,0899 + etc.,

read log ( F Jft^ ) = 0-399,0899 +etc.

p. lx. Table, 4 = 2, B = ". For (21-556) read (31-556).

â€ž A=2, 5 = 8. For (12-202) read (13-202).

p. lxii. Line 11.

â€ž -67449^ , -67449^

For _^. 2ftMld _^z fc ,

read -67449 2,3, and -674492,3,.

p. ixiii. The two solidi have been dropped in the biquadratic :

For ft (8ft- 9ft - 12) (4ft - 3ft) = (10ft- 12ft - 18) 2 (ft +3)*,

read ft (8ft - 9ft - 12)/(4ft - 3ft) = (10ft - 12ft - 18) 2 /(ft + 3) 2 .

p. lxv. Formula (lxxvi)

For S ft 20,^ = 205 -etc.,

read Ftp, 2s 2 ^/s, & = 2ft - etc.

p. lxxv. Table, column Nvi, 3rd line. For 38 read 36.

p. lxxvii. Line 2. For " We look out 5-8 in Table L. " read " We look out 5-8 in Table LI."

p. lxxx. Line 5. For e -<-Â«*i g rea d e -4.560c e_

p. lxxxiii. Line 13 from bottom. For 2-371,76665 read 2-371,6665.

Text of Tables.

p. 13. Table V, H-S41. For X 2 = "03172 read -03072.

pp. 82, 83, 84 and 85. For Vll-JN^ and f*77Â«/Fj, at the top of the Tables read 1-177nA^2,

and 1-177 ViV : 2 2 .

p. 92. Table XLVIII.

>r m = 20\

?rc = 20J 1

51-2195

read ra = 20\

m = 20J 1

51-2195

25-6098

25-6098

2

12-4765

2

12-4765

3

5-9099

3

5-9099

4

2-7154

4

2-7154

5

1-2068

5

1-2068

6

â€¢5172

6

â€¢51 72

7

â– 0839

7

â€¢2130

8

â€¢0315

8

â€¢0839

9

â€¢0112

9

â€¢0315

10

â€¢0037

10

â– 0112

11

â€¢0012

11

â€¢0037

12

â€¢0003

12

â€¢0012

IS

â€¢oooi

13

â€¢0003

U

â€¢0000

U

â€¢0001

15

â€”

15

â€¢0000

p. 126. Table LI V. For r = 2, <Â£Â° = 5, log H(r, Â») = -106,5985,

read log H(r, Â»)- -196,5985.

p. 141. â€ž For r = 45, <Â£Â°=44, log F(r, v ) = -483,7836,

read log F(r, v) = 7-483,7836.

p. 142. â€ž For r = 50, <Â£Â° = 4, \ogH{r, v) = -932,5457,

read log H{r, v) = -392,5457.

p. 142. â€ž For r= 50, </>Â° =31, log ff(r, v)= 933,2995,

read log H(r, v) = -393,2995.

p. 143. For log log e -637 7799 16.

read log loge T-637 7843 11.

The issue of this list of Errata has been intentionally delayed in order to make it as

complete as a wider use of the volume would render possible. The Editor will be as grateful

for further emendations, as he has been for the above.

CONTENTS

Preface

Introduction to the Use of the Tables .

PAGE

V

xiii

TABLES *

TABLE PAGE

I. Table of Deviates of the Normal Curve for each Permille of

Frequency xv 1

II. Tables of the Probability Integral : Area and Ordinate of

the Normal Curve in terms of the Abscissa . . xvii 2-7

III. Tables of the Probability Integral : Abscissa and Ordinate

in terms of difference of Areas ..... xvii 9-10

IV. Tables of the Probability Integral : Logarithms of Areas

for high Values of Deviate xxi 11

V. Probable Errors of Means and Standard Deviations . xxii 12-18

VI. Probable Errors of Coefficients of Variation . . . xxii 18

VII. Abac for Probable Error of a Coefficient of Correlation r xxiii 19

VIII. Probable Error of a Coefficient of Correlation : Table to

facilitate the calculation of 1 â€” r 2 . . . . xxiii 20-21

IX. Values of the Incomplete Normal Moment Functions, First

to Tenth Moments xxiv 22-23

X. Numerical Values and Graph of Generalised Probable Error xxv 24

XL Values of the Functions ^râ€ž i|r 2 , and ^ 3 required to determine

the constants of a Normal Frequency Distribution from

the Moments of its Truncated Tail .... xxviii 25

XII. Tables for Testing Goodness of Fit : 3 to 30 frequency

groupings ......... xxxi 26-28

* The Roman figures to the pages refer to the Introduction, where the Table is discussed, the

Arabic to the Table itself.

B. 6

Tables for Statisticians and Biometriciam

TABLE

XIII. Tables for Testing Goodness of Fit: Auxiliary Table A,

to assist in determining P for high values of x> .

XIVâ€” XVI. Tables for Testing Goodness of Fit : Auxiliary Tables

B â€” D. Numerical values needed in the calculation

and extension of Tables of P .

XVII. Tables for Testing Goodness of Fit: Value of (-log P) for

high values of ^ 2 when the frequency is in a fourfold

Grouping .........

XVIII. Probability of Association on a Correlation Scale : Values

of (â€” log P) for an observed value of the Tetrachoric

Correlation r and the Standard Deviation of r for

zero Association .......

XIX. Probability of Association on a Correlation Scale : Values

of x 2 corresponding to Values of (- log P)

XX. Probability of Association on a Correlation Scale : Values

of log^; 2 corresponding to Values of Tetrachoric r

and a r .........

XXI. Probability of Association on a Correlation Scale : Abac to

determine <r r for a Population of given size, divided

into a fourfold Table with given marginal Frequencies

XXII. Probability of Association on a Correlation Scale: Abac to

determine the Equiprobable Tetrachoric Correla-

tion r,. from a Knowledge of log^ 3 and a cr r .

XXIII. Approximate Values of Probable Error of r from a four-

fold Correlation Table : Values of % r for values of r

XXIV. Approximate Values of Probable Error of r from a four-

fold Correlation Table : Values of ^ a for Values of

marginal |(1 + a)

XXV. Table to determine the Probability that the mean of a

very small sample n (4 to 10), drawn from a normal

Population will not exceed (in algebraic sense) the

Mean of the Population by more than z times the

Standard Deviation of the Sample ....

XXVI. Table to assist the Calculation of the Ordinates of the

Frequency Curve y = i/ e~ px,a (1 + xja) p

XXVII. Tables of Powers of Natural Numbers, 1 to 100

XXVIII. Tables of Sums of Powers of Natural Numbers 1 to 100 .

XXIX. Tables to facilitate the Determination of Tetrachoric Corre-

lation : Tables of the Tetrachoric Functions t, to t 6

for given marginal Frequencies ....

PAGE PAGE

xxxiii 29

xxxiii 30

xxxiv 31

xxxvi 31

xxxvi 32

xxxvi 32

xlii 33

xlii 34

xl 35

xl 35

xliii 36

xlv 37

xlvi 38-39

xlvi 40-41

1 42-51

Contents

XI

TABLE PAOE PAGE

XXX. Tables to facilitate the Determination of Tetrachoric

Correlation : Supplementary Tables for determining

high Tetrachoric Correlations (r = "80 to TOO) for

given positions of the dichotomic lines . . liii 52-57

XXXI. Table of the Logarithms of the Gamma Function,

logF(p) from p-1 to p = 2 . . . . lv 58-61

XXXII. Table to deduce the Subtense from a knowledge of Arc

and Chord in the Case of the Common Catenary lvi 62-63

XXXIII. A, Extension of Table XXXII for very flat Catenaries;

B, Extension of Table XXXII for very narrow

Catenaries lvi 64*

XXXIV. Diagram to find the Correlation Coefficient r from the

Mean Positive Contingency on the Hypothesis of

a Normal Distribution ...... lvii 65

XXXV. Diagram to determine the Type of a Frequency Distri-

bution from a Knowledge of the Constants fa and

fa. Customary Values of fa and fa . . . lxiii 66

XXXVI. Diagram showing Distribution of Frequency Types for

High or Unusual Values of fa and fa, . . . lxiii 67

XXXVII. Probable Errors of Frequency Constants: Table for the

Probable Error of fa for given Values of fa and fa . lxii 68-69

XXXVIII. Probable Errors of Frequency Constants : Table for the

Probable Error of fa 2 for given values of fa and fa . lxii 70-71

XXXIX. Probable Errors of Frequency Constants : Values of the

Correlation of Deviations in fa and fa(Rp v ^) for

given values of fa and fa lxii 72-73

XL. Probable Errors of Frequency Constants : Probable Error

of the Distance between Mean and Mode for given

Values of fa and fa lxii 74-75

XLI. Probable Errors of Frequency Constants : Probable Error

of the Skewness for given Values of fa and fa . lxii 76-77

XLII. Values of the Frequency Constants fa, fa, fa and fa

for given Values of fa and fa on the Assumption that

the Frequency can be described by one or other of

Pearson's Frequency Types ..... lxii 78-79

XLI 1 1. Probable Errors of the Frequency Constants : Probable

Error of the Criterion of Type, /c 2 , for given Values

of fa and fa lxii 80-81

XLIV. Probable Error of the Determination of Frequency Type :

Value of Semi-minor Axis of Probability Ellipse for

given Values of fa and fa lxiv 82-83

6â€”2

Xll

Tables for Statisticians and Biometricians

TABLE

XLV. Probable Error of the Determination of Frequency Type :

Value of Semi-major Axis of Probability Ellipse for

given Values of /3i and /3 2

XLVI. Probable Error of the Determination of Frequency Type :

Value of Angle between Major Axis of Probability

Ellipse and Axis of fi 2 for given Values of /9, and /3 2

XLVII. Probable Error of the Determination of Frequency Type :

Diagram to determine for given Values of & and /9 2 ,

a given Frequency Distribution belongs definitely

to a given Type of Frequency ....

XLVIII. Probable Occurrences in Second Small Samples. Per-

centage Frequency of each number of Successes in

a Second Small Sample of m after p Successes in

a First Small Sample of n .

XLIX. Logarithm of Factorial |_n from n = 1 to 1000 .

L. Table of Fourth Moments of Subgroup Frequencies, i.e.

n x x*, for n = 1 to 400, x = 1 to 19, for Verification

of the Calculation of Raw Moments

LI. General Term of Poisson's Exponential Limit to the

Binomial, i.e. of the so-called " Law of Small

Numbers." Value of e~ m m x j\x for m = - l to 15'0,

and x = up to such figure as makes the Function

significant in the sixth decimal Place .

LII. Table of Poisson's Exponential Limit to the Binomial to

be used in the Determination of the Probable Errors

of Cell Frequencies n = 1 to 30. Percentage of

Occurrences in random Samples when the Population

proportionately sampled would give actually n to

the Cell

LIII. Angles, Arcs and Decimals of Degrees. Table giving

(a) the Arc for each Degree from 1Â° to 180Â°, (b) the

Arc for each Minute and Second of Angle and (c)

the Value in Decimals of a Degree of each Arc and

Second of Angle

LIV. Table of the G(r,v) Integrals. Values of log.F(r, v)

and H(r, v) for r = 1 to 50 and <Â£ = tan -1 v from

0Â° to 45Â°

LV. Miscellaneous Constants in Frequent Use by Statisticians

and Biometricians .......

PAGE PAGE

lxiv 84-85

lxiv 86-87

lxv

88

lxx 89-97

lxxiii 98-101

lxxiv 102-112

lxxvi 113-121

lxxvii 122-124

$c it, its

TABLES FOR STATISTICIANS

AND BIOMETRICIANS

EDITED BY

KARL PEARSON, F.R.S.

GALTON PROFESSOR, UNIVERSITY OF LONDON

ISSUED WITH ASSISTANCE FROM THE GRANT MADE BY

THE WORSHIPFUL COMPANY OF DRAPERS TO THE

BIOMETRIC LABORATORY

UNIVERSITY COLLEGE

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PREFACE

|~ AM very conscious of the delay which has intervened between the announce-

ment of the publication of these Tables and their appearance. This delay has

been chiefly due to two causes." First the great labour necessary, which largely

fell on those otherwise occupied, and secondly the great expense involved (a) in

the calculation of the Tables, and (b) in their publication. This matter of expense

is one which my somewhat urgent correspondents, I venture to think, have entirely

overlooked. It is perfectly true that only one single Table in this volume has

been directly paid for, but a very large part of the labour of calculation has been

done by the Staff of the Biometric Laboratory, whose very existence depends on

the generous grant made to that laboratory by the Worshipful Company of

Drapers. Our staff is not a large one and it has many duties, so that the progress

of calculation has of necessity been slow. Even now I am omitting projected

Tables, which I can only hope may be incorporated in a later edition of this

work, e.g. Tables of the Incomplete B- and T-functions, and the Table needed to

complete Everitt's work on High Values of Tetrachoric r when r lies between

â€” - 80 and â€” TOO. It would only satisfy my ideal of what these Tables should be,

had I been able to throw into one volume with the present special tables,

extensive tables of squares, of square roots, of reciprocals and of the natural

trigonometric functions tabled to decimals of a degree. Logarithmic tables are

relatively little used by the statistician to-day, which is the age of mechanical

calculators, and he is perfectly ready to throw aside the fiction that there is any

gain in the cumbersome notation of minutes and seconds of angle â€” a system

which would have disappeared long ago, but for the appalling 'scrapping' of

astronomical apparatus it would involve. But the ideal of one handy book for

the statistician cannot be realised until we have a body of scientific statisticians

far more numerous than at present. Statisticians must for the time being carry

about with them not only this volume but a copy of Barlow's Tables, and a

set of Tables of the Trigonometrical Functions.

vi Tables for Statisticians and Biometricians

Beside the cost of calculating these Tables, to which I have referred, must be

added the cost of printing them. I had to do this slowly as opportunity offered

in my Journal Biometrika, and the Tables as printed were moulded, in order

that stereos might be taken for reproduction. Even as it is, there are a number

of Tables in this volume, either printed here for the first time (e.g. Tables of the

Logarithm of the Factorial and of the Fourth Moment), or published here for the

first time (e.g. Tables of the G(r, v) Integrals), the setting up of which has

naturally been very expensive.

From the beginning of this work in 1901* when the first of these Tables was

published and moulded, I have had one end in view, the publication, as funds

would permit, of as full a series of Tables as possible. It is needless to say that

no anticipation of profit was ever made, the contributors worked for the sake

of science, and the aim was to provide what was possible at the lowest rate we

could. The issue may appear to many as even now costly ; let me assure those

inclined to cavil, that to pay its way with our existing public double or treble

the present price would not have availed; we are able to publish because of the

direct aid provided by initial publication in Biometrika and by direct assistance

from the Drapers' Company Grant. Yet a few years ago when a reprint of these

Tables in America was only stopped by the threat to prevent the circulation of

the book in which they were to appear entering any country with which we had

a reasonable copyright law, I was vigorously charged with checking the progress

of science and acting solely from commercial ends ! Meanwhile without any leave,

large portions of these tables have been reprinted, sometimes without even citing

the originals, in American psychological text-books. Two Russian subjects have

reissued many of these Tables in Russian and Polish versions, and copies of their

works in contravention of copyright are carried into other European countries.

It does not seem to have occurred to these men of science that there was any-

thing blameworthy in depriving Biometrika of such increased circulation as it

obtained from being the sole locus of these Tables, nor did they see in their

actions any injury to science as a whole resulting from lessening my power to

publish other work of a similar character. It is a singular phase of modern science

that it steals with a plagiaristic right hand while it stabs with a critical left.

The Introduction gives a brief description of each individual table ; it is by no

means intended to replace actual instruction in the use of the tables such as

* When issuing their prospectus in the spring of 1901 the Editors of Biometrika promised to

provide " numerical tables tending to reduce the labour of statistical arithmetic"

Preface vii

is given in a statistical laboratory, nor does it profess to provide an account

of the innumerable uses to which they may be put, or to warn the reader of the

many difficulties which may arise from inept handling of them. Additional aid

may be found in the text which usually accompanies the original publication of

the tables.

In conclusion here I wish to thank the loyal friends and colleagues â€” Dr W. F.

Sheppard, Mr W. Palin Elderton, Dr Alice Lee, Mr P. F. Everitt, Miss Julia Bell,

Miss Winifred Gibson, Mr A. Rhind, Mr H. E. Soper and others â€” whose un-

remitting exertions have enabled so much to be accomplished, if that much is

indeed not the whole we need. I have further to acknowledge the courtesy

of the Council of the British Association, who have permitted the republication

of the Tables of the G (r, v) Integrals, originally published in their Transactions.

To the Syndics of the Cambridge Press I owe a deep debt of gratitude for

allowing me the services of their staff in the preparation of this work. Pages and

pages of these Tables' were originally set up for Biometrika, or were set up afresh

here, without the appearance of a single error. To those who have had experience

of numerical tables prepared elsewhere, the excellence of the Cambridge first proof

of columns of figures is a joy, which deserves the fullest acknowledgement.

Should this work ever reach a second edition I will promise two things,

rendered possible by the stereotyping of the tables : it shall not only appear

at a much reduced price, but it shall be largely increased in extent.

KARL PEARSON.

Biometric Laboratory,

February 7, 1914.

Errata

The reader is requested to make before using these Tables the following corrections on

pp. 82, 83, 84 and 85 :

For 177 VF2i and 177 sfFs 2 at the top of the Tables read 1-177 ViVSi and 1-177 V#2 a .

When you can measure what you are speaking about and express it in

numbers, you know something about it, but when you cannot measure it, when

you cannot express it in numbers, your knowledge is of a meagre and unsatis-

factory kind.

Lord Kelvin.

La theorie des probabilites n'est au fond que le bon sens reduit au calcul ;

elle fait apprÂ£cier avec exactitude ce que les esprits justes sentent par une sorte

d'instinct, sans qu'ils puissent souvent s'en rendre compte.

Laplace.

ERRATA, ANTE USUM DILIGENTER CORRIGENDA.

Introduction.

p. xiii. Equation (i) cancel the + sign which follows A 3 w > or replace by - sign,

p. xiv. For Equation (vii) bis

ff i i(u -u 1 + u_ l + u i ) + 6%(5u 1 -5u -u_ l -u i )+u a -u (6) = 0,

read 2 i (-2Â« -Mi + Â«-i+Â»2) + <5i( 5 "i- 3M o- U-i-u 2 )+u -u (d) = O,

and add : " This equation is most effectively dealt with by finding the value of

%-Â«o(d)

<?o=

and then calculating :

kiZuo+U-i + Uz-bua)

p. xxxiv. Formula (xxxi), For A 7 (ab â€” cdf read N(adâ€”bcf-

p. xxxv. Table (3) was taken from Biometrika, Vol. IX. p. 292. Unfortunately it was not

there noted that Mr Yule's unit was 1000 houses : see his Theory of Statistics, p. 61. He

has drawn the Editor's attention to this regrettable omission. The table for the statistical

constants at the centre of the page should read :

(3) Houses xÂ»- 1439-2998, P=8-730/10 312 , while # 2 , <j> and C 2 remain unchanged.

On p. xxxvi. Lines 3 â€” 9 while correct for the illustration actually given as table (3)

on p. xxv, are of course incorrect for the true unit of 1000 houses. The statement in

Lines 19 â€” 23 with regard to the houses building or built is incorrect ; there is very marked

positive association. We must now include the house-data, and Lines 26 â€” 27 should

read : " If we regard these four tables the order of ascending association judged by either

<j> or <7 2 is (3), (4), (5), (2) as against Mr Yule's (2), (3), (4), (5)."

p. xlvii. Line 6. For 4th -071,162 read 4th -073,116.

p. xlviii. Table column (i), 22nd Line of figures. For 45 read 5-0, and for S(x) at foot

read S(y).

p. xlix. Line 1. For &i = 10e read 6 1 = 10c l .

p. lv. Formula (xlii).

For log CJft/ ) = -0399,0899 + etc.,

read log ( F Jft^ ) = 0-399,0899 +etc.

p. lx. Table, 4 = 2, B = ". For (21-556) read (31-556).

â€ž A=2, 5 = 8. For (12-202) read (13-202).

p. lxii. Line 11.

â€ž -67449^ , -67449^

For _^. 2ftMld _^z fc ,

read -67449 2,3, and -674492,3,.

p. ixiii. The two solidi have been dropped in the biquadratic :

For ft (8ft- 9ft - 12) (4ft - 3ft) = (10ft- 12ft - 18) 2 (ft +3)*,

read ft (8ft - 9ft - 12)/(4ft - 3ft) = (10ft - 12ft - 18) 2 /(ft + 3) 2 .

p. lxv. Formula (lxxvi)

For S ft 20,^ = 205 -etc.,

read Ftp, 2s 2 ^/s, & = 2ft - etc.

p. lxxv. Table, column Nvi, 3rd line. For 38 read 36.

p. lxxvii. Line 2. For " We look out 5-8 in Table L. " read " We look out 5-8 in Table LI."

p. lxxx. Line 5. For e -<-Â«*i g rea d e -4.560c e_

p. lxxxiii. Line 13 from bottom. For 2-371,76665 read 2-371,6665.

Text of Tables.

p. 13. Table V, H-S41. For X 2 = "03172 read -03072.

pp. 82, 83, 84 and 85. For Vll-JN^ and f*77Â«/Fj, at the top of the Tables read 1-177nA^2,

and 1-177 ViV : 2 2 .

p. 92. Table XLVIII.

>r m = 20\

?rc = 20J 1

51-2195

read ra = 20\

m = 20J 1

51-2195

25-6098

25-6098

2

12-4765

2

12-4765

3

5-9099

3

5-9099

4

2-7154

4

2-7154

5

1-2068

5

1-2068

6

â€¢5172

6

â€¢51 72

7

â– 0839

7

â€¢2130

8

â€¢0315

8

â€¢0839

9

â€¢0112

9

â€¢0315

10

â€¢0037

10

â– 0112

11

â€¢0012

11

â€¢0037

12

â€¢0003

12

â€¢0012

IS

â€¢oooi

13

â€¢0003

U

â€¢0000

U

â€¢0001

15

â€”

15

â€¢0000

p. 126. Table LI V. For r = 2, <Â£Â° = 5, log H(r, Â») = -106,5985,

read log H(r, Â»)- -196,5985.

p. 141. â€ž For r = 45, <Â£Â°=44, log F(r, v ) = -483,7836,

read log F(r, v) = 7-483,7836.

p. 142. â€ž For r = 50, <Â£Â° = 4, \ogH{r, v) = -932,5457,

read log H{r, v) = -392,5457.

p. 142. â€ž For r= 50, </>Â° =31, log ff(r, v)= 933,2995,

read log H(r, v) = -393,2995.

p. 143. For log log e -637 7799 16.

read log loge T-637 7843 11.

The issue of this list of Errata has been intentionally delayed in order to make it as

complete as a wider use of the volume would render possible. The Editor will be as grateful

for further emendations, as he has been for the above.

CONTENTS

Preface

Introduction to the Use of the Tables .

PAGE

V

xiii

TABLES *

TABLE PAGE

I. Table of Deviates of the Normal Curve for each Permille of

Frequency xv 1

II. Tables of the Probability Integral : Area and Ordinate of

the Normal Curve in terms of the Abscissa . . xvii 2-7

III. Tables of the Probability Integral : Abscissa and Ordinate

in terms of difference of Areas ..... xvii 9-10

IV. Tables of the Probability Integral : Logarithms of Areas

for high Values of Deviate xxi 11

V. Probable Errors of Means and Standard Deviations . xxii 12-18

VI. Probable Errors of Coefficients of Variation . . . xxii 18

VII. Abac for Probable Error of a Coefficient of Correlation r xxiii 19

VIII. Probable Error of a Coefficient of Correlation : Table to

facilitate the calculation of 1 â€” r 2 . . . . xxiii 20-21

IX. Values of the Incomplete Normal Moment Functions, First

to Tenth Moments xxiv 22-23

X. Numerical Values and Graph of Generalised Probable Error xxv 24

XL Values of the Functions ^râ€ž i|r 2 , and ^ 3 required to determine

the constants of a Normal Frequency Distribution from

the Moments of its Truncated Tail .... xxviii 25

XII. Tables for Testing Goodness of Fit : 3 to 30 frequency

groupings ......... xxxi 26-28

* The Roman figures to the pages refer to the Introduction, where the Table is discussed, the

Arabic to the Table itself.

B. 6

Tables for Statisticians and Biometriciam

TABLE

XIII. Tables for Testing Goodness of Fit: Auxiliary Table A,

to assist in determining P for high values of x> .

XIVâ€” XVI. Tables for Testing Goodness of Fit : Auxiliary Tables

B â€” D. Numerical values needed in the calculation

and extension of Tables of P .

XVII. Tables for Testing Goodness of Fit: Value of (-log P) for

high values of ^ 2 when the frequency is in a fourfold

Grouping .........

XVIII. Probability of Association on a Correlation Scale : Values

of (â€” log P) for an observed value of the Tetrachoric

Correlation r and the Standard Deviation of r for

zero Association .......

XIX. Probability of Association on a Correlation Scale : Values

of x 2 corresponding to Values of (- log P)

XX. Probability of Association on a Correlation Scale : Values

of log^; 2 corresponding to Values of Tetrachoric r

and a r .........

XXI. Probability of Association on a Correlation Scale : Abac to

determine <r r for a Population of given size, divided

into a fourfold Table with given marginal Frequencies

XXII. Probability of Association on a Correlation Scale: Abac to

determine the Equiprobable Tetrachoric Correla-

tion r,. from a Knowledge of log^ 3 and a cr r .

XXIII. Approximate Values of Probable Error of r from a four-

fold Correlation Table : Values of % r for values of r

XXIV. Approximate Values of Probable Error of r from a four-

fold Correlation Table : Values of ^ a for Values of

marginal |(1 + a)

XXV. Table to determine the Probability that the mean of a

very small sample n (4 to 10), drawn from a normal

Population will not exceed (in algebraic sense) the

Mean of the Population by more than z times the

Standard Deviation of the Sample ....

XXVI. Table to assist the Calculation of the Ordinates of the

Frequency Curve y = i/ e~ px,a (1 + xja) p

XXVII. Tables of Powers of Natural Numbers, 1 to 100

XXVIII. Tables of Sums of Powers of Natural Numbers 1 to 100 .

XXIX. Tables to facilitate the Determination of Tetrachoric Corre-

lation : Tables of the Tetrachoric Functions t, to t 6

for given marginal Frequencies ....

PAGE PAGE

xxxiii 29

xxxiii 30

xxxiv 31

xxxvi 31

xxxvi 32

xxxvi 32

xlii 33

xlii 34

xl 35

xl 35

xliii 36

xlv 37

xlvi 38-39

xlvi 40-41

1 42-51

Contents

XI

TABLE PAOE PAGE

XXX. Tables to facilitate the Determination of Tetrachoric

Correlation : Supplementary Tables for determining

high Tetrachoric Correlations (r = "80 to TOO) for

given positions of the dichotomic lines . . liii 52-57

XXXI. Table of the Logarithms of the Gamma Function,

logF(p) from p-1 to p = 2 . . . . lv 58-61

XXXII. Table to deduce the Subtense from a knowledge of Arc

and Chord in the Case of the Common Catenary lvi 62-63

XXXIII. A, Extension of Table XXXII for very flat Catenaries;

B, Extension of Table XXXII for very narrow

Catenaries lvi 64*

XXXIV. Diagram to find the Correlation Coefficient r from the

Mean Positive Contingency on the Hypothesis of

a Normal Distribution ...... lvii 65

XXXV. Diagram to determine the Type of a Frequency Distri-

bution from a Knowledge of the Constants fa and

fa. Customary Values of fa and fa . . . lxiii 66

XXXVI. Diagram showing Distribution of Frequency Types for

High or Unusual Values of fa and fa, . . . lxiii 67

XXXVII. Probable Errors of Frequency Constants: Table for the

Probable Error of fa for given Values of fa and fa . lxii 68-69

XXXVIII. Probable Errors of Frequency Constants : Table for the

Probable Error of fa 2 for given values of fa and fa . lxii 70-71

XXXIX. Probable Errors of Frequency Constants : Values of the

Correlation of Deviations in fa and fa(Rp v ^) for

given values of fa and fa lxii 72-73

XL. Probable Errors of Frequency Constants : Probable Error

of the Distance between Mean and Mode for given

Values of fa and fa lxii 74-75

XLI. Probable Errors of Frequency Constants : Probable Error

of the Skewness for given Values of fa and fa . lxii 76-77

XLII. Values of the Frequency Constants fa, fa, fa and fa

for given Values of fa and fa on the Assumption that

the Frequency can be described by one or other of

Pearson's Frequency Types ..... lxii 78-79

XLI 1 1. Probable Errors of the Frequency Constants : Probable

Error of the Criterion of Type, /c 2 , for given Values

of fa and fa lxii 80-81

XLIV. Probable Error of the Determination of Frequency Type :

Value of Semi-minor Axis of Probability Ellipse for

given Values of fa and fa lxiv 82-83

6â€”2

Xll

Tables for Statisticians and Biometricians

TABLE

XLV. Probable Error of the Determination of Frequency Type :

Value of Semi-major Axis of Probability Ellipse for

given Values of /3i and /3 2

XLVI. Probable Error of the Determination of Frequency Type :

Value of Angle between Major Axis of Probability

Ellipse and Axis of fi 2 for given Values of /9, and /3 2

XLVII. Probable Error of the Determination of Frequency Type :

Diagram to determine for given Values of & and /9 2 ,

a given Frequency Distribution belongs definitely

to a given Type of Frequency ....

XLVIII. Probable Occurrences in Second Small Samples. Per-

centage Frequency of each number of Successes in

a Second Small Sample of m after p Successes in

a First Small Sample of n .

XLIX. Logarithm of Factorial |_n from n = 1 to 1000 .

L. Table of Fourth Moments of Subgroup Frequencies, i.e.

n x x*, for n = 1 to 400, x = 1 to 19, for Verification

of the Calculation of Raw Moments

LI. General Term of Poisson's Exponential Limit to the

Binomial, i.e. of the so-called " Law of Small

Numbers." Value of e~ m m x j\x for m = - l to 15'0,

and x = up to such figure as makes the Function

significant in the sixth decimal Place .

LII. Table of Poisson's Exponential Limit to the Binomial to

be used in the Determination of the Probable Errors

of Cell Frequencies n = 1 to 30. Percentage of

Occurrences in random Samples when the Population

proportionately sampled would give actually n to

the Cell

LIII. Angles, Arcs and Decimals of Degrees. Table giving

(a) the Arc for each Degree from 1Â° to 180Â°, (b) the

Arc for each Minute and Second of Angle and (c)

the Value in Decimals of a Degree of each Arc and

Second of Angle

LIV. Table of the G(r,v) Integrals. Values of log.F(r, v)

and H(r, v) for r = 1 to 50 and <Â£ = tan -1 v from

0Â° to 45Â°

LV. Miscellaneous Constants in Frequent Use by Statisticians

and Biometricians .......

PAGE PAGE

lxiv 84-85

lxiv 86-87

lxv

88

lxx 89-97

lxxiii 98-101

lxxiv 102-112

lxxvi 113-121

lxxvii 122-124

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