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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

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THE LIFE, LETTERS, AND LABOURS OF FRANCIS
GALTON. Vol. I (1822—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 — IX (1902 — 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 — IX
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nimal Kingdom. By Ernest
:., Alice Lee, D.Sc, Edna
arion Radford, and Karl
S. ' [Shorth).

dan. By Karl Pearson,
>, and C. H. Usher. Text,
Part I. Issued. Price

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On the Relation of Fertility in Man

icial Status, and on the changes in this

Relation that have taken place in the last

50 years. By David Heron, M.A., D.Sc.

Issued. Sold only with complete sets.

A First Study of the Statistics of
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Karl Pearson. F.R.S. Issued. Price3*.nef.

A Second Study of th9 Statistics of
Pulmonary Tuberculosis. .Marital Infec-
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irtative Mating by Ethel M. Elderton.
lis nvl.

The Health of the School-Child in re-
lation to r By Karl
Pk arson, F.R.S.

On the Inheritance of the Diathesis
'.ithisis and Insanity. A Statistical
Study base! upon the Family Hisrt

JOMNO,

M.D . B 3




VI. A Third Study of the Statistics of

Pulmonary Tuberculosis. The Mortality
i)f the Tuberculous and Sanatorium Treat-
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8. J. Pkbky, A.I.A. limed. Price 3s. net.

VII. On the Intensity of Natural Selection

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VIII. A Fourth Study of the Statistics of

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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

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PREFACE



|~ AM very conscious of the delay which has intervened between the announce-
ment of the publication of these Tables and their appearance. This delay has
been chiefly due to two causes." First the great labour necessary, which largely
fell on those otherwise occupied, and secondly the great expense involved (a) in
the calculation of the Tables, and (b) in their publication. This matter of expense
is one which my somewhat urgent correspondents, I venture to think, have entirely
overlooked. It is perfectly true that only one single Table in this volume has
been directly paid for, but a very large part of the labour of calculation has been
done by the Staff of the Biometric Laboratory, whose very existence depends on
the generous grant made to that laboratory by the Worshipful Company of
Drapers. Our staff is not a large one and it has many duties, so that the progress
of calculation has of necessity been slow. Even now I am omitting projected
Tables, which I can only hope may be incorporated in a later edition of this
work, e.g. Tables of the Incomplete B- and T-functions, and the Table needed to
complete Everitt's work on High Values of Tetrachoric r when r lies between
— - 80 and — TOO. It would only satisfy my ideal of what these Tables should be,
had I been able to throw into one volume with the present special tables,
extensive tables of squares, of square roots, of reciprocals and of the natural
trigonometric functions tabled to decimals of a degree. Logarithmic tables are
relatively little used by the statistician to-day, which is the age of mechanical
calculators, and he is perfectly ready to throw aside the fiction that there is any
gain in the cumbersome notation of minutes and seconds of angle — a system
which would have disappeared long ago, but for the appalling 'scrapping' of
astronomical apparatus it would involve. But the ideal of one handy book for
the statistician cannot be realised until we have a body of scientific statisticians
far more numerous than at present. Statisticians must for the time being carry
about with them not only this volume but a copy of Barlow's Tables, and a
set of Tables of the Trigonometrical Functions.



vi Tables for Statisticians and Biometricians

Beside the cost of calculating these Tables, to which I have referred, must be
added the cost of printing them. I had to do this slowly as opportunity offered
in my Journal Biometrika, and the Tables as printed were moulded, in order
that stereos might be taken for reproduction. Even as it is, there are a number
of Tables in this volume, either printed here for the first time (e.g. Tables of the
Logarithm of the Factorial and of the Fourth Moment), or published here for the
first time (e.g. Tables of the G(r, v) Integrals), the setting up of which has
naturally been very expensive.

From the beginning of this work in 1901* when the first of these Tables was
published and moulded, I have had one end in view, the publication, as funds
would permit, of as full a series of Tables as possible. It is needless to say that
no anticipation of profit was ever made, the contributors worked for the sake
of science, and the aim was to provide what was possible at the lowest rate we
could. The issue may appear to many as even now costly ; let me assure those
inclined to cavil, that to pay its way with our existing public double or treble
the present price would not have availed; we are able to publish because of the
direct aid provided by initial publication in Biometrika and by direct assistance
from the Drapers' Company Grant. Yet a few years ago when a reprint of these
Tables in America was only stopped by the threat to prevent the circulation of
the book in which they were to appear entering any country with which we had
a reasonable copyright law, I was vigorously charged with checking the progress
of science and acting solely from commercial ends ! Meanwhile without any leave,
large portions of these tables have been reprinted, sometimes without even citing
the originals, in American psychological text-books. Two Russian subjects have
reissued many of these Tables in Russian and Polish versions, and copies of their
works in contravention of copyright are carried into other European countries.
It does not seem to have occurred to these men of science that there was any-
thing blameworthy in depriving Biometrika of such increased circulation as it
obtained from being the sole locus of these Tables, nor did they see in their
actions any injury to science as a whole resulting from lessening my power to
publish other work of a similar character. It is a singular phase of modern science
that it steals with a plagiaristic right hand while it stabs with a critical left.

The Introduction gives a brief description of each individual table ; it is by no
means intended to replace actual instruction in the use of the tables such as

* When issuing their prospectus in the spring of 1901 the Editors of Biometrika promised to
provide " numerical tables tending to reduce the labour of statistical arithmetic"



Preface vii

is given in a statistical laboratory, nor does it profess to provide an account
of the innumerable uses to which they may be put, or to warn the reader of the
many difficulties which may arise from inept handling of them. Additional aid
may be found in the text which usually accompanies the original publication of
the tables.

In conclusion here I wish to thank the loyal friends and colleagues — Dr W. F.
Sheppard, Mr W. Palin Elderton, Dr Alice Lee, Mr P. F. Everitt, Miss Julia Bell,
Miss Winifred Gibson, Mr A. Rhind, Mr H. E. Soper and others — whose un-
remitting exertions have enabled so much to be accomplished, if that much is
indeed not the whole we need. I have further to acknowledge the courtesy
of the Council of the British Association, who have permitted the republication
of the Tables of the G (r, v) Integrals, originally published in their Transactions.

To the Syndics of the Cambridge Press I owe a deep debt of gratitude for
allowing me the services of their staff in the preparation of this work. Pages and
pages of these Tables' were originally set up for Biometrika, or were set up afresh
here, without the appearance of a single error. To those who have had experience
of numerical tables prepared elsewhere, the excellence of the Cambridge first proof
of columns of figures is a joy, which deserves the fullest acknowledgement.

Should this work ever reach a second edition I will promise two things,
rendered possible by the stereotyping of the tables : it shall not only appear
at a much reduced price, but it shall be largely increased in extent.

KARL PEARSON.



Biometric Laboratory,
February 7, 1914.



Errata

The reader is requested to make before using these Tables the following corrections on
pp. 82, 83, 84 and 85 :

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



When you can measure what you are speaking about and express it in
numbers, you know something about it, but when you cannot measure it, when
you cannot express it in numbers, your knowledge is of a meagre and unsatis-
factory kind.

Lord Kelvin.

La theorie des probabilites n'est au fond que le bon sens reduit au calcul ;
elle fait appr£cier avec exactitude ce que les esprits justes sentent par une sorte
d'instinct, sans qu'ils puissent souvent s'en rendre compte.

Laplace.



ERRATA, ANTE USUM DILIGENTER CORRIGENDA.

Introduction.

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

ff i i(u -u 1 + u_ l + u i ) + 6%(5u 1 -5u -u_ l -u i )+u a -u (6) = 0,
read 2 i (-2« -Mi + «-i+»2) + <5i( 5 "i- 3M o- U-i-u 2 )+u -u (d) = O,
and add : " This equation is most effectively dealt with by finding the value of

%-«o(d)



<?o=



and then calculating :



kiZuo+U-i + Uz-bua)



p. xxxiv. Formula (xxxi), For A 7 (ab — cdf read N(ad—bcf-

p. xxxv. Table (3) was taken from Biometrika, Vol. IX. p. 292. Unfortunately it was not
there noted that Mr Yule's unit was 1000 houses : see his Theory of Statistics, p. 61. He
has drawn the Editor's attention to this regrettable omission. The table for the statistical
constants at the centre of the page should read :

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

On p. xxxvi. Lines 3 — 9 while correct for the illustration actually given as table (3)
on p. xxv, are of course incorrect for the true unit of 1000 houses. The statement in
Lines 19 — 23 with regard to the houses building or built is incorrect ; there is very marked
positive association. We must now include the house-data, and Lines 26 — 27 should
read : " If we regard these four tables the order of ascending association judged by either
<j> or <7 2 is (3), (4), (5), (2) as against Mr Yule's (2), (3), (4), (5)."

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

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

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

p. lv. Formula (xlii).



For log CJft/ ) = -0399,0899 + etc.,
read log ( F Jft^ ) = 0-399,0899 +etc.



p. lx. Table, 4 = 2, B = ". For (21-556) read (31-556).
„ A=2, 5 = 8. For (12-202) read (13-202).

p. lxii. Line 11.

„ -67449^ , -67449^

For _^. 2ftMld _^z fc ,

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



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

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

p. lxv. Formula (lxxvi)

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

read Ftp, 2s 2 ^/s, & = 2ft - etc.
p. lxxv. Table, column Nvi, 3rd line. For 38 read 36.

p. lxxvii. Line 2. For " We look out 5-8 in Table L. " read " We look out 5-8 in Table LI."
p. lxxx. Line 5. For e -<-«*i g rea d e -4.560c e_
p. lxxxiii. Line 13 from bottom. For 2-371,76665 read 2-371,6665.

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

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

and 1-177 ViV : 2 2 .
p. 92. Table XLVIII.



>r m = 20\
?rc = 20J 1


51-2195


read ra = 20\
m = 20J 1


51-2195


25-6098


25-6098


2


12-4765


2


12-4765


3


5-9099


3


5-9099


4


2-7154


4


2-7154


5


1-2068


5


1-2068


6


•5172


6


•51 72


7


■0839


7


•2130


8


•0315


8


•0839


9


•0112


9


•0315


10


•0037


10


■0112


11


•0012


11


•0037


12


•0003


12


•0012


IS


•oooi


13


•0003


U


•0000


U


•0001


15





15


•0000



p. 126. Table LI V. For r = 2, <£° = 5, log H(r, ») = -106,5985,
read log H(r, »)- -196,5985.

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

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

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

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

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

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

p. 143. For log log e -637 7799 16.
read log loge T-637 7843 11.

The issue of this list of Errata has been intentionally delayed in order to make it as
complete as a wider use of the volume would render possible. The Editor will be as grateful
for further emendations, as he has been for the above.



CONTENTS

Preface

Introduction to the Use of the Tables .



PAGE
V

xiii



TABLES *

TABLE PAGE

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

Frequency xv 1

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

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

III. Tables of the Probability Integral : Abscissa and Ordinate

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

IV. Tables of the Probability Integral : Logarithms of Areas

for high Values of Deviate xxi 11

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

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

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

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

facilitate the calculation of 1 — r 2 . . . . xxiii 20-21

IX. Values of the Incomplete Normal Moment Functions, First

to Tenth Moments xxiv 22-23

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

XL Values of the Functions ^r„ i|r 2 , and ^ 3 required to determine
the constants of a Normal Frequency Distribution from
the Moments of its Truncated Tail .... xxviii 25

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

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

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

B. 6



Tables for Statisticians and Biometriciam



TABLE

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

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

XIV— XVI. Tables for Testing Goodness of Fit : Auxiliary Tables

B — D. Numerical values needed in the calculation

and extension of Tables of P .

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

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

XVIII. Probability of Association on a Correlation Scale : Values

of (— log P) for an observed value of the Tetrachoric
Correlation r and the Standard Deviation of r for
zero Association .......

XIX. Probability of Association on a Correlation Scale : Values

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

XX. Probability of Association on a Correlation Scale : Values

of log^; 2 corresponding to Values of Tetrachoric r
and a r .........

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

determine <r r for a Population of given size, divided
into a fourfold Table with given marginal Frequencies

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

determine the Equiprobable Tetrachoric Correla-
tion r,. from a Knowledge of log^ 3 and a cr r .

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

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

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

fold Correlation Table : Values of ^ a for Values of

marginal |(1 + a)

XXV. Table to determine the Probability that the mean of a
very small sample n (4 to 10), drawn from a normal
Population will not exceed (in algebraic sense) the
Mean of the Population by more than z times the
Standard Deviation of the Sample ....

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

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

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

XXIX. Tables to facilitate the Determination of Tetrachoric Corre-
lation : Tables of the Tetrachoric Functions t, to t 6
for given marginal Frequencies ....



PAGE PAGE

xxxiii 29

xxxiii 30

xxxiv 31

xxxvi 31

xxxvi 32

xxxvi 32

xlii 33

xlii 34

xl 35

xl 35



xliii 36

xlv 37

xlvi 38-39

xlvi 40-41



1 42-51



Contents



XI



TABLE PAOE PAGE

XXX. Tables to facilitate the Determination of Tetrachoric
Correlation : Supplementary Tables for determining
high Tetrachoric Correlations (r = "80 to TOO) for
given positions of the dichotomic lines . . liii 52-57

XXXI. Table of the Logarithms of the Gamma Function,

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

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

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

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

B, Extension of Table XXXII for very narrow

Catenaries lvi 64*

XXXIV. Diagram to find the Correlation Coefficient r from the

Mean Positive Contingency on the Hypothesis of

a Normal Distribution ...... lvii 65

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

bution from a Knowledge of the Constants fa and

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

XXXVI. Diagram showing Distribution of Frequency Types for

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

XXXVII. Probable Errors of Frequency Constants: Table for the

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

XXXVIII. Probable Errors of Frequency Constants : Table for the

Probable Error of fa 2 for given values of fa and fa . lxii 70-71
XXXIX. Probable Errors of Frequency Constants : Values of the
Correlation of Deviations in fa and fa(Rp v ^) for
given values of fa and fa lxii 72-73

XL. Probable Errors of Frequency Constants : Probable Error
of the Distance between Mean and Mode for given
Values of fa and fa lxii 74-75

XLI. Probable Errors of Frequency Constants : Probable Error

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

XLII. Values of the Frequency Constants fa, fa, fa and fa
for given Values of fa and fa on the Assumption that
the Frequency can be described by one or other of
Pearson's Frequency Types ..... lxii 78-79
XLI 1 1. Probable Errors of the Frequency Constants : Probable
Error of the Criterion of Type, /c 2 , for given Values

of fa and fa lxii 80-81

XLIV. Probable Error of the Determination of Frequency Type :
Value of Semi-minor Axis of Probability Ellipse for

given Values of fa and fa lxiv 82-83

6—2



Xll



Tables for Statisticians and Biometricians



TABLE

XLV. Probable Error of the Determination of Frequency Type :
Value of Semi-major Axis of Probability Ellipse for
given Values of /3i and /3 2

XLVI. Probable Error of the Determination of Frequency Type :
Value of Angle between Major Axis of Probability
Ellipse and Axis of fi 2 for given Values of /9, and /3 2

XLVII. Probable Error of the Determination of Frequency Type :
Diagram to determine for given Values of & and /9 2 ,
a given Frequency Distribution belongs definitely
to a given Type of Frequency ....

XLVIII. Probable Occurrences in Second Small Samples. Per-
centage Frequency of each number of Successes in
a Second Small Sample of m after p Successes in
a First Small Sample of n .

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

L. Table of Fourth Moments of Subgroup Frequencies, i.e.
n x x*, for n = 1 to 400, x = 1 to 19, for Verification
of the Calculation of Raw Moments

LI. General Term of Poisson's Exponential Limit to the
Binomial, i.e. of the so-called " Law of Small
Numbers." Value of e~ m m x j\x for m = - l to 15'0,
and x = up to such figure as makes the Function
significant in the sixth decimal Place .

LII. Table of Poisson's Exponential Limit to the Binomial to
be used in the Determination of the Probable Errors
of Cell Frequencies n = 1 to 30. Percentage of
Occurrences in random Samples when the Population
proportionately sampled would give actually n to
the Cell

LIII. Angles, Arcs and Decimals of Degrees. Table giving
(a) the Arc for each Degree from 1° to 180°, (b) the
Arc for each Minute and Second of Angle and (c)
the Value in Decimals of a Degree of each Arc and
Second of Angle

LIV. Table of the G(r,v) Integrals. Values of log.F(r, v)
and H(r, v) for r = 1 to 50 and <£ = tan -1 v from
0° to 45°

LV. Miscellaneous Constants in Frequent Use by Statisticians
and Biometricians .......



PAGE PAGE



lxiv 84-85



lxiv 86-87



lxv



88



lxx 89-97

lxxiii 98-101



lxxiv 102-112



lxxvi 113-121



lxxvii 122-124



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