HA
33
P4
v.1
cop .2

TABLES FOR STATISTICIANS
AND BIOMETRICIANS

EDITED BY

KARL PEARSON, F.I

LTON PROFESSOR, 1

ISSUED WITH ASSIS: MADE BY

THE WORSHIPFl ' HE

V:

THE LIFE, LETTERS, AND LABOURS OF FRANCIS
GALTON.

BIOMETRIKA.

^^Bt ^

I. Mathen

II. M

in.

IV.

Mathei
Mathei

V. Ms

I.

II.

III.

On the

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to the

IM:

A Firs; Study of the statistics 01

i Of

i

Statistics of
Th«

•N, I
. Pi

VII. On the Intensity of Natural Selection

VIII. A Fouri / of the Statistics of

IX. A Statistical Study of Oral Tem-

itance of the Die

presented to

Xibran?

of tbe

of Toronto

\

TABLES FOR STATISTICIANS AND

BIOMETRICIANS

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TABLES FOR STATISTICIANS
AND BIOMETRICIANS

EDITED BY

KARL PEARSON, F.R.S.

GALTON PROFESSOR, UNIVERSITY OF LONDON

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VI

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PEEFACE

T 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 (6) 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 F-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 Statist i<-lmi* <m<l

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 l>y 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 Incuts 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 uniing their prospectus in the spring of 1'JOl 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 1-77 ViVSi and 177 -JN^ at the top of the Tables read 1-177 V2V2, and M

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.

LOUD KELVIN.

La theVme des probabilites n'est au fond que le bon sens reduit au calcul ;
elle fait apprecier 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%o, or replace by - sign,
p. xiv. For Equation (vii)bi8

read ff*% (-U0- itj + a. i + uj + d^ (5«1-3u0- («_, -ttj)+Wo-»o(0)=0,
and add : " This equation is most effectively dealt with by finding the value of

Up-

'

and then calculating :

p. xxxiv. Formula (xxxi), For N(ab — cd")'* read N(ad-bcY.

l>. 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 X2= 1439-2998, P-8'730/10312, while <p*, <j> and Ct 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
Lilies 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
<t> or <72 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 4'5 read 5-0, and for S(x) at foot
read S(jj}.

p. xlix. Line 1. For 6j = 10c0 read 61 = 10ci.
p. Iv. Formula (xiii).

For Io8 (r^f.-. ) = '0399,0899 + etc.,

read log

p. Ix. Table, A =-2, B = 7. For (21-556) read (31-566).
„ A = 2, B=S. For (12-202) read (13-202).

p. Ixii. Line 11.

•67449 , -67449
for — 20, and — 20, ,

read -6744920, and -67449 20,.

p. Ixiii. The two w>lidi have been dn>p|ied in the bi<|iiadrati< ;

For M8&-9ft-12)(4&-30,) = (10fc-12;S1

read 0, (8ft-90,-12)/(4ft-3ft)-(100,- 120, - 18W(/3,+3)».

p. Ixv. Formula (Issvi)

20» - etc.

p. Ixxv. Table, column A'*,', 3nl line, /or 38 rfad 36.

p. Ixxvii. Line 2. For " We look out 5'8 in Table L. " rwirf " We look out 5-8 in Table LI."
p. Ixxx. Line ft. For «-«•*•»« read «-«•*•»•.
p. Ixxxiii. Lino 13 from bottom. For 2-371,76665 read 2-371,6665.

Text of Tablet.
p. 13. Table V, n-241. For x»= "03172 read -03072.

pp. 82,83, 84 and 85. For 1-77\'.V2, uid 1-77^2, at the top of the Tables read 1-177 ^A' 2,
and

p. 92. Table XLVIII.

For » = 20\ 0
m-20J ;

r.1-2195
25-6098

read n = 20\ 0
ra = 20j 1

51-2195
25-6086

g

12-4765

.'

12-4766

S

5-9099

.:

5-9099

1

2-7154

4

2-7154

5

1-2068

r,

1-2068

6

•5172

6

•5172

7

•0839

7

•2130

#

O81B

8

•0839

9

•0112

9

•0315

70

•0037

in

Ol 12

jj

•0012

11

•0037

j<g

•0003

1?

•0012

13

•0001

IS

•0003

14

•0000

14

•0001

15

—

15

•0000

p. 126. Table I. IV.
p. 141.

p. I '-'- «

p. 142.

For r = 2, <f>° = 5, log H(r,v)=* -106,5985,
read log/f(r, v)=-196,5985.

For r = 45, <f>° = 44, log F(r, v) = '483,7836,
read log/"(r, i/)= 7-483,7836.

For r = 50, c/>° - 4, log H (r, „ ) = -932,5457,
read logJ7(r, v) = -392,6457.

For r— 60, <p° -31, log H (r, v} = -933,2995,
read log H (r, i/) = -393,2995.

p. 143. For log logs -637779916.
read log log e I -637 7843 11.

The istfue of this list of Errata han been intentionally delayed in order to make it as
complete aa a wider UHC of the volume would render possible. The Editor will be as grateful
for further emendation.*, iw ho has been for the above.

CONTENTS

PREFACE

INTRODUCTION TO THE USE OF THE TABLES .

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* . . . . 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

XI. Values of the Functions fa, fa, and fa 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 ami

I II. I •

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

to assist in determining P for high values of %* .

X I V — 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 •)£ 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 tf corresponding to Values of (- log P)

X X. Probability of Association on a Correlation Scale : Values
of \ogjf corresponding to Values of Tetrachoric r
and ,ov

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

determine t<rr 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 rr from a Knowledge of log ^'J and 0<rr .

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 •%* 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 Onlinates of the

Frequency Curve y = y,e~''xltt ( 1 i / u)>'

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 rt
for given marginal Frequencies ....

iii -1'

xxxiii 30

xxxiv 31

x \.\vi .'!!

xxxvi 82

xxxvi 32

xlii -'i:!

xlii :)4

xl :\~>

xl .S.">

xliii :><;

xlv 37

xlvi 38-39

xlvi 40-41

1 42-51

Contents

x

TABLE PAGE PAGE

XXX. Tables to facilitate the Determination of Tetrachoric

Correlation : Supplementary Tables for determining

high Tetrachoric Correlations (»- = '80 to I'OO) for

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

XXX [. Table of the Logarithms of the Gamma Function,

loglXp) from p = \ to p=2 . . . . Iv 58-61
XXXII. Table to deduce the Subtense from a knowledge of Arc

and Chord in the Case of the Common Catenary Ivi 62-63
XXXIII. A, Extension of Table XXXII for very flat Catenaries;
B, Extension of Table XXXII for very narrow
Catenaries ........ Ivi 64

X X X I V. Diagram to find the Correlation Coefficient r from the
Mean Positive Contingency on the Hypothesis of
a Normal Distribution ...... Ivii 65

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

bution from a Knowledge of the Constants ft and

ft. Customary Values of ft and ft . . . Ixiii 66

XXXVI. Diagram showing Distribution of Frequency Types for

High or Unusual Values of ft and ft . . . Ixiii 67

XXXVII. Probable Errors of Frequency Constants: Table for the

Probable Error of ft for given Values of ft and ft . Ixii 68-69

XXXVIII. Probable Errors of Frequency Constants : Table for the

Probable Error of ft for given values of ft and ft . Ixii 70-71
XXXIX. Probable Errors of Frequency Constants: Values of the
Correlation of Deviations in ft and ft(fl^iPi) for
given values of ft and ft ..... Ixii 72-73

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

XLI. Probable Errors of Frequency Constants : Probable Error

of the Skewness for given Values of ft and ft . Ixii 76-77
XLII. Values of the Frequency Constants ft, ft, ft and ft
for given Values of ft and ft on the Assumption that
the Frequency can be described by one or other of
Pearson's Frequency Types ..... Ixii 78-79

XLIII. Probable Errors of the Frequency Constants : Probable
Error of the Criterion of Type, «•„, for given Values
of ft and ft . . . ..... Ixii 80-81

XLI V. Probable Error of the Determination of Frequency Type :
Value of Semi-minor Axis of Probability Ellipse for
given Values of ft and ft ..... Ixiv 82-83

6—2

XII

Tablf* for Statist t'ritnt* <ui<l Ii!<nnrtrii-'nnis

TABLE I'A"K

XLV. Probable Error of the Determination of Frequency Type :
Value of Semi-major Axis of Probability Ellipse for
given Value's of /3, and & Ixiv 84-85

\ I. VI. Probable Error of the Determination of Frequency Type:
Value of Angle between Major Axis of Probability
Ellipse and Axis of & for given Values of /9, and /8, Ixiv 86-87

X L V 1 1. Probable Error of the Determination of Frequency Type :
Diagram to determine for given Values of /9, and j8s,
a given Frequency Distribution belongs definitely
to a given Type of Frequency . . . . Ixv 88

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 Ixx 89-97

XLIX. Logarithm of Factorial |_n from n = 1 to 1000 . . Ixxiii !»8-101

L. Table of Fourth Moments of Subgroup Frequencies, i.e.
n x ar4, for n = 1 to 400, oc = 1 to 19, for Verification
of the Calculation of Raw Moments . . . Ixxiv 102-112

LI. General Term of Poisson's Exponential Limit to the
Binomial, i.e. of the so-called " Law of Small
Numbers." Value of e~1"mx/[x for m = 'l to lo'O,
and a; = 0 up to such figure as makes the Function
significant in the sixth decimal Place . . . Ixxvi 113-121

LII. Table of Poisson's Exponential Limit to the Binomial to
be used in the Determination of the Probable Errors
of Cell Frequencies 7; = 1 to 30. Percentage of
Occurrences in random Samples when the Population
proportionately sampled would give actually n to
the Cell Ixx vii 122-1 24

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 Ixxix 125

LIV. Table of the 0(r,v) Integrals. Values of \ogF(r, v)
and H(r, v) for r = l to 50 and </>=tan~'i> from
0° to 45° Ixxxi 12G-142

LV. Miscellaneous Constants in Frequent Use by Statisticians

and Biometricians . Ixxxiii 143

INTRODUCTION TO THE USE OF THE TABLES

For this introduction to the use of the Tables I have largely drawn on the
prefaces to the original papers in Biometrika, and record here my acknowledge-
ments to the authors of the same.

INTERPOLATION.

(1) A word must first be said as to interpolation. Let a function u be tabled
for the argument x proceeding by differences A# = h. Then the scheme of such
a table with the differences of u is :

*-l

«-,

Au_3

«-l

«-,

AM_.

*,

"<

A«0

a-,

MI

A«,

«»

",

Aitj

a:.

«,

AH.

#4

"i

AH,

a-,

"5

A'-M_3

ASM

A'M_,

A3M_,

AX

A*,

A4H_, etc., etc.

AX
A*«,

where: AH, = M»+, — M,,

AX = AJ/.-H — AM,,
AX = AX+I — ^X etc., etc.

Now there are throe interpolation formulae which it is desirable to remember.
If the function be required for the value xa+ffh and this value be termed M0(0),
then we have :

0(1-0)A1_. , 0(l-0)(2-0)
3!

.(i),
•("),

where <^>=l-0. This is Everett's formula*. And lastly:

K, ( 0) = M0 + 0 1 (A«0 + Au_.) + ~} A««_, - g(I3"|^ i < A*!*., -1- A»«_0 - , . (iii),
where we work with the differences on or adjacent to the horizontal through xu.

* .luiiriiiil <if the Institute of Actuaries, Vol. xxxv, p. 452.

xiv Tables for Statisticians a ml Hinni, triciaii* | INTKHI-OLATION

It is very rarely indei-d that we need go beyond second differences, often the
tiivt will suffice. Not infrequently the inverse problem aris.-s. namely w<- are
given M«(0) and have to determine 6 from it. If we only go '^ ^ar «» second
differences, either (i) or (iii) gives us a quadratic to find 0 and the root will
be obvious without ambiguity. Usually it suthYrs to find

.•ind t.hi-n determine ft from

f) = (M. (ft) - «.)/A«., + ~ A»i/../A« .................... (iv) ;

or to find fP = («„(#) - «„)/£ (Ai/<, + AH_,)

and then

0* A

.(v).

Very often good results an- readily obtained by applying fvigrange's inter-
polation formula which for three value* of n reduces to

0)M, ............... (Vi).

Or, we may use the mean of two such formulae and take

«.(0) = (l-0)(l-{0)«0+i0(5-0)»1-i0(l-0)(H_1 + «a) ... (vii).
The resulting quadratics arc respectively :

* (i («,+»-,)- Wo) + tfi(M,-«-,) + «.-W.(fl)-0 ......... (Vi)b*(

and ^ i («„ - M, + •(_, + «„) + ^ \ (5«! - 5«« - «_, - «.,) + ?/„ - u0 (6) = 0 . . .(vii)1*1.

(2) There are some tables in this book which are of double entry, e.g. those
for the Tetrachoric Functions and for the G (r, v) Integrals. The simplest solid
interpolation formula, using second differences, is :

l)A'Xo) ...... (viii),

where A denotes a difference with regard to x, and A' with regard to y. But if
we consider uXtV to be the ordinate of a surface, and the figure, p. xv, to represent
the xy plane of such a surface, then it is clear that, if P be the point x, y, and
A, B, C, D, &c. the adjacent points at which the ordinates are known from the
table of double entry, only the points A, B, C, D, J, and N are used by the above
formula ; and of these points, not equal weight is given to the fundamental points
A, B, G, D, for C only appears in a second difference. If another point of
the fundamental square other than A be taken as origin, we get a divergent,
occasionally a widely divergent result. If we use only four points — A, B, C, D —
to determine the value of the function at P, then we might take the ordinate

1]

Introduction

xv

at P of the plane which (by the method of least squares) most nearly passes
through the four points of the surface vertically above A, B, G, D. We have then

«*..-/ = i (wo,o + M,,O + "0,i + MI.I) + i (<«,,„ - «(,,„ + «i,i - MO,I) (a - '5)

+ 2 (MO,I - «o,o + MM - M,,0)(?/ - -5) (ix),

but by trial it has been found that this formula gives occasionally worse results
than that for first differences, using only three points. To find by the methods
of simple interpolation (with first or first and second differences) the points
a and 6, and then interpolate P between them, generally gives a fairly good
result; but this result usually differs somewhat from that obtained by first simply

F G II I

© © ©

(0, -1) (1, -1)

(0, 0)

A •*• x -» a
©... . x .

(-1, -1)

(-1,0)
E

1,0)
B

©

(2, -1)

(2, 0)
J

©

I

I

O I' x

©

a

(-1,

©

It

(<>, 1)

0

(1, 1)

©
if

(1, 2)

©
K

(2, 1)

©
L

(2, 2)

© ©

K \

(-1, L' (0, 2)

interpolating e and f and then interpolating between e and f*. Various other
methods for interpolation in n-diinensioned space will be found discussed by Palin
Elderton in Biometrika^. The ideal method can hardly yet be said to be known,
and it may well vary from table to table and from one part of the same table
to another. One or other of the above methods will, however, suffice in practice
for most statistical purposes.

I consider now the individual tables.

TABLE I (p. 1)

Table of Deviates of the Normal Curve for each Permille of Frequency. (Calcu-
lated by Sheppard and published by Galton in Biometrika, Vol. V. p. 405.)

If N be the total number in a population, zSx the frequency between x and
x + Sx, y the standard -deviation, then the frequency curve of the population
assuming its distribution to be Gaussian or normal will be:

N

.(ix)

V27TO-

* B. A. Report, Dover 1899. Tables of G (r, K) -Integrals, Keport of the Committee (Drawn up
by K. Pearson). f Vol. vi. p. 94.

\\l

I'nhli-K for

nmf li!<nn< tr

[I

the origin being the moan. Table I. gives the value of x/tr for each thousandth
of the area of this curve, — each ' pcrmillc ' — reckoned from left to right,

In enterinj; tin- table we enter from the left-hand column ami toji row if the
]ietmille be less than 500. For example, if the frequency below a particular value
were .'is" per thousand, the corresponding deviate would be —0*2871, the number
placet! at the intersection of the '38 row from left and '007 column from top.
The negative sign is always to be given when reading permilles In-low 500,
because the deviate will be in defect of the mean, supposing increasing variun -
to be plotted as usual from left to right.

On the other hand if the permille be greater than 500 we enter the table from
the right-hand column and bottom row. For example, if the permille be 74-8, the
deviate is + 0'6682, the number placed at the intersection of the '74 row fmni
right and '008 column from bottom of the table. The plus sign must be given, as
the deviation is in excess of the mean, if the convention as to plotting variables
has been observed.

Illustration: The following observations were made on the nature of the
degree taken by 1011 Cambridge undergraduates measured at the Anthropological
Society's Laboratory :

Poll

Third Class

4S7
189

Second Class
First Class

182
153

Find the deviates of these on a normal or Gaussian scale.

The sums from the lowest to each class top are 487, 676, 858 and 1011
respectively. If we term with Francis Galtou the one man in a thousand of
surpassing intelligence or special ability a "genius," we have on multiplying by
•0009891197 the reciprocal of 1011, the series for entering Table I. Thus we
find:

II. •!:•'• :

(•481)
(-482)

A
Ax -7

•4817

•MM

and -9990

•047B

(•668)'4344

(•848)1-027!!

(•999) 3-0902

•0451

(-669) -4372

(•849)1-03-22

—

•0025

A -0028

A -0043

—

•00175

Ax '6 -001IIK

AX -7 -00301

—

- -0458 + -4361

-(-1-0309

+ 3-0902

Supposing with Pearson* that 100 units of intelligence ("mentaces") separate
the lowest man of the First Class from the highest man of the Poll, we have
+ r0309 — (— '0458) = 100/tr, where a is the standard deviation of intelligence.
Thus ff = 100/1 '0767 = 92'88 mentaces. Hence we conclude that the range of
Third Class men is from - 4'25 (i.e. 92'88 x (- -0458)) below to + 4>0'50

Hiumttrikti, Vol. V. p. 100.

I — III] Introduction xvii

(i.e. 92-88 x (+ -4361)) above the average undergraduate. The range of Second
Class men is from + 40'50 to + 95'75 mentaces above the average undergraduate,
and the range of First Class men all those with more than 95'7o mentaces above
the average. The "genius" corresponds to an excess of no less than 287'02
mentaces. If we suppose that one individual in 1000 is completely feeble-
minded or practically wanting in all intelligence, we should credit roughly the
average man with 300 mentaces, and we should then have our range of intel-
ligence on a Gaussian scale :

Poll : below 296 mentaces ;

Third Class : above 296 and below 340 mentaces ;

Second Class: above 340 and below 396 mentaces;

First Class : above 396 mentaces ;

" Genius " : above 587 mentaces.

In rough numbers : Poll, below 300 ; Third Class, 300 to 350 ; Second Class,
350 to 400 ; First Class, over 400 ; " Genius," over 600.

Of course there is much that is hypothetical here, but the numbers give us
some appreciation of the distribution of ability, and they serve to illustrate the
construction of a Gaussian or normal scale. When more than three or four
significant figures are needed Tables II and III must be used.

TABLES II AND III (pp. 2—10)

Talks of the Probability Integral: Area and Ordinate of the Normal Curve in
terms of the Abscissa • ; and Abscissa and Ordinate in Terms of Difference of Areas.
(Calculated by Dr W. F. Sheppard, and published in Biometrika, Vol. n. pp.
174_190.)

" Sheppard's Tables" were the first to express the Gaussian* or normal
probability integral in terms of the standard deviation ; they are so familiar to
statisticians that it would almost seem a work of supererogation to explain their
' use," which is further too manifold for full description. We can only give a few
sample illustrations.

It is most important ivlten using these tables to pay attention to the signs of
the differences recorded at the tops of the columns.

Illustration (i). The mean length of cubit in 1063 adult English males is
recorded as 18"'31 ± '019 and of their 1063 adult sons as 18"'52 ± "021. Determine
the odds against these two measurements being really identical, i.e. random
samples from the same population. We assume that the deviation of means and
their differences follow the normal law. The difference is 0"'21 and the probable
error of this difference = V(-019^ + ('021)a = 0"'0283. Since the probable error

* The term is usual, but inaccurate. Laplace had reached the probability integral and suggested
its tabulation several years before Gauss.

B. e

xviii Tables for St<it!xii<-i<ni* • <m<l />i»/i/< trlcians [II — III

= -(J7449 x standard deviation, we have the standard deviation of the difference
«0"-04196. Hence the deviation in terms of the standard deviation

= 0-21/(0'04196) = 5-0048.

Table II, p. 8, gives the area $ (1 + o) of the normal curve up to the abscissa xja.
Noting the remark at the foot of the table, we have

xfa- = 5-00, i (1+ a) = "999,999,7 1 33,
x\a = 5-01 , J (1 + a) = -999,999,7278,

A = 145,

A x 48 70,

x/a = 5-0048, £ (1 + a) = -999,999,7203.
Hence J (1 - a) = -000,000,2797.

Accordingly if we suppose the deviation as likely to be in defect as in
excess, the probability that we shall reach the observed deviation, or exceed it, is
2 x ^ (1 — a), and that we shall not is J (1 + a) — £ (1 — a), or the odds against the
result on a pure random sampling chance are '999,999,4406 to '000,000,5594, or
1,787,629 to 1, i.e. overwhelming odds. Thus we may reasonably argue that sons
in the professional classes in 1900 were substantially differentiated from their
fathers by a longer forearm of about £".

Illustration (ii). Find the value in mentaces of the mean intelligence of Poll-
men, First, Second and Third Class men as given by the numbers in the Illustration
to Table I.

The equation to the normal or Gaussian curve being

we easily find that if there be 'tabled' ordinates zl and za* at the abscissae
Xi and xt, which cut off an area nn, then the mean Xj3 of this area is given by

It will be sufficient to take the values of the abscissae already found, i.e.
XI/<T = - -0458, XI/<T= + -4361,
xt/a = + I -0309, xtl<r = + 3-0902.
We require the z's for these. For example :

x = -04, z = -398,6233

•05, z = -398,4439

0 = -58, A, -1793

A., - 397.

* The nymbol t here used is that of the Tables, i.e. -^— «-*<«/»)'.

V2r

II — III] Introduction xix

Therefore by formula (i) p. xiii :

•KQ v -4,9

*, = -398,6233 - -58 [1793] + -^-^ [397]

= -398,6233]

-1040> = -398,5241.
+ 48j

Or, we might proceed as follows: for the Poll-men ^(1 — a) = -4817, hence
a = -0366. But from Table III, p. 9, which gives z for a:

a = -03, z = -398,6603
a = -04, z = -398,4408
0 = -66,' A,= -2194

A2= -627.
Hence by formula (i):

•fifi x -^4,
*, = -398,6603 - -66 [2194] + - - [627]

z

= -398,0603]

-1448> =-398,5225.

+ 70J

We conclude therefore that z would be correct to five figures with second differ-
ences, and that for four figures, first differences from either Table II or Table III
will suffice.

If we use formula (ii) p. xiii — Everitt's formula — we find from Table II :

*, = -398,6233 - -58 [1793] + " '" [397] +

= -398,6233 v

- 1 040 1

+ M. -398,52!,,

+ 5
and from Table III :

z, = -398,6603 --C6 [2194]+'' -[627] + '^^ ^*[627]

= •398,6603;

-1448

J- = -398,5223.

+ 31 )

Working with formula (iii), Table II gives us ^='398,5242 and Table III
zt = "398,5225 with second differences. We shall not therefore without higher
differences get from any of our formulae closer than '398,522 with a possible error

«9

XX Table* for Statistician.-! ana" />/»////» tri<-ian* [II — III

of 1 or 2 in the lost place. This is, of course, amply sufficient for statistical
purposes, where four figures as a rule would be sufficient.

Using formula (i) p. xiii we obtain :

*, = -39852, *, = -23-l

z, = -36275, *4 = -00337.
Whence :

_
= + - - <r = - -82730- = - 76-84 mentaces,

"TO 1 7

•39852 - -3627.')
= + -- a = + -192oo- = + 17-88 mentaces,

' *

x, = + '36275~'2345° a- = + -7121* = + 66-14 mentaces,
* 1 »S() 1

+ -23450 >- • -00337 ^ = + ] .^^ = + U2'83 mentaces,
•

•f)AQQ7 _ A

x^ = + ff = + 3-3700o- = + 313-01 mentaces.

"0010

Assuming as before the average man to have 300 mentaces of intelligence
we find :

Average Poll-man has 223 mentaces.
Average Third Class man has 318 mentaces.
Average Second Class man has 366 mentaces.
Average First Class man has 443 mentaces.
Average man of "genius" has 613 mentaces.

Thus the average First Class Honours man is twice as able as the average Poll-
man, and the average "genius" has not quite twice the ability of the average
Third Class Honours man.

Illustration (iii). It is required to determine normal curve frequencies corre-
sponding to the following frequencies of the cephalic index in Bavarian skulls.

Here the mean and standard deviation found by moments in the usual way are

m = 83-069, o- = 3'432.

The deviations from the mean were next expressed in terms of the standard
deviation, i.e. these deviations are

-13-569, -12-569, ...- 0-569, +'431, +1-431, +2-431, ... + 14-431,

and they are multiplied on a calculator by the reciprocal of the standard deviation,
whence the column xja is found. Table II gives us £(l+a) knowing x/a- ; this
has been calculated by first differences only. We shall consider as an illustration
to Table XII, whether the normal distribution thus reached is to be considered
a good fit to the observations.

IV]

Introduction

xxi

Index

Observed

mft

4(i + ")

Calculated
Frequency

69-5—70-f>

1

-3-9539

•99996

Under 70-5 -1

70-5—71-5

1

-3-6625 -99988

•2

: 1-5— 72-5

—

-3-3711 -99963

•6

72-5— 7S-5

2-5

-3-0797 -99896

1-5

73-5—74-5

1-5

-2-7883 -99735

3-3

74-5—75-5

3-5

-2-4969 -99374

6-7

75-5 — 76-5

12-5

-2-2055 -98629

12-7

76-5—77-5

17

-1-9141 -97219

22-1

77-5—78-5

37

- 1 -6228 -94768

35-3

78-5—79-5

55

-1-3314 -90846

51-9

79-5 — 80-5

71-5 -1-0400 -85082

70-1

80-5—81-5

82 - -7486 -77294

87-0

81-5—8.'-:,

116 - -4572 -67623

99-4

X. '-5— 83-5

98

- -1658 -56584

104-2

83-5—84-5

107

•1256 -54997

100-5

84-5—8-,:-:

82

•4170 -66165

89-1

85-5—86-5

7 1 -7084 -76064

72-6

86-5—87-5

58 -9998 -841:2!)

54'3

87-5—88-5

34-5 1-2912 -90167

37-4

88-5—89-6

19 1-582'. -94324

23-7

89-5—90-5

10 1-8739 -96953

13-8

90-5—91-5

8

2-1653

•98482

7'4

91-5— !>.'-.'.

3 2-4.r.<i7 -99299

3-6

92-5—93-5

1-5 2-7481 -99700

1-6

9S-5—94-5

2 3-0395

•99882

•7

94-5— [>:.-.-,

1-5 3-3309 '99957

•3

95-5—96-5

3-6223 -99!tx-,

Over 95 -5 -1

96-5—97-5

3-9137

•99995

—

97-5—98-5

1

4-2050

•99999

—

Totals ...

900

—

—

900-2

TABLE IV (p. 11)

Extension of the Table of the Probability Integral F-^(\-a). (Calculated
by Julia Bell, M.A., Drapers' Research Memoirs, Biometric Series, vm, p. 27.)

It has been found needful occasionally to determine probabilities for deviations
exceeding considerably the limit x\a = 6 of Sheppard's Table II.

Illustration. If ac/a = 34 31, determine to two significant figures the probability
of a deviation occurring as large or larger than this.

The table gives us :

33
34
35
36

238-39135
252-95315
267-94888
283-37855

14-56180
14-99573
15-42967

•43393
•43394

\\ii Tables for Slut !*t i<-;<i a* and Binimtrii-litii* (T — VI

Hence using formula (ii) p. xiii :
(- log F)= 252 95315 + -31 [14-9957:}]

•31 x -9039 -69 x -5239

- - x -43393 - x -43394

o o

= 252-953151

+ 4-648G8

>• = 257-55542.

- -04G41
Hence log F = - 257'55542 = 258-44458,

.F=2-7834/10«",
which measures the improbability required.

TABLE V (pp. 12—18) AND TABLE VI (p. 18)

Probable Errors of Means, Standard Deviations and Coefficients of Variation.
(Table V calculated by Winifred Gibson, B.Sc.; Table VI by Dr Raymond Pearl
and T. Blakeman, M.A. Biometrika, Vol. iv. pp. 385—393.)

If m be a mean, a a standard deviation and F=100<r/w a coefficient of
variation, for a population of n, we have

Probable Error of Mean

•-6744898o/Vn-K<r ...................... (xi),

Probable Error of Standard Deviation

= -G744898o-/V2n = ^a ........................... (xii),

Probable Error of the Coefficient of Variation

= -6744898 Fxl + 2 j ^ .................. («"),

= -6744898/^271 x ^

= X«X ^ ................................................... (xiv>-

Table V gives ^ and £, for each value of n up to 1000, Table VI gives -ty for
each value of V proceeding by units from 0 to 50.

When the frequency n is greater than 1000, the tables may still be used by
taking out a square factor, which can be divided out at sight.

Illustration (i). n = 2834 = 4 x 70S-5.

n = 708, x, = -02.530 ; H = 709, x. = '02533.
.-. w = 708-5, X, = -02534, and .-. for n = 2834,
we have x, = '01267.

Illustration (ii). In the case of the 900 Bavarian crania of the Illustration (iii)
to Table II the values

m = 83-069, a = 3-482,

V — VIII] Introduction xxiii

and therefore F=4'1315 were found. It is required to find the probable errors
of these values.

For 900, £! = "02248 and ^2= '01590, hence the probable errors of m and o- are

p.e. of m = ftcr = '077,

p.e. of <r = X,<T = '055.
Next for F =4-1315,

^, = 4-00639 + -1315 [1-00609] - £ (-1315) (-8685) x [299]
= 4-13852.

p.e. of V= x-2 x 1^= '01590 x 4-13852

= -0658.
Hence our results should be recorded as

m = 83-069 ±-07 7,
a = 3-432 ± -055,
F= 4-1315 + -0658.

TABLE VII (p. 19)

Abac for Probable Errors of r. (Calculated by Dr David Heron, drawn by
H. Gertrude Jones, Biometrika, Vol. vn. p. 411.)

The probable error of a coefficient of correlation

To ascertain the value of this function approximately, turn the page horizontal,
enter with the proper frequency on the scale at base, follow the corresponding
vertical until the sloping line with the given correlation is reached, then move
along the horizontal to the left until the scale of probable error is reached, which
will give the required approximate probable error of the correlation in a population
of the given size.

Illustration. r = '671 and n = 415.

Probable error of r = '018. The actual value is -0182.

TABLE VIII (pp. 20—21)

Values of 1 - r*. (Calculated by H. E. Soper, M.A.)
Illustration. As in the last example let

r = -671 and n = 415.
Then probable error of r = ^, x (1 — r3)

= -549,759 (from Table VIII)

x -03311 (from Table V)

= -0182.

The value of 1 — r" for r to four instead of to three figures can be obtained by
interpolation.

xxiv

Tnlilta fur S/,iffxfii-fnnfi mill /if

[IX

TABI.K IX (pp. 22—23)

Values of the Incomplete Normal Moment Functions. (Calculated by Dr Alice
Lee, Biometrika, Vol. vi. p. 59.)

The nth incomplete normal moment function is defined to be

We take

»IB (x) = fin(x)/{(n — I ) (n — 3) (n - 5) ... 11 if n be even)

)• (xvi),

= ,*„(*)/{(« -l)(n-3)(n- 5). ..2} if n be odd J

and mn (.'•) is the function tabled.

In multiple correlation (supposed normal), the frequency surface is

X -ix*

z~ 1

x • (ivii),

where X* = I? I (Rpp^p'/Gp") ~^~ 2<S ^Rpy-l'p^g/O'pO'q)}

and Ji= 1, r,,, ru...rln (xviii),

'•„,, 'V., '«... 1
while Rpp and 7?OT are the usual minors.

%* = constant is the "ellipsoid" of equal frequency in n-dimensional space.
The total frequency, i.e. the volume of the surface, inside any ellipsoid % is

7^= f*zdV
Jo

J r Iff _ \2Tpn-l(X) 'f ] ,}

*' = 2 4. JR ... (n-2}

.(xix).

if « be odd

Thus a knowledge of the incomplete normal moment functions enables us
to predict for multiple variables whether an outlying observation consisting of
a system of n variate values is or is not reasonably probable.

If IXJN=%, we obtain the 'ellipsoidal' contour ^0 within which half the
frequency lies. This x<> 's fche "generalised probable error" of Pearson and Lee.

IX— X]

Introduction

xxv

Values of the "generalised probable error" coefficients are given in Table X
for w=l to 11, and by means of a smooth curve the results may probably be
extended to w = 15. The values found for this extension are :

n = 12

n = 13

« = 14

n = 15

Xo

3-367

3-513

3-654

3-791

Illustration (i). Let us consider long bone data for Frenchmen. 1 = F= femur,
'2 = H = humerus, 3 = T = tibia, 4 = R — radius*, then by formula (xviii) p. xxiv :

R =

1, '84.21, -8058, -7439

•8421, 1, -8G01, -8451

•8058, -8601, 1, -7804

•7439, -8451, '7804, 1

Further in cms :

m, = 45-23, o-, = 2'372,
TO, = 33-01, 0-,= 1-538,
m, = 36-81, a, = 1'799,
TO4= 24-39, o-4=ri70.
What is the chance that the following individual may be considered French ?

/" = 36-97, H' = 26-82, r = 30'56, R' = 20'68.
The deviations in terms of their standard deviations are :

a:, = (F' - ml)lff1 = - 3'482, a;, = (//' - m^/ov, = - 4'059,
x, = (T - TO,)/<7, = - 3-474, xt = (R' - «i4)/o-4 = - 3'171.
Further :

% = 3-7810, % = 6-5496, ^ = 4-3406, % = 3'6508,
K H -K ft

%= 2-0231, ^ = 1-1404, R£ = 0-2130, §= 2-1946,

K Jl ti fi

^ = 2-3175, ^ = 0-6842.
H H

Whence

n is even, hence :

16-741,035 and x = 4'0916,

' For particulars of these length measurements the reader must consult R. S. Proc. Vol. 61,
pp. 343 et teq. and Phil. Tram. Vol. 192, A, p. 180.

B. d

\.\\l

for Xfn/isfiriiiiix it in/

[IX — X

and from the Table, p. 22, we have by formula (i), p. xiii :

m,(4-0916) = -397,7378 + -916[3650] - $ (-91(i) (-084) [1043]

= -398,0682.
Hence IJN = \/27r x "398,0682 = -9978.

Thus the odds are 9978 to 22, say 454 to 1 against a deviation-complex as
great as or greater than this occurring in a French male skeleton, i.e. the bones very
improbably were those of a Frenchman. Actually they were those of a male of
the Aino race.

Illustration (ii). The following are the ordinates of a frequency distribution
for the speed of American trotting horses*. It is assumed that they form a
truncated normal curve, and we require to determine (i) the mean of the whole
population, (ii) its standard deviation, and (iii) what fraction the ' tail ' is of the
whole population.

The values of frequency in an arbitrary scale are :

Seconds

Frequency

Seconds

Frequency

89— 88

92-8

80—19

45-8

28—27

l(K)-4

19—18

38-4

87—86

95-0

18—17

87-a

86—85

71-2

17—16

19-8

85—84

67-6

16—15

10-7

24— ss

61-3

15—14

15-8

:•• :.'

61-4

14—13

7-9

88—81

44-8

13—18

5-0

81—80

44-5

18—11

2-1

11—10

5-6

Taking the working origin at 20 — 19 seconds, we find
i/,' = -3-9214, v,' = 32-545,666

for raw moment coefficients. Hence, if d be the distance from 29 seconds, i.e. the
stump of the tail from the mean, and S the standard deviation of the tail about its
mean:

d = 9-5 - 3-9214 = 5-5786 sees.,

2» = Vf> _ Vi'» = 17-168,288,
and accordingly 2*/dJ = '55l7.

If this value be compared with those for i/r, in Table XI, p. 25, it will be seen
that we have got slightly more than the half of a normal curve, i.e. not a true
tail. We cannot therefore use Table XI, but must fall back on Table IX.

* Gallon, R. S. Proc. Vol. 62, p. 310. See for another method of fitting, Pearson, Jiiomelrika,
Vol. n. p. 3.

IX — X]

Introduction

xxv

Let x be the distance from stump to centre of curve, n equal the area of
truncated portion, and N be whole population. Then

n/N=T + I1'' ^=e-*x'*dx' = $ + m0(x/<r) ............... (xx);

J» JO V27T

I r<*> ro nJ i :•) 'l
nx^NaU + -f=e-**daf\,
(Jo }-&•/%* )

( 1 f" X' -il'2 , ,)

= Na \ -=e • dx'\ ,

(V27T JO V27T )

= No- \-== - TO, («/«•)[ ................................. (xxi);

IV27T

,v...^ir+r £..-»•*[

I Jo J-zlvVZTr

{fz/<r T'i , ,, 1

i+f -J=e-^2cfa'
Jo V27T j

= ^<7S {i + wi, (^/tr)} ....................................... (xxii).

Now d = x + x, and 22 = /*/ — S*.

1 a;

rf \/i^ ~ "^ ^*^ + ff ^ + m° ^*^

Hence -= — - - — - — — - ........................... (xxiii),

a- $ + f«o (xja)

s, {i + m, (*/<r)) (i + m0 (*/«•)) - J^L - m, (*/«r)J

^= {i + ^^M}* ...... (XX1V)>

2, g+OTt)(H>».)--^

and =

say, for brevity.

Here m, and m,, are given by Table IX and J + m» is the £ + J^x of Table II.

Formula (xxv) has not yet been tabled for different values of x, as it occurs
much more rarely than the corresponding function for a true tail.

If we take three values a! = 0, O'l and 0'2, we have, from Tables II and IX,

x = 0, i + m, = -500,0000, \ + m, = -500,0000,

«'=•!, „ =-539,8278, „ ='500,1325,
x ='2, „ =-579,2597, „ ='501,0512,
Whence from formula (xxv) for the three values of x

'5528 and

V2?r

, = -398,9423,

='396,9526,
='391,0427.

\\\iii Tables for Sfafixtiriiiii* <m<l />' nn»< Irii-iniis [IX — XI

But our value of ^/d* is '5517. Thus we find by interpolation

*' = -10GO.
It remains to determine m,, nt, and m, for this value of #', or simpler £ + »i,j,

A + MJ- and -== — »i, from the above values for #' = '1 and a;' = '2. We timl

V2TT

i + m, = -542,194, ^ + m.i = '500,184, -^= - ml = '396,598.

V27T

Whence d/<r = + 'lOfiO = '8375.

•£>42,194

Thus <7 = 5-5786/('8375) = 6'6610 sees.,

* = a;'x<j-=-7061 sees.,

This gives a mean of 28'29 sees, with a variability measured by 6'6G sees.
Those actually registered as trotters are 54 °/0 of the population.

TABLE X (p. 24)
See uuder Table IX, p. xxv.

TABLE XI (p. 25)

Constants of Normal Curve from Moments of Tail about Stump. (Pearson and
Lee, Biometrika; Vol. vi. pp. 65 and 68.)

This Table may be of service in cases of the following kind :

(a) In some cases a record is actually truncated as in the case of the
American Trotters dealt with on p. xxvi. Or, again, we may take a record of
stature obtained by measuring all men who exceed 69", or a record of mental
capacity found by measuring all persons with low intelligence in a community.

(b) When we recognise heterogeneity, e.g. when we have a mixture of male
and female bones, or two strains of trypanosomes, we can occasionally get a rough
approximate analysis by supposing the tails to represent homogeneous material
and then fitting them with normal curves. From the tails we get two components,
and if their compound agrees fairly well with the observed total we have
performed an analysis far more rapidly than by using the nonic equation*.
Difficulties arise owing to deviations from Gaussian frequency being not infrequent;
different dichotomic lines may give different results, and owing to paucity of
material in the 'tails' and corresponding irregularity there will be large probable
errors.

Cases under (a) and (b) will be treated by exactly the same process, but our
Table supposes that the distribution considered is less than half the normal distribu-
tion. The rules for determining the mean, standard deviation and total frequency
of the untruncated population are given on p. 25 under Table XI itself.

* Pearson, Phil. Tram. Vol. 185, A, p. 84.

XI]

Introduction

xxix

Illustration (i). The following distribution represents the 'tail' of a group of
301 mentally defective children measured by G. Jaederholm using Binnet-Simon
test methods* :

Mental Defect in Years.

|

-A.
•
**t*

5-

|

Co

|

|

J

]

i

1

1

1

1

l

1

"3

i

i

i

i

i

4-

g

1

QO

|g

"*t"

-^.

IQ

*O

i

*O

\

i

1

i

1

1

1

I

Frequency ...

34

18

13

3

4

4

1

1

78

We find, with origin at — 3'45,

d = 1-8077 in J years and 2* = 27258.
Hence ^•I = 2V<? = '834.

The Table (p. 25) gives us h' = 2'186 and ^ = 2-833. Hence the mean is at
distance from — 3'45 on left= 2'186 x IT, and for the standard deviation

<r = ^sxd = 2-833x1-8077 = 5-1212 £ years = 2'5606 years,
whence mean is at distance h = 5'597 years from — 3'45.

Now h' = 2-186 corresponds by Table II to$(I+a) = '98559.
.'. n/N= -01441, or N= 78/('01441) = 5413.

Thus the distribution is the tail of a population of 5413 individuals with
a mean at 2'15 years of mental excess, and a standard deviation of 2'56 years.

The example is merely illustrative and of no importance in itself.

Illustration (ii). Assuming that the correlation data follow the normal law,
determine indirectly the 'plural' partial correlation f of habits of mother and
health of baby for the limited universe of unclean homes.

The correlations between the various characters found by tetrachorie tables are
Habits of mother and health of baby: rla = "3060,

Habits of mother and cleanliness of home : rls = '7958,
Health of baby and cleanliness of home : rw = "2578.
There are 947 out of 2931 homes which are not clean. Data from Bradford.
Here n/N =947/2931 = '3231 = J (1 -a) ;

hence a = '3538, and by Table III

x' = x/<r = -459,049.

* Mrndelitm and the Problem of Mental Defect. 11. On the Continuity of Mental Defect. Dulau&Co.,
37, Soho Square, W.

t Biometrika, Vol. ix. p. 289.

\\\ Table* for Statistician* tun/ Ilinun-trii-iaiix \ I

Now on the assumption of a normal distribution :

^'..r* * .-N-

Jx V27TO-

-No* I" a»g

J*

Here Table IX (p. 22) shows that if s be odd, w,(oo ) = '398,9423, i.e.
:in.l (p. 23) if s be even, m, (oo ) = '500,0000.

Hence in obtaining the moment coefficients of the tail, about the mean of the
whole population, m, (x') should be subtracted from '398,9423 or from '500,0000
before the results are multiplied by (a —!)(*— 3) ... 2 or (s — l)(s — 3) ... 1, when
s is odd or even respectively. It is convenient to term p, (oo ) — /*»(X) the com-
plementary incomplete moment function of order a*.

For s=l and a = 2, we have

ji/t,' = ffN {TO, (oo ) - TO, (*')},
«/<,' = <r*N {m, (oo ) - TO* (a;')},

for in this case the multiplying factors to proceed from mt(x') to p,(x') are both
unity.

Now x1 = xj<7 can be found when n is known from Tables II or III. Hence we
have for the distance of centroid of tail from its stump, and for the square of its
standard-deviation about its centroid :

d = /*,' - h'o- = a- P^ {m, (oo ) - wi, (a;1)} - A'l

Of course m, (ao ) — n^x') is the z of Sheppard's Tables II and III.
Returning to our numerical example, we have from Table IX (p. 22) :
(•45905) = '030,6721 + '5905 [162049] - 1 ('5905) ('4095) [26358]

= -039,9222,

TO, (oo ) - TO, (-45905) = '359,02.

Found directly from Sheppard's Tables, it equals '35905.
Similarly from Table IX (p. 23) :

m, ('45905) = -008,1136 + -5905 [73,162] - }(-5905) ('4095) [30661]

= -012,0630,
and TO, (oo)-m, (-45905) = -487,9370.

* It is the function used by Dr Alice Lee and myself, Biometrika, Vol. vi. p. 65 to form Table XI,
P. 2fi, but by an oversight not adequately distinguished in symbol from n, (x') of p. 60 of the same memoir.

XI — XII] Introduction xxxi

We thus reach d = a ('35902/-3231 - -45905)

= -652120-,

and S2 = a" {'48794/-3231 - (M1117)2}

= •274,786<r'.

Or the distance of centroid from stump, and the standard deviation of the tail are
respectively

d = -652120- and 2 = '52426(7.

We now find X = Vl - S'/o" = '85155.

The formula for the ' plural ' correlation coefficient is*

where r,,' = Xr,,, r.,' = Xr^,

= -2195, = -G777.

Thus 3r12 = -2191.

Actually taking out the universe of not clean homes, the correlation of habits of
mother and health of child is '1615. The difference is considerable, but ,fu is
deduced from the entire population of 2931 homes, while '1615 depends on only a
third of this number. The 'singular' partial correlation is 3r,.j=-l723, i.e. the
average relation between habits of mother and health of child for each individual
grade of cleanliness of home.

TABLE XII (p. 26)

Tables for testing Goodness of Fit. (W. Palin Elderton, Biometrika, Vol. I.
pp. 155—163.)

The theory of testing frequency distributions for goodness of fit was first given
by Pearson f and may be summed up as follows:

If a frequency distribution or table contains ri ' cells ' and the contents of these
cells be m,', m?', »»,' ... m^ in number, while m,, m^, m3 ... mn be the numbers that
would occur in these cells on any theory ; then calculate

/square of difference of theoretical and observed frequencies^ . ....
= sum I —i— . ..(xxvm),

\ theoretical frequency /

and the probability that random sampling would lead to as large or larger deviation
between theory and observation is

if n be even

............ (xxix).

• Pearson, Phil. Tram. Vol. 200, A, p. 25. + Phil. Mag. Vol. L. pp. 157—175, 1900.

\ \ \ i i

/',////, x _/;//• Stttffxfiriiiits <i ml

Ml

Short proxisional tallies of /' \\viv gixen in tin- article referred to and wen-
replaced in the following year by the piv.M-nt. standard tables of Palin Elderton.

In using the test for goodness of fit, due regard should be paid to the
conditions under which it is deduced. It is assumed that the frequencies form
a normal system of variates. This is legitimate only when in the binomial (jo-f q)n,
<l is not very small as compared with j>. If </ be not very .-mall as compared with p,
even for n finite, the binomial approaches closely to the normal curve. Accordingly
in using the test it is desirable to club together small frequencies at the tails of
curves or margins of surfaces. The difficulty becomes very obvious when theory
can go by fractions, but observations only by units.

The theory can be extended to cover much ground in all sorts of sampling*.

Illustration. The following data for observed frequencies of cephalic index in
Bavarian crania and for corresponding frequencies of a fitted Gaussian curve have
already been considered on p. xx. Test the goodness of fit

Observed

Gaussian

(m'-m)'

(«')

(m)

m' - m

m

Under 7:,-:.

9-5

12-4

- 2-9

•68

•-, ;<:•-,

12-5

12-7

- 0-2

•00

7H-5 — 77 '5

17

22-1

- 5-1

1-18

77-5—78-5

37

35-3

+ 17

•08

78-5—79-5

55

51-9

+ 3-1

•19

79-5—80-5

7 1 •:.

70-1

+ 1-4

•03

80-5— -81 :',

SL>

87-0

- 5-0

•29

81-5— 8." ,'•

116

99-4

+ 16-6

W7

8S-5—83-6

98

104-2 - 6-2

•87

83-5—84-5

107 100-5 + <;•:>

•42

84-5— 8', -5 82 89-1

- 7-1

•57

85-5— »;-.-, 74

72-6

+ 1-4

•03

86-5—87-5

58

54-3

+ 3-7

•25

87-5—^

84-6

37-4

- 2-9

•22

88-5— #.>•-. 19

23-7

- 4-7

•89

89-5—90-5

10

13-8

- 3-8

1-05

90-5—91-5

8

7-4

+ 0-6

•05

Over 91 -5

9

6-3

+ 2-7

1-16

Totals ...

900

900-2

18 Groups

*»= 10-27

* A word of caution must be given about a recent extension by Slutsky (see Journal of Royal
Rtiitiitical Society, Vol. i.xxvn. pp. 78—84) who has applied it to test the goodness of fit of regression
curves. In such cases the means and standard deviations of each array should, I think, be deduced
from the theoretical surface, and the method would then agree with that illustrated on pp. xxiv — xxvi,
i.e. on the probability of a given complex of variates differing from the run of values of a given
population significantly. Slutnky after assuming that the observed frequencies and standard deviations
of the arrays may replace the theoretical values, deduces his 7' from Elderton's Tables instead of from
the incomplete normal moment tablet). He finds for the fit of a straight regression line, used to predict
the probable price of rye at Samara from the price a month previously, x2 = 2'2'2, giving P=-02, a bad
fit. Had he, however, used the theoretical standard-deviation of an array, i.e. <rv'l -r', instead of the
very irregular observed standard deviations of individual arrays, he would have found x" = H-H4, leading
to /' = -(i4 an excellent tit, which would probably have been still further improved by the use of
theoretical total frequencies for the arrays, based, say, on a Gaussian distribution.

XIII — XVI] Introduction xxxiii

Taking from the column for n = 18 (p. 27) the values for x"=10 and 11, we
interpolate P = '891 for x" = 10'27 by first differences, and conclude that in 89 out
of 100 trials we should get in random sampling a fit as bad or worse than that
observed, if the real distribution were Gaussian. Accordingly we say that a
Gaussian curve describes excellently the distribution of Bavarian cephalic indices.

TABLES XIII— XVI (pp. 29—30)

Auxiliary Tables provided by W. Palin Elderton (Biometrika, Vol. I. pp. 162 —
163), useful for calculating values of P for •£ outside the range of the existing table.

For such cases we must turn back to the fundamental formulae (xxix) of p. xxxi,
and the numerical values of considerable portions of these formulae will be found
evaluated in these auxiliary tables.

Illustration. Find P for n'= 11 and X3 = 78, |x2 = 39, hence by formula
we have

= e-» (40 + 760-5 + 9886 5 + 96393-375)
= e-*> x 107080-375,

where the powers of 39 are taken out of Table XXVII (p. 38).
Hence using Table XIII,

log P = 17-0625,1520 + 5-0297,0988 = f2-0922,2508,
which gives us P = 1-23659/10"1.

As a rule we can select n to be odd, but, if it is necessarily even, there is more
trouble, not in the determination of the series, but in the evaluation of the
integral

A table of the values of F=%I for ^ = 5 to 500 has been given as Table IV
(p. 11). This gives x'=25 to 250000 but the intervals are large.
If greater accuracy be required then Schlomilch's formula*

__ _ ^ __

+ 2) (X3 + *) (X1 + 6) X' (*' + 2) (x3 + 4) (*' + 6) (jf + 8)

129

WTWTIo)+"-} (xxx)

X'(X' + 2)(X1 + 4) (x3 +
must be used.

Here */— ve~^ will be found in Table XIII, and the series converges fairly

V 7T

rapidly.

" ( 'iniiiifHiliiiiii dsr liiiliertn Analytit, Bd. n. 8. 270, Braunschweig, 1879.
I!. «

\ \ \ I \

for Stnllsl iriims it ml /iiimii'tn'i-iiiiix XVII

TABLE XVII (,, :ih

\'n I ties of (—/(«//' i <-<irrc»/>iiiidiny to given values of •%* in a fourfold l<ilil>:
(K. Pearson: On a Novel Method of regarding the Association of two Van
classed solely in Alternate Categories. Drapers' (\niijmny Research Memoirs,
Biometric Series, vm. Dulau & Co.)

If individuals be classed by the characters into -1 :md not-^1, B and not-.fi, we
form a tetrachoric table of the form

A

Not-J

Totals

B ...
N,,t -/;

a

e

b
d

a + 6
c+d

Totals

a+c

b + d

.V

For such a table :

N(ab-cdY

(a + b)(c+d)(b

.(xxxi),

gives a measure of the probability of independence, and, if the two attributes are
highly associated, •£ will De large and P the probability of independence very
small and largely outside Palin Elderton's Table XII. Table XVII provides for
such cases.

Illustrations. The following tables are given by Mr G. U. Yule in his Theory
of Statistics*. His conclusions with regard to them arc:

1. Datura : " No Association."

2. Eye Colour in Father and Son : " Shows the tendency to resemblance."

.3. Houses in course of erection, Urban and Rural : " Distinct Positive
Association."

4. Imbecility and Deaf-Mutism : " High Degree of Association."

5. Developmental Defects and Dullness : " Very high indeed."

It is required to measure the degree of probability that the variates in these
five cases are independent.

(1) Datura. (2) Eye-Colour in Father and Son.

Colour of Flower. Father.

Violet

White

Totals

— '

rrickly ...
Smooth...

47
12

21

I

68
15

Totals ...

59

24

83

Light

Not Light

Totals

§

Light
N"t, Light ...

471
151

148
230

(519
Ul

Totals ...

«22

378

1000

* Pp. 37, 34, 62, 33, 84 and 45 respectively.

XVII]

(3) Houses in course of erection in
Urban and Rural Districts.

Introduction xxxv

(4) Imbecility and Deaf-Mutism.

Built

Building

Totals

Urban .
Rural ...

4960
1749

50
12

5010
1761

Totals ...

6709

n

6771

Imbecile

Non-Imbecile

Totals ...

Deaf Mute
Non-Deaf Mute . . .

451
48,431

14,795
32,464,323

15,246
32,512,754

Totals

48,882

32,479,118

32,528,000

(5) Developmental Defects and Dullness.

With Defects

Without

Totals

Dull
Not-Dull ...

888
1420

1186
22,793

2074
24,213

Totals

2308

23,979

26,287

<£" the menn square contingency = \*JN and <f> is the product-moment coefficient
on the assumption that the 'presence of the character' is to be considered as
a concrete unit*. The coefficient of mean square contingency C'» = V</>2/(! + $2).

The following table gives the values of %*, <£-, <f> and Cit and the values of P
deduced.

x'

**

*

C3

P

(1) Datum -7080

•085,301

•2921

•2803

•8713

(2) Eye-Colour 133-3265 -133,327

•3651

•3430

1 -035/1028

(3) Houses 1-4393 -000,2125

•0146 '0146

•6948

(4) Imbecility and Deaf-Mutism 8014'(i2 -000,2464

•0157

•0157

3-179/101739

(5) Defects and Dullness ... 3256'797 -123,894

•:).-,!!>

•3320

2-846/10™

For example, from Table XVII by Formula (i) we have for %*= 133-3265:

(- ,og ,, , „*. + «£» [,0-770] - 5 (»«•) ("«»)

log P - 28'015,
P = 1 '035/1 Oa.

= 27-985,
and therefore

which leads to

Or again, for x'= 8014'62 :

- * I- .73,,.
= 1738-498,

* Pearson and Heron, Biometrika, Vol. ix. p. 107.

,000]

«9

xxxvi Tabltut Jni- Stalls! n-lmiK >im/ IHniiH-trlriinix I XVI I \\

and therefore log P = 1739 502,

and P = 3179/10"».

In the first and third cases a different treatment must lie ust-d. For x*=l'
we use Table XII.

We have for i»' — 4 :

P = -801253 + -4393 [- 228846] - J (-4393) (-5607) [+ 480G4]

= •6948.
Had we worked from Table XVII by Formula (i), we should have had

P=-6950.
For •£ = '7080, we can use Table XII, remembering that for %' = 0. f

We have

P = 1-000,000 + -708 [- 198,747] - } (-708) (-292) [- 30,099]

= -8624.

Had we worked from Table XVII by Formula (i), we should have had P =
close enough for practical purposes.

The true value of P worked from
P.

=2{ i f%-ix'd +JLe-i^ 1

lV2irJx * V27T

by using Table II is P = '8713. See p. xxxviii.

Examining the values of P we see that having regard to the errors of random
sampling we can only say that there is no relation between rural and urban
districts and houses building or built ; there is clearly no ' distinct association,'
for in 69 out of 100 cases in sampling from independent material we should get
more highly associated results. There is likewise no association on the given
material in the Datura characters. The other three cases have clearly very marked
association, quite independent of any influence of random sampling. If we regard
these three tables the order of ascending association judged by either <f> or Ct is
(4), (5), (2), as against Mr Yule's (2), (4), (5). If we disregard the non-significance
and take merely intensity of association, without regard to random sampling, the
order is (3), (4), (1), (5), (2), as against Mr Yule's order (1), (2), (3), (4), (5).

The best method of inquiry at present for relative association in the case of
four-fold tables is, I hold, first to investigate P and throw out as not associated
those cases like the 'Houses, built and building" above. Then to use either
" tetrachoric rt " or Ct according as we are justified in considering the variates
as continuous or not rp (see p. xxxvii) may be used as control.

TABLES XVIII— XX (pp. 31—32)

Tables for deterniinin;/ t/ie. Hijiiiprobable Tetraclioric Correlation rr. (Pearson
and Bell : On a Novel Method of regarding the Association of two Variates classed

XVIII— XX]

Introduction

xxxvn

solely in Alternate Categories. Drapers' Company Research Memoirs, Biometric
Series, vm. Dulau & Co.)

We have seen under the discussion of the previous Table how to find a measure
of the improbability of two variates being independent, when they are classed in
alternate categories. The difficulty in such cases is to appreciate the relative
importance of very large inverse powers of 10. The object of the present tables is
to enable us to deduce a tetrachoric correlation, rt, of which the improbability
is the same as that of the given system supposing it to arise, when the two
variates have the same marginal frequencies but are really independent. In order
to do this we have to determine Oo> for the given marginal frequencies, i.e. the
standard deviation of rt on the assumption that r is really zero. This may be easily
found from Abac Diagram XXI or from Table XXIV (see below). Table XVIII
then gives us the value of (- logP) for each value of rt and Oo>. If we now turn
to our original table and calculate its •)£, this as we have seen will correspond
to a given (— log P). We now make the (— log P) from our ^2 correspond to the
(— logP) from our rt and (,ov, this gives us a value of rt which has the same degree
of improbability as our observed table. In other words, instead of trying to
appreciate the meaning of inverse high powers of 10, we say that a table of the
same marginal frequency would be as improbable if it had a tetrachoric correlation
rt arising from random sampling of independent variates. Thus we read our
improbability on a scale of tetrachoric correlation. We use our correlation merely
as a scale to measure probability on.

As log^5 provides a more satisfactory basis for interpolation, and as many
readers use logarithm tables and not calculators, log^2 will be the form in which
•)£ will be often presented. Table XX provides the value of rt corresponding to
given 0<7r and given log ^'-'.

We will assume for the present that Oov can be readily found from the marginal
totals : see p. xli below.

Illustration. Obtain the values of rf for the five tables given above on
pp. xxxiv — v.

The values of log ^* and H<rr are as follows :

log X-

o°v

(1)

Datum

1-8500

•1941

(2)
(3)

Ky< '-Colour, Father and Sou
Houses in Course of Erection

2-1249

•1582

•0514
•0634

(4)

Imbecility and Deaf-Mutism

3-9039

•0175

(5)

Developmental Defect* and Dullness

3-5128

•0201

Of the values here recorded for log ^5 and Oa-r, those for the first and third
cases lie beyond the limits of our Table XX. But if the point 00V='0634 and
log x2 = -1582 be noted on the Abac XXII, p. 34, it will be seen that, having due

\\.\viii Tulilts j'ur Stittixticiiiiix mill liiuii<ttri<-i<inx [XVIII — X X

regard tu the spacing* <>t' tin- OOnehttlOB curves, the value of the equi probable
correlation is under '03, say "027. In other words no significant association can be
asserted.

In the case of 0<rr = -1941 we are thrown back on the original formulae*. In
the first place we must find P for the given value of y;-, i.e. -7080 (see p. xxxv).
But for n' = 4 from formula (xxix),

= 2 (-200,0578 + -280,0088 x -841 42)
= •871,3256.
To obtain r we have to use the formula below, where 0<rr=-1941, and

/«= | ( , — 3) , the fa, fj,4, /i, being the normal moment functions of Table IX.
2

128m'

............... (xxxii).

Substituting the values of 0<rr = '1941 and V2m = 4-852,107, we have for

,- = •03, P= -90550,

r = -04, P = -86501.
Whence for P = -87133, we have r = '038.

We now turn to the three cases which fall inside Table XX.

('!) Eye-colour, Father and Son.

log X'= 2-1249 ,,ff, = -0514,

r = 0-5 log v' = 2-0942

_ .QS * /V

r = 06 log •£ = 2-2748

;- = 0'6 log X2= 2-1 239

r = 0-7 log xs =2-293.",.
Linear differences will suffice

rr = -05 r = 0-5 + [1] = 0-517,

l = 0-001.

Hence Oo-, = '0514 giv.

= •517 + -01 2 = '529.

• Drapert' Company Jtetearch ilemoin. Diametric Series VIL " A Novel Method," etc. : Me
pp. 12, 13.

XVIII — XX] Introduction

Interpolating for a<rr first,

xxxix

log %- = 2-0737,
,- = •6 Oov=-0514 log X" = 2-2537.
Hence for log ^2 = 2'124'J :

We conclude tliat the equiprobable correlation is '53.

(4) Imbecility and Deaf-mutism.

log x* = 3-9039 0<r,. = '0175,

r = 0-95, Oov = -01, log x2 = 4'3673 ; Oov = '02, log %> = 3'7660.
Hence: r = Q-<J5, ,<rr = '0175, Iogx3 = 39163.

Again :

r = 0 90, Oar = -01 , log x2 = 4-2207 ; 0<7r = "02, log x2 = 3'6197.
Hence: r = O'iM), 0<rr = '0175, log X3 = 37699.

Interpolating log x- = 3'9039 between 3 9163 and 3 7699, we find

rp = 0-946.

(5) Developmental Defects and Dullness.

log x* = 3-5 1 28, u<rr = -0201 .

r = 0-8, 0<7, = -02, log xs = 3'4097 ; 0<rr = '03, log X3 = 3'0598.
Hence : Oo-,. = -0201, log x2 = 3'4062.

r = 0-!», 0<7r=-02, log x== 3-6197; Oo-r = -03, log X5 = 3'2690.
Hence : log x-' = 3'6162, for Oo-r = '0201.
Thus, by interpolating log x" = 3-5128 between 3'4062 and 3'6162, we find

rf = -851.
We have accordingly the following results :

Oi

P

ft

n

«

(1) Datura

•urn

•8713

•038

-•188 + -140

-•282

(2) Eye-Colour

(3) Houses

•3430
•0146

1 -035/1028
•6948

•529
•027

•550+ -027
- -081 + -043

•581
-•190

(4) Imbecility and ])eaf-Muti>in

•0157

3-179/101"8

•946

•330 +'01 2

•907

(5) Defects and Dullness

•3320

2-846/10708

•851

•652+ '009

•846

It will be Been that equiprobable rp confirms generally the results from P, i.e.
the tables for ' Datura ' and ' Houses ' give no sensible association. rt also confirms
this view and shows that ' Houses ' is even lower in the scale than ' Datura.' The
order of rp is the same as that of Yule's coefficient of association Q, but neither
rp , rt, Cs, P or Q support the conclusions stated to flow from the percentages on

xl Tables for Sfallxtin'mi* <nnl l{!innt'tr'n-i<iiix \\\\\\ \\ IV

p. xx\i\. Until r,. anil (} i;i\e \>v\ hi^'h results (i>r ( 4) ;IIK| (o ), ami tins i- in a>-<-onl-
aiicc with the view elsewhere expressed lli.it I'm- r\tivnn- dichotomies (J is not to
be trusted. It may further he ili>nl>i>-<|, whether lor such dichotomies the theory
of the distribution of deviations on which ;-,. is based can in its turn be accept* d.
(In the whole /•, se,-m-> to mi- i he most .sit isl'ad 01 \ cot-tlien-iit of association, to be
controlled by results for r,. in the casrs win-re iK-itlier the tlichotomies are extreme,
nor the numbers so large or so small as to fall outside the moderate range of
Tables XVIII— XX or Abacs XXI and XXII.

AI-.A.S XXI AND XXII (pp. 33—34).
See after Tables XXIII and XXIV.

TAIIUSS XXIII AND XXIV

Tables fur determining approximately the probable error of a tetracliurir
correlation. (Pearson, Bwmctriku, Vol. IX. pp. 22 — 27. Tables calculated by
Julia Bell, M.A.)

Given a tetrachoric table

a

b

rt+1>

t

d

C + d

U + r

/,+,/

N

so arranged that a + c> b + <l and a \-b >c + d,

then if |(1 + a,) = (a + b)/N, i ( 1 + a,) = (o + c)/N,

and ?-, be the correlation, we have approximately :

Probable error of rt = Xi • Xrt • Xo, • Xo,-
where K = '67449/VF,

and is tabled in Table V, p. 12,

at — ff • a.,-

H and K being found from the z column of Table II, p. 2, and

.1,1-0,)

ll)>

-, snr
~^

sin~'r, being read in degrees. ^a and ^a are tabled in Ta'nle XXIV and \r in
Table XXIII (p. 35).

This value of the probable error is only approximate and may diverge con-
siderably from the true value* for extreme dichotomies. In such cases the full
formula must be used.
• I'liil. Traiu. Vol. 195, p. 14. Xn >» formula (I) should of course not be inclucii-.l un.i. r the radical.

XXIII— XXIVJ

Introduction

xli

When rt is zero in the population and not in the sample, the standard deviation
Oo> of r = 0 is given accurately by ^=xaiXa.2-

Illustration (i). Tetrachoric rt for the Table

22,793
1,186

23,979

1,420
888

2,308

24,213
2,074

26,287

is '652. Find approximately its probable error.
From fable XXIII :

r = -65, xr = '6785 ; r = '66, Xr = "6675.
. -. Xr = -6785 - -01 10 x -2 = -6763.

Now £(l+o,) = '9211, J(l+«s) = -9122.

Hence from Table XXIII,

Xai = 1 8249 + -11 [754] = 1-8332,
Xa, = 1-7623 + -22 [626] = 1-7761,
Xa,Xa, = 3-2559.

X\ cannot be found from Table V in this case as ^V is beyond its range. But it
equals

•67449/V26287 = -67449/162-13 = -00416.

Thus finally p.e. of rt = '00416 x -6763 x 3'2559

= •009.

Illustration (ii). Find the value of Oo-r for the table :

471
151

622

148
230

619
381

378 1000

Here

£(1+0,) = '619 and $(1 +aa) = '622.
Xa, = 1-2712 + -9 [36] = 1-2744,
Xai = 1-2748 + -2 [39] = 1 "2756.
0^ = ^,^/^1000 = '0514.
In a similar manner the values for all the Oo-r's in the table on p. xxxvii were

found.
B.

xlii 7'nl>li* /;//• Statistician* <i/t</ /ii'mnfricinns [XXI. \.\II

ABAC XXI (p. 33)

For determination of the standard-deviation of the correlation coefficients obt<
by random samjding from a four-fold table in which the correlation is zero.
(Drapers' Company Research Memoirs, Biometric Series, vill. G. H. Soper's Abac.)

Method of use: Enter with the total frequency of the sample on the left-hand
scale, and with the first value of £(l + a) on the bottom scale. The horizontal
through the former and the vertical through the latter meet at a point. At this
point pass up the diagonal to the left-hand scale again. Where you meet that
scale pass along the horizontal until you meet the vertical through the second
value of 4 (1 +o). Then from this point pass along the diagonal again to the loft-
hand scale, whence traverse the horizontal to the right-hand scale and there the
required value of 0<rr may be read off.

Illustration (i). Find the value of Oo> for the case just given of

#=1000, 4(1+ a,) = -619, i (! + *)» -622.

The vertical through '619 meets the 1000 horizontal in a point whose diagonal
reaches the left-hand scale almost exactly in 620. Whence passing horizontally
we reach the vertical through '622 in a point about midway between two diagonal
lines. Passing up midway between these two diagonals, we reach almost exactly
the 380 line on the left-hand scale. Passing across to the right-hand scale along
this line, we see that we are slightly above the middle of the division between
•050 and '052, say -0512. The actual value of Oo> is "0514.

Illustration (ii). Let

#=•6771, $ (! + «,) = '7399, £ (1 + «2) = -9908.

A similar process gives first 450 ou left-hand scale and then about 248, whence
crossing to right-hand scale we find Oo> = '0635 instead of '0634 actual.

ABAC XXII (p. 34)

Abac to determine from log ^a and Ofr the value of the equiprobable correlation
rp,for a fourfold table. (Drapers' Company Research Memoirs, Biometric Series,
vin. G. H. Soper's Abac.)

The rule is very simple : Enter the Abac with the proper value of Oo> on the
scale at the foot and rise on the vertical till the horizontal through the proper
value of log ^* on the left-hand scale is reached. Then follow the curve through
the meet of these two lines to the right-hand scale, where the requisite correlation
will be found inscribed.

Illustration. Take the Table for Eye Colour in Father and Son given on
p. xxxiv. Here, as just shewn, Oo>='0514 and (p. xxxvii) logx'=2'1249. If we enter
with the vertical through '0514 ou the scale at the bottom, and the horizontal
through 2-1249 on the left-hand scale, the curve through their point of intersection
reaches the right-hand scale just below the '53 mark, say '529. This agrees with
the correlation found above (p. xxxviii) by interpolation from Table XX.

XXV]

Introduction

xliii

TABLE XXV (p. 36)

Value of the probability that the mean of a small sample of n, drawn at random
from a population following the normal law, will not exceed (in the algebraic sense)
the mean of that population by more than z times the standard deviation of the
sample. ("Student": Biometrika, Vol. vi. p. 19.)

When n is greater than 10, it will be sufficient as a rule to use the approximate
result

vr- 73 fr (n-3)x>
P= = e 2 dx (xxxv)

V27T J-oo

as a measure of the probability. This may be found from Table II.

Illustration (i). Experiments of A. R. Cushney and A. R. Peebles on the
difference in effect of Dextro-hyoscyamine hydrobromide and Laevo-hyoscyamine
hydrobromide*.

Additional Hours of

Patient

Sleep

(Laevo - Dextro)

1

+ 1-2

2

+ 2-4

3

+ 1-3

4

+ 1-3

5

0

6

+ 1-0

7

+ 1-8

8

+0-8

9

+ 4-6

10

+ 1-4

Mean

+ 1-58

Standard Deviation

1-17

Table XXV shows that for z = 135 :

P = -99854,

or the odds are 666 to 1 that leavo- is a better soporific than dextro-hyoscyamine
hydrobromide.

Illustration (ii). Difference in weight of crops of potatoes grown by Dr Voelcker
with (i) sulphate of potash and (ii) kainite as artificial manure.

* Journal of Phyiiology, 1904.

xliv

Tables for Statistician* ami Jii»titrtrician* [XXV

Oain by sulphate o

1904 (a) 10 cwt. 3 qr. 20 Ibs.

(b) 1 ton 10 cwt. 1 (jr. '26 Ibs.

1905 (a) 6 cwt 0 qr. 3 Ib.s.
(6) 13 cwt. 2 qr. 8 Ibs.

Average gain= 15-25 cwt., and the standard deviation = 9 cwt., z= 15'25/9 = 1 '694.
Here n = 4, and Table XXV gives us

P = -9653 + 0-94 x [46] = '9696,

or the odds are about 32 to 1 that the sulphate of potash is a better dressing than
kainite for potatoes.

Illustration (iii). Test whether it is of advantage to kiln-dry barley seed
before sowing. The following table gives price of head corn in shillings per
quarter for 11 sowings, the first seven in 1899 and the last four in 1900.

Not Kiln-dried

Kiln -dried

A

26-5

26-5

0

28

26-5

1-5

29-5

28-5

1

1899 -

30

29

1

27-5

27

0-5

26

26

0

29

26

3

29-5

28-5

1

28-5
30

28
29

0-5
1

28-5

28

0-5

Mean

•91

Standard Deviation

•79

The Gaussian curve gives

0- -91/79 -1-1519,
x' = x/JT/S = 1-1519 x 2'8284 = 3'258,

/-S'258 . „

Here
and if

which evaluated by Table II, p. 6, gives

P = -99944,
or the odds are 2845 to 1 in favour of not kiln-drying seed barley.

XXVI]

Introduction

xlv

If we had actually worked with the non-approximate formula, we should have
found

P = -9976,

or odds of 416 to 1, considerably less than the approximate formula provide, but
not enough difference to vitiate any conclusion likely to be drawn in practice*.

TABLE XXVI (p. 37)
Table for use in plotting Type III Curves, i.e.

X

-p^

.(xxxvi)

(W. P. Elderton, Biometrika, Vol. II. p. 270.)

Rule : Taking p for the curve, multiply the values in the Table by p in
succession on the machine with p on as multiplier. Then subtract the results from
the logarithm of y,, and we have the logarithms of the ordinates of the curve at
the abscissae found by multiplying X in the first column of the Table by a of the
curve. The curve can then be plotted. Its origin will be the mode. It is usually
quite unnecessary to use the whole series of ordinates, either alternate ordinates
will suffice, or we cut off one or both tails at a considerable distance from their
tabulated values.

Illustration. The frequency curve of barometric heights at Dunrobin Castle is
given by the curve

-22-9323-

The range X = - -65 to + '90 is easily seen to be sufficient. Column (i) of
the accompanying table gives aX for these values, the second gives
22-9323 x (log,0(l + X) - X loglo e) ;

* The three illustrations above are drawn from "Student's " original paper. He gives (1. c. p. 19)
the values for P as drawn from the Gaussian for n = 10 to compare with those obtained from the full
formula. They are, — corrected for slips :

z

Full Formula

Gaussian

z

Full Formula

Gaussian

•1

•61462

•60411

1-1

•99539

•99819

•2

•71846

•70159

1-2

•99713 -99925

•3

•80423

•78641

1-3

•99819

•99971

•4

•86970

•86520

1-4

•99885

•99989

•5

•91609

•90691

1-5

•99926

•99996

•6

•94732

•94375

1-6

•99951

•99999

•7

•96747

•96799

1-7

•99968

—

•8

•98007

•98285

1-8

•99978

—

•9

•98780

•99137

1-9

•99985

1-0

•99252

•99592

2-0

•99990

—

Clearly even for n = 10, the Gaussian ascends too rapidly in P, and this must be borne in mind in
deducing conclusions for z = l and upwards when n = Jl to 20, say.

xlvi Table* for St«ti*tic!nn* tuut Rwmetrictaii* [ X X VI— X \\1II

actually these values are negative and must be subtracted from logy,, i.e. 1 '592,621;
the resulting values are given in the third column. In column (iv) are given
the an ti logarithms of the numbers in column (iii), and these must be plotted to
the values in column (i) to obtain the graph of the curve which is a good fit.

(i) (») (i'i) HV)

z = aX

j[fc>gli(l + J[)-jriog.r

logy

y

- 9-77

•2 174,991

- -882,370

•13

- 8-88

1 -923,630

- -331,009

•47

- 7'99

1 -472,368

•120,253

i-aa

- 7'11

1-103,755

•488,866

s-oe

- 6'22

•804,557

•788,064

6-14

- 5-33

•564,456

1-028,165

10-67

- 4'44

•375,287

•217,334

16-49

- 3-65

•230,493

•362. 1:^

L'3-02

- 2'67

•124,683

•467,938

:.'!»-37

- 1-78

•053,386

•1)39,235

34-61

0-88

•012,888

•579,733

38-00

o-oo

•000,000

•592,621

39-14

0-89

•012,039

•580,582

38-07

1-78

•046,713

•545,908

35-ir.

2-67

•101,957

•490,664

3(1-95

3'55

•176,074

1-416,547

26-09

4-44

•267,482

1-325,139

21-14

5-33

•374,828

1 -217,793

16-51

6'22

•496,920

1-095,701

12-47

7'11

•632,702

•959,919

9-12

7'99

•781,189

•811,432

6-48

8'88

•941,609

•651,112

4-48

9'77

1-112,905

•479,716

3-02

10-66

1-294,689

•297,932

1-99

11-55

1-486,196

•106,425

1-28

12-44

1-686,831

- -094,210

•80

13-32

1-896,111

- -303,490

•50

14-21

2-113,510

- -520,889

•30

15-10

2-338,613

- -745,992

•18

15-99

2-570,963

- -978,342

•11

Once the reader is used to the process it will be found to work readily, and the
same multipliers are kept on the mechanical calculator throughout.

TABLES XXVII AND XXVIII (pp. 38—41)

Tables of the Powers and Sums of the Powers of the natural numbers from 1 to
100. (W. Palin Elderton, Biometrika, Vol. II. p. 474.)

These tables can be used in a great variety of ways, for example in finding the
roots of equations, Or in fitting parabolae of various orders to curves.

Illustration (i). Find the positive root of the equation :
<f> (r) = -002,7267-7 + -057,1497- + '017,1927-

+ -083,578^ + '088,331»J + '134,7 17»J + r - -560,386 = 0.

• Actually these values are negative, and are therefore iiibtracted from log y0 to give (iii).

XXVII— XXVIII]

Introduction

xlvii

The positive root is less than '56, but the term in r2 shows that it must be less
than -52. Take "52 and '50 as trials. From Table XXVII we have

1st

2nd

3rd

4th

5th

6th

7th

and

•500,000,
•250,000..
•125,000,
•062,500,
•031,250,
•015,625,
•007,813.

•520,000
•270,400
•140,608
•071,162
•038,020
•019,771
•010,281

Multiply out by the coefficients of <f> (r), retaining the products always on the
arithmometer. We find

<f> (-52) = + -01 6,384.
$ (-50) = - -008,990.

Interpolating r = '52 - £f§ff x 2 = '5071,

which is correct to last figure.

Illustration (ii). Fit a cubic parabola to the data below, giving the average
age of husband to each age of wife in Italy (see Biometrika, Vol. n. p. 20). We
will suppose each observation to be of equal weight, — this is of course not the fact,
but it will illustrate the general method of fitting parabolic curves. In the paper
just cited illustrations are given up to parabolae of the sixth order. The object
here is to show the use of Table XXVII.

Age of
Bride

Probable Age
of Groom

Age of
Bride

Probable Age
of Groom

Age of
Bride

Probable Age
of Groom

15-6

25-0

25-5

27-0

35-5

36-0

16-5 25-2

26-5

27-5

36-5

37-0

17-5 25-4

27-5

28-0

37 '5

38-5

18-5 25-5

H*

29-0

38-5

39-5

19-5

J6-8

29-5

30-0

39-5

41-5

20-5

86*

30-5

32-0

40-5

41-5

21-5

25-75

31-5

33-0

41-5

42-5

22-5 26-0

32-5

33-5

42-5

43-5

23-5 26-0

33-5

34-0

43-5

43-5

24-5 26-8

34-5

34-5

44-5

43-5

^~~

—

45-5

43-5

The ages of groom have been taken as approximate means. Now we can take
our axis of x, the age of bride through 30'5, and the age of groom to be measured
from 32'0. x will accordingly range from —15 to +15, and the age 32 + y of
groom will range from y = - 7 to y= 11 '5. We can now re-arrange the above
table in a form suitable for working on the following table. Then the squares,
cubes, and if necessary, higher powers of x are taken from Table XXVII, p. 38,
and are given as Columns (iii) and (iv) below. The entries in Column (i) are then
multiplied by those in (ii), (iii) and (iv) by continuous process on the machine, and

xlviii

Tables for Statisticians and Biometricians [X XVIII

it is not needful to enter separate products, the sums being reached which are
placed at the foot. Next from Table XXVIII we read off

), 5 (*•)-(>, ,<?(*•) -2 (5(15*)).

These give us :

5 (*>) = 2480, .V (**) = 356,624, 5 (a?) = UO.96,5840.
We have now all the numerical data for a solution. Let the required cubic be

y = cc + c,x + Cja? + c,s*.

Then we must make u • = S(y — c,,— CiX — c^af — c,**)1 a minimum. The resulting
equations are

5 (y) = CoS (1 ) + c, 5 (*) + c,5 (af) + c,S («•),
(*) + c, 5 (a?) + c»S (a?) + c,S («•),
c,5

(i)

(ii) (iii)

(iv)

(v)

(vi)

(vii)

»

i

.r-

f>

xy

••»

**y

- 7-0 -15

225

- 337.-.

- 6-8 -14

196 --2744

—

—

—

- 6-6 -13

169 -:il'.i7

—

—

—

- 6-5 -12

144

-1728

— —

—

i;-:> -11

121

- 1331

—

- 6-5 -10

100

-1000

—

—

- 6-25 , - 9 j 81

- 729

—

—

- 6-0 - 8

64

- 512

—

—

- 6-0 - 7

49

- 343

—

—

- 5-2

- 6

36

- 216

—

—

- 50

- 5

25

- 125

—

—

—

- 4-5

- 4

16

- 64

—

—

- 4-0

- 3

9

- 27

—

—

- 3-0

- 2

4

- 8

—

—

- 2-0

- 1

1

_ i

—

—

0

0

0

0

—

—

1-0

1

1

1

—

—

1-6

1

4

8

—

—

—

2-0

3

9

27

—

—

—

2-5

4

16

64

—

—

—

4O

0

25

125

—

—

4-5

6

36

216

—

—

6-5

7

49

343

—

—

7 -ft

8

64

512

—

—

9-5 9

81

729

—

—

9-5 10

100

1000

—

—

—

10-5 11

121

1831

—

—

—

11-5

12

144

L7S8

1 1 •:.

13 169

2197

1 1 •-. 14 196

L'7 1 I

11-6

15

225

3:175

—

—

—

5(*) = 23-(i.-.

_

—

—

5(j,y)-183:i-|:,

S(*»,y)- 4580-35

S y)- 248,807-85

XXVIII]

Introduction

xlix

ORY/'

Write &„ = c«, 6, = 10c0, 62 = lOOc,, 63 = 1000c3.

Then our equations are

•23650 = b, x -31000 + 62 x -24800,

1-83345 = 6, x -24800 + b, x -35662,

•45603 = 60 x -24800 + 62 x -35662,

2 48808 = 6, x -35662 + b3 x -60966 ;

giving 60 = - -58626, .-. c» = - -58626,

b. = 1-686453, c, = '016,8645,

6, = 9-59613, c, = -959,613,

63 = - 1-532,144, c, = - -001,532,144,

and the required cubic is

y = - -58626 + -959,6133; + '016,8645^ - -001,532,144^.

45, — , — , — , — i — i — , — , — • — i

K)

25 •• *

15

20

25 3O 35 40 45

Age of Bride.

The graph of the cubic and the observations are given in the accompanying
diagram. If X and Y be the actual ages of bride and groom, then

Y = 61-30457 - 4-344.941Z + -157.0553Z' - -001,53214Z3.

For higher parabolic curves fitted to the same data, see Biometnka, Vol. n.
pp. 21—22.

B. g

Tablet fa- Stutixtii-iaiix <tn</ /iiiinntri<-i<tn* [XX IX

TAIU.K \.\I.\ (pp. 42—51)

Tables of the Tetraclmric functions. (P. F. Kvcritt, llinmvtnfoi. Vol. vn.
pp. 437—451.)

The purpose of these tables is to expedite the calculation of tetrachoric /•,, the
correlation coefficient from a four-fold table, when we suppose the variates to be
Gaussian in the law of their frequency.

Let the table be

(I

b

a + b

c

d

c+d

a+c

b+d

N

where o is the quadrant in which the mean falls, then b + d and
each less than N. Let

then

are clearly

T,,Trl'/-" + ............ (xxxvii)

is the equation to determine r the tetrachoric correlation, and Table XXIX gives
the values for given TO, Le. ^ (1 — a) of the following six tetrachoric functions
TI, T.J ... T,, and further of h, the ratio of the abscissa of the dichotomic line to tin-
standard deviation of the corresponding variate.

It is occasionally needful to go beyond the first six tetrachoric functions. In
this case the following finite difference formula is available :

where

(xxxviii),

(»- 2). \'n (H - 1 ) .................. (xxxix).

The following table gives the values of pn and */„ from n = 7 to 24.

n

P.

</n

n

P«

9.

7

•37796

•77152

(0

•25000

•90370

8

•35355

•80178

n

•24254

•90951

f>

•33333

•82406

ts

•23570

•91466

in

•31623

•843L>7

19

i'L'!)42

•91925

11

•30151

•85812

so

j:t(51 -9233S

13

•28868

•87039

.•!

•21822 -92711

IS

•smo

•88070

.'.'

•818W

•93048

u

•26726

•68951)

.'.;

•20851

•93356

/.-, -25820

•89709

-"/

•20412

••13638

XXIX]

Introduction

li

Illustration (i). Find the correlation between dullness and developmental
defects as indicated in the following table for 26,287 children.

Without Defects

With Defects

Totals

Not Dull
Dull

22,793
1,186

1,420
888

24,213

2,074

Totals

23.979

2,308

26,287

= -078,898,

Here

Whence by interpolation from Table, p. 43 :

T,= '14712,

T,= -14694,

T:! = -05977,

T4=- -04262,

TS=- -06702,

T6=- -00752,

h = 1-41253,

T/= -15945,

T/ = -15268,

TS'= -05431,

T/ = - '05137,

T5' = - -06755,

T,' = -00017,

k = 1-35442.

Proceeding to apply the difference formula (xxxviii) for four further functions
we have

T/ = -05221,
T,' = -02480,
T,' = - -03185,
T,; = - -03460.

T. = -04770,
TS = -02985,
T, = - -02530,
T]0 = _ -03647,
Hence the equation for r is

•026,854 = -023,458?- + -022,435r2 + "003,246^

+ •002,189r4 + -004,527?-' - -OOO.OOb-6
+ •002,490?-' + 000.742?-8 + -OOO.SOe?-9
+ -0()l,262r10.

Whence we find ?•= '652 + '009.

Illustration (ii). Fjnd the tetrachoric correlation for the four-fold table given
for Houses in course of Erection on p. xxxv. Here

i (1 - a,) = TO = Jff j = -260,080 ; $ (1 - «,) = TO' = ^T = -009,157.

By simple linear interpolation,

T, = -32442,
T2= -14753,
T3=--0776G,

T,' = -02468,
T2' = -04116,
T,; = -04599,
T/ = -03048.

lii

Tal>l,s fur Statistician* and lii"iii'tri<-i<nis [XXIX

tlir • .(nation for r:

- -000,6093 = -008,007 r + -006,072^ - -003.572^

- •003,357r«.

Whence r = --081 ±'043.
Or, the association is not definitely significant.
Illustration (iii). Find the tetrachoric r for the Table of Bradford Parents :

Mother's Habits.

I

a

CO

~l.

a

I

Good

I',.,,!

Totals

<;,,<Hl

:i!»i <;T

1061

Bad

i.,n 171;

8M

Totals ...

1153 543

Hi9fi

Here a brief experience will show the reader that to proceed by tetraclmiic
functions will require a very large amount of labour.

We have

J (1 - a,) = TO = 543/1696 = "32017 ; i(l -«I)-T.'- 635/1690 = "37441,
djN = 476/1 696 = '28066.

We have accordingly the following series of tetrachoric functions — the first
6 from the table, the remaining 18 from the difference formula.

dlN

4(1 -a)

T\

Ti

T3

Tt

Ti

»"fl

•28066

•32017

•:i7iil

•35769
•37901

•11817
•08581

-•11415

-•13887

- -09489
- -07178

•05674
•08288

•08012
•06325

r-.

Tg

r«

TIU

rn

i-u

TIJ

TH

-•02963
-•06629

-•06913
- -05709

•01368
•04034

•ooon

•05223

-•00324
-•02957

-•05294
-•04819

-•(10401
•0217G

O46M

•(U.V.S

i-u

TU

T\;

TIS

rn

»•»>

•

'"ii

T-:

•00922
- -01575

-•04103
- -04245

"

-•01304
•01103

•OM00

•03966

•01586
-•00723

- -03167
-•03714

-•01793
•00411

•02768
•03484

r-a

TJ4

A

—

—

—

—

—

•01944
-•00151

- -02407
-•03272

•46732
•32020

—

—

—

—

—

XXIX — XXX] Introduction liii

Considering only the equation as far as Everitt's Tables extend, we have
£ (r) = - 16079 + -13557r + -01014r2 + •01585r3-f "OOGSlr4 + •00470r5 + "00507^ = 0.

This leads to r = "9365, but the series indicates that the terms are far from
converging rapidly.

The first 12 tetrachoric functions were then used, the last six being found
by the table of pn and qn above, and the value of r was found to be '9152.
Then 18 functions were used and gave r— '9114.

Lastly 24 tetrachoric functions were used, and the equation below obtained,
which led to r = '9105.

$ (r) = _ -16079 + -13557r + -01014r2 -I- 'OlSSSr3 + -00681r< + •00470r5
+ -005077* + -00167r7 + -003957-" + -00055r» + -00315r">
+ 000107-" + -00255?-12 - -00009r's + -00212r" - 'OOOlor15
+ 00174r16 - -00014r" + -00143r's - 'OOOllr19 + -00118j«
- -00057?-21 + -00096/---' - -00003/-2' + -00079r«.

It will be seen that even with this very large amount of labour we cannot be
sure of having reached a final result*. To obviate this the following table was
constructed by Everitt, and there is no doubt that the extension of this table to
the whole range of correlation would much simplify the discovery of tetrachoric »•<•
At present the calculation of high values of r,, for negative correlations is in hand.

TABLE XXX (pp. 52—27)

Supplementary Tables for determining High Correlations from Tetrachoric
Groupings. (P. F. Everitt, Biometrika, Vol. VHI. pp. 385 — 395.)

Using the notation of p. 1,

N 2irVl-
iii the case of a tetrachoric table, or

d 1

where F=-^=| e~*Aa\ iff- A -'/

vl -;

.(xli).

* Mr H. E. Soper working out this example draws my attention to the fact that convergence ia closely
given by a form : rn=rco (1 + a .c"), where n is the number of terms used and a and c are constants.
Hence (r. - r^ ) (rn+2ra - ra) = (rn+m - r. )\

or r -

••
In oar case take >i = 6, m = 6, and we find

,,g- =

The valne r24 is -9105. In this case a= '1567 and c = '7574, but we cannot assert that these would
be constants for all tables. If we use r12, r^ and r^,, we find r^ = -9102.

liv

Table* f»r Sl<ifi.*tician8 ami

Hence r, h being known, }' is a tabled integral for each value of Y. Accordingly
by aid of Table II we know -==«"** , and usim: a i|u:idraturr formula, d/ff can

be found for each value of h,k and r.

Table XXX gives, for r = '80, '85, '90, '95 and TOO, and values of It and /
proceeding by '1, the values of d/N. For given valiu-s of h, k and d/N, we can
then find r by interpolation from these tables. The process is far shorter than
that required by Table XXIX when we have to proceed to many terms. Un-
fortunately opportunity has not yet arisen for fully completing similar tables
for r negative and over '80.

Illustrutiun. Determine the correlation in habits between Mother and Father
in Bradford. The data are

Mother.

Habits Good

11. il.it-* Bad

Total*

$

1

Habits Hood ...
Habits Bad

!»!M
150

67

I7(i

10G1

<;:).->

Totals

1153

M3

1(596

Here (6 + d)/N= -32017, (c+d)/N = '37441, and therefore h = -4G722, A- = -32020
from Table II. Also d/N = 476/1696 = -28066.

Inspection of Table XXX shows that r will be likely to lie between -90 and '95.
\Ve extract from the Table for d/N:

r=-90

»-«

I-*

ft.*

•2943

•2728

/- -.

•2784

•2602

r = -95

* = ,

>-.,

*=-3

•3130

•L'silM

t. -4

•2980

•2787

Hence :

r=-90

h = -4

;f=-5

t-= -32020

•2911

4709

r=-95

;i=-4

A = -5

k- -32020

•3104

•2876

Thus :

r=-90

^=•32020

•2771

; '.'-. h = -40722

-32020

XXXI] Introduction Iv

We have now the desired h and k and have to interpolate d/N = '28006 between
•2771 and "2951. There results r = "9099.

This is in excellent agreement with the value "9105 deduced from 24 terms, or
from the final value '9102, which can be deduced from the 12, 18 and 24 term
values on the logarithmic rate of decrease hypothesis : see footnote p. liii.

TABLE XXXI (pp. 58—61)

The r-Function. (J. H. Duffell : Biometrika, Vol. vn. pp. 43—47.)

It is well known that T(x + l) = xT'(x), and this property enables us to raise
or lower the argument of the T-function at will. As a rule in most statistical
investigations we require T(x + l)/xxe~x. The following formula due to Pearson
will then be found to give r(x + l)/xxe~x with great exactness:

log ( F r^l = '0399,0899 + J log a + -080,929 sin — — . . .(xlii).

\ 3j & / 3C

For values of x+1 less than 6 and often for values less than 10, we find
log T(a;+ 1) or \agT(p) from Table XXXI by reduction to p between 1 and 2.

The reader's attention must be especially drawn as to the rules, given on
the Table itself, as to (i) characteristic, (ii) change of third figure of mantissa
at a bar, and (iii) the sign of the differences on the facing pages of the tables.
The difference tabled under 1*144, say, is the drop from 1*144 to T145.

Illustration (i). Find T ('2346).

By the reduction formula T(-2346) = r(l-2346)/'2346.

Hence log T (-2346) = log T (1-2346) -1-370,3280.

log T (1-234) = 1-958,9685 A = - 1069,

log T (1-235) = 1-958,8616 -6A = - [641-4].
. • . log F (1-2346) = 1-958,9685 - [641 ]

= T'958,9044.

log T (-2346)= 1-958,9044
-1-370,3280

•588,5764
Or T (-2346) = 3-87772.

Ivi Table* /«/• Xtnti.«ti<>i«nx and Blometric'uin* [ X X X 1 1 XXXIII

Illustration (ii). Find F (87614).

F (87614) » 77(il4 x 67614 x 57614 x 4 7614 x 37614 x 27U14

x 17614 r(17614).

log T (87614) = -889.9401 + log T(17614)
•830,0366
760,5280
•677,7347
•575,3495
•441, 1 2! i:;
•245.8580

= 4-420,5762 + log T (17614).
log F (1761 4) = 1-964.5473 + '4 [1113]

= 1-964,5918.
. •. log T (87614) = 4-385,1680.

Hence F (87614) = 24275-49.

TABLE XXXII (pp. 62—63)

TABLE XXXIII, A and B (p. 64).

Subtense from Arc and Chord in the case of the Common Catenary. (Julia Bell
and H. E. Soper: see Bwmetrika, Vol. vm. pp. 316, 338, and Vol. ix. pp. 401—2.)

If c be the parameter of the common catenary, then we know that

y = c cosh u ................................. (xliii),

where u = x/c is its equation.
If the chord be 2#, then

subtense/chord = (y — c)/(2x)\

_ (sinh $uY \ ..................... (xliv),

"' }

sinh M
arc/chord = — .............................. (*lv),

arc — chord _ sinh « — u $ / i *\

"^hord ~iT 'loo'

Mllitrlisi- (sinh JH)-' _ a , , ..v

chord" ~iT =K)0
Corresponding values of a and ft are given in the Tables XXXII and XXX 1 1 1

XXXIII A AND B] Introduction Ivii

Illustration (i). A cable of 132-5 is suspended over the gap between two
towers of the same height, 115 feet apart. What will be the droop of the cable?

ft = 100 032-5 -115)= n.52

1 J.O

Table XXXIII A, gives us a = 21-62 = 100 subtense/chord.

.-. subtense = '2162 x 115

= 24-86.
Thus the droop is 24'86 ft.

Illustration (ii). A catenary arch is to have a rise of 50 ft., centre line
measurement, and a span of 200. What is the length of the centre line ?

3=100x50/200=25-0,
but a = 25 by Table XXXII gives £ = 15-1.

100 (arc — chord)/chord = 15*1.
.-. arc = 230-2 ft. «

Illustration (iii). For some races the shape of the nasal bridge is very ap-
proximately a catenary. Thus if the nasal chord from dacryon to dacryon be
measured and also the tape measure from dacryon to dacryon, we obtain the
mesodacryal index y9. The tables enable us to pass to the mesodacryal index a,
and thus ascertain the nasal subtense, which is slightly harder of direct measure-
ment than the arcual or tape measure.

In the skull of a male gorilla the mesodacryal chord was 22'6 mm., and the
mesoilacryal arc 30 mm. Determine the mesodacryal subtense

- 100 - - _ 30-74

~~ ~

Hence, from Table XXXII :

a = 38-84 = 100 subtense/22'6.
.-. subtense = 22'6 x -3884 = 8'8 mm.

The actual value of the mesodacryal subtense measured on the skull was
87 mm.

ABAC XXXIV (p. 65)

Diagram to find the Correlation Coefficient r from Mean Contingency on the
Hypothesis of a Normal Frequency Distribution. (Pearson : Drapers' Company
Research Memoirs, No. 1, "On the Theory of Contingency.")

If npg be the frequency in the cell of the ^)th column and qth row of a correlation
or contingency table, and mp be the total frequency in the pth column, nq the

* Should there be any use for this table for constructional purposes, which there ought to be when the
value of the catenary arch is more fully recognised, I will in a later edition of this work give the value
of u corresponding to each /3, so that the parameter c can be at once read off and the form of the arch
readily plotted. It might also be desirable to give the values of o and /9 to two decimal places. We
have these data in our MS. copies.

B. ft

hiii Ta&fa Jbr Statutieiaiu tmd />inin>triri<ins [X.\M\

total frequency in the gth row, and N the whole population, then if the two
variates are independent, the frequency to be expected in the p, </th cell will be

N X ' N
and the observed excess over this, i.e. npq-n<l™p , is termed the 'contingency ' in

this cell. The total contingency must be of course zero, i.e. the sum of all the
cell contingencies. If, however, we take only the positive excess contingencies and

divide them by N, i.e. •^• = -^2+[np, AT) • we ODtain tne so-called 'mean

contingency.' On the assumption of normal frequency distribution it is possible
to deduce the actual correlation from -v/r, provided that the cells are sufficiently
.small fur summation to replace integration. As in practice our cells are hanlly
likely to exceed 8x8, and may be smaller and unequal in area, we shall generally
find a value below that of the true correlation, even if the system be accurately
normal. A corrective factor corresponding to the class-index correlation has not
yet been theoretically deduced. But experience seems to show that to add half the
correction due to class-index correlations gives good results. That is to say, that,
if r+ be the correlation found from the Abac, p. 65, and rxp and rxc be the class-
index correlations for x and y, we should take for the true correlation :

7>

*[•

(xlviii).

r«o.*v^J

It is clear that this is the same thing as taking the mean of the crude mean
contingency correlation and its value as corrected for the class-index correlations.
The following illustrations may indicate the method of procedure.

Illustration (i). Find the correlation from the table on p. lix by mean
contingency. The first number in each cell is the frequency reduced to 1000, the
second number is that to be expected on the basis of independent probability, and
the third is the mean contingency of the cell.

The sum of the positive contingencies is 94136, hence the mean contingency
is "094. Entering the diagram with "094 on the base scale, we pass up the vertical
to the curve, and then along the horizontal to the left hand scale and find r^, = '285.

The class-index correlation for the vertical marginal frequency is ryC ='9645,
and that for the horizontal marginal frequency is '9624*. Hence

and r = i ('307 + -285) = "296.

The table is actually a true Gaussian distribution with correlation equal
to -300.

* Biometrika, Vol. «. p. 218.

XXXIV]

>

s

Introduction

First Variate A.

lix

1

2

.;

4

5 + 6

7

8

Totals

1

4-04
[1-884

2-810

17-16

(10-948)
6-212

7-55

(8-976)
- 1 -426

3-30

(6-120)
-2-820

091

(2-346)
-1-436

0-92

(3-434)
-2-514

0-12

(0-952)
-0-832

34

s

17-41

(10-836)

<;•:,; >,

123-59

(96-922)

.',;-i;r,x

7976

(79-464)
0-296

44-64

(54-180)
- 9-o40

14-61

(20-769)
-6-159

17-67

(30-401)
- 12-731

3-32

(8-428)
-5-108

301

3

8-86

(10-224)
-1-364

9300

(91-448)
1-552

78-31

(74-976)
S-334

52-04

(51-120)
0-920

19-20

(19-596)
-0-396

26-40

(28-684)
.':.'* 4

6-19

(7-952)
-1-762

284

4

2-83

(4-932)
-2-102

o; -73

(44-114)
-6-384

37-24

(36-168)
1-072

27-51

(24-660)
2-850

1095

C9-4.r>3)
'1-497

16-31

(13-837)
2-473

4-43

(3-836)
0-594

137

5+6

162

(3-780)
-2-160

25-21

(33-810)
-8-600

27-75

(27-720;
0-030

22-09

(18-900)
3-190

9-26

(7-245)
2-015

14-64
(10-605)
4-035

4-43

(2-940)
1-490

105

7

1-02

(3-528)
-2-508

1950
(31-ftM
-1*066

24-47

(25-872)
-1-402

21-39

(17-640)
3-750

9-58

(6-762)

1-X1X

16-36

(9-898)
6-462

5-68

(2-744)

..'•;». -v;

98

8

0-22

(1-476)
-1-256

5-81

(13-202)
-7-392

8-92

(10-824)

-1-904

903

(7-380)
1-650

4-49

(2-829)
1-661

8-70

(4-141)
4-559

3-83

(1-148)
2-682

41

Totals

36

322

264

180

69

101

28

1000

Illustration (ii). Find r^ by mean contingency for the table on p. Ix:

The sum of the positive contingencies is 169'846, or we have mean contingency
^r = -170, whence the diagram leads us to r^ = '480. The marginal frequencies are
the same as in Illustration (i). Thus we have

The table gives actually a true Gaussian distribution with correlation -500.
It will be seen from Illustrations (i) and (ii), that if the distribution be Gaussian,
even if the marginal frequencies are in fairly irregular groupings, r# will be
reasonably close to the true contingency, and corrected as suggested above will

give excellent results.

/» 2

Ix Tables for Statisticians a // <l />i»inttn'citin.« \\\\ — XLVI

First Variate A.

1

/

S

;

.-, + .;

7

»

Totals

1

738
L-8M

6-156

1985
(10-948)
8-901

494
8-976

;•"•'•

138

(6-120)
-4-740

026

(2-346)

r-oM

018

(3-434)

-••<•:••',

0-01

(0-952)
- 0-95/

34

•

2058

(10-836)

•-.-;;

145-47

(96-922)
48-548

7894
-0-5*4

3538

(54-180)

-/.s-."'c

9-72
(20-769)

- ll •<>',:>

9-27
(30-401)

- ///.,/

1-04

(8-428)
-7-388

301

s

601

(10-224)

-4-914

9363

(91-182)

.'•i*:

85 41

(74-976)
W-434

5434
(51-120)
3-990

1859
19-fi06

-1-006

22-33
(88-684

-6:154

369

(7-952)

;•.'•;.-

284

4

1-26

(4-932)
-3-672

3181

41-114

- /."..•«;

39-49

(36-168)

..".;.'.'

31-03
(24-660)
6-S70

12-29

(9-453)

fsn

1736
(13-837)
3-699

3-76

(3-836)

_-,;;.;

137

5+6

053

(3-780)
-S-250

1811

(33-810)
- 75-700

27-79

(27-720)
0-070

25-14

(18-90)
6-240

11-09

(7-245)
3-845

17-62

(10-605)
7-015

4-72

(2-940)
1-780

105

7

0-22
(3-528)
-$•808

1102
(21-556)

- to-sse

21-59
(25-872)
-4-98*

23-66
(17-640)

<;-n.'n

11-86
6-768)

5-098

21-89

(9-898)
11-992

7-76

(2-744)
5-016

98

S

0-02

(1-476)
-1-456

2-11

(12-202)
-11-092

5-84

ao-824)
'-4-984

847

(7-380)
1-090

5-19
(8-829

.'-.till

12-35

(4-141)
8-209

7-02

(1-148)

.•--,*;.<

41

Totals

36

322

264

180

69

101

L'S

1000

TABLES XXXV— XLVI (pp. 66—87)

Criteria for Frequency Types and Probable Errors of Frequency Constants.
(A. J. Rhind: Biometrika, Vol. vn. pp. 127—147 and pp. 386—397.)

It in desirable to consider all these tables under one heading, namely the
general investigation of frequency type and of the probable errors of frequency
constants.

The main lines of Pearson's theory of frequency are involved in the following
statements:

XXXV— XLVI] Introduction Ixi

If the differential equation to the uni-modal frequency distribution be

1 du x — a

-^"-TTT

y dx f(x)

we may suppose f(x) expanded in a series of powers of x, and so

_ _

ydx c0+c1x + c*a;i+ ... + cnxn+ ..."

then a, c0, clt ct, ... cn... can be uniquely determined from the 'moment co-
efficients ' of the frequency distribution. These constants are functions of certain
other constants /9,, /32— 3, $,, /94 — 15, ... which vanish for the Gaussian curve, and
are small for any distribution not widely divergent from the Gaussian. Further
c0, c,, c2...cn... converge, if, as usual, these constants are less than unity, the
factors of convergence being of the order V/3-constant. As a matter of fact cn
involves the (n + 2)th moment coefficient, and thus we obtain values of the
c-constants subject to very large errors, if we retain terms beyond ca. If we stop
at cs then our differential equation is of the form

1 dy _ x — a

.(li),

y dx c0 +

and we need only /9, = fi^jn? an^ & = /*4//i,2, where ft,2, p,, /j.t are the second, third
and fourth moment coefficients about the mean.

If we take the form - ^ = '.we reach the Gaussian, in which each con-
y dx c0

tributory cause-group is independent, and if the number of groups be not very
large, each cause-group is of equal valency and contributes with equal frequency

results in excess and defect of its mean contribution. If we take - -?• =

y dx c0 + c^x

then each contributory cause-group is still of equal valency and independent, but
does not give contributions in excess and defect of equal frequency.

Finally if we take -fim— , then contributory cause-groups are

-

ydx

not of equal valency, they are not independent, but their results correlated, and
further contributions in excess and defect are not equally probable. The use of this

form —fm -- - was adopted to allow of this wide generalisation of the
y dx c0 + c,x + ctx*

Gaussian hypothesis.

If we adopt it, every ^-constant is expressible by means of the formulae :

/8n(even) = (n + l)[i/3n_1 + (l+ia);9n_J)/(l-Hn-l)a) ............ (Hi),

/S. (odd) =(n+l){iA/8»-, + (l+i«)/8^}/(l-i (»-!)«) ...... 0"i),

where a = (2& - 3/3, - 6)/(& + 3) ........................... (liv),

in terms of lower /3-constants.

Ixii Table* for Sttitidician* ami liimm-trinn,,* [XXXV— XLVI

Table XLII, (a)— (d) gives the values of £„ /8«, y9, and & in terms of tf, :inil
/3,. Hence as soon as /3, and & are calculated we can find the numerical values of

......... (iv),

theoretically. Although these values will not be those which would be absolutely
deduced from the data themselves, they will, considering the large probable errors
of ft,, fit, ft, and fr be reasonable approximations to them. The values of the
probable errors of £, and & are determinate by formulae involving £,, /9, ... /9,.

From these formulae, Tables XXXVII and XXXVIII, giving the values of
VJ^S^ and ViVSp, have been constructed. Hence multiplying by ^, from Table V,
we obtain

•67449 v -67449 _

- S and

the probable errors of /9, and y8,.

If we add to the standard deviations of $, and $,, the correlation between
deviations in /3, and /98, namely Rp^, which correlation is given in Table XXXIX,
we can find the probable errors of any functions of /Si and /92. Two such important
functions are the distance d from mean to mode and the skewness sk of the
distribution. The probable errors of d and sk can be found from Tables XL and
XLI respectively, the former by multiplying the tabulated value V./VSj/o- by a x ^,
(from Table V), and the latter by multiplying the tabulated value V^S^ by ^,
(from Table V).

Thus far we have only been concerned with the constants which describe
certain physical characters of the frequency distribution without regard to the
type of curve suited to the distribution. We now turn to the latter subject.

It is known that the type of frequency depends upon a certain criterion *,.
Hence near the critical values of «, more than one type of curve may describe the
frequency within the limit of the probable error of *,. Table XLIII gives the
probable error of *,, if the entries in that table be multiplied by the ^, of
Table V.

The following are the series of Type curves which arise according to the value
of the criteria

*,=2& -3/9, -6 ....................................... (Ivi),

A (A (, ..

/9, is by necessity >f&. Hence for our curves all possible values of /3,, & lie in
the positive quadrant between the lines /9, = j/8, and & = J^/9, + f , the latter being
if we go to & the limit of failure of Type IV, for its ft, becomes infinite. Beyond
the latter line distributions are heterotypic.

Introduction

Ixiii

Type

VII

Equation to Curve
I

XXXV— XLVI]

Criterion

«2 = 0 ft = 0, ft>3 .__ „ ,

(l +— J (Iviii).

^ = 0 ft = 0, ft = 3 Normal y = y<,e ^2 (Hx).

«» = 0 ft = 0, ft<3 II. y = y0(l-^T (ix).

\ w- /

^ = 0 ft = 0, ft < 1-8 II,, y = y, L__ (lxj).

-rtea-ijj

*.,>()<! IV y = </0-, -r^- (Ixii).

*2=1* V y = y0e ->/**-* (Ixiii).

/c2 > 1 < <» VI y^y0(x — a)mi/x»^ (Ixiv).

*.,= », i.e. 2ft-3ft-6 = 0 III y = yae~P«(l + -

\ t*1

Kt < 0, i.e. negative. Below /= 0 Ix ij = ya

K.,<0. Inside /=0

K., <0. Above /=0 Ur y = y,, ^

For K, < 0,/=0 represents the biquadratic

ft (8ft - 9ft - 12) (4ft - 3ft) = (10ft- 12ft - 18)"(ft + 3)2 ...(Ixix).

Type I is thus divided into three subclasses, limited range curves, J-shaped
curves and U-shaped curves.

Diagram XXXV enables the reader at once to find the type appropriate to his
distribution, and Diagram XXXVI gives the same figure on a much larger scale to
indicate the changes that occur with large values of ft and ft.

Knowing the values of ft and ft the computer can fix his point on the
Diagram XXXV, but he may come so near a critical point or line, that one curve
may appear as reasonable as another. It is clear, for example, that in the
neighbourhood of the Gaussian point G, he might possibly use II, Ij, III, VI, V,

Ji y=y°(1 + ir) \~^> •••(ixvii)-

1

- ...(Ixviii).

* The branch of the cubic «r2=l with which we are concerned passes through the Gaussian point,
at which jp= oo , and along this branch p is always >5.

Ixiv Table* for Statistical H* <nnl />//////< •Y/Vw/u- (XXXV — XLVI

IV or VIII, and as all these types at that point transform into each other, the
forms actually deduced will be almost identical, however different their equations.
But there will be other occasions when doubt as to the use of the simpler of two
curves may arise; for example if /9, = *8, $,= 4*15, are we justified in using
Type III as simpler than Type I ?

Now we have to remember that the variates /8,, /9, form a frequency surface, of
which the equation is

M i__i /A* . ftf «*».». A AA

" "

and that the contours of this surface projected onto the 8,, y9, plane of
Diagram XXXV form a series of similar and similarly placed ellipses. Within
any one of these ellipses a certain amount of the volume of the y9,, /3,-frequency
lies, and therefore if this system of contours were properly placed round the y9,, /8S
point on Diagram XXXV we could tell at once the probability that the given point,
owing to random sampling, should fall outside a given elliptic contour.

The ellipse which has for principal semi-axes 1*1772, and 1*1772..,, where 2, and
2, are the principal axes of the ellipse :

1=i- -fsTl+v1! v'% ' '"') (lxxi).

covers an area on which stands just one half the frequency, i.e. it is the ellipse
determined by the generalised probable error of two variates (see Table X, p. 24).

The semi-minor axis 1*1772, and the semi-major axis 1*1772, of this "Probability
Ellipse " multiplied by ^N are given in Tables XLIV and XLV respectively, and
Table XLVI gives the angle in degrees between the major axis of this ellipse and
the axis of /9... It is thus possible to construct from Tables XLIV — XLVI the
" probability ellipse" round a given point /8,, y8,, and to test the area within which
half the frequency lies. If the probability required be not J, but much less, then
we note that the probability, that a point will lie outside the ellipse with semi-
axes X2, and X2S is P = e ~ *x*.

Let X2a =1-177 v/FS.,x'(iT44!l ...(Ixxii),

or X5 = q x -630,672,

an/1 /-' — p ~ *7 X 'SlB.SSo

• III1 1 I ^~ O .

Hence logP = -(j x *136,949.

Accordingly 9 = 10 : /' = *0427,

9=12: P= -02*27,

9=15: P=*0088,

a = 20 : P = *0018.

XXXV— XLVI] Introduction Ixv

Hence we select the grade of working probability we require, roughly 1 in 23,
1 in 44, 1 in 114 or 1 in 555, and this determines q. Divide i^the total frequency
by q and look up in Table V, %t for N/q, multiply this by the 1177 VJV2,j of
Table XLV, p. 84, and we obtain the semi-major axis of the required ellipse-
Multiply the same ^, by 1'177 ViVS, of Table XLIV and we have the semi-minor
axis. We can then construct round the point Qi, /82 this ellipse and ascertain if
it cuts critical boundaries on Diagram XXXV, p. 66, the orientation being given by
Table XLVI, p. 86. Less accurately, but for practical purposes effectively, we may
work on Diagram XL VII, p. 88. We proceed just as before, to select our q and so
determine our \2, and X2i. Then we take the ratio of 2,/2j. We now pick out
of the ellipses on p. 88 the set having the nearest 2,/S2 value and out of this set
the ellipse with the nearest XS2 value of its semi-major axis. This ellipse or if
necessary an interpolated one is transferred to tracing paper and placed with its
centre at the given point (&, /?,), and its major axis touching the dotted curve. If
this ellipse does not cut a critical line, we can be certain that to the given degree of
probability the curve is of the type into the area of which its /9j, /3a point falls.

It would be impossible in an Introduction to these tables to give the whole
theory of frequency curves*. But one or two formulae may be usefully placed
here for reference.

a v'/Si (0, + 3)
Distance a from mode to mean = 7^^ -^-3 -- ^ .................. (Ixxiii),

2 (o#2 - o#, - y)

Skewness sk - &l ^* + ^ ClxxnA

~2(5&-6/3,-9) ' f*

Stf = & - 4/3,& + 4/3,' - ft + 160,0, - 80, + 16/8.) ............... (Ixxv bis),

2,, 2,,Eft „, = 2/3a - 3&0, - 40,0, + 60,«0, + 3& ft, - 6/9, + 1 2ft' + 24/3, (Ixxvi).

It is from the above formulae that the Tables now under discussion have been
calculated.

Illustration. The following percentages of black measured with a colour top
are stated to occur with the recorded frequencies in the skin colour of white
and negro crosses f.

Discuss the type of frequency curve suited to the data and determine the chief
physical constants of the distribution and their probable errors.

* The general theory is given in " Skew Variation in Homogeneous Material," Phil. Trans. Vol. 186
(1895), A, pp. 343—414: Supplement, Vol. 197 (1901), A, pp. 443—459; "On the Mathematical Theory
of Errors of Judgment," Phil. Tram. Vol. 198 (1902), pp. 274—279 ; "Das Fehlergesetz und seine
Verallgemeinerungen durch Fechner und Pearson," A Rejoinder, Siometrika, Vol. iv. pp. 169 — 212.
"Skew Frequency Curves," A Rejoinder to Professor Kapteyn, Ibid. Vol. v. pp. 168—171, and " On the
curves which are most suitable for describing the frequency of Random Samples of a Population,"
Ibid. Vol. v. pp. 172—175.

t Extracted from C. B. Davenport, Heredity of S/c.'n Color in Negro-White Croiiei, Carnegie
Institution of Washington, 1913.

B. <

Ixvi Tablet ft »•

«n<l

[\\.\V \ LVI

The working origin was taken at 20, the centre of the group 18—22. The
centre of the first group at 1'47 % is | (20- 1'47) = 3706 on the negative side of
the working origin and may be taken to contribute — 56, + 206', — 763, + 2830

I1, 1.. H1.1I.V

Frequency

Percentage

Frequency

0— 11*

15

43-47

10

.*— 7

120

48— 5S

L'l

8— IX

139

53—57

14

13—17

1.-.7

68— m

0

18— XI

Ififi

63—e7

:i

:•: .':
.'- • '

188

117

7* — 77

3
2

SS—S7

92

78- K.'

1

S8—4*

50

8S—87

—

to the first, second, third and fourth moments respectively. The working unit
being b°/Q, the raw moment coefficients are:

»,'= -567,2191, i//= 7-703,4990,
vt' = 28-982,5042, «/«' = 253-268,8730.
Whence transferring to mean and correcting, we have

Mean = 22-8361, o- = 270156,
/<., = 7-298,428, /u3=16'238,780, /x4= 198-909,921.

These lead to

& = -678,295, ft = 3-734,20:2,

*, = 2/8, - 3/8, - 6 = - -566,483, *, = - 1-052,180,
sk = -495,087, Distance from Mean to Mode = d = 1-337,508.

These values, except the mean, are all in working units. Therefore in per-
centages of black :

<r= 13-5078 and d = 6-6875.

We can now find the probable errors of these constants. We first want ^,
from Table V, but 1086 is outside the limit of n. We therefore take ^, for 543
and have ^i = '02047, and we find ^ = '014,47. We can repeat our constants with
their probable errors

Mean = 22-8361 ± '2765, <r = 13'5078 ± '1955.
Then from Table XXXVII,

A = 37 : v^ 2,, = 4-70 + {j}§ [2] = 47 1 ,
A -3-8:

Hence for & = 37342 : VFS,, = 471 + -

VJV^ = 4-83.
* 4 at 0 and 11 at 3, giring a mean at 1-47 "/„.

XXXV— XLVI] Introduction Ixvii

Similarly from Table XXXVIII :

& = 3-7 : V^S3a = 12'02 - ff| [66] = H'65.
ft = 3-8 : VF2A = 13-60 - f§£ [72] = 1319.
Hence for £, = 37342: VFSft = 11-65 + f$&[l'54],

V#2ft = 12-18.
Thus we find, multiplying by % :

& = -6783 ±-0989,
& = 37342 ±-2493.

It is clear that the /9, and j9s are significantly different from the Gaussian
ft = 0 and & = 3.

We next turn to the skewness, using Table XLI :

& = 37 : VF2* = 1-98 + |§} [21] = 2-10,
& = 3-8: VF2* = 1-88 + ff§ [16] =1-97.
Hence for & = 37342 : VJV2,* = 2'10 - ^ [13]

= 2-06.

Thus the skewness = '4951 ± '0422, or the distribution is significantly skew.
Passing to Table XL for the probable error of d, we have

& = 37 : Vtf 2d/<7 = 214 + {ft [20] = 2'25,
/92 = 3-8: VF2d/<r = 2-03 + ^[17] = 2-13.
Hence for #,= 37342: V^2d/o-=2-25-^y[12]

= 2-21.

Thus Probable Error of d = *i x a x 2-21 = -6111,

and d = 6-6875 ±-6111.

The probable error of AC, is to be found from the relation :

(VNSJ = 4 (VJVS^ + 9 (VF2ft)' - 12 (V^a) (V#2fc) x RM. (Ixxvii)
Thus we require RM . Table XXXIX, p. 72, will provide this:
& = 37 : Rfltt = -892 + ffift [5] = '895,
A = 3-8 : flftft = -893 + |fj| [5] = '896.

Hence for y82= 37342 we may take -R/3,/3., = "895. Accordingly
(VJV"2,,)3 = 593-4096 + 209-9601 - 631-8278

= 171-5419.

Or, V2VX, = 13-0974.

Hence p.e. of AT, = x, x VF2., = '2681 ,

or, «, = - -566,483 ± -2681.

It

Ixviii Taltlfxfin- St<it!*ti<-i(iiix dint Iliuinftricians [.\.\\V--.\I.VI

It would look therefore as if /c, were significantly negative, but it is just possible
that *, might be zero. Such a big probable error for *, suggests our being
in the neighbourhood of a critical limit. This is verified on examining Table
XLIII to find the value of V3rS.,. We see that it is over 80, and we thus
conclude that the probable error of *, may lie between 1 and 2. Thus we cannot
be definitely certain of the sign or magnitude of *,, when we are even relatively in
the neighbourhood of AC, = 0.

If we turn to Diagram XXXV (p. 66) we see that the point £, = '678 and
$.= 3734 is not very close but is approaching the line along which Type III is
applicable, and this is the source of the disturbance noted.

We accordingly try to measure the probability that Type III would be as
satisfactory as Type I within the area of which our /8,, &. values actually lie.
We must take 1-177VJVX from Table XLIV and M77VJV2, from Table XLV.
\Vc have

£, = •6789, & = 3-7 then l-mVlVS, = 2'3

„ A = 3-8 „ „ =2-5.

Hence for £,= 37342, M77VF2, = 2-3 + -jtffo [2] = 2-37.

Again: &=37 1-177^2,- 16 -||ft[l] = 15-43,

A = 3-8 1-mVtfS, = 18 -ffJKl]- 17-43,
& = 37342 1-177^2,= 15-43 + flfo [2] = 16-11.

Thus: 2,/22 = 2-37/16-11 = '147 = -15, say. Or, if we turn to Diagram XLVII
(p. 88), our system of ellipses is half-way between the 3rd (£,/£„ = '14) and the
4th (2i/2s = '16). Now if such a system of ellipses be traced off and centred at
the point £, = -678, $, = 3734 on Diagram XLVII to the right and then the
major-axis be brought into parallelism with the dotted lines, we find that the
biggest of these ellipses \2, = *5 fails to reach the critical line III. But the
semi-major axis of the probability-ellipse is ri772v> = Ifrll/i/N = '493. Hence
we must conclude that it is more probable that the curve is of Type I than of
Type III. This is readily determined and is usually sufficient guide. Actually
the value of XSj must be about "6 before we get an ellipse to approximately touch
the Type III line. But 2, = '493/1-177 = '419, and accordingly \ = '6/-419 = 1-432,
which gives P = e~*XJ = '36 nearly, or the odds are 16 to 9 that the point would
not lie outside this contour. But if it did lie outside this contour, the chance
of ite being on or over the Type III line corresponds to only a very small section
of the total frequency outside this contour. If we invert the problem and put the
system of ellipses on the nearest point of the Type III line we find that the odds
are very much in favour of the point /9, = '678, /8« = 3734 lying outeide such a
system. On the whole it is reasonable to conclude that Type I is properly used
although we should probably not get bad results from a Type III curve. In
some respects a suitable fit would be obtained by using Type I, and fixing its

XXXV— XLVI] Introduction Ixix

start at zero *, but the vagueness of what is meant by ' percentage of black ' as a
factor, when the entire pigmentation of the skin probably arises from a single
melanin pigment, only varying in concentration in the pigment granules and in
the density of granules themselves. We have therefore contented ourselves by
fitting a Type I curve, as further illustration of the use of the tables in the
present work. The theory of fitting is given in the paper cited below f. Following
the usual notation we find :

r = 6 08, - ft - l)/(3& - 2& + 6) = 21-7755,
« = r»/{4 + JA (r + 2)V(r + 1)) = 57-764,468,
b- = /*,?•- (>• + l)/e = (36-9391)'.

Hence: m, = 2-0917, m.2=17'6838,

a, = 3-9071, «, = 33-0320,

and :

x

l-

V \ 33-03207
To find j/0 since m, is large, we use the approximation to the formula :

N (»i, + m. + 1) <r(""+1 ,

...... (lxxvm))

i— l i\

namely w-- (»*, + »i1+l) /OT, +ma I9\mi+mt m,)

~ b r (mi + 1)/(.-, ,»,»") v-~^-

the evaluation of the two P-functious for m, + wi, + 1 and ma + 1 following easily
by Stirling's Theorem. If we write Z= r(3-0917)/{e-8'0917 (2-0917)2-0917} we have

logZ= log 2-0917 -2-0917 log 2 0917,
+ log 1-091 7,
+ log T( 1-0917),
+ 2-0917 loge.

From Table XXXI (p. 58) we find log F (1-0917) = 1-979,8897 and loge is
given by Table LV (p. 143). Hence we determine, log Z= -576,5176. Evaluating
the rest of the expression for log ya we have :

log y0 = 2-233,3936, .

2/0=171-157.
Thus our curve is

(
1

* For method, see Phil. Traru. Vol. 186, A, pp. 370, 371.

t Phil. Traru. Vol. 186, A, pp. 367—370. See also Palin Elderton : Frequency Curves and Corre-
lation, Lay ton Brothers.

Ixx TcM*for8tati$tici<in*an<l Biomrtrician* [XXXV— XLVI

with origin at the mode = 16-1486 iu actual percentages, ~ 3-2297 in working
units. To calculate // we take the origin at 0 and have

logy = 26-134,8705 + 2-0917 log (*•+ -6774)+ 17'6838 log (3G 2017 -<r),
\vln'iv ./ ni:iv In- put 0, 1, 2, 3, 4, 5... working units c<>rrrs|><>ii<ling to 0, 5, 10, I "•
20,25... actual percentages. The curve is shewn in the acrmn|>anying diagram,
and considering the nature of the data is a reasonable graduation.

auu

190
180
170
160

/

i

/*

150
14O

/

N

130
12O

i

/

\

\

110
100

7

\

Of)
BO
70
60

/

\

\

4O
30

1

1

\

20 i

""•

\

7

10 15 20 25 30 35 40 45 5O 55 60 65 70 75 80
Per cent, of Black in Skin Colour.

TABLE XLVIII (pp. 89—97).

Percentage frequency of Occurrences in a Second Sample of m after p Occur-
rences in a First Sample n. (M. Greenwood, Biometrika, Vol. IX. pp. 69 — 90.)

If we assume the tnith of Bayes' Theorem then an event having occurred
p times and failed q times in n trials, the chance that it will occur s times and fail
m — s times in a second series of m trials is :

/•I

I x»(\-xyidx

Jo

These results can be evaluated as all the indices are integers and the series
C,+ C, + Ct + ... + Ct+ ... expressed iu the usual hypergeometrical form :

m (m - 1) (m - 2)

\q \n + m +

n+l [i + ™ P + 1 + ^_(^il) (P
n+l\ ll^y + TO (2 (q +

5T+ U«X).

XL VIII]

Introduction

Ixxi

If « be large and p not widely different from q, then results may be obtained from
the Gaussian curve, using as S. D. A/m- -, but if either p/n or q/n be very

small and m or n are commensurable, this no longer holds*. The case, however,
of p and q widely different and n and m commensurable and themselves small
numbers frequently arises, especially in laboratory work or in the treatment of
rare diseases f. The present table gives the evaluation of the hypergeometrical
series, formula (Ixxx) above, for a series of values of m, n, p and q. It is not
sufficiently comprehensive to allow of very accurate interpolation in certain of its
ranges, but it has involved a large amount of work, and will undoubtedly be
of help till a more complete table can be calculated. Meanwhile if the reader
feels in doubt as to any interpolation, it is not a very arduous task to calculate
the result required from formula (Ixxx) by aid of Table XLIX.

Illustration (i). In a batch of 79 recruits for a certain regiment four were
found to be syphilitic. What number of syphilitics may be anticipated in a
further batch of 40 recruits ?

Here n = 79, p = 4, q = 75 and m = 40. We must first interpolate in the
p = 4> column on p. 97 between n = 100, m = 25, and n= 100, TO =50 for m = 40,
ie. we must go £$ towards the m=5Q series, or we must add 0'4 times the first
series to 0'6 times the second series. We then repeat the same process for the
series for p = 4 and n = 50, m = 25 and n = 50, m = 50 on p. 95. There results :

Occurrences

n=50, m = 40, p=4

n=100, w = 40, p = 4

0

6-8654

20-7406

1

13-5880

26-7023

.'

16-3802

21-5249

-;

15-7086

14-1138

;

13-2702

8-2460

5

10-3867

4-4566

6

7-7259

2-2637

7

B-M78

1-0886

S

3-8221

•4977

9

2-5583

•2171

10

1 -6581

•0906

11

1-0409

•0362

12

•6332

•0139

IS

•3734

•0052

lit

•2137

•0019

15

•1188

•0007

16 and over

•1311

•0003

* For a full discussion of the subject : see Pearson, " On the Influence of Past Experience on Future
Expectation," Philosophical Magazine, 1907, p. 365.

t Tables recently published by Boss and Stott ("Tables of Statistical Error," Annals of Tropical
Medicine and Paraiitology, Vol. v. No. 3, 1911), appear to be designed to meet such cases, but being
based on the Gaussian curve are, I think, very likely to lead the user to fallacious conclusions.

Ixxii Tnbloi _t'ur Statistician* and .Bioim-trici'im* [XhVIII

\V. must interpolate between these two series for n = 79, that is we must take
042 times the first series and 0'58 times the second series. The results are
given below, and set against the direct calculation from formula (Ixxx), using
Table XLIX.

Rj Interpolation. Direct Calculation.

« = 70, y> = 4, »i = 40

n = 79,j> = 4, m = 40

II

1 l-.H»i

12-6143

/

-M-I943

21-937!)

.'

!'.»•:«; IL' 22-5152

3

1 1-7828

17-6667

k

10-3562

11-6727

5

6-9472

6-8143

6

1-M78

3-6137

7

2-9529

1-7713

8

1 -8940

•8118

9

1-2004

•3507

10

•7489

•1436

11

•4582

•0560

1.'

•2740

•0208

IS

•1598

•0074

14

•0908

•0025

15 -0503

•0008

16

•0271

)

17

•0142

V0003

18 and over

•0140

)

The interpolation does not give a result very close to the actual series. For
example, not more than three syphilitics might be anticipated in 70 °/0 of samples
of 40 by the interpolated series ; actually not more than 3 are to be expected
in 75 % of samples. At the same time the result is much better than the
normal curve theory provides. In the latter case we have

Mean = 40x^ = 2025,
Standard-Deviation = \/40 x ^ x \ {} = 1'387.
Hence (3'5 - 2'025)/1'387 = 1-064

and by Table II this value of x corresponds to ^(1 + a) = '86, i.e. in 86 °/0 per cent,
of samples of 40, we should have not more than 3 cases. It will be seen therefore
that (i) the values at the latter end of the Table are not close enough to obtain
very accurate results by interpolation, but (ii) that the Gaussian gives a still
poorer approximation.

I /! /nitration (ii). Of 10 patients subjected by a first surgeon to a given
operation only one dies. A second surgeon in performing the same operation
on 7 patients, presumably equally affected, loses 4 cases. Would it be reason-
able to assume the second surgeon had inferior operative skill ?

XLVIII— XLIX]

Introduction

Ixxiii

On p. 91, we have the series for p = 2 when n = 10 for the values m = 5 and
m = 10. Taking "6 of the first series and -4 of the second we have :

Interpolation from
Table.

Actual Value from
formula.

71 = 10, HI = 7, p=l

1
n = 10, m = 7,p = l

0

3V9762

35-9477

1

30-6704

31 -4542

2

17-3366

18-8725

S

8-2114

8-9869

4

3-5248

3-4565

5

1-4446

1-0370

6

•5676

•2200

7

•1996

•0251

8

[-05611

9

[•0114]

10

[-0012]

The chance that if the two surgeons are of equal skill 4 or more patients will
die out of the second surgeon's 7 operations is -058 by interpolation and -047
actually. Hence the odds against the occurrence are 16 to 1 by the table and
20 to 1 actually. It will be observed that interpolation gives small values at
impossible numbers of deaths, but these have to be reckoned in to obtain the
total number 100. That all seven patients should die under the second surgeon,
if of equal skill, involves odds of 500 to 1 about in the interpolation result, but
4000 to 1 about actually. On the Gaussian hypothesis in the original problem the
mean = 7 x ^ = -7 and the S. D. = V7 x •& x & = -7937, and (3'5 - -7)/-7937 = 3'52
roughly, or this corresponds to odds of about 4545 to 1 — which are wholly un-
reasonable. Thus the Table gives by interpolation odds of approximately the
right value, which may serve many useful purposes, for those who are unable to
work out the values required from formula (Ixxx). At the same time it is clear
that a much larger Table with closer values of the quantities involved is desirable.

TABLE XLIX (pp. 98—101)

The Logarithms of Factorials. (Calculated by Julia Bell, published here for
the first time.)

This table was obtained by adding up in succession consecutive logarithms in
a table of logarithms to 12 figures. Not until the work was completed did we
realise the existence of the splendid table of C. F. Degen*, which was then used
to confirm our own results. De Morgan in his Treatise on the Theory of Probabilities
of 1837 published an abridgement to six decimals of Degen's Table of Factorials.
His values cannot, however, be trusted to the sixth figure of the mantissa. The

* Tabularum ad faciliorem et breviorem probabilitatis computationem utilum. Havniae, MDCCCIXIV.
This gives the logarithms of the factorials up to 1200 with 18 figures in the mantissa.

B. *

l\\iv Talk* for Statisticians <nnl Rionicfrfrimu \\IAX-L

use of a factorial table is extremely varied, especially in problems in probability
involving high numbers.

Illustration. In a certain district the number of children born per month
is 662 and the chance of a birth being male is -51 and of its being female '49.
Evaluate the chance that in a given month there should be an equal number of
boys and girls born, and compare it with the chance of the most probable numbers
(338 boys and 324 girls) being born.

The chance of equal numbers of boys and girls being born is :

Therefore

. n 111 \ 1-707,5702] +1581714,6156
1 1+1-690,19611 -1383-941,41 14
where the logs of the factorials are found from Table XLIX. Hence

logCT.- 260-660,6453) _g

+ 197773,2042} "

or C, = '027155, or once in about 36'8 months, say once in three years the records
may be expected to show equal numbers of boys and girls born in the month*.

The chance of the most probable number of boys and girls is given by

log CM= 338 x 1707,5702+ I581714,6ir>(>

+ 324 x 1-690,1961 - 709-645,9652

• 674-359,6453

Or Cm = '030993, or the most probable numbers will only be born once in
32'3 months, or say once in two years and eight months.

We have Ce/Cm='876, or the chance of equal boys and girls is 88% of the
chance of the most probable numbers of boys and girls.

TABLE L (pp. 102—112)

Tables of Fourtli- Moments of Subgroup Frequencies. (Calculated by Alice Lee
and P. F. Everitt ; published here for the first time.)

In the usual method of determining the raw moments of a frequency, we take
moments about an arbitrary origin, which is towards the apparent mode and

• Actually of course the problem is more complex, because the number of children born per month
is not constant.

Introduction

Ixxv

multiply by plus and minus abscissae increasing by units — the ' working unit.'
Thus an error made in an early moment may be carried on to the later moments.
To control the results Table L was calculated a number of years ago, and from it
the fourth moments for such frequencies as most usually occur can be read off at
sight, and the raw fourth moment column thus tested before proceeding further.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Head
Length

Frequency

Abscissa

AV

AV2'

AV

AV

Table L

171

1

-20

- 20

+ 400

- 8,000

+ 160,000

160,000

2

1

19

19

361

6,859 130,321

130,321

fi

2

18

38

648

11,664 209,952

209,952

4

0

17

—

—

— —

—

5

3

16 48

768

12,288 196,608

196,608

6

3

15 45

675

10,125 151,875

151,875

7

5

14

70

980

13,720 192,080

192,080

8

7

13

91

1,183

15,379 199,927

199,927

9

12

12

144

1,728

20,736 248,832

248,832

180

13

11

143

1,573

17,303 190,333

190,333

1

17

10

170

1,700

17,000 170,000

170,000

2

28

9

252

2,268

20,412

183,708

183,708

S

24 8

192

1,536

12,288

98,304

98,304

4

43 7

301

2,107

14,749

103,243

103,243

5

53 6

318

1,908

11,448

68,688

68,688

8

:>7 5 285

1,425

7,125

35,625

35,625

7

55

4

220

880

3,520

14,080

14,080

8

88

3

204

612

1,836

5,508

5,508

9

83

2

166

332

664 1,328

1,328

190

85

- 1

- 85

+ 85

85

+ 85

85

1

96

0

-

—

—

—

2

102

+ 1

+ 102

+ 102

+ 102

+ 102

102

3

79

2

158

316

632

1,264 I 1,264

4

&3

3

249

747

2,241

6,723

6,723

5

66

4

264

1,056

4,224

16,896

16,896

e

66

5

330

1,650

8,250

42,250

41,250

7

50 0

336

2,016

12,096

72,576

72,576

8

43

7

301

2,107

14,749

103,243

103,243

9

to

8

280

2,240

17,920

143,360

143,360

200

30

9

270

2,430

21,870

196,830

196,830

1

20

10

200

2,000

20,000

200,000

200,000

2

24

11

264

2,904

31,944

351,384 351,384

8

14

12

168

2,016

24,192

290,304 290,304

4

13

13

169

2,197

28,561

371,293 371,293

5

8

14

112

1,568

21,952

307,328 307,328

6

3

15

45

675

10,125

151,875

151,875

7

6

16

96

1,536

24,576

393,216

393,216

8

0

17

—

—

—

—

—

9

1

18

18

324

5,832

104,976

104,976

210

1

+ 19

+ 19

+ 361

+ 6,859

+ 130,321

130,321

Totals

1306

—

+

+

+

+

—

Ixxvi Table* for Stufixtictans am/ /iinmt'fn'fitmx [L- LI

The multiplication can therefore be done very rapidly and it suffices to re-examim-
not the whole of the arithmetic l.ut only those rows which do uot agree with the
table.

lUu»tmtwn. Calculate the first four raw moments of the distribution of head
lengths in 1306 non-habitual criminals on the previous page and test whether
they are correct.

This was an actually worked out case, and it will be seen that in this instance
only one slip was made — that of a wrong multiplication by 5 in the contribution
to the fourth moment of the frequency of head lengths 196. Often far more
serious blunders are found. Correction would be made and the columns then
added up on the adding machine. Two points should be noticed. First it is not
in practice necessary to copy out the results from Table L, — they are merely
compared on the table itself with the items in column (vii) and any divergence
noted. Secondly in actual practice, it would be quite sufficient to take 20 instead
of 40 sub-groups in this case. Sheppard's corrections would fully adjust for the
difference.

TABLE LI (pp. 113—121)

Tables of the General Term of Poisson's Exponential Expansion ("Law <>{'
Small Numbers"). (H. E. Soper, Biometrika, Vol. X. p. 25.)

The limit to the binomial series

(Ixxxi),

when q is very small, but nq = m is finite, was first shewn by Poisson to be

(Ixxxii).

The present table provides the value of the terms of this series, i.e. e~mmx/a;l
to six decimals for m = O'l to m = 15 by tenths.

A previous table for m ~ O'l to m = 10 to four decimals has been published by
Bortkewitsch*, but his values are not always correct to the fourth decimal.
Poisson's exponential limit to the binomial has been termed the " Law of Small
Numbers " by Bortkewitsch, but there are objections to the term. The approxi-
mation depends on the smallness of q (or, of course, p) and the largeness of n, so
that the mean m is finite. Thus 100 murders per annum might be quite a "small
number," if they occurred in a population of 40,000,000, for H would be large and
q would be small. It is therefore space and time which has limited the present
table to m = 15, not the idea of m being small of necessity.

Illustration (i). The number of monthly births in the Canton Vaud being
taken as 662, and one birth in 114 being that of an imbecile, find the chance of 12
or more imbeciles being born in a month.

" Dot Qttetz der klrinen /.ahlrn, Leipzig, 1898.

LI— LII]

Introduction

Ixxvii

/113 1 X882
The binomial is I =— ^ + -— j . n is accordingly large and q small, while

749 = 5 "8 nearly. We look out 5'8 in Table L and sum the terms for 12 and beyond.
We find the chance of 12 or more = '01595. Actually worked from the binomial,
it is "01564. Or about once in five years, we might expect in Canton Vaud a
month with 12 imbecile births*.

Illustration (ii). Bortkewitsch (loc. cit. p. 25) gives the following deaths from
kicks of a horse in ten Prussian Army Corps during 20 years, reached after
excluding four corps for special reasons:

Annual
Deaths

Frequency
Observed

Frequency
Poisson's Series

ii

109

108-72

1

65

66-22

:

•2-2

20-22

s

3

4-12

4

1

•63

5

—

•08

6 and over

—

•01

Totals ...

200

200

The mean m of the observed frequency is -61, whence using Table LI (p. 113)
and taking '9 the series for 0'6 and -1 times the series for 0'7, we reach figures,
which multiplied by 200 give us the column headed " Frequency, Poisson's Series "
above. Such good agreement, however, is very rare. A good fit to actual data
with the Exponential Binomial Limit is not often found. Its chief use lies in
theoretical investigations of chance and probable error : see Whitaker, Biometriku,
Vol. x. p. 36.

TABLE LII (pp. 122—124)

Table of Poisson's Exponential for Cell Frequencies 1 to 30. (Luoy Whitaker,
Biometrika, Vol. x. pp. 36—71.)

Given a cell in which the frequency is n, corresponding to the population N.
Then if n, and N are very large (or we suppose, without this, the individual to be
returned before a second draw), the number in this sth cell will be distributed in
M samples of m according to the binomial law

* See Eugenia Laboratory Memoirs, XIII.
p. 22.

A Second Study of the Influence of Parental Alcoholism, "

Ixxviii Tablf* for Statixticiniix an*l /ii>nti<fn'ri,iits 1. 1 1

The mean will be mn,/N and tin- standard deviation A/ m £ f 1 — "'} . If

we

only have a single sample of m and do not know the distribution in the actual
population we are compelled to give n,jN the value m,/m, where m, is the number
found in the «th cell of the sample. If n,/N or m,/m be very small and »i large-
the binomial will approach Poisson's Exponential Limit, and in such cases the
deviations in the samples for the sth cell will be distributed very differently from
those following a Gaussian law, and the usual rule for deducing the probability of
deviations of a given size by means of the probability integral fails markedly.
It is not till we get something like 30 out of 1000 in a cell that we can trust the
Gaussian to give us at all a reasonable approach. The present table endeavours
to provide material in the case of cell frequencies 1 to 30, which will supply the
place of the probability integral.

Illustration (i). Suppose the actual number to be expected in a cell is 17,
what is the probability that the observed number will deviate by more than 5
from this result? Looking at p. 123 we see that in 8'467 °/0 of cases there will be
a deviation in defect of 6 or more and in 9'526 °/0 of cases a deviation in excess of
6 or more. Hence in 17'993 % say 18 "/„ of cases we should get values less than
12 or greater than 22. Thus once in every 5 or 6 trials we should get values
which differ as widely as 6 or more from the true value.

Now look at the matter from the Gaussian standpoint. The standard
deviation is

Here TO is supposed large compared with 17, so that the S. D. = Vl7 = 4-123
nearly. But suppose HI = 800, we should have

S. D. = Vl7 (1 - •<)21-25) = Vl7x-9787r) = 4-079.

Now we want deviations in excess of 5, i.e. we must take 5>5/4'079 = T348.
If we turn to Table II we find for this argument

£(1 + a) = '9102 or £ (1 -a) = '0898.

Hence we should conclude that in not more than 17'96°/0 of cases would deviations
exceed ± 5. Actually such occur in 17'99 "/„ of cases. Thus the actual per-
centages are very close, but the Poissou series tells us that 8P47 °/0 of cases will
be in defect and 9'53 °/0 in excess, while the Gaussian gives 8'98 % in both excess
and defect. We may further ask the percentage of times that 17 itself would
occur; according to the Gaussian it will occur in 9'76 % °f trials, actually it will
occur in 9'63 °/0- With values of cell-frequency less than 17, say in the single
digits, far greater divergences will be encountered.

LII— LIII]

Introduction

Ixxix

Illustration (ii). Consider the fourfold Table below and discuss the relative
probabilities that it has arisen from a population which shews 0, 1, 2, 3, etc. indi-

A

Xot-.l

Totals

a

N"t-fl ...

127 -5
863-5

0

87

127-5
950-5

991

87

1078

viduals for this size of sample in the cell B, not-.4. On the assumption that 0
is really the population of this cell, the probability is unity. Hence we have the
following result.

Population)
of cell (

0

1

j

3

4

t?

6

7

8

9

10

11

12

13 &
over

Probability)
ofO [

1

•36788

•13534

•04979

•01832

•00674

•00248

•00091

•00034

•00012

•00005

•00002

•00001

•00000

occurring )

Sum = 1-58200.

Whence taking the a priori probabilities proportional to the probability of 0
occurring on the separate possibilities we have:

Probabilities that the Table arose from a population
with x in the B, not-A cell.

X

Probability

X

Probability

II

•632,110

7

•000,575

1

•232,541

8

•000,215

!

•085,550

9

•000,076

s

•031,473

10

•000,032

4

•011,580

11

•000,013

5

•004,260

12 -000,006

6

•001,568

18 and over

•000,000

The " association " of such a Table cannot therefore be considered " perfect," for
in 37 °/0 of cases it would arise from a Table with a unit or more in the B, not-J.
cell. The above is actually a Table of the correlation of stature in father and son.
Grave caution is therefore needful in discussing such "perfect association" tables.

TABLE LIII (p. 125)

Angles, Arcs and Decimals of Degrees. (Based on Button's Mathematical
Tables.)

This Table gives degrees in radians for the first two quadrants; it then gives
minutes and seconds from 1 to 60 in fractions of a degree and in radians. The

Ixxx Table* for Statuticiaiu am/ /*»o/w«ric/a//*

need of such a table is very obvious, au.l ari«,s in too great a variety of circum-

stances to be specified.

Illiutnttwii. It is rv.,uire.l to plot th.- .-urve»:

* = 14-99 17 tau 0,

Here log y = log 235'323 + 32-8023 log cos 6 - 4'5696 log « x
To cover the whole range of observations we must proceed fimn, ' - • - «Mo
rouehlv It will be found sufficient to take 0 by ^te*

*

and multiplying by the logarithmic cos.nes of 36 9 rt^
obtained by multiplying (taking the third factor from Tabl

4-5696 x log e x -017,4533 = '216,7955
on the machine and multiplying the result in succession by 3, 6, 9, etc.

jj j i i OQKQ91 — 9-171 fifi44 and the second column
The first column is added to Iog23532-i- «m,uc

Another problem sometimes arises given x to find y For

« = - 2-0885. Required to find the maximum ordmate yma.
tan 6 = - 2-0885/14-9917 = - '139,3104,

whence by a table of natural tangents

0 = - 7° 55' '851265,
= - r 55' 51".

The log cosine of this value of Q is

1-995,8962.

Table LIII gives us :

r= -122,1730 m are

.,., = -015,9989 „ „

•851,265' = -851,265 x "000,2909= -000.2*76 „ „

Hence 0= -'138,4195 „ „

• See PWI. Tran,. Vol. 186, A, p. 387. Pearson's Type IV frequency curve fitted to the .Utur.
of 2192 St Louis School OirU aged 8.

LIII — LIV] Introduction Ixxxi

Hence logymo= 2-371,6644

+ 32-8023 (- -004,1038)

+ 1-984,5521 x -138,4195

2-511,7510.
Hence ymo = 324-901.

TABLE LIV (pp. 126—142)

Tables of the 0 (r, v) Integrals. (Calculated by Alice Lee, D.Sc. Transactions
British Association Report, Dover, 1899, pp. 65 — 120.)

The purpose of this table is to obtain the value of the integral

0

(r, v) = f sinr 0e'°d6 . . . .(Ixxxiii).

Jo

In order to obtain small differences in tabulated values two additional
functions F (r, v) and H (r, v) are introduced.

The relations between the three functions are then expressed by the following
series of equations :

F(r, v) = e~*n Q(r, v) .............................. (Ixxxiv),

.....................

vr — 1

G(r>v) = e*"F(r,v) ................................. (Ixxxvi),

^^,.,) ............... (l-xvii),

where tan tf> = v/r.

Pearson's Type IV Skew Frequency Curve is of the form

-Flan-'-

Hence if N be its total area, i.e. the entire population under discussion,

^r f ' *i
Jo

a

o
N 1

B. I

Ixxxii Tables for Stati«ti<-i<in* and Bioinetrici>n<* [Ll\

The function H (r, v) is introduced because, as a rule, its logarithms have far
smaller differences and it is thus capable of more exact determination from a table
of double entry. Its physical relation to the curve may be expressed as follows;
let the origin be transferred to the mean, then if y, be the ordinate at the mean,

r,")

where a is the standard-deviation of the curve

a

.(xcii).

Vr — 1 cos <f>
The distance of the mean from the origin is given by

/*,' = — « tan <f> (xciii).

When r is fairly large :

- (xciv).

/

Hence --A/ -*/ «~'? (xcv)'

// (/, v) V ,- - 1 V 2ir

/l-4co.sa<4
where q = . / - J- ,

and thus the evaluation if $ be > 60° may be made by aid of Table II*.

Illustration. In the curve fitted to the statures of St Louis School Girls,
aged 8 (p. Ixxx), we have

N~ 2192, a =14-9917,

r = 30-8023, v = 4'56967.
Find y,.

We have tan <f> = vfr = '148,3548.

Hence <f> = 83 26''31315 = 8°'43855.

Turning to the Tables, p. 136, we see the large differences of \ogF(r, v) at
this value of <f>, and accordingly settle to work with log H(r, v).
We have for log H (r, v),

r = 30 r = 31

<f> = 8° -388,2032 -388,5583,

<j> = 9° -388,2278 •:{,S8,5822,

log H(r, v) = '388,2032 + ('4386) [24«] - i ("4386) x ('5614) [28]
= •388,2137.

• For a fuller dincnssion of these integrals see Phil. Train. Vol. 186, A, pp. 876—881, B. A. 7'nni».
Report, Liverpool, 1896, Preliminary Bcport of Committee..., and the /*./(. Trant. Report, Dover, 1899,
already cited.

— LV] Introduction Ixxxiii

</> = 8°-4386, r = 31:

log H(r, v) = -388,5583 + (-4386) [238] - £ (-4386) (-5614) [27]

= "388,5684.
<f> = 8°-4386, r=32:

log H (r, v) = -388,8910 + ('4386) [231] - } (-4386) (-5614) [26]

= -388,9008.
Hence <f> = 8H386, r = 30 8023 :

log H (r, v) = -388,2137 + -8023 [3547] - ± (-8023) (1977) [- 223]

= -388-5001.
Hence by formula (Ixxxv) :

log F (r, v) = v<f> log e + r+ I log cos <j> - \ log (»•-!) + log H (r, v).
Or, using Tables LIII and LV, we have

logF(r, v)= -292,2901 - -737,1249
+ 1-849,6578
+ -388,5001

•.-..•{0,4480
- -737,1249

\ogF(r, i/)- 1-798.8231
Finally from formula (xci) :

log y0 = log N - log a - 1-793,3231

= 3-340,8405 -1-175,8509

-1793,3231

•969,1740
= 2-371,76665.

Or y0 = 235-324*.

TABLE LV.

This table contains some miscellaneous constants in frequent statistical or
biometric use and requires no illustration. It has already been used in the
illustrations to previous tables.

I have had the generous assistance of my colleagues Miss E. M. Elderton and
Mr H. E. Soper in the preparation of the Illustrations to these Tables. I can
hardly hope that arithmetical slips have wholly escaped us in a first edition,
and I shall be grateful for the communication of any corrections that my readers
may discover are necessary.

* The value 235-323 obtained in Phil. Trant. Vol. 186, A, p. 387, was found by the approximate
formula (xciv) before tables were calculated.

Every reader may now see in what way the higher branches of mathematics
are concerned in our present subject. They are the abbreviators of long and
tedious operations, and it would be perfectly possible, with sufficient time and
industry, to do without their use When both the ordinary and the mathe-
matical result are derived from the same hypothesis, the latter must be the more
correct : and in those numerous cases in which the difficulty lies in reducing the
original circumstances to a mathematical form, there is nothing to show that we
are less liable to error in deducing a common sense result from principles too
indefinite for calculation, than we should be in attempting to define more closely,
and to apply numerical reasoning. — DE MORGAN.

Tables of the Probability Integral

TABLE I.

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

Permille

•000

•001

•002

•003

•004

•005

•006

•007

•008

•009

•010

•0!)

00

3-0902

2-8782

2-7478

2-6521

2-5758

2-5121

2-4573

2-4089

2-365<i

2-3263

•99

•ni

2-32G3

2-2904

2-2571

2-2262

2-1973

2-1701

2-1444

2-1201

2-0969

2-0749

2-0537

•98

•ii.'

2-0537

2-03:55

2-0141

1-9954

1-9774

1-9600

1-9431

1-9268

1-9110

1-8957

1-8808

•97

•a.;

1-8808

1-8663

1-8522

1-8384

1-8250

1-8119

1-7991

1 -7866

1-7744

1-7624

1-7507

•96

•04

1-7507

1-7392

1-7279

1-7169

1-7060

1-6954

1-6849

1-6747

1-6646

1-6546

1-6449

•95

•05

1-6449

1-6352

1-6258

1-6164

1-6072

1-5982

1-5893

1 -5805

1-5718

1-5632

1-5548

•94

•06

1-5548

1 -5464

1 -5382

1-5301

1-5220

1-5141

1-5063

1-4985

1-4909

1-4833

1-4758

•93

•07

1-4758

1-4684

r-Niii

1-4538

1-4466

1 -4395

1 -4325

1-4255

1-4187

1-4118

1-4051

•92

•08

1-4051

1-3984

1-3917

1-3852

1-3787

1-3722

1-3658

1-3595

1-3532

1-3469

1-3408

•91

•09

1 -3408

1-3346

1-3285

1-3225

1-3165

1-3106

1-3047

1-2988

1-2930

1-2873

1-2816

•90

•10

1-2816

1-2759

1-2702

1-2646

1-2591

1-2536

1-2481

1-2426

1-2372

1-2319

1 -2265

•89

•11

1-2265

1-2212

1-2160

1-2107

1-2055

1-2004

1-1952

1-1901

1-1850

1-1800

1-1750

•88

•12

1-1750

1-1700

1-1650

1-1601

1-1552

1-1503

1-1 455

1-1407

1-1359

1-1311

1-1264

•87

•13

1-1264

1-1217

1-1170

1-1123

1-1077

1-1031

1-0985

1-0939

1-0893

1-0848

1-0803

•86

•u

1-0803

1-0758

1-0714

1-0669

1-0625

1-0581

1-0537

1-0494

1-0450

1-0407

1-0364

•85

•J.5

1O3C4

1-0322

1-0279

1-0237

1-0194

1-0152

1-0110

1-0069

1-0027

0-9986

0-9945

•84

•Ji:

0-9945

0-9904

0-9863

0-9822

0-9782

0-9741

0-9701

0-9661

0-9621

0-9581

0-9542

•83

•17

0-9542

0-9502

0-9463

0-9424

0-9385

0-9346

0-9307

0-9269

0-9230

0-9192

0-9154

•at

•18

0-9154

0-9116

0-9078

0-9040

0-9002

0-8965

0-8927

0-8890

0-8853

0-8816

0-8779

•81

•10

0-8779

0-8742

0-8705

0-8669

0-8633

0-8596

0-8560

0-8524

0-8488

0-8452

0-8416

•80

•20

0-8416

0-8381

0-8345

0-8310

0-8274

0-8239

0-8204

0-8169

0-8134

0-8099

0-8064

•79

•21

0-8064

0-8030

0-7995

0-7961

0-7926

0-7892

0-7858

0-7824

0-7790

0-7756

0-7722

•78

0-7722

0-7688

0-7655

0-7621

0-7588

0-7."'l

0-7521

0-7488

0-7454

0-7421

0-7388

•77

0-7388

0-7356

0-7323

0-7290

0-7257

0'7:!:!.->

0-7192

0-7160

0-7128

0-7095

0-7063

•76

14

0-7063

0-7031

0-6999

0-6967

0-6935

0-6903

0-6871

0-6840

0-6808

0-6776

0-6745

•75

•25

0-6745

0-6713

0-6682

0-6651

0-6620

0-6588

0-6557

0-6526

0-6495

0-6464

0-6433

•74

•26

0-6433

0-6403

0-6372

0-6341

0-6311

0-6280

0-6250

0-6219

0-6189

0-6158

0-6128

•73

•ft

0-6128

0-6098

0-6068

0-6038

0-6008

0-5978

0-5948

0-5918

0-5888

0-5858

0-5828

•72

•28

0-5828

0-5799

0-5769

0-5740

0-5710

0-5681

0-5651

0-5622

0-5592

0-5563

0-5534

•71

•29

0-5534

0-5505

0-5476

0-5446

0-5417

0-5388

0-5359

0-5330

0-5302

0-5273

0-5244

•70

•so

0-5244

0-5215

0-5187

0-5158

0-5129

0-5101

0-5072

0-6044

0-5015

0-4987

0-4959

•69

•SI

0-4959

0-4930

0-4902

0-4874

0-4845

0-4817

0-4789

0-4761

0-4733

0-4705

0-4677

•68

•S2

0-4677

0-4649

0-4621

0-4593

0-4565

0-4538

0-4510

0-4482

0-4454

0-4427

0-4399

•67

•S3

0-4399

0-4372

0-4344

0-4316

0-4289

0-4261

0-4234

0-4207

0-4179

0-4152

0-4125

•66

•34

0-4125

0-4097

0-4070

0-4043

0-4016

0-3989

0-3961

0-3934

0-3907

0-3880

0-3853

•65

•S5

0-3853

0-3826

0-3799

0-3772

0-3745

0-3719

0-3692

0-3665

0-3638

0-3611

0-3585

•64

•86

0-3585

0-3558

0 •:}-).•? 1

0-3505

0-3478

0-3451

0-3425

0-3398

0-3372

0-3345

0-3319

•63

•S7

0-3319

0-3292

o-3:J66

0-3239

0-3213

0-3186

0-3160

0-3134

0-3107

0-3081

0-3055

•62

•S8

0-3055

0-3029

0-3002

0-2976

0-2950

0-2924

0-2898

0-2871

0-2845

0-2819

0-2793

•61

•so

0-2793

0-2767

0-2741

0-2715

0-2689

0-2663

0-2637

0-2611

0-2585

0-2559

0-2533

•60

•4"

0-2533

0-2508

0-2482

0-2456

0-2430

0-2404

0-2378

0-2353

0-2327

0-2301

0-2275

•59

•41

0-2-275

0-2-2M

0-2224

0-2198

0-2173

0-2147

0-2121

0-2096

0-2070

0-2045

0-2019

•58

•-'/-'

0-2019

0-1993

0-1968

0-1942

0-1917

0-1891

0-1866

0-1840

0-1815

0-1789

0-1764

•57

•4-1

0-1764

0-1738

0-1713

0-1687

0-1662

0-1637

0-1611

0-1586

0-1560

0-1535

0-1510

•56

•44

0-1510

0-1484

0-1459

0-1434

0-1408

0-1383

0-1358

0-1332

0-1307

0-1282

0-1257

•55

•45

0-1257

0-1231

0-1206

0-1181

0-1156

0-1130

0-1105

0-1080

0-1055

0-1030

0-1004

•54

•¥>

0-1004

0-0979

0-0954

0-0929

0-0904

0-0878

0-0853

0-0828

0-0803

0-0778

0-0753

•53

•47

0-0753

0-0728

0-0702

0-0677

0-0652

0-0627

0-0602

0-0577

0-0552

0-0527

0-0502

•52

•48

0-0502

0-0476

0-0451

0-0426

0-0401

0-0376

0-0351

0-0326

0-0301

0-0276

0-0251

•51

•40

0-0251

0-0226

0-0201

0-0175

0-0150

0-0125

o-oioo

0-0075

0-0050

0-0025

o-oooo

•50

•010

•009

•008

•007

•006

•005

•004

•003

•002

•007

•000

Permille

2

Tablfti for Statisticians and
TABLE II. Area and Ui-dinate in terms of Abscissa.

•

!(!+«)

A

?

i

A

*

,„,

01

•".

•03
•04

•5000000
B0898M

•50797-3
•5119665
•5159.'.:tl

30804
30800
30882
30870

0

4

12
16

•3989423
•3989223
•3088625

•3987(12-
•39802:13

109

.vis

097
1886

Ml

300
300
398
398

•05

•5190388

39831

20

•3984 139

179:1

2191

397

•06
•07
•08
•09
•10

•5270032
•5318814
•5358564
•5398278

30810
397
39750
30714
30675

24
28
32
36
40

•3982248
•3979061
•3976677
•3973298
•3969525

1688

29-1

3379
3773
4166

397
3!H1
395
394
393

•11
'12
•13

•14
•IS

•5437953
•5477584
•55171 < is
•5556700
•5596177

39(131
395- 1
30532
30477
39418

44
48
51

55
50

•3966360
•3060802
•3955854
•3950517
•3944793

4558
4948
5337
5724
6110

392

390
3s!)
387
380

•16
•17
•18
•19
•20

•5635595
•5674919
•5714237
•57634.V1
•5792507

303:.:.
30288
39217
39143
39065

63
67

71
74

78

•39381 is}
•3932 190
•3925315
•3918060
•3910427

6493
6875
7255
7(133
8008

384
382
380
378
375

•XI

•j.'

•~">
•25

•5831662
•5870644
•5909:.41
•594-
•5987063

38983
38897
38808
38715
38618

82
86
89
88
97

•3902419
•3894038
•3885286
•3870 Kit!
•3866681

8381
8752
9120
9485
9847

373

371
308
365
362

•:•:
•30

•6025681
•6064199
•6102612
•6140919
•6179114

3s;,i8
38414
38306

38195
38081

100
104
107
111
114

•3856834
•3846027
•3836003
•3825146
•3813878

10207
10564
10017
11208
11615

360
357
354
350
347

•31
•32
•S3
•SJL
•35

•62551.-.-
•6293000
•6330717
•6368307

37063
37842
37717
37
37458

118
121
12:.
128
131

•38022(11
•3790305
•3778007
•3765372
•3752403

11058
1229S
12635
12008
13297

344

340
337
333
329

•36
•37
•38

•at
•40

•6405764
•6443088
•6480273
•6517317
•6554217

37323
37185
37044

3(1900
307 53

135

138
141
144
147

•3739106
•3725483
•3711539
•3697277
•3682701

13623
13044
14262
14575

14885

325
322
318
313

309

41
42
•43
•44
•45

•6590970
•6627573
•6664022
•6700314
•673(1 lis

36602
3(1149
3(1293
36133
35971

150
153
156
159
162

•3667817
•3652(127
•3637136
•3621349
•3605270

15190
16491

15787
16079
16367

305
301
896

292
288

•46

•47

;-

(1772419
•0808SSB

•6879331
•6914625

35806
35638
35467
35294

165
168
171
173
176

•3588003
•3572253
•355532:.
•3538124
•3520653

16650
16028
17202
17470

283
278
274
269
264

•

*<!+«)

A

+

A*

•50
•51

•liin i.:--.
•69497 13
•WHU';^1

35118
3493!)

178
179

181

•53

•7IH9I l'i

347:>s

34">7 1

184

•54
•55

•7084016

•7088 |i (3

343^
34200

186

180

•:,<;
•67
•58
•59
•60

•71221 i< i3
•7156612

•7100127
•7224047
•7257 1 C9

34009
33815
33620
33422
33222

191
188

106
L86

200

•61
•62
•63
•64
•65

•7290691
•7323711
•7356527
•7380137
•7421539

33020
32816
32610
32402
32102

202
204
200

ion

2111

•66
•67
•68

•7453731
7486711

•751747s

31080

31767

11 \ - 1

212
214
215

•69

•70

•7549029
•7580363

31OOJ

31334
31116

217
219

•71
•72
•73
•74
•75

•7611470

•764237:.
•7673049
•7703.500
•7733726

30806
30674
30451
3022(i
30001

220

m

223
225
220

•76
•77
•78
•79
•80

•7763727
•7793501
•7823046
•7852361
•7881446

29773
20545
29316
29085
28853

227
228
230
231
232

•81
•8S
•83
•84
•85

•79102!i:t
•7938919
•70673011
•709.ri i:>s
•802337.'.

28620
28387
28152
27917
27680

233
234
235

23.'.
236

•86
•87
•88
•89
•90

•8051055
•80784!»8
•8105703

•8132(171
•8 1593! 19

27443
27205
26007
26728
26480

237
238
238
239
239

•91

•!>..'

•:>.!

•94
•95

•8185887
•8212131!
•8238145
•8263912
•8289439

26240
26008
25768
25527
25285

240
240
241
241
241

•96
•91

•98
•99
1-00

•8314724
•8339768
•8364569
•838!) 1 2! i
•8413447

25044
84809

24560

2 13 IS

242
242
848

212
212

Tables of the Probability Integral
TABLE II.— (continued).

z

A

A>

•3520653
•3502919
•3484925
•3466677
•3448180
•34294:J9

17734

17994
18248
18497
18741
18981

264

259
254
249
244
239

•3410458
•3391243
•3371799
•3352132
•3332246

19215
19444

19667
19886
20099

234
229
224
219
213

•3312147
•3291840
•3271330
•3250623
•3229724

20307
20510
20707
20899
21086

208
203
197

192

187

•3208638
•3187371
•3165929
•3144317
•3122539

21267
21442
21613
21777
21936

181
176
170
MB
159

•3100603
•3078513

•:t< 150274
•3033893
•3011374

22090
22239

22381
2251!)
22650

154
148
143
137
132

•2988724
•2965948
•2943050
•2920038
•2896916

22777
22897
23013
23122
23227

126
121
115
110
104

•2873689
•28503(11
•2M:>6945
•2803438
•2779849

23325
23419
23507
23589
23666

99
93

88
83
77

•27.-.0182
•2732444
•2708640
•2684774
•2660852

23738
23805
23866
23922
23972

72
66
61
56
51

•2636880
4613863

•2588805
•2564713
•2540591

24017
24058
2401 <:;
24122
24147

45
40
35
30
25

•2516443
•2492277
•2468095
•2443901
•2419707

24167
24182
24191
24196

20
15
10
5
0

X

*(! + «)

A

+

A2

z

A

A2

+

1-00
1-01
1-02
1-0.1
1-04
1-05

•8413447
•8437524
•8461358
•8484950
•8508300
•8531409

24076
23834
23592
23351
23109
22868

242
242
242
242
242
241

•2419707
•2395511
•2371320
•2347138
•2322970
•2298821

24196
24191
24182
24168
24149
24125

0
5
10
14
19
24

106
1-07
1-08
1-09
1-10

•8554277
•8576903
•8599289
•8621-134
•8643339

22626
22386
22145
21905
21665

241

241
240
240
240

•2274696
•2250599
•2226535
•2202508
•2178522

24097
24064
24027
23986
23940

28
33
37
41
46

1-11
1-12
1-13
1-14
1-15

•8665005
•8686431
•8707619
•8728568
•8749281

21426
21188
20950
20712
20475

239
239
238
237
237

•2154582
•2130691
•2106856
•2083078
•2059363

23890
23836
23778
23715
23649

50
54
58
62
66

1-16
1-17
1-18
1-19
1-20

•8769756
•8789995
•8809999
•8829768
•8849303

20239
20004
19769
19535
19302

236
235
235
234
233

•2035714
•2012135
•1988631
•1965205
•1941861

23578
23504
23426
23344
23259

70
74
78
82
85

1-81

1 ' ! .'
1-2X
1-24
1-25

•8868606
•8887670
•8906514
•8925123
•8943502

19070
18839
18609
18379
18151

232
231
230

22!)
228

•1918602
•1895432
•1872354
•1849373
•1826491

23170
23077
22981
22882
22779

89
93
96
99
103

I-;-;
1-27
14*
149

1-30

•8961653
•8979.-.T7
•8997274
•9014747
•9031995

17924
17697
17472
17248
17026

227
226
225
224
223

•1803712
•1781038
•1758474
•1730022
•1713686

22673
22564
22452
22337
22218

106
109
112
115
118

1-S1

1 '• 1 !

;•.;.;
1-84

1-35

•9049021
•9065825
•9082409
•9098773
•9114920

16804
16584
16366

16147
15930

222
220
219
818
217

•1691468
•1669370
•1647397
•1625551
•1603833

22097
21973
21847
21717
21585

121

124
127
129
132

J-M

1-S7
1-SfS
1-S9
1-40

•9130850
•9146565
•9162067
•9177356
•9192433

15715
15501
15289
15078
14868

215

214
212
211
210

•1582248
•1560797
•1539483
•1518308
•1497275

21451
21314
21175
21033
20890

134
137
139
142
144

1-41

]•>,'
I-',:
1-44

I'!,-

•9207302
•11221902
•9236415
•9250663
•9264707

14600
14453
14248
14044
13842

208
207
205
204
202

•1470385
•1455041
•1435046
•1414600
•1394306

20744
20596
20446
20294
20140

146

148
150
152
154

l-J,c,

1-47
1-lfi

1-49
1-50

•9278550
•92921S1
•9305034
•9318879
•9331928

13642
13443
13245
13049

201
199
197
196
194

•1374165
•1354181
•1334353
•1314684
•1295176

19985
19828
19669
19508

155
157
159
160
162

1—2

Table* for Sfatittirfuiix <tml Biometrici
TABLE II. Area and Uidinate in terms of Abscissa.

*

*(!+«)

A

A*

t

A

A1

1-50
1-51
1-52
1-53

•9331928
•9344783
•9357445
•9369916

12855
19661

12471

194
193
191
189

•1295176
•1275830
•1256646

•1237628

19346
19183

11*01*

162
163
165
166

1-54
1-55

•9388198

•

122-2
12094
11908

188
186

•121877:,
•1200090

[8660
18517

167
168

1-56
1-57
1-58
1-59
1-60

•9406201
•9417!i2 1
4498466
•9440826
•9452007

11724
11541
11360
11181
11004

184
188
181
179
177

•1181573

•1145018
•112701-2
•1109208

18348
18177
18006
17834
17661

169
170

171
172
173

1-61

;•<;.
1-64
1-65

•9463011
•9473-3! »
•9484493
•9494974
•9505285

10828
10654
10482
10311
10142

176
174
172
170
169

•1091548
•10740(il
•1051,7 l*i
•103!)<ill
•1022649

171-7
17312
17137
16962
16786

174
171
175
176
176

1-66
1-67
1-68
1-V9

•9516428
•9525403
•9535213
•9544860

9975
9810
9647

167
165
163
162

•1005864
•0989255
•0972823
•0956568

16609
16432
16255

177
177
177
178

1-70

•9554345

9325

160

•0940491

16077

15.-

178

1-71
1-72
1-73
1-74
1-75

•9563671
•9572838
•9581849
•9590705
•9599408

9167
9011
8856
8704
8653

168
156
165
153
151

•0924591
•0908870
•0893326
•08771"; 1
•0862773

16722
15544
15366
15188
15010

178
178

178
178
178

1-76
1-77
1-78
1-79
1-80

•9607961
•9616364
•9624620
•9632730
•9640697

8403
8256
8110
7966
7824

149
147
146
144
142

•0847764

•081827-
•0803801
•078951 ii

1483-2
14051
14477
14300
14123

178
178
177
177
177

1-81

1 •.* .'
1-83
1-84

i M

•9648521
•9666205
•9663750
•9671159
4878481

7684
7645
7409
7273
7140

140
139
137
185
133

•0775371)
•0761433
•0747i;i>'3
•0734068
•0720649

13946
13770
13594
13419
1324.r)

176
176
176
175
175

1-86
1-87
1-88
1-89
1-90

•9686572
•9692681
•9699460
•9706210
•9712834

7009
6879
6751
6624
6500

132
130
128
196
125

•0707101
•0694333
•0681436
•066871 1
•0656158

13071

1 2*117
1 2725
12553
12382

174
173
173

171

1-91

i •' ;
1-94

•9719334
•9725711
•9731 9(!0
•9738102
H7U119

6377
(i-255
6136
6018
6902

123
121
120
118
116

•0643777

•0631 5i ;c
•0619524
•0607<;52

12211
19041

11873
11705
11638

170
170
168
168
167

i •".

(•M

1-99
KM

•9750021
4768
•9761 1*2
•976701.-.
4778499

6787
6674
5568

6453

116
113
111
110
108

•0584409
•0573038

0661881

•05507*n

11372

11201;
11042
10879

166
165
164
163

'

*(!+«>

A

A«

.'•IX 1

.'•Hi

,.,,:

."".;
2-05

4779499

•9777

•9788217
•979324S
•9798178

6345
5989

5134
6031
4929

108
106
105
103

102
100

KM

MS

2-10

•9803007

•9807738

4819879

•9816911
•9821356

4731
4634
4589

4445
4352

95
94
92

2-11
2-12
2-13
2-14
2-15

•9825708
•9829970
4884149
4688996

•9842224

4262
4172
4084
8996

3913

91
89
88
86
85

2-16
S-17
2-18
2-19

>• 'n

•9846137
•984:-
•9853713
•9857379
•98609C6

3829
3747
8666

3587
3509

84
82
81
79
78

.".'!
;.'•;.'.-,

•9864474
•9867906
•98712C3
•9874545

•98777:..-.

3432
3357
3283
3210
3138

77

74
73

71

2-26
2-27

•9880894

3068
9989

70
69
68

>. >i,
2-30

•9881I-!':',
•9892759

2932
2865
1800

66
65

2-31
3-32
2-33
3-34
S-35

469
4900969
4903

2736
9674

2612
2552
2492

64
63
62
60
59

2-SG
2-37
MS

•9908626
•9911060
•9913437
•991575-
•9918025

2434

2377
2321

9967

2213

68
67

56
54

'-']''

•HH20237
•11112231)7
•9924506
•9926564

2160
2108
2058
2008

53
59

51

50

(( ,

•99285/2

1960

4»

•"'<'•

•9930531

48

.';-;.;

4989449

I88&

47

.. ,',,

4984809

•9937903

1820

1775

46
46
44

Tables of the Probability Integral
TABLE II. — (continued).

z

A

+

•0539910
•0529192
•0518636
•0508239
•0498001
•0487920

10717
10557
10397
10238
10081
9924

162
161
160
159
157
156

•0477996
•0468226
•0458611
•0449148
•0439836

9769
9616
9463
9312
9162

155
154
153
151
150

•0430674
•0421661
•0412795
•0404076
•0395500

9013
8866
8720
8575
8432

149
147
146
145
143

•0387069
•0378779
•0370629
•0362619
•0354746

8290
8149
8010
7873
7737

142
140
139
138
136

•0347009
•0339408

7608

135
133

•0331939
•032400:!

73:',7

132
130

•03173!»7

7077

129

•0310319
•0303370

i >i;'.)0546

6950
6824
6699

127
126
125

•0283270

6678

6455

122

O876818
•0870481

•i 1204265
•0258166
•0252182

0335
0210
6099

.-,'.. s 1
6870

120
119
117
116
114

•0246313
•0240556
•0234910
•0229374
•0223945

5757
5646
6536
5428
6322

113

111
110
108
107

•OS18884

0213407
•0208294

5217
5113

105
104

102

•(•203281
•0198374

5011
4910
4811

101
99

•0193563
•0188850
•0184233
•0179711
•0175283

4713
4617
4522
4428

98
96
95
93
92

X

Ki-HO

A

A2

z

A

A2

2-50
2-51
2-52
2-53
2-54
2-55

•9937903
•9939634
•9941323
•9942969
•9944574
•9946139

1731

1688
1646
1605
1565
1525

44
43
42
41
40
39

•0175283
•0170947
•0166701
•0162545
•0158476
•0154493

4336
4246
4157
4069
3982
3897

92
91
89
88
86
85

2-56
2-57
f-fiS

2-59
2-60

•9947664
•9949151
•9950600
•9952012
•9953388

1487
1449
1412
1370
1341

39
38
37
36
35

•0150596
•0146782
•0143051
•0139401
•0135830

3814
3731
3650
3571
3493

84
82
81
80
78

2-61
2-62
2-6S
2-64

•9954729
•9956035
•9957308
•9958547
•9959754

1306
1272
1239
1207
1176

35
34
33
32
32

•0132337

•0128921
•0125581
•0122315
•0119122

3416
3340
3266
3193
3121

77
76
74
73
72

.'•<;,;

2-68
2-69
2-70

•9960930
•9962074
•9963189
•9964274
•9965330

1145
1115

1085
1050
1028

31
30
29
29

28

•0116001
•0112951
•0109969
•0107056
•0104209

3051
2981
2913

2847
2781

70
69
68
67
66

2-71
2-73

•9966358
•996735!)
•9968333

1001
974

27
27
26

•0101428
•0098712
•0096058

2717
2654

64
63

62

2-74
2-76

•9909280
•9970202

948
922
897

26
25

•0093466
•0090936

2592
2531
2471

61
60

2-76

2-77
2-78
2-79
2-80

•9971099
•9971972
•9972821
•9973646
•9974449

873
849
825
803
781

24
24
23
23
22

•0088465
•0086052
•0083697
•0081398
•0079155

2413
2355
2299
2244
2189

59
57
56
55
54

2-81
S-82
2-83
2-84

•9975229
•9975988
•9976720
•9977443

759
738

717

22
21
21
20

•0076965
•0074829
•0072744
•0070711

2136
2084
2033

53
52

51
50

2-85

•9978140

697

20

•00687 28

1983

49

678

1934

2-86

Mr

2-88

•9978818
•9979476
•9980116

658
640

19
19

18

•0066793
•0064907
•0063067

1886
1839

48
47
46

S-89
2-90

•9980738
•9981342

622
604

587

18

17

•0061274
•0059525

1793
1748
1704

46

44

2-91
2-92

'•'!.',
t-i^l

•9981929
•9982 11)8
•9983052
•9983589

570
553
537

17
10
16
16

•0057821
•(K 156 160
•0054541
•0052963

1661
1019
1578

43

42
41

40

®'(]K

•9984111

522

15

•0051426

1537

40

507

1497

2-9G
2-97

•9984618
•9985110

492

15
11

•004992!)
•0048470

145!)

39
38

2-98

•9985588

478

14

•0047050

1421

37

2-99
3-00

•998f;i i:, 1
•9986501

400

14
13

•0015000
•0044318

1384
1347

36

35

0

Tables for Srittixtii-iiiii* <ni<f Ou
TABLE II. Area and Onlinnte in terms of Abscissa.

t

i(i+«)

A

A»

s

A

A'

SDO

•9986601

417

13

1818

35

3-01

•9986938
•9987301

*••> i
424

111

13

•IMI 13007
•00417 >

1877

36
34

31)4

•9987772
•9988171
4088668

** I I

387
37.".

12
12
12

•0041

•0038098

1210
1178
1146

33
32
32

3D6
$-07
MS
3D9
3-10

•9988933
•9989297
•0080660

•9989992

•9990321

884

353
342

332

11
11
11
10
10

•0036951
•0035836
•00347".!
•696

•0032068

1115
1085
1056
1087

999

31
30
89
29
28

s-n

3-12

•9990646
•9990957

31 2
1O-'

10
10

•0031609

•0030<;!.s

B71

27

27

s-is

•9991200

OVi

108

9

•00297.Y4

Ql ft

26

3-14
3-10

•9991553
•9991836

2*1
275

9

B

•0028835
•0027D13

y j o

893
868

26

2.',

3-16
3-17
3-18

•9992112
•9992378

•9992030

887

258

-'".II

9
8
8

•0027075
8831

•002." 1 1 2

843
820

24
24
23

3-19

!.!<!)2886
•9993129

•DU

242
235

8
8

•0024015
•0023* 1 1

771
752

23

22

3-21
3-24

s-ss

•9993.. HO
:!H10

•!)!)!• 102 I

•9994230

227
22H
213
200
200

7
7
7
7
7

•002-
4031649

•(K 120900
•0020290

731
710
689

860

21
21
20
20
19

3-27

•999-4 12!)
•9994023

193

187

6
6

•0019641

•on 11)010

631

19
18

3-28

•9991*10

lot

1H1

6

•001831)7

_'( ~

18

3-29

.,-:,,

•999-1!)!) 1
•9995166

AO1

175
169

C
6

•0017803
•0017220

677
500

17
17

••:!

.;-.;.;

•9995335
•9995499
4008868

164
159
153

6
5
6

•0016066
•0016122
•0015595

643
687

17
16
10

-. .'.

•9995811

148
143

6
6

•0016084
•0014687

496
481

16

15

3-36
3-37
3-38

•9996103
•9096212
•9996376

139
134
iin

5
6

4

•0014106
•0013039
•0013187

467
453

15
14
14

3-39

•9996605

lOvl

4

•00127-48

49ft

13

3-40

•9996631

121

4

•0012322

fi£U

413

13

3-41

•99907:. 2

U7

4

•0011910

400

13

•9996869

l

4

•0011510

1H>.

12

••-;••

4

•0011122

3 "(I

12

3-44

•9997091

IIH:

4

•0010717

i '>

12

3-45

•9997197

IVQ

102

4

•00103*3

868

11

3-40

•9997299

99

3

•0010030

142

11

3-47

•9997398

95

3

•0009689

O^£

331

11

3-48

•999741)3

3

•0001)3:.^

10

3-49

•9997.~.*."i

„

3

•0009037

10

•9W97674

3

•0008727

10

X

»a+4

A

I*

3-50

•9997674

VI'

3

3-51

gQ
. .1

3

s-r>»

•9997*12

oo

3

3-63

17928

•?•?

3

3-54

•9997ii!)l)

/ t

3

3-55

•9998074

"" -,

3

72

3-56

•9998146

3

3-57

16816

69

2

3-58

•9998282

ftf.

2

3-59

•OBOB847

DO

c->

2

3-60

0008408

oz

2

60

3-61

999*11;')

2

.••'•'.'

Kft

2

S-63

uU

•2

S-V4

•999.S037

54

2

,;•<;.-,

•9998689

60

2

3-66

•999873!)

2

3-67

•999*7*7

A*7

•2

3-68

*998K>,I

47

AK

2

•9998879

4O

i«>

2

s-w

9998989

4.5

4-2

2

S-71

•9998964

Af\

2

MSI

•11! 11)1)004

4U

.;-;.;

•9999048

3!)

•11! i! 11)080

37

3-75

•in 11)9116

30

35

.;•:<;

•ll!i!)9150

•9998

33

•>.»

3-78

•: in: »92 16

ai

1

•9999217

31

1

8-80

•9999277

30
on

1

29

3-81

•9999305

S-82

•9999333

0-

,:-x.;

•H9f)!l3.".!l

c

8-94

•mi! in:',*:,

5

3-85

•9999409

24

3-86

•!)!)!)!) 133

23

, i'",^/

•0800466

,•;-.«

•9999478

22
21

S-89

•9999408

• c( i

S-90

•9999519

•u

19

3-91

•9991):,:',!)

iii

3-92

•8999667

Uf

18

,;•'.!.:

1678

.:•'.<',

•9809698

3-95

•9999609

16

S-9G

•999902.-.

1 K

3-07

•9809641

to
15

.:-!iS

•99! 56

.:.''.'(

•9009670

14

1

4-00

4009688

1

Tables of the Probability Integral
TABLE II.— (continued).

z

A

A2

•0008727

10

•0008426
•0008135
0007853
•0007581
•0007317

301
291
282
273
264

10
9
9
9
8

256

•0007061
•0006814
•0006575
O006343
•0006119

247
239
232
224
217

8
8
8
8

7

•0005902
•0005693
•0005490

•00052!)!
•0005105

210
203
196

189

7
7
7
8
6

183

•0004921
O004744

177

6
6

•0004.-)73
•0004 ins

171
165

6
6

•0004248

160
155

5

•0004093
•0003944

149

5
5

•0003800

144

5

•0003661

139

5

•0003.-.^';

135
130

5

•0003396

4

•0003271

125

4

•IK)* 13149

121

4

OOo:;
•0002919

117
113

4
4

108

•0002810

4

•0002705

1 05

4

•0002*;' > 1

102

4

•OU0250I!

98

3

•0002411

M

3

91

•0002:):;i I

3

•00022:52

88

3

•0002147

85

3

00020*;:,

82

3

•0001987

79

3

76

O001910

3

•00018:57

73

3

•00017*;';

71

3

OOOlons

08

2

O001633

66

2

63

•0001 5C9

, . i

2

•0001 50N

ol

2

•00014!!)

59

2

•0001 393

57

2

•oo. 1 1 :j:;x

BC

2

X

,<!+.)

A

A2

~

A

A2

4-00

•9999683

13

1

•0001338

2

4-01

•9999696

1

•0001286

2

to''

•9999709

1 9

0

•0001235

2

•99997i'l

1 £i

•0001186

40

2

4-04

•9999733

•0001140

47

2

4-05

•9999744

11

•0001094

43

2

4'06

•9999755

in

•0001051

4.9

2

407

•9999765

1U

10

•0001009

9m
4ft

2

4-08

•9999775

IV

•0000! Mi!)

rtV

2

4-00

•9999784

•0000930

37

1

4-10

•9999793

9

•0000893

/

30

1

4-11

•9999802

•0000857

OR

1

4-12

•999981 1

•0000822

OJ

1

I/-;

•999! M9

8
a

•0000789

33

09

1

4008696

•0000757

ofl

1

4-16

•9999834

7
7

•0000726

81

30

1

4-16

•9999841

7

•0000697

90

1

4-17

•9999848

I

•0000668

•D

1

4-18

•9999^5 1

7

•0000641

27

1

4-1$

•9999861

g

•U000615

25

1

4-.'"

•9999867

6

•0000589

24

1

4-21

•9999872

•0000565

00

1

)/•'.'

•!)! 199878

5

•0000542

•0

99

1

'/'-' ;

•9999883

•0000519

ilfl

1

'i'-"i

•9999^

5

•0000498

91

1

•9999893

5

•0000477

& i
20

1

4-26

•9999898

•0000457

in

1

4-37

•99999U:;

4

•0000438

1«7

IS

1

•999!)! 107

•0000420

10

10

1

.'/•.'.''

•9! I! )!)!)! 1

4

•0000402

10

1

4'30

•9999915

4

•0000385

16

1

4-si

•9999918

A

•0000369

1C

1

•9999922

"±

3

•0000354

10

15

1

if •. :. :

•99999:;.-,

3

•0000339

4-34

•9999929

•0000324

1 J.

4-35

•9999932

3

•0000310

1*4

13

4-36

•9999935

3

•0000297

13

4-37

•9999938

Q

•0000284

19

4-38

•9999941

o

•0000272

1 a
1 9

0

4-39

•9999943

•O000261

131

4-40

•9999946

2

•0000249

11

4-41

•9999948

•0000239

10

4-42

•9999951

•0000228

in

4-43

•9999953

•0000218

i\i

q

4-44

•9999955

•0000209

i/

9

4-45

•9999957

2

•0000200

9

4W

•9999959

9

•0000191

g

4-47

•9999961

•

•0000183

8

4-48

•9999963

•0000175

g

4'4'J

•9999964

•0000167

4-50

•9999906

•0000160

8

TII Men for Statistician* and /iio»t<fn'<-itinx
TABLE II. Area and Urdinate in terms of Abtcissa*.

X

id+«)

*

4-M

ma

159837

4-51

67686

152797

4-5S

69080

146051

4-53

70508

139590

4-'-4

71KT:i

133401

4-55

73177

LS7473

4-56

71423

1217H7

4-57

75t;i i

11631;:!

4-58

76751

111159

4-50

77838

106177

4'tiO

78875

101409

4-61

79867

96845

4-68

80813

92477

4-63

81717

88297

4-64

82580

84298

4-65

83403

80472

4-1 ;i;

84190

76812

4-67

84940

73311

4-68

85C56

69962

.}-<;.9

8IS340

66760

4-70

86992

63698

4-71

87(il I

60771

;..'

882t>8

67972

4-73

•--774

ne

7-74

80314

527:'.: i

4-75

89*j:»

502!i:.

476

90320

47960

4-77

90789

4572H

4"78

91:>3.->

435! 1C

4-79

inc.tii

41559

4-80

92067

39613

4-81

92453

37755

.'/•*.'

92822

35980

4-8S

93173

34285

w

93508

32667

4-85

93827

3112:!

4-86

94131

29647

4-87

94420

28239

4-88

94696

26895

; 89

94958

25613

4-90

!i">:i08

24390

4-91

95446

23222

,..,,

95673

22108

4-M

95HMI

21046

4-94

90

20033

',-•'•

962MI

19066

4-01;

96475

18144

4-97

96652

17265

96821

16428

.i,-:>'.i

96981

15«:fi»

X

i(l+«)

1

."<•<»/

97133

14867

5-U1

97278

14141

:.-»:

97416

13450

97.VJ8

12791

:,•<!',

97672

12162

6-06

97791

11564

,"<•"';

97904

10994

5-07

98011

10461

5-08

98113

9934

6-00

98210

9441

5-10

98302

8972

c-ii

98389

8526

.-,•/.'

98472

8101

..•I.;

98551

7696

5-14

98626

7311

5-15

9861)8

6944

5-16

98765

6595

5-17

98830

6MH

5-18

98891

5947

5-19

98949

6647

5-M

99004

5361

5-~'/

99056

5089

6-Xt

99105

4831

;'>•:.;

99152

4585

99197

4351

s*e

99210

4128

6-26

99280

3917

5-27

99318

3710

e-u

99354

8685

5-29

99388

3344

5-30

99421

3171

6-31

99452

3007

.'.-.:;

09481

2852

.-,-.:.:

99509

2704

6-94

99535

2563

5-35

99560

2430

5-36

99584

2303

„•,;;

99606

2183

5-38

99628

2069

5-39

99648

1960

6-40

99667

1857

5-41

996.H.-,

1760

997(1:;

1667

,•;,;

997 IN

1579

e-44

!I!I734

1 l!i:,

5-45

99748

1416

5-41;

99762

1341

6-47

9977.r.

1270

.,-J.v

99787

1202

::-',:i

99799

1138

X

i(l-H')

1

5-.TO

99810

M77

C-51

1019

99831

! M l.'i

10

913

s-86

99857

H17

5-56

99865

773

5-57

99873

731

5-58

99880

691

6-59

99886

654

5-60

99893

618

5-61

99899

.',•>:.'

99905

553

.",•»;.;

99910

e/a

5-64

99915

494

5-65

99920

467

8-66

99924

t 11

6-91

99929

417

B-68

99933

3! II

6-69

!i!»:i:;i;

37 -2

6-70

99940

Wl

6-71

99944

332

99947

313

.',•',.:

90960

ne

99953

880

.-,-;.-,

99955

264

6<n

999.-."

249

s-rr

99960

235

8-78

99963

222

S'7fl

99965

210

5-80

99967

198

5-81

99969

187

.r.sv

99971

176

S-88

99972

166

5-8J,

99974

157

C-85

99975

148

8-88

99977

tag

5-87

99978

131

5-88

!>!W79

liM

8-89

99981

117

5-90

99982

110

8-91

99983

104

fi-'.i.1

99984

98

99985

92

•"''•'"/

99986

87

5-95

99987

82

5-96

99987

77

6-97

99988

73

5-.W

999NI

68

99990

65

l-.-IHI

99990

61

Prefix W999 to each entry.

Tables of the Probability Integral 9

TABLE III. Abscissa and Ordinate in terms of difference of Areas.

a

X

A
+

A2
+

A3

+

z

A

A2

A3

•00
•01

•OS
•03

•04

•05

•ooooooo

•0125335
•0250689
•0376083
•0501536
•0627068

125335
125354
125394
125453
125532
125631

0
20
39
59
79
99

20
20
20
20
20
20

•3989423
•3989109
•3988169
•3986603
•3984408
•3981587

313

940
1567
2194
2821
3449

627
627
627
627
627
628

0
0
0
0
0
1

•06
•07
•08
•00
•10

•0752699
•0878448
•1004337
•1130385
•1256613

125750
125889
126048
126228
126429

119

139
159
180
201

20
20
20
21
21

•3978138
•3974060
•3969353
•3964016
•3958049

4078
4707
5337
5967
6599

628
629
630
631
632

1
1
1
1
1

•11
•12
•IS

•14
•IS

•1383042
•1509692
•1636585
•1763742
•1891184

126650
126893
127157
127443
127751

221
243
264

286
308

21
21
22
22
22

•3951450
•3944218
•3936352
•3927852
•3918715

7232
7866
8501
9137
9775

633

634
635
636
638

1
1
1
1

2

•16
•17
•18
•19

•20

•2018935
•2147016
•2275450
•2404260
•2533471

128081
128434
128811
129211
129635

330
353
376

400
424

23
23
24
24
25

•3908939
•3898525
•3887469
•3875769
•3863425

10415
11056
11699
12344
12991

640
641
643
645
647

2
2
2
2
2

•21

• it

•23

•24
•25

•2663106
•2793190
•2923749
•3054808
•3186394

130084
130559
131059
131586
132140

449
474
500
527
554

25
26
27
27
28

•3850434
•3836794
•3822501
•3807555
•3791952

13641
14292
14946
15603
16262

649
652
654
657
659

2
2
3
3
3

•26

•27
•28
•29

•so

•3318533
•3451255
•3584588
•3718561
•3853205

132722
133333
133973
134644
135346

582
611
640
671
702

29
30
30
31
32

•3775690
•3758766
•3741177
•3722919
•3703990

16924
17589
18258
18929
19604

662
665
668
672
675

3
3
3
3

4

•SI
•32
•33
•34
•35

•3988551
•4124631
•4261480
•4399132
•4537622

136081
136849
137652
138490
139366

735
768
803
839
876

34
35
36
37
39

•3684386
•3664103
•3643138
•3621487
•3599146

20283
20965
21651
22342
23036

679

682
686
690
695

4
4
4
4
4

•36

•37
•38

•so
•40

•4676988
•4817268
•4958503
•5100735
•5244005

140281
141235
142231
143271
144355

914
954
996
1039
1085

40
42
43
45

47

•3576109
•3552374
•3527935

•3502788
•3476926

23735
24439
25148
25861
26580

699
704
709
714
719

5
5
6
5
6

•41
•42
•43

•44
•45

•5388360
•5533847
•5680515
•5828415
•5977601

145487
146668
147900
149186
150529

1132
1181
1232
1286
1342

49
51
54
56
59

•3450346
•3423041
•3395005
•3366233
•3336719

27305
28035
28772
29514
30264

725
730
736
743
749

6
6
6

7
7

•46
•47'
•48
•49
•50

•6128130
•6280060
•6433454
•6588377
•6744898

151930
153394
154923
156521

1402
1464
1529
1598
1670

62
65
69

72

•3306455
•3275435
•3243652
•3211098
•3177766

31020
31783
32554
33333

756
763
771
779

787

7
7
8
8

B.

10

Tables for Statisticians and

TABLE III. Abscissa and Ordinate in terms of difference of Areas.

a

*

A

+

A1

+

As

+

f

A

A1

A»

•50
•51
•5t
•53
•54
•56

•6744898
•6903088
•7063026
•7224791
•7388468
•7654150

168191
159937
161765
163678
165682
167782

1670
1747
1828
1913
2004
2100

76
81
86
91
96
102

•3177766
•31 43646
•3108732
•3073013
•3036481
•2999125

31119
34915
3.V719
36532
37356
38189

787

71)5

MM

814
823
834

9
9
9
10

10
11

•66
•57
•58
•59
•60

•7721932
•7'-!tl917
•8064212
•8238936
•8416212

169984
172296
174724
177276
179961

2203
2312

2428
2552
2685

109
116
124
133
143

•2960936
•2921902
•MttOU

•2841256
•2799619

39034
39889
40757
41637
42530

844
856
867
880
893

11
12
12
13
14

•61
•6*
•&J
•64
•65

•8596174
•8778963
•8964734
•9153651
•9345893

182789
185771
188917
192242
195760

2828
2981
3147
3325
3518

153
165

178
193
209

•2757089
•2713653
•2<i(i9295
•2624000
•2577753

43437
44358
45295
46247

47217

907
921
937
953
970

16
15
16
17

18

•66
•67
•68
•69
•70

•9541653
•9741139
•9944579
1-0152220
1-0364334

199486
20344(1
207641
212114
216882

3727
3954
4201
4472
4769

227
248
271
297
326

•2530535
•2482330
•2433117
•2382877
•2331588

48205
49213
50240
51289
62362

988
1007
1028
1049
1072

19
20
22
23
25

•71
•72
•7S
•74
•75

1-0681216
1-0803193
1-1030626
1-1263911
1-1503494

221977
227432
233286
239583
246374

5095
5455
5854
6297
6792

360
399
443
495
555

•2279226
•2225767
•2171185
•2115451
•2058535

53469
64582

55731
66916
68130

1097
1123
1152
1182
1215

26
28
30
33
36

•76
•77
•78
•79
•80

1-1749868
1-2003589
1-2265281
1-2535654
1-2815516

253721
261693
870373
279861

7347
7972
8681
9488
10414

625
709
808
926

•2000405
•1941024
•1880356
•1818357
•1754983

69380
60669
61999
63374

1250
1288
1330
1375
1425

38
42
45
60

Tables of the Probability Integral

11

TABLE IV.

Extension of Table of the Probability Integral F=$(I — a).

i r°°

F=-f=\ e~te*dx. The table gives (—logF) for x.
\2

irJ *

X

-logF

X

-log*'

X

-logF

5

6-54265

SO

197-30921

50

544-96634

6

9-00586

31

210-56940

60

783-90743

7

11-89285

32

224-26344

70

1066-26576

8

15-20614

S3

238-39135

80

1392-04459

9

18-94746

34

252-95315

90

1761-24604

10

23-11805

35

267-94888

100

2173-87154

11

27-71882

SC

283-37855

150

4888-38812

IS

32-75044

37

299-24218

ZOO

8688-58977

IS

38-21345

38

315-53979

850

13574-49960

14

44-10827

39

332-27139

300

19546-12790

15

60-43522

40

349-43701

350

26603-48018

1(1

57-19458

41

367-03664

400

34746-55970

17

64-38658

-(-'

385-07032

450

43975-36860

18

72-01140

43

403-53804

600

54289-40830

19

80-06919

44

422-43983

SO

88-56010

45

441-77568

N.B. To obtain any thing

SI

97-48422

46

461-54561

but a rough apprecia-

ss

106-84167

47

481-74964

tion after x = 50, the

S3

116-63253

48

502-38776

table would require

24

126-85686

49

523-45999

much extension, but

for many practical

25

137-51475

BO

544-96634

problems it suffices to

26

148-60624

take after # = 50:

27

160-13139

88

172-09024

J-. * I.-4*

29

184-48283

>/2ir*

30

197-30921

From each of the values in this table -30103 must be subtracted, if we wish to
obtain the probability ZF, then given by ( - log 2/1), that the value is greater than x.
without regard to sign.

a— 2

12 Table* for Statitfictdit* "//'/ /ti

TABLE V. Probable Errors of Means and Standard Deviations.

N

Xi

X.

1

•67449

•47694

*

•47694

•33724

3

•38942

•J7536

4

•337-'4

•23847

6

•30164

•21329

6

•27636

•19471

7

•26493

•18026

8

•23847

•16M;:!

9

•22483

•15898

10

•21329

•16082

11

•20337

•14380

It

•19471

•13768

IS

•18707

•13228

14

•18026

•11747

15

•17415

•12314

16

•16862

•11923

17

•163.59

•11567

18

•15898

•11241

19

•15474

•lOiM:!

SO

•15082

•10665

»1

•14719

•10408

tx

•14380

•10168

gs

•14064

•09945

-•;

•13768

•09736

KB

•13490

•09539

96

•13228

•09353

«r

•12981

•09179

S8

•12747

•09013

S9

•12525

•08856

30

•12314

•08708

31

•12114

•08566

3g

•11923

•08431

33

•11741

•08302

34

•11567

•08179

36

•11401

•08062

36

•11241

•07949

37

•11088

•07841

38

•10942

•07737

39

•10800

•07637

40

•10665

•07541

41

•10534

•07448

4*

•10408

•07359

43

•10286

•07273

44

•10168

•07190

4S

•10055

•07110

4e

•09946

•07032

47

•09838

•06957

48

•09736

•06884

49

•09636

•06813

60

•09539

•06746

n

Xt

X*

SI

•0941'!

•06678

•09363

•06614

S3

•09266

•06551

64

•0917!)

•06490

•09095

•06431

u

•09013

•06373

67

•08934

•06317

68

•08856

•06262

a

•08781

•06209

60

•08708

•06157

Cl

•08636

•06107

c,!

•o-i.->06

•06057

6S

•08 198

•06009

84

•08431

•05902

U

•083CC

•05916

C6

•08302

•05871

67

•08240

•058->7

68

•08179

•05784

M

•08120

•05742

70

•08062

•05700

71

•08005

•05660

•07949

•05621

73

•07894

•05582

74

•07841

•05544

75

•07788

•05507

76

•07737

•05471

77

•07687

•05435

78

•07637

•05400

79

•07589

•05366

80

•07541

•05332

81

•07494

•05299

88

•07448

•05267

83

•07403

•05235

84

•0735!)

•05204

85

•07316

•05173

KG

•07273

•05113

87

•07231

•O.M13

88

•07190

•O.V)84

89

•07150

•05056

90

•07110

•06027

91

•07071

•05000

9S

•07032

O407I

>i.;

•06994

•04946

94

•00957

•04819

95

•06920

•04893

86

•06884

•04868

97

•06848

•04843

98

•06813

•04818

99

•06779

•04793

100

•06748

•04769

n

X,

*1

101

•06711

•04746

/".'

•06678

•047:!:!

lot

•06646

•04699

104

•06614

•01077

105

•06582

•04654

106

•06551

•04632

107

•06521

•04611

108

•06490

•04

109

•06460

•01508

110

•06431

•04547

111

•06402

•04527

11!

•06373

•01507

113

•06345

•04487

114

•06317

•04IH7

115

•06290

•04447

116

•06i(y

•04 li'H

117

•06236

•04409

118

•06209

•04391

119

•06183

•0437:!

l.Jn

•06157

•04354

1S1

•00 132

•04330

1 .'2

•OOloT

•04318

i .'.;

•06082

•04300

124

•06057

•04283

US

•06033

•04266

i .•<;

•06009

•04249

127

•05985

•04232

1 .'.f

•05962

•ot216

; .".>

OMtt

•04199

I. »i

•05916

•04183

131

•05S'(3

•04167

133

•05871

•04161

133

•0584!)

•04136

134

•05S:J7

•04120

us

•05805

•04105

136

•05784

•04090

i.;:

•05763

•04076

1;:S

•05742

•04060

1S9

•05721

•04046

140

•05700

•04031

141

•05680

•04017

142

•05060

•04002

143

05610

OM68

144

•O.MIiJl

•03!)7 1

145

•06601

•03961

m;

•05582

•03917

147

•05563

•o:i!»:u

14$

•i '.-,544

•o:i()20

149

•055-20

•03907

150

•05507

•OMM

Tables for Facilitating the Computation of Probable Errors 13
TABLE V. Probable Errors of Means and Standard Deviations.

n

*i

X;

151

•05489

•03881

152

•05471

•03868

153

•05453

•03856

154

•05435

•03843

155

•05418

•03831

156

•05400

•03819

157

•05383

•03806

158

•05366

•03794

159

•05349

•03782

160

•05332

•03771

161

•05316

•03759

162

•05299

•03747

163

•05283

•03736

164

•05267

•03724

165

•05251

•03713

166

•05235

•03702

167

•05219

•03691

168

•05204

•03680

1G9

•05188

•03669

170

•05173

•03658

171

•05158

•03647

1 7 .'

•05143

•03637

173

•05128

•03626

nit

•05113

•03616

175

•05099

•03605

176

•05084

•03595

177

•05070

•03585

178

•05056

•03575

179

•05041

•03565

180

•05027

•03555

181

•05013

•03545

182

•05000

•03535

183

•04986

•03520

184

•(14972

•03516

185

•04959

•03507

186

•04946

•03497

187

•04932

•03488

188

•04919

•03478

189

•04906

•03469

190

•04893

•03460

191

•04880

•03451

19S

•04868

•03442

19S

•04855

•03433

194

•04843

•03424

196

•04830

•03415

196

•04818

•03407

197

•04806

•03398

198

•04793

•03389

199

•04781

•03381

200

•04769

•03372

n

*i

*2

201

•04757

•03364

202

•04746

•03356

203

•04734

•03347

304

•04722

•03339

SOS

•04711

•03331

206

•04699

•03323

207

•04688

•03315

308

•04677

•03307

209

•04666

•03299

210

•04654

•03291

211

•04643

•03283

212

•04632

•03276

21S

•046:!:;

•03268

214

•04611

•03260

S15

•04600

•03253

216

•04589

•03245

217

•04579

•03238

218

•04568

•03230

219

•04558

•03223

220

•04547

•03216

221

•04537

•03208

.'.'.'

•04527

•03201

.'.'.:

•04517

•03194

224

•04507

•03187

225

•04497

•03180

«M

•04487

•03173

227

•04477

•03166

228

•04467

•03159

MB

•04457

•03152

,.'.:<>

•04447

•03145

231

•04438

•03138

232

•04428

•03131

233

•04419

•03125

MU

•04409

•(•3118

235

•04400

•03111

S36

•04391

•03105

237

•04381

•03098

238

•04372

•03092

',:.;'.)

•04363

•03085

2JtO

•04354

•03079

241

•04345

•03172

2.1,2

•04336

•03066

243

•04327

•03060

244

•04318

•03053

245

•04309

•03047

246

•04300

•03041

247

•04292

•03035

S48

•04283

•03029

249

•04274

•03022

250

•04266

•03016

n

Xi

*2

251

•04257

•03010

252

•04249

•03004

253

•04240

•02998

254

•04232

•02993

255

•04224

•02987

256

•04216

•02981

257

•04207

•02975

258

•04199

•02969

259

•04191

•02964

260

•04183

•02958

261

•04175

•02952

262

•04167

•02947

263

•04159

•02941

264

•04151

•02935

2G5

•04143

•02930

266

•04136

•02924

267

•04128

•02919

268

•04120

•02913

269

•04112

•02908

270

•04105

•02903

271

•04097

•02897

272

•04090

•02892

273

•04082

•02887

274

•04075

•02881

275

•04067

•02876

S76

•04060

•02871

277

•04053

•02866

278

•04045

•02860

279

•04038

•02855

280

•04031

•02850

281

•04024

•02845

282

•04017

•02840

283

•04009

•02835

284

•04002

•02830

285

•03995

•02825

286

•03988

•02820

287

•03981

•02815

288

•03974

•02810

289

•03968

•02806

290

•03961

•02801

291

•03954

•02796

292

•03947

•02791

293

•03940

•02780

294

•03934

•02782

295

•03927

•02777

296

•03920

•02772

297

•03913

•02767

298

•03907

•02763

299

•03901

•02758

300

•03894

•02754

14 Tablr* for Statisticians ami liiimn-trii-!nnx

TAULK V. Probable Errors of Means and Standard Deviations.

n

Xt

xt

301

•03888

•02749

••:

•03881

•02744

•O:^::.

•02740

.;

•03M;^

•02735

u

•03862

•02731

.;<«;

•OH

•027 2( i

0)

•03850

•02722

SOS

•03843

•02718

,.,

•03837

•02713

310

•03831

•02709

..11

•038:!.-.

•02704

::

•03819

•0271* i

.;/.:

•03812

•02696

314

1H8M

•02692

315

•03800

•02687

316

•03794

•02683

317

•03788

•02679

318

•03782

•02675

sin

•0377t;

•02670

•03771

•02666

.11

•0371.-:.

•02662

• ::

•037:.: i

•02658

<087M

•02654

••:',

•03717

•02650

.,:.-.

•037 1 1

•02646

.

•03736

•02642

•03730

•096*7

as

•03721

•02633

•03719

•02629

330

•(13713

•02625

331

•03707

•02621

•03702

•02618

333

•03696

•02614

••',

•03691

•02610

.,

•03685

•02606

336

•03680

•02602

337

•03674

•02598

338

•03669

•02594

339

•03663

•02590

340

•03668

•02687

341

•03653

•02583

',•

•03647

•02579

•;••

•03642

•02575

•'44

•03637

•OJ571

;•

•03631

•i '2568

34C

•03626

•02664

347

•03621

•02660

348

•03616

•02657

349

•03610

•02563

.....

•03606

•02549

n

Xi

*»

.;.-,!

•03600

•02546

•03595

•02542

A'tS

•03590

•02538

•O358.r>

•02535

•03580

•02531

.;.-,<;

•03575

•02528

.-..-

•03570

•02524

sn

•03:.(;:,

•02521

::.-,!>

•03560

•02517

.:,:,!

•03555

•02614

SGI

•03560

•02510

.'..'

•03545

•02507

3ti3

•03540

•02503

3ti4

•03535

•02500

3V5

•03530

•02196

SGG

•03526

•02493

307

•03521

•02490

368

•03516

•02486

309

•03511

•02483

970

•03507

•02479

S71

•03502

•02476

378

•03497

•02473

373

•034!i2

•02469

374

•03488

•02466

375

•03483

•02463

376

•03478

•02460

377

•03474

•02456

378

•03469

•02453

S79

•03465

•02450

S80

•03460

•02447

S81

•03456

•02443

0M

•03451

•02440

MB

•03446

•02437

S84

•03442

•02434

385

•03438

•02431

386

•03433

•02428

387

•03429

•02424

388

•03424

•02421

389

•03420

•02418

390

•03415

•02415

391

•03411

•02412

39S

•i 13-107

•02409

MS

•i (3402

•02406

394

•03398

•02103

395

•03394

•02400

396

•03389

•02397

89)

•03385

•02394

m

•03381

•02391

399

•03377

•02388

400

•03372

•02385

n

Xi

xt

401

•03368

•02382

4i>2

•03364

•02379

403

•03360

•02376

.',"4

•03356

•0071

405

•033:.-'

•( 12370

406

•03347

•02367

407

•03343

•02364

408

•03339

•02361

4<i9

•03335

•02358

410

•03331

•02355

411

•03327

•02353

412

•03323

•osaeo

4is

•033 1!)

4t3«7

414

•03315

•U234 1

415

•03311

•02341

416

•03307

•02338

417

•03303

•02330

418

•03299

•02333

419

•03295

•02330

420

•03291

•02327

',-•!

•03287

•02324

',.'-'

•03283

•02322

1*9

•0327!)

•02319

',.",

•03271!

•02316

435

•0327-2

•02313

4S6

•03268

•0231 1

427

•032i ;i

•02308

', .'X

•032< ;o

•02305

429

•03256

•02303

',•<>

•03253

•02300

4S1

•03249

•02297

|M

•03245

•02295

4SS

•03241

•089M

4*4

•03-238

•02289

.',-••-'

•03234

•02287

436

•03230

•02284

4-97

•03227

•02281

438

•032-23

•1.2279

439

•03219

•02276

440

•03216

•02274

w

•03212

•02271

44*

•03208

•02269

443

•03205

•02266

444

•03201

•022(13

445

•03197

•02261

446

•03194

•02258

447

•03190

•0225<;

44*

•03187

•02253

449

•03183

•02251

450

•03180

•02248

Tables for Facilitating the Computation of Probable Errors 15
TABLE V. Probable Errors of Means and Standard Deviations.

n

*1

X,

j,r>i

•03176

•02246

453

•03173

•02243

453

•0316!)

•02241

4*4

•03166

•02238

455

•03162

•02236

456

•03159

•02233

H67

•03155

•02231

458

•03152

•0222!)

•','•••

•03148

•02226

!,<;<>

•0314.-,

•02224

.{•;/

•03141

•02221

.'/'•'-'

•03138

•0221!)

463

•0313')

•02217

.{'<•;

•03131

•02214

465

•03128

•02212

466

•03125

•02209

41:7

•03121

•02207

}U

•03118

•0220.->

/'•'•'

•03115

•022i >i'

470

•03111

•02200

471

•03108

•02198

(.' •'

•0310.-,

•02195

473

•03101

•02193

474

•03098

•02191

475

•03095

•02188

476

•03092

•02186

477

•03088

•02184

478

•03085

•02181

470

•03082

•02179

4X0

•oso; <)

•02177

481

•03075

•02175

482

•0307:2

•02172

;,,;

•03069

•02170

484

•03066

•02168

485

•03063

•02166

486

•03060

•02163

487

•03056

•02161

488

•03053

•02159

48!)

•03050

•02157

490

•03047

•02155

491

•03044

•02152

492

•03041

•02150

'/•'•••

•03038

•02148

494

•03035

•02146

495

•03032

•02144

496

•03029

•02142

-','•>•

•03026

•02139

498

•03022

•02137

yi'.i

•03019

•o:>13.-|

800

•03016

•02133

n

*i

*2

501

•03013

•02131

502

•03010

•02129

503

•03007

•02127

504

•03004

•02124

505

•03001

•02122

506

•02998

•02120

507

•02996

•02118

508

•02993

•02116

5'/.9

•02990

•02114

510

•02987

•02112

511

•02984

•02110

513

•02981

•02108

513

•02978

•02106

514

•02975

•02104

515

•02972

•02102

5i<:

•02969

•02100

517

•02966

•02098

518

•02964

•02096

519

•02961

•02094

,', .'"

•02958

•02092

621

•02955

•02089

-, • i

•02952

•02087

523

•0294!)

•02085

SM

•02947

•02084

525

•02944

•02082

536

•02941

•02080

527

•02938

•02078

528

•0293.-,

•02076

539

•02!m

•02074

:,;n

•02930

•02072

Ml

•02927

•02070

•02924

•02068

.-,.;.;

•02922

•02066

.-,.;;

•02919

•02064

•02916

•02062

•02913

•02060

•02911

•02058

r,.;8

•02908

•02056

SB

•02905

•02054

540

•02903

•02052

541

•02900

•02051

543

•02897

•02049

543

•02895

•02047

544

•02892

•02045

545

•02889

•02043

546

•02887

•02041

547

•02884

•02039

548

•02881

•02037

549

•02879

•02036

650

•02876

•02034

n

Xi

*2

551

•02873

•02032

553

•02871

•02030

55S

•02868

•02028

554

•02866

•02026

555

•028U3

•02024

656

•02860

•02023

557

•02858

•02021

558

•02855

•02019

559

•02853

•02017

560

•02850

•02015

561

•02848

•02014

563

•02845

•02012

.-,<;.:

•02843

•02010

264

•02840

•02008

5tio

•02838

•02006

£M

•02835

•02005

567

•02833

•02003

568

•02830

•02001

569

•02828

•01999

570

•02825

•01998

571

•02823

•01996

572

•02820

•01994

573

•02818

•01992

574

•02815

•01991

575

•02813

•01990

576

•02810

•01987

577

•02808

•01986

578

•02806

•01984

579

•02803

•01982

580

•02801

•01980

581

•02798

•01978

582

•02796

•01977

583

•02793

•01975

584

•02791

•01974

585

•02789

•01972

586

•02786

•01970

587

•02784

•01969

588

•02782

•01967

680

•02779

•01965

590

•02777

•01964

591

•02774

•01962

592

•02772

•01960

593

•02770

•01959

594

•02767

•01957

595

•02765

•01955

596

•02763

•01954

597

•02761

•01952

598

•02758

•01950

599

•02756

•01949

600

•02754

•01947

16 Tahlt-tt for Sttiti.tfli'iiinx and fiiometrieians

TABLE V. Probable Errors of Means and Standard Deviations.

n

Xi

x,

601

•02761

•01945

60S

•O2749

•01944

60S

•02747

•01942

604

•02711

•I'l'.ill

605

•02742

•01939

999

•02740

•01!):!7

901

•02738

•01936

608

•O273.r>

•01934

609

•02733

•01933

610

•02731

•01931

611

•027-2!)

•01929

61,'

•02726

•01928

613

•02721

•01926

614

•02722

•0102.'.

615

•02720

•01923

616

•02718

•01922

617

•0271.')

•01920

618

•02713

•01919

619

•02711

•01917

>.."'

•02709

•01915

621

•02707

•01914

Off

•027' > 1

•01912

•02702

•01911

0*4

•0271 » '

•01909

6S5

•02698

•01908

6t6

•02696

•01906

•026!)4

•01905

•02H1I1-'

•01903

999

•02*Mi

•01902

630

•02687

•01900

631

•02685

•01899

999

•02683

•01897

<..:.:

•02681

•01896

634

•02679

•01894

635

•02677

•01893

,..,,,

•02676

•01891

637

•02672

•01890

•02670

•01888

639

•02668

•01887

640

•02666

•01885

641

•02664

•01884

'•',-•

•02662

•01822

-•; •

•02660

•01881

*JU

•02668

•01879

645

•02656

•01878

646

•02654

•01876

647

•02652

•01875

648

•02650

•01874

'•',''

•02648

•01872

650

•02646

•01871

n

X,

X,

651

•02644

•01869

65S

•02642

•01868

653

•02639

•01866

•02637

•01865

655

•02636

•01864

656

•i 12633

•01862

657

own

•01861

9U

•02629

•01859

650

•02627

•01858

660

•02625

•01856

661

•02623

•01865

<;•;.'

•02621

•01854

998

•02620

•0181

<••'•'',

•02618

•01851

665

•02616

•01849

666

•0261 1

•01848

997

•02612

•01847

668

•02610

•01845

999

•02608

•01844

670

•02606

•01843

671

•02604

•01841

an

•02602

•01810

673

•02600

•01 83*

674

•02598

•01837

675

•02596

•01836

676

•02594

•01834

677

•02592

•01833

678

•02590

•01832

679

•02588

•01830

680

•02587

•01829

681

•02585

•01828

HXJ

•02583

•01826

989

•02581

•01825

684

•02579

•01824

685

O2677

•01822

686

•02575

•01821

687

•02573

•01820

989

•02571

•01818

680

•i i2f>70

•01817

690

•02568

•01816

691

•02566

•01814

>;.•*.'

•02564

•01813

695

•02562

•01812

694

•02560

•01810

896

•02558

•01809

696

•02557

•01808

697

•02555

•01807

698

•02553

•01805

699

•02551

•01804

700

•02549

•01803

n

X,

X,

701

•02548

•01801

70S

•02546

•01800

70,1

02544

•01799

704

•02642

•01798

705

•02540

•01796

706

•02538

•017!i.-i

707

•02537

•01794

708

•02535

•01792

709

•02533

•01791

710

•02531

•01790

711

•02630

01789

712

•02528

•01787

71S

•02526

•01786

714

•02524

•01785

715

•02522

•01784

716

•02521

•01782

717

•02519

•01781

718

•02517

41780

710

•025 ].-•

•0177i>

'; .'a

•02514

•01777

7X1

•02512

•01776

:.'.'

•02510

•01775

: .'•:

•02508

•(M771

7S4

•02507

•01771

725

•02505

•01771

726

•02503

•01770

727

•02502

•01769

728

•02500

•01768

7S9

•02498

•01766

730

•02496

•01765

7S1

•02495

•01764

?.;?

•02493

•017<!3

;.;.;•

•02491

•01762

794

•02490

•01760

735

•02488

•01769

7S6

•02486

•01758

737

•02485

•01757

738

•02483

•01756

739

•02481

•01754

740

•02479

•01753

741

•02478

•01752

743

•02476

•01751

743

•02474

•01750

744

•02473

•01749

745

•02471

•01747

746

•024fi!)

•01746

747

•02468

•01745

748

•02466

•017H

749

•02465

•01713

750

•02463

•0171:!

Tables for Facilitating the Computation of Probable Errors 17
TABLE V. Probable Errors of Means and Standard Deviations.

n

xt

x»

751

•02161

•01740

7.-: 1

•02460

•01739

7SS

•02458

•01738

76A

•02456

•01737

755

•02405

•01 730

756

•02453

•01735

757

•02451

•01733

758

•02450

•01732

7,7:>

•02448

•01731

760

•02447

•01730

7>:i

•03445

•01729

76i

•02443

•01728

769

48449

•01727

784

•02440

•01725

765

•02439

•01724

766

•02437

•01723

767

iiL'!:)5

•01722

768

•IIL' 134

•01721

769

132

•01720

770

•i.'2131

•01719

771

129

•01718

772

•02428

•01717

778

•' 12420

•01715

774

•02 124

•ol714

775

•02423

•01713

77<;

•02421

•01712

777

•02420

•01711

77tf

•02418

•01710

77.''

•02417

•01709

780

•02415

•01708

781

•02414

•01707

78*

•02412

•01706

783

•02410

•01704

7.S1;

•02409

•01703

•02407

•01702

788

•02406

•01701

787

•02404

•01700

•02403

•01099

•08401

•01698

790

•02400

•01697

7'.n

•023!l«

•0169G

•02397

•01600

793

•08398

O16W

794

•I 123!) i

•01093

795

•02392

•016M

706

•02301

•010!) '

797

•023S!)

•01088

798

Dtt

•01688

799

•02:.

•01687

800

•023M5

•01686

n

*i

*3

801

•02383

•01685

802

•02382

•01684

809

•02380

•01683

804

•0237!)

•01082

806

•02377

•01681

806

•02376

•01680

807

•02374

•01679

808

•02373

•01678

809

•02371

•01677

810

•02370

•01676

811

•02368

•01675

81S

•02367

•01674

813

•02366

•01673

814

•02364

•01672

815

•02363

•01671

816

•02361

•01670

817

•02300

•01669

813

•02358

•01668

819

•02357

•01667

830

•02355

•01666

821

•02354

•U1665

822

•02353

•01664

823

•02351

•01602

824

•02350

•01601

us

•02348

•01600

CM

•02347

•01G59

837

•02345

•01658

82S

•02314

•01657

839

•02343

•01056

980

•02341

•01655

8S1

•02340

•01654

832

•0233S

•01653

833

•02337

•01652

834

•02336

•01051

835

•02334

•01651

886

•02333

•01650

837

•02331

•010 !!»

83S

•02330

•01648

839

•02329

•01017

840

•02327

•01646

841

•0232(5

•01645

N.~> !

•02324

•01644

849

•02323

•01643

844

•02322

•01012

845

•02320

•01641

846

•02319

•01040

847

•02318

•0163'.)

848

•02316

•01638

8!fl

•02315

•01637

850

•02313

•01036

n

xt

*2

S51

•02312

•01635

8u2

•02311

•01634

853

•02309

•01633

854

•02308

•01632

855

•02307

•01631

856

•02305

•01630

857

•02304

•01629

858

•02.303

•01628

859

•02301

•01627

860

•02300

•01626

861

•02299

•01625

862

•02297

•01624

863

•02296

•01624

864

•02295

•01623

865

•02293

•01622

866

•02292

•01C21

867

•02291

•01020

868

•02289

•01619

869

•02288

•01618

870

•02287

•01617

871

•02285

•01616

872

•02284

•01615

873

•02283

•01014

874

•02281

•01613

875

•02280

•01612

876

•02279

•01611

877

•02278

•01610

878

•02276

•01610

879

•02275

•01609

8SO

•02274

•01008

881

•02272

•01607

88*

•02271

•01606

883

•02270

•01605

884

•02209

•01604

885

•02267

•01603

880

•02266

•01602

887

•02265

•01601

88S

•02263

•01600

8S9

•02262

•01600

890

•02261

•01599

891

•02260

•01598

892

•02258

•01597

899

•02257

•01596

894

•02256

•01595

895

•02255

•01594

896

•02253

•01593

897

•02252

•01592

S98

•02251

•01592

899

•02250

•01591

900

•02248

•01590

B.

18 Tablm for Stutlsliriuii* and J3ioiiuli-icn(ias

TABLE V. TABLE VL

Probable Errors of Afeans and Standard- Deviations. Probable Errors of Coejjicit nt «f Variation.

M

x,

Xi

;>/)/

147

•01589

146

••'1588

1 15

•01587

904

•02213

416

9or>

M9

•01585

906

08841

411

B :

•02210

41684

•MS

J38

•01 :

..'37

•oi

910

OS

•015*1

911

188

•01

91*

•022:1:1

•ol. ',79

91

•02232

•01578

914

131

•01578

915

49880

•01577

916

•08

41676

917

•i 12227

•01575

918

•02226

•01574

919

•0222.-.

•01573

•02224

•01572

921

•02223

•01572

•02221

•01571

9SS

•02220

•01570

•02219

•01569

•02218

•01568

916

•02217

•01567

OS7

•02215

•01566

918

•02214

•01566

919

•02213

•01565

0*0

•02212

•01564

931

•02211

•01563

. •

•01562

933

•02208

•01561

934

•02207

•01

935

•02206

•01560

936

•02205

•01559

9 :

•02203

•01558

938

•02202

•01557

939

•02201

•01556

'••',"

•02200

•01556

941

•02199

•01555

94*

•021HH

•01551

943

OB1M

41668

944

48196

41669

945

481M

•01551

046

•021! 13

•01551

••'..

1199

•01660

948

•0211)1

•01549

949

•0218!!

•015 is

49

•01517

«

Xi

x,

•02187

•01647

•• :

•02186

•015 Hi

053

•".'185

•01545

964

•' HI 84

•oi. -.11

OSS

•02183

•01543

an

•02181

•01648

•02180

•01648

•02 17!)

•01541

069

•08178

•01540

000

•02177

•01539

061

•02176

•01539

•02175

•016!

063

•02174

•015:17

•02172

•01536

9G5

•02171

•01 5:i:.

900

•02170

•01535

967

•02169

•01534

988

•02168

•01533

•"2167

•01532

970

•02166

•01531

971

•02 1C,.-)

•01681

972

•02163

•01531)

97S

48169

•01688

974

•02 Mil

•01688

975

•02160

•01527

976

•02159

•01 r.27

977

•021. IS

•01526

978

08167

•01525

979

•08166

•01524

980

•02155

•01524

981

•01163

•01523

98*

•02152

•01522

us.:

•02151

•01521

984

•02150

•01520

985

•02149

•01520

986

•02148

•0151'.)

981

•02147

•01518

988

•02146

•01517

989

•"2145

•01517

D'JO

•02144

•oi. MI;

991

•02143

•01615

•••-'.'

•(12142

•01. '.I 1

•02140

•0151 1

994

•02139

•01513

995

•02138

•01512

996

•021:17

•01611

997

•08186

•01510

998

•i 12 1 :!:•

•01510

999

•O2i::i

•01509

1000

•oam

•01508

V

*

J

A»

0

1

3

f

o-ooooo

1-00010
'080
840870

1-OUIIO

I -00070
l-ooi'.xi
1 40374

0

60
120

180

-'*'' 1

60
60
60
60

4
6

61)1

140

J.>.l

898

69

r,

7
10

(i -02 156

748489
846104
841
1048

1-00908
L 41866
141688
148167

1-02690

358
417
475
533
590

69
66

57

11

1 _•
IS

14

i.-,

11-18880

12-17157
13-21787
14-27176
15-33379

14)
148987

1 -o i •
L 46889

1-06202

617
703
75: i
814
B68

57
66
56
55
54

17

I'.i
20

16-40449
17-48440
18-57405
19-67395
20-78461

1-07070
1-07991
1 •( IS905
1-09!)! HI
1-11066

981

974
1025
1076
1180

53

53

51
50

SI

• >

33

24
25

21-90653
23-04021
24-18612
25-34473
26-51650

1-121D2
1-13368
1-14591
1-15861
1-17177

1175

1 223
1270

1816
1369

49
48

•17

46

15

S6
27
88

29
SO

27-70190
28-90135
30-11530
31-34416
32-58834

1-1853!)
1-19946
L -81896

1-22886
1-24418

1406
1449
1491
1533
1573

44
43

12
41
40

SI
32
33

34

35

33-84825

35-12428
36-41681

37-72621
39-05285

1-25901
1-27603
1-29253
1-30940
1-32664

1612
1660

1687
1723
1758

39
38
37
86

35

36
37
38
39

40

40-39707
41-75922
43-13962
41-53861
45-95050

1-34122

1-36215
1 -38041
1-39899
1-41789

1793
1826

I 858
1890
1920

34

88
32
31
30

41

'i'
44
45

47-3935!!
48-85017
50 '326.-, 1
61 -82296
53-33971

1-4370!)
1-46668
1-47686
1-49649

1-51675

1950
1978

8069

30
29

28
27
26

46
47
48

64-87706
;:i524
68-01151

1 -53734
1-55818

1-57! 127

9084

2109

2i:i2

25

21

21

49

so

11610
81-83784

1-6221 1

2155
2177

23
22

Probable Error of a Coefficient of Correlation
TABLE VII. Abac for Probable Errors of r.

19

Or***? ^ ? "9

Scale of Correlation

* <0 <D C1 •< O

•^ •> l> <*> «> «0 ®

Abac for determining the Probable Errors of Correlation Coefficients.

3-2

•JO

or Stutixticianx ami liiuni, t,-'«-i<n<x

TABLE VIII. Values of l-r*fnr
Values of 1 - r».

-001 to •!>:>:>.

r

•000

•001

•00*

•005

•004

•005

•OUti

•008

•oust

•000

1-000000

•999999

'.*'. '.' !' " •

•900091

499084

•999 975

4JJ8 :•:.!

•999936

•999919

•010

•999900

400 B70

•000 M6

B00 -:>.|

•999804

•9'J!) 744

•999711

•999 676

•999039

•0*0

•999800

WO 580

•999 516

•999 471

•999 424

•090 878

•999 324

•999271

•'.ilr.i -JUi

•999 159

•030

•999 100

ttO 088

we vn

•998911

•998 M i

•998 776

•998704

•998631

•998 -I7!t

•040

•998400

•998319

•998 236

•998 151

•998064

•:i:»7 :C5

•997 884

•997 791

•997 599

•050

•997600

•997399

•997 296

•997 191

•997 084

B96 *<;4

•'.I'M :.-,i

•996 519

•060

•996400

•996 279

•996 156

•996 031

•99r> 904

•991 77.r>

•995 644

•995511

•!i!i:, 37(i

•J39

•070

•995100

.<:•; U9

•994 816

•994 671

•994 524

•sun 375

•994 224

•994 071

•!i:i:: !tu;

•993 759

•oso

•993600

•993439

•993 276

•993 111

•992 944

•992 770

•:i:i:> cu-i

•992 431

•992 079

•090

•991900

•991 719

•991 536

•991351

•991 164

•00978

•990 784

4JOO 891

•990 396

•990 199

•100

•990000

•989799

•989596

•989 391

•989184

•988 975

•988 764

•988551

•988 336

•988119

•110

•987900

•987 679

•987 456

•987 231

•987 004

•986 775

•986 544

•986311

•980 076

•986 839

•ito

•985600

•985 359

•985 116

•984 871

•984 624

•984 376

•984 124

•983 871

•983 616

•983 359

•ISO

•983100

•982 839

•982 576

•982311

•982044

•981 775

•981 504

•981 231

•980 n.v,

•980679

•no

•980400

•980 119

•979836

•979 651

•979 264

•978 975

•978 684

•978 391

•978096

•977 799

•150

•977500

•977 199

•976 896

•976 591

•976 284

•975 975

•975 664

•975 351

•975 036

•974 719

•160

•974400

•974 079

•973 766

•973 431

•973 104

•972 775

•972 444

•'.i--2 in

•071 77ti

•971 439

•170

•971 100

•970 769

•970 416

•970 071

•969 724

•969 375

•969024

•968 671

•968 316

•967 959

•180

•967600

•967 239

•088876

•90G511

•966 144

•965 775

•965 404

•965031

•904 656

•964 279

•190

•963900

•963519

•963136

•962 751

•962 304

•961 975

•961 684

•961 191

•960796

•960399

•too

•960000

•959599

•9.19 196

•968 Tin

•958 384

•957 975

•957 664

•957 151

•956 736

•956 319

•HO

•955900

•955 479

•955 056

•954 631

•954 204

•953 776

•953 344

•952 911

•952 476

•952 039

.,,,,

•951 600

•951 159

•950 710

•950 271

•949 824

•949 375

•948 924

•948 471

•948 016

•9 47 559

•230

•947 100

•946 639

•946 176

•945 711

•945244

•944 776

•944 304

•943 831

•943 356

•942 879

•HO

•942400

•941 919

•941 436

•940 951

•940 464

•939 975

•939 484

•938 991

•938 496

•937 999

•937 500

•936 999

•936 496

•935 991

•935 484

•934 975

•934 464

•933 951

•933 436

•932 919

•:•:,!

•932400

•931 879

•081 :',.,(]

•930 831

•930 304

•929 775

•9-29 244

•928 711

•927 639

•gro

•927100

•926 659

•926 016

•925 471

•924 924

•924 375

•923 824

•923 271

•'.}-2-2 7n;

•922 159

•980

•921600

•921 039

•920 476

•919 911

•919 344

•918 775

•918 204

•917 631

•917 056

•916 479

•S90

•915900

•915319

•914 736

•914 151

•913 664

•912 975

•912 384

•911 791

•911 196

•910 699

•300

•910000

•909399

•908796

•908 191

•907 684

•906975

•906364

•905 751

•905 136

•904519

•1910

•903900

•903279

•!luj 1 ;.-,(!

•902 031

•901404

•900775

•900144

•899511

•898 ^7t;

•898 239

•sso

•897600

•806 BOO

•896316

•895 671

•895 024

•894 376

•893 724

•893 071

•892 416

•891 759

•3.10

1 100

•890 439

•889 776

•889 111

•888 444

•887 775

•887 104

•886 431

•885 766

•885 079

•340

•884400

•883 719

•883 036

•882 351

•881664

•880 976

•880 284

•879 591

•878 896

•878 199

•877500

•876 799

•876096

•875 391

•874 684

•873 !I7.-)

•873 204

•872 551

•871 836

•871 119

•870400

•869 679

•868 956

•868 231

•867 504

•866 775

•866 044

•865 311

•864 576

•863 839

•370

•863100

•862359

•861 616

•860 871

•860 124

•859 375

•858 624

•857 871

•857 116

•856 359

•sso

•865600

•854 839

•854 076

•853 311

•852 544

•861 775

•851004

•850 231

•849 456

•848 679

•390

•847900

•847 119

•846336

•846 551

•844 764

•843 975

•843 184

•842 391

•841 696

•840 799

•400

•840000

•839199

•838 396

•837 591

•836 784

•835 975

•835 164

•834 351

•833 536

•832 719

•410

•831 900

•831 079

•830 2f»6

•829 431

•828 604

•827 775

•vji; si 1 1

•826 111

•825 276

•824 439

•420

•823600

•822 759

•821 916

•821 071

•819 375

•818 524

•817 671

•816 816

•815 959

•4SO

•815 100

•814 239

•813 376

•812511

•811 644

•810 775

•809 904

\S09 031

•808 156

•807 279

•440

•806400

•805 619

•804 636

•803 751

•80S M;I

•801 975

•801 084

•800191

•7:1:1 i::n;

•798 399

•450

•797500

•706 690

•708 c,:iii

•794 791

798 884

•792 975

70s ">;i

•791 151

•790 236

•789319

•460

•788400

•787 479

786606

•785 631

•784 704

•783 775

788844

•781 911

•780976

•780 039

•470

•779100

•778 169

777 216

•776 271

•775 324

•774 375

•773 424

•772 471

•771 516

•770 559

•480

•769600

•788 630

•767 676

•766 711

•765 744

•764 775

•763804

•762 831

•761 856

•760 879

•490

•759900

•768 919

•757 936

766 :•:.!

760 9M

•754 975

•753 984

788 :i'M

•751 996

•750 999

Probable Error of a Coefficient of Correlation

21

TABLE VIII. Values of I- 1* for r = -001 to '999.
Values of 1 — r*.

r

•000

•001

•00.2

•003

•004

•005

•006

•007

•008

•009

•500

•750000

•748 999

•747 996

•746 991

•745 984

•744 975

•743 964

•742 951

•741 936

•740 919

•510

•739 900

•738 879

•737 856

•736 831

•735 804

•734 775

•733 744

•732711

•731 676

•730 639

•729 609

•728 559

•727 516

•726 471

•725 424

•724 375

•723 324

•722 271

•721 216

•720 159

•530

•719 100

•718 039

•716 976

•715911

•714 844

•713 775

•712 704

•711 631

•710 556

•709 479

•540

•708 400

•707 319

•706 236

•705 151

•704 064

•702 975

•701 884

•700 791

•699 696

•698 599

•550

•697 500

•696 399

•C95 296

•694 191

•693 084

•691 975

•690 864

•689 751

•688 636

•087 519

•560

•686 400

•685 279

•684 156

•683 031

•681 904

•680 775

•679 644

•678 511

•677 370

•076 239

•570

•675 100

•673 959

•672 816

•671 671

•670 524

•669 375

•668 224

•667 071

•665 916

•664 759

•663 600

•662 439

•061 276

•660 111

•658 944

•657 775

•656 604

•655 431

•654 256

•653 079

•59U

•051 900

•650 719

•649 536

•648 351

•647 164

•645 975

•644 784

•643 591

•642 396

•641 199

•600

•640000

•638 799 -637 596

•636 391

•635 184

•633 975

•632 764

•631 551

•630 336

•629 119

•610

•627 900

•626 679 '625 456

•624 231

•623 004

•621-775

•620 544

•619 311

•618 076

•616 839

•620

•615600

•614 350

•613 116

•611 871

•610 624

•609 375

•608 124

•606 871

•605 616

•604 359

•6SO

•603 100

•601 839

•600576

•599311

•598 044 '596 775

•595 504

•594 231

•592 956

•591 679

•G40

•590400

•589119 -587836

•586 551

•585 264

•583 975

•582 684

•581 391

•580 096

•578 799

•650

•577 500

•576 199 -574 896

•573 591

•572 284

•570 975

•569 664

•568 351

•567 036

•565 719

•660

•564400

•563 079

•561 756

•560 431

•559 104

•557 775

•556 444

•555 111

•553 776

•552 439

•670

•551 100

•549 759

•548 416

•547 071

•545 724 -544 375

•543 024

•541 671

•540 316

•538 959

•680

•537 600 -536 239 '534 876

•533 511

•532 144 '530 775

•529 404

•528 031

•526 656

•525 279

•690

•523900

•522 519

•521 136

•519 751

•518 364 -516 975

•515 584

•514 191

•512 79U

•511 399

•TOO

•510000

•508 699

•507 196

•505 791

•504 384 -502 975

•501 564

•500 151

•498 736

•497 319

•710

•495900

•494 479

•493 056

•491 631

•490 204 -488 775

•487 344

•485 911

•484 476

•483 039

•720

•IM IMO

•480159

•478 716

•477 271

•475 824 -474 375

•472 924

•471 471

•470 016

•468 559

•;.;<>

•467 100

•465 639

•464 176

•462711

•461 244

•459 775

•458 304

•450 831

•455 356

•453 879

•740

•452400

•450 919

•449 436

•447 951

•446 464

•444 975

•443 484

•441 991

•440 496

•438 999

•750

•437500

•435 999

•434 496

•432 991

•431 484

•429 975

•428 464

•426 951

•425 436

•423 919

•760

•422400

•410879

•419 356

•417 831

•416 304

•414 775

•413 244

•411 711

•410 176

•408 639

•770

•407 100

•405 559

•404 016

•402 471

•400924

•399 375

•397 824

•396 271

•394 716

•393 159

•780

•391600

•390 039

•388 476

•386911

•385 344

•383 775

•382 204

•380 631

•379 056

•377 479

•790

•375900

•374 319

•372 736

•371 151

•369 564

•367 975

•366 384

•364 791

•363 196

•361 599

•800 -360000

•358 399

•356 796

•355 191

•353 584

•351 975

•350 364

•348 751

•347 136

•345 519

•810

•343900

•342 279

•340 656

•339 031

•337 404

•335 775

•334 144

•332 511

•330 876

•329 239

•820

•327600

•325 959

•324 316

•322671

•321 024

•319 375

•317 724

•316 071

•314 416

•312 759

•8SO

•311 100

•309439

•307 776

•306 111

•304 444

•302 775

•301 104

•299 431

•297 756

•296 079

•840 -294 400

•2!>2 719

•291 036

•289 351

•287 664

•285 975

•284 284

•282 591

•280 896

•279 199

•850

•277 500

•275 799

•274 096

•272 391

•270 684

•268 975

•267 264

•265 551

•263 836

•262 119

•860

•260400

•258 679

•256 956

•255 231

•253 504

•251 775

•250 044

•248 311

•240 576

•244 839

•870

•243 100

•241 359

•239 616

•237 871

•236 124

•234 375

•232 024

•230 871

•229 116

•227 359

•880

•225600

•223 839

•222 076

•220311

•218 544

•216 775

•215 004

•213 231

•211 456

•209 679

•890

•207 900

•206 119

•204 336

•202 551

•200 764

•198 975

•197 184

•195 391

•193 596

•191 799

•900

•190000

•188 199

•186 396

•184 591

•182 784

•180 975

•179 164

•177 351

•175 536

•173 719

•910

•171900

•170 079

•168 256

•166 431

•164 604

•162 775

•160 944

•159 111

•157 276

•155 439

•920

•153 600

•151 759

•149 916

•148 071

•146 224

•144 375

•142 524

•140 671

•138 816

•136 959

•9SO

•135 100

•133 239

•131 376

•129511

•127 644

•125 775

•123 904

•122 031

•120 156

•118 279

•940

•116400

•114 519

•112636

•110751

•108 864

•106 975

•105 084

•103 191

•101 296

•099 399

•950

•097 500

•095 599

•093 696

•091 791

•089 884

•087 975

•086 064

•084 151

•082 236

•080 319

•960

•078400

•076 479

•074 556

•072 631

•070 704

•068 775

•066844

•064 911

•062 976

•061 039

•970

•059 100

•057 159

•055 216

•053 271

•051 324

•049 375

•047 424

•045 471

•043 516

•041 559

•980

•039600

•037 639

•035 676

•033 711

•031 744

•029 775

•027 804

•025 831

•023 856

•021 879

•990

019900

•017 919

•015 936

•013 951

•Oil 96-:

009 975

•007 984

•005991

•003 996

•001 999

,!'<»' Statisticians inuf Ilinnn tficinns

TABLE IX. Values of the Incomplete Normal Moment Function fin(x).
A. Odd Moments mn (x) - /*,, (*) ( ,i - 1) (;« - tt) (« - •>) ... 2J.

X

«

•i(*)

•ife)

'»iW

•»«

*(•)

o-o

•ooooooo

•ooooooo

•ooooooo

•ooooooo

•ooooooo

0-1

•(K)19897

•0000050

•ooooooo

•ooooooo

•ooooooo

•0078996

•0000787

•0000005

•OtKXXXX)

•ooooooo

0-3

•0175545

•0003920

•0000059

•0000001

•ooooooo

0-4

•0306721

•0012105

•0000321

•0000006

•ooooooo

0-5

•0468770

•0028688

•0001183

•0000037

•0000001

0-6

•0657 177

•0057:i7^

•0003300

•0000151

•0000005

0-7

•<!Nil!-S3

•OlOlMil

•00081 hi

•0000493

•0000024

0-8

•1092507

•01651!) L

•00171 1-2

•0001350

•0000086

0-9

•1328570

•OJ.-KW25

•00327H2

•0003242

•0000259

1-0

•1569716

•08B8

•0057399

•0006988

•0000687

1-1

•1810901

•0492895

•0094199

•0013795

•0001634

j-g

•20-17

•ocj9i.':j

•0146092

•0025293

•0003r>!!(

1-3

•2275737

•0827672

•0215865

•004353U

•0007135

1-4

•2492148

•1024819

•0305828

•0070957

•0013414

1-5

•2004247

•1237174

•0417570

•Oil'

•0023771;

1-6

•2880214

•1460428

•0551764

•01640IJH

•0040005

1-7

•30481)32

•16891)23

•0708039

•OL-U5098

•0064248

1-8

•8188681

•1920!i±i

1945

•03MU8

•00981111

1-9

•3333265

•21 IS899

•1080009

•0436894

•0146688

g-0

•3441(513

•236yii!U

•1:N'.)874

•0569995

•0210056

M

•3549587

•^r>79749

•1510502

•0724606

•0291380

s-i

•3C34677

•2776192

•17374S8

•0899486

•0392533

g-3

•37061. -.2

•29.r>'

•1!)06019

•1092390

•05 14703

*-4

•3765478

•31-20:) 15

•21H1769

•1300173

•0658224

g-B

•3814140

•3266380

•2410506

•1518971

2459

g-G

•3863593

•3394489

•2618602

•1744437

•1006767

g-7

•3885213

•3505370

•2813106

•1972006

•1206653

g-s

•3910268

•3599983

•2991823

•2197161)

•141&391

g-9

•3929897

•3679593

•3153329

•^115682

•1640231

3-0

•3945104

•3745671

•3296946

•2623860

•1866< i37

3-1

•3956755

•3799784

•34221 ;<!2

•2818638

•2093055

s-g

•3965582

•3843517

•3531dL'!>

•2997718

•2315079

s-s

•3972197

•3878403

•362301!)

•3159582

•2528687

3-4

•3977101

•3905878

•3700046

•3303476

•2730432

3-5

•3980696

•3887944

•3763548

•3129335

•2917671

3-(i

•3983304

•3943653

•3815183

•3537687

•3088145

3-7

•3985175

•395C099

•3856585

•3629529

•3240979

3-8

•3986503

4886430

•38«'.i:Ml

•3706199

•3375646

3-9

•3987436

•3972329

•31)14881

•3769253

•34U2:!7(i

4-0

•3988085

•3977378

•3934552

•3820351

•3591947

4-1

•3988.-.:',i i

1028

•3941)111!)

•38611 (If)

•3675554

•3988833

•3883635

: •

•3883304

•3744677

4-3

•3989037

•3888

4861

•381

•3800964

4-4

•3!»S9173

•3986759

•3975073

•31)37367

•3846117

4'-'

4880168

•:i!is7645

•:i:i79452

•31)51801

•:{^1809

4-0

•3!»S9321

SMS

•3982573

•3!)(i2557

•3909614

4-7

•3980369

•:i! ISM ;.-,<;

1770

•3970466

•3930967

4-8

•3989383

•3988!)i'7

•30862! IS

•3976 2' i:.

•31)17135

4"J

•3989398

•:t!»s9100

•3867348

•3980316

4800807

o-o

•3989408

M88

•3988061

0381

•3968097

oo

•3989423

•3!)8!)423

•3868483

•3081(123

•3981) i 2:5

Incomplete Normal Moment Functions

TABLE IX. Values of the Incomplete Normal Moment Function.
B. Even Moments mn (x) = /*„ (x)/{(n - 1) (n - 3) (n - 5) . . .1 }.

23

—

X

mt(x)

mt(x)

•iW

mt(x)

n>io (x)

o-o

•ooooooo

•ooooooo

•ooooooo

•ooooooo

•ooooooo

o-i

•0001325

•0000002

•ooooooo

•ooooooo

•ooooooo

o-s

•0010512

•0000084

•ooooooo

•ooooooo

•ooooooo

0-3

•0034951

•0000626

•OOOOoi is

•ooooooo

•ooooooo

0-4

•0081136

•0002572

•00000.-)8

•0000001

•ooooooo

0-5

•0154298

•0007604

•0000270

•0000008

•ooooooi

0-6

•0258121

•0018200

•0000925

•0000037

•0000001

0-7

•0394585

•0037:>75

•0002588

•0000139

•0000006

0-8

•0563914

•0069507

•0006223

•0000437

•0000025

0-9

•0764632

•0118045

•0013297

•0001177

•0000086

1-0

•0993740

•0187171

•0025857

•0002812

•0000251

1-1

•1246965

OS804S8

•0046525

•0006094

•0000658

1-2

•1519070

•0400559

•0078427

•0012100

•0001558

1-3

•1804203

•0549-' 14

•0125028

•0022617

•0003386

1-4

•2096248

•0720741

•0189894

•0039577

•0006842

1-5

•2389164

•0932091

•0276408

•0065653

•0012964

1-6

•2677274

•1168830

•0387442

•0103*69

•0023209

1-7

•2955511

•1415300

•o.-, 25059

•0157516

•0039494

1-8

•3219594

•1684803

•0690258

•0229926

•0064207

1-9

•3466134

•1965937

•0882796

•0324204

•0100147

t-o

•3692680

•2252921

•1101113

•0442938

•0150415

g-1

•3897700

•2539927

•1342371

•0587910

•0218224

S-2

•4080525

•2821413

•1602593

•0759866

•0306667

»'3

•4241237

•3092387

• 18761(03

•0958345

•0418437

S-4

•4380556

•3348616

•2159821

•1181613

•0555560

2-5

•4499005

•3586763

•2445598

•1426700

•0719132

2-fi

•4600231

•3804450

•2728554

•1689546

•0909136

2-7

•4683965

•4000247

•3003387

•1965228

•1124320

2-8

•4752816

•4173616

•3265431

•2248263

•1362197

9-0

•4808719

•4324798

•3510MJ

•2532933

•1619132

3-0

•1S.-.3546

•4154679

•3736720

•2813629

•1890538

3-1

•4889053

•4564647

•3941138

•3085150

•2171145

3-2

•4916833

•4656432

•4123121

•3342962

•2455315

3-3

•4038321

•4731975

•4282552

•3583379

•2737379

S-4

•4954736

•4793298

•4420056

•3803672

•3011962

3-5

•4967130

•4842409

•4536843

•4002102

•3274261

3-6

•4976381

•4881218

•4634555

•4177877

•3520261

3-7

•4983205

•4911484

•4715111

•4331061

•3746880

3-8

•4988183

•4934784

•4780568

•4462441

•3952025

3-9

•4991771

•4952491

•4833001

•4573366

•4134583

4-0

•4994330

•4965779

•4874418

•4665592

•4294345

4-1

•4996133

•4975627

•4906683

•4741120

•4431886

!>•:

•4997391

•4982835

•4931479

•4802063

•4548407

4-3

•4998258

•4988045

•4950279

•4850521

•4645574

4-4

•4998849

•4991766

•4964343

•4888500

•4725352

4-5

•499! CM 7

•4994392

•49747--".)

•4917846

•4789861

4-6

•4999512

•4996222

•4982298

•4940207

•4841246

4-7

•4999688

•4997483

•4987744

•4957010

•4881574

4-8

•4999802

•4998342

•4991613

•4969464

•4912765

4-9

•4999876

•4998919

•4994326

•4978572

•4936544

5-0

•4999923

•4999303

•4996200

•4985144

•4954417

00

•5000000

•5000000

•5000000

•5000000

•5000000

Tnhlis t'm- Mutisliritinx ami BioautriotO

'J'AI'.l.i; X. l>ii<<min <;/' lii-neralised ' ProbiMe Error.'
Table of Generalised ' Probable Errors.'

Number of
Variables

I'robable Error

J

01:71

1-177,4062

3

1-888,1061

4

1 -882,109

r,

2-086,0146

6

2-312,5982

7

2-519,0869

8

2-7lo

9

1-888,3989

10

3-o:.(i,4366

11

M16.740I

Diagram for Value of Probable Error for n Variables.

40-

•

- —

^-^^

3-4

3-2

^

^

^

^^

OR

^

^

^

^

£2-4

-
v 9-2

^

^

/

~

I 2-0

^

/

^

y/

•^ »•

-^ ir

/

1

i. <-4

/

f

««

/

1-0

/

•8

/

•ft

t

/

• 4

/

•2

/

0
-2.

/

2 3 4

~i 6 » <

9 10 11 12 13 14 IS

Kumbtr of Variabltt

Determination of Normal Curve from Tail 25

TABLE XI. Constants of Normal Curve from Moments of Tail

about Stump.

Values of the Functions ^r, and tyt required to determine the Constants of a
Normal Frequency Distribution from the Moments of its Truncated Tail.

K

*i

ft

*3

V

ft

*

*»

o-oo

•571

1-253

2-000

1-1

•734

1-977

7-371

o-oi

•573

1-259

2-016

1-2

•746

2-051

8-690

0-02

•574

1-265

2-032

1-S

•757

2-126

10-331

0-03

•576

1-271

2-049

1-4

•767

2-202

12-383

0-04

•578

1-276

2-066

1-5

•777

2-280

14-968

0-05

•580

1-282

2-083

1-6

•787

2-358

18-248

0-06

•581

1-288

2-100

1-7

•796

2-437

22-439

0-07

•583

1-294

2-118

1-8

•804

2-517

27-832

0-08

•585

1-300

2-136

1-9

•813

2-598

34-823

0-09

•587

1-305

2-155

2-0

•820

2-679

43-956

o-i

•588

1-311

2-173

2-1

•828

2-762

55-977

0-2

•605

1-371

2-377

8-8

•835

2-845

71-925

0-3

•622

1-432

2-617

2-s

•842

2-929

93-248

0-4

•638

1-495

2-902

2-4

•848

3-013

121-988

0-5

•653

1-560

3-241

2-5

•854

3-098

161-038

0-6

•668

1-626

3-646

2-6

•860

3-184

214-537

0-7

•682

1-693

4-133

2-7

•866

3-270

288-434

0-8

•696

1-762

4-720

f*

•871

3-357

391-374

0-9

•709

1-833

5-433

2-9

•876

3-445

535-963

1-0

•722

1-904

6-303

s-o

•880

3-532

740-796

1-1

•734

1-977

7-371

s-s

•888

3-969

[4299-226]

Let d equal distance of centroid of tail from stump, 2 = standard deviation of
the tail about its mean, and n = its area.

(i) Find ^r, from >Jr, = S*/d'. Hence from table determine h'.

(ii) From this value of h' find i/ra, then o-=dx^2 gives the standard
deviation of the uncurtailed normal curve.

(iii) A = A'x<r gives the origin of the uncurtailed normal curve.

(iv) Knowing h', Table II gives J (1 + at) and therefore the ratio £(! — «)
of tail to total area of curve N, or N = n/±(l-a.). For many purposes it is
sufficient to use N = n x trt.

28

Tables for Statixticuuis and BioHietriciam

TABLE XIL Test far Goodness of Fit. Values of P.

X1

•'=3

•r-4

ri' = 5

n' = 6

«' = 7

n' = 8

«' = 3

n'=10

n' = ll

1

•006531

•801253

•909798

•962566

481611

494880

996MO

•900498

•999828

e

•367879

•672407

•736759

•849146

•19609

•959840

•981012

•991468

•996340

s

•223130

•391625

•567825

400980

•808847

•888002

•934357

694800

•981424

4

•13633.)

•261464

•406006

•549416

•676676

•77!»77S

•867123

•911413

•947347

6

069060

•171797

•287298

•415880

•543813

•659963

•Tr.T.'.Ti;

•834308

•891178

6

•049787

•111610

•199148

•306219

•423190

•539750

•647232

•739019

•815263

7

•030197

071897

•135888

•OOMQ

•320847

MOhS.KO

•536632

•637119

•725444

8

•018316

•046012

O91578

•156236

•238103

•332594

•433470

•534146

•628837

9

•011109

•029291

•061099

•109064

•173578

•252656

•342296

•437274

•532104

10

•006738

•018566

O40428

•075235

•124652

•188573

•265026

•350485

•440493

11

•004087

•01172G

OM6M

O51380

•088376

•138619

•201699

•275709

•367518

12

•002479

•007383

•017351

•034787

•061969

•100558

•151204

•213308

•285057

IS

•001603

•004637

O11276

•023379

•043036

•072109

•111850

•162607

•-'•23672

14

•000912

•002905

•007295

•015609

•029636

•051181

•081765

•122325

•172992

IS

•000563

•001817

O04701

•010363

•020256

O36000

•059145

•090937

•132061

16

•000335

•001134

•003019

•006844

•013754

025116

•042380

086881

•099632

17

•000203

•000707

•001933

•004500

•009283

•017396

•030109

•048716

•074364

18

•000123

•000440

•001234

•002947

•006232

•011970

•021226

•035174

•054964

19

•000075

•000273

•000786

•001922

•004164

•008187

014880

•025193

•040263

20

•000045

•000170

•000499

•001250

O02769

•005570

•010336

•017913

O292,r)3

SI

•000028

•000105

•000317

•000810

•001835

•003770

•007147

•012650

•021093

22

•000017

•000065

•000200

•000524

•001211

•<M I:-:, 11

•004916

•(K)ssso

•015105

23

•000010

•000040

•000127

•000338

•000796

•001705

•003364

•006197

O10747

*4

•000006

•000025

•000080

•000217

•000522

•001139

•002292

•004301

•007600

M

•000004

•000016

•000050

•000139

•000341

000759

•001554

•002971

•005345

36

•000002

•000010

•000032

•000090

•000223

•000504

•001050

•002043

•003740

S7

•000001

•000006

•000020

•000057

•000145

•000333

•000707

001399

•002604

M

•000001

•000004

•000012

•000037

•000094

•000220

•000474

•000954

•001805

n

•000001

•000002

•000008

O00023

•000061

•000145

•000317

•000648

•001246

so

•oooooo

•000001

•000005

•000016

•000039

•000095

•000211

•000439

•000857

40

•oooooo

oooooo

•oooooo

•oooooo

•000001

000001

•000003

•000008

•000017

50

•oooooo

•OOOOOO

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

60

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

70

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

•oooooo

Tables for Testing Goodness of Fit

27

TABLE XII.— (continued).

X1

n'=12

n'=13

ii* =14

n'=15

n'=16

n' = 17

n'=18

n' = 19

n' = 20

1

•999950

•999986

•999997

•999999

1-

I-

1-

1-

1-

s

•998496

•999406

•999774

•999917

•999970

•999990

•999997

•999999

1-

3

•990726

•995544

•997934

•999074

•999598

•999830

•999931

•999972

•999989

4

•969917

•983436

•991191

•995406

•997737

•998903

•999483

•999763

•999894

6

•931167

•957979

•975193

•985813

•992127

•995754

•997771

•998860

•999431

6

•873365

•916082

•946153

•960491

•979749

•988095

•993187

•996197

•997929

7

•799073

•857613

•902151

•934711

•957650

•973260

•983549

•990125

•994213

8

•713304

•785131

•843001

•889327

•923783

•948867

•966547

•978637

•986671

9

•621892

•702931

•772943

•831051

•877517

•913414

•940261

•959743

•973479

10

•530387

•615960

•693934

•762183

•819739

•866628

•903610

•931906

•952946

11

•443263

•528919

•610817

•686036

•752594

•809485

•856564

•894357

•923839

12

•362642

•445680

•527643

•606303

•679028

•743980

•800136

•847237

•885624

IS

•293326

•369041

•447812

•526524

•602298

•672758

•730186

•791573

•838571

U

•232993

•300708

•373844

•449711

•525529

•598714

•667102

•729091

•783691

15

•1M4M

•241436

•307354

•378154

•451418

•524638

•595482

•661967

•722598

16

•141130

•191230

•249129

•313374

•382051

•452901

•523834

•592547

•657277

17

•107876

•149597

•199304

•256178

•318864

•385597

•454366

•523105

•589868

18

•081581

•115691

•157520

•206781

•262666

•323897

•388841

•455653

•522438

19

•061094

•088529

•123104

104949

•213734

•268663

•328532

•391823

•456836

20

•045341

•067086

•095210

•130141

•171932

•220220

•274229

•332819

•394578

SI

•033371

050380

•072929

•101632

•136830

•178510

•226291

•279413

•336801

®<9

024374

•037520

•055362

O78614

•107804

•143191

•184719

•231985

•284256

23

•017676

•027726

•041677

O60270

O84140

•113735

•149251

•190590

•237342

24

•012733

•020341

•031130

O4J822

06BQM

O89504

•119435

•155028

•196152

95

009117

•014822

•023084

O34.->66

O49943

O69824

O94710

•124915

•160542

S6

•006490

•010734

•017001

O25887

038023

O54028

O74461

O99758

•130189

S7

•004595

•007727

•012441

•019254

028736

O41483

O58068

•078995

•104653

S8

•003238

•005532

•009050

O14228

O2 1569

O31620

O44938

O62055

O83428

S9

•002270

•003940

O06546

O10450

O16085

O23936

O34526

•048379

O65985

SO

•001585

•002792

•004710

O07632

O11921

O18002

O26345

•037446

•051798

40

•000036

•000072

•000138

•000255

O00453

O00778

O01294

•002087

O03272

BO

•000001

•000001

•000003

O00006

O00012

O00023

O00042

•000075

•000131

60

•000000

•000000

•oooooo

OOOOOO

OOOOOO

O00001

O00001

O00002

•000004

70

•000000

•oooooo

OOOOOO

OOOOOO

OOOOOO

OOOOOO

OOOOOO

OOOOOO

OOOOOO

4—2

Table* for Stati*tlrltuni ontJ

TABLE XII. Test for Goodness of Fit. Values of P.

x«

«' = 21

n -J-

n' = 23

n' = 24

n'=25

n'^20

n' = 27

«' = 28

n' = 29

n' = 30

1

l«

1-

1-

I-

1-

1-

I-

1-

I-

1-

t

1-

1-

1-

1-

1-

1-

1-

1-

1-

1-

3

•099996

.(,•!' .... ,*,
,'.'. '-'.

499889

1-

I-

1-

1-

1-

1-

1-

4

•999954

988880

•999992

•999997

•999999

1-

V

1-

1-

1-

s

•999722

888888

•999939

•999972

•999987

•999994

•999998

489989

1-

I-

6

•818898

986487

•91)9708

•999855

•999929

•999966

498964

489998

489899

•999999

1

•996086

•998142

•998980

•999458

•999711

Mean

•999924

•999962

•999981

•999991

8

•991868

•9!t:.l-J3

•997160

•996871

•999085

•999494

•99!)7i'0

489818

•999924

•999060

9

•888802

•869814

•9!I3:»1

•9!»:.!i;.7

•997595

498598

•999194

•999546

•999748

•999863

10

•968171

•978918

•986304

•9!)1:!77

•994547

•996653

•997981

•998803

•999302

•999699

11

440888

•962787

•974749

•983189

•989012

•992946

•995549

•997239

•998315

•998988

If

•in CUT*;

•939617

•957379

•970470

•979908

•986567

•991173

•994294

•996372

•997728

13

•877384

906684

•933161

•951990

•966121

478601

•983974

•989247

488900

•995384

14

•830496

•869599

•901479

•926871

•94oo:.o

•961732

•973000

•981254

•987189

•991377

IS

•776408

•822952

862238

•894634

•920759

•941383

•957334

•969432

•978436

•985015

16

•716624

•769650

•815886

•855268

•888076

•914828

•936203

•952947

•965819

•975536

17

«08874

•711106

•763362

•809251

•848662

•881793

•909083

•931122

•948589

•902181

18

•587408

•649004

•705988

•757489

•803008

•842390

•875773

•903519

•926149

•944272

19

•621826

•585140

•645328

•701224

•751990

•797120

•836430

•870001

•898136

•921288

to

•457930

•521261

•583040

•641912

•696776

•746825

•7:11550

•830756

•864464

•892927

tl

•397132

•458944

520738

•581087

•638725

•692009

741904

•786288

•825349

•859149

Sg

•340511

•399510

•459889

•620252

•579267

•635744

•688697

•737377

•781291

•820189

S3

•288795

•343979

•401730

•460771

•519798

•677564

•632947

•685013

•733041

•776543

*4

•242392

•293058

•347229

•403808

•461597

•519373

•575965

•630316

•681535

•728932

S5

•201431

•247164

•297075

•350285

405760

•462373

•518975

•574462

•627836

•678248

96

•165812

406448

•251682

•300866

•353165

•407598

•463105

•518600

•573045

•625491

zr

•135264

•170853

•211226

•255967

•304453

•86588 1

•409333

•463794

•518247

•571705

S8

•109399

•140151

•175681

•215781

•260040

•307853

•358458

•410973

•464447

517913

S9

•087759

•114002

•144861

•180310

•220131

•263916

•311082

•360899

•412528

•466066

SO

089854

•091988

•118464

•149402

•184752

•224289

•267611

•314154

•363218

•414004

40

•004995

•007437

•010812

•015369

•021387

•i I:!!) 104

•039012

•051237

•OW12H

•083937

50

•000221

000365

•000586

•000921

•001416

•002131

•003144

•004551

•006467

•009032

60

•000007

•000013

•000022

•000038

•000064

•000104

•000168

•000264

•000407

•000618

70

•000000

•000000

•000001

•000001

•000002

•000004

•000007

•000011

•000019

•000030

Tables for Testing Goodness of Fit

29

TABLE XITI. Auxiliary Table A.

xa

M'x/^'*'}

log*-**1

Xs

log (x N/L-*j

loge-i*a

1

T-68479282

T-78285276

51

TT-68121586

12-92549071

s

1-61816058

1-56570552

62

11-46828520

12-70834347

.-;

1-48905897

1-34855828

63

11-25527422

12-49119623

4

1-33438109

1-13141104

64

11-04218593

12-27404899

5

1-16568886

2-91426380

56

12-82902315

12-05690175

6

2-988132:2 i

2-69711655

56

12-61578858

13-83975450

7

2-80445839

2-47996931

57

12-40248475

13-62260726

8

2-61630713

2-26282207

58

12-18911408

13-40546002

9

2-42473615

2-04567483

59

13-97667885

13-18831278

10

2-23046765

3-82852759

60

13-76218123

14-97116554

11

2-03401675

3-61138035

61

13-54862328

14-75401830

12

3-83576379

3-39423311

62

13-33500696

14-53687106

13

3-63599760

3-17708587

63

13-12133415

14-31972382

14

3-43494271

4-95993863

64

14-90760662

14-10257658

15

3-23277708

4-74279139

65

14-69382607

T5-88542934

16

3-02964420

4-52564414

66

14-48099412

15-66828209

17

4-82566143

4-30849690

67

14-26611232

15-45113485

18

4-62092598

4-09134966

68

14-05218213

15-23398761

1'J

4-41551928

5-87420242

69

15-83820498

15-01684037

20

4-20951024

5-65705518

70

15-62418221

16-79969313

21

4-0029571 ::>

6-43990794

71

15-41011512

16-58254589

22

6-79591210

6-22276070

72

15-19600496

16-36539865

S3

5-58841744

5-00561346

73

lg-98185290

16-14825141

*4

5-38051190

6-78846622

74

16-76766009

17-93110417

5-17222904

6-57131898

75

16-55342762

17-71395693

2G

6-96359847

6-35417173

76

16-33916654

17-49680968

27

6-75464644

6-13702449

77

16-12484787

17-27966244

28

6-54539033

7-91987725

78

17-91050256

17-06251520

29

6-33586907

7-70273001

79

17-69612157

18-84536796

SO

6-12608346

7-48558277

80

17-48170578

18-62822072

31

7-91605644

7-26843553

81

17-26725605

18-41107348

38

7-70580334

7-05128829

82

17-05277323

18-19392624

3S

7-49533808

8-83414105

83

18-83825810

19-97677900

34

7-28467333

S-61699381

84

18-62371146

T9-75963176

35

7-0738^1 M;:,

8-39984657

85

18-40913404

19-54248452

36

8-86279064

S-l 8269932

86

18-19452656

19-32533727

37

H65159301

9-96555208

87

19-97988972

19-10819003

38

8 44023670

5-74840484

88

1976522419

20-89104279

39

8'228729!>7

5-53125760

89

19-56053062

20-67389555

40

8-01708044

9-31411036

90

19-33580963

20-45674831

41

9-80529511

9-09696312

91

15-12106183

20-23960107

42

§•59338058

10-87981588

92

20-90628780

20-02245383

43

9-38134293

10-66266864

93

20-69148812

21-80530659

M

9-16918780

10-44552140

94

20-47666333

21-58815935

45

TO-95692047

10-22837416

95

20-26181397

21-37101211

46

"1074454-.S!)

10O1122691

96

20-04694054

21-15386486

47

10-53206*W

11-79407967

97

21-83204355

22-93671762

48

l'i:j 1949311

J 1 07693243

98

21-61712348

22-71957038

49

TO-10G823i".»

11-35978519

99

21-40218080

22-60242314

60

"1 1-89406301

li-1 4263795

100

21-18721596

22-28527590

Tablfn for Shitixticlnmt tunl lti<nn< (n'ri

TABLES XIV— XVI. Auxiliary Tables B, C and D.
TABLE XIV (B>

of colorj [N] : — [n] = n (n - 2) (n - 4)

•
odd DOS.

colog [it]

n
even nos.

colog [n]

1

•00000000

5

T-69897000

S

T-52287875

4

1 -09691001

C

2-82390874

(i

2-31875876

1

1-97881070

8

3-41566878

9

:i( (-2456819

10

4-41666878

11

I8S17U1

IS

5-33648753

1.1

i, -Mi:i2:i215

14

6-1903594!)

ir,

7-69314089

it;

8-98623951

17

g-46269197

18

!»-7309670l

19

9-18393s:i7

00

M 12993701

ei

TI-86171908

..-.'

11-08761433

23

12-49999124

S4

13 70730309

15-10205 123

Sti

11 29232974

27

18-67068747

88

](i,S4517171

:•'

Hi-LllSiH'117

SO

17-36805045

SI

18-71692778

S3

15-86290048

33

19-19841984

34

SS14S166

S5

IT464S4&79

36

L'2-77511906

37

».,.u

S8

L'.'i 19533546

39

24-49507946

40

25-59327547

41

56"-88229f.<U

42

27-97002618

43

§7-24882715

44

§8-32657350

45

L'!iT,!»-,Gl Kit

Ift

30-66381567

47

31-92351678

48

H2-SJ8257443

49

23332070

50

33-28360443

61

H4S6700H

62

35-567601(1!)

63

V6014746B

64

Buaona

65

.-',706111196

56

38O8701930

67

•4080711

68

40-32359131

69

•1 1 53438510

60

42-54544006

61

43-7490.-)526

6S

44-76304H:i7

63

45-94971471

64

U686S59

ti5

4013680135

66

47- 12732446

67

38-31072665

68

49-29481554

69

80-47187746

70

f. 1-44971760

71

52-62061911

72

r,:!-59238501

73

H-75790680

74

BB-7nifiU9

76

MASS22:i!'0

76

67-84233970

77

BS-99574426

78

BBD00MM8

79

68-09811717

80

80-04715511

81

ill -18963215

8i

62-13334125

83

i:: 1-27065406

84

84-209061 !)7

85

:i4113514

86

66-274563.^

87

67-40161688

88

68-33008084

89

89-45222588

90

70-37683>':«

91

71-49318448

98

72-412050.-) 1

93

75-52470154

94

74-43892265

95

75-64697793

96

76-45665142

97

77-66020620

98

78-46642534

99

75-66457100

100

SO-46A42534

TABLE XV (C).

X1

</!/>»*'<*

1

•3173106

'

•157M99

.;

•0832646

4

04M009

5

OS6M74

6

•0143060

7

•0081506

8

•0046776

9

•0026998

10

O016654

11

•0009112

1 >

•0005321

IS

•0003115

14

oooiaaa

16

•0001076

16

•0000634

17

•0000374

18

•0000981

19

•0000132

n

•0000078

81

•0000046

22

0000027

as

•0000016

24

O000011

S5

•0000007

S6

•0000004

27

•0000003

S8

•0000002

29

O000001

SO

•ooooooo

TABLE XVI (D).

Function

Log. Function

V ir

1-7828527590
1-9019400616

Probability of Association on Correlation- Scale

31

TABLE XVII.

Values of (—logP) corresponding to given values of ^ in a fourfold table.
(Extension of Table XII for ri = 4.)

X1

-logP

X1

-logP

x'

-logP

X1

-logP

Xa

-logP

X3

-logP

1

0096

36

5-021

50

10-097

1100

237-439

2600

562-973

13500

2929-521

2

0-242

27

5-230

60

12-231

1150

248-287

2700

584-680

HOOO

3038-086

S

0-407

28

5-440

70

14-370

1200

259-135

2800

606-387

14500

3146-652

4

0-583

29

5-o:>i>

80

16-513

1250

209-983

2900

628-094

15000

3255-219

r,

0-765

SO

6-860

90

18-659

1300

280-832

3000

649-801

15500

3363-785

6

0-952

SI

6-071

100

20-809

1350

291-681

5500

758-311

16000

3472-352

7

1-143

,; }

6-281

150

31-579

1400

302-531

4000

866-886

16500

3580-919

8

1-337

8t

6-492

too

42-37'>

1450

313-381

4500

975-434

17000

3689-486

9

1-533

34

6-703

M0

53-184

1500

324-231

5000

1083-995

17500

3798-053

10

1-731

S5

6-914

500

64-002

1550

335-081

5500

1192-538

18000

3906-621

11

1-931

$6

7-120

550

74-826

1600

345931

6000

1301-092

1S500

4015-188

12

2-132

S7

7-337

400

85-655

1650

350-782

6500

1409-649

19000

4123-756

IS

2-334

38

7-549

450

96-487

1700

367-633

7000

1518-206

19500

4232-324

14

2-537

88

7-761

500

107-321

1750

378-484

7500

1626-765

20000

4340-892

15

2-741

40

7-972

650

118-158

1800

389-335

8000

1735-324

20500

4449-461

16

2-945

41

8-184

600

128-997

1850

400-187

8500

1843-885

21000

4558-029

17

3-151

42

8-397

650

139-837

1900

411-038

9000

1952-446

21500

4666-597

1ft

3-357

43

8-609

700

150-678

1950

421-890

9500

2061-008

32000

4775-166

7.9

3-564

44

8-821

750

161-520

2000

432-742

10000

2169-570

22500

4883-735

M

3-770

45

9-034

800

172-364

2050

443-594

10500

2278-133

23000

4992-304

SI

3-978

46

9-246

850

183-208

2100

454-446

11000

2386-697

23500

5100-873

22

4-186

47

9-459

900

194O53

2200

476-151

11500

2495-261

24000

o209-442

ttt

4-394

48

9-672

950

204-899

2300

497-856

12000

2603-825

24500

5318-011

94

4-602

40

9-880

1000

215-745

2400

519-561

12500

2712-390

25000

5426-580

J5

4-811

50

10-OU7

1050

226-592

2500

541267

13000

2820-955

26

5021

1100

237-439

1GOO

562-973

13500

2929-521

TABLE XVIII.

Values of (- log P), entering with r and
Values of Oo-r.

•01

•OS

•03

•04

•05

•06

•07

•08

0-05

6-248

1-907

1-020

0-675

0-498

0-392

0-322

0-273

0-075

13-228

3-760

1-908

1-217

0-874

0-674

0-545

0-456

0-1

22-924

6-267

3-076

1-910

1-343

1-019

0-814

0-675

0-15

50-687

13-329

6-298

3-784

2-586

1-916

1-498

1-218

o-e

90-035

23254

10-771

6-343

4-259

3-100

2-384

1-924

0-5

206-348

62-453

23-836

13-758

9-057

6-478

4-906

3903

9-4

380-266

96-013

43-254

24-726

16-112

11-407

8-552

6-686

0-6

626-428

157-607

70-669

40-177

26-025

18-312

13-642

10-597

0-6

970-879

243753

108-980

61-747

39-845

27-922

20-713

16-020

0-7

1463-946

367033

163-781

92-579

59-584

41-634

30-792

23-740

0-8

2220-267

656-100

247-801

139-832

89-819

62-625

46-209

35-539

it".,

3607 -r>i' 4

902-0 4!J

401-907

226-479

145-241

101-085

74-442

57-134

0-95

6056-547

1265-013

562-757

316-904

203-069

141-207

103-886

79-671

32

Tables for Stall xtirimis and Bi

TABLE XIX.

Values of •£ corresponding to the values of (- log P) »n Table X VI II.

Values of o<7r.

•01

•09

•...;

•04

•05

•06

•or

•08

0-05

31-84

10-88

636

4-51

3-52

2-91

1-48

2-l!l

0-075

84-60

19-95

10-89

7-38

5-58

4-51

3-78

3-28

o-i

109-82

31-93

16-64

10-90

803

6-35

5-M

4-51

0-15

238-45

65-13

88-08

20-07

14-24

10-93

8-82

7-39

o-s

422-29

1 1 1 -35

53-16

88-89

22-35

16-75

13-25

IO!I7

0-3

956-68

246-62

114-05

G7-11

45-11

32-93

25-46

20-64

0-4

1758-21

447-81

204-07

118-18

78-13

56-14

42-73

33-92

o-s

2892-33

731-95

330-80

189-82

124-22

88-38

66-60

52-34

0-6

4479-02

1129-10

507-65

289-58

188-28

133-02

99-55

77-70

0-7

6750-09

1697-24

760-43

431-96

279-58

196-57

1 Hi-35

113-fil

0-8

10233-49

2568-34

11 17-76

649-98

419-81

293-64

'21 7-7-1

168-84

0-9

16624-37

4166-12

1857-93

1049-48

674-92

471-22

348-23

868-86

0-95

L'.-!-J!)5-86

5833-82

2599-00

1466-24

941-56

656-32

484-15

372-37

TABLE XX.

Values of log ^* corresjiunding to values of r and a<rr in Tables
XVII and XVII I.

Values of ,<rr.

•01

•OS

•OS

•04

•05

•06

•07

•08

0-05

1-5030

1-0366

0-8035

0-6542

0-5465

0-4639

0-3945

0-341 ' I

0-075

1-8108

1-2999

1-0370

0-8681

0-7406

0-6542

<i-.-»77B

0-5159

0-1

2-0407

1-5042

1-2212

1-0374

0-9047

0-8028

0-7210

0-6522

0-15

2-3774

1-8138

1-5062

1-3025

1-1.

1-038H

0-9I.V>

0-8686

0-2

2-6256

2-0467

1-7256

1-6091

1-3493

1-2210

1-1222

1-040 i

o-s

2-9808

2-3920

2-0571

1-8270

1-6543

1-617C

I '4067

1-31 is

0-4

3-2451

2-6511

2-3098

2-0725

1-8928

1-741)3

1-63H7

1-630B

0-6

3-4(112

2-864f>

2-M96

2-2783

2-0912

1-9464

1-823.-.

1-7188

0-6

3-6512

3-0527

2-7056

2-4618

2-2748

8-1 23i»

1-11980

i s;x)4

n-7

3-8293

3-2-'! "7

2-881 1

2-6354

2-4465

2-2935

2-1654

MM

0-8

4-0100

3-4097

3-0598

2-8129

2-62:! t

2-4678

2-3379

2-2-2B2

0-9

4-2207

3-6197

3-2690

3O210

2-8293

2-6732

2-54 lit

8-4886

4-3673

3-7660

3-4148

3-1662

2-9738

8-8171

2-6850

2-5710

Abaca for Determining the EqmprobcMe Tetrachoric r. 33
XXI. Abac to determine Oov.

XX \ \

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x

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s\\

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s\V

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xXV

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99

L.Q1

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7«0-70 -8O -84 -88 -9O -92 -94 95 «6 <7 «80 985

Scale of KI + OI) and
B.

34

Tables for Statistician* ami Biotuctricians

XXII. Abac to determine rr.

4-4

43

01

-OS

06

•07

OS

Value of

Approximate Values of Probable Error of r

35

Approximate values of Probable Error of r from a four-fold Correlation

Table (to be used with x, of Table V).
TABLE XXIII. Values of XT far Values of r.

r

Xr

r

Xr

r

Xr

r

Xr

r

Xr

•00

1-0000

•20

•9717

•40

•8845

•GO

•7298

•80

•4843

•01

•9999

•21

•9688

•41

•8785

•ea

•7200

•81

•4687

•03

•9997

•22

•9657

•42

•8723

•62

•7099

•82

•4526

•OS

•9994

•23

•9625

•43

•8659

•63

•6997

•83

•4362

•04

•9989

•24

•9591

•44

•8594

•64

•6892

•84

•4192

•05

•9982

•25

•9556

•45

•8527

•65

•6785

•85

•4018

•06

•9975

•26

•9520

•46

•8458

•66

•6C75

•86

•3838

•07

•99G6

•87

•9482

•47

•8388

•67

•65<>3

•87

•36i>2

•08

•9955

•28

•9442

•43

•8315

•68

•6448

•88

•3461

•09

•9943

•29

•9401

•49

•8241

•69

•6331

•89

•3262

•10

•9930

•so

•9358

•50

•8165

•70

•6211

•90

•3057

•11

•9915

•si

•9:514

•51

•8087

•71

•6088

•91

•2843

•13

•9899

•32

•9268

•5S

•8007

•72

•5962

•92

•2620

•IS

•9881

•S3

•9881

•53

•7926

•73

•5834

•93

•2387

•u

•9802

•34

•9172

•54

•7842

•74

•5702

•04

•2142

•15

•9841

•35

•9122

•55

•7756

•75

•55C8

•95

•1882

•16

•9819

•36

•9070

•56

•7669

•76

•5430

•96

•1605

•17

•VIM

•37

•9016

•57

•7579

•77

•5288

•97

•1305

•18

•9771

•38

•8961

•58

•7488

•78

•5144

•98

•0972

•19

•9748

•39

•8904

•59

•7394

•79

•4995

•99

•0585

1-00

•0000

TABLE XXIV.
Values of X' for Values of % (1 + a).

i(l + a)

Xa

*(! + «)

Xi

i(l + a)

Xa

4(i + »)

Xa

•BO

1-2533

•65

1 -2877

•80

1-4288

•95

2-1132

•ul

1 -:>f)35

•66

1-2928

•81

1-4457

•96

2-2740

•52

1-2539

•67

1-2984

•83

1-4641

•97

2-5071

•S3

1-2546

•68

1-3044

•83

1-4844

•98

2-8915

•54

i -^r,56

•00

1-3109

•84

1-5067

•985

3-2097

•55

1 '2569

•70

1-3180

•85

1-5315

•990

3-7333

•56

1 ^r.85

•71

1-3256

•86

1-5590

•Ml

3-8854

•57

1 -:>i;0 I

•72

1-3338

•87

1-5897

•<J'J2

4-0639

•58

l-L'020

•73

1-3427

•88

1-6245

•993

4-278 I

•68

1-2652

'~4

1-3523

•80

1-6640

•904

4-5419

•60

1-2680

•75

1 -3626

•90

1-7094

•995

4-8779

•1:1

1-2712

•76

1-3738

•91

1-7623

•9'M

5-3278

•i:!

1 -2748

•77

1 -38.')!)

•92

1-8249

•997

5-9776

•6,9

1-2787

•78

1-3800

•93

1-9003

•998

7-0465

•04

1-2H30

•79

1-4133

•94

1-9937

•999

9-3870

6-2

36

Tables J'or titulistlciaiw and

TABLE XXV.

< n n

Values of y ••

I 42V,
„ I ...-.— t/ n be

n -2 n- 4 :; -n- J

!n^S'^5-" 3 1 ., ,

I • • • 5 • o V n °e oaa
\ » m i

Or, </«« probability that the mean of a sample of n, drawn at random from a
normal population, will not exceed (in algebraic sense) the mean of the popu-
lation by more titan z times the standard deviation of the sample.

(

n=4

n = 5

n=6

n=7

n = 8

n = 9

ii = Kl

•1

•5G33

•5745

•5841

•5928

•6006

•60787

•61462

•g

•0-241

•6468

•6634

•6703

•6930

•70705

•71846

•3

•G804

•7098

•7340

•7649

•7733

•7890 1

•80-123

•4

-7309

•7657

•7939

•8175

•8376

•854G5

•86970

•5

•7749

•8131

•8488

•8667

•8863

•90251

•91009

•6

•8125

•8518

•8813

•9040

•9218

•93600

•94732

•7

•8440

•8830

•9109

•9314

•9168

•95851

•96747

•8

•8701

•9076

•9332

•9512

•9640

•97328

•98007

•9

•8915

•9269

•9498

•9669

•9756

•9827!)

•98780

1-0

•9092

•9419

•9622

•9751

•9834

•98890

•99252

1-1

•9236

•9537

•9714

•9821

•9887

•99280

•99539

V*

•9354

•9818

•9782

•9870

•9922

•99528

•99713

1-3

•9451

•9700

•9832

•9905

•9946

•99688

•99819

1-4

•9531

•9766

•9870

•9930

•9969

•99791

•99885

1-5

•9598

•9800

•9899

•9948

•9973

•99859

•09926

1-6

•9653

•9836

•9920

•9961

•9981

•99903

•99951

1-7

•9699

•9864

•9937

•9970

•99933

•99968

1-8

•9737

•9886

•9950

•9977

•9990

•99953

•99978

1-9

•9770

•9904

•9969

•9983

•9999

•99967

•99986

e-o

•9797

•9919

•9967

•9986

•9994

•9997«

•999DO

e-i

•9821

•9931

•9973

•9989

•9996

•99983

•91W93

s-s

'9841

•9941

•9978

•9992

•9997

•99987

•99996

2-3

•9668

•9960

•9982

•9993

•9998

•!»;il)91

•99996

S-4

•9679

•9957

•9985

•9995

•9998

•99993

•99997

i-S

•9886

•9!)63

•9987

•9996

•9998

•99995

ft

•9886

•9967

•9989

•9998

•9999

•99990

*7

•9806

•9972

•9991

•9997

•9999

•99997

•981

S-8

•9916

•8975

•9992

•9998

•<l!)998

•91).

s-o

•99M

•0978

•9998

•9998

•91)99

•99998

•9!i:»:ia

3-0

•U931

•9981

•9994

•9998

^~

•99999

•^~

Table for Type III Curves

37

TABLE XXVI.
Table for use in plotting Type III Curves, i.e.

X

logio(l + X)-Xlog,0«

X

log,0(l + X)-Xlog10e

-•96

•888 450

+ -80

•092 163

-•90

•609 135

+ -85

•101 979

-•85

•454 759

+ -90

•112111

-•80

•351 534

+ '95

•122 545

-•75

•276 339

+ 1-00

•133 205

-•70

•218 873

+ 1-05

•144 255

-•65

•173 641

+ 1-10

•155 505

-•60

•137 363

+ 1-15

•167 000

-•55

•107 926

+ 1-20

•178 731

-•50

•083 883

+ 1-25

•190 685

-•45

•064 205

+ 1-30

•202 855

-•40

•048 131

+ 1-35

•215 230

-•35

035 084

+ 1-40

•227 801

-•30

•024 614

+ 1-45

•240 561

-•25

•016 365

+ 1-50

•253 502

-•20

•010 051

+ 1-55

•266 616

-•15

•005 437

+ 1-60

•279 898

-•10

•002 328

+ 1-65

•293 340

-O5

•000562

+ 1-70

•306 937

OO

OOOOOO

+ 1-75

•320 683

+ O5

•000525

+ 1-80

•334 572

+ •10

•002 037

+ 1-85

•348 600

+ •15

•004446

+ 1-90

•3G2 761

+ •20

•007678

+ 1-95

•377 052

+ •25

•Oil 664

+2-00

•391 468

+ •»>

•016 345

+ 2-05

•406 004

+ •3.-,

•021 669

+ 2-10

•420 657

+ •40

•027590

+ 2-15

•435 422

+ •45

•034 065

+ 2-20

•450 298

+ •50

•041 056

+ 2-25

•465 279

+ •55

•048 530

+ 2-30

•480 363

+ 60

•056 457

+ 2-35

•495 547

+ •(>:>

•064808

+ 2-40

•510 828

+ •70

•073 557

+ 2-45

•526 202

+ •75

•082683

+ 2-50

•541 608

38

Tables for Statisticians awl

TABLE XX Ml.
Prnvers of Natural Numbert.

«

n«

ft»

*«

M*

n«

n'

•

1

1

1

1

1

1

1

1

S

4

8

1C

32

64

128

g

s

9

27

81

243

no

2187

S

4

16

64

256

1024

4006

16384

4

5

25

125

625

3125

16625

78UO

5

6

3G

216

1296

777fi

46656

279936

6

7

l:i

343

2401

16807

117649

8235 13

7

8

64

512

4096

32768

202144

2097 1. 52

8

9

M

789

6561

59049

531441

47*2: Mill

9

10

100

1000

10000

100000

1000000

1000UOOO

10

11

121

1331

14641

161051

1771561

19487171

11

IS

144

1728

20736

248832

2985984

358:} 1808

IS

IS

in

2197

28561

371293

4826809

62748517

13

14

190

2744

38416

537824

7529536

105413504

H

15

225

3375

50625

759375

11390G25

170859375

15

16

250

4006

65530

1018576

16777216

268435456

16

17

2S9

4913

83521

1419857

24137569

410338673

17

18

884

5832

104976

1889568

34012221

612220032

18

19

361

08M

130321

247C099

47045^1

893871739

19

to

400

8000

160000

3200000

64000000

1280000000

go

tl

441

9261

194481

4084101

88U1

1801088541

ei

Si

484

10648

234256

5153632

113379904

2494357888

S2

.'•:

529

12167

279841

6436343

148035889

3404825447

S3

*4

576

13S24

331770

7962624

191102976

4586471424

•»

95

625

15C25

390625

9765625

244140625

6103515025

S6

96

676

17576

456976

11881376

308915776

8031810176

S6

g7

729

19683

531441

14318907

387420489

104603632H3

g7

g8

784

21952

614656

17210368

481890304

13492928512

S3

S9

841

24389

707281

20511149

694823321

1724987G309

S9

SO

900

27000

810000

24300000

729000000

21870000000

SO

SI

961

29791

923521

28629151

887503681

27512614111

31

• ::

1024

32768

1048576

3355 1 132

1073741824

34359738368

3g

SS

1089

35937

1185921

39135393

1291467969

42018442977

...;

84

1150

39304

1330336

45435424

1544804416

52523350144

34

35

an

42875

1500025

52521875

1838265G25

64339296875

35

SC

1296

46Gf)G

1679616

604GG176

2176782336

78364164096

36

S7

1309

60053

18741(11

69343! 157

2565726409

94931877133

37

38

1444

64872

2085136

79235168

3010936384

1144155S2592

38

S9

1881

59319

2313441

90224199

3518743761

1372:51000079

39

40

1600

64000

2560000

102400000

4096000000

103840000000

40

41

1681

68991

2M25761

115850201

4760104241

HH7.VI 27:1^1

41

42

1784

74088

3111096

13009123:3

6489031711

2305393332 IS

»-'

4*

184!)

79507

3418SU1

147008 1 i:!

6321363049

271HKS01 11(17

43

44

1996

85184

3748096

164910224

72503 13S50

31 927 7S( 19001

44

¥•

1OU

91125

4100625

184528125

8303765625

373009 1531 25

¥'

t*

2IH;

9733C,

4477I-.0

205902970

9474296896

495817667816

.',<:

4~

no0

1U3S23

4879081

229315007

10779215329

f,oi;ii:!3i2o|i;:{

M

48

2»)4

110.VJ2

5308416

25480

12230590464

5870(^312272

48

-*9

2401

117649

6764801

2824752 1!)

13841287201

878883078849

60

HOO

125000

6260000

312500000

15625000000

781250000000

Tables of Powers and Sums of Powers

39

TABLE XXVII.— (continued).
Powers of Natural Numbers.

n

n*

»3

»<

n'

tf

nr

n

SI

2601

132651

6765201

345025251

17596287801

897410677851

51

52

2704

140608

7311616

380204032

19770609664

1028071702528

52

S3

2809

148877

7890481

418195493

22164361129

1174711139837

53

54

2916

157464

8503056

459165024

24794911296

1338925209984

54

55

3025

16G375

9150625

503284375

27680640625

1522435234375

55

56

3136

17')616

9834496

550731776

30840979456

1727094849536

56

57

3249

185193

10556001

601692057

34296447249

1954897493193

57

58

33G4

195112

11316496

656356768

38068692544

2207984167552

58

59

3481

205379

12117361

714924299

42180533641

2488651484819

50

60

3600

216000

12960000

777600000

46656000000

2799360000000

CO

61

3721

226981

13815811

844596301

51520374361

3142742836021

61

62

3844

ij:!-328

14776336

916132832

56800235584

3521614606208

62

6S

3969

250047

15752H61

992436543

62523502209

3938980639167

63

64

4096

262144

16777216

1073741824

68719476736

4398046511104

64

65

48M

274625

17850625

1160290625

75418890625

4902227890625

65

M

4356

287496

18974736

1252332576

82653950016

5455160701056

66

6,v

4489

300763

20151121

1350125107

90458382169

6060711605323

67

6S

4624

314432

21381376

1453933568

98867482624

6722988818432

68

69

4761

328509

22667121

1 v: 1031349

107918163081

7446353252589

69

70

4900

343000

24010000

1680700000

117649000000

8235430000000

70

71

5041

357911

25411681

1804229351

128100283921

9095120158391

71

78

5184

373248

26873856

1934917632

i:J93 14069504

10030613004288

73

73

5329

389017

28398241

2073071593

151334226289

11047398519097

73

74

5476

405224

29986576

2219006624

164206490176

12151280273024

74

75

SC25

421875

31640625

2373046875

177978515625

13348388671875

75

76

577»;

438976

33362176

2535525376

192699928576

14645194571776

76

77

69S8

456533

35153041

2706784157

208422380089

16048523266853

77

78

6084

474552

37015056

2887174368

225199600704

17565568854912

78

79

8M1

493039

38950081

3077056399

243087455521

19203908986159

79

80

6400

612000

40960000

3276800000

262144000000

20971520000000

80

81

6561

531441

43046721

3486784401

282429536481

22876792454961

81

M

6724

551368

45212176

3707398432

304006671424

24928547056768

82

83

6881)

571787

47458321

393! Kill Hi 13

326940373369

27136050989627

83

84

7056

592704

49787136

4182119424

351298031616

29509034655744

84

85

7225

614125

52200625

4437053125

377149515625

32057708828125

85

86

7396

636056

54700816

4704270176

404567235136

34792782221696

86

87

7569

658503

572*9761

4984209207

433626201009

37725479487783

87

88

7744

681472

fi!t!Mi9536

5277319168

464404086784

40867559636992

88

89

7921

704969

62742241

5584059449

496981290961

44231334895529

89

90

8100

729000

65610000

5904900000

531441000000

47829690000000

90

91

8281

753571

ns-,7i96i

6240321451

567869252041

51676101935731

91

M

8464

778688

7l63!l2!)'i

6590815232

606355001344

55784660123648

92

93

8649

8043r>7

74805201

-83693

646990183449

60170087060757

93

94

MM

830584

78074896

7339040224

689869781056

64847759419264

94

95

9025

857375

81450625

7737809375

735091890625

69833729609375

95

96

9216

884736

84934656

8153726976

782757789696

75144747810816

96

97

9409

912073

88529281

8587340257

832972004929

80798284478113

97

98

MM

941192

92236816

9039207968

885842380864

86812553324672

9S

99

060]

9702:t!»

96059601

9509900499

941480149401

93206534790699

99

100

10000

1000000

100000000

10000000000

1000000000000

100000000000000

100

40

T,il,tc for

<tn<l ttinm, trin'diis

TABLE XXVIII.
£IH»I* of Powers of A'utural Numbers.

ft

S(n)

,?(»«)

•$(»')

*(«•)

*w

•w

»(«•»)

n

1

1

1

1

1

1

1

1

1

t

3

5

9

17

33

66

129

:

s

6

14

3G

M

270

794

2316

3

4

10

30

100

354

1300

4890

18700

4

6

15

55

225

979

4425

20515

•em

5

6

21

91

441

2275

12201

67171

176781

6

7

U

140

784

467*.

2! 1008

184820

1200304

7

8

36

MM

1296

8772

<il77C

44(5964

3i.'!)7456

8

9

45

l!N-i

2025

1533:5

120825

97H405

8080425

9

10

65

385

3025

25333

220826

1978405

18080426

10

11

66

MM

4356

88674

381876

3749966

37567596

11

It

78

650

MM

60710

<;:u>708

6735950

73399404

12

a

91

819

M'SJ

80971

100!

11562759

136147921

13

14

105

1015

11025

127687

1531

19092295

241561425

14

15

120

1240

14400

178312

-J-:i!liOO

30482920

412420800

15

16

136

1496

18 190

243848

3:1, 17776

47260136

680856256

16

17

153

1785

23109

32731 lit

4767633

71397705

10911!)l'.i-J!i

17

18

171

2109

29841

432345

8657201

ld.r) :

1703414981

18

19

1'JO

2470

361 00

668

9133300

1.VJ 155810

^v. '7286700

19

HO

810

2870

44100

l-2333:;i HI

21G455810

3877286700

SO

XI

231

3311

533(51

917147

16417401

302221931

6678376241

21

.'.'

858

64009

1151403

21571033

415001835

8172733129

.'.'

:•

270

43M

7817fl

1431 :>41

2800737(1

603637724

11677558576

S3

*4

300

4900

90000

17<;:;o;!o

3.MITOOOO

754740700

16104030000

•">

£5

MB

66SB

I0o(iir>

2153U45

4573.M Il'.-i

998881325

22267545626

£5

»;

351

6201

123201

2610C21

57617001

1307797101

30299355801

M

27

378

6930

142884

814S

719G5fX)8

1695217590

40759709004

i;

S8

406

771 1

164Niti

3756718

8917ii^7«;

2177107894

64252637516

98

20

435

8550

189225

44G3999

109687425

2771931215

71602513825

SO

30

465

9455

216225

5273999

133987425

3500931215

93372513825

30

SI

49G

10416

246016

6197520

16261(5576

4388434896

120885127936

31

3S

MB

11440

278784

7246096

196171008

5462176720

155244866304

32

S3

561

125:.'!)

314721

8432017

135306401

6753644689

197863309281

33

34

595

18686

354025

970*353

180741825

8298449105

250386659425

34

35

630

14910

396900

11208978

333263700

10136714730

314725956300

35

36

see

16206

443556

12948594

393729876

12313I!'7

393090120396

36

37

703

17575

4!U.'03

1482L'755

•463073VS3

14879223475

488021997529

37

38

741

19019

5 1'J 61

16907891

542309001

17890159859

602437580121

u

39

780

80MO

608400

!!):>:> 1332

C3-J533200

21408903620

739668586800

39

40

820

22140

<i7i'400

21781332

734933200

25504903620

903508586800

40

41

861

nan

741381

24607093

850789401

30256007861

1098262860681

41

42

903

M5409

i'77 18789

98l4W)ii33

35744039605

132S802193!U!l

42

4S

946

27434

s:il!»16

3M37.'!H)

1128489076

42065402654

1600620805036

44

MO

980100

31 —

1293405300

49321716510

1919898614700

44

46

1080

31 :!'.).-,

1071225

3*! 1*1:3 ii

1477933 J25

67625482135

2293568067825

',--•

4<;

1001

33:, 11

1168561

434037C7

1683896401

67099779031

2729385725041

46

47

use

3.->7:>0

1272384

48343448

1913241408

77878994360

3236008846604

47

48

1176

1382976

63651M64

2168045376

90109584824

3823077187776

48

49

1 I'!':,

40lL'-|

1500625

59416665

2450521 K'c'5

1039.'>087-2025

4501300260626

49

BO

1276

taou

1625625

00666668

2763020625

119575872025

5282560260625

50

Tables of Powers and Snins of Powers

TABLE XXVIII.— (continued).
Bums of Powers of Natural Numbers.

41

n

S(n)

«(*»)

8(*>)

,S>«)

aw

£(»•)

«P)

n

51

1326

45526

1758276

72431866

3108045876

137172159826

6179960938476

51

52

1378

48230

1898884

79743482

3488249908

156942769490

7208032641004

52

53

1431

51039

2047761

87633963

3906445401

179107130619

8382743780841

53

54

1485

53955

2205225

96137019

4365610425

203902041915

9721668990825

54

55

1540

56980

2371600

105287644

4868894800

231582682540

11244104225200

55

56

1596

60116

25472-16

115122140

5419626576

262423661996

12971199074736

56

•',7

1053

63365

2732409

125678141

6021318633

296720109245

14926096567929

57

58

1711

66729

2927521

136994637

6677675401

334788801789

17134080735481

58

.59

1770

70210

3132900

149111998

7392599700

376969335430

19622732220300

59

60

1830

73810

3348900

162071998

8170199700

423625335430

22422092220300

60

61

1891

77531

3575881

175917839

9014796001

475145709791

25564835056321

61

62

1953

81375

3814209

190694175

9930928833

531945945375

2908644966252!)

62

63

2016

85344

4064256

206447136

10923365376

594469447584

33025430301690

63

64

2080

89440

4326400

223224352

11997107200

663188924320

37423476812800

64

65

2145

93065

4601025

241074977

13157397825

738607814945

42325704703425

65

66

2211

98021

4888521

260049713

14409730401

821261764961

47780865404481

66

67

2278

102510

5189284

280200834

15759855508

911720147130

53841577009804

67

68

2346

107134

5503716

301582210

17213789076

1010587629754

60564565828236

68

69

2415

111895

5832225

324249331

18777820425

1118505792835

68010919080825

69

70

2485

116795

6175225

348259331

20458520425

1236154792835

76246349080825

70

71

2556

121836

6533136

373671012

22262749776

1364255076756

85341469239216

71

72

2628

127020

6906384

400544868

24197667408

1503569146260

95372082243504

72

7o

2701

132349

7295401

428943109

26270739001

1654903372549

106419480762601

73

74

2775

137825

7700625

458929685

28489745625

1819109862725

118570761035625

74

75

2850

143450

8122500

490570310

30862792500

1997088378350

131919149707500

75

7<;

2<I26

149226

8561476

523932486

33398317876

2189788306926

146564344279276

76

77

3003

155155

9018009

669085527

36105102033

2398210687015

162612867546129

77

78

3081

161239

9492561

596100583

38992276401

2023410287719

180178436401041

78

79

3160

167480

9985600

835060684

42069332800

2866497743240

199382345387200

79

80

3240

173880

10497600

676010604

45346132800

3128641743240

220353865387200

80

81

3321

180441

11029041

719057385

48832917201

3411071279721

243230657842161

81

8S

3103

187165

11580409

764269561

52540315633

3715077951145

268159204898929

82

83

848a

194054

12152l!n;

811727882

56479356270

4042018324514

295295255888556

83

84

0)70

201110

1274 III i •

861515018

60661475700

4393316356130

324804290544300

84

85

8666

208335

13359025

913715643

65098528825

4770465871755

356861999372425

85

86

3741

215731

13!)! (5081

968416459

69802799001

5175033106891

391654781594121

86

87

8888

22331 K)

14653584

1025706220

74787008208

5608659307900

429380261081904

87

88

3!) Hi

231044

15335056

1085675756

80064327376

0073063394684

470247820718896

88

89

4005

238965

16040025

1148417997

85648386825

6570044685645

514479155614425

89

90

4005

247065

16769025

1214027997

91553286825

7101485685645

562308845614425

90

91

4186

255346

1 75225! M;

1282602958

97793608276

7669354037686

613984947550156

91

M

427K

263810

18301284

135424225-1

104384423508

8275709939030

669769607673804

92

93

4371

272459

19105641

1429047455

111341307201

8922700122479

729939694734561

93

04

4465

281295

10936225

1507122351

118680347425

9612569903535

794787454153825

94

95

450(1

M0390

20793600

1588572976

126418156800

10347661794160

864621183763200

95

96

4656

299536

21 678336

1673507632

134571383776

11130419583856

939765931574016

96

97

4753

308945

22591009

1762030913

143159224033

11963391588785

1020564216052129

97

98

4851

318549

23532201

1854273729

162198432001

12849233969649

1107376769376801

98

M

4950

328350

24502500

1950333330

161708332500

13790714119050

1200583304167500

99

100

5050

838800

25502500

2050333330

171708332500

14790714119050

1300583304167500

100

B.

(•_' Tables for St«ti*ti<-i<iu* <u/</ lii<>in>tri<-i<ii<.<

TABLE XXIX. Tetrachoric Function* for Fourfold Correlation Tables.

i(l-a)

n

^t

«•»

»"«

»•»

T«

A

•001

•00337

•00736

+ •01175

+ •01391

+ •0113-1

+ •00415

3-09023

•on

•00634

•01290

+ •01885

+ -01968

+ •01 269

+ -IKK >:,:(

2-87816

i i

•00916

•01778

+ •02446

+ -02.-5:!:.

+ •01.

- -oo:»2*

2-74778

•004

•01185

•02222

+ •02918

+ •02587

+ •011 11

--006S7

2-65207

•005

•01446

•02634

+ '03326

+ •02764

+ •O09.r>2

- -01017

2-57583

•006

•01700

•03020

+ -03686

+ -Ois-^7

+ -00770

- 01318

2-51211

•007

•01949

•03386

+ -04008

+ -02970

+ -(H

- -01592

2-45726

•008

•02192

•037:5 1

+ -04298

+ -03021

+ -00371

- -01841

2-40892

•009

•02431

•04066

+ •04561

+ -03047

+ •00164

-•02067

2-36562

•010

•02665

•04384

+ •04800

+ •030:.:;

-•001144

-•02271

2-32635

•Vll

•02896

•04690

+ •06020

+ •03041

- •002.-.3

- '021. -.7

2-29037

•012

•03123

•04986

+ -05221

+ •03014

-•00460

- -iini-.-,

2-26713

•01S

•03348

•05270

+ -05 106

+ -02975

-O0664

- -02777

|**MS]

•014

•03569

•06545

+ -05577

+ -02926

-•00866

- -02914

2-lU7j:»

•015

•037h7

•05811

+ -or,7:5.j

+ •02867

- -01064

- -03037

2-17009

•016

•04003

•06069

+ -05880

+ -02801

- -01259

- -03147

2-14441

•017

•04216

•06320

+ •06015

+ -02727

- -01449

- '03246

2-12007

•018

•04427

•06564

+ -06139

+ -02IU7

- -01636

- -0:5:5:54

2-09693

•019

•04636

•06801

+ -06254

+ -02562

-•01818

-•034 11

2-07485

-,,.-.}

•04842

•07031

+ •06361

+ •0247:!

-•01996

-03479

2-05375

•Oil

•05046

•07256

+ •06459

+ •02378

-•02170

- -03638

2-03352

.,,,_,

•0:>249

•07475

+ -06549

+ •02280

-•02:510

- -03589

2-01409

•OH

•05449

•07688

+ -06633

+ •02 179

-•02

- -03632

1-99539

-„.'.;

•05648

•07897

+ -06709

+ -02074

- -02666

- -03667

1-97737

•025

•05845

•08100

+ -06780

+ -01968

- -02823

- -03696

1-95996

•OM

•06040

•0829!)

+ -06844

+ •01858

- -02976

- -03718

1-94313

•Of?

•062:53

•08493

+ •06903

+ -01747

- -03125

-•03734

1-92684

•028

•06425

•08682

+ -06956

+ -01634

- -03270

-•03744

1-91104

•tt>9

•06615

•08868

+ -07005

+ •01 520

- -03411

- -03749

1-89570

•030

•06804

•09049

+ O7048

+ -01404

- -03547

-•03749

1-88079

•031

•06992

•09227

+ -07087

+ •01 287

- -03680

-•03744

1-86630

•03i

•07177

•09400

+ -07122

+ •01168

- -03810

- -03734

1-85218

•OSS

•07362

•09570

+ -07153

+ •01049

- -03i«:.

- -03720

1-83842

•034

•07* i:.

•09737

+ •07 1 79

+ -00929

-•04o:, 7

- -03702

1-82501

•035

•07727

•09900

+ •07202

+ -00809

-•041 76

- -03680

1-81191

•036

•07908

•10060

+ -07221

+ -00688

- -04291

- -03654

1-79912

•037

•08087

•10216

+ •07 2:57

+ -00566

- -04402

-•03624

1-78661

•038

•08265

•10370

+ -07249

+ -00444

- -04510

-•03592

1-77438

•039

•08442

•10520

+ -07258

+ -00322

- -04615

- -O3.v><;

1-76241

•040

•08617

•10668

+ -07264

+ -00200

- -04717

- -03617

1-75069

•041

•08792

•10812

+ -07268

+ -00077

- -04815

- -03476

1-73D20

•04S

•08966

•100

+ -07268

- -00045

- -04910

- -03431

1-72793

•043

•09137

•11093

+ -07266

- -00167

- -05003

- -03384

1-71689

•044

•09309

•111

+ -07261

-•00290

-•05092

- -03335

1-70604

•045

•09479

•li:5<;:5

+ -O72.r>3

-•00412

- -05178

- -032s:i

1-69540

•046

•09648

•11490

+ -07243

-•00534

- -05261

- -0322!)

1-68I1I I

•047

•09816

•1UU

+ -07231

-•00656

- -05342

-•0317:5

1-67466

•048

oeoa

•11760

+ •0721 7

-•00778

- -05420

-•03116

1-66456

•040

•1014!)

•11874

+ -07200

-•00899

- -05495

-•03or,5

1-65463

•Ofiii

•1031 1

•1199*;

+ -07181

--0102H

-•05567

- -02! KM

1 -i! 1485

TI, TJ »nj A are essentially positive.

Tables of the Tetraclwric Functions
TABLE XXIX.— (continued).

43

*(!-«>

TI

TJ

T3

*1

1-5

1-6

h

•051

•10478

•12115

+ •07160

- -01140

- -05637

- -02931

1-63523

•052

•10641

•12232

+ •07138

- -01260

- -05704

- -02866

1-62576

•OSS

•10803

•12347

+ •07113

- -01380

- -05769

- -02799

1-61644

•054

•10964

•12460

+ -07087

- -01499

- -05831

- -02732

1 -60725

•055

•11124

•12571

+ -07058

-•01618

- -05891

- -02662

1-59819

•056

•11284

•12680

+ -07028

- -01736

- -05949

- -02592

1-58927

•057

•11442

•12787

+ -06997

- -01854

-•06004

- -02r)20

1-58047

•058

•11600

•12892

+ -06964

- -01971

- -06057

- -02447

1-57179

•059

•11766

•12995

+ -06929

- -02087

- -06107

- -02373

1-56322

•060

•11912

•13096

+ -06893

-•02203

- -06155

- -02298

1-55477

•061

•12067

•13196

+ •06855

- -02318

- -06202

- -02222

1 -54643

•Mi

•12222

•13293

+ -00816

- -02433

- -06246

- -02145

1 -53820

•on

•12375

•13389

+ -06775

- -02547

- -06288

- -02068

1-53007

•06^

•12528

•13483

+ -06734

- -02660

- -06328

- -01989

1-52204

•065

•12679

•13675

+ •06690

- -02773

- -06365

- -01910

1-51410

•066

•12830

•13666

+ •06646

- -02884

- -06401

- -01830

1-50626

•067

•12981

•13754

+ •06601

- -02996

- -06435

- -01749

1-49851

•068

•13130

•13842

+ -06554

- -03106

- -06467

- -01668

1-49085

•069

•13279

•13927

+ -06506

- -03216

- -06498

- -01586

1 -48328

•070

•13427

•14011

+ -06457

- -03325

- -065-JU

- -01504

1-47579

•071

•13674

•14094

+ -06407

- -03433

- -06552

- -01421

1-46838

•072

•13720

•14175

+ -06356

- -03541

- -06.")77

- -01337

1-46106

an

•13866

•14254

+ •06304

- -03648

- -IHJ600

- -01253

1-45381

•074

•14011

•14332

+ '06251

- -03754

- -06621

-•01 169

1 -44663

•075

•14156

•14409

+ -06197

-•03859

- -06641

- -01085

1-43953

•076

•14299

•14484

+ -06142

- -03963

- -06659

- -01000

1-43250

•077

•14442

•14668

+ •06086

- -040(>7

- -06675

-•0091 5

1-42554

•078

•14584

•14630

+ •06029

-•04170

- -06690

- -00829

1-41865

•079

•14726

•14701

+ -05971

-O4272

- -06703

- -00743

1-41183

•080

•14867

•14771

+ •05913

- -04374

- -06715

- -00668

1-40507

•081

•16007

•14839

+ •06854

- -04474

- -06725

- -00572

1-39838

•082

•16146

•14906

+ •05794

- -04574

- -06733

- -00485

1 -39174

•083

•15285

•14971

+ -05733

- -04673

- -06741

- -00399

1-38517

•084

•15423

•16036

+ -05671

-•04771

- -06746

- -00312

1-37866

•085

•16661

•16099

+ -05600

- -04869

- -06751

- -00226

1-37220

•086

•15698

•16160

+ •05546

-•04965

- -06753

- -00139

1-36581

•087

•15834

•15221

+ -05483

- -05061

- -06755

-•00053

1-35946

•088

•15970

•16280

+ -05418

- -05156

- -06755

+ -00034

1-35317

089

•16105

•15339

+ -05353

- -05250

- -06754

+ -00120

1-34694

•090

•16239

•15396

+ -05288

- -05344

-•06751

+ -00207

1 -34076

•091

•16373

•15451

+ -08222

- -05436

- -06748

+ -00294

1-33462

•n:i.'

•16506

•15506

+ •05156

- -05528

- -06743

+ -00380

1-32854

•09S

•16639

•155t;n

+ •05088

- -05619

- -06736

+ -00467

1-32251

•094

•16770

•16612

+ •05020

- -05709

- -06729

+ -00553

1-31652

•OH

•16902

•15663

+ •04952

- -05798

- -06720

+ -00639

1-31058

•096

•17033

•16713

+ •04883

- -06887

- -06710

+ -00725

•30469

•097

•17163

•16763

+ -04813

- -05975

- -06699

+ -00811

•29884

•098

•17292

•15811

+ -04744

- -06061

- -06687

+ -00897

•29303

•099

•17121

•16858

+ -04673

- -06148

- -06674

+ -00982

•28727

•inn

•17550

•16904

+ •04602

- -06233

- -06660

+ -01068

•28155

6-2

44 Tables for Statisiin'tmx and

TABLE XXIX. Tetrachoric Functions for Fourfold Correlation Tables.

1(1--)

tl

ft

TJ

••4

*6

rt

h

•101

•17678

•15948

+ -04631

-•06317

-•00644

+ •01153

I -27587

•lOt

•17808

•159U2

+ •04459

-•06401

- -06628

+ •01-238

1 L'7024

•103

•17932

•16036

+ •04387

-•06484

- -08610

+ •01 322

I -88464

•104

•18058

•16077

+ -04315

- -06566

- -06592

+ •01407

1-25908

•106

•18184

•16118

+ -04242

-•08647

-•06572

+ •01491

1-25357

•106

•18309

•16158

+ -041B9

-•06727

- -06651

+ •01676

1-24808

•107

•18433

•16197

+ •04095

-•00807

- -06530

+ •01669

1-24264

•108

•18567

•16235

+ •04021

- -0688C

- -06507

+ •01742

•23723

•109

•18681

•16272

+ •03947

-•069GI

- -06484

+ -01826

•23186

•110

•18804

•16308

+ -03872

- -07041

-•06459

+ •01908

•22653

•111

•18926

•16343

+ -03797

-•071 17

-•06434

+ •01990

22123

•in

•19048

•16378

+ -03721

- -07193

-•06408

+ -02072

•21596

•us

•19169

•16411

+ -03646

- -07268

-•08381

+ -02154

•21073

•114

•19290

•16413

+ -03570

- -07342

-•06353

+ -02235

•20553

•115

•19410

•16476

+ •03493

- -07415

- -06324

+ -02316

•20036

•116

•19530

•16506

+ •0341 7

-•07488

- -06294

+ -02397

•19522

•117

•19649

•16536

+ -03340

- -07559

- -06264

+ •02477

•19012

•118

•19768

•16565

+ -03263

-•07630

- -06233

4 -02567

•18604

•119

•19886

•16593

+ -03186

- -07700

- -06201

+ -02636

1-18000

•ieo

•20004

•16620

+ -03108

- '07770

-•06168

+ -02716

1-17499

•Itl

•20121

•16647

+ •03030

-•07838

- -06134

+ -02794

1-17000

•1 :.'

•20238

•16672

+ -02952

-•07906

-•06100

+ -02873

1-16505

•US

•20354

•16697

+ -02874

- -07973

-•06065

+ -02950

1-16012

•Itk

•20470

•1G721

+ '02796

- -08039

- -06029

+ -03028

i •loan

•iss

•20586

•16745

+ •02717

- -08105

- -05992

+ -03105

1-15035

•ISO

•20700

•16767

+ •02638

-•08169

- -05955

+ -03181

1-14551

•Ii7

•20814

•16789

+ -02669

- -08233

- -05917

+ -03257

1-14069

•1 M

•20928

•16810

+ •02480

- -08297

- -05878

+ -03333

1-13590

•129

•21042

•16830

+ -02401

-•08359

- -06839

+ -03408

1-13113

•ISO

•21165

•16849

+ •02321

- -08421

- -05799

+ •03483

1-12639

•131

•21267

•16868

+ •02241

- -08482

- -05758

+ -03557

1-12168

•13*

•21379

•16886

+ -02H12

- -08642

- -nr.TiT

+ -03631

1-11699

'ISS

•21490

•16903

+ -02082

-•08601

- -05676

+ -03704

1-11232

•UA

•21601

•16919

+ -02001

-•08660

- -05632

+ -03777

1-10768

•ISS

•21712

•16935

+ •01921

-•08718

- -05589

+ •03860

1-10306

•136

•21822

•16950

+ O1841

- -08776

- -06546

+ -03921

1-09847

•1S7

•21932

•16964

+ •01760

-•08831

- -05501

+ -03993

1-09390

•138

•22041

•16978

+ •01680

-•08887

- -05456

+ -04064

1 -08935

•139

•22149

•16990

+ -01699

- -08942

-•054 11

+ -04134

1 -08482

•140

•22258

•17003

+ •01518

- -08996

- -05365

+ -04204

1-08032

•141

•22365

•17014

+ •01437

-•09050

- -05318

+ •04273

1-07684

•14*

•22473

•17026

+ -oi3:><;

- -09103

- -06271

+ -04342

1-07138

•143

•22580

•17036

+ -01276

- -09155

- -05224

+ •04410

1 -06694

•144

•22686

•17044

+ •01194

-•09206

- -06176

+ •0117*

1-06252

•Itf

•22792

•17053

+ •01113

- -09257

- -05127

+ -04545

1-05812

•146

•22898

•17061

+ •01032

-•09307

-•06078

+ •04612

1-05374

•147

•23003

•17069

+ •00950

- -09356

- -05028

+ -04678

1 -04!t:i'.»

•148

•23108

•17076

+ •00869

-•09405

- -04978

+ •04744

L-OtfOb

•149

•23212

•17082

+ •00788

-O9452

- -04928

+ -04809

1-04(17:'.

•ICO

23316

•17087

+ •00706

- -09499

- 04877

+ •04874

1 03043

Tables of the Tetrachoric Functions
TABLE XXIX.— (continued).

45

4(i-«)

TI

TS

*3

Tt

Tfi

^8

h

•161

•23419

•17092

+ -00625

- -09546

- -04825

+ '04938

1-03215

•152

•23522

•17097

+ -00543

- -O9.p)92

- -04774

+ -05002

1-02789

•153

•23625

•17100

+ •00462

- -09637

- -04721

+ •05065

l-0236f>

•154

•23727

•17103

+ -00380

- -09681

- -04669

+ -05127

1-01943

•155

•23829

•17106

+ -00298

- '09725

- -04615

+ -05189

1-01522

•156

•23930

•17108

+ •00217

-•09768

- -04562

+ -05250

1-01103

•157

•24031

•17109

+ -00135

- -09810

- -04508

+ -05311

1-00686

•158

•24131

•17110

+ •00053

- -09852

- -04454

+ -05371

1-00271

•159

•24232

•17110

-•00028

- -09892

- -04399

+ -05431

•99858

•1GO

•24331

•17109

-•00110

- -09933

- -04344

+ -05490

•99446

•161

•24430

•17108

- -00191

- -09972

- -04288

+ -05549

•99036

•162

•24529

•17107

- -00273

- -10011

- -04232

+ -05607

•98627

•16S

•24628

•17104

- -0035.1)

- -10049

- -04176

+ -05664

•98220

•164

•24726

•17102

- -00436

-•10087

- -04120

+ -05721

•97815

•165

•24823

•17098

- -00518

- -10124

- -04063

+ -05778

•97411

•166

•24921

•17094

-•00599

-•10160

- -04006

+ •05834

•97009

•167

•25017

•17090

-•00681

- -10196

- -03948

+ -05889

•96609

•168

•25114

•17085

-•00762

- -10231

- -03890

+ '05943

•%210

•169

•25210

•17080

-•00844

- -10265

- -03832

+ -05998

•95812

•170

•26305

•17073

-•00925

- -10299

- '03774

+ -06051

•95417

•171

•25401

•17067

- -01007

- -10332

- -03715

+ -06104

•95022

•na

•^.-.495

•17060

-•01088

- -10364

- O3656

+ -06156

•94629

•17S

•25590

•17052

-•01169

- -10396

- -03597

+ -06208

•94238

•174

•25684

•17044

- -01251

- -10427

- -03537

+ -06260

•93848

•175

•25778

•17035

-•01332

-•10458

- -03478

+ -06310

•93459

•176

•25871

•17026

-•01413

- -10487

- -03417

+ •06360

•93072

•177

•25964

•17016

- -01494

- -10517

- -03357

+ -06410

•92686

•178

•2GO.W

•17006

- -O157.r>

- -10545

- -03296

+ -06459

•92301

•179

•26148

•16995

- -01656

- -10573

- -03236

+ -06507

•91918

•180

•26240

•16984

-•01737

-•10601

- -03176

+ -06555

•91537

•181

•26331

•16972

- -01817

- '10627

-•03113

+ •06603

•91156

•182

•26422

•16960

-•01898

- -10653

- -03052

+ -06649

•90777

-us

•26513

•16948

- -01979

- -10679

- -02990

+ •06695

•90399

•184

•26603

•16934

- -02069

- -10704

- -02928

+ -06741

•90023

•185

•26693

•16921

- -02140

- -10728

- -02806

+ •06786

•89647

•186

•26782

•16907

- -02220

- -10752

- -02803

+ •06830

•89273

•187

•26871

•16892

-•02300

- -10776

- -02741

+ -06874

•88901

•188

•26960

•16877

-•02380

- -10798

- -02678

+ •06917

•88529

•189

•27049

•16861

-•02460

- -10819

- -02615

+ •06960

•88159

•190

•27137

•16845

- -02540

- -10841

- -02552

+ -07002

•87790

•191

•27224

•16829

- -02620

- -10861

- -02489

+ •07044

•87422

•19S

•27311

•16812

- -02700

- -10882

- -02425

+ -07085

•87055

•193

•27398

•16795

- -02779

-•10901

- -02362

+ -07125

•86689

•194

•27485

•16777

- -02859

-•10920

- -02298

+ •07165

•86325

•195

•27671

•16759

- -02938

- -10939

- -02234

+ -07204

•85962

•196

•27657

•16740

- -03018

- -10956

- -02170

+ -07243

•85600

•197

•27742

•16721

-•03097

- -10974

- -02106

+ •07281

•85239

•198

•27827

•16701

- -03176

- -10990

- -02041

+ -07319

•84879

•199

•27912

•16681

- -03255

- -11007

- -01977

+ -07356

•84520

•200

•27996

•16661

- -03334

- -11022

- -01912

+ -07392

•84162

46 Table* for Statistician* <m<1 Ilium, triciunn

TABLE XXIX. Tetrachoric Functions for Fourfold Correlation Tables.

KI-.)

TI

f|

TI

*

»•«

r»

A

•SOI

•28080

•10640

-•03412

-•11037

-•01848

+ D7428

•83805

•sot

•28164

•16619

3491

- -11051

- -01783

+ -07n;i

•83450

m

•282 1 7

•16597

- -03569

- -11065

-D1718

+ -07498

•83095

:•",

•28330

•16675

-•03648

- -11079

- D1653

+ D7632

•82742

•284 1 3

•16553

-•03726

- -11091

- DU.-7

+ -07666

S23H9

m

-I95

•l(>.r,30

-•03804

-•11104

- D1522

+ •07599

•82038

m

9577

•16506

OHM

-•iin:«

-•01 457

+ •07(132

•81687

IN

•28058

•16483

_ •<>:.

-•imo

- -01391

+ -07664

•81338

•509

•28739

•164.V.I

- -041 I'M

-•11137

- D1326

+ -07695

•80990

•no

•28820

•16434

1114

-•11147

- -01260

+ -07726

•80642

:ii

•28901

•16409

-O41 '.1-2

-•11157

- -01194

+ -07756

•80296

•:l :

•28981

•l<;:isi

•04841

-•11166

-•01 in

+ D7786

•79960

•//.;

•29060

•M8M

' ' 1346

-•11174

- -OH",:',

+ D7816

•79606

•///,

•29 1 40

•1633-2

- 1)44:23

-•11182

- -00997

+ D7844

•79262

•SIS

21)219

•163"-.

- 1)4 I!)!)

-•11189

- -00931

+ -07872

•78919

•S1G

21)298

•16279

- '04576

-•11196

-•00865

+ -07899

•78677

•m

•29376

•10251

-•04652

- -11203

-D0799

+ D7926

•78237

•utt

•29454

•1622 I

- -04728

-•11208

- -00733

+ -07952

•77897

•sin

•29532

•16196

- -04804

-•11214

- -00667

+ -07978

•77657

..,,,

•29609

•16167

- -04880

-•11218

-D0600

+ D8004

•77219

•.•:/

•29686

•1C139

- -04956

-•11223

- -00534

+ -08028

•76882

•::.'

•297C3

•16110

- -0503 1

- -11226

- -00468

+ D8052

•76546

•::.;

•29840

•16080

- -05107

-•11230

-•Oll|(i2

+ -08076

•76210

•:.",

•299 Ui

•16050

- -05182

-•11233

- -00335

+ -08099

•75876

•::.-,

•29991

•16020

- -05257

-•11235

- -00269

+ -08122

•75541

at;

•30067

•15090

- -05332

-•11237

- -00203

+ -08144

•75208

.-.-,-

•3(>l42

•15959

- -05406

-•11238

-•00136

+ D8 165

•74876

•/.'.•>•

•30216

•16927

- -05481

-•11239

-•00070

+ •08186

•74545

-Mi

•30291

•15896

- D5555

-•11239

-D0004

+ •08207

•74214

•sso

•30365

•16864

- -05629

-•11239

+ •00063

+ -0822C

•73886

•/.;/

•3043!)

•16832

- -05703

-•11238

+ D01 29

+ -08246

•73556

-.•.;.•

•30512

•16799

- -05777

-•11237

+•00190

+ D8266

•73228

•&SS

•30585

•15766

- -05851

-•11236

+ •002112

+ -08283

•72900

•S»4

•30058

•157:i::

- -O.r>i'i! I

-•11233

+ -00328

+ •08301

•72574

•?.ti

•30730

•16699

- -05997

-•11230

+ -00394

+ 1)8318

•72248

•«c

•30802

•15665

- -06070

- -11227

+ •00461

+ •08334

•71923

•237

•30874

•15631

- -06143

-•11224

+ •oo:.-27

+ 1)8351

•71699

•sss

•30945

•16596

- -06215

-•11220

+ 1'Hi.-i!»3

+ 1)8366

•71275

•iS9

•31017

•15561

- -06288

-•11215

+ -00659

+ -08381

•70952

•gjtO

•31087

•15526

- -06360

-•11210

+ -00726

+ -08396

•70630

•S41

•3H68

•15490

- -06432

-•11205

+ -0071)2

+ D8410

•70309

•-",-'

•3 1 228

•15454

- -06504

-•11199

+ -OOH5H

+ -08423

•69988

••',•:

•31298

•15418

- -06576

-•111!»2

+ •001)21

+ •0813(1

•(>:.>

•-".',

•31 367

•15382

-1)6047

-•11185

+ UOUDO

+ D8449

(11)349

•-",-•

•31436

•15345

-1)6718

-•11178

+-010M

+ -08IC1

•CD031

•*46

•31505

•16308

- -06789

-•11170

+ D1 122

+ 1)8172

•68713

•tlfi

•31574

•16270

- -06860

-•11162

+ •01188

+ •081*3

•683! Mi

*4t

•31042

•15232

- -0«i!>:!l

-•M154

+ •01253

+ -OM:II

•68080

146

31710

•16194

-•07001

-•11145

+ •0131!)

+ D8.-i(H

•67764

•SM

•31778

•15156

-D7071

-•11135

+ •013X5

+ •08513

•67449

Tables of the Tetrachoric Functions
TABLE XXIX.— (continued).

47

4<l-a)

TI

TS

*-s

>"4

T6

i-o

h

•251

•31845

•15117

- -07141

-•11125

+ -01450

+ -08522

•67135

•252

•31912

•15078

-•07211

-•11115

+ -01516

+ -08530

•66821

•253

•31979

•15039

- -07280

-•11104

+ -01581

+ "08538

•66508

•254

•32045

•14999

- -07350

- -11093

+ -01647

+ -08:)46

•66196

•255

•32111

•14959

- '07419

- -11081

+ •01712

+ -08553

•65884

•256

•32177

•14919

- -07488

- -11069

+ "01777

+ -08559

•65573

•257

•32242

•14879

- -07557

- '11056

+ -01842

+ -08565

•65262

•258

•32307

•14838

- -07625

- -11043

+ •01907

+ -08571

•64952

•32372

•14797

- -07693

- -11030

+ -01972

+ '08575

•64643

•360

•32437

•14756

- -07761

-•11016

+ -02037

+ -08580

•64335

•261

•32501

•14714

- -07829

-•11002

+ -02102

+ -08584

•64027

•262

•32565

•14672

- -07897

- '10987

+ -0216<>

+ -08587

•63719

•263

•32628

•14630

-•07964

- -10972

+ •02231

+ •08590

•63412

•264

•32691

•14588

-•08031

- -10956

+ -02295

+ -08593

•63106

•265

•32754

•14545

-•08098

- -10940

+ -02360

+ '08595

•62801

•JM

•32817

•14502

- -08165

- -10924

+ -02424

+ -08596

•62496

•tw

•32879

•14459

- -08231

- -10907

+ -02488

+ -08597

•62191

•32941

•1 1 115

- -08208

- -10890

+ -02552

+ -08598

•61887

•33003

•14372

- -08364

- -10873

+ -02616

+ -08598

•61584

•270

•33065

•1432K

- -08429

- -10855

+ -02680

+ -08598

•61281

•371

•33126

•14283

- -08495

- '10837

+ -02743

+ -08597

•60979

• n :

•33187

•14239

-•08560

- -10818

+ -02807

+ O8596

•60678

•37S

•33247

•14194

--OV

- -10799

+ -02870

+ -08594

•60376

•374

•33307

•14I4!I

-•08690

-•10779

+ -02933

+ -08591

•60076

•275

•33367

•14104

- -08755

- -10759

+ •02997

+ -08589

•59776

•276

•33427

•14058

-•08819

- -10739

+ -03060

+ -08586

•69477

•277

•334H6

•14012

-•08883

-•10718

+ •03122

+ -08582

•59178

•278

•33545

•1396li

- -08947

- -10697

+ •03185

+ -08578

•58879

•/;.'<

•33604

•UMO

-•09011

-•10676

+ •03248

+ -08573

•58581

•280

•33662

•13H7:i

-•09074

- -10654

+ -03310

+ -08568

•68284

•281

•33720

•13826

-•09137

- -10632

+ •03372

+ -08563

•57987

•>•/

•33778

•13779

-•09200

- -10609

+ -03434

+ -08557

•57691

4M

•33836

•13732

-•09263

- -10587

+ -03496

+ -08551

•57395

•-'•-•{

•33*: i.'.

•13i

- -09325

- -10563

+ -03558

+ -08544

•5710(1

•asr,

•33950

•13637

- -09388

- -10540

+ -03620

+ -08536

•56805

•34007

•13589

- -09450

- -10516

+ -03681

+ -08529

•56611

•287

•34063

•I3.r. n

- -09511

- -10491

+ -03743

+ •08521

•56217

•34119

•13492

- -09573

- -10466

+ -03804

+ •08512

•55924

•34175

• 13443

- -09634

- -10441

+ -03865

+ -08503

•55631

•34230

•13394

- -09695

- -10416

+ -03926

+ -08494

•55338

-.".a

•34286

•13345

-•09756

- -10390

+ •03987

+ -08484

•65047

•JM

•34341

•132!)6

- -09816

- -10364

+ -04047

+ -08473

•64755

•«M

•34395

•132 l«;

-•09876

- -10337

+ •04107

+ -08463

•54464

•344 »:»

•13196

- -09936

- -10310

+ -04168

+ -08451

•54174

•3 »:>( >:5

•13146

- -09996

- -10283

+ -04228

+ -08440

•53884

•296

•34557

•13096

- -10056

- -10256

+ •04287

+ -08428

•53594

•..'.''7

•34611

•13046

-•10115

- -10228

+ -04347

+ -08415

•53305

•208

•31664

•12995

-•10174

- -10199

+ -04407

+ •08402

•53016

•299

•:vi717

•12944

- '10233

- -10171

+ -04466

+ -08389

•52728

•300

•34769

•12893

- -10291

- -10142

+ -04525

+ -08375

•52440

48 Table* fur M«ti.*t it-inn* and

TABLE XXIX. Tetrachoric Functions for Fourfold Correlation Tables.

1(1-.)

n

Tt

*•»

»-4

id

ik

k

•sot

•34822

•18841

- -10349

-•10113

+ •04684

+ -ON3H1

•52153

•sot

•34874

•12790

-•10407

-•10083

+ •04643

+ •08347

•51866

•SOS

•34925

•12738

-•10466

-•10063

+ -04701

+ •OKtt-2

•51579

•304

•34977

•12686

-•10622

-•10023

+ -04759

+ •08316

•51293

•SOS

•36028

•12634

-•10680

- -09992

+ •04818

+ •08301

•51007

•S06

•36079

•12581

- -10636

-•09961

+ O4876

+ O8:!*!

•60722

•SOT

•36129

•12529

- -10693

-•09930

+ -04933

+ -08268

•50437

•SOS

•35180

•12476

- -10750

-•09889

+ -04991

+ •08251

•50153

•309

•35230

•12423

-•10806

-•09867

+ •05048

+ -08233

I! (869

•310

•35279

•12370

- -10862

- -09834

+ -05105

+ -08216

•49686

•311

•35329

•12316

-•10917

- -09802

+ •05162

+ •08197

•49302

•31S

•35378

•12263

- -10973

- -09769

+ •05219

+ •08179

•49019

•SIS

•35427

•12209

- -11028

- -09736

+ -05276

+ •08160

•48736

•314

•35476

•12155

-•11082

- -09703

+ '05332

+ •08140

•48464

•315

•35524

•12101

-•11137

-•09669

+ •05388

+ •08121

•48173

•316

•35572

•12040

-•11191

- -0963!)

+ •06444

+ •08101

•47891

•317

•35620

•11992

- -11245

-•09600

+ -05500

+ •08080

•47610

•318

•35667

•11937

-•11299

- -09566

+ -05655

+ •08059

•47330

•319

•35714

•11882

- -113.r>3

- -09531

+ -05610

+ •08038

•47050

•sto

•36761

•11827

-•11406

-•09495

+ •05665

+ •08016

•46770

•SSI

•35808

•11771

- '11459

-•09460

+ •06720

+ •07994

•46490

•.;.-.-

•35854

•11716

-•11512

- -09424

+ -06776

+ -07972

•46211

•sts

•35900

•11660

-•11664

-•09388

+ -0582!)

+ -07949

•45933

•sn

•36946

•11604

-•11616

- -09351

+ -05883

+ -07920

•45654

•3S5

•35991

•11548

-•11668

-•09315

+ -05937

+ -07902

•45376

•326

•36037

•11492

--1172U

-•09278

+ -05991

+ -07878

•46099

•ssrr

•36082

•11436

-•11771

- -09240

+ 06044

+ -07854

•44821

•sss

•36126

•11379

-•11822

- -09203

+ •06097

+ -07829

•44544

•Si9

•36171

•11322

-•11873

-•09165

+ -06160

+ O7804

•44268

•8K

•36215

•11265

-•11923

- -09127

+ •06203

+ -07779

•43991

•331

•36259

•11208

-•11974

- -09088

+ •06255

+ -07763

•43715

•36302

•11151

-•12024

-•09049

+ •06308

+ -07727

•43440

• ...

•36346

•11093

- -12073

-•09010

+ •06360

+ -07701

•13164

•334

•36389

•11036

-•12123

- -08971

+ •00412

+ -07674

•42889

36431

•10978

-•12172

-•08932

+ •06463

+ -07647

•42616

•336

•36474

•10920

-•12221

-•08892

+ •06514

+ •07620

•42340

•337

•36516

•10862

- -12270

-•08852

+ •06565

+ •07692

•42066

•SS8

•36558

•10804

- -12318

-•08811

+ •06616

+ •07564

•41793

•339

•36GOO

•10745

- -12366

-•08771

+ •06667

+ -0753.-.

•41519

•340

•36641

•10687

- -12414

-•08730

+ -06717

+ '07507

•41246

•341

•36682

•10628

- -12461

- -08689

+ -06767

+ -07477

•40974

•30

•36723

•10569

- -12509

-•08647

+ •06817

+ -07448

•40701

•848

•36764

•10510

-•12555

-•08606

+ •06867

+ •07418

•40429

•344

•36804

•loi.-.i

-•moot

-•08564

+ •06916

+ •07388

•40157

•345

•36844

•10391

-•12649

-•08522

+ •06965

+ •07368

•39886

•346

•36884

•10332

- '12695

-•08479

+ -07014

+ -07327

•39614

•347

•36923

•10272

-•12741

- '08437

+ •07062

+ -07296

•39343

;;.s

•36962

•10212

- '12786

- -08394

+ •07110

+ •07264

•39073

•349

•37001

•10152

-•ias:n

- -08351

+ •07158

+ -07232

•38802

•350

-97040

•10002

- -12876

-•08307

+ -07206

+ •07200

•38532

Tables of the Tetrachoric Functions
TABLE XXIX.— (continued).

i(l-a)

n

rj

1-3

>•«

n

"6

h

•SSI

•37078

•10032

- -12921

- -08264

+ -07254

+ -07168

•38262

•S52

•37116

•09971

- -12966

- -08220

+ -07301

+ -07135

•37993

•S5S

•37154

•09911

-•13010

- -08176

+ -07348

+ -07102

•37723

•364

•37192

•09850

- -13054

- -08131

+ -07395

+ -07069

•37454

•355

•37229

•09789

-•13097

-•08087

+ -07441

+ -07035

•37186

•356

•37266

•09728

- -13140

-•08042

' + -07487

+ -07002

•36917

•357

•37303

•09667

- -13183

- -07997

+ -07533

+ -06967

•36649

•358

•37340

•09606

- -13226

- -07952

+ •07579

+ -06933

•36381

•359

•37376

•09544

- -13269

- -07906

+ -07624

+ -06898

•36113

•360

•37412

•09483

- -13311

- -07861

+ -07669

+ -06863

•35846

•861

•37447

•09421

- -13353

- -07815

+ -07714

+ -06827

•35579

•362

•37483

•09359

- -13304

- -07768

+ -07758

+ -06792

•35312

•,;.;.;

•37518

•09297

- '13436

- -07722

+ -07803

+ -06756

•35045

•364

•37553

•09235

- -13477

- -07675

+ -07847

+ •06719

•34779

•.:<;:,

•37588

•09173

-•13517

- -07629

+ -07890

+ -06683

•34613

•.,<.',

•37622

•09111

- -13558

- -07682

+ -07934

+ '06646

•34247

•367

•37656

•09048

-• 13598

- -07534

+ -07977

+ -06609

•33981

•368

•37690

•08985

-• 13638

- -07487

+ -08020

+ •06571

•33715

•Ml

•37724

•089--:'.

- -13677

- -07439

+ -08062

+ -06534

•33450

•370

•37757

•08860

- '13717

- -07391

+ -08105

+ -06496

•33185

•371

•37790

•08797

- -13766

-•07343

+ -08147

+ -06458

•32921

•37823

•08734

- -13794

- -07295

+ •081 88

+ -06419

•32656

•373

•37855

•08671

-•13833

- -07246

+ -08230

+ -06380

•32392

•374

•37888

•08607

-•13871

- -07198

+ -08271

+ •06341

•32128

•375

•37920

•08544

-•13909

- -07149

+ •08312

+ -06302

•31864

•376

•37951

•08480

- -13946

-•07100

+ -08352

+ -06262

•31600

•377

•37983

•08416

- -13984

- -07050

+ -08392

+ -06222

•31337

•378

•38014

•08353

- -14021

- -07001

+ -08432

+ -06182

•31074

an

•38045

•08289

- -14057

- -06951

+ -08472

+ •06142

•30811

•380

•38076

•08225

- -14094

-•06901

+ -08512

+ -06101

•30548

•381

•38106

•08160

- -14130

- -06851

+ -08551

+ •06061

•30286

•S8S

•38136

•08096

- -14166

- -06801

+ -08589

+ -06019

•30023

•383

•38166

•08032

- -14201

- -06750

+ -08628

+ -05978

•29761

•384

•38196

•07967

- -14236

- -06700

+ -08666

+ -05936

•29499

•385

•38225

•07903

- -14271

- -06649

+ -08704

+ -05895

•29237

•386

•38254

•07838

- -14306

- -06598

+ -08742

+ -05853

•28976

•387

•38283

•07773

- -14340

-•06547

+ -08779

+ -05810

•28715

•388

•38312

•07708

- -14374

- -06495

+ -08816

+ -05766

•28454

•380

•38340

•07643

- -14408

-•06444

+ -08853

+ -05725

•28193

•390

•38338

•O7678

- -14442

- -06392

+ -08889

+ -06682

•27932

•391

•38396

•07513

-•14475

- -06340

+ -08925

+ -05638

•27671

•392

•38423

•07447

-•14508

-•06288

+ •08961

+ -05595

•27411

•393

•38151

•07382

- -14540

- -06236

+ -08997

+ -05551

•27151

•394

•38478

•07316

- -14573

-•06183

+ •09032

+ -05507

•26891

•395

•38504

•07251

-•14604

- -06131

+ •09067

+ -05463

•26631

•396

•38531

•07185

- '14636

- -06078

+ -09101

+ -05419

•26371

•397

•38557

•07119

- -14668

- -06025

+ -09136

+ -05374

•26112

•398

•38583

•07053

- -14699

- -05972

+ -09170

+ -05329

•25853

•399

•38600

•06987

- -14730

- -05919

+ -09203

+ -05284

•25594

•400

•38634

•06021

- -14760

- -OSSOfi

+ -09237

+ •05239

•25335

B.

50 Tables for Statistician*! and Biometrlcians

TABLE XXIX. Tetniciioric Function* for Fourfold Cotrclation Tables.

1(1 -a)

n

ri

TI

••4

r»

'•

k

•401

•38659

•06855

- -14790

-•06812

+ -09270

+ -06193

•25076

T^

•;•'-•

•38684

•08789

- -14820

-•06758

+ -09303

+ -05148

•24817

•403

•38709

•06722

-•14850

- -05705

+ -09335

+ -05102

•24559

•404

•38734

•06666

- -14879

- -06651

+ •09367

+ -06056

•24301

•405

•38758

•08589

- -14908

- -05596

+ -09399

+ •05010

•24043

•406

•38782

•06622

- -14937

- -05542

+ •09430

+ •04963

•23785

•407

•3880S

•06456

- -14965

-O5488

+ -OIMI;::

+ 04916

•23527

•408

•38829

•06389

- -14993

- -06433

+ -o:>

+ •04869

•23269

•409

•38852

•06322

-•16021

- -05378

+ -09523

+ •04822

•23012

•410

•38876

•06265

-•16049

- -05323

+ •09553

+ -04775

•227.-. 1

•411

•38897

•06188

-•16076

-•06268

+ -09583

+ -04728

•224U7

•41*

•38920

•06121

- -16103

- -05213

+ •09613

+ -04680

•22240

•1,18

•38942

•06053

- -15130

- -05168

+ •09642

+ -04632

•21983

•414

•38964

•05986

- -15156

- -05102

+ -09671

+ -O4.'is I

•21727

•415

•3898ft

•05919

-•15182

-•05047

+ -09700

+ -04536

•21470

•41li

•39007

•05851

-•15208

- -04991

+ -09728

+ -04488

•21214

•417

•39028

•05784

- -16233

- -04935

+ -09756

+ O4439

•20i»:>7

•418

•39049

•05716

- -15258

- -04879

+ -09784

+ •04390

•20701

•419

•39069

•05648

- -16283

- -04823

+ -09811

+ -04341

•20445

•4*0

•39089

•05680

-•15308

- -04767

+ -09838

+ -04292

•20189

•4S1

•39109

•05513

- -15332

- -04711

+ -09865

+ •04243

•19934

'4t*

•39129

•05445

- -16356

- -04654

+ -09891

+ -04194

•19678

•4*3

•39149

•06377

-•16380

- -04598

+ •09918

+ -04144

•19422

•4*4

•39168

•05309

- -16403

- -04541

+ -09943

+ -04094

•19167

•425

•39187

•05240

- -15426

- -04484

+ •09969

+ -04044

•18912

:',:••>

•39206

•05172

- '15449

- -04427

+ •09994

+ -03994

•18657

•427

•39224

•05104

- -16471

- -04370

+ •10019

+ -03944

•18402

•4*8

•39243

•05036

-• 15493

- -04313

+ -10043

+ -03894

•18147

•420

•39261

•04967

-•1651 5

- -04256

+ -10067

+ -03843

•17892

•430

•39279

•04899

- -16537

- -04198

+ -10091

+ -03793

•17637

•4Si

•39296

•04830

- -16558

- -04141

+ •101 15

+ -03742

•17383

•43S

•39313

•04761

- '15579

-•04083

+ •10138

+ -03691

•17128

•4S3

•39330

•04693

- -16599

-•04026

+ •10161

+ -03640

•16874

-434

•39347

•04624

- -15620

-•03968

+ •10183

+ -03589

•16<W '

•4SS

•39364

•04555

-•15640

- -03910

+ -10205

+ -03537

•163' ;•;

•4S6

•39380

•04486

- -15659

- -03852

+ • 10227

+ -03486

•16112

•437

•39396

•04418

- -15679

- -03794

+ •10249

+ -03434

•15858

•438

•39411

•04349

- -15698

- -03735

+ -10270

+ -03382

•15604

•4S9

•39427

•04280

-•15717

- -03677

+ -10291

+ -03330

•15351

•440

•39442

•04211

- -15735

- -03619

+ •10311

+ -03278

•16097

•441

•39457

•04141

- -15753

- -03560

+ •10331

+ -03226

•14843

•44*

•39472

•04072

- -15771

- -03502

+ '10351

+ O3174

•14690

•44S

•39486

•04003

- -16789

- -03443

+ •10371

+ •03121

•1 43:i7

•444

•39501

•03934

- -15806

- -03384

+ • 10390

+ •03069

•14084

•445

•39514

•03864

- -16823

- -03325

+ -10409

+ •03016

•13830

•446

•39528

•03795

- -15840

- -03:!t;i;

+ -10427

+ •02963

•13677

•447

•39542

•03726

- -16856

-•03207

+ • 10446

+ •02910

•13324

•448

•39555

•03666

- -16872

-•03148

+ -10463

+ •02868

•13072

•449

•39568

•03587

- -16888

- -03089

+ -10481

+ •02804

•12819

•460

•39680

•03617

- '16904

-•03030

+ -10498

+ •02761

•ItMfl

Tables of the Tetrachoric Functions
TABLE XXIX.— (continued).

51

4(i-o)

n

T»

rs

r«

r«

«H

h

•481

•39593

•03447

- -15919

- -02970

+ -10515

+ -02698

•12314

•452

•39605

•03378

- -15934

- -02911

+ -10532

+ -02644

•12061

•45S

•39617

•03308

- -15948

- -02851

+ •10548

+ -02591

•11809

•454

•39629

•03238

- -15962

- -02792

+ -10564

+ -02537

•11556

•455

•39640

•03168

- -15976

- -02732

+ -10579

+ •02484

•11304

•456

•39651

•03099

- -16990

- -02673

+ • 10594

+ •02430

•11052

•457

•39662

•03029

- -16003

- -02613

+ •10609

+ -02376

•10799

•458

•39673

•02959

- -16016

- -02553

+ •10624

+ '02322

•10547

•459

•39683

•02889

- -16029

- -02493

+ -10638

+ -02268

•10295

•460

•39694

•02819

- -16041

- -02433

+ -10652

+ •0221 4

•10043

•461

•39703

•02749

- -16053

- -02373

+ •10665

+ •02159

•09791

•462

•39713

•02679

- -16065

- -02313

+ -10678

+ -02105

•09540

•463

•39723

•02609

- -16077

- -022r)3

+ -10691

+ -02051

•09288

•4G4

•39732

•02539

- -16088

- -02193

+ • 10704

+ -01996

•09036

•465

•39741

•02469

-•16099

- -02132

+ •10716

+ -01941

•08784

•466

•39749

•02398

- -16109

- -02072

+ -10727

+ -01887

•08533

•467

•39758

•02328

- -16120

-•02012

+ -10739

+ -01832

•08281

•468

•39766

•02258

- -16130

- -01961

+ -10750

+ '01777

•08030

•469

•39774

•02188

- -16139

- -01891

+ -10761

+ -01722

•07778

•470

•39781

•02117

- -16149

- -01830

+ -10771

+ -01668

•07527

•471

•39789

•02047

- -16158

- -01770

+ -10781

+ -01613

•07276

•472

•39796

•01977

- -16166

-•01709

+ •10791

+ -01558

•07024

•473

•39803

•01906

- -16175

- -01648

+ -10801

+ -01502

•06773

•474

•39809

•01836

- -16183

-•01588

+ -10810

+ -01447

•06522

•475

•39816

•01765

- -16191

- -01527

+ •10818

+ -01392

•06271

•476

•39822

•01695

- -16198

- -01466

+ -10827

+ •01 337

•06020

•477

•39828

•01C25

-•16206

- -01405

+ -10835

+ -01281

•05768

•478

•39834

•01554

-•16212

- -01344

+ -10842

+ -01226

•05517

•479

•39839

•01484

-•16219

-•01 284

+ •10850

+ •01171

•05266

•480

•39844

•01413

- -16225

- -01223

+ -10857

+ •01115

•05015

•481

•39849

•01342

- -16231

-•01 162

+ -10864

+ •01060

•04764

•489

•39864

•01272

-•16-237

-•01101

+ -10870

+ -01004

•04513

•483

•39858

•01201

- -16242

-•01040

+ -10876

+ -00949

•04263

•484

•39862

•01131

- -16247

-•00979

+ •10882

+ -00893

•04012

•486

•39866

•01060

- -16252

-•00918

+ •10887

+ -00837

•03761

•486

•39870

•00980

- -16267

-•00857

+ •10892

+ -00782

•03510

•487

•39873

•00919

- -16261

-•00796

+ -10896

+ -00726

•03269

•488

•39876

•00848

- -16265

-•00734

+ •10901

+ -00670

•03008

•489

•39879

•00778

- -16268

-•00673

+ •10906

+ -00614

•02758

•490

•39882

•1)0707

- -16271

- -00612

+ •10908

+ -00559

•02507

•491

•39884

•00636

- -16274

- -00551

+ •10912

+ -00503

•02256

•JM

•39886

•00566

-•16277

-•00490

+ -10914

+ -00447

•02005

•493

•39888

•00495

- -16279

- -00429

+ •1091 7

+ -00391

•01755

•494

•39890

•00424

- -16281

- -00367

+ •10919

+ -00335

•01504

•495

•39891

•00354

- -16283

-•00306

+ -10921

+ -00279

•01253

•496

•39892

•00283

- -16284

- -00245

+ -10923

+ -00224

•01003

•497

•39893

•00212

- -16285

-•00184

+ -10924

+ -00168

•00752

•498

•:{!I894

•00141

- -16286

- -00122

+ -10925

+ -00112

•00501

•490

•39894

•00071

- -16287

-•00061

+ -10925

+ -00056

•00251

•500

•39894

•ooooo

- -16287

•OOOOO

+ -10925

•OOOOO

•OOOOO

7—2

•VJ

(iinf

TABLE XXX. Supplementary Tables for determining High
r = -80.

A-

0

•1

•*

•a

•4

•5

•6

•7

•8

•9

1-0

1-1

1-S

* o-O

•3976

•3766

•3538

•3294

•3039

•2778

•2515

•2254

•2001

•1759

•1531

•1320

•1127

o-i

•3766

•8068

•8880

•3162

•2930

4688

•2445

•2200

•1960

•1788

•1509

•1304

•1116

o-t

•BM

•3380

•MM

•3011

•88M

•2.1SC,

•2361

•2134

•1808

•1690

•1481

•1284

•1102

0-3

•3294

•3162

•3011

•2843

•2661

•2466

•2263

* >:.<;

•IMS

•1643

•1446

•1258

•1083

0-4

•3039

•sen

•2804

••661

•2503

•2332

•2162

•1965

•1775

•1587

•1402

•1226

•1060

0-5

•2778

•am

•8868

•2466

•2332

•2186

•2028

•1862

•1692

•1520

•1351

•1187

•1031

0-6

•2515

•2445

•2361

•2263

•2152

•2028

•1893

•1748

•me

•1444

•1291

•1140

0988

0-7

•2254

•2200

•2134

•2056

•1965

•1862

•1748

•1625

•1494

•1359

•1222

•1086

00M

0-8

•2001

•1960

•1000

•1848

•1776

•1692

•1598

•1494

•1383

•1266

•1146

•10M

0808

09

•1759

•1728

•1690

•1643

•1587

•15-20

•1444

•1359

•1266

•1167

•1064

•0958

0668

1-0

•1531

•1509

•1481

•1446

•1402

•1351

•1291

•1222

•1146

•1064

•0076

0686

•0794

1-1

•1320

•1304

•1284

•1258

•1226

•1187

•1140

•1086

•1025

•0958

0686

0608

•0731

1-t

•1127

•1116

•1102

•1083

•1060

•1031

•0995

•0954

•0906

•0888

•0794

•0731

•0665

rs

•0953

•0946

•0936

•0923

•0006

•0885

•0859

•0828

•0791

•0749

•0702

06M

•0597

1-4

•0798

•0793

•0787

•0778

•0766

•0751

•0733

•0710

•0(182

•0650

•0614

•0574

•0530

1-5

•0662

0608

•0655

•0649

•0641

•0631

•0618

•0601

•0581

•0557

OM8

•0498

•0464

1-6

•0545

•0543

•0540

•0536

•0531

•0524

•0515

•0503

•0489

•0471

•0451

•0427

•0401

1-7

•0444

•0443

•0441

•0438

•0435

•0430

•0424

•0416

•0406

•0394

•0379

•0362

•0342

1-8

•0358

•0357

•0357

•0355

•0353

•0350

•0346

•0341

•0334

•0325

•0316

•0302

•0287

1-9

•0287

•0286

•0286

•0880

•0283

•0281

•0279

•0275

•0271

•OMB

•0258

•0249

•0238

:<>

•0227

•0227

•0227

•0226

•0225

•0224

•0223

•0220

•0217

•0213

0808

O80S

•0195

g-1

•0178

•0178

•0178

•0178

•0177

•0177

•0176

•0174

•0172

•0170

•0167

•0163

•0158

g-t

•0139

•0139

•0139

•0139

•0138

•0138

•0137

•0137

•0136

•0134

•0132

•0129

•0126

2-S

•0107

•0107

•0107

•0107

•0107

•0107

•0106

•0106

•0105

•0104

•0103

•0101

•0099

'•4

•0082

•0082

0061

4061

•0082

•0082

•0082

•0081

•0081

•0080

•0079

O078

•0077

t-5

•0062

ooa

•0062

•0062

•(XMi2

•0062

ooa

•0062

•0061

•0061

•0061

0060

•0059

t-e

•0047

•0047

•0047

•0047

•0047

•0047

•0048

•0046

•0046

•0046

•0046

•0045

•0045

r = '85.

A=

0

•1

*

•3

•4

•5

•6

•7

•8

•9

1-0

1-1

1-2

1=0-0

•4117

•3905

•3670

•3417

•3149

•2873

•SMB

•2319

•20.12

•1798

•1560

•1341

•1141

0-1

•3905

•3723

•3518

•3292

•3050

•2796

•2537

•2277

•ma

•1777

•1546

•1332

•ii, -it;

0-2

•3670

•3518

•3342

•3145

•2930

•2702

•2464

•2222

•1983

•1749

•1887

•1319

•1127

0-3

•3417

•3292

•3145

•2978

•2791

•2588

•2374

•2154

•1931

•1712

•1501

•1301

•1110

0-4

•3149

•3050

•2930

•2791

•2632

•2457

•2268

•8070

•1867

•1665

•1467

•1277

•1099

0-K

•2873

•2796

•2702

•2588

•2467

•2309

•2146

•1972

•1790

•1606

•1423

•1246

•1078

0-6

•2595

•2637

•2464

•2374

•2268

•2146

•2008

•1859

•1700

•1535

•1370

•120C

•1049

0-7

•2319

•2277

•2222

•2154

•2070

•1972

•1859

•1733

•1597

•1453

•1306

•1158

•1014

0-8

•2052

•2022

•1983

•1931

•1867

•1790

•1700

•1597

•1483

•13(i()

•12:52

•1101

•0971

0-9

•1798

•1777

•1749

•1712

•1665

•1606

•103,1

•14,13

•1360

•1258

•1M!I

•1035

•0920

1-0

•1560

•1546

•1527

•1501

•1467

•1 123

•1370

•13(16

•1232

•1149

•1058

•0962

•OSI12

1-1

•1341

•1332

•1319

•1301

•1277

•1246

•1206

•1158

•1101

•1035

•0962

oea

•0798

1-2

•1141

•1136

•1127

•1116

•1099

•1078

•1049

•1014

•0971

0080

066S

•0798

•0729

1-3

•0963

•0959

•0954

•0947

•0936

•0921

•0901

•0876

•0845

•0807

•0763

•0712

•0656

1-4

•oxo.1

•0803

•0800

•0795

•0788

•0778

•0765

•0748

•0725

•0686

066fi

0686

•0583

1-0

•0666

•0665

•0664

O661

•0656

•0650

•0642

•0630

•0(il5

•0595

0671

•0543

•0510

l-(i

•0547

•0547

•0546

•0544

•0541

•0538

•0532

•0525

•d.114

•0501

•0484

•0164

•0439

1-7

•0445

•0445

•0444

•0443

•0442

•0440

•0436

•or(2

•0426

•0416

•0405

•0390

•0373

1-8

•0359

•0359

•0359

•0368

•0357

•O:MO

•0354

•0801

OS47

•0341

•0334

•0384

•0312

1-9

•0287

•0287

•0287

•0287

•0286

O866

•0284

•0283

•0280

•0276

•0272

•0265

•0267

-••a

•0227

•0227

•(1227

•0227

•0227

0887

•0226

•0225

•0221

•0218

•0214

•0209

g-1

•0179

•0179

•0179

•0178

•0178

•0178

•0178

•0177

•0176

•0176

•0173

•0171

•0167

t-t

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0138

•0138

•0137

•0136

•0136

•0133

t-3

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0106

•0106

•0105

•0104

t-4

0061

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0081

•0081

•0081

0080

t-6

ooa

ooa

ooa

•0062

•<X*;2

ooa

ooa

•0062

ooa

•0062

•0062

•0061

•0061

t-6

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0046

•0046

•0046

•0046

Tables for High Fourfold Correlation

Correlations from Tetrachoric Groupings.

r=-80.

A=

1-S

1-4

1-5

1-6

1-7

1-8

1-9

2-0

2-1

2-2

S-3

f-4

2-5

2-6

k = 0-0

•0953

•0798

•0662

•0545

•0444

•0358

•0287

•0227

•0178

•0139

•0107

•0082

•0062

•0047

o-i

•0946

•0793

•0659

•0543

•0443

•0357

•0286

•0227

•0178

•0139

•0107

•0082

•0062

•0047

0-2

•0936

•0787

•0655

•(1540

•0441

•0357

•0288

•0227

•0178

•0139

•0107

•0082

•0062

•0047

o-s

•0923

•0778

•0649

•0536

•0438

•0355

•0285

•0226

•0178

•013!)

•0107

•0082

•0062

•0047

0-4

•0906

•0766

•0641

•0531

•0435

•0353

•0283

•0225

•0177

•0138

•0107

•0082

•0062

•0047

0-5

•0885

•0751

•0631

•0524

•0430

•0350

•0281

•0224

•0177

•0138

•0107

•0082

•0062

•0047

0-6

•0859

•0733

•0618

•0515

•0424

•0346

•0279

•0223

•0176

•0137

•0106

•0082

•0062

•0046

0-7

•0828

•0710

•0601

•0503

•0416

•0341

•0275

•0220

•0174

•0137

•0106

•0081

•0062

•0046

0-8

•0791

•0682

•0581

•0489

•0406

•0334

•0271

•0217

•0172

•0135

•0105

•0081

•0061

•0040

0-9

•0749

•0650

•0557

•0471

•0394

•0325

•0265

•0213

•0170

•0134

•0104

•0080

•0061

•0046

1-0

•0702

•0614

•0529

•0451

•0379

•0315

•0258

•0209

•0167

•0132

•0103

•0079

•0061

•0046

1-1

•0652

•0574

•0498

•0427

•0362

•0302

•0249

•0202

•0163

•0129

•0101

•0078

•0060

•0045

1-2

•0597

•0530

•0464

•0401

•0342

•0287

•0238

•0195

•0158

•0126

•0099

•0077

•0059

•0045

1-S

•0541

•0484

•0427

•0372

•0319

•0271

•0226

•0186

•0151

•0121

•0096

•0075

•0058

•0044

1-4

•0484

•0436

•0388

•0341

•0285

•0252

•0212

•0176

•0144

•0117

•0093

•0073

•0057

•0043

1-5

•0427

•0388

•0348

•0309

•0270

•0232

•0197

•0165

•0136

•0111

•0089

•0070

•0055

•0042

1-6

•0372

•0341

•0309

•027(i

•0243

•0211

•0181

•0153

•0127

•0104

•0084

•0067

•0053

•0041

1-7

•0319

•0295

•0270

•0243

•0216

•0190

•0164

•0140

•0117

•0097

•0079

•0063

•0050

•0039

1-S

•0271

•0252

•0232

•0211

•0190

•0168

•0146

•0126

•0107

•0089

•0073

•0059

•0047

•0037

1-9

•0226

•0212

•0197

•0181

•0164

•0146

•0129

•0112

•0096

•0081

•0067

•0055

•0044

•0035

s-o

•0186

•0176

•0165

•0153

•0140

•0126

•0112

•0098

•0085

•0072

•0060

•0050

•0040

•0032

S-l

•0151

•0144

•0136

•0127

•0117

•0107

•0096

•0085

•0074

•0064

•0054

•0045

•0037

•0030

:.".'

•0121

•0117

•0111

•0104

•0097

•0089

•0081

•0072

•0064

•0055

•0047

•0040

•0033

•0027

2-3

0086

•0093

•0089

•0084

•0079

•0073

•0007

•0060

•0054

•0047

•0041

•0035

•0029

•0024

2-4

•0075

•0073

•0070

•0067

•0063

•0059

•0055

•0050

•0045

•0040

•0035

•0030

•0025

•0021

2-5

•0058

•0057

•0055

•0053

•0050

•0047

•0044

•0040

•0037

•0033

•0029

•0025

•0022

•0018

2-G

•0044

•0043

•0042

•0041

•0039

•0037

•0035

•0032

•0030

•0027

•0024

•0021

•0018

•0016

r='85.

A=

1-S

1-4

1
1-5

1-G

If

1-8

1-9

s-o

2-1

2-2

2-3

»-4

2-5

s-o

t=<fO

•0963

•0805

•0666

•0547

•0445

O359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

o-i

•0959

•0803

•0665

•0547

•0445

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

o-s

•0954

•0800

•0664

•0546

•0444

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

o-s

•0947

•0795

•0661

•0544

•0443

•0358

•0287

•0227

•0178

•0139

•0107

•0082

•0062

•0047

0-4

•0936

•07W

•0656

•0541

•0442

•0357

•0286

•0227

•0178

•0139

•0107

•0082

•0062

•0047

0-5

•0921

•0778

•0650

•0538

•0440

•0356

•0285

•0227

•0178

•0139 , -0107

•0082

•0062

•0047

0-6

•0901

•0765

•0642

•0532

•0436

•0354

•0284

•0226

•0178

•0139 -0107

•0082

•0062

•0047

0-7

•0876

•0748

•0630

•0525

•0432

•0351

•0283

•0225

•0177

•0138

•0107

•0082

•0062

•0047

0-8

•0445

•0725

•0615

•0514

•0425

•0347

•0280

•0224

•0176

•0138

•0107

•0082

•0062

•0047

0-9

•0807

•0698

•0595

•0501

•0416

•0341

•0276

•0221

•0175

•0137

•0106

•0081

•0062

•0046

1-0

•0763

•0665

•0571

•0484

•0405

•0334

•0272

•0218

•0173

•0136

•0106

•0081

•0062

•0046

1-1

•0712

•0626

•0543

•0464

•0390

•0324

•0265

•0214

•0171

•0135

•0105

•0081

•0061

•0046

1-S

•0656

•0583

•0510

•0439

•0373

•0312

•0257

•0209

•0167

•0133

•0104

•0080

•0061

•0046

1-19

•o.i!»7

•0535

•0473

•0411

•0352

•0297

•0247

•0202

•0163

•0130

•0102

•0079

•0060

•0046

1-4

•11.135

•0485

•0432

•0380

•0329

•0280

•0234

•0194

•0157

•0126

•0100

•0078

•0060

•0045

1-5

•0473

•0432

•0390

•0346

•0302

•0260

•0220

•0183

•0150

•0121

•0097

•0076

•0058

•0045

1-G

•0411

•0380

•0346

•0311

•0274

•0239

•0204

•0172

•0142

•0116 -oo'.tt

•0073

•0057

•0044

1-7

•0352

•0329

•0302

•0274

•0245

•0216

•0186

•0159

•0133

•0109

•0088

•0070

•0055

•0043

1-8

•0297

•0280

•0260

•0239

•0216

•0192

•0168

•0144

•0122

•0102

•0083

•0067

•0053

•0041

1-9

•0247

•0234

•0220

•0204

•0186

•0168

•0149

•0129

•0111

•0093

•0077

•0063

•0050

•0039

•••a

•0202

•0194

•0183

•0172

•0159

•0144

•0129

•0114

•0099

•0084

•0070

•0058

•0047

•0037

2-1

•0163

•0157

•0150

•0142

•0133

•0122

•0111

•0099

•0087

•0075

•0063

•0053

•0043

•0034

2-2

•0130

•0126

•0121

•0116

•0109

•0102

•0093

•0084

•0075

•0066

•OOJ6

•0047

•0039

•0032

S-3

•0102

•0100

•0097

•0093

•0088

•0083

•0077

•0070

•0063

•0056

•0049

•0042

•0035

•0029

2-4

•0079

•0078

•0076

•0073

•0070

•0067

•0063

•0058

•0053

•0047

•0042

•0036

•0031

•0025

2-5

•0060

•0060

•0058

•0057

•0055

•0053

•0050

•0047

•0043

•0039

•0035

•0031

•0026

•0022

;•<:

•00 Hi

•0045

•0045

•0044

•0043

•0041

•0039

•0037

•0034

•0032

•0029

•0025

•0022

•0019

54

fur Shi finf it-id nit timl /ilium frit-in H.-<

TABLE XXX. Supplementary Tables for determining lliyh
r = -90.

A-

0

•1

•*

•J

•4

•5

•6

7

•8

•9

1-0

I'l

1'g

i

•4282

•4067

•3888

•3562

•3266

•Ml

•2670

•2377

•SOM

•1827

•1679

•1353

•1149

o-i

•4067

•:<--:

•8678

•3441

•3183

•2910

•2630

•2350

•2077

•1817

•1674

•1350

•1147

•38H

•3678

*MM

•3302

•3076

•2830

•2573

•2311

-2< >:,-2

•1801

•1664

•1345

•1144

0-3

•86M

•3441

;3302

•3135

•2943

•2728

•2498

•2258

•2016

•1778

•1560

•1336

•1140

0-4

4806

•3183

•3076

4MI

•2784

MO)

•2401

•2187

•1966

•1744

•IBM

•1322

•1132

0-5

•8088

•2910

•8880

•me

•2602

•2453

•88M

•8007

•1800

•1608

•1497

•1302

•1119

,,-,,

•8870

•2630

•2573

•2498

•2401

•2284

•2145

•1988

•1817

•1637

•1454

•1274

•1101

0-7

•M77

•2350

•2311

•ssoe

•2187

•2097

•1988

•1860

•1717

•1561

•1399

•1236

•1075

pt

•2094

•2077

•sou

•2016

•1966

•1900

•1817

•1717

•HiOO

•1470

•1331

•1186

•1041

0-9

•1817

•1817

•1801

•1778

•1744

•1698

•1637

•1561

•1470

•13G5

•1249

•1124

•0997

1-0

•1679

•1574

•1564

•1650

•1528

•1497

•1464

•1399

•1331

•1249

•1155

•1052

•0942

1-1

•1353

•1350

•1346

•1336

•1322

•1302

•1274

•1236

•1186

•1124

•1052

0088

•0878

Vi

•1149

•1147

•1144

•1140

•1132

•1119

•1101

•1075

•1041

•0997

•0942

•0878

0808

1-3

0067

O0M

OMB

•0962

0068

•0950

•0939

•0923

O900

•08G9

•0830

•0783

•0727

1-4

0607

•0807

O808

•0805

•llsill!

•0798

•0792

•0782

•0767

•0747

•0720

•0686

•0645

1-5

0068

0668

0807

•0667

•0665

•0663

•0660

•0654

•0645

•0632

•0614

•0591

•0562

r>;

•0548

•0548

•0548

•0547

•0547

•0546

•0544

•0540

•0535

•I..', I'M

•0516

•0501

•0481

1-7

•0446

•0446

•0446 ! '0445

•0445

•0445

•0444

•0442

•0439

•0435

•0428

•0418

•oi or,

1-8

O868

•0359

•(1359

•0359

•0359

•0359

•0358

•0357

O868

•0353

•0350

•0344

•0338

1-9

O887

0881

•0287

•0287

•0287

•0287

•0287

•0286

•0286

•0284

•0282

•0279

•0274

8-0

O887

•0227

•0227

•0227

O887

•0227

•0227

•0227

•0227

O888

O880

•0223

•0220

2-1

•0179

•0179

•0179

•0179

•0179

•0179

•0179

•0178

•0178

•0178 -0177

•0176

•0175

t-s

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139 -0138

•0138

•0137

8-3

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0106

s-4

•0082

•IKIS-2

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0068

•0082

8-6

•0062

ooa

•0062

•0069

ooa

•0062

•0062

O069

•0062

•0062

•0062

•0062

0081

t-6

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

r=-95.

h=

0

•1

4

•3

•4

•s

•6

•7

•8

•9

1-0

I'l

1-2

t=o-o

•4MB

•4271

•4005

•3705

•3385

•3055

•2729

•2414

•2116

•1840

•1586

•1357

•1151

o-i

•4271

•4009

•3880

•3622

•u:m

•3026

•2713

•2407

•2113

•1839

•1586

•1356

•1151

o-g

-4O06

•:t--"

•3712

•:•,:,' HI

•3252

•2976

•8888

•2392

•2106

•1835

•1686

•1356

•1150

0-3

•3705

•3622

•3500

•3338

•3135

•8888

•2637

•2366

•2092

•1829

•1688

•1355

•1150

0-4

•3385

•3333

•3252

•3135

•2980

•2787

•8664

•2320

•2067

•1816

•1576

•1352

•1149

0-5

•3055

•3026

•2976

•8888

•2787

•2640

•2459

•2250

•2024

•1792

•1563

•1346

•1147

0-6

1788

•2713

•2685

•2637

•2564

•2459

•2321

•2153

•1960

•1753

•1642

•1335

•1141

0-7

•2414

•2407

•2392

•2365

•2320

•2250

•2153

•2025

•1870

•1694

•1506

•1315

•1131

0-8

•2116

•2113

•2106

•8009

•2067

•2024

•1960

•1870

•1763

•1611

•1452

•1283

•1113

0-9

•1840

•1839

•1835

•1829

•1816

•1792

•1753

•1694

•1611

•1606

•1377

•1234

•1084

1-0

•1586

•1686

•1686

•1682

•1576

•1563

•1542

•1506

•1452

•1377

•1281

•11G7

•1041

1-1

•1357

•1356

•1356

•1355

•1352

•1346

•1335

•1315

•1283

•1884

•1167

•1082

•0981

i-e

•1151

•1151

•1160

•1150

•1149

•1147

•1141

•1131

•1113

•1084

•1041

•0981

OB08

1-3

OD8B

O868

0868

OB68

•0967

•0966

•0964

•0959

•0950

•0934

•0908

•0870

•0818

1-4

O806

0808

oeoe

oeoe

•0807

•0807

0808

•0804

•0800

•0788

•0778

•0766

•0721

IT,

O888

0868

O868

•0668

•0668

O008

O868

•0667

•<Hi(i.->

•0661

•0664

•0642

•0888

1-0

•0548

•1C, IS

•ii.-, is

•0548

•0548

•0648

•0.148

•0648

O647

•0546

O648

O688

•0526

1-7

•0446

•0446

•0446

•0446

•0446

•0446

•0446

•0445

•our,

•oil".

O448

•0440

O486

1-8

•• .:•,-,:.

O359

•0359

•0359

•0359

•0359

•0359

•0359

•0359

•0359

•0358

•0357

•i M.-.ri

1-9

•MI;-?

•0287

•0287

•0287

•0287

•0287

•0287

•0287

•i I-'*?

•0287

O887

•0286

•0285

s-o

•0227

•0227

•0227

•0227

•0227

•0227

•0227

•0227

•0^27

•0887

•0227

•0227

•0227

g-1

O178

•0179

•0179

•0179

•0179

•0179

•0179

•0179

•0179

•0179

•0179

•0179

•0178

t-t

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0189

g-3

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

*-4

•0068

OOU

0088

•0082

•0082

ooa

•IKIS2

•0082

•0082

•0082

•0082

•0088

•0082

S-B

ooa

•0062

ooa

•0062

•0082

•0062

•0062

•0062

•0062

•0062

•0062

ooa

•0062

g-e

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

Tables for Hiyh Fourfold Correlation

55

Correlations from Tetrachoric Groupings.

r = -90.

A =

1-3

1-4

1 -5

1-6

1-7

1-8

1-9

2'0

2-1

S-2

2-S

2-4

S-S

S-6

t=o-o

•0967

•0807

•0668

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-1

•0966

•0807

•0668

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-2

•0965

•0806

•0667

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-3

•0962

•0805

•0667

•0547

•0445

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-4

•0958

•0802

•0665

•0547

•0445

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-5

•0950

•0798

•0663

•0546

•0445

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-6

•0930

•0792

•0660

•0544

•0444

•0358

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-7

•0923

•0782

•0654

•0540

•0442

•0357

•0286

•0227

•0178

•0139

•0107

•0082

•0062

•0047

0-8

•0900

•0767

•0645

•0535

•0439

•0356

•0286

•0227

•0178

•0139

•0107

•0082

•0062

•0047

0-9

•0869

•0747

•0632

•0528

•0435

•0353

•0284

•0226

•0178

•0139

•0107

•0082

•0062

•0047

1-0

•0830

•0720

•0614

•0516

•0428

•0350

•0282

•0225

•0177

•0138

•0107

•0082

•0062

•0047

1-1

•0783

•0686

•0591

•0501

•0418

•0344

•0279

•0223

•0176

•0138

•0107

•0082

•0062

•0047

1-2

•0727

•0645

•0562

•0481

•0405

•0338

•0274

•0220

•0175

•0137

•0106

•0082

•0062

•0047

1-3

•0664

•0596

•0526

•0456

•0388

•0325

•0267

•0216

•0173

•0136

•0106

•0081

•0062

•0046

1-4

•0596

•0543

•0485

•0426

•0367

•0310

•0258

•0211

•0169

•0134

•0105

•0081

•0062

•0046

1-5

•0526

•0485

•0439

•0391

•0341

•0292

•0246

•0203

•0164

•0131

•0103

•0080

•0061

•0046

1-6

•0456

•0426

•0391

•0353

•0312

•0271

•0231

•0193

•0158

•0127

•0101

•0079

•0060

•0046

1-7

•0388

•0367

•0341

•0312

•0281

•0247

•0214

•0181

•0150

•0122

•0098

•0077

•0059

•0045

1-8

•0325

•0310

•0292

•0271

•0247

•0221

•0194

•0167

•0140

•0116

•0094

•0074

•0058

•0044

1-9

•0267

•0258

•0246

•0231

•0214

•0194

•0173

•0151

•0129

•0108

•0088

•0071

•0056

•0043

2-0

•0216

•0211

•0203

•0193

•0181

•0167

•0151

•0134

•0116

•0099

•0082

•0067

•0053

•0042

2-1

•0173

•0169

•0164

•0158

•0150

•0140

•0129

•0116

•0102

•0088

•0075

•0062

•0050

•0040

w

•0136

•0134

•0131

•0127

•0122

•0116

•0108

•0099

•0088

•0078

•0067

•0056

•0046

•0037

2-3

•0106

•0105

•0103

•0101

•0098

•0094

•0088

•0082

•0075

•0067

•0058

•0050

•0042

•0034

2-4

•0081

•0081

•0080

•0079

•0077

•0074

•0071

•0067

•0062

•0056

•0050

•0044

•0037

0031

2-5

•0062

•0062

•0061

•0060

•0059

•0058

•0056

•0053

•0050

•0046

•0042

•0037

•0032

•0027

2-6

•(•'Mi; -0046

•0046

•0046

•0045

•0044

•0043

•0042

•0040

•0037

•0034

•0031

•0027

•0024

r = -95.

A=

1-3

1'4

1-5

1-6

1-7

1-8

1-9

s-o

2 -1

2-2

2-S

2-4

2-5

2-6

1 = 0-0

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

o-i

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-2

•oeea

•0808

•0668

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

o-s

•0968

•0808

•0888

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-4

•0967

•0807

•0668

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

o-s

•0966

•0607

•0668

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-6

•0964

•0806

•0668

•0548

•0446

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-7

•0959

•0804

•0667

•0548

•0445

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-8

•0950

•0800

•0665

•0547

•0445

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

0-9

•it!»34

•0792

•0661

•0546

•0445

•0359

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

1-0

•0006

•0778

•0654

•0542

•0443

•0358

•0287

•0227

•0179

•0139

•0107

•0082

•0062

•0047

1-1

•0870

•0755

•0642

•0536

•0440

•0357

•0286

•0227

•0179

•0139

•0107

•0082

•0062

•0047

1-2

•0818

•0721

•0622

•0525

•0435

•0355

•0285

•0227

•0178

•0139

•0107

•0082

•0062

•0047

1-S

•0752

•0676

•0593

•0508

•0426

•0350

•0283

•0226

•0178

•0139

•0107

•0082

•0062

•0047

1-4

•0676

•0619

•0554

•0483

•0411

•0342

•0279

•0224

•0177

•0139

•0107

•0082

•0062

•0047

1-6

•0593

•0554

•0505

•0450

•0390

•0330

•0273

•0221

•0176

•0138

•0107

•0082

•0062

•0047

1-6

•0508

•0483

•0450

•0409

•0362

•0312

•0263

•0215

•0173

•0137

•0106

•0082

•0062

•0047

1-7

•0426

•0411

•0390

•0362

•0328

•0289

•0248

•0207

•0169

•0135

•0105

•0081

•0062

•0047

1-8

•0350

•0342

•0330

•0312

•0289

•0261

•0229

•0195

•0162

•0131

•0104

•0080

•0062

•0046

1-9

•0283

•0279

•0273

•0263

•0248

•0229

•0205

•0179

•0152

•0125

•0101

•0079

•0061

•0046

2-0

•0226

•0224

•0221

•0215

•0207

•0195

•0179

•0160

•0139

•0117

•0096

•0077

•0060

•0046

SI

•0178

•0177

•0176

•0173

•0169

•0162

•0152

•0139

•0124

•0107

•0090

•0073

•0058

•0045

!' !

•0139

•0139

•0138

•0137

•0135

•0131

•0125

•0117

•0107

•0095

•0082

•0068

•0055

•0043

2-3

•0107

•0107

•0107

•0106

•0105

•0104

•0101

•0096

•0090

•0082

•0072

•0062

•0051

•0041

2-4

•0082

•0082

•0082

•0082

•0081

•0080

•0079

•0077

•0073

•0068

•0062

•0054

•0046

•0038

S-B

O06S

•0062

•0062

•0062

•0062

•0062

•0061

•0060

•0058

•0055

•0051

•0046

•0040

•0034

SI-G

•0047

•(•047

•0047

•0047

•0047

•0046

•0046

•0046

•0045

•0043

•0041

•0038

•0034

•0030

Tables for Statisticians and JJioinclriciaita

TABLE XXX. Supplementary Tables for determining IHyh
r=l-00.

A-

0

•1

•£

•s

•4

•5

•6

•7

•8

•9

1-0

1-1

1-2

t=OD*

•BOOO

•4609

•4207

•3821

•3446

•3085

•2743

•2420

•2119

•1841

•1587

•1357

•1151

o-i

•4602

•4602

•4207

•:?S:M

•34 16

•3086

•2743

•2420

•2119

•1841

•1587

•1357

•1151

o-t

•4'2(l~

•4207

•4207

•8881

•3446

•BOBS

•2743

•2420

•2119

•1841

•1587

•1357

•1151

0-3

•3821

•3821

•3821

•3821

3446

•3085

•2743

•2420

•L'll!)

•1841

•1587

•1357

•1151

0-4

•3446

•3446

•3446

•3446

•3446

•3085

•2743

•2420

•2119

•1841

•1587

•1357

•1151

0-5

•3085

•3d85

•3085

•3085

•3085

•3085

•2743

•2420

•2119

•1841

•1587

•1357

•1151

0-6

•2743

•2743

•2743

-21 115

•2743

•2743

•2743

•2420

•2119

•1841

•1587

•1357

•1151

0-7

•2420

•24i()

•8490

•8480

•2420

•2420

•2420

•2420

•2119

•1841

•1587

•1357

•1151

0-8

•2119

•2119

•2119

•2119

•2119

•2119

•2119

•2119

•Jl l!l

•1841

•1587

•1357

•1151

0-9

•1841

•1841

•1841

•1841

•1841

•1841

•1841

•1841

•1841

•1841

•1587

•1357

•1151

1-0

•1587

•1587

•1587

•1587

•1587

•1587

•1587

•1587

•1887

•1587

•1587

•1357

•1151

1-1

•1357

•1357

•1357

•1357

•1357

•1357

•1357

•1357

•1357

•1357

•1357

•1357

•1151

r:

•1151

•1151

•1151

•1161

•1151

•1151

•1151

•1151

•1161

•1151

•1151

•1151

•1151

1-8

•0968

•0968

•0968

•0968

•0068

•0968

0068

•0968

•0968

•0968

0868

•0968

•lines

1-4

•0808

•0808

•0808

•0808

•0808

•0808

•0808

•1 INI IS

•osos

0808

•0808

OBoe

0806

1-5

•0688

•0668

•0668

•IliiliH

•0668

•0888

0868

•0668

•0888

•0888

•o<;<;,x

0868

O8S6

1-6

•0548

•0648

•0548

•0548

•0548

•0548

•0548

•0548

0648

•05 IS

•0548

•0548

•0548

1-7

•0446

O446

•0446

•mi*;

•0446

•0446

•0448

•0446

•0446

•0446

•0446

•0446

•044<;

1-8

•0359

•0359

•0359

•0359

•0359

•0359

•0368

•0359

•0359

•0359

•0359

•0359

•0359

1-9

•0287

•0287

•0287

•0287

•0287

•0287

•0287

•0287

•0287

•0287

•0287

•0287

•0287

2-0

•0228

•0228

•0228

•0228

•0228

•0228

•0228

•0228

•0228

•0228

•OL'-JS

•0228

O888

g-l

•0179

•0179

•0179

•0179

•0179

•0179

•0179

•017!)

•0179

•0179

•0179

•0179

•0179

2-2

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

s-s

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

2-4

•0082

•0082

•0082

•0082

•0081

•0082

«OB9

•0082

•OOH2

•0082

•0082

•0082

O08I

2-5

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

2-6

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

1

Tables for High Fourfold Correlation

57

Correlations from Tetrachoric Groupings.

r-roo,

A =

i-s

1-4

1-5

1-6

1-7

1-8

1-9

2-0

2'1

S-2

2-3

2-4

2-5

2-6

i- = ii-ii

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

o-i

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

w

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

o-s

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

0-4

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

0-5

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

0-6

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

0-7

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

0-8

•0968

•0808

O668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

0-9

•0968

•(1808

•06M

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

1-0

•0968

•0806

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

1-1

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

1-2

•<>!»;s

•0606

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

rs

•0968

•0808

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

1'4

•0606

•0808

•0668

•0548

•0446

•0359

•0287

•0228

0179

•0139

•0107

•0082

•0062

•0047

1-5

•0668

•0668

•0668

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

1-6

•().-) 18

•0548

•0548

•0548

•0446

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

l'~

•0446

•0446

•0446

•0446

•0446

•0359

•0287

•0228

•0179

•(H39

•0107

•0082

•0062

•0047

1-8

•0359

•0359

•0359

•0359

•0359

•0359

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

l-'.i

•0287

•0287

•0287

•0287

•0287

•0287

•0287

•0228

•0179

•0139

•0107

•0082

•0062

•0047

;n

•0228

•0228

•0228

•0228

•0228

•0228

•0228

•0228

•0179

•0139

•0107

•0082

•0062

•0047

2-1

•0179

•0179

•0179

•0179

•0179

•0179

•0179

•0)79

•0179

•0139

•0107

•0082

•0062

•0047

.'•.'

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0139

•0107

•0082

•0062

•0047

.'•.:

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0107

•0082

•0062

•0047

-'"'/

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0082

•0062

•0047

S-5

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0062

•0047

!-ii

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

•0047

B.

68 Tablets for St<itixtici<inx <nnl />i<i/iiifri<'i«nx

TABLE XXXI. The Y-Function.

Loo F(j>), Negatire Charaoteriotic, I

r

0

1

.'

3

4

S

6

7

8

0

1-00

•999,9996

7497

5001

2612

c •• <:v '

•886,7550

5087

2ti:!7

Of78

-7727

1-01

•997,6287

•>:,:,

0430

-soil

580TJ

•996,3196

0798

_S|,,S

.ffiffl

3C.4H

IDS

•996,1279

suit;

-.;:,. ;i

- I2I2

- [87B

•993,9536

7207

I—.;

nn

OMB

1-03

•992,7064

6671

3384

1104

-W.'il

•991,6564

4305

8061

-9801

-TUB

1-04

•990,5334

3108

<l--:i

*ti77

-r.171

•989,4273

B080

-9*95

77it;

- R 1 1

1-05

•988,3379

1220

:«»;s

6922

- I7M

•987,2651

0686

mt

VSR

- ll-s

1-06

•986,2089

-HB9

-7111(1

5H3o

- :i7.->7

•986,1690

-!M;:tn

- '-,--

-BJSO

-34«!i

1-07

•984,1455

-•.M2s

7107

- r,:!:)-

- :«;ts i

•983,]::-:!

-!t:iM7

-7OT

-5115

-3439

1-08

•982,1469

-SB0B

-75J9

.-.:,: i! i

-37555

•981,1717

- !»7s:,

Tan

5041

- I'UO

1-09

•980,2123

0223

%SR

-i;it2

- I.-.H1

•979,2686

0818

-8!»:.t;

-7J.KJ

1-10

•978,3107

1670

-9738

-7914

iVf.i:,

•977,4283

2476

0676

-^^

TM:.:,

I'll

•976,5313

3538

1768

0005

-ma

•975,6497

4753

3014

1281

I..-,.-,.-,

ria

•974,7834

8120

4411

1708

ioia

•973,9323

7(538

G860

4ua

8688

1-13

•973,0962

-HOB

-7858

mi7

^ j.-.^i

•972,2751

1126

-:..-,os

-7*i«;

BBS

1-14

•971,4689

3094

1600

- 9922

- s:{ i.-,

•970,6774

5Ll)0

8660

8086

(1549

1-15

•969,9007

7471

5941

4417

2JMW

•969,1386

-ON79

-8878

.;^s:{

5S3

1-16

•968,3910

2432

0960

-0493

-hira

•967,6578

6129

86M

K48

0816

1-17

•966,9390

7869

6554

6145

3742

•966,2344

0952

-BBU

-Ms:,

-Win

1-18

•966,5440

4076

27 IK

1366

0019

•!Mi4,8677

7341

6011

4687

3368

1-10

•964,2054

0746

-Kill

-8147

-(isnti

•963,5570

4290

3016

1747

0483

i-to

•962,9225

7973

6725

5484

4248

•962,3017

17!»2

<::,-:\

-S35H

1-S1

•961,6946

5748

4AM

3369

2188

•961,1011

-IIS 11

- BB7B

75IB

-8381

J-f2

•960,5212

4068

1880

1796

0669

•959,9546

8430

7318

88U

5111

1 ::

•969,4015

2925

1840

0760

-9RB

•958,8(516

7553

84M

6441

4393

1-S4

•958,3350

2313

1280

0253

:pj:!i

•967,8215

7204

6198

5197

4201

•957,3:! 11

ma

1846

0271

-93(1]

•956,8337

7377

6423

5474

4530

•956,3592

8858

1730

0806

-1IKSS

•955,8975

8(Hi7

7186

C,L'tl7

.-.37 1

1-S7

•955,4487

3604

2727

1855

0988

•955,0126

-!.L'.;S

-sin;

-757(1

(1728

1-18

•964,6891

5068

4232

3410

2593

•954,1782

0975

0173

o:i7<!

-S-.K5

l-t9

•953,7798

7016

• ;•_>:',: i

5467

4700

•953,3938

3181

2429

1682

0940

1-30

•953,0203

-9470

-8743

so21

-7:!o:i

•952,6590

5883

6180

4482

3789

1-31

•952,3100

2417

1739

1065

0396

•951,97:<2

9073

8419

7770

7126

1-U

•951,6485

B8BI i

6ttO

4098

3975

•951,33:-)9

2748

2142

1541

0944

1-33

•951,0353

-97t;<i

-OIHI

-B99B

-6034

•950,7466

6903

6344

5791

5242

1-34

•950,4698

4158

3624

3094

8B68

•950,2048

1532

1021

0514

0012

1-35

•949,9515

9023

B68B

8052

7573

•949,7100

6630

6166

5706

5251

r •••

•949,4800

4355

3913

3477

3044

•949,2617

2194

1776

IMS

0953

1-37

•949,0549

0149

- 97:. i

- ii.'ft;:;

-8177

•948,8.V).-)

8218

7846

7478

7116

1-38

•948,6756

6402

6052

5707

B808

•948,5030

4698

4371

4049

3731

/-..•

•948,3417

3108

2803

25(53

ojMS

•948,1916

1630

1848

1070

0797

1-40

•948,0628

0263

0003

-574R

-5J57

•947,9250

9008

8770

8537

8308

1-41

•947,8084

7864

7648

7437

7230

•947,7027

6829

8686

6446

6261

1-42

•947,6081

5905

5733

5565

5402

•947,6243

5088

4888

4703

4652

1-43

•947,4515

43H2

4254

4130

4010

1)47,3894

3783

3676

3574

3476

1-44

•947,3382

3292

3207

3125

3049

•947,2976

2908

2844

2784

2728

1-45

•947,2677

1680

1687

2549

J.M 1

•947,2 IM

2469

2437

2419

2406

J, la

•O47 91*17

M *>(»•>

•ifi

«7*i /) J*O«7f

•MM

2392

2396

2404

•947,2416

2432

2452

2477

2506

•947,2539

2576

2617

2662

2712

•947,2706

2824

2886

2952

3022

1-48

•947,3097

3175

3258

884B

3436

•! 117,3531

3630

3731

3841

3953

1.43

•947,4068

4188

4312

4440

4572

'.M7,4708

4848

4992

6141

6293

1.50

447,6448

6610

6774

5943

6116

•947,6292

6473

6658

6847

7040

A horizontal bar m
lowered one unit.

that the third figure of the mantUaa has changed, a negative sign that it must be

Tables of the T- Function

59

DIFFERENCES : — NEGATIVE down to rule

0

1

9

S

4

5

6

7

8

9

P

2503

2496

2489

2482

2475

2468

2460

2454

2446

2440

i-oo

2432

2425

2419

2411

2404

2398

2390

2383

2377

2369

1-01

2363

2355

2349

2342

2335

2328

2321

2314

2307

2301

V02

2293

2287

2280

2273

2267

2259

2253

2246

2239

2233

1-03

2226

2219

2212

2206

2198

2193

2185

2179

2172

2165

1-04

2159

2152

2146

2139

2132

2126

2119

2112

2106

2099

1-05

2093

2086

2080

2073

2067

2060

2053

2047

2041

2034

1-06

20-27

2021

2015

2008

2002

1995

1989

1983

1976

1970

1-07

1963

1957

1950

1944

1938

1932

1925

1919

1912

1906

V08

1900

1894

1887

1881

1875

1868

1862

1856

1850

1843

1-09

1837

1832

1824

1819

1812

1807

1800

1794

1787

1782

1-10

1775

1770

1763

1757

1751

1744

1739

1733

1726

1721

1-11

1714

1709

1702

1696

1690

1685

1678

1672

1666

1660

1-12

1654

1649

1642

1636

1630

1625

1618

1612

1607

1600

1-13

1595

1589

1583

1577

1571

1565

1559

1554

1547

1542

1-14

1536

1530

1524

1519

1512

1507

1501

1495

1490

1483

1-15

1478

1472

1467

1460

1455

1449

1443

1438

1432

1426

1-16

1421

1415

1409

1403

1398

1392

1386

1381

1375

1370

1-17

1364

1358

1352

1347

1342

1336

1330

1324

1319

1314

1-18

1308

1302

1297

1291

1286

1280

1274

1269

1264

1258

1-19

1-25-2

1248

1241

1236

1231

1225

1219

1215

1208

1204

1-20

1198

1192

1187

1181

1177

1170

1166

1160

1154

1149

1-21

1144

1138

1134

1127

1123

1116

1112

1106

1101

1096

1-22

10!»0

1085

1080

1075

1069

1063

1059

1053

1048

1043

1-2S

1037

1033

1027

1021

1017

1011

1006

1001

996

990

V24

985

980

975

970

964

960

954

949

944

938

1-25

934

928

924

918

913

908

902

898

893

887

l-2(j

883

877

872

867

862

858

852

846

842

837

1-27

832

827

822

817

811

807

802

797

791

787

V28

782

777

772

767

762

757

752

747

742

737

1-29

733

727

722

718

713

707

703

698

693

689

i-so

683

678

674

669

664

659

654

£49

645

640

1-31

G35

630

625

620

616

611

606

601

697

591

1-32

587

582

678

572

568

563

659

553

649

544

1-33

640

534

530

526

620

516

511

507

602

497

1-34

492

488

483

479

473

470

404

460

455

451

1-35

445

442

436

433

427

423

418

414

409

404

1-30

400

395

391

386

382

377

372

368

363

359

1-37

354

350

345

341

336

332

327

3:22

318

314

1-38

309

305

300

295

292

286

282

278

273

269

V39

265

260

255

251

247

242

238

233

229

224

1-40

220

216

211

207

203

198

193

190

185

180

1-41

176

172

168

163

159

154

150

146

141

137

1-42

133

128

124

120

116

111

107

102

98

94

V43

>»>

85

82

76

73

68

64

60

56

51

1-44

47

43

-38

-35

-30

-25

-22

-18

-13

- 9

1-45

- 4

_ i

+ 4

-4- H

4-19

4-lfl

4-9O

+ 25

4-99

+ 33

l'4ti

+ ••'>-

+ 41

^
45

T O

50

T \£t

54

T 1U

58

T "V

62

66

-f &&

70

75

1-47

78

83

87

91

95

99

104

107

112

115

1-48

120

124

128

132

136

140

144

149

152

156

1-49

161

MM

169

173

176

181

185

189

193

197

1-50

* Differences change sign at horizontal rule.

8—2

GO

or Statist idaiix miff /tiomt fr
TABLE XXXI. The V-Function.

r

Loo r(j>), Negative Characteristic, I

0

1

f

3

4

5

6

7

8

9

1-61

•947,7237

7437

7642

7*51

BOM

•947,8281

B009

8727

B8QQ

9189

i-st

•947,9426

9667

9912

HMOI

+ 0414

•948,0671

0932

1196

1465

1738

1-53

•948,2015

nee

2580

isaa

3161

•948,3457

3758

4062

4370

4689

1-54

•948,4998

5318

5642

5970

6302

•948,0688

6977

7321

7668

8019

1-65

•948,8374

8733

Booe

9463

9834

•949.020S

0687

0969

1355

1745

1-56

•949,2139

2537

BUM

3344

3753

•949,4166

4583

6004

5429

B867

1-57

•949,6289

6786

7165

7609

BOM

•949,8508

8963

9422

B88B

+ 0351

1-58

•990,0822

1296

1774

2255

2741

•960,3230

3723

4220

4720

6888

1-59

•950,5733

6845

6760

7280

7803

•950,8330

8860

tan

9933

+ 0175

1-60

•951,1020

1669

21-2-2

2679

3240

•951,3 "i

4372

4SI43

55111

6088

1-61

•951,0680

7887

7857

8451

9048

•951,9649

+ <i-:, i

+0BH

+ 1475

+ 2H91

1-Gg

•952,2710

3333

3900

4591

5225

•952,6863

6604

7149

7798

8451

1-63

•952,9107

976C

+0430

+IW7

+ 17-67

•953,2442

8180

3-ol

44S(i

5175

1-64

•953,5867

6563

7263

7966

8673

•953,9383

+3097

+ OH 5

+1599

+BKI

1-66

•954,2989

37-.' 1

4456

6195

5938

•954,01 ;M

7434

8187

8944

9704

1-66

•955,0468

1836

2007

27*2

3560

•955,4342

6127

6916

6708

7504

1-67

•955,8303

9106

9913

+ 0723

+ 1536

•956,2353

3174

3! OS

4825

6666

1-68

•956,6491

7S30

8170

9015

DM; i

•957,0716

1571

2430

3293

4188

1-69

•957,5028

5901

6777

7657

8540

•957,9427

+ 0:117

+ 1211

-t L'|OS

4-SRJ8

1-70

•958,3912

1880

5731

6645

7663

•958,8484

0406

+IBJB

+ -j-ji a

1-71

•959,3141

4083

6088

5977

6929

•959,7884

6848

9805

+ 0771

+ 17.10

!•::

•990,8719

MM

4667

5650

6636

•960,7625

8618

9614

+ 0(111!

+ 'Kii(i

1-7S

•961,2022

3632

4645

5661

6681

'961,7704

8730

B760

+ 0793

+ 1^:10

1-74

•962,2869

3912

4959

6009

7062

•962,8118

9178

+0241

+ 130S

+ 237H

J-75

•963,3451

4527

6607

6690

7776

•963,8866

ll!»5!»

+ lo:,5

+2155

+H58

1-70

•964,4364

5473

6586

7702

8821

•964,99*4

+ 1070

+ 3331

+ 4407

1-77

•965,5606

6749

7894

9043

+ 01!);-,

•966,1350

2509

3(i7l

4836

6004

1-78

•966,7176

835 1

9529

+ 0710

+ 181)5

•967,3082

4274

5468

6665

7866

1-79

•967,9070

+ 0277

+ 1488

+ 2701

+ 391H

•968,5138

6361

7688

8818

-iOo.M

1-80

•969,1287

2526

3768

5014

6263

•969,7515

8770

+ 1 N IL'!>

+ 1291

+ 1-.-.55

1-81

•970,3823

5095

6369

7646

8i>27

•971,0211

1498

2788

4082

5378

1-82

•971,6678

7981

!)L'S7

+ (*.-,! Mj

+ 1908

'972,3224

4542

5804

718!)

8517

1-83

•972,9848

+ 1182

+ 2520

+ 3800

+ 5204

•973,6551

7900

925 t

+ 0610

+ 1909

1-84

•974,3331

4697

6065

7437

8812

•975,0190

1671

2955

4342

5733

1-85

•975,7126

Ben

9922

+ 1325

+3719

•976,4139

6551

6066

S3S1

1-86

•977,1230

M67

4087

669 1

(!!)-, 7

•977,8397

9839

-i- 1 - v>

+ -J73 1

+2189

1-87

•978,6640

7098

8559

+ 001'.-;

+ 1490

•979,2960

4433

rape

7389

ss7 1

1-88

•980,0356

1844

3335

4830

6327

•980,7827

9331

+ OS37

. wn

+ 385!)

1-89

•981,5374

6893

8414

9939

+ 1468

•982,2996

4!>30

6066

760*i

9148

1-90

wajoon

2242

3793

53 is

0905

•983,8465

4-OQ3B

+ 15: ir,

+ 31(11

+ 1730

1-91

•984,6311

7890

9471

+ T055

+ 2012

•985,4232

5825

7421

9020

+ (Mi21

1-9S

•986,2226

:MI

51 15

705V

8675

•987,0294

1917

3542

5170

1-93

•987,8436

+ 0073

+ 17i:i

+ 3350

+ 5002

•988,6651

8302

9967

+ Kil 1

+ 3275

1.94

•989,4938

MOO

8274

9946

+IBS1

•990,3299

4980

6663

8350

+ 0039

1-95

•991,1732

3127

51-J5

0820

s53o

•992,0237

1947

3(i5<)

5375

70H3

1-9G

•992,8815

+ 053!)

+ 2200

+ 39i»5

+ 5728

•993,7464

9202

+ 0:113

+ 208H

+ 1 1.".5

1-97

•994,6185

7937

9693

+ ii-,]

+ 32 1:1

•995,4977

6744

8513

-1 O^Mi

+ L'IH)L'

1-98

•996,3840

5621

7405

9192

+ 0982

•997,2774

4569

6368

81(19

9972

1-99

•998,1779

3588

6401

7216

9034

•999,0854

2678

4604

6333

8105

A horizontal bar mean* that the third figure of the mantissa baa changed, a positive sign that it must be
raited one unit.

Tables of the T- Function

61

DIFFERENCES : — on this page, POSITIVE

f

0

1

2

S

4

5

6

7

8

9

200

205

209

213

217

221

225

229

233

237

1-51

241

245

249

253

257

261

264

269

273

277

1-52

280

285

288

293

296

301

304

308

312

316

1-53

320

324

328

332

336

339

344

347

351

355

1-54

359

363

367

371

374

379

382

386

390

394

1-BH

398

401

406

409

413

417

421

425

428

432

1-56

436

440

444

447

452

455

459

463

466

471

1-57

474

478

481

486

489

493

497

500

505

508

1-58

612

515

520

523

527

530

535

538

542

545

1-59

549

553

557

561

564

568

571

576

579

582

1-60

587

590

594

597

601

605

608

613

616

619

1-G1

623

627

631

634

638

641

645

649

653

656

1-62

659

664

667

670

675

678

681

685

689

692

1-6S

696

700

703

707

710

714

718

721

724

729

1-04

732

735

739

743

746

750

753

757

760

764

1-G5

768

771

775

778

782

785

789

792

796

799

1 -(Jij

803

807

810

813

817

821

824

827

831

835

1-67

838

841

845

849

852

855

859

863

866

869

1-68

873

876

880

883

887

890

894

897

900

904

1-69

908

911

914

918

921

925

928

931

935

938

1-70

942

945

949

952

955

959

962

966

969

972

1-71

976

979

983

986

989

993

996

999

1003

1006

1'72

1010

1013

1016

1020

1023

1026

1030

1033

1037

1039

1-73

1043

1017

1050

1053

1056

1060

1063

1067

1070

1073

1-74

1070

1080

1083

1086

1090

1093

1096

1100

1103

1106

1-7S

1109

1113

1116

1119

1123

1126

1129

1132

1136

1139

1-76

1143

1145

1149

1152

1155

1159

1102

1165

1168

1172

1-77

1175

1178

1181

1185

1187

1192

1194

1197

1201

1204

1-78

1207

lill

1213

1217

1220

1223

1227

1230

1233

1236

1-79

1239

1242

1246

1249

12r>2

1255

1259

1262

1264

1268

1-80

1272

1274

1:277

1281

1284

1287

1290

1294

1296

1300

1-81

1303

1306

1309

1312

1316

1318

1322

1325

1328

1331

1-82

1334

1338

1340

1344

1347

1349

1354

1356

1359

1362

1-83

1366

1368

1372

1375

1378

1381

1384

1387

1391

1393

1-84

1396

1400

1403

1405

1409

1412

1415

1418

1421

1425

1-85

14:27

1430

1434

1436

1440

1442

1446

1449

1452

1454

1-86

1458

1461

1464

1467

1470

1473

1476

1480

1482

1485

1-87

1488

1491

1495

1497

1500

1504

1606

1509

1513

1515

1-88

1519

1521

1525

1527

1530

1534

1536

1540

1542

1545

1-89

1549

1551

1555

1557

1560

1563

1567

1569

1572

1575

1-90

1579

1581

1584

1587

1590

1593

1596

1599

1601

1605

1-91

1608

1611

1613

1617

1619

1623

1625

1628

1632

1634

1-92

1637

1640

1643

1646

1649

1651

1655

1657

1661

1663

1-93

1667

1609

1672

1675

1678

1681

1683

1687

1689

1693

1-94

1695

1698

1701

1704

1707

1710

1712

1716

1718

1722

1-95

1724

1727

1729

1733

1730

1738

1741

1745

1747

1750

1-9G

1752

1756

1758

1762

1764

1767

1769

1773

1776

1778

1-97

1781

1784

1787

1790

1792

1795

1799

1801

1803

1807

1-98

1809

1813

1816

1818

1820

1824

1826

1829

1832

1835

1-99

62

T<il>1e» for

TABLE XXXII. Subtense f ruin .. d Chord

Table to paxs from measured index /9 = 100 (arc — chord)/chord of a curve to the index
and may be closely represented by a common catenary. Suggested toe: to past

Values of a for given values of /8 as argument.

ft

•o

•/

•a

•3

•4

•5

•6

•7

•8

•9

1.1

23-1

23-2

23-2

23-3

23-4

23-5

23 -6

23-7

23-8

23-9

14

24-0

24-1

24-2

24-3

iM-4

24-5

24-6

24-7

M-7

24-8

15

24-9

25-0

25-1

L'J'J

25-3

25-4

25-5

25-6

25-8

25-7

Id

25-8

25 -9

26-0

M-]

26-2

M-a

26-4

26-4

26-5

L'I; i;

i;

-•; 7

2(i-8

26-9

27-0

27-0

L'7'l

27-2

27-3

27-4

27-6

18

W6

27-7

27-7

27-8

27-9

28-0

28-1

28-2

28-3

28-3

IS

28-4

B*f

28-6

28-7

28-7

K-fl

28-9

29-0

29-1

mt

to

._,,,.._,

29-3

29-4

29-6

29-6

29-6

29-7

29-8

29-9

30-0

SI

»CO

30-1

30-2

30-3

30-4

30-4

30-5

30-6

307

30-8

-•

30-8

30-9

31-0

31-1

31-2

31-2

31-3

31-4

31-5

31-6

ts

31-8

31-7

31-8

31-9

31-9

32-0

32-1

8M

32-3

32-3

• •>

32-4

32-5

:«-G

32 -G

32-7

32-8

32-9

32-9

33-0

33-1

to

33-2

33-3

33-3

33-4

335

33-6

33-6

33-7

33-8

33-9

go

33-9

34O

34-1

Ml

34-2

34-3

34-4

34-5

34-5

34-6

tr

34-7

34-8

34-8

34-9

35-0

35-1

35-1

35-2

353

35-3

S8

35-4

35-5

35-6

35-6

35-7

35-8

35-9

35-9

36-0

36-1

S9

36-2

36-2

36-3

36-4

36-4

:{(]•->

36-6

36-7

36-7

36-8

SO

36-9

36-9

37-0

37-1

37-2

37-2

37 -3

37-4

37-5

37-5

SI

37-6

377

37-7

37-8

37-9

38-0

38-0

38-1

38-2

38-2

Si

38-3

38-4

38-4

38-5

38-6

38-7

38-7

38-8

38-9

38-9

S3

39-0

39-1

39-2

39-2

39-3

39-4

39-4

39-5

39-6

39-6

34

39-7

39-8

39-8

39-9

40-0

40-1

40-1

40-2

40-3

40-3

S5

40-4

40-5

40-5

40-6

40-7

40-7

40-8

40-9

41-0

41-0

Sli

ll'l

41-2

41-2

41-3

41-4

41-4

41-5

41-6

41-6

41-7

S7

41-8

41-8

41-9

42-0

42-0

42-1

42-2

42-2

42-3

liM

38

4>-4

42-6

42-6

42-6

42-7

42-8

42-9

42-9

43-0

43-1

39

43-1

43-2

13-3

43-3

43-4

43-5

43-5

43-6

437

43-7

40

43-8

43-9

43-9

44-0

44-1

44-1

44-2

44-3

44-3

44-4

41

44-5

44-5

44-8

44-6

44-7

44-8

44-8

44-9

45-0

45-0

42

45-1

45-2

45-2

45-3

45-4

45-4

45-5

45-6

45-8

45-7

43

45-8

45-8

45-9

46-0

40-0

46-1

4G-2

46-2

46-3

46-4

44

46-4

46-5

46-5

46-6

46-7

46-7

4G-8

46-9

46-9

47-0

45

47-1

47-1

47-2

47-3

47-3

47-4

47-5

47-5

47-6

47 -G

40

47-7

47-8

47-8

47-9

48-0

48-0

48-1

48-2

48-2

48-3

47

48- J

48-4

48-5

48-5

48-6

48-7

48-7

48-8

48-9

48-9

48

49-0

49-1

49-1

494

49-2

49-3

49-4

49-4

49-5

49-0

49

49-6

49-7

49-8

49-8

49-9

49-9

50-0

50-1

50-1 "

60-2

to

50-3

50-3

60-4

50-6

60-6

50-6

50-0

BO-7

50 '8

50-8

61

50-9

51-0

51-0

51-1

61-1

51-2

51-3

51-3

51-4

51-5

5S

61-5

51 -0

51-6

51-7

51-8

51-8

51-9

52-0

52-0

62-1

S3

U'l

Ml

Ml

52-3

52-4

52-5

62-5

52-6

n-6

52-7

64

52-8

M-fl

U-9

53-0

53-0

53-1

53-1

53-2

53-3

53-3

65

53-4

53-4

53-5

53-6

53 -G

53-7

53-8

53-8

HI

5:M)

66

54-0

54-1

54-1

64-2

54-3

54-3

54-4

54-4

54-5

54 -0

67

54-6

54-7

54-7

54 -H

64-9

54-9

65-0

55-0

55'1

56-i

68

615-2

r..v3

55-4

55-5

55-5

55-0

55-7

55-7

55-8

69

55-8

55-9

56-0

wo

56-1

66-1

6G-2

56-3

56-3

:.ii-i

CO

50-5

50-5

5G-6

50-6

56-7

56-8

56-8

56-9

57-0

61

67-1

57-1

672

57-2

57-3

57-4

57-4

67-5

57-5

57-0

CX

57-7

57-7

57-8

67-8

67-9

58-0

68-0

68-1

58-1

681

C3

583

68-3

58-4

68 '4

58-5

68-8

58-8

68-7

58-7

58-8

C4

.13-9

f>8-0

59-0

59-0

59-1

69-2

59-2

59-3

59-3

59 '4

Tables of Catenary Indices

63

in the case of the Common Catenary,

a=100 subtensejchord, on the assumption that the curve is symmetrical about the subtense
from callipers and tape measurements of the nasal bridge to the ratio of" rise" to " span."

Values of a for given values of /3 as argument.

/3

•o

•1

•2

•3

•4

•5

•6

•7

•8

•9

65

59-5

59-5

59-6

59-6

59-7

59-8

59-8

59-9

59-9

60-0

66

60-1

co-i

60-2

60-2

GO'S

60-4

60-4

60-5

60-5

60-6

67

60-7

60-7

60-8

60-8

60-9

ci-o

61-0

61-1

61-1

61-2

68

61-3

61-3

61-4

61-4

61-5

61-6

61-6

61-7

61-7

61 '8

69

61-9

61-9

62-0

62-0

62-1

62-1

02-2

62-3

62-3

62-4

70

62-4

62-5

62-6

62-6

62-7

62-7

62-8

62-9

62-9

63-0

71

63-0

63-1

63-1

63-2

63-3

63-3

63-4

63-4

63-5

63-6

72

63-6

63-7

63-7

63-8

63-9

63-9

64-0

64-0

64-1

64-1

73

64-2

64-3

64-3

64-4

04-4

64-5

64-6

64-6

W7

64-7

74

64-8

64-9

64-9

65-0

65-0

65-1

65-1

65-2

65-3

65-3

75

65-4

65-4

65-5

65-6

65-0

65-7

65-7

65 -8

65-8

65-9

76

66-0

66-0

66-1

66-2

66-2

66-3

66-3

66-4

66-4

66-5

77

66-5

66-6

66-7

66-7

66-8

66-8

66-9

86 -fl

67-0

67-1

78

67-1

67-2

67-2

67-3

67-4

67-4

67-5

67-5

67-6

67-6

79

67-7

67-8

67-8

67-9

67-9

68-0

68-0

68-1

68-2

68-2

80

68-3

68-3

68-4

68-5

68-5

68-6

68-6

68-7

68-7

68-8

81

68-9

68-9

69-0

69-0

69-1

69-1

69-2

60-3

C!)'3

69-4

82

69-4

69-5

69-5

69-6

69-7

69-7

69-8

G9'8

69-0

70-0

8S

70-0

70-1

70-1

70-2

70-2

70-3

70 '4

70-4

70-5

70-5

84

70-6

70-6

70-7

70-8

70-8

70-9

70-9

7TO

71-0

7M

85

71-2

71-2

71-3

71-3

71-4

71-4

71-5

71-6

71-6

71-7

86

71-7

71-8

71-8

71-9

72-0

72-0

72-1

72-1

72-2

72-2

87

72-3

72-4

72-4

72-5

72-5

72-6

72-6

72-7

72-8

72-8

88

72-9

72-9

730

73O

73-1

73-2

73-2

73-3

73-3

73-4

89

73-4

73-5

73-6

73 •«

73-7

737

73-8

73-8

73-9

73-9

90

74-0

74-1

74-1

74-2

74-2

74-3

74-3

74-4

74-5

74-5

91

74-6

74-6

74-7

74-7

74-8

74-9

74-9

75-0

75-0

75-1

92

75-1

75-2

75-3

75-3

75-4

75-4

75-5

75-5

75-6

75-6

93

75-7

75-8

75-8

75-9

75-9

76-0

76-0

76-1

76-2

76-2

94

76-3

7<J'3

76-4

76-4

76-5

76-6

76-6

76-7

76-7

76-8

95

76-8

76-9

76-9

77-0

77-1

77-1

77-2

77-2

77-3

77-3

96

77-4

77-5

77-5

77-6

77-6

77-7

77-7

77-8

77-8

77-9

97

78-0

78-0

78-1

78-1

78-2

78-2

78-3

78-3

78-4

78-5

98

78-5

78-6

78-6

78-7

78-7

78-8

78-9

78-9

79-0

79-0

99

79-1

79-1

79-2

79-2

79-3

79-4

79-4

79-5

79-5

79-6

100

79-6

79-7

79-8

79-8

79-9

79-9

80-0

80-0

80-1

80-1

101

B01

80-3

80-3

80-4

80-4

80-5

80-5

80-6

80-6

80-7

1'H

80-8

80-8

80-9

80-9

81-0

81--0

81-1

81-1

81-2

81-3

1CS

8 1 -3

81-4

81-4

81-5

81-5

81-6

81-6

81-7

81-8

81-8

104

81-9

81-9

82-0

82-0

82-1

82-1

82-2

82-3

82-3

82-4

105

82 '4

82-5

82-5

82-6

82-6

82-7

82-8

82-8

82-9

82-9

106

83-0

83-0

83-1

83-1

83-2

833

83-3

83-4

83-4

83-5

107

83-5

83-6

83-6

83-7

83-8

83-8

83-9

83-9

84-0

84-0

108

84 -I

84-1

84-2

84-3

84-3

84-4

84-4

84-5

84-5

84-6

109

84-6

84-7

84-8

84-8

84-9

84-9

85-0

85-0

85-1

85-1

110

85-2

85-3

85-3

85-4

85-4

85 '5

85-5

85-6

85-6

85-7

111

85-8

8ft -8

85-9

85-9

86-0

86-0

86-1

86-1

86-2

86-2

112

86-3

86-4

86-4

86-5

86-5

86-6

86-6

86-7

86-7

86-8

113

80-9

86-9

87-0

87-0

87-1

87-1

87'2

87-2

87-3

87-4

114

87-4

87-5

87-5

87-6

87-6

87-7

87-7

87-8

87-8

87-9

115

88-0

88-0

88-1

88-1

88 '2

88-2

88-3

88-3

88-4

88-5

in,

88 -5

88-0

88-6

88-7

88-7

88-8

88-8

88-9

88-9

89-0

Tables for

mi'l

TABLE XXXIII, A and I1.. Hupplmn-ttt'try Tables of Subtense from

Arc and Chord.

TABLE XXXIIL

Supplementary Tables for Subtense Index a. as calculated from the
urcttal value /3 on the Catenary Hypotltesis.

(A) Values of a. for low /3.

ft

•o

•1

•J8

•s

•4

•6

•e

•7

•8

•9

6

15-3

15-4

i.vi;

15-7

15-8

16-0

16-1

16-2

163

16-5

7

16-6

16-7

16-8

17-0

17-1

17-2

17-3

17-4

17'li

17-7

8

17-8

17-9

18-0

18-1

18-3

18-4

18-5

18-6

18-7

18-8

9

18-9

19-1

19-2

19-3

19-4

19-5

19-6

19-7

19-8

19'!»

10

20-0

20-1

20-2

20-3

20-4

20-6

20-7

20-8

20-9

21-n

11

21-1

21-2

21-3

21-4

21-5

21-6

21-7

21-8

21-9

22-0

It

22-1

22-2

22-3

22-4

22-5

22-6

22-7

22-8

22-9

23-0

(B) Values of a. for high /9.

0

a

0

a

0

a

0

a

0

a

0

a

101

80-2

126

94-0

151

107-5

176

120-8

201

133-9

147-0

102

80-8

127

94-5

152

108-0

177

121-3

909

134-5

147-5

103

81-3

128

95-1

U8

108-5

178

121-8

KM

13.1-0

998

148-0

104

81-9

129

95-6

154

109-1

179

122-4

204

135-5

229

148-6

105

82-4

130

96-2

155

109-6

180

122-9

205

136-0

980

149-1

106

83-0

131

96-7

156

110-2

181

123-4

908

136-6

981

149-6

107

BS-8

!.:.'

97-2

157

110-7

182

124-0

207

137-1

150-1

108

84-1

133

97-8

158

111-2

183

124-5

208

137-6

988

150-6

109

84-6

134

98-3

159

111-8

184

125-0

209

138-1

234

151-2

110

85-2

135

98-9

160

112-3

185

125-5

S10

138-7

235

151-7

111

86-8

136

99-4

Itil

112-8

186

126-1

211

139-2

236

152-2

112

86-3

137

99-9

a;:

113-4

187

126-6

212

139-7

237

152-7

113

86-9

tM

100-5

163

113-9

188

127-1

213

140-2

238

153-2

114

87-4

139

101-0

164

114-4

189

127-7

214

140-8

989

lf.3-8

115

88-0

140

101-6

165

114-9

190

128-2

916

141-3

240

M4-3

116

88-5

141

102-1

166

115-5

191

128-7

916

141-8

,.",1

1.11-8

117

89-1

US

102-7

167

116-0

192

129-2

217

142-3

•-' '/ -'

155-3

118

89-6

143

103-2

168

116-6

193

129-8

218

142-8

"/•'

155-8

119

90-1

144

103-7

169

117-1

194

130-3

219

143-4

w

156-3

190

90-7

13ft

104-3

170

117-6

195

130-8

990

143-9

•"<•'

156-9

121

91-2

146

104-8

171

118-1

196

131-3

991

144-4

346

1.17-4

122

91-8

147

105-3

172

118-7

197

131-8

.' : :

144-9

947

157-9

198

92-3

148

105-9

173

119-2

198

132-4

2S3

145-4

948

158-4

124

92-9

149

106-4

174

119-7

199

132-9

99A

146-0

,.",'.>

159-0

126

!M-4

150

106-9

176

120-2

900

133-4

998

146-5

250

159-5

Diagram of Mean Contingency

65

.<=

•8

o

fe;

tej

g

H

X

M

•
-*

06 Talk* for Statisticians and tiioinetricians

XXXV. Diagram to determine the type of a Frequency Distrilnillnn firm a knowledge
of the Constant* & and /3,. Customary Values of /9, and /3».

1 -2 -3 -4 -5 -6 7 -8 -9 1-0 11 1-2 13 1-4 1-5 16 17 18

Frequency Type from /8, and /J,

67

XXXVI. Diagram showing Distribution of Frequency Types for High Values

for /9, and /3a.

A

tf— 2

08

/,////•>/,,/• Stnfiiiit\-i<tn#

TA11LK .\.\XV1I. To find the Probable Error of £,.
Vttlites of

'

u-oo

(/•W5

o-io

0-16

o-to

0-£B

0-50

0-S5

0-40

0-45

0-50

0-65

u-co

U-(Ja

0-70

0-75

t-o

o-oo

0-58

0-93

1-15

1-37

1-57

1-77

1-97

2-17

2-38

2-B8

1-90

8-09

3-24

3-46

3-71

s-i

ooo

0-59

0-0B

1-12

1-30

1-50

1-70

1-90

2-10

2-48

2-69

2-91

3-1 •_•

3-34

3-57

ft

o-oo

o-eo

0-97

1-13

1-30

1-48

r«;7

1-86

a-oe

2-22

2-41

2-61

2-81

3 Ml

:i-J2

3-44

t-s

o-oo

0-62

0-99

1-16

1-32

1-4!)

1-07

1-84

2-09

2-19

2 -36

2-55

2-74

3-12

3-32

tl-4

o-oo

0-64

1O2

1-20

1-37

1-54

1-70

1-80

BOS

2-18

2-34

2-51

2-08

3-03

3-22

g-B

o-oo

0-66

1O5

1-26

1-46

1-61

1-76

1-91

2 OS

2-19

2-33

2-49

2-65

2-81

8-97

3-14

t-6

o-oo

0-69

1-10

1-34

1-54

1-72

1-86

2-00

2-13

2-25

2-37

2-50

2-(i4

2-92

308

t-7

ooo

0-73

1-15

1-42

1-64

1-83

1 -90

2O9

2-22

I'M

2-42

2-53

••60

8-77

8-90

8-06

t'8

o-oo

0-77

I'M

1-51

i •-:,

I'M

2-07

2-20

2-32

2-41

2-50

2-60

2-70

8-80

8-98

305

£•9

ooo

0-81

1-30

1-61

1-87

2-06

1-M

2-33

2-44

2-53

2-62

2-70

2-79

2-89

2-99

3-09

s-o

o-oo

0-87

1-40

1-73

2-01

2-84

2-47

2-57

2-07

2-76

2-84

2-92

3-00

3-09

3-18

3-1

o-oo

0-94

1-53

1-86

2-17

2-35

2-51

2-64

2-75

2-85

2-94

3-00

3-08

3-15

3-23

3-30

s-e

ooo

1-02

1-67

2-02

2-33

2-52

2-71

2-84

2-95

3O5

3-14

3-22

3-27

3-33

3-40

3-46

3-3

000

1-12

1-82

2-20

2-50

2-71

2-92

3-06

3-18

3-28

3-37

3-44

3-50

3-53

3-60

3-OJ

3-4

ooo

1-24

1-99

2-38

.,.,;.

2-93

3-14

3-30

3-43

3-53

3-63

3-70

3-75

3-79

3-83

3\i7

3-6

o-oo

1-37

2-16

2-57

2-89

3 '17

3-39

3 •.-)<;

3-69

3-81

3-91

3-98

4-1)2

4-06

4-10

4-12

3-6

000

1-50

2-33

2-78

3-11

3-43

3-65

3-84

3-99

4-12

4-22

4-29

4-33

4-37

4-40

441

3-7

ooo

1-64

2-50

2-99

3-36

3-70

3-93

4-14

4-31

4-44

4-54

4-61

4-66

4-70

4-72

4-74

3-8

ooo

1-78

2-67

3-20

3-62

3-97

4-23

4-46

4-64

4-77

4-87

4-95

6-00

505

5O7

5-09

3-9

ooo

1-93

2-86

3-43

3-89

4-25

4-54

4-79

4-97

5-n

5-23

5-32

6-38

5-43

5-46

5-48

4-0

ooo

2-10

3-07

3-69

4-17

4-55

4-87

6-13

5-32

5-48

5-62

5-72

5-79

6-84

5-88

:. 98

M

—

—

3-29

3-87

•1-17

4-87

6-21

6-49

5-69

5-87

6-03

G-15

(i--:i

0-2S

0-32

u:J3

*'*

—

—

3-53

4-19

4-7!)

5-21

6-58

5-88

6-10

6-30

C-4G

0-60

C-C9

6-75

6-80

6-81

4-3

—

—

3-78

4-52

5-13

6-58

6-97

6-29

G-54

6-75

6-93

707

7-18

7-25

7'2S>

7-31

4-4

—

—

405

1*80

5-49

5-98

6-40

6-74

701

7-24

7-42

7-57

7-68

7-76

7-80

7-83

4-5

—

—

433

6-18

0-88

6-42

6-87

7-23

7-62

7-7B

7-95

8-10

8-21

8-29

8-34

8-37

4-6

—

—

—

—

—

—

7-37

7-76

8-07

8-30

8-51

8-66

8-76

8-85

8-91

8-95

4-7

—

—

—

—

—

—

7-90

8-31

8-64

8-90

9-11

9-25

9-35

9-44

9-50

9-54

4-8

—

—

—

—

—

—

8-46

8-88

9-24

9-54

9-75

9-89

9-99

10-08

ln-14

10-18

4-9

—

—

—

—

—

9O5

9-47

9-86

10-21

10-42

10-58

10-69

10-78

10-84

10-80

s-o

—

—

—

—

—

—

9-60

10O8

10-50

10-90

11-19

11-33

11-44

11-53

11-60

11-04

5-1

—

—

—

—

—

—

—

—

12-26

i2-:w

12-42

12-43

5-2

—

—

—

—

—

13-10

13-20

13-29

13-29

6-3

—

—

—

—

—

—

13-98

14-15

14-18

14-18

6-4

—

—

—

_

—

—

—

—

1 l-!i 1

15-05

15-10

15-11

—

—

—

—

—

—

__f

15-90 15-98

16-05

16-07

5-6'

_

—

—

—

5-7

—

6-8

_

—

__

—

—

6-9

—

—

—

—

^_

—

6-0

—

—

—

—

—

—

6-1

—

_

—

—

.

__

(i-t

—

—

—

—

—

^_

—

—

—

G-3

—

—

—

' —

—

—

—

—

—

—

—

—

C-4

—

—

—

—

—

—

—

—

—

_.

—

—

—

—

—

6-5

—

—

—

_

—

—

—

—

—

—

—

6-6

—

—

_

—

—

_

_

_ _

_

—

6-7

—

—

—

—

__

__

^_

__

6-8

—

__

—

—

__

6-9

—

—

_

_

_

^^

^_

__

7-n

—

•"•

—

~*

—

—

—

—

—

—

—

—

—

—

—

Probable Errors of Frequency Constants

TABLE XXXVII.— (continued).
Values of

A

0-80

0-85

0-90

0-95

1-00

1-05

1-10

1-15

1-20

1-25

1-SO

1-55

1-40

1-4S

1-50

3-96

4-21

4-47

4-73

5-00

5-27

5-55

5-83

6-12

6-41

6-71

7-01

7-31

7-62

7-94

2-0

3-80

4-03

4-27

4-53

4-80

5-07

5-34

5-62

5-90

6-18

6-48

6-77

7-07

7-37

7-69

2-1

3'6ti

3-88

4-11

4-36

4-63

4-88

5-15

5-42

5-69

5-96

6-25

6-54

6-84

7-14

7-45

8-2

3-52

3-74

3-96

4-20

4-46

4-71

4-96

5-22

5-48

5-75

6-02

6-31

6-61

6-91

7-21

3-3

3-41

3-62

3-83

4-05

4-29

4-54

4-78

5-03

5-28

5-55

5-82

6-10

6-38

G-68

6-97

a-it

3-32

3-51

3-71

3-92

4-15

4-38

4-61

4-85

5-10

5-36

5-62

5-89

6-16

6-45

6-74

2-5

3-25

3-42

3-60

3-80

4-01

4-23

4-45

4-68

4-92

5-17

5-43

5-68

5-94

6-22

6-51

2-6

3-20

3-35

3-51

3-69

3-89

4-10

4-32

4-54

4-76

5-00

5-24

5-48

5-73

6-00

6-28

2-7

3-18

3-32

3-47

3-63

3-80

4-00

4-21

4-41

4-a

4-84

5-07

5-30

5-53

5-79

6-06

2-8

3-19

:i-32

3-45

3-60

3-75

3-92

4-11

4-30

4-49

4-70

4-91

5-12

5-34

5-59

5-85

2-9

3-27

3-38

3-49

3-61

3-74

3-87

4-03

4-21

4-39

4-58

4-78

4-98

5-19

5-42

5-68

3-0

3-38

3-47

3-57

3-67

3-77

3-89

4-02

4-1(5

4-31

4-48

4-<J<J

4-85

5-05

5-28

5-53

s-i

3-53

3-60

3-68

3-76

3-84

3-93

4-03

4-15

4-28

4-43

4-5!)

4-75

4-92

5-15

5-40

3-2

3-70

3-75

3-81

3-88

3-95

4-02

4-10

4-19

4-28

4-42

4-56

4-69

4-84

5-04

5-28

8-3

3-90

3-93

3-97

4-03

4-08

4-14

4-20

4-26

4-34

4-45

4-56

4-66

4-78

4-97

5-18

3-4

4-14

4-17

4-19

4-M

4-26

4-30

4-34

4-39

4-45

4-52

4-60

4-68

4-79

4-93

5-12

3-5

4-42

4-44

4-45

4-47

4-49

4-51

4-54

4-57

4-62

4-67

4-74

4-78

4-81

4-95

6-09

3-6

4-74

4-75

4-76

4-70

4-77

4-78

4-79

4-81

4-84

4-87

4-90

4-92

4-95

5-04

5-13

3-7

5-10

5-10

5-09

5-08

5-08

6-07

5-07

5-06

5-06

5-08

5-09

5-11

5-14

5-18

5-22

3-8

5-49

5-48

5-46

5-44

5-42

6-40

5-37

5-35

5-33

5-32

0-32

5-33

5-34

5-35

5-37

3-9

5-89

5-88

5-86

5-83

5-80

5-76

6-72

5-69

5-65

5-62

5-60

5-58

5-57

5-57'

5-59

4-0

6-33

6-32

6-30

6-26

6-21

6-16

6-11

6-06

6-02

5-98

5-94

5-91

5-88

5 -80

5-86

4-1

6-80

§•79

6-76

6-71

6-65

6-60

6-:>l

6-48

G-42

6-36

6-31

6-27

6-24

6-21

6-18

4-2

7-30

7-28

7-25

7-19

7-13

7-07

7-01

6-93

6-87

6-80

6-74

6-67

6-62

6-57

6-53

4-s

7-83

7-80

7-76

7-71

7-65

7-58

7-51

7-44

7-37

7-28

7-20

7-12

7-05

6-98

6-SJ2

4-4

8-38

8'3fi

8-32

8-28

8-21

8-14

8-07

7-99

7-90

7-81

7-71

7-61

7-51

7-42

7-34

4-5

8-96

8-95

8-91

8-86

8-79

8-72

8-64

8-55

8-45

8-35

8-24

8-13

8-00

7-90

7-80

4-o

9-57

9-57

9-53

9-47

9-40

9-33

9-24

9-14

9-04

8-93

8-82

8-69

8-55

8-42

8-31

4-7

10-20

10-23

10-16

10-10

10-05

'9-97

9-88

9-77

9-67

9-;")5

9-42

9-28

9-14

9-00

8-87

4-8

10-1)1

10-92

10-87

10-80

10-74

10-66

10-57

10-44

10-32

10-18

10-04

9'90

9-76

9-63

9-50

4-9

11-66

11-65

11-61

11-55

11-48

11-39

11-29

11-17

11-04

10-90

10-77

10-62

10-46

10-30

10-14

5-0

12-45

12-43

12-38

12-32

12-24

12-14

12-03

11-91

11-78

11-64

11-50

11-34

11-15

10-96

10-82

5-1

13-28

13-25

13-20

13-13

13-04

12-92

12-80

12-67

12-54

12-40

12-24

12-07

11-88

11-72

11-54

5-2

14-16

14-12

14-07

13-98

13-87

13-76

13-63

13-47

13-35

13-20

13-02

12-84

12-66

12-48

12-30

5-3

15-0!)

15-06

15-00

14-90

14-78

14-65

14-51

14-36

14-22

14-05

13-81

13-07

13-48

13-29

13-11

5-4

16*06

16-02

15-96

15-87

15-76

15-63

15-49

15-33

15-17

15-00

14-81

14-61

14-40

14-18

13-97

5-5

—

—

17-02

16-91

16-79

16-67

16-51

16-34

16-18

15-95

15-70

15-50

15-30

15-07

14-84

5-6

—

—

18-14

17-99

17-88

17-75

17-58

17-40

17-23

16-94

16-70

16-47

16-26

16-04

15-77

5-7

—

—

19-34

19-13

19-02

18-87

18-69

18-48

18-26

17-98

17-74

17-50

17-26

17-01

16-76

5-8

—

—

20-57

20-36

20-20

20-03

19-84

19-62

19-39

19-11

18-84

18-59

18-32

18-05

17-78

5-9

—

—

21-86

21-65

21-45

21-25

21-03

20-79

20-54

20-29

20-02

19-76

19-47

19-18

18-90

6-0

—

—

—

22-36

22-18

21-92

21-61

21-31

20-97

20-61

20-30

20-13

6-1

—

—

-

23-77

23-61

23-32

23-00

22-63

22-22

21-82

21-50

21-29

6-2

—

—

—

25-33

25-09

24-74

24-38

24-00

23-55

23-13

22-78

22-50

6-3

—

—

26-95

26-64

26-27

25-86

25-43

26-00

24-52

24-12

23-82

6-4

_

—

__

28-61

28-18

27-73

27-30

26-89

26-46

26-06

25-65

25-24

6-5

—

—

—

27-67

27-21

26-75

6-6

29-40

28-90

28-35

6-7

__

31-15

30-61

29-94

6-8

—

.

33-02

32-41

3; -72

6-9

~~

—

—

—

—

—

—

—

—

—

—

—

34-89

34-16

33-59

TO

7U

Tuble* for Stutixtiriums and Bnnu< triciants

TABLE X X X V 1 1 1. T» ./i«</ Probable Error of &.
Values of V-tf 2,,.

A

•

"• •

0-10

0-16

o-to

0-S5

0-30

0-55

0-40

0-45

0-50

0-55

0-GO

0-65

0-70

0-75

t-o

1-41

i-eo

1-74

1-93

2-11

2-28

2-44

2-60

2-77

2-94

3-12

8-M

3-46

8-ea

386

405

g-1

\ -.-.;

1-76

1-89

2-00

2-10

2-20

2-35

2-51

2-68

8-86

8-Ofi

8-M

3-44

3-64

8-84

I 404

g-g

1 -7:.

1-94

2-07

2-16

2-20

2-28

2-40

2-53

2 Us

2-85

3O4

3-43

3-(l3

4O3

g-3

1-95

2-16

1 H

2-35

8-41

2-49

2-58

2-67

2-78

ra

309

327

3-45

3-64

403

*-4

2-18

2-39

•- :.:>,

2-60

2-72

2-82

2-90

2-99

3O2

3-10

3-22

3-35

3-50

3 (is

3-86

4O3

t-5

8-46

2-68

2-83

2-97

3-09

3-19

3-27

3-31

3-32

3-36

347

3-53

3-63

3-7.-.

8-88

4O3

g-6

2-78

3-03

3-24

3-38

3-52

3-GO

3-66

369

3-69

3-70

3-75

3-78

3-83

3-87

3-95

407

g-7

3-17

3-48

3-71

3-87

3-98

4-03

4O8

4-12

4-11

409

4O7

406

4O6

4-06

4O7

4-15

g-8

3-64

402

4-26

4-42

4-52

4-58

4-60

4-59

4-57

4-52

4-44

4-39

4-34

4-M

4-31

4-34

g-9

4-22

4-65

4-94

6-11

5-20

5-22

5-18

6-13

507

4-99

4-90

4-80

4-70

4-88

4-80

4-G1

3-0

4-90

.-. ; -

6-76

6-89

5-95

6-93

5-86

5-76

5-65

6-53

5-41

5-30

5-20

B-U

5O5

5OO

s-i

5-75

6-41

C-72

8-88

6-90

6-82

6-70

6-54

6-33

<;-22

607

5-92

6-79

5-(i:i

5-53

6-77

7-55

7-90

8-00

7-97

7-83

7-63

7-42

7-21

7O3

6-86

6-70

6-53

6-39

(i-2i;

6-14

3-3

8-00

8-83

!»-22

9-30

9-22

9 O2

8-80

8-53

8-29

8O5

7-83

7-60

7-38

7-20

7-01

6-84

3-4

9-37

10-28

10-68

1076

10-67

10-46

10-20

9-91

9-62

9-31

9O1

8-73

8-44

8-18

7*8

7-66

S-B

10-85

11-75

12-31

12-52

12-46

12-25

11-95

11-60

11-24

10-86

10-45

1003

B-88

9-26

8-90

8-54

3-6

12-67

13-74

14-40

14-78

14-5;}

14-21

13-80

13-38

12-95

12-55

12-10

11-60

1106

1054

10O2

it:,:,

3-7

14-78

15-98

16-78

1709

16"!)3

16-53

1605

15-58

15O9

14-61

14O8

13-49

12-74

1202

11-36

10-80

3-8

17-50

18-83

19-83

20O3

19-78

19-36

18-76

18-22

17-64

16-98

16-25

15-30

14-42

i:' tio

12-88

12-27

3-9

20-80

22-50

88-88

23-81

23-34

22-67

21-98

21-14

20-29

19-45

18-58

17-.-.1

16-50

15-54

14-77

14 '06

4-0

24-74

88-89

28-47

28-05

27-24

M-M

25-25

24-18

23O3

22-02

2101

20-01

1904

18-12

17-23

1636

4-1

—

3500

34-17

32-88

31-36

29-77

28-13

26-60

25-12

23-82

88-64

21-54

20-53

19-5(5

18-62

4'*

—

—

43-3

41-4

39-2

37-2

35-2

33 '2

31-2

29-2

27-4

880

84-7

88-4

22-3

21-3

JL-g

55-3

51-6

48-0

44-6

41-2

38-6

36-2

338

31-8

30-1

28-5

20-9

25-6

84-a

4-4

—

—

72-7

660

59-

54-1

49-5

45-7

42-1

39-2

36-8

34-8

32-9

31O

29-2

87-7

4-5

—

—

: »;-.".

82-7

72-7

65-3

59-8

54-7

50-8

47-2

440

41O

86-fl

36-2

31 1

32O

4-0

—

—

—

—

—

—

75O

68O

62-2

56-9

52-2

48-2

45-1

a-i

396

37-2

4-7

—

—

—

—

—

—

101-3

87-2

76-8

68-3

62O

58D

r,2-7

49*1

45-9

428

4-8

—

—

—

—

—

—

140O

115-2

96-2

826

72-7

66-1

eon

5(i"7

49-3

4-9

—

—

—

—

—

—

204-5

150-8

122-3

102-5

89-1

802

72-1

6C-6

61-4

56-7

5-0

—

32-1-7

206O

154-2

126-8

110-1

96-9

86-6

78-1

71-2

65-6

5-1

103 '6

'i i- 1

85 '9

7S-O

if A

'ID -4

- > i *
1 1 fi-.i

1O-1-O

t O V

ii | .(i

..".

1 0" ^

|- ,•-'

1 1 O t

1 I 1-S

lv^ v/

i •' ;• i

• ' 1 ' '
!• i'1-Ci

5-1

L i ** **

224"4

1*4** O

1 ."i 1 •()

1 \to \)

rt-'-i!

^ -T

6-6

i •>— 'i

5-8

__

_^

_ _

5-7

_

_ _

—

—

—

—

6-8

—

—

—

—

—

—

6-9

_

—

—

—

—

—

—

—

—

—

—

6-0

—

_i

—

—

—

—

6-1

_

—

__

—

—

—

—

—

—

G-g

—

—

—

G-3

_

—

—

•

—

—

—

—

—

—

c-4

—

—

—

—

—

—

—

—

—

—

—

6-5

—

—

—

—

—

—

—

—

6-6

._

__.

_._

—

—

6-7

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

6-8

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

7-0

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

Probable Errors of Frequency Constants

71

TABLE XXXVIII.— (continued).
Values of

0-80

0-85

0-90

0-95

1 -00

1-05

1-10

1-15

1-20

1-25

1-30

1-35

1-40

1-43

1-50

4-24

4-43

4-62 4-81

5-00

5-19

5-38

5-56

5-75

5-94

6-12

6-30

6-49

6-67

6-84

2-0

4-23

4-41

4-.-.H 4-77

4-96

5-15

5-34

5-53

5-72

5-90

6-08

6-27

6-47

6-65

6-83

2-1

4-22

4-39

4-56 4-74

4-!W

5-12

5-31

5-50

5-69

5-87

6-05

6-24

6-44

6-63

6-82

2-2

4-20

4-36

4-r,:j 4-71

4-1X)

5-08

5-27

5-46

5-65

5-84

6-02

6-21

6'4l

6-61

6-80

2-3

4-19

4-35

4-51

4-69

4-87

5-05

5-23

5-42

5-61

5-80

5-99

6-18

6-38

6-58

6-78

2-4

4-18

4-34

4-50

4-67

4-85

5-03

5-21

5-39

5-58

5-77

5-96

6-15

6-35

6-54

6-74

2-5

420

4-35

4-50 1 4-67

4-84

5-01

5-20

5-36

5-54

5-72

5-91

6-11

6-30

6-49

6-68

2-6

4-26

4-38

4-52

4-68

4-84 5O1

5-18

5-34

5-51

5-67

5-85

6-05

6-25

6-44

6-62

2-7

4-40

4-50

4-60

4-72

4-86

5-03

5-19

5-34

5-49

5-65

5-83

6-02

6-21

6-39

6-58

2-8

4-63

4-67

4-73

4-82

4-93

5-05

540

5-35

5-50

5-66

5-82

6-00

6-18

6-36

6-54

2-9

4-98

4-97

4-99

5-03

5-10

5-18

5-28

5-39

5-52

5-66

5-82

5-98

6-15

6-33

6-51

8-0

5-47

5-42

5-38

5-34

5-36

5-37

5-41

5-48

5-58

5-70

5-83

5-97

6-14

6-32

6-52

s-i

6-03

5-92

5-83

5-75

5-67

5-62

5-60

5-62

5-68

5-78

5-90

6-03

6-18

6-34

6-52

S-2

6-67

6-51

6-35

6-22

BOB

6-00

.VIM

5-94

5-92

5-95

6-01

6-12

6-25

6-37

6-53

3-3

7-41

7-17

6-95

6-77

6-61

6-48

6-29

6'26

6-24

6-22

6-22

6-26

6-34

6-47

6-61

3-4

8-22

7-92

7-04

7-38

7-17

6-99

6-84

6-72

6-63

6-57

6-54

6-53

6-56

6-61

6-71

3-5

9-14

8-80

8-51

s-sa

7-98

7-70

7-53

7-40

7-22

709

6-99

6-98

6-95

6-92

6-93

8-6

10-34

9-94

9-58

9-25

8-96

8-66

8-36

8-14

7-90

7-75

7-61

7-51

7-42

7-34

7-23

3-7

11-77

11-89

10-88

10-37

9-98

9-62

9-31

9-03

8-73

8-51

8-29

8-11

7-94

7-78

7-60

3-8

13-42

12-85

12-31

ll-7:i

11-30

10-86

10-41

10-02

9-64

9-34

9-03

8-77

8-52

8.30

8-10

3-9

15-58

14-84

14-10

13-12

12-79

LMO

11-64

11-13

10-65

10-21

9-83

9-51

9-20

8-92

8-67

4-0

17-72

16-85

16-01

15-21

14-44

13-70

13-00

1234

11-73

11-17

10-67

10-24

9-87

9-58

9-32

4-1

20-2

19-2

18-3

17-3

16-4

15-5

14-7

14-0

13-3

12-6

12-0

11-5

11-0

10-5

10-3

4-2

23-1

22-0

20-9

19-8

18-7

17-6

16-7

15-8

15-0

14-2

13-5

12-8

12-3

11-8

11-3

4-3

26-3

25-0

23-8

22-5

21-3

20-1

19O

18-0

17-1

16-1

15-3

14-6

139

13-2

12-6

4-4

30-1

28-4

26-8

25-3

23-9

22-6

21-4

20-3

19-3

18-3

17-3

16-4

15-6

14-8

14-1

4-5

347

32-5

30-5

28-8

27-3

25-6

24-2

22-9

217

20-6

19-5

18-4

17-5

16-7

16-1

4-6

40-0

37-4

35-0

32-8

:•,•••:»

29-2

27-6

26-1

24-7

23-3

22O

20-9

19-8

18-8

18-1

4-7

46-1

43-1

40-3

37-7

35-3

33-2

31-4

29-7

28-0

26-4

25-0

236

22-3

21-2

20-3

4-8

52-4

48-8

46-8

43-1

40-2

37-8

35-6

33-6

31-6

29-8

28-1

26-6

25-1

23-8

22-7

4-9

00-6

50-1

BS-3

48-8

45-5

42-6

40-0

376

35-4

33-4

31-5

29-8

28-1

26-6

25-2

5-0

71-2

6.3-1

60-6

56-5

52-6

49-1

458

43-1

40-5

38-0

35-6

33-6

31-7

30-0

28-4

5-1

83O

76-4

70-8

65-4

60-6

56-3

52-5

19-3

46-3

43-4

40-4

38-0

35-6

33-6

31-7

5-2

98-8

89-6

81-9

75-6

70-2

65O

60-2

M-l

52-5

48-9

45-5

42-6

39-9

37-4

35-2

5-3

1 1--I

105-8

96-0

87-6

80-4

74O

68-3

63-4

58-8

54-7

51-0

47-7

44-5

41-5

38'9

5-4

111-4

1*4-0

111-2

99-6

91-2

84-0

77-4

71-2

65-7

61-2

56-9

52-9

49-4

46-2

43-5

5-5

—

1814

117-4

105-2

960

87-3

79-3

72-8

67-8

63-2

58-6

54-8

51-4

48-8

5-6

—

IflOO

142-4

126-4

113-4

102-2

9:5O

84-4

77-3

71-1

65-6

60-8

57-2

54-7

5-7

—

i:i!J-2 175-8

154-8

134-2

1196

107-0

97-2

88-4

80-6

74-4

69-4

64-9

61-5

5-8

— ^(io-o L'-I-I;

192-8

163-6

142-8

128-0

114-6

104-0

94-6

86-0

79-6

74-4

70-2

5-9

— :J78-1 284-0

231-5

198-2

171-6

151-5

136-2

123-8

112-8

103-4

94-8

87-5

81-4

6-0

—

—

206-3

186-3

167-5

150O

134-2

121-5

111-0

101-8

92-8

6-1

—

264

232

205

280

160

141

128

116

107

6-2

—

350

297

251

216

188

164

148

132

122

6-3

510

376

308

263

225

196

172

152

138

6-4

889

524

387

313

264

229

200

177

161

6-5

—

—

237

204

184

6-6

___

286

249

220

6-7

__

363

305

268

6-8

_.

485

392

333

6-9

1

—

—

—

—

—

—

—

—

—

—

—

747

510

416

7-0

Tables for Statistician* and Biomctriciana

TAULE XXXIX. To find tie con-elation in errors of fr and &.

Values of -K0,*,.

A

o-oo

0-05

o-io

6-75

0-20

0-25

0-30

0-35

0-40

0-45

0'50

0-55

0-60

0-65

070

0-75

t-o
t-1
ft
e-s

f-4
t-5
f6
t-7
f8
f9
3-0
3-1
• •:
3-3
S-4
3-6
3-6
3-7
3-8
3-9

4-0
4-1
4-2
4-s

;-;
;••

4-ti
4-7
4-s
4-9
5-0

5*|

o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo
o-oo

•570
•557
•5;>1
•650
•551
•554
•557
•557
•556
•550
•549
•534
•524
•51-2
•501
•490
•477
•40::
•450
•438
•422

•706
•685
•07::
•603
•6CO
•659
•662
•.;>:-
•071
•680
•684
•687
•688
•688
•6S6
•681
•676
•670
•662
•654
•645
•630
•608
•580
•540
•481

•770
•765
•728
•719
•712
•706
•706
•710
•710
•724
•738
•744
•746
•747
•748
•747
•745
•741
•736
•720
•713
•702
•682
•658
•628
•590

•823
•798
•771
•765
•752
•745
•742
•744
•750
•760
•774
•781
•786
•788
•790
•790
•788
•784
•779
•770
•760
•748
•733
•712
•688
•657

•863
•838
•814
•799
•787
•776
•773
•773
•7711
•7-7
•796

«oe

•811
•814
•816
•815
•813
•810
•803
•796
•788
•780
•770
•753
•732
•709

•894
•870

•847

•814
•805
•799
•800
•803
•810
•816
•825
•830
•832
•833
•833
•832
•831
•828
•822
•816
•807
•793
•784
•770
•749
•716
•674
•615
•532
•362

•917
•895
•874
•859
•843
•834
•825
•825
•826
•830
•835
•840
•842
•845
•848
•849
•850
•848
•845
•841
•837
•830
•822
•811
•796
•780
•754
•723
•681
•620
•534

•935
•914
•896
•880
•867
•858
•851
•846
•842
•844
•847
•850
•852
•855
•858
•860
•860
•859
•858
•856
•663
•849
•842
•832
•819
•804
•784
•759
•787
•680

•en

•949
•936
•919
•900
•880
•878
•871
•863
•858
•857
•857
•858
•860
•863
•865
•867
•867
•867
•866
•866
•865
•862
•857
•848
•837
•824
•808
•788
•761
•728
•687

•960
•948
•934
•919
•905
•893
•888
•876
•871
•868
•867
•867
•868
•870
•872
•873
•874
•874
•874
•875
•873
•871
•867
•860
•851
•841
•828
•812
•791
•766
•731

•968
•959
•948
•935
•880
•908
•886
•869
•883
•878
•875
•874
•875
•876
•878
•879
•880
•881
•882
•881
•881
•880
•877
•871
•863
•853
•842
•830
•815
•795
•707

•976
•969
•960
•948
•93(5
•924
•913
•902
•893
•887
•883
•882
•882
•882
•883
•884
•886
•887
•888
•889

•KNS

•887
•884
•878
•872
•865
•856
•846
•834
•818
•798
'768

•983
•977
•888

•957
•945
•933
•921
•911
•901
•895
•890

•SMI

•889
•888
•889
•890
•891
•881
•893
•894
•894
•892
•890
•885
•NSI i

•874
•868
•860
•849
•835

•7U!J

•060
•988

•975
•965
•954
•941
•930

•910
•903
•898
•897
•896
•886
•890
•835
•888
•887
•898
•899
•899
•887
•894
•890
•887
•882
•877
•870
•861
•860
•837
•8''0

•881
•888

•ll-u
•i)71
<88I

•949
•938

•:ii>

•91!)

•:)!•-•
•906
•903
•:>< .-
•901
•900
•900
•900
•901
•901
•903
•903
•901
«89
•897
•8M
•880
•886
•879
872
•889
•851

.QQ7

5-t
1-*

—

—

—

—

—

—

—

—

—

—

—

—

•730

•fi7!l

•768
•729

•799
•7fi<»

•810

.711.1

5.L

'60S

•686

•736

•774

•f

•496

•601

'674

•724

1-K

5*f

1-K

,--,

t;./.

6.1

, •

. •

fi'i

° •?

, -

6*1?

fi'7

/J.O

fi-Q

7'f)

Probable Errors of Frequency Constants

TABLE XXXIX.— (continued).
Values of RW,.

A

73

0-80

0-85

0-90

0-95

1-00

1-05

1 -10

1-15

1-20

1-25

1-SO

1-S5

1-40

1-45

i-so

•993

•995

•997

•999

i-ooo

1-000

i-ooo

1-000

i-ooo

1-000

1-000

i-ooo

i-ooo

1-000

1-000

2-0

•989

•991

•994

•996

•998

•998

•999

•999

•999

1-000

1-000

1-000

i-ooo

i-ooo

1-000

•983

•986

•989

•992

•99.')

•996

•997

•998

•998

•999

i-ooo

i-ooo

1-000

1-000

1-000

2-2

•976

•980

•984

•988

•992

•993

•994

•995

•997

•998

•999

•999

•999

1-000

i-ooo

2 '3

•968

•973

•978

•983

•987

•988

•991

•993

•995

•996

•998

•998

•999

•999

i-ooo

2 '4

•958

•965

•972

•977

•982

•985

•988

•990

•992

•994

•996

•997

•998

•999

1-000

2-5

•947

•956

•964

•970

•976

•980

•984

•986

•988

•991

•993

•995

•997

•998

•999

2-6

•937

•947

•957

•!W3

•968

•973

•977

•980

•983

•986

•989

•992

•995

•997

•998

2'7

•928

•949

•!t.V>

•960

•965

•970

•974

•978

•981

•985

•989

•992

•994

•996

2-8

•921

•932

•949

•947

•952

•957

•963

•968

•972

•976

•980

•984

•988

•990

•992

2-0

•915

•923

•931

•937

•943

•948

•954

•960

•966

•971

•975

•979

•983

•986

•988

3-0

•909

•918

•989

•989

•936

•942

•947

•953

•959

•965

•970

•974

•978

•981

•984

3-1

•907

•912

•918

•!'!' I

•930

•936

•941

•946

'952

•958

•963

•968

•973

•977

•980

3-2

•906

•909

•914

•919

•925

•930

•935

•940

•946

•961

•956

•961

•966

•971

•975

3-3

•905

•908

•912

•916

•920

•930

•935

•940

•945

•950

•954

•958

•964

•974

3-4

•904

•907

•910

•914

•918

•989

•926

•931

•936

•940

•944

•948

•952

•958

•965

3-5

•904

•907

•910

•914

•918

'921

•984

•928

•932

•935

'938

•942

•946

•952

•959

3-6

•900

•910

•914

•917

•920

•923

•927

•930

•933

•935

•937

•940

•946

•953

S-7

•900

•908

•911

•914

•917

•920

•922

•925

•928

•930

•932

•934

•938

•941

•948

3-8

•906

•909

•911

•914

•917

•919

•921

•924

•987

•929

•931

•933

•935

•939

•944

3-9

•906

•909

•912

•914

•917

•919

•921

•923

•926

•928

•930

•932

•934

•936

•940

4-0

•905

•908

•911

•914

•917

•919

•921

•923

'02j

•927

•930

•931

•932

•933

•934

•905

•907

•910

•913

•916

•919

•921

•ijj3

'924

•986

•929

•929

•930

•930

•929

4-2

•903

•906

•910

•913

•916

•918

•920

•989

'924

•998

•929

•928

•928

•927

•924

4-S

•900

•904

•9t>8

•912

•916

•918

•990

•999

•923

•926

•928

•987

•927

•925

•922

4'4

•897

•902

•906

•910

•915

•918

•990

'988

•923

•926

•928

•927

•926

•923

•920

4-5

•893

•898

•903

•908

•913

•916

•919

•920

•988

•925

•927

•926

•925

•923

•920

4-6

•887

•894

•900

•905

•910

•913

•917

•919

"921

•984

•926

•925

•925

•923

•922

4-7

•881

•890

•896

•901

•906

•910

•914

•917

•920

•923

•925

•926

•926

•925

•925

4-8

•874

•884

•890

•895

•901

•907

•911

•915

•919

•922

•925

•926

•927

927

•928

4-9

•863

•875

•883

•889

•896

•903

•908

•913

•918

•922

•925

•927

•928

•930

•932

5-0

•851

•864

•875

•882

•890

•898

•905

•911

•917

•922

•925

•928

•931

•933

•936

5-1

•837

•852

•866

•875

•884

•8i>2

•901

•909

•916

•921

•924

•928

•933

•937

•941

5-2

•820

•839

•853

•865

•876

•885

•895

•904

•913

•918

•923

929

•935

•940

•945

5-S

•798

•818

•837

•853

•867

•877

•888

•898

•908

•915

•921

•928

•935

•941

•947

5-4

•764

•792

•817

•837

•854

•867

•880

«90

•900

•910

•918

•925

•933

•940

•947

5-5

—

—

•789

•815

•835

•852

•868

•880

•890

•904

•911

917

•926

•935

•944

5-6

—

•750

•786

•811

•835

•854

•869

•880

•892

•901

•909

•917

•927

•938

5-7

—

—

•701

•748

•783

•811

•835

•852

•866

•879

•890

•897

•905

•915

•928

5-8

—

•640

•700

•748

•781

•810

•828

•846

•861

•875

•883

•892

•901

•913

5-9

•544

•639

•703

•746

•778

•802

•825

•842

•857

•867

•879

•886

•893

6-0

—

—

•741

•769

•796

•820

•837

•852

•866

•872

•873

6-1

—

—

—

—

—

•691

•727

•762

•792

•815

•836

•852

•858

•856

6-2

—

—

•628

•678

•724

•761

•790

•818

•838

•845

•842

6-3

—

—

—

—

—

•526

•606

•675

•724

•763

•793

•818

•831

•834

6-4

—

...

—

•354

•526

•619

•680

•726

•761

•791

•814

•831

6-6

—

—

—

—

—

761

•790

•832

6-6

—

•721

•760

•837

6-7

__

—

•670

•727

•845

6-8

__

•600

•683

•857

6-9

—

—

—

—

—

—

—

—

—

—

— —

—

•468

•602

•876

r-o

B.

10

74 Tables fur Statist ici<ins and Biometrieiaiu

TABLE XL. To Jind the Probable Error of tite distance from Mean, to Mode,

Values of - - iLj.
a

A,

o-oo

0-05

o-io

0-15

0-20

0-K

0-30

0-S5

0-40

O-^S

0-50

0-55

0-60

0-65

0-70

075

0-80

*0

3'54

2.1|t

4.-\f>

—

—

—

—

—

—

—

—

3-03

2-44

8-10

1-80
3 -in

1-68

2.RQ

1-4:!

•'•it;

1-30

1 -KN

: -

1-87

1 >AA

2-76

1 >AA

9'65
o-nft

—

—

—

—

—

—

—

—

—

—

3'91

3-17

2-60

3'7K

s-4

fB

:•'•
f-7
N
::'
3-0
3-1

s-t

3.3
34
3-6
3-6
37
3'*
3-9
4-0
4-1
4*
4-S
4-4
4-6
4-6

V
4-8
4-9
5-0

K-1

1-46
1-35
1-28
1-25
1-23
1-22
1-23
1-26
1-27
1-29
1-30
1-31
1-32
1-31
1-30
1-29
l-L'7

•58
•46
•37
•30
•28
1-27
1-26
1-27
1-28
1-29
1-29
1-30
1-31
1-31
1-31
1-33
1-37

2-07
1-67
1-58
1-50
1-43
1-38
1-34
1-32
1-30
1-28
1-28
1-29
1-30
1-31
1-32
1-35
1-40
1-47
1-58
1-75
1-98
•1-Z1

2-08
1-98
1-83
1-71
1-60
•51
•44
•38
•32
•29
•27
•26
•26
•28
•33
•39
•48
1-C2
1-77
1-97
2-20

2-87
2-60
2-34
2-11
1-90
1-73
1-58
1-48
1-39
1-31
1-25
1-22
1-22
1-25
1-30
1-39
1-50
1-64
1-78
1-95
2-15

4-04
3-42
2-98
2-60
2-27
1-98
176
1-60
1-47
1-37
1-30
1-26
1-25
1-27
1-32
1-40
1-51
1-65
1-79
1-94
2-11

5-21
4-43
3-75
3-17
2-69
2-29
2-00
•75
•55
•45
•37
•32
•30
•32
•36
•42
•53
•65
•78
•93
2-10
2-44
2-93
3-74
5-44
10-66

6-72
5-06
4-12
3-20
2-63
•2-2:}
1-92
1-68
1-54
1-46
1-40
1-37
1-38
1-41
1-46
1-55
1-65
1-76
1-90
2-09
2-40
2-85
3-52
4-64
6'83

-

7'48
5-28
3-84
3-06
2-54
2-13
1-83
1-63
1-54
1-60
1-46
1-46
1-48
1-51
1-57
1-65
1-75
1-88
2-09
2-38
2-78
3-33
4-16
5-53

7-45
4-78
3-58
2-94
2-37
2-03
1-79
1-66
1-61
1-57
1-55
1-56
l -:,s

1-61
1-66
1-75
1-89
2-10
2'3b
2-71
3-16
3-87
4-84

6-65
4-28
3-42
2-72
2-27
2-00
1-83
1-74
1-69
1-65
1-64
1-65
1-66
1-70
1-78
1-92
2-12
2'34
2'67
3-07
3-63
4-37

5-18
3-93
3-12
2-57
2-24
2-03
1-89

•as

•76
•73
•73
•75
•77
1-83
1-96
2-15
2-3U
8-68
3-00
3-45
4-04

6-43
4-52
S-M
8-90

2-51
2-26
2-08
1-97
1-88
1 M

1 -83

!•- .
1-85
1-90
2-01
2-18
2-38
2-64
2-95
3-32
3-79

4'J.fi

8-24
5-50
4-21
3-30
2-88
2-55
2-31
2-14
2-03
1-96
1-94
1-94
1-94
T97
2-07
2'22
2'40
2-63
2'90
3-24
3-62

4*91

10-89
6-76
6-07
4-04
3-36
2-89
8-86
2-34

g-ao

2-11

2 -cm

2-03
2-03
2-05
2-13
2-26
2-42
2-62
2-85
3-11
3-47

3'<l')

8-66
6-22
4-66
3-86
3-18
2-86
8-61
2-43
2-27
2-19
l!-13
2-12
2-13
2-20
2-32
2-45
2-61
2-81
3-05
3-37

I'RI

8-00
B-M
4-50
3-61
8-84
8-95
2-69
2-49
2-36

B-se

2-22
2-23
2'29
2-38
2-48
2-61
2-80
3-03
3-31

3'7d

A-*

6-'*s

51/UL

4*7'*

4-<17

d'9.1

6-3

e.jt

—

—

—

—

—

—

—

—

—

—

—

—

6-84

'*•-'!

6-19

7-(U3

5-60
•7 'fin

5-27

R.OA

4-92
6 -74

»•»

5'S

U*81

10 '8y

8 '87

7-64

6'81

K-fi

5.7

K-g

6-9

K-fl

6'1

6'S

K-H

K-i

o 4
6-5

R'fi

fi-7

6-8

i •,

7'0

Probable Errors of Frequency Constants

TABLE XL— (continued).
Values of - - ^

76

0-85

0-90

0-95

1 -00

1-05

1-10

1-15

1-20

1-S5

1-SO

1-35

1-40

1 -45

1-50

1-20

1-13

1-07

1-02

•97

•92

•87

•83

•80

•76

•72

•68

•64

•60

2-0

1-64

1-49

1-38

1-29

1-21

1-14

1-08

1-03

•99

•94

•90

•86

•82

•78

2-1

2-22

1-97

1-80

1-66

1-54

1-45

1-38

1-32

1-26

1-20

1-14

1-08

1-02

•96

2-2

3-10

2-63

2 -46

2-15

1-98

1-85

1-75

1-67

1-58

1-50

1-41

1-32

1-24

1-16

2-3

—

3-72

3-20

2-81

2-56

2-36

2-22

2-09

1-96

1-84

1-73

1-62

1-51

1-39

2-4

—

—

—

3-94

3-40

3-08

2-87

2-66

2-47

2-29

2-12

1-96

1-80

1-64

2-5

—

—

—

—

—

4-32

3-82

3-48

3-14

2-86

2-60

2-34

2-10

1-91

2-6

—

—

—

—

—

—

4-82

4-20

3-66

3-18

2-78

2-46

2-19

2-7

—

—

—

—

—

5-73

4-82

3-88

3-28

2-87

2-49

2-8

—

—

—

—

__

6-63

4-81

3-94

3-31

2-80

2-9

—

—

—

—

6-15

4-80

3-79

3-12

3-0

4. 29

3 "48

S'l

3-88

S'2

6-80

8'S

5-38

6 '55

S'A

4-21

4-95

c-oo

7-33

9-36

12-16

_

_

_

—

<j jf
S-5

3 '74

4-34

5-12

6-03

7-17

8-80

11-52

. ,

—

—

S'6

3-34

3-80

4-32

5-01

5-78

6 -65

8-28

11-12

—

—

3-7

3-00

3-3')

3-74

4'23

4-72

5-41

6-36

8-00

10-22

—

—

8-8

2-74

3-00

3-32

3-6(5

4-04

4-50

5-18

6-16

7-43

9-32

—

—

—

3-9

2T)5

2-77

3-02

3-29

3-60

3-98

4-50

5-12

5-92

6-91

8-00

9-23

11-08

—

4-0

2-42

2-60

2-79

3-00

3-27

3-61

4-06

4-61

5-20

5-90

6-80

7-86

9-48

11-58

4-1

2-35

2-50

2-67

2-86

3-09

3-38

3-71

4-13

4-60

5-15

5-86

6-72

7-70

9-10

4-2

2-35

2-48

8-«S

2-77

2-96

3-21

3-49

3-82

4-18

4-59

5-14

5-82

6'C6

7-59

4-s

&

2-39

2-50

2-61

2-73

2-89

3-10

3-32

3-55

3 '83

4-16

4-60

5-16

5-82

6-60

4'4

8-48

2-53

2-63

2-74

2-87

3-02

3-20

3-39

3-61

3-87

4-21

4-66

5-18

5-86

4-5

2-52

2-60

2-69

2-79

2-89

3-01

3-15

3-31

3-48

3-67

3-95

4-31

4-77

5-36

4-0

2-64

2-69

2-77

2-85

2-93

3-02

3-13

3-25

3-39

3-55

3-78

4-10

4-50

4-99

4"?

2-81

2-82

2-86

2-93

3-00

3-08

3-17

3-27

3-38

3-50

3-68

3-94

4-26

4-66

4'8

3-02

3-03

3-04

3-08

3-12

3-1C

3-22

3-30

3-39

3-50

3-63

3-81

4-07

4-35

4'9

3'28

3 -20

3-25

3-26

3-28

3-31

3-34

3-39

3-44

3-51

3-61

3-73

3-90

4-09

5'0

3-64

3'55

3-51

3-49

3-47

3-47

3-48

3-49

3-52

3-56

3-61

3-68

3-77

3-90

5-1

4-04

3-90

3-81

3-74

3-68

3-65

3-63

3-62

3-61

3-62

3-63

3 '65

3-69

3-76

5-2

4-60

4-35

4-18

4-05

3-95

3-88

3-81

3-75

3-72

3-70

3-69

3-68

3-69

3-70

5-S

5-33

4-98

4-71

4-52

4-37

4-24

4-12

4-01

3-92

3-85

3-79

3-75

3-73

3-72

5-4

6-21

5'74

5-36

6-08

4-86

4-66

4-48

4-32

4-20

4-09

3-99

3-92

3-87

3-82

5-5

—

6-69

6-27

5-83

5-49

6-19

4-94

4-73

4-56

4-42

4-30

4-18

4-11

4-04

5-6

—

8-11

7-48

6-82

6-32

5-90

5-55

5-27

5-03

4-84

4-68

4-54

4-43

4-35

5-7

—

10-18

9-11

8-12

7-45

6-85

6-35

5-95

5-62

5-37

5-16

5-00

4-87

4-76

5-8

—

13-53

11-44

9-84

8-71

7-94

7-32

6-82

6-45

6-13

5-85

5-61

5-43

5-29

5-9

—

19-95

14-26

11-92

10-48

9-38

U'KA

8-55

1 A.OO

7-89
9 .01

7-38

8-r-i

6-95

8.f\n

6-62

17 .CO

6-33

7.1 r.

6-10
6 -ft?

5-90
6 "60

6-0
S'l

—

__

__

_

O4

14-83

1U iO

12-55

•jl

11-19

bz

10-24

Uo

9-40

( •'•'

8-64

It)

8-08

01
7-68

7-36

6-2

—

—

—

—

19-65

15-85

13-69

12-21

11-01

10-02

9-19

8-65

8-22

6-3

—

—

—

28-03

20-85

17-09

14-56

12-84

11-45

10-45

9-69

9-11

0-4

—

—

—

—

47-99

28-04

21-30

17-44

15-07

13-20

11-90

10-83

10-07

6'5

W-9

12"9

12-4

6'6

—

_

_

_

_

_

_

^_

3)

17-2

15-6

14-7

6-7

21 '8

19'6

18-3

6 '8

9Q-4.

26 "0

24'1

6'9

2(7 ^t

4TO

37 "8

34'9

TO

•±0 v

10—2

70

Tables for Statistician* (ti«l li'«>in< // i< i

TABLE XLL Tu find the Probable Error of the tikewneu sk.
Values of

A

o-oo

0-05

o-io

0-75

o-to

0-*5

0-50

0-55

0-40

0-45

0-60

0-M

0-60

0-66

0-70

0-75

3*41

2 "-ii

S.^O

2.10

1 VI

1*79

, "

3-fi7

3-ii-'

S'&X

9.r.(\

6'ri7

4.1 A

S x

1 •» • 1

1.70

2-AA

1 -1C

1 • " 1

2*02

* 4
t-5
t-6
t-7
t-8
t-9
3-0

s-i
s-e

3-3

3-4
3-B
3-8
3-7
3-8
3-9
4-0
4-1
•>•-
4-3

4-4
4-r>
4-a

4-7
4-*
4-9
5-0

- . ;

1-35
1-28
1-25
1-23
1-22
1-23
1-25
1-27
1-29
1-30
1-31
1-32
1-31
1-30
l-ffl
1-27

1-41
1-30
1-25
1-22
1-20
1-21
1-22
1-23
1-25
1-27
1-29
1-30
1-31
1-32
1-34
1-30

1-62
1-43
1-31
1-24
1-flO
1-20
1-21
1-22
1 -L":i
1-24
1-25
1-26
148
1-30
1-33
1-38
1-46
1-58
1-75
1-95
2-26

2-02
1-75
1-52
1-36
1-28
1-2.J
1-23
1-22
1-21
1-21
1-21
1-22
1-24
1-27
1-30
1-30
1-45
1-67
1-74
1-94
2-19

2-80
2-18
1-84
1-69
1-43
1-34
1-27
1-23
1-21
1-20

I'L'O

1-20
1-20
1-23
1-88
1-35
1-45
1-67
1-73
1-92
2-13

4-06
2-82
2-29
1-89
1-63
1-48
1-36
1-2'J
I'M
1-23
1-22
1-22
1-23
1-25
1-28
1-35
1-44
1-66
1-71
1-90
2-09

6-08
.'{ -.;;,
2-85
2-21
1-8C
1-62
1-50
1-40
1-33
•29
•28
•26
•27
•28
•30
•36
•44
•55
•70
•88
2-07
2-48
3-11
3-78
5-48
11-12

4-90
3-G8
8-70

2-19
1-84
1-G7
1-53
1 ••! 1
1-38
1-35
1-32
1-32
1-33
1-34
1-39
1-46
1-56
1-C9
1-85
2-OC
2-42
2-93
3-53
4-6G
6-96

4-84
3-36
2-CO
2-12
1-88
1-70
1-68
1-49
1-43
1-40
1-39
1-38
1-39
1-43
1-60
1-59
1-70
1-86
2-05
2-37
2-82
3-36
4-17
5-52

4-36
3'33
2'49
2-16
1-90
•74
•61
•54
•60
•47
•46
•46
•48
1-54
1-62
1-72
1-87
2-06
2-35
2-73
3-21
3-87
4-82

4-30
3-00
2-60
2-17
1-94
1-78
1-68
1-61
•56
•54
•53
•66
•60
•(J7
1-76
1-90
2-07
2-33
2-65
3-08
3-63
4'36

3-76
3-06
2-58
2-^7
2-04
1-87
1-75
1-67
1-63
1-61
l-t;:;
1-67
1-72
1-81
1-93
2-08
2-31
2-60
2-97
3-44
4-02

4-62
3-76
3-17
2-73
2-36
2-09
1-91
1-80
1-70
1-71
1-72
1-74
1-78
1-86
1-97
2-11
2-31
2-57
2-89
3-30
3-77

4.JA

(j-(l-2
4-72
3'97
3-28
2-75
2-37
2-13
1-98
1-88
1-82
1-81
1-82
1-85
1-92
2-01
2-15
2-32
2'54
2-82
3-17
3-68
4-n;

H-lsi

c-io

4-87
3-88
3-18
2-68
^•39
:M!»
^'04
1-96
1-92
1-91
1-93
1-98
2-06
2-19
2-33

•2-;>-2

•2--G
3-07
3-45

4-Oft

8-08
.VS{
4-63
3-63
2-98

•^•o;.

2-40
2-22
2-12
2-04
2-03
2-02
2-06
2-12
2-23
2-35
2-50
2-71
3-01
3-37

3.Q7

- - /

6.QQ

1-O9

4-79

4 -4ft

r.e

6-K 1

6.1 U

1-fift

r.oft

yr.i

9*91

?-7fi

6 .on

6iM

' -/

T'/I

Id 'Kit

in-f:7

8*87

7*71

T-/I

£•7

r.o

fi-Q

6n

i'-l

fi*9

/;•.<

/?. i

*» 4
fi-%

r.'ti

fi"T

6 -ft

/?*0

7*n

Probable Errors of Frequency Constants

77

TABLE XLI— (continued).
Values of

0-80

0-8ij

0-90

0-95

1-00

1-05

1-10

1-15

1-20

1-25

1-SO

1-S5

1-40

1-4B

1-60

1-59

1-48

1-39

1-30

1-24

1-19

1-14

1-10

1-06

1-02

•99

•95

•91

•87

•83

2-0

2-20

1-95

1-80

1-68

1-58

1-52

1-47

1-42

1-37

1-32

1-26

1-20

1-15

1-10

1-05

2-1

3-22

2-65

2-29

2-08

1-98

1-91

1-84

1-78

1-72

1-66

1-59

1-52

1-45

1-38

1-31

2-2

6-23

3-80

3-04

2-75

2-53

2-40

2-30

2-21

2-12

2-03

1-94

1-85

1-76

1-67

1-58

2-3

—

—

4-29

3-64

3-31

3-12

2-94

2-78

2-63

2-49

2-36

2-23

2-10

1-98

1-86

1-4

—

—

—

—

4-77

4-27

3-84

3-55

3-29

3-06

2-85

2-66

2-49

2-32

2-15

2-5

—

—

—

—

—

—

*n

5-00

4-39

3-94

3-54

3-20

2-93

2-67

2-44

2-6

—

—

—

—

—

—

—

6-25

5-20

4--Ui

3-88

3-42

3-08

2-74

8-7

—

—

—

—

—

—

—

—

7-05

5-08

4-78

4-03

3-54

3-05

2-8

—

—

—

—

—

—

—

—

—

7-45

6-00

4-85

4-11

3-37

S-'J

—

—

—

—

—

—

—

—

—

7-80

6-05

4-77

3-70

3-0

5*r»7

J.-1O

0.7

7-00

..' /

*± i\j

4.T.Q

<j j.

O.i)

i \t\j
d-30

6-24

_.

_^ .

_

_

. _

—

\JO

O i*

3-3

4-16

4-71

5-60

—

—

—

—

—

—

—

—

—

—

—

3-4

3-38

3-89

4-59

5-58

6-85

8-38

11-48

—

—

—

—

—

—

—

S-5

2-95

3-38

3-99

4-66

5-50

6-48

7-55

9 -US

—

—

—

—

—

—

s-a

2-G7

3-05

3-49

3-97

4-52

5-22

6-00

7-36

9-42

—

—

—

—

—

S-7

2-44

2-75

3-08

3-43

3-80

425

4-78

6-64

7-08

9-OS

—

—

—

—

i'vs'

2-29

8*3

2-79

3-07

3-38

3-70

4-15

4-77

5-62

6 -89

8-76

—

—

—

$-y

2-20

2-38

2-58

2-82

3 -06

3-33

3-fi!)

4-14

4-77

5-50

6-42

7-40

8-57

10-12

4'0

2-16

2-31

2-47

2-65

2-85

3-09

3-34

3-72

4-15

4-61

5-14

5-84

6-80

8-44

11-00

4'1

2-14

2-26

2-40

2-55

2-71

2-88

3-10

3-37

3-67

4-03

4-44

5-04

5-94

7-12

8-67

4-2

2-16

2-27

2-38

2-50

2-65

2-80

2-9G

3-14

3-37

3-65

4-01

4-55

5-28

6-18

7-21

4-s

2-20

2-30

2-41

2-52

2 '64

2-76

2-89

3-04

3-23

3-48

3-78

4-22

4-78

5-44

6 -26

4-4

2-29

2-35

2-44

2-53

2-«3

2-75

2-88

3-02

3-20

3-40

3-G5

3'97

4-40

4-91

5-50

4-6

2-39

a -43

8-48

2-56

2 -06

2-77

2-89

301

3-16

3-34

3-55

3-79

4-16

4-55

6-10

W

2-52

2-54

2-58

2-63

2-71

2'80

8-90

3-01

3-14

3-29

3-47

3-67

3-90

4-28

4-72

4'7

2-70

2-72

2-74

2-78

2-83

2-89

2-97

3 -06

3-10

3-28

3-41

3-58

3-80

4-06

4-42

4-s

2-98

2 '97

2-96

2-97

3-00

3-04

3-08

3-14

3-21

3-30

3-40

3-53

3-68

3-88

4-10

4-9

3-31

3-25

3-21

3-20

3-21

3'22

3-24

3-26

3-31

3-36

3-43

3-51

3-60

3-73

3-96

o-o

3-75

3-64

8<M

3-49

3-45

3-42

3-41

3-41

3-43

3-44

3-4C

3-51

3-57

3-65

3-78

5-1

4-28

4-10

3-96

3-85

3-76

fr«

3-63

3-60

3-57

3-55

3-53

3-53

3-55

3-60

3-07

5-2

4-93

4-69

4-48

4-29

4-13

4-02

3-92

3-84

3-70

3-08

3-63

3-60

3-58

3-59

3-62

r.»o
o o

5-78

.VI 2

B-08

4-80

4-56

4'40

4-26

4-13

4-00

3-89

3-80

3-73

3-68

3 -CO

3-03

5-4

6-tt

G-32

5-82

6-40

5-07

4-84

4-64

4-46

4-30

4-17

4-00

3-95

3-87

3-80

3'75

5-5

—

—

6-75

G-22

5-79

6 -40

5-19

4-97

4-77

4 -CO

4-44

4-30

4-19

4-10

4-01

S-(i

—

—

8-15

7-30

6-73

6-26

5-91

5-61

5-34

6-10

4-90

4-73

4-59

4-47

4-38

5-7

—

—

10-20

8-76

7-98

7-20

6-7C

U-34

5-68

5-44

5-24

5 -06

4-91

4-78

5-8

—

—

13-53

10-83

9-66

8-71

7-90

7-28

6-78

6-40

6-10

5-84

5-62

5-44

6-28

5-9

—

—

19-96

14-30

12-02

10-51

9-39

8-56

7-90

7-39

6 -90

6-61

6-33

6-10

5-89

o-o

—

—

—

—

—

—

11-64

10-26

9-31

8-56

8-03

7-53

7-15

6-82

6-54

6-1

—

—

—

—

—

—

14-83

12-55

11-19

10-13

9-30

8-64

8-08

7-63

7-24

6'»

—

—

—

—

19-65

15-85

13-09

12-21

11-01

10-02

9-19

8-55

s-oi

6-S

—

—

—

—

_

28-03

20-85

17-09

14-56

12-84

11-45

10-45

9-60

8-92

0-4

—

—

—

47-99

28-04

21-30

17-41

15-U7

13-20

11-90

10-83

10-07

t>-5

14*2

12-9

12'4

6-ii

17 -2

15-6

14'7

G'7

21-8

19-6

18-3

<J-8

29-4

26-0

24'1

(i-9

—

—

—

—

—

—

—

—

—

--

—

—

43-0

37-8

34-9

r-o

-

78

Tables for Statisticians mtd Biometridans

TABLE XL1I. To give values of ft, ft, ft and ft in terms

TABLE XLII(«X

Values of ft.

ft

t-o

ft

3-0

5-5

4-0

4-5

6-0

6-5

6-0

6-6

TO

o-o

0

0

0

0

0

o-i

0-48493

0-48871

0-94286

1-80688

1-64906

2-14375

o-i

0-91585

1 -32958

1-78182

2-37049

3-10000

4-01176

0-3

149673

1-85270

2-53043

3-3C094

4-38305

5-65000

7-2368

0-4

1*63801

2-34286

3-20000

4-84478

6-52277

7-09474

9-0462

0-6

1-94118

2-78126

S-8000 '

6-53846

8-37500

10-6364

in:

'••> 3-17350

4-33846

5-80178

7-44706

9-51429

12-0414

18*1686

"•;

2-44<il5

3-62441

4*88988

8*38987

^-M-202

10-53182

13-2885

16-6624

0-8

8-66638

3*88880

5-25714

6*96686

8-99462

11-44375

14-4000

17-8988

0-9

2-83917

4-12064

5-64828

7-47438

9-66454

12-L't

15-3940

19-1768

23-7791

1-0

3-00000

4-36842

6-00000

7-94121

10-24999

13-00000

164857

20-2308

25-0000

1-1

3-13980

4*69061

6-31613

8-36246

10-78796

13-66538

17-0877

21-1750

26-0857

32-0328

1 :

3-26038

4-78812

6-60000

8-74286

11-27443

14-26667

17-8105

22-0225

27-054'' 33-H.IS2

/-.

3-36330

4*96860

6-86464

9-08619

11-71461

14-81071

18-4639

22-7851

87-9817

34-063:)

1-4

3-45000

5-11589

7-08235

9-39582

12-11304

15-30345

19-0536

88-4787

88-7000

34-9164

42-3013

i-r.

3-62174

5-25000

7-88671

9-67501

12-47368

15-75000

19-5864

24-0937

29-4000

35-6786

43-1538

TABLE XLII(6).

Values of ft.

ft

s-o

S-5

s-o

3-5

4-0

4-5

6-0

5-6

6-0

6-5

7-0

o-o

5-00000

8-92856

15-0000

23-7288

31-0000

o-i

5-27356

9-41054

15-7973

25-7430

41-7660

69-3682

0-8

5-44361

9-75086

16-2648

26-4018

42-5000

69-4796

0-3

5-53293

9-86224

16-4907

26-6520

42-5613

68-6776

H4-i7:i-J

0-4

6-66898

9-91072

16-5385

26-6144

42-1807

67*0886

]o'.l-|.-.3.l

0-5

5-53802

9-87751

16-4545

26-3742

41-5076

65-2679

104 ••i<;;>:>

o-o

5-47791

9-81734

16-2732

26-1077

10-6463

63-2707

99-6442

162-125

0-7

5-38824

9-03895

16-0200

26-6026

39-6623

61-1946

95-0525

161-863

0-8

5-27513

8-46991

15-71 13

84-9478

38-6061

59-1016

90-7 143

141-707

0-9

5-14137

15-3706

84-3468

37 •:>' KID

57-0279

86-6331

133-840

210-995

1-0

6-00000

9-02746

16-0000 L'3-7495

36-3971

65-0000

82-80^-2

186-664

196*000

1-1

4-84537

8-78075

14-6111

23-0744

:i5--s:{5

53-0316

79-2091

118-839

181-299

286-374

1-2

4-68319

8-53522

14-2106

22-4107

34-1811

51-1309

76-8392

112-653

169-394

2»il--136

1-S

4-51562

8-27700

21-7535

33-0971

49-3447

72-6772

107-016

158-930

240-845

1-4

4-34440

8-01454

13-3931

21-1002

32-0367

47-5471

69-7076

101-850

149-643

883-304

3i::-i 17

1T>

4-17097

7-76000

12-9832

20-4546

31-0037

45-9038

66-9117

97-1 1 -1

141-333

208-129

313*704

\

ft

Probable Errors of Frequency Constants

79

of & and /3a on the assumption that the Frequency falls into
one or other of Pearsons Types.

TABLE XLII(c).

Values of y8f.

&

2-0

8-5

5-0

5-5

4-0

4-5

6-0

6-5

6-0

6-5

7-0

o-o

0

0

0

0

0

o-i

1-99086

4-39480

9-3207

19-9714

45-9387

128-529

0-2

3-59438

8-03374

16-5960

34-7825

76-6000

193-361

0-3

4-86677

10-68765

22-2196

46-7142

97-2263

228-104

668-284

0-4

5-85929

12-85477

26-5187

53-7090

111-0237

246-506

650-398

0-5

6-51704

14-51540

29-7545

59-4655

120-0543

255-295

614-633

0-6

7-17383

16-7892

32-1362

63-9266

125-6629

258-147

581-205

1618-635

0-7

7-56616

16-6546

33-8306

66-2045

128-8283

257-225

550-107

1368-373

0-8

7-81963

17-2668

34-9714

67-8804

130-2010

253-872

521-257

1196-612

0-9

7-95777

17-6667

35-6658

68-7533

130-2587

248-937

495-375

1068-877

2769-42

1-0

8-00000

17-8291

36-0000

69-0644

129-3434

243-000

469-637

968-318

2280-00

1-1

7-96881

17-8472

36-0437

68-7730

127-7158

236-441

446-547

886-541

1945-69

5313-80

1-2

7-86015

17-7503

35-8535

68-1357

125-5684

229-524

425-062

818-040

1700-98

4135-56

1-8

7-70375

17-5396

35-4754

67-2181

123-0362

222-562

405-663

759-486

1512-94

3388-18

1-4

1-60868

17-2423

34-9407

66-0678

120-5142

.,, (.ggg

386-347

708-620

1363-20

2870-08

7265-31

l-o

7-26808

16-8768

34-2983

64-7210

117-2460

207-227

368-843

663-926

1240-65

2488-62

5719-68

A

TABLE XLII (d).
Values of /S«.

&

2-U

2-5

s-o

S-5

4-0

4-5

6-0

6-5

6-0

6-5

7-0

o-o

14-0000

390649

105-000

290-678

868-015

o-i

16-4616

45-7741

124-835

355-608

1243-832

10228-33

m

17-72M

50-2472

132-998

369-894

1190-700

6204-69

O'J

18-1764

51-0927

134 ">. 15

361-909

1089-739

4485-38

107697-95

0-4

1H-W67

60-2458

131-337

344-886

977-606

3471-87

25413-18

0-6

17-5474

48-78M

126-107

323-447

877-884

2792-19

13737-63

0-6

16-8060

M-8668

119-601

303-252

784-431

2303-07

9048-43

119230-33

0-7

15-9787

44-3106

112-492

277-658

701-500

1934-79

6534-78

40994-77

0-8

15-0148

41-7081

105-200

255-716

628-450

1648-52

6045-80

22660-09

U".l

14-0113

3!)-H!Mii;

97-984

235-072

664-277

1420-51

4024-45

14836-90

137288-7

1-0

13-0000

36-4119

91 -000

216-137

507-894

1235-50

3286-65

10612-25

57584-9

1-1

12-0030

33-7916

84-339

198-263

456-575

1083-04

2741-39

8135-91

33078-5

797653-2

1-2

11-0354

31-3418

78-047

181-987

414-455

955-78

2322-13

6314-06

21891-8

155693-9

1-3

10-1070

28-9776

72-146

167-142

375-834

848-97

1994-05

6108-55

15690-2

75009-7

1-4

9-2240

26-7355

66-637

153-582

342-057

765-79

1726-18

4219-50

11846-9

44891-9

565740

rr,

B-M09

24-6268

61-512

141-477

310-076

676-32

1608-92

3544-82

9281-2

30280-3

180793

A

80

Titbit* for tifolftfiioJiMU and Biomttrieiaiu

TABLE XLIIL

Probable Error of Criterion *,. Values of i/N'S,., for values of 0,, ft,

A

•00

•05

•10

•15

•to

•K

•30

•35

'40

•45

•50

•55

•>;n

•65

•70

t-o

•000

•242

•332

•399

•4M

•645

•582

•631

•671

•716

•768

£06

•854

•899

t-1

•000

•271

•367

•430

•488

•521

•557

-,,,,

•678

•717

•760

•NX)

•H43

•890

ft

•000

•310

•415

•480

•527

•597

•626

•656

•691

•728

•767

•sir,

•900

••3

•355

•477

•650

•606

•660

•678

•700

•725

•7.r,3

•790

«M

•895

*4

,., ,

•417

•MO

•C42

•681

781

•7.-,!)

•770

•7-7

•806

•830

•867

•884

•914

t-5

•000

•500

•ti!i7

•771

•818

•841

•855

•807

>7:f

•881

«ei

me

1117

•

•000

,;,..

•840

•946

roi

1-03

1 iM

1-02

101

n»

1-00

1-00

1-00

1-00

1-Oti

5-7

•000

1-04

1-30

i •:»:>

1-34

1-33

1-30

1-27

1-24

1-19

MB

H2

M<)

1-09

108

t-s

•000

1-83

1-97

1-98

1 -!»3

1-84

1-74

1 '(II

1-55

1-47

1-40

1 33

147

1-23

1-20

g-9

•000

3-:>l

3-71

3-42

3-00

2-66

2-42

2-H-2

H»6

1 -S!l

1-73

1-61

1 -:i I

1-46

1-39

3-0

—

18-8

9'89

6-94

5-ao

4-30

••a

3-10

2-73

2-43

2-18

2-00

I'M

1-71

1-61

3-1

•ooo

—

62-0

20-5

8-47

6-3(3

5-tt

4-42

378

3-34

2-96

2-fi<5

2-41

2-20

2-01

3-S

•000

7-82

91-7

—

49-7

2<> -2

12-2

8-48

(Ml

5-09

4-28

3-71

8-81

1-86

I-M

3-3

•Ol>0

2-99

11-7

70-2

—

142

32-4

15-4

10-8

8-56

8-98

5-62

1-68

3-98

3-4i;

3-4

•ooo

1-82

!•-,,

13-8

65-5

—

344

182

28-9

1(5-8

11-8

8-69

6-88

5-60

4-76

3-5

•ooo

1-43

2-89

6-43

15-8

60 -G

380

—

127

16-5

25-1

16-1

11-5

8-55

6-91

3-6

•ooo

1-17

2-18

4 'OH

8-00

17-2

40-5

215

—

277

76-4

3.V4

22-4

15-8

11-5

3-7

QQ ,

1-04

1-79

3-08

5-18

9-36

L'l-i;

44-3

155

—

767

133

54-6

28-6

19-2

3-8

too

•979

1-54

2-54

4-09

6-33

14-3

26-2

44O

126

h:V,

—

230

80-0

40-8

3-9

•ooo

•920

1-41

2-20

3-32

4-91

8-84

13-1

20-5

38-2

105

448

—

—

125

4-0

, 10

•869

1-35

2-00

2-83

4 -OS

5-98

9-08

137

22-3

41-4

98O

296

—

—

4-1

—

—

1-30

1-94

2-60

3-61

5-08

7-25

10-4

15-8

24-4

40-5

78-7

216

—

4'*

—

—

1-33

1-94

2-58

3-43

4-54

6-09

8-49

11-8

16-8

25-3

39-4

67-2

169

4-3

—

—

1-41

2-01

2-63

3-37

4-20

5-41

7-19

952

12-8

17-8

262

45-2

72-2

4'4

—

—

1-58

2-15

2-74

3-39

4-20

5-18

6-58

8-1 56

109

14-6

20-2

29O

JO-4

—

—

1-81

2-32

2-94

3-69

4-36

5-29

6-45

7-76

9-85

12-4

16 1

21-1

29-9

4*

—

—

—

—

—

—

4-90

5-63

6-60

7-84

9-20

in

136

17-2

141

4-7

—

—

—

—

—

—

5-87

6-46

7-11

7-97

8-96

10-2

12-2

15-3

20-0

4-8

—

—

—

—

—

—

7-43

7-61

7'94

8-41

9-16

10-1

12-0

14-3

1 7-6

4-0

—

—

—

—

—

—

10-1

9-45

9-08

9-36

9-80

10-6

12-0

13-7

15«

-•••>

—

1.V1

11-3

10-4

10-4

108

11-6

12-4

13-6

1 5-4

5-1

—

—

—

—

—

—

—

—

—

—

—

—

13-5

140

15-3

c.m

16'3

15'6

[g-o

O *
f .•

18'3

17'8

[77

O v
t'i

22'8

20-6

20-0

O 4

5-5

__

._

__

—

—

30-6

260

23-3

5-6

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

s-r

—

—

—

—

—

—

—

—

—

—

—

— '

—

—

—

5-8

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

(i-0

—

—

—

—

—

—

—

—

—

—

—

—

—

—

6-1

—

—

—

—

—

—

—

—

• -

—

—

—

—

—

—

—

—

—

—

—

—

6-3

—

—

—

—

—

—

—

6-4

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

6-6

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

ti-7

—

—

—

—

—

—

—

6-8

—

—

—

—

—

—

—

—

—

—

6-9

—

—

—

—

—

—

—

7-0

— —

~~

—

—

—

•—

—

—

—

—

—

~~~

^~

"~~" *

~

Probable Errors of Frequency Constants 81

TABLE XLIII— (continued).
Probable Error of Criterion *„. Values of ViVS«,/or values of ft, /93.

A

•75

•so

•8->

•90

•95

1-00

1-05

1-10

1-15

1-20

1-25

7-30

1-S5

1-40

1-45

1-50

•949

1-00

1-06

1-12

1-18

1-25

1-31

1-39

1-46

1-56

1-64

1-75

1-86

1-97

2-11

2-25

2-0

•937

•992

1-05

1-10

1-16

1-22

1-28

1 -3")

1-42

1-49

1-58

1-67

1-77

1-88

2-00

2-12

2-1

•936

•OS"

1-04

1-09

1-14

1-19

1-25

1-31

1-37

1-43

1-51

1-59

1-69

1-79

1-89

1-99

%-2

•939

fles

1-04

1-08

1-12

1-17

1-22

1-21

1-32

1-38

1-45

1-53

1-62

1-70

1-79

1-87

2-3

•9.~>o

•990

1-03

1-07

1-11

1-15

1-19

1-24

1-29

1-34

1-40

1-47

1-55

1-62

1-69

1-77

2-4

•978

•998

1-03

1-07

1-10

1-14

1-18

1-22

1-27

1-32

1-37

1-43

1-49

1-55

1-61

1-68

2-5

1-01

1-03

1-06

1-09

1-12

1-15

1-18

1-21

1-26

1-31

1-36

1-41

1-46

1-51

1-57

1-63

8-fi

1-07

1O8

1-09

1-11

1-1 I

1-17

1-19

1-21

1-25

1-30

1-35

1-39

1-43

1-48

1-53

•59

2-7

1-17

1-15

1-16

1-17

1-18

1-19

1-21

1-23

•27

1-31

1-35

1-39

1-43

1-47

1-52

•58

%-s

1 33

1-28

1-26

1-26

1 -2.r.

1-25

1-25

1-27

•29

1-32

1-35

1-39

1-43

1-47

1-52

•57

%•:>

1-63

1-47

1-42

1-38

1-36

1-34

1-34

1-34

•35

1-36

1-38

1-40

1-43

1-46

1-51

•55

S-<>

1 -.--..-,

1-72

i-ei

1-56

1-52

1-49

l'4(i

1-44

•43

1-43

1-43

1-44

1-45

1-47

1-61

•57

3-1

2-34

2-13

i-oc

1-83

1-75

1-68

1-62

1-57

•54

1-52

1-51

1-50

1-50

1-51

1 -54

•58

3-2

3-03

UN

2 -44

2-23

2-06

1-94

1-85

1-77

•72

1-67

1-63

1-60

1-59

1-58

1-59

•61

s-s

4-08

3-50

B-oa

2-74

2-49

2-29

2-14

2-02

•92

1-84

1-78

1-73

1-70

1-68

1-68

1-69

S-J,

6-72

4-75

4-12

3-56

3-15

2-83

2-59

2-38

2-i.'2

2-09

1-99

1-91

1-85

1-81

1-79

1-79

s -a

8-85

6-90

5-71

4-79

4-15

3-68

3-22

2-90

2-67

2-49

2-31

2-20

2-09

2-00

1-06

1'92

3-6

14-2

10-4

8-22

6-69

5-70

4-81

4-17

3-67

3-31

3-05

2-81

2 -62

2-45

2-29

2-19

2-10

3-7

:>4-9

18-1

13-8

10-1

8-16

6-71

6-76

4-92

4-28

3-84

3-48

3-16

2-91

2-68

2-50

2-36

S-S

30-5

L'1'7

15-8

12-2

10-0

8-17

6-75

.-.-:.-,

5-00

4-43

3-93

3-52

3-18

2-91

2-72

8-9

242

86-6

48-5

30-0

20-9

15-4

12-0

9-61

7-86

6-60

5-67

4-90

4-34

3-88

3-47

3-19

4-0

—

374

127

62-9

38-3

26O

18-3

14-4

11-5

9-15

7-53

6-45

5-66

4-98

4-38

3-80

4-1

—

—

—

200

91-0

51-6

32-7

23-4

17-4

13-4

10-6

8-70

7-50

6-48

5-52

4-61

4-2

144

478

—

—

314

112

70-0

42-9

29-1

21-3

16-1

12-7

10-3

8-55

7-24

6-25

4-ts

62-8

13.r)

—

—

—

580

192

93-9

S5-8

37-1

26-5

19-8

1.V4

12-1

9-74

8-26

4-4

68-3

119

280

—

—

—

180

126

72-8

46-4

32-2

23 '9

18-3

14-6

11-8

4-5

323

44-7

62'7

99-6

240

742

—

—

—

181

91-0

58-9

41-0

30-0

22-3

16'9

4-6

26-2

33'H

i<;-i

68-0

105

240

532

—

—

260

128

76-7

50-4

36-0

26-6

4-7

21-6

27'1

35-0

47'3

66-8

104

182

413

—

—

—

408

172

99-3

63-0

44-8

4-8

189

22-6

26'9

33-7

44-0

61-5

84-2

115

337

—

—

—

—

249

140

80-8

4-0

17-8

207

24-6

30-1

37-8

48-9

66-0

97-0

K.7

286

—

—

—

—

—

172

5-0

17-2

20O

23-0

27-1

32-5

40-6

51-5

69-2

9!) -8

147

253

669

—

—

5-1

17-3

19-3

21-7

24-8

29 -r,

35-4

43-1

54-7

70-6

94-6

138

216

—

—

—

—

5-2

18-0

19-7

21-4

23-9

27-1

31-5

37-0

44-2

54-5

69-3

93-2

132

205

380

—

—

5-3

19-6

21-6

23-3

25-6

28-6

32-8

380

44-8

54-0

68-6

94 '0

130

185

—

5-4

2:!3

220

tt-fi

23-6

25-2

27-5

30-5

34-3

39-4

45-4

54-7

67-3

80 -5

116

169

275

5-5

—

—

—

24-5

25-8

27-6

29-8

32-2

35-6

39-6

44-8

51-4

63-2

83-0

118

168

5-6

—

26-8

27-8

28-8

30-0

31-5

33-8

36-5

39-8

44-4

53-1

67-0

86-3

116

5-7

—

—

—

30-7

30-8

31-0

31-3

32-0

33'4

35-3

38-2

42-2

48-4

56-6

69'!)

87-9

5-8

—

—

38-4

n-o

34-4

33-5

33'1

84-1

36-0

38-7

42-2

47-0

53-1

61-9

74-8

5-9

—

—

—

MH

42-5

38-3

36-8

3<J-4

36-5

37-9

40-0

43-0

46-6

51-3

57-8

66-2

6-0

—

—

—

—

41-0

41-0

41-6

42-7

44-6

47-2

50-3

55-0

61-5

6-1

—

.

48 -S

47-0

46-1

46-0

46-7

48-0

50-0

53-5

58-5

6-2

57-3

54-0

51-6

49 '8

49-1

49-4

50-0

52 -5

56-8

6-3

—

—

79-4

66 -C

58-6

54-2

51-8

51-2

50-8

52-5

55-5

6-4

—

_„

m

84-0

66-9

59-9

55-9

53-9

53-1

53-6

55-4

6-5

__

__

_

—

—

—

—

57-4

57-0

56-7

6'6

.

—

—

—

63-5

61-5

59-2

6-7

._

73-6

67-8

63-5

6-8

__

__

^_

97-2

777

70-5

6-9

—

—

—

—

—

—

—

—

—

—

—

—

—

134

96-0

82-0

7-0

A

B.

11

.

82 Tables for Statistician* ami Biomclriclans

TABLE XLIV. To find probable Frequency Type.

Values of T77 V^VS1 for given values of /9I( /3, {Semi-Minor Axis
of Probability Ellipse).

A

0

•05

•1

•75

*

•S3

*

•36

•4

•45

•5

•55

•6

•65

•7

•75

t-o

t-1
t-t

t-s
s-4

o-o
o-o
o-o
o-o
o-o

0-5
0-6
0-5
0-5
0'6

0-7
0'7
0-8
0-8
0-9

0-8
0-ff
0-9
0-9
1-0

0-8
0-9
0-9
1-0
1-0

0-8
0'9
0-9
1-0
1-0

0'8
0'8
0-9
I'O
I'O

0'8
0'8
0-9
I'O
I'O

0'7
0'8
0'9
I'O
I'O

0-7
0-8
0-8

0-!)

ro

0-7
0-7
0-8
0-9
1-0

0-6
0-7
0'8
0-9
0'9

0-fl

0-7
0'7
0'8
0'9

0-6
0-6
0'7
0-8
0'9

0'5
0-6

'••7
(I'M
(>•'.)

0'4
0'5
0'6

0'7
0'8

S-5

o-o

0'6

0'9

I'O

I'l

I'l

•1

•1

•1

i

I'O

I'O

I'O

I'O

I'O

t-6
S-7
g-8

0^0

o-o
o-o

0'6
0-7
0-7

1O
1-0
1-1

I'l
1-2
1'3

I'l

1-2
•3

1-3

•2
•3
•3

'2
•3
•3

•1
•2
•3

i

1-2

I'l
1-2
•3

1 '1
•3

I'l

I'l
1'2

0

1-1

1-2

I'O
•1
•2

10

•1

•2

S'9

O'O

0'8

1'2

1'3

•3

1'4

•4

'4

'4

i ^

'4

'4

1 3

'3

S'O

s-i

O'O

o-o

0'8
0-9

1*2
1-3

1'4
1'5

• 1
•5

1-6

•5

•6

'5

Hi

1'5

•6

'5
•6

re

4

'4
•5

•5

3-3
3-4
3-5
3-6
3-7
3-8
3-9
4-0

4-1
4-S
4-s
4-4
4-5
4-1;
4-7
4-8
4-0

5-0

5. 1

O'O
•00
•00
•00
•00
•00
•00
•00
•00

I'O
1-1
1'2
1-4
1-6
1'6
1-8
2-0
2'2

1-6
17
1-8
2-0
2-1
2-3
2-5
2-8
3-0
3-3
3-6
4-0
4-4

1'6
1'8
1'9
2-0
2-1
2-3
2-5
2-7
3-0
3-3
3-6
4-0
4-4
4-9

'6
•8
1-9

2'2
2'4
2'6
2-9
3-2
3'5
3'8
4'2
4'7
5'2

1-9
2-0
2-1
2'3
2-5
2-7
3-0
3'3
3-6
3-9
4-3
4'8
5-3

'7
•9
2-0
2-1
2-3
2'5
2-7
3-0
3-3
3-6
4-0
4-4
4'8
5-3

6'7
7'6
8'5
11

2'0

2'3
2-5
2'7
3'0
3-3
3'6
4-0

4'8
5'3
5'9
6'7
7'6
8'5
10

•7
1'8
2-0
2-1
2-3
2-5
2-7
3-0
3-3
3-6
3'9
4-3
4-7
5-2
5-8
6'6
7'4
8-4

7

2-0
2-1
2-3
2-5
2-7
2-9
3-2
3-5
3-8
4-2
4-6
5-1
5-7
6-4
7'2
8-1

'7
•8
•9

2'3

2-6
2'9
3-1
3-4
3'8
4-1
4-5
5-0
5'6
6'2
6'9
7'8
8'9

'7
•8
•9

2'2
2'4
2-6

2'8

3'4
3-7
4'0
4.4

4-9

6-0
6'6
7'5
8'5

2-2
2'4
2'6
2'8
3'0
3'3
3'6
3'9
4'3
4'7
5-2
5-8
6'4
7-2
8-1
9-1

ID

•2-n

2'3
2'5
2-7
2'9
3'2
3'5
3'8
4'2
4'6

5-7
6'3
7'0
7'7
8 '6

'6

•7

20
2-1
2'3
2'5
2-7
2-9
3-2
3'5
3-8
4-1
4-5
5-0
5-6
6'2
6'8

7'4
8.1

D

•7
•8
•9
2-1
2-3
2-4
2-6
2-8
3-1
3-4
3-7
4-0
4-4
4-8
6-3
5-9
6-5
7-1
7-7

5- >

9'5

8-9

8-6

19

in

S-l

13

12

..

_ 5

1 ft

i "i

.„

'• ri-K

S-7

5-X

5'0

6-1

6'i

K-1

fi-i

6-5

(S-7

6 -ft

fi-<t

7-U

Probable Errors of Frequency Types

TABLE XLIV— (continued).

Values of 177 V./V2, for given values of &, ftt (Semi-Minor Axis
of Probability Ellipse).

83

•8

•85

•9

•05

1-0

1-05

1-1

1-15

1-2

1-25

1-3

1-35

1-4

1-45

1-5

0-4

0-3

0-3

0-2

0-1

O'l

o-o

o-o

o-o

o-o

o-o

00

00

OO

00

2-0

0-5

0-5

0-4

0-3

0-2

0-2

0-2

0-1

0-1

0-1

00

00

00

00

OO

s-i

0-6

0-6

o-:>

o-.r.

0-4

0-4

0-3

0-3

0-3

0-2

o-i

OO

00

OO

00

& ''%

0-7

0-7

0-6

0-6

0-3

0-5

0-4

0-4

0-4

0-3

0-2

0-1

0-1

00

OO

2-3

0-8

0-8

0-7

0-7

OH

0-6

0-5

0-5

0-4

0-4

0-3

0-2

0-2

0-1

00

2-4

0-9

0-8

O-S

0-7

0-7

0-7

0-6

0-6

0-5

0-5

0-4

0-3

0-3

0-2

o-i

9'B

1-0

0-9

0-9

0-8

0-8

0-8

0-7

0-7

0-7

0-6

0-6

0-5

0-4

0-3

0-2

2-G

1-1

1-0

1-0

0-fl

(>•<)

0-9

0-8

0-8

0-8

0-7

0-7

0-6

0-5

0-4

0-3

2-7

ri

1-1

1-1

1-0

1-0

1-0

0-9

0-9

0-9

0-8

0-8

0-7

0-6

0-5

0'5

2-8

1-2

1-2

1-2

1-1

1-1

1-0

1-0

1-0

1-0

1-0

0-9

0-8

0-7

0-7

0-7

2-9

1-3

1-3

•3

1-2

•J

1-2

1-1

1-1

1-1

1-0

10

0-9

0-9

0-8

0-8

3-0

1-4

1-4

•4

1-3

•3

1-3

1-2

1-2

1-2

1-1

1-1

10

1O

0-9

0-9

3-1

1-5

1-5

•5

1-4

• 1

1-4

1-3

1-3

1-2

1-2

1-2

1-1

1-1

10

1O

#•»

1-7

1-6

•6

m

•4

1-4

13

13

1-3

1-2

1-2

1-2

1-1

1-1

10

S-fi

1-8

1-7

•7

1-6

•6

1-6

1-5

1-5

1-4

1-4

1-3

1-3

1-2

1-2

1-1

s-4

1-9

1-9

•8

1-8

•7

1-7

1-6

1-6

1-5

1-5

1-4

1-4

1-3

1-3

1-2

3-5

2-0

2-0

2-0

1-9

•8

1-8

1-7

1-7

1-6

1-6

1-5

1-5

1-4

1-4

1-4

3-6

2-2

2-2

2-1

2-1

2-0

20

1-9

1-8

1-7

1-7

1-6

1-6

1-6

1-5

1-5

3-7

2-4

2-3

2-2

2-2

2-1

2-1

2-0

2-0

1-9

1-9

1-8

1-7

1-7

1-6

1-6

S-8

2-6

•2-':

2-4

2-4

2-3

2-3

2-2

2-1

2-0

2-0

1-9

1-9

1-8

1-7

1-7

3-9

2-8

2-7

2-6

2-6

2-5

2-5

2-4

2-3

2-2

2-2

2-1

2-1

2O

2O

1-9

4-0

3-0

2-9

2-8

2-8

2-7

2-7

2-6

1-6

2-4

2-4

2-3

2-3

2-2

2-2

2-1

4'1

3-2

3-2

3-1

3-0

2-9

2-9

2-8

2-7

•!•(>

2-6

2-5

2-4

2-3

2-3

2-2

4- a

3-5

3-5

3-4

3-3

3-2

3-1

3-0

2-i)

2-8

2-8

2-7

2-6

2-5

2-4

2-4

4-3

3-9

3-8

3-7

3-6

3-5

3-4

3-3

3-1

30

3-0

2-9

2-8

2-7

2-7

2-6

4-4

4-2

4-1

4-0

3-9

37

3-6

3-5

3-4

3-3

3-2

3-1

3-1

3O

30

2'9

4-5

4-6

4-5

4-4

4-2

4-1

3-9

3-8

3-7

3-5

3-4

3'3

3-3

3-2

3-2

3-1

4-0

5-1

4-9

4-8

4-6

4.4

4-3

4-2

4-0

3-8

3-7

3-6

3-5

3-4

3-4

3-3

4-7

5-6

5-4

5-2

6-0

4-8

4-7

4-6

4-4

4-2

4-0

3-9

3-8

3-7

3-6

3-5

4-8

<;~2

6-0

5-7

6-6

6-3

5-1

6-0

4-8

4-6

4-4

4-2

4-1

4O

3-9

3'8

4-9

6-8

6-6

6-3

6-1

6-8

5-6

6-4

5-2

5-0

4-8

4-6

4-4

4-3

4-2

4-1

5-0

7-4

7-2

7-0

6-7

6-4

<;•:;

5-9

6-7

5-4

6-1

4-9

4-7

4-6

4-4

4-3

5-1

8-3

8-0

7-7

7-4

7-1

6-8

6-5

6-2

5-8

5-5

5-2

5O

4-9

4-7

4-5

6-2

:»•:!

8-9

8-6

8-2

7-8

7-6

7-1

6-7

6-3

5-9

5-6

5-4

6-2

5O

4-7

5-3

10

10

9-6

9-1

8-6

8-2

7-8

7'3

6-9

6-5

6-1

5-8

5-6

5-3

5O

5-4

12

11

11

10

9-5

9-0

8-5

8-0

7-5

7-1

6-7

6-4

6-1

5-8

5-5

6-5

—

—

12

11

10

10

9-5

8-9

8-4

7-9

7'4

70

6-6

6-3

60

B-fi

—

—

14

13

12

11

10

10

9-4

8-8

8-3

7-8

7-4

7O

6-7

5-7

—

—

16

14

13

12

12

11

10

10

9-4

8-8

8-3

7-8

7-4

5-8

_

—

18

16

15

14

14

13

12

11

10

10

9-5

8-9

8-4

5-9

—

—

•i-l

20

18

16

15

14

13

12

12

11

11

10

9-5

6-0

—

—

—

—

—

—

18

16

15

14

13

13

12

12

11

6-1

—

—

—

—

—

—

21

19

17

16

15

14

13

13

12

6-2

—

—

—

—

-

—

24

21

19

18

17

16

15

14

13

6-S

—

—

—

—

—

•>.l

24

22

20

19

18

17

16

15

6-4

—

—

—

—

—

—

32

28

25

23

21

20

19

18

16

6-5

6 '6

6'7

t;-R

u o

i:-'i

U if

7-0

11-2

84

Tables for Skttixtic'ums and Biomctrtcidna
TABLE XLV. To find probable Frequency Type.

Values of 177 ^Jf^for values of f)t, ftt (Semi-Mnjur Axis
of Probability Ellipse).

A

0

'OB

•1

•IB

•g

•S5

•3

•35

•4

•45

•5

•55

•6

•65

•7

•75

g-o
t-1
g-g

g-3
g-4
g-5
g-6
g-7
g-8
g-9
3-0

s-i

3-g
3-3

3-4
3-5
3-6
3-7
3-8
3-9

4-0

4'1

'. •
4-3

4-4
4-5
4-6
4-7
4-8
4-9
5-0
?•/

1-0
1-7
1-9
2-1

2-4
2-8
3-3
3-8
ll
5-1
6-8
6-7
7-8
9-2
11
13
15
18
21
25
29

1-9
2-1
2-3
2-5
8-8
3-3
3-8
4 '4
5-0
:••:
6-5
7-5
8-7
10
11
13
16
19
23

n

32

2-2
2-4
2-0
2-8
31
3-5
4-0
1 -<:
5'3
8-0
6-9
8-0
9-2
11
12
14
17
20
24
28
34
43
55
69
87
113

2-5
2-7
2-9
3-1
3-3
3-7
4-2
17
6-4
6-2
7-1
8-3
9-6
11
LI
14
17
20
24
28
33
41
51
62
76
98

2-3
2-9
3-1
3-3
8-fl
3-8
4-3
4-9
5-6
0-3
7-2
8-4
9'8
11
13
15
17
20
24
28
32
39
48
57
69
86

3-1

3-2
3-4
3-5
3-7
4-0
4-5
5O
5-7
6-4
7-3
8-4
9-8
11
13
15
17
20
23
27
31
37
45
53
64
78

3-4

:;•:,
3-7
3-8
3-1)
4-1
4-6
5-1
5-8
6-5
7-:i
8-4
9-7
11
13
14
10
19
22
2G
30
36
42
50
60
71
90
110
140
200
384

3-7

3-8
3-9
4-0
4-1
4-3
4-7
6-2
5-9
6-6
7-3
8-3
9-5
11
12
14
16
19
22
25
29
34
40
48
56
<;<;
79
99
124
160
234

4-1
4-1
4-1
4'2
i-:i
4-5
4-8
5-3
6-9
G-5
7-2
8-2
9-4
11
12
14
16
18
21
24
28
32
38
45
63
62
73
90
110
135
182

4-4
4'4

4-4
4-4
4-5
4-7
4-9
5-4
6-9
6-5
7-1
8-0
9-2
11
12
14
16
18
21
24
27
31
36
43
50
57
67
80
97
119
153

4-6
4-6
4-6
4-7
4<fl
5-0
5-4
5-8
6-4
7-0
7-9
9-0
10
11
13
15
17
20
23
26
30
34
40
46
63
61
72
86
105
130

5-0
4-9
4-9
4-9
4-9
60
6-1
6-4
6-8
64
6-9
7-8
8-8
10
11
12
14
17
19
22
24
U
32
37
43
49
57
87
79

115

nr.

5-4
5-3
5-2
6-2
6-1
6-2
6-3
.vr,
5-8
6-3
6-8
7-7
8-6
9-6
11
12
14
16
18
21
23
26
30
35
40
46
63
62
73
86
102
124

6-7
5-6
6-5
;,:,
5-4
5-4
6-5
5-6
6-9
6-3
6-8
7-6
8-4
9-4
11
12
14
16
18
20
22
25
29
33
38
43
50
58
68
79
M
110

6-1
li-tl
5-9
5-8
6-7
6-7
5-7
,V8
6-0
6-4
6-8
7-5
8-3
9-2
10
11
13
16
17
19
21
24
27
31
36
41
48
65
64
74

Be

iui

G-4
6-3
(J-2
6-1
6-0
6-9
6-0
6-1
6-2
6-6
6-8
7-4
8-1
8-9
10
11
13
14
16
is
M
U
26
29
34
M
46
52

a

69
79

QQ

<;•»

1 -1H

111

i - i

TOO

/;•*

IRS

166

1 If!

19Q

K-l

94 "i

20'*

1 r\A

o 4

«r.r

Af\R

•'ir

K-fi

K-7

t.a

:- -,,

I':'!!

6'1

fi-t

6-3

fi-L

6-5

6'fi

6"7

6-8

G'9

7-0

Probable Errors of Frequency Types

TABLE XLV— (continued).

Values of 1'77 VjV 2., for values o//3,, /32 (Semi-Major Axis
of Probability Ellipse).

85

•s

•85

•9

•95

1-0

1-05

1-1

1-15

1-2

1-25

1-S

1-85

1-4

1-45

1-5

6-8

7-2

7-6

7-9

8-3

8-7

9-1

9-5

9-9

10

11

11

11

12

12

2-0

<;•;

7-0

7-4

7-8

8-1

B-fi

8-9

9-2

9-6

10

11

11

11

11

12

2-1

B-fl

6-8

7-2

7-G

8-0

8-3

8-6

9-0

9-4

9-8

10

10

11

11

12

2-%

6-4

6-7

7-1

7-4

7-8

8-1

8-5

8-8

9-2

9-G

9-9

10

11

11

12

2-3

6-3

6-6

6-9

7-3

7-6

8-0

8-3

8-7

9-0

9-4

9-8

10

10

11

11

*4

C-2

6-5

e-a

7-1

7-6

7-8

8-2

8-5

8-9

9'2

9-6

10

10

11

11

2-5

6-2

6-5

6-8

7-1

7-4

7-7

8-0

8-3

8-7

9-0

9-4

9-9

10

11

11

2-G

6-3

6-5

6-8

7-0

7-3

7-6

7-9

8-2

8-5

8-8

9-2

9-7

10

10

11

2-7

6-4

<;i;

6-a

7-0

7-2

7-5

7-7

8-0

8-3

8-7

9-1

9-5

9-8

10

10

2'8

6-6

6-7

6-8

7-0

7'2

7-4

7-6

7'9

8-2

8-0

8-9

9-3

9-6

10

10

2-9

6-9

7-0

7-1

7-2

7-3

7-5

7-7

7-9

8-2

8-5

8-8

9-1

9-4

9-7

10

3-0

7-3

7-3

7-3

7-4

7-5

7-6

7-8

8-0

8-2

8-5

8-7

9-0

9-3

9-6

9-9

S-l

7-9

7-9

7-8

7-8

7-8

7-9

8-1

8-2

8-3

8-5

8-7

8-9

9-2

9-5

9-8

3-2

8-7

8-5

8-4

8-3

M

8-3

8-3

8-4

8-5

8-7

8-9

9-1

9-3

9-5

9-7

s-s

9-5

9-3

9-1

9-0

8-8

8-8

8-8

8-8

8-9

9-0

9-1

9-2

9-4

9-5

9-7

3-4

11

10

10

9-8

9-6

9-5

9-4

9-3

93

9-3

9-3

9-4

9-5

9-6

9-8

3-5

12

11

11

11

10

10

10

10

9-9

9-9

B-8

9-7

9-7

9-8

9-9

3-6

13

12

12

12

11

11

11

11

11

10

10

10

10

10

10

3-7

15

14

14

13

13

13

12

12

12

11

11

10

10

10

11

38

17

16

16

16

14

14

14

13

13

12

12

11

11

11

11

S'9

19

18

18

17

16

16

15

14

14

13

13

12

12

12

12

4-0

22

21

•20

19

18

17

17

16

If)

15

14

14

13

13

13

4-1

24

23

22

21

20

19

18

18

17

16

15

15

14

14

14

4-2

28

26

25

23

22

21

20

20

19

18

17

16

16

15

15

4-s

32

30

29

27

25

24

23

22

21

20

19

18

17

17

17

4'4

37

35

33

31

29

27

26

25

24

23

22

21

20

19

18

4-5

43

40

37

35

33

31

29

28

27

25

24

23

22

21

20

4-6

49

45

42

39

37

35

33

32

30

28

27

26

24

23

22

47

56

52

48

45

42

40

38

36

M

32

31

29

27

26

25

4-s

64

59

55

51

48

46

42

40

33

36

35

33

31

29

28

4-9

73

68

63

59

55

52

49

46

43

41

39

37

35

33

32

r,-o

85

78

72

67

63

59

55

52

49

46

43

41

39

37

36

5-1

99

90

82

77

72

67

63

58

55

52

49

46

43

41

40

5-2

114

104

95

88

82

76

71

66

62

58

55

51

48

46

44

5-3

136

123

112

102

95

87

80

76

70

65

61

57

54

51

49

5-4

167

147

132

119

108

100

92

86

79

74

69

64

60

57

54

5-5

160

141

126

115

105

96

89

83

78

73

68

64

60

5-6

—

206

169

148

132

120

110

102

95

88

82

7G

72

67

5-7

—

258

206

175

150

13G

126

116

108

100

93

87

81

75

5-8

318

255

215

190

168

150

13C

125

115

107

99

92

85

5-9

—

_

446

332

273

228

200

178

161

147

134

123

113

104

98

c-o

—

264

215

190

171

157

144

130

120

112

3-1

345

268

230

207

184

167

150

138

127

6-2

._

180

364

294

250

215

194

174

159

144

6-3

__

G80

477

370

299

252

224

201

181

165

6V,

__

1047

680

456

3G8

312

2G8

237

212

191

6-5

__

—

—

—

280

248

223

6-6

338

297

2G6

6-7

412

362

320

6-8

525

446

390

G-9

809

584

491

7-0

,

80

Table* for

<tml Iti<nnrt>-i<-lana

TABLE XLVL To find probable Frequency Type.

Angle ottii-eeii Major- Axis and Axis of ftt (Probability KU>
measured in degrees.

A

0

•US

•1

•15

•.'

•/-.

•5

•ss

•;

•4S

•6

•55

•6

•05

•7

•75

g-o
g-1
g-t

g-3

S'4
g-5
g-6
g-7

g-s

g-9
3-0
3-1
3-g
S-S
•'-',
3-5
S-G
3-7
3-8
3-9
4-0
4-1
4-i
4'S
4-4
4-5
4-c
4-7
4-8
4~n
5-u

K-1

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

IJ

11

10
10
9
8
7
7
6
6
5
5
6
4
4
3
3
3
3

2

2

H

21
in
18
17
10
14
13
12
11
10
9
9
8

Q

7
6
6
5
4
4
3
3
2
2
1

28
25
23
22
20
18
17
16
15
14
13
12
12
11
10
9
8
7
7
6
6
5
4
4
3
2

31

28
26
ii.-i
•j:',
21
20
19
17
16
15
14
14
13
12
11
10
9
9
8
7
6
5
5
4
3

33

30
28
27
25
23
22
21
19
18
17
16
16
15
14
13
12
11
10
9
8
7
6
5
5
4

H

::•_•
30
28
26
25
24
•23
21
20
19
18
17
16
15
14
13
U
11
10
9
8
7
6
6
5
4
3
3
2
1

36
34

32
30
29
27
26
25
23
22
21
20
19
18
17
15
14
13
12
11
10
9
8
7
7
6
5
4
4
3
2

37
35
33
32
31
29

n

M

25
23
22
21
20
19
18
17
15
14
13
12
11
10
9
9
8
7
6
6
4
3
2

38
37
35
34
33
31
30
28
27
25
24
23
21
20
19
18
16
15
14
13
12
11
10
10
9
8
7
6
5
4
3

89
88

36
35
34
33
31
29
28
2G
25
24
22
21
20
19
18
1C
15
14
13
12
11
10
9
8
7
6
5
5
4

40
3'J
38
37
35
34
33
31
30
28
27
26
24
22
21
20
19
17
16
15
14
13
U
11
10
9
8
7
6
6
5

41

1<>
:;:•
38
36
35
34
32
31

n

28
27
25
24
22
21
20
18
17
16
15
14
13
12
11
10
9
8
7
6
6

41

40
39
38
37
3(i
35
33
32
30

n

28
26
25
23
22
21
19
18
17
16
15
14
13
12
11
10
9
8
7
6

42
41

40
39
38
37
35
34
33
31
30
29
27
26
24
23
22
20
19
18
17
16
16
14
13
12
11
10
9
8
7

42
41
40
3!)
38
37
36
35
33

;<2

31
30
•2*
27
25
24
23
21
20
19
18
17
16
15
14
13
12
11
10
9
8

B'S

K.t

° 4

6'5

5-6

6'7

5-8

6-9

i,n

6-1

>,- -

6 -ft

G-1

>:•',

G'6

f!-7

G-S

G-9

7-0

Probable Errors of Frequency Types

TABLE XLVI— (continued).

Angle between Major-Axis and Axis of /3, (Probability Ellipse)
measured in degrees.

A

•8

•85

•0

•95

1-0

1-05

1-1

1-15

1-S

1-S5

1-3

1-S5

1-4

1-45

1-5

43

43

44

44

45

45

46

46

46

46

47

47

48

48

49

2-0

42

4-2

43

44

44

44

45

45

45

46

46

47

47

48

48

2-1

41

41

42

42

43

43

44

44

45

45

46

46

46

47

47

2-2

40

10

41

41

42

42

43

43

44

44

45

45

46

46

47

2-o'

39

39

40

40

41

41

42

42

43

43

44

44

45

45

46

2-4

38

38

39

39

40

40

41

41

42

42

43

43

44

44

45

2-5

37

37

38

39

39

39

40

40

41

41

42

42

43

43

44

2-6

36

36

37

38

38

39

39

40

41

41

42

42

43

43

44

2-7

34

35

36

36

37

38

38

39

40

40

41

41

42

42

43

2-8

33

34

35

35

36

37

M

38

39

39

40

40

41

41

42

2 '9

32

33

34

34

35

36

37

37

38

38

39

39

40

40

41

3-0

30

31

32

33

34

35

35

36

37

3S

38

39

39

40

40

3-1

29

30

31

32

33

34

34

35

36

37

38

38

39

39

39

3-2

28

29

30

31

32

!!2

33

34

35

36

37

37

38

38

38

3-3

26

27

28

29

30

31

32

33

34

35

36

36

37

38

38

3-4

25

26

27

28

29

30

31

•A-l

33

34

35

35

36

36

37

3-5

24

25

26

27

28

29

30

31

32

33

34

34

35

35

36

3-6

22

23

24

25

26

27

28

29

31

32

33

33

34

34

35

3-7

21

22

a

24

25

26

27

28

29

30

31

32

33

33

34

3-8

20

21

22

23

24

25

26

27

28

29

30

31

32

32

33

3-9

19

20

21

n

23

24

25

26

27

28

29

30

31

31

32

4-0

18

19

20

21

22

23

24

25

26

27

28

29

30

30

31

4-1

17

18

19

19

20

21

22

23

25

26

27

28

29

30

30

4'2

16

17

18

19

19

20

21

22

23

24

25

26

27

28

29

4-s

15

Hl

17

18

18

19

20

•1\

22

23

24

25

26

27

28

4-4

14

If)

16

16

17

18

19

20

21

22

23

24

25

25

26

4-5

13

14

15

15

16

17

18

19

20

21

22

23

24

25

25

4-6

12

13

14

14

15

16

17

18

19

20

21

22

23

24

24

4-7

11

12

13

14

15

15

16

17

18

19

20

21

22

23

23.

4-8

10

11

12

13

14

14

15

16

17

18

19

20

21

21

22

4-0

0

10

11

12

13

13

14

15

16

17

18

19

20

20

21

5-0

8

9

10

11

\-2

13

13

15

16

16

17

18

19

20

20

5-1

7

8

9

10

11

12

13

14

15

15

16

17

18

19

19

5-2

6

7

8

9

10

11

12

13

14

14

15

16

17

18

18

5-3

(i

7

8

8

9

10

11

12

13

13

14

15

16

17

17

5-4

5

6

7

7

8

9

10

11

12

12

13

14

15

16

17

5-5

—

—

6

7

7

8

9

10

11

12

13

13

14

15

16

5-6

—

—

5

6

6

7

8

9

10

11

12

12

13

14

15

5'7

—

—

4

5

5

6

7

8

9

10

11

11

12

13

14

5-8

—

—

3

4

5

5

6

7

8

9

10

10

11

12

13

5-9

—

—

•1

3

4

4

6

6

7

8

9

9

10

11

12

6-0

—

—

—

—

—

—

6

7

8

9

9

10

11

6-1

—

—

—

—

—

—

—

—

5

6

7

8

8

9

10

6-2

—

—

—

—

—

—

—

—

4

5

6

6

7

8

9

6-3

—

—

—

—

—

—

—

—

3

4

1

6

6

7

8

6-4

—

—

—

—

—

—

—

^

3

4

6

6

6

7

6-5

5

6

7

6'6

4

5

6

6 '7

;j

4

5

6-g

2

3

4

6-9

J

3

4

7'0

88

Tables for Statisticians and JJivinctrn:i«ns

Diagram XLVII detenu ininy tlie probability of a given Type <</" Frc>/nenci/.
8,

Probable Occurrences in Second Small Samples

89

TABLE XL VIII. Percentage Frequency of Successes in a Second Sample
" m " after drawing " p " Successes in a First Sample " n ".

Successes

p = 0

p=l

p = 2

p = 3

j> = 4

n = 6) 0

68-3333

31-8182

15-9091

7-0707

m = 5\ 1

26-5151

31-8182

26-5151

17-6768

10-6060

21-2121

26-5151

25-2525

3

3-53o4

10-6060

18-9394

25-2525

4

•8838

3-7879

9-4697

17-6768

6

•1263

•7576

2-6515

7-0707

»=6| 0

53-8462

26-9231

12-2378

4-8951

m = 6( /

26-9231

29-3706

22-0280

13-0536

2

12-2378

22-0280

24-4755

20-3963

S

4-8951

13-0536

20-3962

23-3100

4

1-6317

6-1189

13-1119

20-3963

6

•4079

2-0979

6-1189

13-0536

6

•0582

•4079

1-6317

4-8951

n = 7| 0

61 -5385

35-8974

19-5804

9-7902

m = 5( 1

25-6410

32-6340

29-3706

21-7560

2

9-3240

19-5804

26-1072

27-1950

3

2-7972

8-7024

16-3170

23-3100

4

•6216

2-7195

6-9930

13-5975

6

•0777

•4662

1-6317

4-3512

n = 7| 0

57-1429

30-7692

16-3846

6-9930

wi=6f 1

26-3736

30-7692

25-1748

16-7832

S

10-9890

20-9790

25-1748

23-3100

3

3-9960

11-1888

18-6480

23-3100

4

1-1988

4-6620

10-4895

17-4825

B

•2664

1-3986

4-1958

9-3240

6

•0333

•2331

•9324

2-7972

»=7| 0

53-3333

26-6667

12-3077

5-1282

m=7| 1

26-6667

28-7179

21-5385

13-0536

t

12-3077

21-5385

23-4965

19-5804

S

6-1282

13-0536

19-5804

21-7560

4

1-8648

6-5268

13-0536

19-0365

s

•6594

2-6107

6-8531

13-0536

6

•1243

•7615

2-6107

6-5268

7

•0156

•1243

•5594

1-8648

n=8| 0

64-2857

39-5604

23-0769

12-5874

6-2937

m=5( /

24-7253

32-9670

31-4685

25-1748

17-4825

.'

8-2418

17-9820

LT.-1748

27-9720

26-2238

S

2-2478

7-1928

13-9860

20-9790

26-2238

4

•4495

1 -9980

5-2448

10-4895

17-4825

B

•0499

•2997

1-0489

2-7972

6-2937

n=8| 0

60-0000

34-2857

18-4615

9-2308

4-1958

m = 6 ) 1

25-7143

31 -6484

27-6923

20-1398

12-6874

.'

9-8901

19-7802

25-1748

25-1748

20-9790

• :

3-2!J<;7

9-5904

16-7832

22-377(i

24-4755

4

•8991

3-5964

8-3916

14-6853

20-9790

B

•1798

•95!)0

2-9370

6-7133

12-5874

6

•0200

•1399

•5594

1-6783

4-1958

n = K| '>

56-2500

30-0000

15-0000

6-9231

2-8846

m=7| /

26-2500

30-0000

24-2308

16-1538

9-1783

.'

11-2500

20-7692

24-2308

22-0280

16-5210

3

4-3269

11-5386

18-3566

22-0280

21-4161

4

1-4423

6-2448

11-0140

i7-1329

21-4161

B

•3934

1-8881

6-1399

10-2797

16-5210

tf

•0787

•4896

1-7132

4-4056

9-1783

7

•0087

•0699

•3147

1-0489

2-8846

B.

12

90

Table* for Statist iciiiiix ami

TABLE XLVIII— (contlnufd). Percentage Frequency of Succeues in a Second
Sample "m" after drawing "p" Successes in a First tiample "n".

, = 8

«=8I

0

62-9419

26-4706

12-3529

5-2941

2*0362

i« B|

j

86-4706

28-2353

21-1766

13-0317

B-7873

*

12-3629

21-1765

22-8054

19-00 1:,

12-!>"i7(i

5

6-2941

13-0317

19-0045

IH-1 107

4

2-0362

6-7873

12-9576

1--1407

20-1563

6

•6787

2-9617

7-2563

12-9000

18-1407

e

•1851

1-0366

3--2250

7-S668

18-9376

7

•0370

•2633

1-0366

2-9617

6-7873

8

•0041

•0370

•1851

•6787

2-0362

n = 9)

0

66-6667

42-8571

26-3736

15-3846

8-3916

».ftj

1

23-8095

32-9670

32-9670

27-9720

20-9790

S

7-3260

16-4835

23-9760

27-9720

27*9720

3

1-8316

5-9940

1 1 -!)880

18-6480

24*4755

4

•3330

1-4985

3-9960

8-1585

13*9860

6

•0333

•1998

•6993

1-8648

4-1958

n=9(

0

62-6000

37-5000

21-4286

11-5385

6*7692

Ih ''•<

1

25-0000

32-1429

29-6703

23-0769

15*7343

•

8-9286

18-5439

24-7253

26 "2238

23-6014

3

2-7472

8-2418

14-9850

20-9790

14*4750

jf

•6868

2-8097

6-7433

12-2378

18-3566

S

•1249

•6743

2-0979

4-8951

9-4405

6

•0125

•0874

•3496

1-0489

2-6224

« = 9|

0

58-8235

33-0882

17-6471

8-8235

4*0724

ni = 7 |

1

25-7353

30-8824

26-4706

19-0045

11-8778

.'

10-2941

19-8529

24-4344

23-7557

19-4364

3

3-67(1:,

10-1810

16-9683

21-5961

22-6759

4

1-1312

4-2421

9-2:,:. i

15-1172

20-1563

5

•2828

1-3883

3-8873

8*06:25

13-6055

6

•0514

•3239

1-1518

3'0i!34

6-4788

7

•0051

•0411

•1851

•6170

1-6968

T! = 0|

0

65-5555

29-4118

14-7059

6-8627

2-9412

M • > |

1

26-1438

29-4118

23-5294

15-6863

9-0498

S

11-4379

20-5882

23-5294

21-1161

15*8371

3

4-5752

11-7647

18-0995

21-1161

20-1563

4

1-6340

6'65(il

11-3122

16-7969

20-1563

6

•6027

2-2624

6-7589

10-7500

16*1250

6

•1257

•7199

2-3036

5*9700

10-0782

7

•Q229

•1645

1-9197

4*5249

8

•0023

•0206

•1028

•3771

1*1312

*-9l

0

52-6316

26-3158

12-3839

6-4180

2*1672

».»!

1

26-3158

27-8638

20-8978

13-0031

6*9659

S

12-3839

20-8978

22-2910

18-5759

12-8602

3

6-4180

13-0031

18-5759

20-0047

17*5042

4

2-1672

6-9659

12-8602

17-5042

19-0955

6

•7740

3-2151

7-5018

12-7303

17-1859

6

•2381

1-2503

3-6372

7-6382

12*7303

7

•0595

•3897

1 -4029

3-6372

7-501K

8

•0108

•0877

•3897

1-2503

3*2150

6

•0011

•0108

•0595

•2381

•7740

«-6|

ii

:. I -5454

27-2727

12-1212

4-6454

• 5j

1

27-2727

30-3030

22-7273

12-9870

g

H-lttS

22-7273

26-9740

21-6450

S

4'-r> 1*1 1

112-9870

21-6450

26-9740

4

1 --2987

5-4112

12-9870

22-7:!73

e

•2165

1-2987

4-54:. I

12-1212

Probable Occurrences in Second Small Samples
TABLE XLVIII— (continued).

91

Successes

j> = 0

p=l

i>=2

P

= 3

p=4

p=5

P

= 6

P

= 7

w=10l 0

68-7500

45-8333

29-4643

18-

1318

10-5769

5-7692

»- 5} 1

22-9167

32-7381

33-9972

30'

2198

24-0385

17-3077

6-5476

15-1099

22-6648

27-

4725

28-8461

26-9230

s

1-5110

6-036G

10-3022

16'

4835

22-4359

26-9230

4

•2518

1-1447

3-0907

6'

4103

11-2179

17-3077

5

•0229

•1373

•4807

1

2820

2-8846

5-7692

« = 10| 0

52-3809

26-1905

12-4060

6

5138

2-2704

•8514

m-lOf 1

26-1905

27-5689

20-6767

12

•9736

7-0949

3-4056

S

12-4060

20-6767

21-8930

18'

2441

12-7709

7-6625

§

5-5138

12-9736

18-2441

19

•4604

17-0278

12-5744

4

2-2704

7-0949

12-7709

17

0278

18-3377

16-5039

6

•8514

3-4056

7-6625

12-5744

16-5039

18-0043

e

•2838

1-4190

3-9295

7

•8590

12-5030

16-5039

7

•0811

•4990

1-6841

4

•0826

7-8590

12-5744

8

•0187

•1403

•5741

1

•6841

3-9295

7-60^5

9

•0031

•0284

•1403

•4990

1-4190

3-4056

JO

•0003

•0031

•0187

•0811

•2838

•8514

n = 15| 0

76-1905

57-1429

42-1053

30

•4094

21-4654

14-7575

9-8383

6

•3246

»i= 5f 1

18-0478

30-0752

35-0877

35

•7757

33-5397

29-5149

24

•5958

19

•4604

2

4-0100

10-0251

16-5119

22

•3598

26-8318

29-5149

80

•2717

n

•1906

S

•6683

2-3588

6-1599

8

•9439

13-4159

18-1631

22

•7038

98

•5369

_£

•0786

•3685

1-0320

2

•2360

4-1280

6-8111

10-3199

14

•5953

6

•0049

•0295

•1032

•2752

•6192

1-2384

2

•2704

3

•8921

n=15l 0

61-5384

36-9231

21-5385

12

•1739

6-6403

3-4783

1

•7391

•8238

m = 10| ^

24-6154

30-7692

28-0936

22

•1344

15-8103

10-4348

a

•4073

3-6613

S

9-2308

18-0602

22-9857

23

•7154

21-3439

17-2997

u

•8146

8

•7226

g

3-2107

8-7565

14-5941

18

•9723

20-9694

20-6034

18

•0913

1 1

•5376

4

1-0216

3-6485

7-6619

12

•2322

16-3095

18-9958

19

•7873

18

•6566

5

•2919

1-3135

3-3874

6

•5238

10-3613

14-2469

17

•4128

1!)

•1897

6

•0730

•4032

1-2546

2

•8781

5-3965

8-7064

u

•4377

U

•9914

7

•0153

•1024

•3795

1

•0279

2-2614

4-2644

7

•1073

10

•6609

8

•0025

•0203

•0889

•2827

•7269

1-5991

1

•1094

6

•4516

9

•0003

•0028

•0145

•0538

•1615

•4146

•9423

1

•9383

10

•oooo

•0002

•0012

•0054

•0189

•0565

•1508

•3661

n = 15| 0

61-6129

25-8066

12-4583

5

•7842

2-5707

1-0876

•4351

•1631

i p f *

25-8065

26-6963

20-0222

12

•8538

7-4156

3-9155

1

•9033

•8512

S

12-4583

20-0222

20-7638

17

•3032

12-4583

7-9941

4

•6342

2

•4375

s

5-7842

12-8538

17-3032

17

•9953

15-7459

12-0490

8

•2152

g

•0297

4

2-5707

7-4156

12-4583

15

•7459

16-4305

14-7874

11

•7361

8

•2991

6

1-0876

3-9155

7-9941

12

•0490

14-7874

15-4916

14

•2006

11

•5313

6

•4351

1-9033

4-6342

8

•2162

11-7361

14-2006

14

•9480

19

•8803

7

•1631

•8512

2-4375

5

•0297

8-2991

11-6313

13-8803

11

•6968

g

•0567

•3482

1-1607

2

•7663

5-2415

8-3282

11

•4309

13

•7783

9

•0181

•1289

•4965

1

•3589

2-9443

5-3344

a

•3350

11

•4309

10

•0052

•0426

•1882

•5889

1-4548

3-0006

6

•3344

8

•3282

11

•0013

•0122

•0618

•2204

•6200

1-4548

2

•9443

5

•2415

IS

•0003

•0029

•0169

•0689

•2204

•5889

1

•3589

2

•7664

IS

•0001

•0006

•0037

•0170

•0618

•1882

•4965

1

•1607

14

•OOOO

•0001

•0006

•00-29

•0122

•0426

•1290

•3482

15

•oooo

•oooo

•OOOO

•0003

•0013

•0052

•0181

•0567

12—2

M

Table* for st<itlxti<-inii* tn«l Biomttricia»u

TABLE XLVIII— (continued). Percentage Frequency of Successes in a Second
Sample "m" after drawing "p" Successes in a First Sample "n".

ButeiMM

r,o

J> = 1

P=a

p = 3

j> = 4

p-6

n-201 0

80-7608

64-6164

51-1538

40-0334

30-9349

23-5695

m- 5|' 1

16-1038

26-92:!!

33-3612

36-3940

:{i;-273

35-3542

»

2-6923

7-0234

12-1313

17-3305

22-0964

•6-0006

a

•3518

1-2770

2-8884

6-1992

8-1408

11-6780

A

•0319

•1520

•4333

"J-.77

1-8090

3-0647

6

•0010

•0091

•0319

•0851

•1916

•3331

• -SOI 0

67-7419

45-1613

29-6884

19-0211

11-9763

7-3700

w-lOf 1

22-6806

31-1457

31-7019

28-1796

23-0313

17-6880

g

7-0078

15-0167

21-1346

24-3861

24-8738

23-2155

s

2-0022

6-9326

10-8882

15-6071

19-3463

21-5333

4

•5191

1-9965

4-.r)'.^l

7-9661

11-7760

15-4158

6

•1198

•5750

1 -o932

3-3250

6-7809

8-8091

6

•0940

•1398

•4618

1-1335

2-2940

4-0375

7

•0040

•0278

•1079

•3084

•7210

1-4671

8

•0005

•0043

•0193

•0636

•1707

•3946

9

•0000

•0004

•0024

•0089

•0274

•0722

10

•0000

•0000

•0002

•0006

•0023

•0068

n=»20l 0

58-3333

33-3333

18-6275

10-1604

5-3977

1-7868

»j = l.»| 1

25-0000

29-4118

25-4011

19-0508

13-0590

8-3.^78

g

10-2941

18-7166

22-2259

21-5090

18-2826

14-1218

S

4-0553

10-1381

15-5343

18-6411

19-1232

17-4841

4

1-5207

4-9056

9-3206

13-4987

16-3913

17-4841

6

•5396

2-1584

4-9495

8-4849

12-0203

14-7942

6

•1799

•8683

2-3569

4-7138

7-7053

10-8491

7

•0558

•3190

1-0101

2-3310

4-3590

6-9744

8

•0159

•1063

•3885

1-0256

2-1795

3-9421

9

•0041

•0318

•1330

•3989

•9581

1-9511

10

•0010

•0084

•0399

•1353

•3658

•8362

11

•0002

•0019

•0102

•0391

•1188

•3041

It

•0000

•0004

0089

•0093

•0317

•0907

IS

•0000

•0001

•0004

•0017

•0065

•0209

14

•0000

•0000

•0000

•0002

•0009

•0033

IS

•0000

•0000

•0000

•0000

•0001

•0003

«-20l 0

M-SOj 1

51-2195
25-6098

25-6098
26-2664

12-4765
19-6998

6-9099
12-7783

2-7154
7-5427

1-2068
4-1377

g

12-4765

19-6998

20-2323

16-8602

12-2839

8-o:»29

3

6-9099

12-7783

16-8602

1V3419

1.V1742

11-7715

4

2-7154

7-5427

12-2839

15-1742

1.V6340

14-0706

6

1-2068

4-1377

8-0929

11-7715

14-0706

14-5245

6

•5172

2-1297

4-9048

8-2768

11 -3-173

13-3141

7

•0839

1-0326

2-7689

5-3399

8-3213

11-0186

§

•0315

•4719

1-4462

3-1817

6-6954

8-3131

9

•0112

•2030

•7070

1-7554

3-4638

6-7473

10

•0037

•0819

•3218

•8965

1-9757

3-6474

11

•0012

•0308

•1358

•4226

1-0362

2-1221

l:

•0003

•0107

•0528

•1829

•4974

1-1274

IS

•0001

«OM

•0188

•0720

•2168

•5429

14

•0000

•0010

•0080

•0255

•0848

•2345

15

—

•0003

•0017

•0080

•0293

•0893

n

—

•0000

•0004

•0022

•0087

•0293

17

—

_J

•0001

•0005

•0022

0080

18

—

—

•0000

•IK Mi

•0004

•0017

19

_

__

•0000

•0001

•0003

to

—

—

—

. —

4000

•0000

Probable Occurrences in Second Small Samples

93

TABLE XLVIII— (continued).

Successes

n = 25l 0

83-8710

69-8925

57-8421

47-5131

38-7144

31-2693

H. »[ 2

13-9785

24-1008

30-9868

35-1949

37-2254

37-5232

«

1-9281

5-1645

9-1813

13-5365

17-8682

21-8885

5

•2065

•7651

1-7656

3-2488

5-2115

7-6134

4

•0153

•0736

•2119

•4738

•9064

1-5573

5

•0006

•0035

•0123

•0329

•0741

•1483

N = 25| 0

72-2222

61-5873

36-4146

25-3798

17-4486

11-8200

n»=10[ 1

20-6349

30-3455

33-1041

31-7248

28-1430

23-6401

#

5-4622

12-4141

18-6211

23-0261

25-3287

25-6780

5

1-3242

4-1380

8-0091

12-2806

16-3035

19-5642

It

•2897

1-1680

2-8032

5-1875

8-1618

11-4125

5

•0561

•2803

•8119

1-7786

3-2607

6-2673

6

•0093

•0564

•1933

•4940

1-0451

1-9313

7

•0013

•0092

•0368

•1086

•2628

•5518

8

•0001

•0011

•0053

•0179

•0493

•1170

9

•0000

•0001

•0005

•0020

•0062

•0165

10

•0000

•0000

•0000

•0001

•0004

•0012

n = 25l 0

63-4146

39-6341

24-3902

14-7625

8-7777

6-1203

M-lfif 1

23-7805

30-4878

28-8832

23-9392

18-2869

13-1666

2

8-536«

16-8485

21-8575

23-2742

21 -9443

18-9753

S

2-9204

7-8930

13-1550

17-2894

19-5777

19-9337

4

•9472

3-2888

6-7654

10-6788

14-2383

16-8191

6

•2894

1-2403

3-0643

6-6953

8-8100

11-9361

6

•0827

•4256

1-2381

2-6697

4-7366

7-2943

7

•0218

•1326

•4477

1-1072

2-2329

3-8807

8

•0053

•0373

•1444

•4060

•9240

1-8017

9

•0012

•0094

•0412

•1307

•3337

•7266

10

•0002

•0020

•0102

•0364

•1038

•2515

11

•0000

•0004

•0022

•0086

•0272

•0732

U

•0000

•0001

•0004

•0016

•0058

•0173

IS

•0000

•0000

•0001

•0002

•0009

•0031

14

—

•0000

•0000

•0000

•0001

•0004

is

—

—

•0000

•0000

•0000

•0000

n = 25l 0

56-5217

31-4010

17-1278

9-1014

4-7988

2-4579

tn=20( 1

25-1208

28-5463

23-8993

17-4502

11-7044

7-3738

:.'

10-8476

18-9202

21-6231

20-2168

16-6788

12-5733

S

4-5409

10-8116

16-8218

18-1951

17-9618

15-8820

4

1-8380

5-C036

10-0864

13-8796

16-0711

16-4186

6

•7172

2-6897

6-7932

9-3504

12-5094

14-5943

6

•2690

1-2069

3-0491

6-6MH1

8-6871

11-4669

7

•0965

•5082

1-4833

3-1589

6-4604

8-0943

8

•0330

•2009

•6695

1-6133

3-1317

5-1816

9

•0107

•0744

•2806

•7592

1-6449

3-0226

10

•0033

•0257

•1089

•3290

•7916

1-6088

11

•0009

•0082

•0390

•1308

•3482

•7800

12

•0002

•0024

•0128

•0476

•1393

•3429

IS

•0001

•0006

•U' His

•0156

•0502

•1357

14

•0000

•0002

•0010

•0046

•0162

•0477

15

—

•0000

•0002

•0012

•0045

•0147

16

—

—

•0001

•0002

•0011

•0039

17

—

—

—

•0000

•0002

•0008

18

_

—

—

_

•oooo

•0001

19

—

_

•0000

HO

—

—

—

—

—

—

1)4

Table* for Slatixtiriaitu nml n'nnnetriciai>*

TABLE XLVIII— (continued). Percentage Frequency of Successes in a Second
"m" Hjicr (Inuring " p" Successes in a First Sample "»".

Sacfetfft

p=0

, = 1

j»=i

p = 3

p=4

f = 6

H--S5I 0

60-9804

25-4902

12-4860

6-9824

2-8003

1-2784

•-ttf ;

85-4902

26-0104

19-6078

12-7285

7-6094

4-2613

t

12-4850

19-6078

19-9229

16-C"^I

12-1751

8-1352

3

6-9824

12-7285

16-6024

16-9713

14-8499

11-6037

4

2-8003

7-6094

12-1751

14-8499

15-1953

13-6767

5

1-2784

4-2613

S-l.'i.-.L'

11-6037

13-6757

14-0093

6

•5682

2-2598

6-04.-) 1

8-2883

11-1185

12-8418

7

•MBS

M411

1*9844

6-4870

8-2990

10-7251

8

•1027

•5502

1-0103

3-3951

6-7456

8-2555

9

•0416

•2535

•8365

1-9732

3-7128

6-9003

10

•0162

•1115

•4118

1-0801

2-2477

3-9335

11

•0061

•0468

•1921

•5573

1 -2771

1-4681

IS

•0022

•0187

•0848

•2709

•6811

1-4*14

IS

•0007

•0070

•0353

•1238

•340G

•TMi^

U

•0002

•0026

•0138

•0531

•1592

•3971

ir,

—

•0008

•0051

•0212

•0693

•1879

1C

—

•0003

•0017

•0079

•0280

•0822

17

—

•0001

•0005

•0027

•0104

•0330

18

—

—

•0002

•0008

•0035

•0120

19

—

—

•oooo

•0002

•0010

•0039

to

—

—

—

•0001

4001

•0011

XI

—

—

—

•0001

•0003

«

—

—

—

—

—

•0001

ts

—

—

_

*4

—

—

—

—

—

—

K>

—

—

—

—

—

—

n = 50l 0

91-0714

82-7922

75-1263

68-0389

61-4967

55-4670

m- 5| /

8-2792

15-3319

21-2621

26-1688

30-1454

33-2805

1

•6133

1-7357

3-2711

5-1311

7-231!)

9-5087

3

•0317

•1335

•3207

•6157

1-033/i

1-5848

4

•0013

•0065

•0192

•0440

•0861

•1517

6

•0000

•0001

•0005

•0014

•0033

•0066

n=50l 0

83-6065

69-6721

57-8633

47-8869

39-4857

32-4340

w=10( 7

13-9344

23-6177

29-9293

33-6048

35-2551

36-3833

2

2-1256

6-4972

9-4513

13-5019

17-3070

20-6402

3

•2932

1-02H7

2-2503

3-9278

6-9827

8-3080

4

4880

•1607

•4296

•8910

1-5803

2-5164

6

•0039

•0210

OMB

•1614

•8888

•5921

6

•0003

•0023

•0084

•0233

•0536

•1085

7

•(MX)

•0002

4008

•0026

•0067

•0152

8

•0000

•OOOO

•0001

•0002

•0006

•0015

9

•oooo

•oooo

•oooo

•oooo

•OOOO

•0001

10

•oooo

•oooo

•oooo

•oooo

•oooo

•oooo

»-50l 0

77-2727

59-4406

45-5092

34-6737

26-2849

19-8214

w-16f 1

17-8322

27-8628

32-5066

33-5552

32-3176

29-7321

t

3-9008

9-2876

14-6804

19-2530

22-6222

24-6927

3

•8049

2-5965

6-2143

8-31^1

11-6306

14-7589

4

•1558

•6385

1-5643

2-9695

4-8127

6-9910

B

•0281

•1405

•4083

•9011

1-6718

2-74<;:>

6

•0047

•0278

•0939

•2371

•4976

•9155

7

•0007

•0049

•0191

•0544

•1279

•2616

8

•0001

•IK HIS

•0034

•0109

•0284

•0642

9

—

•0001

•0005

•0019

•0054

•0132

10

—

—

4001

•0003

•0009

•0024

11

—

—

—

•0001

•0003

IS

_

—

•oooo

13, 14, 15

—

—

—

—

—

—

Probable Occurrences in Second Small Samples
TABLE XLVIII— (continued).

95

Successes

p = 0

, = 1

j>=2

p = 3

p = 4

p=5

n = 50l 0

71-8310

51-3078

36-4360

25-7195

18-0421

12-5748

»n = 20| 1

20-5231

29-7437

32-1494

30-7099

27-3365

23-2150

2

5-6513

12-4661

18-2340

22-1018

23-9720

24-1218

S

l-4!t:,9

4-4055

8-2882

12-2410

15-7316

18-3785

4

•3796

1-4377

3-2515

5-6902

8-4901

11-3384

6

•0920

•4247

1-1380

2-3122

3-9438

5-9480

€

•0212

•1161

•3613

•8391

1-6163

2-7262

7

•0046

•0295

•1049

•27ol

•5926

1-1089

8

•0010

•0070

•0279

•0820

•1959

•4039

9

•0002

•0015

•0068

•0222

•0585

•1323

10

•0000

•0003

•0015

•0055

•0158

•0390

11

—

•0001

•0003

•0012

•0039

•0103

12

—

—

•0001

•0002

•0008

•0024

IS

—

—

—

•oooo

•0002

•0005

14

—

—

—

—

•oooo

•0001

15—20

—

—

—

—

—

•oooo

n = 50| 0

67-1053

44-7368

29-6230

19-4782

12-7149

8-2378

fn = 25| 1

22-3684

30-227(1

30-43 1C,

27-0530

22-3854

17-6525

2

7-2546

14-!»06H

20-2898

22-8617

23-0250

21-4900

S

2-2857

6-3492

10-954H

15-0234

17-90*3

19-3831

4

•6984

2-4592

5-1643

8-3826

11-5877

14-3204

6

4066

•8853

2-2004

4-1420

6-5376

9-1130

6

•0590

•2994

•8629

1-8648

3-3018

5-1406

7

•0163

•0956

•3146

•7627

1-5167

2-6162

8

•0043

•0289

•1073

•2904

•6398

1-2147

9

•0011

•0083

•0343

•1029

•2494

•5181

10

•0003

•0022

•0103

•0340

•0901

•2038

11

•0001

•0000

•0029

•0105

•0302

•0741

12

•0000

•0001

•0008

•0030

•0094

•0249

IS

•oooo

•oooo

•0002

4008

•0027

•0077

U

—

•oooo

•OOOO

•ooos

•0007

•0022

15

—

—

•oooo

•oooo

•0002

•0006

16

•oooo

•oooo

•0001

17

—

—

•OOOO

•OOOO

18—25

—

—

—

—

—

•oooo

n = 50| 0

50-4950

25-2475

12-4963

6-1206

2-9657

1-4210

wi = 50( /

25-2475

25-5026

W-196B

12-6198

7-7231

4-4H75

2

12-4!i»;.',

19-1 269

1 9-3:2 41

16-1034

11-9504

8-1873

S

6-1206

12-619H

Ki-1034

16-272H

14-2388

11-2686

4

2-96.',7

7-7231

1 1 -9504

14-238*

14-3919

12-9527

6

1-4210

4-4875

8-1873

11-26HII

12-9527

13-0951

6

•6731

2-5063

5-2821

8-2C.77

10-6753

12-0038

7

•3151

1 -3552

3-2480

5-7108

8-2014

10-1734

8

•1457

•7126

1-9185

3-7517

6-9437

8-0780

9

O66C

•MM

1-0942

2-3606

4-0975

6-0662

10

•0300

•1831

•6049

1-429H

2-7034

4-3381

11

•0133

•0898

•3250

•8367

1-7147

2-9694

12

•0058

•0431

•1699

•4743

1-0490

1-9531

IS

•0025

•0909

•0866

•2610

•6205

1-2381

14

•0011

•0083

•0431

•1396

•3557

•7582

I',

•0004

•0042

•0209

•0726

•1978

•4493

16

•0002

•0019

•0099

•0888

•1068

•2580

17

•0001

•0008

•0046

•0182

•0561

•1437

18

—

•0003

•0021

•0088

•0286

•0777

I'.i

•0001

•0009

•0041

•0142

•0408

to

•0004

•0019

•0069

•0208

SI

—

•0002

•0008

•0032

•0103

tf

•0001

•0004

•0015

•0050

S3

•0002

•0007

•0023

*4

—

—

—

•0001

•0003

•0010

So

—

—

•00<»)

•0001

•0005

S6

•OOOO

•0002

27

__

•OOOO

•0001

S8-50

—

—

—

—

—

•oooo

90

T(tbl<* for Statisticians and Biometricinns

TABLK X F.\' 1 1 1 —(continued). Percentage Frequency of Successes in a Second
" m " after drawing "p" Successes in a First Sample " n ".

SoOOtMM

M-1001 0
w- lOf 1

p-0

P04MO

- ITU
MO

82-7191
15-1778
1 -8978

76-1308
80-8696
3-6107

68-1737
6-4097

61-8023 56-9719 60-6412
29-1520 31-9839 34-0855
7-4962 9-6874 11 '9 134

45-7719
36-5510
14-1158

41-3280

su •!>;.-.!>

16-2472

37-2763

:«;•;»: '72
18-2690

*

060fl

•1891

•4416

•8243

1-3455 2-0065 2-8031

3-7269

4-76M

5<90M

4

•0033

•0166

•0442

•0971

•1829

4008

4857

•7174

1-0109

1-3708

S

•0008

•0011

•0036

oooo

•0194

OM

OM1

•1044

•1609

•2374

6

•OOOO

•0001

•0002

•0007

•0016

•0034

•0065

•0115

•0194

•0309

7

—

•oooo

•oooo

•OOOO

•0001

•0002

•0005

•0010

•0017

oon

8

—

—

—

—

•( 1 ft 1. 1

•oooo

•oooo

oooi

•0001

•0002

9

__

—

—

—

—

—

—

•oooo

•oooo

•oooo

10

—

—

—

—

—

—

—

—

—

—

p = W p = 15 p=20 p = 25 p = 30 p = 35 p = 40 p=« p = 50

0

n*BM

19-605G

11-0992

6-0712

3-1946 1

•6083

•7697

•3473

•1463

1

36-9441

33-0200

25-8982

18-5708

12-3788 7

•7198 4

•5082

2-4580

1 -2-J.-J4

e

20-1613

2i;-*727

28-8081

26-8613

22-5639 17

•3696 12

•3485

8-1229

4-9313

3

7-1284

13-8698

20-0784

24-1644

25-4567 24

•1112 20

•8229

16-5036

12-0167

4

1-8005

6-0127

9-6930

14-9554

19-6711 22

•8554 23

•9308

22-8255

19-9224

6

•3376

1-3220

3-3813

6-6468

10-8709 15

•4516 19

•5797

22-4513

23-4799

6

•0474

•2671

•8619

2-1464

4-3483 7

•5418 11

•5470

15 '9030

19-9224

7

•0049

•0363

•1683

•4968

1-2424 2-6232 4-8456

8-0093

12-0167

8

•0003

•0036

•0200

•0788

•2425

•6221 1-3845

2-7446

4-9313

9

•oooo

•0008

•0015

•0077

•0292

•0908

•2432

•5778

1-2434

10

—

•oooo

•0001

•0004

•0017

•0062

•0199

•0567

•1463

Suocewes p=0 p = l p = 2 p = 3 j>=4 p = 5 p = 6 p = 7 p = 8

p=9

71 = 10

n-1001

0

95-2830

90-7467

86-3830 82

•1896 78

•1607 74-2914

70-5768

67-0123

63-5933

60-3153

57-1739

*- »r

1

4-5373

8-7256

12-5800 16-1156 19

•3467 22-2874

24-9514

27-3520

29-5020

31-4142

33-1007

S

•1745

•6083

•9867 1

•5956 2

•3216 3-1517

4-0737

5-0756

6-1463

7-2749

8-4512

3

•0051

•0199

•0488

•0957

•1642 -2573

•3780

•5287

•7117

•9287

1-1813

4

•0001

•0005

•0015

•0034

•0067 -0119

•0197

•0306

•0454

•0649

•0899

6

•oooo

•oooo

•oooo

•0001

•0001 -0003

•0004

•0008

•0013

•0020

•0030

1<"I

0

87-0690

75-7121

65-7500 57-0221 49

•3852 42-7116

36-8873

31-8110

27-3928

23-5527

20-2198

m- 15 f

1

11-3568

19-9243

26-1836 30-5476 33

•3684 34-9458

35-5336

35-3456

34-5611

33-3293

31-7739

.:

1-3947

3-7027

6-5459 9

•6321 12

•7407 16-7096

18-4248

20-8110

22-8233

24-4415

25-6636

3

•1604

•6730

1-2777 2

•2767 3

•5456 5-0126

6-7156

8-5076

10-3011

12-2207

14-0301

4

•0172

•0774

•2091

•4386

•7879 1 -2724

1-9006

2-6738

3-5865

4-6273

6-7796

5

•0017

•0093

•0295

•0715

•1458 -2641

•4381

•6787

•9959

1 -3!I73

1-8884

6

•0002

•0010

•0037

•0100

•0229 -0461

•0842

•1428

•2278

•3459

•5036

7

•oooo

•0001

•0004

•0012

•0031 -0068

•0137

•0252

•0435

•0711

•1112

8

•oooo

•oooo

•0001

•0004 -0009

•0019

•0037

•0070

•0122

•0804

9

—

—

—

•oooo

•OOOO -0001

•0002

•0005

•0009

•0017

•0031

10

—

—

—

—

— -0000

•oooo

•0001

•0001

•OIK 12

•0004

;/_/5 _ _ _ _ _ _

•oooo

•oooo

•oooo

•oooo

n-1001

0

83-4711

69-5592

57-8686 48

•0604 39

•8449 32-9751

27-2403

22-4613

18-4859

15-1848

12-4488

m- SOf

1

13-9119

23-3813

29-4247 32

•8618 34

•3491 34-4088

33-4530

31-8036

29-7094

27-3600

24-8976

g

2-2212

6-6472

9-5567 13

•4663 17

•0252 20-0718

22-4994

24-2787

25-4270

25-9920

26-0397

3

•3388

1-1584

2-4716 4

•2124 6

•2724 8-5261

10-8479

13-1236

15-2562

17-1690

16-soor,

4

•0492

•2122

•5480 1

•0993 1

•8873 2-9118

4-1635

6-6775

7-1382

8-7832

10-4578

5

•0068

•0354

•1077

•2490

•4853 -8394

1-3291

1-9649

2-7495

3-6775

4-7351!

e

•0009

•• K « 1

•0191

•0500

•1093 -2099

•3658

•6913

•8994

1-3010

1-8040

7

•0001

•0008

•0031

•0090

•0219 -0462

•0881

•1547

•2545

•3965

•6888

8

•oooo

•0001

•0004

•0015

•0039 -0090

•0187

•0356

•0630

•1053

•1675

9

—

•oooo

•0001

•0002

•0006 -0016

•0035

•0072

•0137

•0245

•0416

10

—

—

•oooo

•i . « i . .

•0001 -0002

•0006

•0013

•0026

•0050

O09I

11

—

—

—

•oooo -oooo

•0001

•0002

•0005

•0009

•0017

l:

—

—

—

—

— —

oooo

•oooo

•0001

•0001

•0003

73_fO — — — - — — —

•oooo

•oooo

•oooo

Probable Occurrences in Second Small Samples 97

Successes

p=0

p = l

TABLE XLVIII— (continued).
p=3 p=4 p=5 p = 6

p=9

»=--100J 0

80-1587

64-1270

51-1981

40-7920

32-4330

25-7320

20-3711

16-0915

12-6823

9-0724

7-8232

m= 25 j /

16-0317

25-8576

31-2184

33-4361

33-5052

32-1649

29-9575

27-2737

24-3890

21-4922

18-7076

J

3-1029

7-5681

12-2826

16-5799

20-1031

22-7047

24-3723

25-1757

25-2300

24-6693

23-6306

3

•5802

1-9024

3-8912

6-3556

9-0661

11-8013

14-3734

16-6391

18-5020

19-9086

20-8423

4

•1046

•4323

1-0701

2-0562

3-3806

4-9929

6-8150

8-7536

10-7117

12-5970

14-3291

5

•0182

•0908

•2644

•5855

1-0922

1-8077

2-7378

3-8700

5-1757

6-6134

8-1327

6

•0030

•0178

•0597

•1501

•3138

•5764

•9606

1-4841

2-1565

2-9790

3-9431

7

•0005

•0033

•0125

•0351

•0815

•1647

•3000

•5035

•7910

1-1761

1-6693

«

•0001

•0005

•0024

•0075

•0193

•0426

•0844

•1531

•2589

•4127

•6260

9

•0000

•0001

•0004

•0015

•0042

•0100

•0215

•0421

•0763

•1299

•2099

JO

—

•0000

•0001

•0003

•0008

•0022

•0050

•0105

•0203

•0369

•0634

11

—

—

•0000

•0001

•0001

•0004

•0011

•0024

•0049

•0095

•0173

12

—

—

—

•0000

•0000

•0001

•0002

•0005

•0011

•0022

•0043

IS

—

—

—

—

—

•0000

•oooo

•0001

•0002

•0005

•0009

14

—

—

—

—

—

—

—

•0000

•0000

•0001

•0002

15—25

—

^~

~

" '

—

"

~

"

-

•0000

•0000

n=100l 0 (S6-8874

44-5916

29-6280

19-6185

12-9456

8-5121

5-5769

3-6405

2-3676

1-5339

•9900

m= 60) 1 22-2958

29-9273

30-0284

26-6919

22-1671

17-6113

13-5550

10-1832

7-5029

5-4395

3-8892

J

7-3322

14-8625

20-0189

22-3956

22-4728

20-9746

18-5789

15-8126

13-0370

10-4710

8-2261

S

2-3780

6-4708

10-9693

14-8274

17-4789

18-7745

18-8406

17-9434

16-3894

14-4635

12-3988

4

•7603

2-6038

5-3333

8-4691

11-4896

13-9817

15-7005

16-5656

16-6252

16-0095

14-8876

6

•2396

•9912

2-3852

4-3589

6-6996

9-12:28

11-3492

13-1572

14-4085

15-0512

15-1066

6

•0743

•3614

1-0008

2-0720

3-5636

5-3759

7-3484

9-2958

11-0430

12-4505

13-4281

7

•0227

•1271

•3987

•9237

1-7600

2-9173

4-3512

5-9710

7-6559

9-2753

10-7081

8

•0068

•0433

•1520

•3901

•8167

1-4771

2-3900

3-5398

4-8771

6-3248

7-7895

9

•0020

•0143

•0557

•1572

•3590

•7044

1-2301

1-9578

2-8874

3-9946

5-2324

10

•0006

•0046

•0197

•0607

•1504

•3185

•5978

1-0184

1-6022

2-3574

3-2752

11

•0002

•0014

•0068

•0226

•0603

•1373

•2758

•5012

•8386

1-3088

1-9239

IS

—

•0004

•0022

•0081

•0232

•0566

•1213

•2345

•4161

•6871

1-0664

IS

•0001

•0007

oosa

•0086

•0224

•0510

•1046

•1965

•3425

•5601

H

—

—

•0002

•0009

•0031

•0085

•0206

•0447

•0886

•1627

•2797

15

—

—

•0001

•0003

•0011

•0031

•0080

•0183

•0382

•0738

•1332

i>;

—

—

•0001

•0004

•0011

•0030

•0072

•0158

•0320

•0606

17

—

—

•0001

•0004

•0011

•0027

•0063

•0133

•0264

18

—

•0001

•0004

•0010

•0024

•0053

•0110

19

—

—

—

—

•0001

•0003

•0009

•0020

•0044

go

—

—

—

—

—

•0001

•0003

•0008

•0017

91

—

—

—

—

—

—

—

—

•0001

•0003

•0006

gg

—

—

—

—

—

—

—

—

—

•0001

•0002

IS

—

—

—

—

—

—

—

—

—

—

•0001

£4—60

—

—

—

—

—

—

—

—

—

—

—

13

98

I—WO

Table* for Statisticians and Biometricians

TABLE XLIX. Logarithms of Factorials.

log \n from n= 1 to n = 1000.

*

tag IS

1

•0000000

S

•3010300

s

•778 1613

4

1-3802112

S

2-079 1812

6

2-857 3325

7

3-702 4305

8

4-605 5205

9

5-559 7630

10

6-559 7630

11

7-601 1657

It

8-680 3370

13

9-794 2803

14

10-940 4084

15

12-116 4996

16

13-320 6196

17

14-551 0685

18

15-806 3410

19

17-0850946

SO

18-386 1246

SI

19-708 3439

S3

21-050 7666

S3

22-412 4944

14

23-792 7057

S5

25-190 6457

S6

26-605 6190

S7

B8OM96S8

S8

29-484 1408

S9

30 946 5388

SO

32-423 6601

31

33-915 0218

3S

36-420 1717

S3

:'.«; :i:(- 88ft7

34

38-470 1646

So

40-014 2326

36

41-670 5361

37

43-138 7369

38

44-718 6205

39

46-3096861

40

47-911 6451

41

49-624 4289

4S

61-147 6782

43

62-781 1467

44

M 1^1 :.:o:;

45

56077 8119

46

67-740 6697

47

59-412 6676

48

61-0939088

49

62-784 1049

60

61 1^30749

n

log|»

61

66-190 6460

69

67-906 6484

63

89-630 :':M:»

64

71-363 3180

55

73-103 6807

66

74-851 8687

67

76-607 7436

68

78-371 1716

59

80-142 0236

60

81-920 1748

61

83-705 6047

6S

85-497 8964

63

87-297 2369

64

89-103 4169

66

90-916 3303

66

92-735 8742

67

94-561 9490

68

96-394 4579

69

98-233 3070

70

100-078 4050

71

101-929 6634

7S

103-786 9959

73

105-650 3187

74

107-519 5505

76

109-394 6117

76

111-275 4253

77

113-161 9160

78

115-0540106

79

116-951 6377

80

118-8547277

81

120-763 2127

M

122-677 0266

83

124-596 1047

84

126-520 3840

85

128-449 8029

86

130-384 3013

87

132-323 8206

88

134-26H 3033

89

136-217 6933

90

138-171 9358

91

140-130 9772

9S

142-094 7650

93

144-063 2480

04

146-036 3758

95

148-014 0994

96

149-996 3707

97

151-983 1424

98

153-974 3685

99

155-970 0037

100

157-970(1037

n

logli

101

159-974 3250

lot

161-982 92/i2

103

163-995 7624

104

166O12 7958

we,

168-033 9851

106

170-059 2909

107

172-088 6747

10S

174-122 0985

109

176-159 5250

110

178-2009176

111

180-246 2406

112

182-295 4586

113

184-348 5371

114

186-405 4419

115

188-466 1398

116

190-530 5978

117

192-598 7836

118

194-670 6656

119

196-746 2126

in

198-825 3938

181

200-908 1792

i-;:

202-994 5390

198

205-084 4442

1*4

207-177 8658

125

209-274 7759

1 M

211-375 1464

1X7

213-478 9501

128

215-586 1601

189

217-696 7498

ISO

219-810 6932

131

221-927 9645

132

224-048 5384

133

226-172 3900

134

228-299 4948

1SS

230'429 8286

136

232-563 3675

137

234-700 0881

138

236-839 9672

us

238-982 9820

140

241-129 1100

HI

243-i7H 3291

<i-

245-430 6174

247-585 9535

,44

249-744 3160

145

251-905 6840

146

254-070 0368

147

256 237 3542

148

258-407 6169

149

260-580 8022

150

262-756 8934

n

log|n

151

264-935 8704

152

267-117 7139

153

269-302 4054

154

271-489 9261

155

273-680 2578

156

275-873 3824

157

278-069 2820

158

280-267 9391

159

282-469 3363

160

284-673 4562

161

286-880 2821

162

289-089 7971

163

291-301 9847

164

293-516 8286

165

295-734 3126

166

297-954 4206

167

300-177 1371

168

302-402 4464

169

304-630 3331

170

306-860 7820

171

309-093 7781

172

311-329 3066

173

313-567 3527

174

315-807 9019

175

318-050 9400

176

320-296 452G

177

:WL'-:>14 4259

178

324-794 8459

179

327-047 6989

180

329-302 9714

181

331-560 6500

/.s'.'

333-820 7214

183

336-083 1725

184

338-347 9903

185

340-615 1620

186

342-884 6750

187

345-156 5166

188

347-430 6744

189

349-707 1362

190

351-985 8898

191

354-266 9232

ran

356-550 1844

193

358-835 7817

194

:ili T1235835

195

363-413 6181

196

365-705 8742

197

368-000 3404

198

370-297 0056

199

372 595 8586

no

374-896 8886

N

log|»

201
KM

203
204
205

377-200 0847
379-505 4361
381-812 9321
384-122 5623
386-434 3161

106

m

.'US

210

388-748 1834
391O64 1537
393 382 2170
395-702 3633
398-024 5826

til
212
213
214
215

400-348 8651
402-675 2009
405-003 5805
407-333 9943

409-666 43i>8

216
217

U8

220

412-000 8865
414-337 3463
416-675 8027
419-0162469
421-358 6695

>:.<
MS
996

426-049 4148
4^8-397 7197
i:;i )-747 9677
133-100 1502

SS6
227
999
999

230

435-454 2586
437-810 2845
440-16S -IW
442-528 0548
444-889 7827

231
233
988

234
235

447-253 3946
449-618 8826
451-9862385
454-35.r, l :,il
456-726 5223

236
237
988

239
240

469-099 4343
461-474 1826
463-850 7596
466-229 1575
168-6093687

S41

242

S43
244

245

470-991 3857
473-375 2011
47;V7i!08074
478-148 1972
480-537 3633

250

482-928 2984
485-320 9954
487-715 4470
490-111 Mm
492-509 5864

Logarithms of Factorials
TABLE XLIX— (continued).

251—500

•

login

251

494-909 2601

252

497-310 6607

253

499-713 7812

254

502-1186149

255

504-525 1551

256

506-933 3950

257

509-343 3282

258

511-7549479

259

514-168 2476

260

516-583 2210

261

518-999 8615

Ml

521-418 1628

26S

523-838 1185

264

526-259 7225

528-682 9683

266

531-107 8500

:•..'

533-534 3612

635-962 4960

,:><•>

538-392 2483

270

540-823 6121

271

543-2S6 5814

272

545691 1503

919

548-127 3129

550-565 0635

Z75

6631)04 3902

276

655-445 3052

277

657-887 7850

278

560-331 8298

9M

562-777 4340

280

565-224 5920

281

567-673 2984

282

570-123 5475

MS

r.72-:.75 3339

284

675-028 6523

ttt

577-483 4971

M

r,->>-'.>:>,'.> sr,:;i

ts7

582-397 7450

288

584-857 1375

289

587-318 0354

•W

689-780 4334

291

592-244 3264

,:'.<:

594-709 7092

flM

597-1765768

984

599-644 '.>•! \-l

295

002-1147462

604-586 0379

607-058 7943

298

609-533 0106

tap

01 •!•( x>S 6818

500

614 485 8030

•

log[»

SOI

616-964 3695

SOS

619-444 3765

303

021-9258191

304

624-408 6927

305

026-892 9925

306

629-378 7140

307

631-865 8523

308

634-354 4031

309

036-844 3615

310

639-335 7232

311

641-828 4836

644-H22 6382

313

646-818 1825

314

049 3)5 1122

315

0518134227

316

054-313 1098

317

656-814 1691

318

059-31 6 5962

319

061-820380:>

320

064-325 5369

S2i

066-832 0419

322

609-339 8978

323

671-849 1003

8*4

674-359 6453

M8

676-871 5287

326

679-384 7463

Stf

681-899 2940

684-415 1679

329

686-932 3638

330

689-450 8777

SSI

691-970 7057

332

694-491 8438

S33

697-014 2880

334

699-538 0345

Mfl

702-063 0793

556

704-589 4186

557

707-117 0485

an

709-645 9052

712-176 1649

540

714-707 6438

341

717-2103982

341

719-774 4243

343

722-309 7184

944

724-846 2768

146

727-384 0959

346

729-923 1720

347

732-463 5015

348

735-005 0807

349

737-547 9062

350

740091 9742

n

log[n

351

742-637 2813

352

745-183 8240

353

747-731 5987

354

750-280 6020

355

752-830 8303

S56

755-382 2803

357

757-934 9485

358

760-488 8310

359

763-043 9260

360

765-600 2285

361

768-157 7357

362

770-716 4443

363

773-276 3509

364

775-837 4523

365

778-399 7452

366

780-963 2262

367

783-627 8923

368

786-093 7401

369

788-660 7665

370

791-2289682

371

793-798 3421

372

796-368 8851

373

798-940 5939

374

801-513 4655

S75

804O87 4868

376

806-662 6846

377

809-239 0200

378

811-8165178

379

814-395 1670

380

816-974 9406

381

819-555 8655

382

822-137 9289

SU

824-721 1277

384

827-305 4589

385

829-890 9190

386

832-477 5069

387

835-065 2179

388

837-654 0496

389

840-243 9992

390

842-835 0638

391

845-427 2406

39J

848-020 5267

393

850-614 9192

394

853-210 4154

395

855-807 0125

396

858-404 7077

397

861-003 4982

398

863-603 3813

898

806-204 3542

400

868-806 4142

•

log[n

401

871-409 5586

402

874-013 7846

403

870-019 0890

404

879-225 4710

405

881-832 9200

406

884-441 4521

407

887-051 0465

408

889-601 7066

409

892-273 4300

410

894-886 2138

411

897-500 0556

412

900-114 9528

413

902-730 9029

414

905-347 9032

415

907-965 9513

416

910-586 0447

417

913-205 1807

418

915-826 3570

419

918-448 5710

420

921-071 8203

421

923-696 1024

-}--•

926-321 4149

•','••

928-947 7552

m

931-575 1211

425

934-203 5100

426

936-832 9196

427

939-403 3475

428

942-094 7913

429

944-727 2486

430

947-360 7170

431

949-995 1943

432

952-630 6780

433

955-267 1659

4*4

957-904 6557

435

960-543 1449

436

963-182 6314

437

905-823 1128

438

968-464 6869

439

971-107 0515

440

973-750 5041

441

976-394 9427

',',.'

979-040 3650

443

981-686 7087

444

984-334 1517

445

986-9825117

446

989-631 8460

447

992-282 1541

448

994-933 4321

449

997-585 6784

450

1000-238 8910

n

log [n

451

1002-893 0675

452

1005-548 2059

453

1008-204 3041

454

1010-861 3600

455

1013-519 3714

456

1016-178 3362

457

1018-838 2524

458

1021-499 1179

459

1024-160 9306

460

1026-823 6884

461

1029-487 3893

462

1032-162 0313

463

1034-817 6123

464

1037-484 1303

465

1040-151 5832

466

1042-819 9092

467

1045-489 2860

468

1048-159 5319

469

1050-830 7047

470

1053-502 8026

471

1056-175 8235

7/.V

1058-849 7655

473

1061-5240266

474

1064-200 4050

475

1066-877 0986

476

1069-554 7056

477

1072-233 2239

478

1074-912 6518

479

1077-592 9873

480

1080-274 2286

481

1082-956 3737

482

1085-639 4207

483

1088-323 3678

484

1091-008 2132

485

1093-693 9549

486

1096-380 5912

487

1099-068 1202

488

1101-7565400

489

1 104-445 8488

490

1107-1360449

491

1109-827 1264

493

1112-5190915

493

1115-211 9384

494

1117-905 0654

495

1120'000 2706

496

1123-2957523

497

1125-992 1086

498

1128-689 3380

499

1131-387 4385

500

1134-086 4085

13—2

100
601—750

Tables for Statisticians ami Biometricians
Table of log \n from n = 1 to n = 1000.

n

Mfc.

50*
505
504
606

1136-7862463
1139-4869500
1142-1886180
1144-8909485
1147-594 2399

606
507
608
609
510

1150-298 3904
1153-0038984
1155-7092621
1168-4159798
1161-1235500

577
57*
575
574
575

1163-8319709
1166-641 2409
1169-261 3583
1171-9623214
1174-674 1286

576
577
518
519
5*0

1177-386 7783
1180-1002088
1182-8145986
1186-529 7660
1188-245 7693

5*7
5**
5*5
5*4
5*5

1190-962 6070
1193-6802775
1196-398 7792
1199-118 1105
1201-838 2698

5*6
5*7
5S8
5*9
550

1204-559 2556
1207-2810662
1210-0037001
1212-727 1558
1215-451 4316

557
55*
555
554
555

1218-176 5262
1220-902 4378
1223-629 1660
1226-356 7063
1229-085 0600

•
557
• ••
559
540

1231-8142248
1234-544 1991
1237-2749814
1240-006 5702
1242-738 9639

547

I ; ."••
54*
545

1245-472 1612
1248-208 1005
1250-940 9603
1253-676 6592
1256-412 9557

546

•

1259-150 1483
1261
1264-626 9162
1267 -306
1270

•

*t

557

1272-848 0029

.

1276-6899419

555

1278-332 6671

554

1281-076 1768

£55

1283-820 4698

556

1286-665 5446

557

1289-311 3998

55«

1292-058 0340

559

1294-805 4458

560

1297-553 6338

567

1300-302 5967

56*

1303-052 3330

565

1305-802 8414

564

1308-554 1205

565

1311-306 1690

566

1314-058 9864

567

1316-812 5684

56S

1319-566 9168

569

11522-322 0290

570

1325-077 9039

577

1327-8346400

57*

1330-591 9360

575

1333-350 0907

574

1336-1090026

575

1338-868 6704

576

1341-6290929

577

1344-390 2687

57S

1347-152 1965

579

1349-914 8751

5SO

1362-678 3031

581

1355-442 4792

58S

1358-207 4022

583

1360-973 0708

584

1363-739 4836

585

1366-506 U395

at

1369-274 5371

r,87

1372-043 1752

588

1374-812 5525

589

1377-582 6678

590

1380-353 5198

591

1383-125 1073

M

1385-897 4-290

595

1388-670 4837

594

1391-444 2702

595

1394-218 7871

596

1396-994 0334

597

1399-770 0077

59«

1402-5467089

599

1405-321 13.-.7

600

1408-102 2870

n

!•«[•

601
60*
605
604
605

1410-881 1614
1413-660 7679
1416-441 0752
1419-222 1122
1422-0038676

606
607
60S
609
610

1424-786 3402
1427-569 6289
1430-353 4324
1433-138 0497
1435-923 3796

611
61 S
613
614
616

1438-709 4208
1441-496 1722
1444-283 6327
1447-071 8011
1449-860 6762

616
617
618
619
620

1452-650 2569
1455-440 6420
1458-231 5305
1461-0232212
1463-815 6129

6S1
6SS

ess

6S4

ess

1466-608 7046
1469-402 4948
1472-196 9829
1474-992 1675
1477-788 0475

6S6

6S8
6*9
650

1480-584 6218
1483-381 8894
1486-179 8490
1488-978 4997
1491-7778402

631

633
633
634
636

1494-577 8696
1497-378 6866
1500-179 9904
1502-982 0796
1505-784 8633

636
637
638
639
640

1508-588 3105
1611-392 4499
1514-197 270(1
1617-002 7714
1519-808 9514

641
642
643

'• ; >

1622-615 8094
1526-4233445
1528-231 B
lf.31-0404413
1533-8500010

646

(147
1:48
<;.&

U50

1536-660 2335
1539-471 1378
1542-282 7128
1M.V0949575
1647-907 8709

ft

log^

65;

65*
653

654
655

1650-721 4519
1553-530 6995
155IJ-3506126
1659-166 1904
1661-982 4317

656
657
658
659
660

1564-799 3355
1567-6169009

I.-.TII-I.T, 1268
K>73-2r>l U122
1576-073 6561

661

M)

665
664
665

1578-893 7676
1681-7146156
K.M-536 12!ll
1587-3582972
1590-181 1188

666
667
668
669
670

1593-004 5931
1595-828 7189
1698-653 4954
1601-4789215
1604-3049963

671
67S
673
674
675

1607-131 7188
1609-959 0881
Hi 12 -787 1031
1015-615 7630
1618-446 0668

676
677
678
679
680

1621-2750135
1624-105 6022
1626-936 8319
1629-768 7016
1632-601 2106

681
682
888

984

685

1635-434 3577
1638-268 1420
1641-1025627
1643-937 6189
1646-773 3094

686
687

690

1649-609 6335
1652-446 5903

1655-284 1787
1658-122 3979
1660-961 2470

891

694

695

1663-800 7251
1666-6408312
1669-481 5644
Hi72-:i22 9239
1675-164 9087

696
697
898

699
700

1678-007 5179
1680-850 7507
1683-694 6061
1686 639 0833
1689-384 1813

n

log|n

701

1692-i'J'.'

TO*

1695-076 2365

r<a

1697-923 1918

704

1700-770 7ti II

705

1703-618 9536

1(M

1706-467 7

707

1709-31 7 1777

708

1712-1U7 2!<l!)

709

1715-017 K.72

710

1717-869 1155

711

1720-720 9851

713

1723 -573 4651

713

1726-4265540

714

172!l-2802529

715

1732-134 5589

7 n;

1734-989 4719

717

1737-844 9911

718

1740-701 1155

719

1743-557 >-

7*0

1746-415 1769

721

1749-273 1122

?..'.'

1752-131 6494

1 7: VI "990 7877

17:.7 -850 5262

7»S

1760-710 8642

1703-.VT1 8009

l7ili;-4333353

; tt

1769-295 4667

7 .'!>

1772-158 1942

730

1775-021 5170

7S1

1777-885 4344

75*

1780-749 9455

755

1783-6150495

754

1786-480 7465

755

1789-347 0329

756

1792-213 9107

757

I7H.VOM1 3782

75S

1797-9494345

759

1800-8180790

740

1SU3-6S7 3107

741

1806-657 1289

742

IN ill -427 5328

743

IM 2-298 5216

744

1 N15- 17(1 0946

745

1818-042 2508

746

I*.1' I -9 14 9897

747

lh2.T7*s 3103

748

1826-662 2119

749

l-2'.l -536 6937

'.v.<o

|.'-:'.2-411 7549

Logarithms of Factorials
TABLE XLIX— (continued).

101

751—1000

n

log [n_

751
752
753
754
755

1835-287 3949
1838-163 6127
1841-0404077
1843-917 7790
1846-795 7260

756
757
758
759
760

1849-674 2478
1852-553 3437
1855-433 0129
1858-313 2546
1861-1940682

761
762
763
764
765

1864-075 4529
1866-957 4079
1869-839 9324
1872-7230258
1875-606 6872

766
767
768
7C,9
770

1878-490 9160
1881-3757113
1884-261 0726
1887-146 9989
1890-033 4896

771
772
773
774
775

1892 -92C 5440
1895-808 1613
1898-696 3408
1901-5850817
1904-474 3835

776
777
77S
779
780

1907-364 2452
1910-254 6662
1913-145 6458
1916-037 1832
1918-929 2778

781
782
783
784
785

1921-821 9289
1924-715 1356
1927-6088974
1930-503 2135
1933-398 0831

786
787
788
789
790

1936-293 5057
1939-189 4804
1942-086 0066
1944-983 0836
1947-880 7107

791
78*

793
794
795

1950-778 8872
1953-677 6124
1956-576 8866
1959-476 7061
lWii-377 0732

796
797
798
799
800

1!)>:.V277 9863
l'JOS-179 4440
1971-081 4475
l:»7:: :IH:>, 9943
1970-887 0842

n

log [n_

801
802
803
804
805

1979-790 7168
1982-6948911
1985-5996067
1988-504 8627
1991-4106586

806
801
808
808

810

1994-316 9936
1997-2238672
2000-131 2785
2003-039 2271
2005-947 7121

811
812
813
814
815

2008-856 7329
2011-7662890
2014-676 3795
2017-5870039
2020-498 1615

816
817
818
819
8*0

2023-409 8517
2026-322 0737
2029-234 8270
2032-148 1109
2035-061 9248

821
822
823
824

825

2037-976 2679
2040-891 1398
2043-806 5396
2046-722 4«OS
2049-638 9208

826
827
*tt

829
830

2052-555 9008
2055-473 4063
2058-391 4367
2061-3099912
2064-229 0693

831
88*

833
834

835

2067-148 6703
2070-068 7936
2072-989 4386
2075-9106047
2078-832 2912

836
887
888

8S9
840

2081 -754 4974
2084-677 2229
2087-6004669
2090-524 2289
2093-448 5082

841

* y -
843

844
845

2096-373 3042
2099-298 6162
2102-224 4438
2105-160 7863
2108-077 6430

846
847
848
849
850

-

2111-0050133
2113-932 8967
2116-861 2926
21 19-790 2U03
2122-7196192

n

log [n_

851

2125-649 5488

852

2128-5799884

853

2131-5109374

854

2134-4423953

855

2137-3743614

856

2140-306 8352

857

2143-239 8160

858

2146-173 3033

859

2149-107 2964

860

2152-041 7949

861

2154-976 7980

86*

2157-9123053

86S

2160-848 3161

864

2163-784 8298

865

2166-721 8459

866

2169-659 3638

867

2172-597 3829

868

2175-535 9027

869

2178-474 9224

870

2181-4144417

871

2184-354 4598

872

2187-294 9763

879

2190-235 9906

874

2193-177 5020

875

2196-1195101

876

2199-0620142

877

2202-005 0138

878

2204-948 5083

879

2207-892 4971

880

2210-836 9798

881

2213-781 9557

882

2216-727 4243

883

2219-673 3850

884

2222-619 8373

885

2225-566 7805

886

2228-514 2143

887

2231-462 1379

888

2234-410 5509

889

2237-359 4526

890

2240-308 8426

891

2243-258 7203

892

2246-209 0852

893

2249-159 9366

894

2252-111 2742

895

2255-063 0972

896

2258-015 4052

897

2260-968 1976

898

2263-921 4740

899

2266-875 2337

900

2269-829 4762

n

log jit

901
902
90S

904
905

2272-784 2010
2275-739 4075
2278-695 0953
2281-651 2637
2284-607 9123

906
907
908
909
910

2287-565 0405
2290-522 6478
2293-480 7336
2296-439 2975
2299-398 3389

911
912
913
914
915

2302-357 8573
2305-317 8521
2308-278 3229
2311-2392691
2314-200 6902

916
917
918
919
920

2317-162 5856
2320-124 9550
2323-087 7977
2326-051 1132
2329-014 9010

921
922
923
924
925

2331-979 1606
2334-943 8915
2337-909 0932
2340-874 7652
2343-840 9069

926
927
918
929
930

2346-807 5179
2349-774 6977
2352-742 145G
2355-710 1614
2358-678 6443

931
932
933
934
935

2361-647 5940
2364-617 0099
2367-5868915
2370-557 2384
2373-528 0500

936
937
938
939
940

2376-499 3259
2379-471 0655
2382-443 2683
2385-415 9339
2388-3890618

941
942
943

944
945

2391-3626514
2394-336 7023
2397-311 2140
2400-286 18CO
2403-261 6178

946
947
948
949
950

2406-237 5089
2409-213 8589
2412-190 6672
2415-167 9334
2418-145 6570

n

log |n_

951

2421

•123 8376

952

2424

•102 4745

953

2427

•081 5674

954

2430-061 1158

955

2433-041 1192

956

2436-021 5771

957

2439

•002 4890

958

2441

•983 8545

959

2444

•965 6731

960

2447

•947 9443

961

2450

•930 6677

962

2453-913 8428

963

2456

•897 4691

964

2459

•881 5461

965

2462

•866 0734

966

2465

•851 0506

967

2468-836 4770

968

2471

•822 3524

969

2474

•808 6762

970

2477

•795 4479

971

2480

•782 C671

972

2483

•770 3334

973

2486

•758 4462

974

2489

•747 0052

975

2492

•736 0098

976

2495

•725 4596

977

2498

•715 3542

978

2501

•705 6930

979

2504

•696 4757

980

2507

•687 7018

981

2510

679 3708

982

2513

•671 4823

983

2516

•664 0358

984

2519

•657 0309

985

2522

650 4672

986

2525

644 3441

987

2528

638 6612

988

2531

633 4182

989

2534

628 6145

990

2537

624 2497

991

2540

620 3233

992

2543

616 8350

993

2546

613 7842

994

2549

611 1706

995

2552

608 9937

996

2555

607 2530

997

2558

605 9482

998

2561

605 0787

999

2564

604 6442

1000

2507

004 6442

102

Table* for Statiatii-iaii* and

TABLE L. Table of Fourth- Moments of Subgroup-Frequencies.
Ordinate 2 — 11. Frequency 1 — 50.

•

x='.'

* = 3

-r = 4

*=5

1 = 6

«=7

x=8

z=9

* = 11

n

1

16

81

•M

625

1296

2401

4096

6561

14641

1

t

32

162

512

1 2:.0

2592

4801

8192

13188

2lt:>2

t

3

48

243

708

1875

8888

7203

12288

19683

43923

3

4

64

324

1024

2500

6184

9604

16384

96844

66064

4

6

80

405

1280

3125

6480

12005

20480

32805

73205

5

6

96

488

1536

3750

7776

14406

24576

80666

87846

6

7

112

567

1 7H:!

4375

9072

16807

28672

45927

102487

7

8

128

648

2048

5000

10368

19208

32768

BM88

117128

t

9

144

2304

5625

11664

21609

36804

59049

131769

S

10

160

810

MOO

6250

12960

24010

40960

65610

146410

10

11

176

891

2816

6875

14256

26411

45056

72171

161051

11

It

192

972

3072

7500

15552

28812

49152

78732

175688

111

13

208

1053

3328

8125

16848

31213

53248

86983

190333

13

14

224

1134

3584

8750

18144

33614

57344

91854

2or.(71

14

15

240

1215

3840

9375

19440

36015

61440

98415

219615

15

16

256

1296

4086

10000

20736

38416

65536

104976

234256

16

\ 17

272

1377

4352

10625

29033

40817

60689

111537

148887

17

18

888

1458

4608

11250

23328

43218

73728

118088

263538

18

19.

304

1539

4864

11875

84624

45619

77884

124659

278179

19

n

320

1620

5120

12500

25920

48020

81920

131220

898680

to

si

336

1701

5376

13126

27216

60421

88016

137781

307461

21

•-•

352

1782

5632

13750

88619

f.2H;>2

90112

144342

322102

£2

23

368

1863

5888

14375

89606

55223

94208

150903

336743

iS

24

384

1944

6144

15000

31104

57624

08904

157464

351384

*4

25

400

8086

6400

15025

32400

60086

102400

164025

366025

S5

se

416

2106

6656

16250

33696

62426

106496

170586

S80666

26

ft

432

2187

6912

16875

34992

64827

110592

177147

395307

ft

98

448

8966

7168

17500

36288

67228

114688

183708

409948

28

£9

464

2349

7424

18125

37584

69629

118784

190269

484688

u

SO

480

2430

7680

18750

88880

72030

122880

196830

439230

SO

31

496

2511

7936

19375

40176

74431

120976

203391

453871

31

32

512

2592

8192

lit* KM)

41472

76832

131072

209952

468512

38

33

528

2673

Mis

20625

42768

79233

135168

216513

483153

33

34

544

2754

8704

21250

44064

81634

139264

223074

497794

34

35

560

2835

8960

21876

45360

84035

143360

229635

512435

35

36

576

2916

9216

22500

46606

86436

147456

236196

527076

36

37

592

2997

9472

23125

47952

88837

151552

242757

641717

37

38

608

3078

9728

23750

49248

91238

155648

249318

556358

38

39

624

3159

9984

24375

50544

93639

169744

255879

570999

39

40

640

3240

10240

26000

51840

96040

163840

262440

685640

40

41

656

3321

10496

25625

53136

98441

167936

868001

800181

41

49

072

3402

10752

86860

54432

100842

172032

275562

614922

4*

43

888

3483

11008

26875

55728

103243

176188

888183

688668

43

44

704

3564

11264

27500

67024

105644

180224

188684

011201

44

45

720

3645

11520

28125

58320

108045

184320

2!i:>2 15

Ui-.ss 1:.

45

46

736

3726

11776

28750

69616

110446

188416

301806

673486

4*

47

762

3807

12032

29375

60912

112847

192512

308367

688127

47

48

768

8888

12288

30000

62208

115248

196608

3ll!»2S

702768

48

49

784

8868

12544

aoett

63504

117649

200704

321489

717409

49

50

800

MM

12800

31250

64800

120050

204800

328050

732060

50

i

Verification of the Fourth Moment

103

TABLE If— (continued).
Ordinate 12 — 19. Frequency 1 — 50.

n

ar=12

* = 13

i=14

ar=15

x=ie

z = 17

2 = 18

i = 19

n

1

20736

28561

38416

50625

65536

83521

104976

130321

1

2

41472

57122

76832

101250

131072

167042

209952

260642

2

S

62208

85683

115248

151875

196608

250563

314928

390963

S

4

82944

114244

153664

202500

262144

334084

419904

521284

4

5

103680

142805

192080

253125

327680

417605

624880

651605

5

6

124416

171366

230496

303750

393216

501126

629856

781926

6

7

145152

199927

26S912

354375

458752

584647

734832

912247

7

8

100888

228488

307328

405000

524288

668168

839808

1042568

8

9

186624

257049

345744

455625

589824

751639

944784

1172889

9

10

207360

285610

384160

506250

655360

835210

1049760

1303210

10

11

228096

314171

422576

556875

720896

918731

1154736

1433531

11

12

248832

342732

460992

607500

786432

1002252

1259712

1563852

12

IS

269568

371293

499408

658125

851968

1085773

1364688

1694173

IS

U

290304

399854

537824

708750

917504

1169294

1469664

1824494

14

IS

311040

428415

576240

759375

983040

1252815

1574640

1954815

15

16

331776

456976

614656

810000

1048576

1336336

1679616

2085136

16

17

352512

485537

653072

860625

1114112

1419857

1784592

2215457

17

18

373248

514098

691488

911250

1179648

1503373

1889568

2345778

18

19

393984

542659

729904

961875

1245184

1586899

1994544

2476099

19

n

414720

571220

768320

1012500

1310720

1670420

2099520

2606420

20

21

43.->456

599781

806736

1063125

1376256

1753941

2204496

2736741

21

n

4561!t2

628342

845152

1113750

1441792

1837462

2309472

2867062

22

,'.>•

476928

656903

883568

1164375

1507328

1920983

2414448

2997383

23

24

4970 U

685464

921984

1215000

1572864

2004504

2519424

3127704

24

25

518400

714025

960400

1265625

1638400

2088025

2624400

3258025

25

ft

539136

742586

998816

1316250

1703936

2171646

2729376

3388346

26

27

559872

771147

1037232

1366875

1769472

2255067

2834352

3518667

27

28

580608

799708

1075648

1417500

1835008

2338588

2939328

3648988

28

29

601344

828269

1114064

1468125

1900544

L'122109

3044304

3779309

20

SO

622080

856830

1152480

1518750

1966080

2505630

3149280

3909630

SO

SI

642816

885391

1190896

1569375

2031616

2589151

3254256

4039951

SI

S2

663052

913952

1229312

1620000

2097152

2672672

3359232

4170272

S2

S3

684288

942513

1267728

1670625

2162688

2756193

3464208

4300593

S3

34

705024

971074

1306144

1721250

2228224

2839714

3569184

4430914

34

35

725760

999635

1344560

1771875

2293760

2923235

3674160

4561235

35

36

746496

1028196

1382976

1822500

2359296

3006756

3779136

4691556

SO

37

767232

1056757

1421392

1873125

2424832

3090277

3884112

4821877

37

38

787968

1085318

1459808

1923750

2490368

317::798

3989088

4952198

38

39

808704

1113879

1498224

1974375

2555904

3257319

4094064

5082519

39

40

829440

1142440

1536640

20250'W

2621440

3340840

4199040

5212840

40

41

850176

1171001

1575056

207f>>;.'.->

2686976

3424361

4304016

5343161

41

42

870912

1199562

1613472

2126250

2752:>12

3507882

4408992

5473482

4%

43

891648

1228123

1651888

2176875

2818048

3591403

4513968

5603803

43

U

912384

1256684

1680804

2227500

2883584

3674924

4618944

5734124

44

45

933120

1285245

1728720 2278125

2949120

3758445

4723920

5864445

45

46

953856

1313806

1767136

2328750

3014656

3841966

4828896

5994766

46

47

974592

1342367

1805552

2379376

3080192

3925487

4933872

6125087

47

48

995328

1370928

1843968

2430000

3145728

4009008

5038848

6255408

48

49

1016064

1399489

18823S1

2480625

3211264

4092529

5143824

6385729

49

60

1036800

1428050

1920800

2531250

3276800

4176050

5248800

6516050

50

104

Tables for Statisticians aiul Jiionietricians

TABLE L— (continued).
Ordinnte 2—11. Frequency 51—100.

i=6

2 = 6

x = 7

« = 8

*=9

x = ll

61

816

4131

13056

31875

880M

122451

208896

334611

746691

51

••:

m

4212

13312

32500

67392

184801

212992

341172

761332

5S

53

848

4293

13568

33125

88888

127253

217088

347733

775973

53

•*4

864

4374

13824

33750

88084

180664

221184

354294

790614

54

55

880

4455

14060

34375

71280

132055

225280

360855

805255

65

56

BM

IBM

14336

86000

72576

131151!

229376

367416

819896

66

912

4617

14092

35625

73872

136857

233472

373977

834537

57

58

928

UM

14848

86860

75168

139258

237568

380538

849178

58

59

944

4771)

15104

36875

76464

141659

841664

387099

863819

59

60

960

4860

15360

37500

77760

144060

840780

393660

878460

60

61

976

4941

15616

38125

79056

146461

249856

400221

893101

61

6g

992

sou

15872

88700

80352

148868

853952

406782

907742

6S

63

1008

5103

16123

39375

81648

151263

258048

413343

922383

63

64

1024

5184

16384

40000

82944

153664

262144

419904

937024

64

65

1040

5265

16640

40625

84240

166060

266240

426465

951665

65

66

1056

5346

16896

41250

85536

158466

270336

433026

966306

66

67

1072

6427

17152

41875

86832

160867

274432

439587

980947

67

68

[086

17408

42500

88128

lt!32(W

:!7S5:>s

446148

995588

68

69

1104

5680

17664

43125

89424

Ifi5ti(i!)

'2W>-2\

452709

1010229

69

70

1120

5670

17920

43750

90720

168070

286720

459270

1024870

70

71

1136

5751

18176

44375

92016

170471

290816

4IJ5SJ1

1039511

71

7g

1152

5832

18432

45000

93312

172872

294912

472392

1054152

7S

73

1168

5913

18888

45625

94608

175273

880008

478953

1068793

73

74

1184

5994

18944

46250

95904

177674

303104

485514

1083434

74

75

1200

6075

18900

46875

97200

180075

307200

492075

1096075

75

76

1216

6156

19468

47500

98496

182476

311296

498636

1112716

76

77

1232

6237

19712

48125

99792

184877

315392

505197

1127357

77

78

IMS

6318

18088

48750

101088

187278

319488

511758

1141998

78

7.9

1264

8300

20224

49375

102384

189679

323584

518319

1156639

79

80

1280

6480

10480

50000

103680

192080

327680

684880

1171280

80

81

1296

6561

20736

5UW5

104976

194481

331776

531441

1185921

81

83

1312

6642

20992

51250

106272

19f!S82

335872

538002

1200562

u

8S

1328

6723

21248

51875

107568

199283

3.-WWS

544563

1215203

83

84

1344

6*il.|

21504

52500

108864

201684

344064

551124

1238844

84

86

1360

6886

21760

53125

110160

204085

348160

557686

1244485

85

86

1376

6088

22016

53750

111456

206486

352256

564246

1259126

86

87

1392

7047

22272

54375

112752

L'llSSH/

356302

570807

1273767

87

88

i KM

7128

88088

55000

114048

211288

360448

677368

1288408

88

89

1424

7209

88784

55625

115344

213689

364544

583929

1303049

89

90

1440

7290

23D40

56250

116640

216090

368640

590490

1317690

90

91

1456

7371

18806

56870

117936

218491

372736

597051

1332331

91

M

1472

7462

23552

57500

119232

220892

376832

603612

1346972

.".'

93

1 !.-s

7533

23808

58125

120528

223293

SSOill'S

C10173

1361613

93

94

1504

7614

24064

58750

121824

880894

3*50- 1

616734

1376254

94

95

1520

TOM

24320

59375

123120

888008

389120

623295

1390895

95

96

1536

7776

24576

80000

12441G

230496

393216

888608

1405536

96

97

1552

7857

24832

00080

125712

232897

397312

636417

1420177

:>,

98

1568

7988

16068

61250

127008

886886

401408

848078

1434818

98

1584

8019

25344

81870

128304

237699

(06004

649539

1449459

99

100

MOO

8100

25600

62500

129600

240100

409600

656100

1464100

100

Verification of the Fourth Moment

105

TABLE L— (continued).
Ordinate 12—19. Frequency 51 — 100.

n

1 = 12

1 = 13

x=U

*=15

* = 16

*=17

x=18

*=19

n

SI

1057536

1456611

1959216

2581875

3342336

4259571

5353776

6646371

51

52

1078272

1485172

1997632

2632500

3407872

4343092

5458752

6776692

52

53

1099008

1513733

2036048

2683125

3473408

4426613

5563728

6907013

S3

54

1119744

1542294

2074464

2733750

3538944

4510134

5668704

7037334

54

55

1140480

1570855

2112880

2784375

3604480

4593655

5773680

7167655

55

56

1161216

1599416

2151296

2835000

3670016

4677176

5878656

7297976

56

57

1181952

1627977

2189712

2885625

3735552

4760697

5983632

7428297

57

58

1202688

1656538

2228128

2936250

3801088

4844218

6088608

7558618

58

59

1223424

1685099

226t>:> 1 l

2986875

3866624

4927739

6193584

7688939

59

60

1244160

1713660 2304900

3037500

3932160

5011260

6298560

7819260

60

61

1264896

1742221

2343376

3088125

3997696

5094781

6403536

7949581

61

62

1285632

1770782

2381792

3138750

4063232

5178302

6508512

8079902

62

63

1306368

1799343

2420208

3189375

4128768

5261823

6613488

8210223

63

64

1327104

1827904

2458624

3240000

4194304

5345344

6718464

8340544

64

65

1347840

1856465

2497040

3290625

4259840

5428865

6823440

8470865

65

66

1368576

1885026

2535456

3341250

4325376

6512386

6928416

8601186

66

67

1389312

1913587

2573872

3391875

4390912

5595907

7033392

8731507

67

68

1410048

1942148

2612288

3442500

4456448

6679428

7138368

8861828

68

69

1430784

1970709

2650704

3493125

4521984

5762949

7243344

8992149

69

70

1451520

1999270

2689120

3543750

4587520

5846470

7348320

9122470

70

71

1472256

2027831

2727536

3594375

4653056

5929991

7453296

9252791

71

72

1492992

2056392

2765952

3645000

4718592

6013512

7558272

9383112

72

73

1513728

2084953

2804368

3695625

4784128

6 >: 17033

7663248

9513433

73

74

1534464

2113514

2842784

3746250

4849664

6180554

7768224

9643754

74

75

1555200

2142075

2881200

3796875

4915200

6264075

7873200

9774075

75

711

1575936

2170636

2919616

3847500

4980736

6347596

7978176

9904396

76

77

1596672

2199197

2958032

389S125

5046272

6431117

8083152

10034717

77

78

1617408

2227758

2996448

3948750

5111808

6514638

8188128

10165038

78

70

1638144

2256319

3034864

3999375

5177344

6598159

8293104

10295359

79

80

1658880

2284880

3073280

4050000

5242880

6681680

8398080

10425680

80

81

1679616

2313441

3111696

4100625

5308416

6765201

8503056

10556001

81

82

1700352

2342002

3150112

4151250

5373952

6848722

8608032

10686322

82

83

1721088

2370563

3188528

4201875

5439488

6932243

8713008

10816643

83

84

1741824

2390124

ttMM4

4252500

5505024

7016764

8817984

10946964

84

85

1762560

2427685

•MDMO

i::<i.U25

5570560

7099285

8922960

11077285

85

86

1783296

MMMfl

3303776

l:C)3750

5636096

7182806

9027936

11207606

86

87

1804032

248-1807

3342192

4404375

6701632

7266327

9132912

11337927

87

88

1824768

2513368

3380608

4455000

5767168

7349848

9237888

11468248

88

89

1845504

2541929

M190M

l.,ii:,G25

5832704

7433369

9342864

11598569

89

90

1866240

2570490

34.')7440

4556250

6898240

7516890

9447840

11728890

90

91

1886976

2599051

3495856

4606875

5963776

7600411

9552816

11859211

91

M

1907712

2627612

UMsra

4657500

6029312

7683932

9657792

11989532

92

95

1928448

2656173

3572688

4708125

6094848

7767453

9762768

12119853

93

94

1949184

2684734

3611104

4758750

6160384

7850974

9867744

12250174

94

95

1969920

2713295

3649520

4809375

6225920

7934495

9972720

12380495

95

96

1990656

2741856

36^7936

4860000

6291456

8018016

10077696

12510816

96

97

2011392

2770417

3726352

4910625

6356992

8101537

10182672

12641137

97

98

2032128

2798978

3764768

4961250

6422528

8185058

10287648

12771458

98

99

2052864

2827539

3803184

5011875

6488064

8268579

10392624

12901779

99

100

2073600

2856100

3841600

5062500

6553600

8352100

10497600

13032100

100

14

Tubl&s Jor &((tti*ti<-itnts aiui

TABLE L— (continued).
Ordinate 2—7. Frequency 101—150.

•

x«a

*=a

i = 4

*=a

1 = 6

i = 7

it

101

1616

8181

25856

63125

130696

242601

J"l

109

1632

BM

2(3112

63750

132192

244903

JOS

1648

8343

26368

64375

133488

247303

103

104

1664

8424

26624

65000

134784

2 IU704

104

105

1680

BMW

26880

65625

136080

252105

106

1696

B686

27136

«6250

137376

254506

lOti

107

1712

8667

273U2

66S7.">

138672

256907

107

108

1728

8748

27648

67500

139968

259308

108

109

1744

Ban

27904

681.'.-.

141264

2(11709

109

110

1760

8910

28160

68750

142160

264110

no

111

1776

8991

28416

69375

143856

266511

in

113

1792

9072

28672

70000

145152

268912

11 .'

113

1808

9153

28928

70625

146448

271313

113

114

1824

9234

29184

71250

147744

273714

114

115

1840

9315

294 1 >

71875

149040

276115

115

116

1856

9396

29696

72500

150336

278516

116

117

1872

9477

29952

73125

151632

280917

117

118

1888

9558

30206

73750

152928

283318

118

119

1904

9639

304U4

74375

154224

285719

ll!t

ISO

1920

9720

30720

75000

l.V.;>20

288120

UK

Igl

1936

9801

30976

75625

156816

290521

1S1

Iti

1952

9882

31232

76250

158112

2H292S

its

123

1968

raea

31488

76875

lf>9408

2:i:.323

1~'J

1S4

1984

10044

31744

77500

160704

21I7724

1-",

IK

2000

10125

32000

78125

162000

300125

125

1 ."',

2016

10206

32256

78750

Ki3296

302526

1S6

1ST

2032

10287

32512

79375

101592

304927

vn

its

2048

10368

32768

80000

165888

307328

uu

1 :•>

2064

10449

33024

80625

167184

309729

1S9

ISO

2080

10530

33280

81250

1^480

312130

ISO

131

MM

10611

33536

81875

169776

314531

131

/.;.'

2112

10692

33792

82500

171072

316932

138

133

2128

10773

34048

83125

172368

319333

133

134

2144

10854

34304

83750

173364

321734

134

/..';

2160

10935

34560

8437.')

174960

32413.-,

135

136

2176

11016

34816

85000

1762.->«

326536

136

137

2192

11097

35072

Basse

177552

328937

137

138

2208

11178

35328

86250

178848

331338

us

139

HM

11269

35584

86875

180144

333739

139

l'i"

2240

11340

35840

87500

181440

336140

140

141

2256

114-J1

36096

88125

182736

338541

141

14S

2272

11502

36352

88750

184032

340942

149

143

Ugg

11583

36608

89375

185328

343343

143

144

2304

11664

36864

90000

186G24

345714

144

145

UK

11748

37120

90625

187920

348145

145

146

OK

11826

37376

91250

189216

350546

146

147

uu

11907

37632

91875

1905 12

3521)17

147

148

068

11988

37888

92500

11)1808

355348

148

149

12069

38144

93125

I!i3104

357719

149

160

MOO

12150

38400

93750

194400

•150

150

Verification of the Fourth Moment

107

TABLE L— (continued).
Ordinate 8—14. Frequency 101—150.

•

x=8

1 = 9

x=ll

x = 12

* = 13

1 = 14

n

101

413696

662661

1478741

2094336

2884661

3880016

101

102

417792

669222

1493382

2115072

2913222

3918432

103

103

421888

675783

1508023

2135808

2941783

3956848

103

104

425984

682344

ir, 2 2664

2156544

2970344

3995264

104

105

430080

688905

1537305

2177280

2998905

4033680

105

106

434176

695466

1551946

2198016

3027466

4072096

106

107

438272

702027

156G587

2218752

3056027

4110512

107

108

442368

708588

1581228

2239488

3084588

4148928

108

109

446464

715149

1595869

2260224

3113149

4187344

109

110

450560

721710

1610510

2280960

3141710

4225760

110

111

454656

728271

1G25151

2301696

3170271

4264176

111

112

458752

734832

1639792

2322432

3198832

4302592

113

113

462848

741393

1G54433

2:; 13168

3227393

4341008

lid

114

466944

7479M

1669074

2363904

3255954

4379424

114

115

471040

7:.4515

1683715

2384640

3284515

4417840

115

116

475136

761076

1698356

2405376

3313076

4456256

11G

117

479232

767637

1712997

2426112

3341637

4494672

117

118

483328

774198

1727638

2446848

3370198

4533088

118

119

487424

780759

1742279

2467584

3398759

4571504

119

120

491520

787320

1756920

2488320

3427320

4609920

no

in

495616

793881

1771561

2509056

3455881

4648336

121

122

499712

800442

1786202

2529792

3484442

4686752

122

zts

503808

807003

1800843

2550528

3513003

4725168

123

1X4

607904

813564

1815484

2571264

3541664

4763584

124

1S5

512000

820125

1830125

2592000

3570125

4802000

125

126

616096

826686

1844766

2612736

3598686

4840416

126

1X7

520192

833247

1859407

2633472

3627247

4878832

127

1X8

524288

839808

1874048

2G54208

3655808

4917248

128

129

528384

846369

1888688

2674944

3684369

4955664

129

ISO

532480

852930

1903330

2t;:i:,680

3712930

4994080

ISO

1S1

536576

85911)1

1JJ17971

2716416

3741491

6032496

131

1S2

540672

860052

1932G12

2737152

3770052

6070912

tat

1SS

544768

872613

1947253

2757888

3798613

5109328

133

134

548864

879174

1961894

2778624

3827174

5147744

134

135

552960

885735

1976535

2799360

3855735

6186160

135

136

557056

892296

1991176

2820096

3884296

6224576

136

137

561152

898857

2005817

2840832

3912857

5262992

137

138

565248

905418

2020458

2861568

3941418

6301408

138

139

569344

911979

2035099

2882304

3969979

5339824

139

140

573440

918540

2049740

2903040

3998540

5378240

140

141

577536

92.-.101

2064381

2923776

4027101

6416656

141

142

581632

931W2

2079022

2944512

4055662

6455072

142

143

585728

9382-23

:;ii63

2965248

4084223

6493488

143

w

589824

944784

2108304

2985984

4112784

5531904

144

145

593920

951345

21221145

3006720

4141345

5570320

145

146

598016

067906

2137586

3027456

4169906

5608736

146

147

602112

D6M07

2152227

3048192

4198467

6647152

147

148

606208

97102-s

2166868

3068928

4227028

5685568

148

149

610304

977589

2181509

3089664

4255589

5723984

149

ISO

614401)

984150

2196150

3110400

4284150

6762400

150

14—2

108

Tables J'oi- Statisticians and

TAHLE L— (continued).
Ordinate 2—12. Frequency 151—200.

*

x = a

x-8

* = 4

«=6

x = 0

i=7

1 = 8

«»9

j- 11

1=12

N

161

2416

12231

M006

94376

195696

362551

618496

990711

2210791

3131136

161

15t

I ; :_•

12312

ism

86000

1808M

864969

622592

997272

2225432

3151.S72

if, :

163

2448

12393

39168

95625

196886

367353

686688

1003833

2240073

3172608

/.•-,.;

164

£464

12474

39424

86880

188684

369754

630784

1010394

2254714

3193344

154

166

1480

L8BM

89680

80876

•00680

37:2155

684880

1016955

22693:.:.

3214080

165

166

L 180

LMtt

39936

87600

202176

374556

638976

1023516

2283996

3234816

15ti

167

2512

12717

40192

98125

203472

376957

643072

1030077

2298637

3255552

157

1

MM

12798

40448

98750

204768

379358

647168

1036638

2313278

3276288

158

159

2544

12879

40704

99376

206064

381769

651264

104311)11

2327919

3297024

15U

160

2560

1-2960

10860

LOOOOO

207360

384160

655360

1049760

2342560

3317760

160

161

1876

13041

41216

100625

908666

886661

659456

1056321

2357201

3338496

mi

2592

13122

41472

101250

U98M

888968

663552

1062882

2371842

3359232

16S

Moe

13203

41728

101875

211248

391363

667648

1069443

2386483

3379968

1G3

164

2624

13284

41984

102500

212544

393764

671744

1076004

2401124

3400704

164

165

M80

13365

42240

103125

213840

396165

675840

1082565

2415765

3421440

tu

166

2656

13446

42496

103750

216136

888660

679936

1089126

2430406

3442176

190

167

2672

13527

42752

104375

216432

400967

684039

1095687

2445047

3462912

i6r

168

M86

LM06

43006

105000

217728

403368

088188

1102248

2459688

3483648

168

169

2704

13G89

43:264

105625

219024

405769

692224

1108809

2474329

3504384

teg

170

2720

13770

43520

106250

220320

408170

<;w32u

1115370

2488970

3525120

170

171

2736

13851

43776

106875

221616

410571

700416

1121931

2503611

3546866

171

17S

2752

13932

44032

107500

222912

412972

7046U

1128492

2518252

3566592

ns

17S

2768

14013

44288

108186

224208

415373

708608

1135053

2532893

3587328

173

174

2784

14094

44544

108750

225504

417774

712704

1141614

2547534

3608064

174

175

1800

14175

44800

109376

22IJHK)

420175

716800

1148175

2562175

3628800

175

176

2816

14256

45056

110000

228096

422.-.7G

720896

1154736

2576816

3649536

176

177

2832

14337

45312

110625

229392

4:! 11177

724992

1161297

2591457

3670272

177

178

2848

14418

45568

111250

830688

427378

729088

1167858

260* ;

3691008

178

179

2864

14499

45824

111875

231984

429779

733184

1174419

2620731*

3711744

179

180

1880

14580

46080

112500

838880

432180

737280

1180980

2635380

3732480

ISO

181

2896

14661

40386

113125

234576

434581

741376

1187541

2650021

3753216

181

18S

2912

14742

46592

113750

235872

436982

745472

1194102

2664662

3773952

189

183

2928

14823

40848

114375

237168

439383

749568

1200663

2679303

371*4688

18S

184

2944

14904

47104

115000

238464

441784

753664

1207224

2<;i (31)44

3815424

184

185

HMO

14985

47360

115625

239760

444186

757760

1213785

2708585

3836160

186

186

2976

16060

47616

116250

241056

446586

761856

1220346

272322C.

3856896

18ti

187

torn

15147

47872

116875

242352

448987

765952

1226907

2737867

3877632

187

188

3008

15228

48128

117500

243648

451388

770048

1233468

2762508

3898368

188

189

3024

15309

48384

118125

244944

453789

774144

1240029

2767149

3919104

189

190

:; •;<!

15390

48040

118760

246240

456190

778840

1246590

2781790

3939840

190

191

3056

15471

48896

119375

247536

458591

782336

1253151

2796431

3960576

191

191

3072

15552

49152

120000

248832

460992

786432

1259712

2811072

3981312

192

193

3088

16631

49408

120625

160188

4113:11*3

790688

1266273

2825713

4002048

193

194

3104

15714

49664

121250

251424

465794

794624

1272834

2840354

4022784

194

195

3120

I->T:».->

10980

121875

252720

468195

798720

1279395

2851

4043520

195

190

3136

15876

50176

122500

254016

470596

802816

1286956

2869636

4064266

196

197

3152

15957

50432

123125

255312

472997

806912

1292517

2884277

4084992

197

198

3168

16 08

B0688

123750

866606

475398

811608

1299078

2898918

4105728

198

199

3184

16119

50944

188376

257904

477799

816104

1305639

2913559

41264114

199

too

3200

16200

51200

125000

U0800

480200

819200

1312200

2928200

4147200

200

Verification of the Fourth Moment

109

TABLE L— (continued).
Ordinate 2—11. Frequency 201—250.

N

*=2

x = 3

2=4

z = 5

1=6

x = 7

*=8

z=9

i=U

n

SOI

3216

16281

51456

125625

260496

482601

823296

1318761

2942841

SOI

202

3232

16362

51712

126250

261792

485002

827392

1325322

2957482

SOS

too

3248

16443

51968

126875

263088

487403

831488

1331883

2972123

203

204

3264

16524

5^224

127500

264384

489804

835584

1338444

2986764

204

205

3280

16605

52480

128125

265680

492205

839680

1345005

3001405

205

206

3296

10888

52736

128750

266976

494606

843776

1351566

3016046

206

,:nr

3312

16767

52992

129375

268272

497007

847872

1358127

3030687

207

808

3328

16848

53248

130000

269568

499408

851968

1364688

3045328

208

209

3344

16929

53.-.04

130625

270864

501809

856064

1371249

3059969

209

210

3360

17010

53760

131250

272160

504210

860160

1377810

3074610

210

Ul

3376

17091

54016

131875

273456

506611

864256

1384371

3089251

211

212

3392

17172

54272

132500

274752

509012

868352

1390932

3103892

212

SIS

3408

17253

54528

133125

276048

511413

872448

1397493

3118533

213

• 14

3424

17334

54784

133750

877344

513814

876544

1404054

3133174

214

215

3440

17415

55040

134375

878840

516215

880640

1410615

3147815

215

316

3456

17496

66898

135000

279936

518616

884736

1417176

3162456

•JIG

217

3472

17577

65552

135625

281232

621017

888832

1423737

3177097

217

S18

M86

17658

55808

136250

282528

523418

892928

1430298

3191738

218

S19

3504

17739

66084

136875

283824

525819

897024

1436859

3206379

219

.'.'ii

3680

17820

56320

137500

285120

628890

901120

1443420

3221020

220

.-.'/

3536

17901

56576

138125

286416

5:i(X!21

905216

1449981

3235661

221

.'.'.'

3552

17982

66889

138750

287712

533022

909312

1456542

3250302

222

2SS

3568

18063

57088

139375

289008

635423

913408

1463103

3264943

22S

824

3584

18144

57344

140000

290304

537824

917504

1469664

3279584

224

225

3600

18225

57600

140625

291600

540225

921600

1476225

3294225

225

S26

8616

18306

57856

141250

292896

542626

925696

1482786

3308866

226

227

3032

18387

58112

141875

294192

545027

929792

1489347

3323507

22"?

m

3648

18468

68888

142500

295488

647428

933888

1495908

3338148

228

3664

18549

58624

143125

296784

549829

937984

1502469

3352789

229

2SO

8880

18630

68880

143750

898080

552230

942080

1509030

3367430

230

.'.11

3696

18711

59136

144375

299376

654631

946176

1515591

3382071

231

232

3712

18792

59392

I 16000

300672

557032

950272

1522152

3396712

232

2SS

3728

18873

59648

145625

301968

559433

954368

1528713

3411353

2SS

234

3744

18954

59904

1 |i^.-,M

303264

561834

958464

1535274

3425994

234

^S5

3760

19035

80160

140875

304560

564235

962560

1541835

3440635

S.'SB

i.'S6

3776

19116

60416

147500

305856

566636

966656

1648396

3455276

236

SS7

3792

19147

80678

148125

307152

569037

970752

1554957

3469917

237

238

3808

19278

60998

148750

308448

571438

974848

1561518

3484558

238

2S9

3824

19359

61184

149375

309744

573839

978944

1568079

3499199

239

S40

3840

19440

61440

150000

311040

576240

983040

1574640

3613840

240

241

3856

19521

61698

150625

312336

578641

987136

1681201

3528481

241

24%

3872

1980S

61952

151250

313632

581042

991232

1587762

3543122

242

n>

8888

19888

C2208

151875

314928

583443

995328

1594323

3557763

24ft

•44

3904

19764 | 6L'l*;i

152500

316224

585844

999424

1600884

3572404

244

-",-

8990

19845

62720

163186

317520

588245

1003520

1607445

3587045

245

240

3936

19926

68978

153750

318816

590646

1007616

1614006

3601686

246

247

3952

20007

<W.U

154375

320112

593047

1011712

1620567

3616327

247

248

8888

- . 188

68488

155000 321408

595448

1015808

1627128

3630968

248

249

3984

20169

63744

155625

322704

697849

1019904

1633689

3645609

249

£60

4000

20250

64000

156250

324000

600250

1024000

1640260

3660250

250

no

or

TABLE L— (continunl).
Ordinate 2—9. Frequency 2.r>l— 300.

•

*-a

* = 3

x-4

* = 5

1 = 6

* = 7

.r = 8

x = y

•

SSI

4016

20331

64256

166875

M6896

602651

1028096

1646811

SSI

S5S

20412

64512

157500

326592

605052

10321!*2

L658378

1048

20493

64768

158125

887888

607453

1036288

1659933

ess

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80674

86084

158750

329184

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1040384

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60880

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330480

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1044480

1673065

tss

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10786

66686

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331776

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160625

333073

617057

1052672

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£58

4128

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162500

330960

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338256

626661

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1073152

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Verification of the Fourth Moment

111

TABLE L— (continued).
Ordinate 2—8. Frequency 301—350.

•

x = 2

••a

x = 4

x = 5

x = 6

*=7

2 = 8

n

SOI

4816

24381

77056

188125

390096

722701

1232896

SOI

302

4832

24462

77312

188750

391392

725102

1236992

302

.;<>.;

4848

24')43

77568

189375

392688

727503

1241088

303

304

4864

24624

77824

190000

393984

729904

1245184

304

305

4880

24705

78080

190625

395280

732305

1249280

305

SOS

4896

24786

78336

191250

396576

734706

1253376

306

307

4912

24867

78592

191875

397872

737107

1257472

307

,,VAS'

4928

24948

78848

192500

399168

739508

1261568

308

800

4944

25029

79104

193125

400464

7419(19

1265664

309

S10

4960

25110

79360

193750

401760

744310

1269760

810

311

4976

85191

79616

194375

403056

746711

1273856

311

312

4992

.'.-.272

79872

195000

404352

749112

1277952

819

.11.;

5008

25353

80128

195625

40r»648

751513

1282048

SIS

314

5024

26434

80384

196250

406944

753914

1286144

314

315

5040

25515

80640

196875

408240

756315

1290240

S15

316

5056

25596

80896

197500

409536

758716

1294336

Slti

317

5072

25677

81152

198125

410832

761117

1298432

317

318

5088

25758

81408

198750

412128

763518

1302528

318

319

5104

25839

81664

199375

413424

765919

1306624

S19

320

5120

25920

81920

200000

414720

768320

1310720

320

• :.'!

5136

•2'iOOl

1176

200625

416016

770721

1314816

321

322

5152

26082

82432

201250

417312

773122

1318912

322

an

5168

2f>i63

82688

201875

418608

775523

1323008

323

SS4

5184

26244

82944

202500

419904

777924

1327104

S%4

SS5

5200

26325

83200

203125

421200

780325

1331200

325

890

5216

26406

83456

203750

422496

782726

1335296

896

897

5232

26487

83712

204375

423792

785127

1339392

327

898

5248

26568

83968

205000

425088

787528

1343488

328

890

5264

:!(;<;49

84224

205625

426384

789929

1347584

329

sso

5280

26730

84480

206250

427680

792330

1351680

330

SSI

5296

26811

84736

206875

428976

794731

1355776

SSI

.::.'

5312

26892

84992

207500

430272

71)7132

1359872

332

333

5328

26973

85248

208125

431568

7!i!>533

1363968

333

334

5344

27054

85504

208750

432864

801934

1368064

334

335

5360

27135

85760

209375

434160

80433.~>

1372160

335

336

5376

27216

86016

210000

435456

806731 !

1376256

336

.;.;;

5981

272:»7

86272

210625

436752

809137

1380352

.:.;;•

888

5408

27378

86528

211250

438048

811538

1384448

338

5424

27459

86784

211875

439344

813939

1388544

3.19

340

5440

27540

87040

212500

440640

816340

1392640

340

341

5456

27621

87296

213125

441936

818741

1396736

341

••'/-'

5472

27702

87552

213750

443232

821142

1400832

342

.;;.;

5468

27783

87808

214375

444528

823543

1404928

343

344

5504

27«; i

88064

215000

445*2 i

825944

1409024

344

S45

5520

27945

88320

215625

447120

828345

1413120

345

346

5536

28026

88576

216250

448416

830746

1417216

346

347

5552

28107

88832

216875

449712

833147

1421312

S47

348

5668

28188

89088

217500

451008

835.-.48

1425408

348

349

6584

28269

89344

218125

452.304

837949

1429504

349

6600

88350

89600

218750

453000

840350

1433600

350

112

or Stat tali dun* and

TABLE L— (continued).
Ordinate 2—7. Frequency 351—400.

•

i = -i

J = 8

z = 4

x = 5

1=6

*=7

n

361

;...;,,

28431

89856

219375

454896

842751

sss

:- :•;•_•

28512

90112

220000

456192 815152

ass

5648

28093

90368

220t;i'.->

4f,7488

M75B3

Sag

354

:,.,;)

28674

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221250

458784

849954

354

355

5680

28755

90880

321876

460080

862355

3i-,5

5606

. -::•;

91136

222500

461376

864756

356

357

6712

28917

913U2

223125

462672

857167

357

6786

28998

91648

223750

463968

859558

358

5744

29079

91904

224375

465264

861959

359

.....

5700

29160

92160

225000

466560

864360

360

56';

6776

29841

92416

225625

467856

866761

SHI

S6g

6792

29322

92672

2262.M i

469152

869162

5808

29403

92928

226875

470448

871563

363

364

6884

29484

93184

227500

471744

873964

364

..:.-.

6840

29565

93440

228125

473040

876365

365

.,,.

68M

29646

!!3(i96

228750

474336

878766

366

....;

6878

29727

93952

229375

475632

881167

S67

sets

5888

29808

94208

230000

476928

883568

368

966

6904

29889

94464

230625

478224

885969

S69

6990

29970

94720

231250

479520

888370

370

..;/

5B8Q

30051

94976

231875

480816

890771

371

871

5952

30132

95232

232500

482112

893172

gn

arts

5988

30213

95488

233125

483408

895573

978

374

5984

30294

95744

233750

484704

897974

S74

375

6000

30375

96000

234375

486000

900376

375

;•;

6016

30456

96256

235000

487296

902776

m

377

6032

30537

96512

235625

488592

905177

377

378

6048

30618

96768

236250

489888

907578

978

..;.••

6064

30699

97024

236875

491184

909979

979

n

8080

30780

97280

237500

492480

912380

.Ml

981

6096

30861

97536

238125

493776

914781

381

-•_•

6112

30942

97792

238750

495072

917182

..X,

6128

31023

98048

239375

496368

1(11)583

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6144

31104

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6160

31185

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498960

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385

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6176

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602848

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604144

933989

389

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6240

31590

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505440

936390

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6256

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6272

31752

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245000

608032

941192

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6288

31833

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6304

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6320

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101120

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948395

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6336

32076

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96

397

6851

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248125

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987

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6368

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101888

248760

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998

399

6384

32319

102144

249375

6171"!

957999

399

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6400

32400

102400

260000

618400

960400

400

Poisson's Exponential Binomial Limit

113

TABLE LI. Tables of e~mmx/x !: General Term of Poisson's Exponential
Expansion ("Law of Small Numbers").

*

.

X

o-i

0-2

o-s

0-4

o-s

0-6

0-7

0-8

0-9

1-0

0

•904837

•818731

•740818

•670320

•606531

•548812

•496585

•449329

•406570

•367879

0

1

•090484

•163746

•222245

•268128

•303265

•329287

•347610

•359463

•365913

•367879

1

2

•004524

•016375

•033337

•053626

•075816

•098786

•121663

•143785

•164661

•183940

2

•:

•000151

•001092

•003334

•007150

•012636

•019757

•028388

•038343

•049398

•061313

3

4

•000004

•000055

•000250

•000715

•001580

•002964

•004968

•007669

•011115

•015328

4

6

—

•000002

•000015

•000057

•000158

•000356

•000696

•001227

•002001

•003066

5

6

—

—

•000001

•000004

•000013

•000036

•000081

•000164

•000300

•000511

6

7

—

—

—

—

•000001

•000003

•000008

•000019

•000039

•000073

7

8

—

—

—

—

—

—

•000001

•000002

•000004

•000009

8

9

~^~

—

—

~—

-~-

—

~~

—

—

•000001

9

X

1-1

1-2

1-S

1-4

1-S

1-6

1-7

1-8

1-9

2-0

X

0

•332871

•301194

•272532

•246597

•223130

.201897

•182684

•165299

•149569

•135335

0

1

•366158

•361433

•354291

•345236

•334695

.323034

•310562

•297538

•284180

•270671

1

2

•201387

•216860

•230289

•241665

•251021

.U84SB

•263978

•267784

•269971

•270671

2

S

•078843

•086744

•099792

•112777

•125510

•137828

•149587

•160671

•170982

•180447

S

4

•020307

•026023

•032432

•039472

•047067

•055131

•063575

•072302

•081216

•090224

4

5

004467

•006S46

•008432

•011052

•014120

•017642

•021615

•026029

•030862

•036089

5

6

•000819

•001249

•001827

•002579

•003530

•004705

•006124

•007KO!)

•009773

•012030

6

7

•000129

•000214

•000339

•000516

•000756

•001075

•001487

•002008

•002653

•003437

7

8

•000018

•000032

•000055

•000090

•000142

•000215

•000316

•000452

•000630

000859

8

9

•000002

•000004

•000008

•000014

•000024

•00003S

•000060

•000090

•000133

•000191

9

10

—

•000001

•000001

•000002

0000 l!

• 00006

•000010

•000016

•000025

•000038

10

11

—

—

—

—

—

•000001

•000002

•000003

•000004

•000007

11

12

~^

~~

"^~

^~

~"~

—

~~"

—

•000001

•000001

12

*

2-1

s-t

2-S

/ •:

S-5

2-6

:-r

2-8

2-9

5-0

X

0

•122456

•110803

•100259 -090718

•082085

•074274

•067206 -060810

•055023

•049787

0

1

•257159

•243767

•230595

•217723

•205212

•193111

•181455

•170268

•159567

•149361

1

2

•270016

•268144

•265185

•261268

•256516

•251045

•244964

•238375

•231373

•224042

2

3

•189012

•196639

•203308

•209014

•213763

•217672

•SKM68

•222484

•223660

•224042

S

4

•099231

•108151

•116902

•125409

•133602

•141422

•148816

•155739

•162154

•168031

4

5

•041677

•047587

•053775

•060196

•066801

•073539

060360

•087214

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5

6

•014587

•017448

•020614

•024078

•027834

•031867

•036162

•040700

•046457

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6

7

•004376

•005484

•006773

•008255

•009941

•011836

•013948

•016280

•018832

•021604

7

8

•001149

•001508

•001947

•002477

•003106

•003847

•004708

•005698

•006827

•008102

8

9

OOOM8

•000369

000496

•i NKir.no

•000863

•001111

•001412

•001773

•002200

•002701

9

10

•000056

•000081

•000114

•000158

•000216

•000289

•000381

•000496

•000638

•000810

10

11

•000011

•000016

OOOOM

O0008E

•000049

•000068

•000094

•000126

•000168

•000221

11

12

-,:., „„,;.>

•000003 i -000005

O00007

•000010

•000015

•000021

•000029

•000041

•000055

12

IS

—

O00001 -000001

•000001

•000002

•000003

•000004

•000006

•000009

•000013

13

H

—

— —

—

—

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•000001

•000001

•000002

•000003

14

15

~ •

—

^~

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15

B.

15

114 Tables for Statisticians and Biometriciaiu

TABLE LT — (continued).

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3-8

3-3

3-4

3-5

3-6

6-7

3-8

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Poisson's Exponential Binomial Limit

115

TABLE LI— (continued).

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5-1

5-2

5-3

5-4

5-5

5-6

5-7

5-8

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10

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18

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15—2

110 Tables for Stnt!iitifi<tnx and Bi>nntri<-i<tns

TABLE LI— {continued).

m

JT

7-1

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7-5

7-4

7-5

7-6

7-7

7-8

7-9

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9

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10

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14

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14

15

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15

16

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10

17

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17

18

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8-1

8-2

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8-4

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0

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3

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5

0

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6

7

•137778

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7

8

•189600

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8

9

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9

10

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10

11

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14

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009640

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16

16

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16

17

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17

18

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19

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79

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•000184

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go

Poisson's Exponential Binomial Limit
TABLE LI— (continued).

117

m

X

8-1

M

8-3

9-4

8-5

8-0

8-7

8-8

8-9

9-0

X

si

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•000113

•000131

•000152

•000175

•000201

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..-.'

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•000037

•000043

•000051

•000059

•000069

•000081

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83

S3

•000009

•000011

•000013

•000016

•000019

•000022

•000020

•000031

•000030

•000042

23

*4

•000003

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•000005

•000006

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•000011

•000013

•000016

24

26

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25

S6

—

—

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26

27

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9-1

9-»

9-3

8-4

9-5

9-0

9-7

9-8

9-9

wo

X

0

•000112

•000101

•000091

•000083

•000075

•000068

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1

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5

6

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6

7

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7

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•130236

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8

9

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9

10

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10

11

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11

12

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14

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14

15

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15

16

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16

17

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17

18

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18

19

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19

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28

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10-3

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118

Tablet for Stnti.*t!i-i<inx nml /ifinnetririans
TABLE LI— (continued).

*

m

*

10-1

10-i

10-3

10-4

10-5

10-6

10-7

10-8

10-9

11-0

4

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4

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7

8

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8

9

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9

10

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10

11

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11

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096687

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O76060

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13

14

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15

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15

16

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17

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17

18

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18

19

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006897

19

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sis

S3

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11-1

11-g

11-3

11-4

11-5

11-6

11-7

11-8

11 -9

12-0

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0

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1

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8

9

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9

10

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10

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11

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16

17

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17

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18

19

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19

Poissoiis Exponential Binomial Limit
TABLE LI— (continued).

119

m

X

X

11-1

11-2

11-S

11-4

11-5

11-6

11-7

11-8

11-9

12-0

20

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21

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21

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2®

23

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23

24

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24

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25

26

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26

27

•000023

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•000035

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27

28

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•000017

•000019

•000022

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•000029

•000033

28

29

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•000005

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•000010

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29

30

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SO

31

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31

32

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82

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/ '•[

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12 -S

12-4

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12-7

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0

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0

I

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1

2

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S

S

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S

4

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4

5

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6

6

•024233

•023037

•021892

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•018740

•017781

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•015988

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6

7

•041889

•040151

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•036836

•035258

•033733

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•028141

7

8

•063358

•061230

•059142

•057095

•055091

•053129

•061819

•049339

•047511

•045730

8

9

•085181

•083000

•080828

•078665

•076515

•074381

•072266

•070171

•068100

•066054

9

10

103069

•101261

•099418

•097544

•095644

•093720

•091777

•089819

•087849

•085870

10

11

•113376

•112308

•111168

•109959

•108686

•107352

•105961

•104516

•103023

•101483

11

12

•114321

•114180

•113947

•113624

•113215

•112720

•112142

•111484

•110749

•109940

12

IS

•106406

•107153

•107811

•108380

•108860

•109251

•109554

•109769

•109897

•109940

13

14

•091965

•093376

•094720

•095994

•097197

•098326

•099381

•100360

•101263

•102087

lit

15

•074185

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•077670

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•(iHii!i!)7

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15

16

•056103

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•071886

16

17

•039932

•041558

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17

18

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029521

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18

19

•017095

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19

20

•010342

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•014099

•014942

•015816

•016721

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20

11

•006968

•006409

006884

•007383

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•008459

•009036

•009640

•010272

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21

•003278

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•004845

•005216

•005609

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22

.'•!

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23

24

400669

•000958

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•001159

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191
*f

25

•000421

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25

•000196

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•000274

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•000340

•000378

•000420

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26

17

•000068

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•000112

•000126

•000142

•000159

•000178

•000199

•000222

•000248

27

28

•000038

•000043

•000049

•000056

•IKHXX;:?

•000071

•000081

•000091

•000102

•000115

28

•000016

•000018 '000021

•000024

•000027

•000031

•0<XH)35

•000040

•000046

•000052

29

M

•000006

•000007 -000009

•000010

•000011

•000013

•000015

•000017

•000020

•000022

SO

31

•000002

•000003

•000003

•000004

•000006

•0()fMK).-)

•000006

•000007

•000008

•000009

SI

Si

•000001

000001

•000001

•000002

•000002

•000002

•<mxi:>

•000003

•000003

•000004

S2

S3

__

•000001

•000001

•000001

•000001

•000001

•000001

•000002

S3

34

—

—

—

—

—

—

•000001

34

Tables for Statisticians and Biomelricians
TABLE LI— (continued).

m

X

f

13-1

1S-*

13-3

13-4

13-5

13-6

13-7

13-8

13-0

14-0

0

•000002

•000002

•000002

•000002

•000001

•000001

•000001

•000001

•000001

•000001

0

1

- tn

Jl

400089

•000090

•000019

•000017

•000015

•000014

•000013

•000012

i

»

•000175

•000161

•000148

•000136

•000125

•000115

•000105

•000097

•000089

•000081

£

s

•000766

•000709

•000657

•000608

000661

•IK m:, 20

•000481

•000445

•000411

•000380

3

4

,, 8610

•002341

008188

•009030

•001897

•001768

•001648

•001535

•001429

•001331

4

.-.

006570

•I.HilN)

^00807

006458

•005123

•004810

•004514

•004236

•003974

•003727

5

6

•014356

•013596

•012872

•012183

•011526

•oloiH-io

•010308

•009743

•009206

008688

6

7

096867

086686

•024458

•023322

•022230

•021181

•020173

•019207

•018280

•017392

7

8

•043994

0488M

•040661

•039064

•037512

•036007

•034547

•033132

•031762

•030435

8

9

•064036

069048

060068

•058161

066960

•054410

•O.r>25.ss

•060809

•049054

•047344

9

to

•• 39687

•081901

•079916

•077936

•(i7f>!)t;3

•073998

•072046

•070107

•068185

•OfitL"--

10

11

'.<•:••]

0B6961

086686

•094940

•093227

•091489

088790

•087953

•086162

•084359

11

12

•109059

•108109

•107094

•106017

•1018SO

•103687

•102441

•101146

•0'.)!IS0.1

•098418

IS

13

•108698

•109773

•109566

•109279

•108914

•108473

•107957

•107370

•106713

•105989

13

14

109888

•103500

104067

•104595

•105024

•105373

•105644

•105836

•io.v.if,i

•105989

14

15

•089807

•091080

•092291

•093439

•094522

•095539

•096488

•097369

•oasisi

•098923

15

16

•073530

•075141

•076717

•078255

•079753

•081208

•082618

•083981

•085295

•OM;;,r,s

If:

17

•O.>6661

•058345

•060019

•061683

•063333

•(Hii'.Mii;

•ofioriN i

•068173

•069741

•071283

17

18

•041237

•042786

•044348

•045920

•047500

•04!«08i;

•ON*;;.-,

•ori:>2fiti

•053856

•055442

18

19

•028432

•088780

•031043

•032385

•033750

•035135

•036539

•037962

•039400

•040852

19

SO

•018623

•019619

•020644

•021698

•022781

•023892

080080

•026193

•027383

•OL'sr.ii7

20

SI

•011617

•012332

•013074

•013846

•014645

•016473

•016329

•017213

•018125

•019064

SI

S2

006017

•007399

•007904

•OOS433

•IK IS! 1ST

•009565

•010168

•010797

•011462

•012132

n

S3

•003940

•004246

•004571

•004913

•005275

•005656

•ixx;o.-,7

•006478

•<x>r,i»2i

•007385

£3

24

•002151

•002336

•002533

•002743

•002967

•003205

•003457

•003725

•004008

004806

*4

H

•001127

•001233

•001348

•001470

•001602

•001744

•001895

•002056

•002229

•002412

•2:,

M

•000568

•000626

•000688

•000758

•000832

•000912

•000998

•OOKKI1

•001191

•001299

S>:

17

•000275

•000306

•000340

•000376

•000416

•000459

•000507

•000558

•000613

•000674

*7

M

•000129

•000144

•000161

•IX Nil SO

•000201

•000223

•000248

•000275

•000305

•000337

S8

n

•000058

•000066

•000074

•000083

IKXIO:I:J

•000105

•000117

•000131

•000146

•000163

29

50

000086

•000029

•000033

•000037

•000042

•000047

•000053

•OIXXHIO

•OOOOGS

•000076

SO

31

•000011

•000012

•000014

•000016

•000018

•000021

•000024

•Of XX 127

•000030

•000034

31

33

•000004

•000005

•000006

•(XXXK17

•000008

•000009

•000010

•000012

•000013

•000015

38

33

•000002

•000002

•000002

•(.XKXXI3

•000003

•000004

•000004

•000005

•000006

•000006

SS

34

•000001

•000001

•000001

•OIXXXIl

•000001

•OOCXXH

•000002

•CKXXXI2

•ooooo:;

•000003

34

S5

—

—

—

—

—

•000001

•000001

•000001

•000001

•000001

35

X

w

14-S

14-3

14-4

14-r,

14-6

14-7

14-8

14-9

15-0

X

•000001

•000001

•000001

•000001

•000001

_

_

_

0

1

•000011

•000010

•000009

•IMXXI*

•000007

•000007

•000006

•000006

•000005

•000005

1

000070

•000068

•000069

•000058

•ixxior.3

•000049

•000045

•000041

•000038

•000034

H'

.•*

000869

•000325

•000300

•000277

•000256

•(XX)237

•000219

•000202

•000186

•000172

3

4

00U88

•001163

•001073

•OOO'.l'.lll

•Om!IL>:)

•OOOS04

•000803

•000747

•(XXXJ94

•ixxmir,

4

6

•003494

•003275

•003070

•002876

•002694

•002523

•002362

•002211

•(X 121 Hi!)

•001936

5

6

00BU9

•007752

•007316

•006902

tXHolO

•006139

•005787

•005454

•005138

•tx».|s:«)

6

7

016641

016786

•014946

•014199

•013486

•012804

•012162

•011530

•010937

•010370

7

8

•029153

•027913

•016710

•026000

•H24443

•023367

•022330

•021331

•020370

•019444

g

9

046673

•044040

"1-JII7

0408M

•o:!;,:iso

•037907

•036472

•035078

•033723

•032407

9

10

•064399

•062637

•060700

068887

•057101

•055343

•053614

•051915

•050247

048611

10

11

•082647

•080730 -078910

•077089

•076270

•073456

•071648

•069850

•068062

086887

11

Poisson's Exponential Binomial Limit
TABLE LI— (continued).

121

X

m

X

14*

14-2

14-3

14-4

14-5

14-6

74-7

14-8

14-9

15-0

12

•096993

•095530

•094034

•092507

•090951

•089371

•087769

•086148

•084510

•082859

IS

IS

•105200

•104349

•103437

•102469

•101446

•100371

•099247

•098076

•096862

•095607

IS

14

•105951

•105839

•105654

•105396

•105069

•104672

•104209

•103681

•103089

•102436

14

16

•099594

•100195

•100723

•101181

•101567

•101881

•102125

•102298

•102402

•102436

15

16

•087768

•088923

•090021

•091063

•092045

•092967

•093827

•094626

•095361

•096034

16

17

•072795

•074277

•075724

•077135

•078509

•079842

•081133

•082380

•083581

•084736

17

18

•057023

•058596

•060158

•061708

•063243

•064761

•066259

•067735

•069187

•070613

18

19

•042317

•043793

•045277

•046768

•048264

•049763

•051263

•052762

•054257

•055747

19

20

•029834

•031093

•032373

•033673

•034992

•036327

•037678

•039044

•040422

•041810

SO

SI

•020031

•021025

•022045

•023090

•024161

•025256

•026375

•027517

•028680

•029865

SI

S3

•012838

•013570

•014329

•015114

•015924

•016761

•017623

•018511

•019424

•020362

22

n

•007870

•008378

•008909

•009462

•010039

•010640

•011264

•011911

•012584

•013280

23

S4

•004624

•004957

•005308

•005677

•006065

•006472

•006899

•007346

•007812

•008300

24

25

•002608

•002816

•003036

•003270

•003518

•003780

•004057

•004348

•004656

•004980

25

M

•001414

•001538

•001670

•001811

•001962

•002123

•002294

•002475

•002668

•002873

26

27

•000739

•000809

•000884

•000966

•001054

•001148

•001249

•001367

•001473

•001596

27

ts

•000372

•000410

•000452

•000497

•000546

•000598

•000656

•000717

•000784

•000855

S8

29

•000181

•000201

•000223

•000247

•000273

•000301

•000332

•000366

•000403

•000442

29

SO

•000085

•000095

•000106

•000118

•000132

•000147

•000163

•000181

•000200

•000221

SO

SI

•000039

•000044

•000049

•000055

•000062

•000069

•000077

•000086

•000096

•000107

31

32

•000017

•000019

•000022

•000025

•000028

•000032

•000035

•000040

•000045

•000050

S3

S3

•000007 -000008

•000009

•000011

•000012

•000014

•000016

•000018

•000020

•000023

S3

•14

•000003 -000003

•000004

•000005

•000005

-000006

•000007

•000008

•000009

•000010

S4

35

•000001

•IKMJOl

•000002

•000002

•000002

•000002

•000003

•000003

•IHXXMU

•000004

S5

S6

^m

•000001

•000001

•000001

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•000001

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•000002

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36

37

—

•^

—

— —

_.

•000001

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37

B.

16

Table* for Statin! i<-ian* an<l lii<nn<tricinnx

TABLE LII.

Table of Poisson- Exponential for Cell Frequencies 1 to 30.
Coll Frequencies

jr

i

*

3

4

B

•,

7

8

9

10

er

s

*;

•

«>

I

16

is

11

'4

13

"si

> ^

It

t j

^ 1

11

10

•005

-—

•111 :!

•050

f =

•034

•277

- £

f

•091

•302

•tn

1-H33

E §

I
Q

•248

•730

1-37 5

I-9M

s a

•R74

1 -735

2-964

5-4!)6

6-708

*$

j-832

\ji t
4-043

6-197

8-177

9-969

1 1 -5(i9

13-014

8

g

4-979

9-168

12-465

15-120

17-299

W678

u
•

g

13-534

19-915

23-810

26-503

88-806

30-071

31 -337

32-390

83-888

CU

1

36-788

40-601

42-319

43-347

44-049

44-568

44-971

45-296

46-565

45-793

Actual

36-788

27-067

22-404

19-537

17-547

16-062

14-900

13-959

13-176

12-M1

z

26-424

32-332

35-277

37-116

38-404

39-370

40-129

40-745

41 -259

41-696

i

*

8-030

14-288

18-474

21 -487

23-782

25-602

27-091

28 -338

29-401

30-323

S

1-899

5-265

8-392

11-067

13-337

15-276

16-950

18-411

19-G9D

20-845

4

•366

1 -656

3-351

5-113

6-800

8-392

9-852

11-192

12-422

LS-8M

—

B

•059

•453

1-191

2-136

3-183

4-262

5-335

6-380

7 '180

8-346

1

6

•008

•110

•380

•813

1-370

2-009

2-700

3-418

4-146

4-878

s

7

•001

•024

•110

•284

•545

•888

1-281

1-726

2-203

2-705

8

•000

•005

•029

•092

•202

•363

•'i~-i

•823

l-llo

1-428

9

•001

•007

•027

•070

•140

•241

•372

•532

•719

^

10

^m

•000

•002

•008

•023

•051

•096

•159

•242

•346

JS

11

•000

•002

•007

•018

•036

•065

•105

•160

bo

IS

^_

__

•001

•002

•006

•013

•02!)

•044

•071

11

13

m^m

•000

•001

•002

•005

•009

•017

•030

te-g

14

__

^m

—

•000

•001

•002

•003

•007

•013

~ ^

IB

__

—

—

•000

•001

•001

•002

•006

i z

16

—

•000

•000

•001

•002

"SǤ

17

__

—

—

•000

•001

-

18

__

—

•001

"5

19

—

—

—

—

—

—

•000

8

to

M»

—

—

—

—

—

—

—

•

tl

—

—

—

—

—

|

tl

—

—

—

—

—

—

—

B

•

__

—

—

—

—

*

*4

__

—

—

—

—

—

—

§

es

—

—

—

—

—

—

—

—

—

—

S

17

^_

^^

~

_

__

_

__

._

_

es

—

—

—

^"

—

••

—

—

—

Table of Poissoris Exponential

123

TABLE LII— (continued).
Cell Frequencies

*

11

IS

IS

14

IS

16

17

18

19

SO

§

SI

so

>*

1Q

bo

j.y
1A

ll

A.O

17

•000

•000

16

•000

•000

•om

•002

fe t>

15

•000

•000

•001

•002

\nj L
'004

•007

Ss

•000

•001

•002

•004

•008

•015

•026

_3 o

19

'000

•oni

•018

•O7R

40

•001

•003

\AJI
•009

•021

•040

•067

'104

•Til

wl O

•209

•Sj

11

•002

•008

•022

•047

•086

•138

•206

•289

i«Jl

•387

•500

g|

10

•020

•052

•105

•181

•279

•401

•543

•706

•886

1-081

— —

9

•121

•229

•374

•553

•763

1-000

1-260

1-538

1-832

2-139

-"" ~

8

•492

•760

1-073

1-423

1-800

2-199

2-612

3'037

3-467

3-901

11

7

1-510

2-034

2-589

3-162

3-745

4-330

4-912

5-489

6-056

6-613

la

6

3-752

4-582

6-403

6-206

6-985

7-740

8-467

9-167

9-840

10-486

5

7-861

8-950

9-976

10-940

11-846

12-699

13-502

14-260

14-975

15-651

d

4

14-319

15-503

16-581

17-568

18-475

19-312

20-087

20-808

21-479

22-107

ti

,-;

23-198

24-239

25-168

26-004

26-761

27-451

28-084

28-665

29-203

29-703

1

|

34-051

34-723

35-317

35-846

36-322

36-753

37-146

37-505

37-836

38-142

1

45-989

46-150

46-311

46-445

46-665

46-674

46-774

46-865

46-948

47-026

Actual

11-938

11-437

10-994

10-599

10-244

9-922

9-629

9-360

9-112

8-884

1

42-073

42-404

42-695

42-956

43-191

43-404

43-597

43-776

43-939

44-091

a

31-130

31-846

32-486

33-064

33-588

34-066

34-503

34-909

35-283

35-630

H

s

21-871

22-798

23-639

24-408

26-114

25-7fi">

26-367

26-928

27-451

27-939

a

4

14-596

16-559

16-450

17-280

18-053

18-776

19-451

20-088

20-686

21-251

B

9-261

10-129

10-953

11-736

12-478

13-184

13-852

14-491

15-099

15-677

i

6

6-593

6-297

6-983

7-650

8-297

8-923

9-526

10-111

10-675

11-219

2

7

3-219

3-742

4-266

4-791

5-311

5-825

6329

6-826

7-313

7-789

t,

8

1-769

2-128

2-501

2-884

3-275

3-669

4-064

4-461

4-856

5-248

,,

9

•929

1-160

1-407

1-671

1-947

2 -232

2-523

2 '824

3-127

3-433

>^

10

•467

•607

•762

•933

1-117

1-312

1-516

1-732

1-954

2-182

je

11

•225

•305

•396

•502

•619

•746

•882

1-030

1-185

1-348

0*3

H

•104

•148

•201

•261

•331

•411

•497

•595

•699

•809

S 2

IS

•047

•069

•097

•131

•172

•219

•272

•333

•400

•473

S<«

14

•020

•031

•046

•IK;:;

•086

•114

•144

•182

•223

•269

"° s

15

•008

•014

•021

•030

•042

•057

•074

•096

•121

•149

1 I

16

•003

•006

•009

•013

•020

•028

•036

•050

•064

•081

»~ •—

17

•001

•002

•004

•006

•009

•014

•017

•025

•033

•042

J

18

•001

•000

•002

•002

•004

•006

•008

•012

•017

•022

0

19

•000

•000

•001

•001

•002

•(.03

•003

•006

•008

•on

§

.''I

•001

•000

•001

•002

•002

•003

•004

•005

i

SI

•000

•000

•000

•001

•001

•002

•002

•003

t

•000

•000

•001

•001

•001

1

iS

.

_

—

•000

•000

•001

0

14

—

— —

—

—

—

—

—

•000

•g

—

— —

—

—

—

—

—

—

8

86

__

—

—

—

—

1

27

—

—

—

—

—

—

—

—

PH

S8

—

—

—

—

—

~—

™~

~

~

16—3

I-J4

or Stntixtieiatu aiul liionu tn'ritnus

TABLE LII— (continued).
Cell Frequencies

j-

»i

«*

ts

*4

ts

16

X7

ts

t9

50

•ooo

1

SI

__

—

—

—

•000

•ooo

•ooo

001

H

-•'

—

—

—

000

•ooo

•001

•001

•001

oos

£

19

—

—

•ooo

•ooo

•001

•001

•002

•003

•004

006

u

ooo

•ooo

•001

•001

•002

•004

•006

•009

•012

•017

•C"3

17

•001

•002

•003

•006

•008

•on

•016

•023

O8I

•041

_T 1

18

•003

•006

•010

•016

•022

•031

•043

•056

•073

081

^3 ^

IS

•012

•020

•030

•043

•059

•078

•102

•129

•160

•195

la

14

•039

•088

•081

•109

•142

•180

•224

•273

•328

•387

^ -

IS

•111

•150

•198

•252

•314

•384

•460

•543

•632

•727

•g-S

It

•277

•355

•443

•640

•647

•762

•884

1-014

1-161

1-293

"sl

11

<OM

•763

•912

1-072

1-240

1-417

1-601

1-791

1-987

8-187

10

1-290

1-512

1-743

1-983

t-tt9

8*488

2-739

3-000

3-263

3-528

a"°

9

2-455

2-778

3-107

3-440

3-775

4-111

4-446

4-781

5-114

6-444

8.S

8

4-336

4-769

5-200

5-CJ88

6-048

6-463

6-872

7-274

7-669

«-o:.7

~ ;

7

7-157

7-689

8-208

8-713

9-204

9-682

10-147

10-599

11-038

11-465

: 1

6

11-107

11-704

12-277

12-827

13-358

13-867

14-357

14-830

15-285

15-724

.J

5

16-292

16-900

17-477

18-025

18-549

19-048

19-688

19-981

20-417

IO-8M

4

tt-OM

23-250

23-771

24-2(13

24-730

25-172

25-591

25-990

26-371

26-734

8

3

30-168

30-603

31O10

31 -391

31-753

32-094

32-416

32-721

33-011 33-287

1

t

38-426

38-691

38-938

39-168

39-387

39-693

39-786

39-970

40 143

40-806

PN

1

47-097

47-164

47-226

47-283

47-340

47-392

47-440

47-486

47-530

47-S72

Actual

8-671

8-473

8-288

8-116

7-952

7-799

7-654

7-617

7-387

7-184

;

44-232

44-363

44-486

44-603

44-708

44-810

44-906

44-997

45-083

45-165

e

15-866

36-258

36-542

36-812

37-062

.'{7 -.".'it

37-525

37-739

37-942

38-135

8

8

18-897

28-828

29-235

29-620

89-968

30-326

30-653

30-965

31-262

31-546

4

21 -785

22-290

22-770

23-227

23-660

24-074

24-469

24-847

25-208

25-566

6

16-230

16-758

17-264

17-748

18-211

18-655

19-083

19-493

19-888

20-269

§

6

11-744

12-251

12-740

13-213

13-669

14-110

14-638

14-951

15-351

16-738

a

7

8-254

8-709

9-153

9-585

10-007

10-418

10-819

11-210

11-591

11-962

s

8

5-637

6-022

6-402

6-777

7-146

7-609

7-866

8-218

8-662

8-901

H

9

3-742

4O52

4-362

4-670

4-978

6-284

5-688

6-890

6-188

6-484

>>

10

2-416

2-654

2-895

3-138

3-385

3-632

8-880

4-129

4-377

4-625

A

11

1-517

1-692

1-873

2-057

2-246

2-438

2-633

2-831

3-030

3-880

%•*

It

•927

1-051

1-181

1-316

1-456

1-599

1-747

1-899

2-053

2-210

:EI

13

•652

•637

•727

•821

•921

1-025

1-134

1-247

1-362

1-481

11

14

•320

•376

•437

•500

•570

•643

•720

•801

•885

•973

•o-<

m

15

•181

•217

•256

•298

•345

•394

•448

•504

•564

•626

Si

16

•100

•122

•147

•173

•204

•237

•272

•311

•352

•395

t«

17

•054

•067

•082

•098

•118

•139

•162

•188

•21: (i

•246

>
h.*

18

•028

•036

•045

•055

•067

•080

•095

•111

•129

•149

«

19

•016

•019

•024

•030

•037

•045

•054

•065

•076

•089

8

to

•007

•010

•013

•016

•020

•025

•031

•037

•044

•O.V2

1

tl

•004

•005

•007

•009

•on

•014

•017

•021

•025

•030

g

:

•002

•002

•004

•005

•006

•007

•009

•on

•014

•017

§

S3

•001

•001

•002

•003

•003

•004

•005

•006

•007

•010

8

*4

•001

•001

•001

•002

•002

•002

•003

•003

•004

•006

<i

#!

•000

•000

•001

•001

•001

•001

•001

•002

•002

•003

te

•000

•IK HI

•000

001

•001

•001

•001

•002

-

t7

_

__

ooo

•000

•000

•000

•001

du

18

—

—

—

—

—

— .

—

—

•000

i

TABLE LIII. Angles, Arcs and Decimals of Degrees 125

LENGTHS or CIRCULAR ARCS

Deg

Arc

Deg.

Arc

Deg.

Arc

/

Deg.

Arc

>i

Deg.

Arc

1

•017 4533

61

1-0646508

Ul

2-1118484

1

•01667

•000 2909

1

•00028

•0000048

S

•0349066

62

1-0821041

122

2-1293017

2

•03333

•000 5818

S

•00056

•0000097

s

•052 3599

6S

1-0995574

123

2-146 7550

3

•05000

•000 8727

S

•00083

•0000145

4

•069 8132

64

1-1170107

124

2-164 2083

4

•06667

•001 1636

4

•00111

•0000194

S

•087 2665

65

1-1344640

125

2-181 6616

5

•08333

•001 4544

5

•00139

•0000242

6

•104 7198

66

1-151 9173

126

2-199 1149

6

•10000

•001 7453

6

•00167

•0000291

7

•122 1730

67

1-1693706

127

2-216 5682

7

•11667

•002 0362

7

•00194

•0000339

»

•1396263

68

1-1868239

128

2-234 0214

8

•13333

•002 3271

8

•00222

•0000388

9

•157 0796

69

1-2042772

129

2-251 4747

9

•15000

•002 6180

9

•00250

•000 0436

10

•174 5329

70

1-221 7305

ISO

2-268 9280

10

•16667

•0029089

10

•00278

•0000485

11

•191 9862

71

1-2391838

131

2-2863813

11

•18333

•003 1998

11

•00306

•0000533

12

•209 4395

72

1-2566371

132

2-3038346

12

•20000 : -0034907

12

•00333 -0000582

IS

•226 8928

73

1-2740901

133

2-321 2879

IS

•21667

•003 7815

IS

•00361

•0000630

14

•244 3461

74

1-291 5436

134

2-338 7412

14

•23333

•0040724

14

•00389

•0000679

15

•261 7994

75

1-308 9969

ISo

2-356 1945

15

•25000

•004 3633

15

•00417

•0000727

16

•279 2527

76

1-3264502

IStl

2-3736478

16

•26667

•004 6542

16

•00444

•0000776

17

•296 7060

77

1-3439035

137

2-391 1011

17

•28333

•004 9451

17

•00472

•0000824

18

•314 1593

78

1-361 3568

138

2-408 5544

18

•30000

•005 2360

18

•00500

•0000873

19

•331 6126

79

1-378 8101

139

2-4260077

19 -31667

•005 5269

19

•00528

•0000921

to

•349 0659

80

1-3962634

140

2-4434610

90 -33333

•005 8178

20

•00556

•0000970

SI

•3665191

81

1-4137167

141

2-460 9142

SI

•35000

•0061087

21

•00583

•0001018

X?

•383 9724

82

1-431 1700

14-'

2-478 3675

22

•36667

•0063995

22

•00611

•0001067

2s

•401 4257

83

1-4486233

143

2-495 8208

23

•38333

•0066904

23

•00639

•0001115

114

•418 8790

84

1-4660766

144

2-5132741

24

•40000

•0069813

24

•00667

•0001164

S5

•436 3323

85

1-483 5299

Itf

2-530 7274

•41667

•007 2722

26

•00694

•0001212

26

•453 7856

86

1-5009832

140

2-548 1807

26

•43333

•007 5631

M

•00722

•000 1261

27

•471 2389

87

1-5184364

147

2-565 6340

S7

•45000

•007 8540

27

•00760

•0001309

U

•488 6922

88

1-535 8897

148

2-583 0873

28

•46667

•0081449

28

•00778

•000 1357

29

•506 1455

89

1-5533430

149

2-600 5406

29

•48333

•0084358

29

ooeoe

•0001406

SO

•523 5988

90

1-5707963

150

2-617 9939

30

•60000

•0087266

SO

•00833

•0001454

SI

•541 0521

91

1-5882496

151

2-635 4472

SI

•51667

•0090175

31

•00861

•000 1503

an

•558 5054

92

1-6057029

i:< .'

2-652 9005

32

•53333

•0093084

32

•00889

•000 1551

ss

•576 9587

93

1-6231562

153

2-6703038

33

•55000

•0095993

S3

•00917

•0001600

34

•5934119

91

1-6406095

154

2-687 8070

34

•56667

•0098902

34

•00944

•000 1648

S5

•6108662

95

1-6580628

156

2-705 2603

35

•58333

•0101811

SS

•00972

•0001697

SG

•628 3185

9G

1-6755161

156

2-722 7136

36

•60000

•0104720

36

•01000

•0001745

S7

•645 7718

97

1-6929694

157

2-740 1669

37

•61667

•010 7629

37

•01028

•000 1794

S8

•663 2251

98

1-710 4i!-J7

168

2-757 6202

M

•63333

•0110538

38

•01056

•0001842

S9

•6806784

99

1-727 8760

169

2-7750735

39

•65000

•Oil 3446

39

•01083

•0001891

40

•698 1317

100

1-745 3293

160

2-792 5268

40

•66667 -Oil 6355

40

•01111

•0001939

41

•7155850

101

1-7627825

161

2-809 9801

41

•68333

•Oil 9264

41

•01139

•000 1988

*

•7330383

102

1-7802358

16S

2-827 4334

it:

•70000

•0122173

42

•01167

•0002036

V

•7604916

103

1-797 6891

163

2-844 8867

«a

•71667

•0125082

4S

•01194

•0002085

44

767 9449

104

1-815 1424

164

2-862 3400

44

•73333

•012 7991

44

•01222

•000 2133

45

•785 3982

105

1-832 5957

165

2-879 7933

45

•75000

•0130900

45

•01250

•0002182

•',>•>

•8028515

106

1-8500490

166

2-897 2466

46

•76667

•0133809

46

•01278

•000 2230

4'

•8203047

107

1-8675023

167

2-914 6999

If!

•78333

•0136717

47

•01306

•000 2279

ta

•8377580

108

1-8849556

168

2-932 1531

4*

•80000

•0139626

48

0-1333

•000 2327

w

•8552113

109

1-9024089

169

2-949 6064

49

•81667

•014 2535

49

•01361

•000 2376

so

•872 6646

110

1-9198622

170

2-967 0597

60

•83333

•014 5444

60

•01389

•000 2424

SI

•890 1179

111

1-9373155

in

2-9845130

61

•86000

•014 8353

51

•01417

•0002473

ftf

•907 5712

lie

1-9547688

172

3-001 9663

62

•86667

•016 1262

62

•01444

•000 2521

53

•925 0245

us

1-972 2221

173

3-0194196

53

•88333

•0154171

53

•01472

•000 2570

54

•942 4778

114

1-9896753

174

3-0368729

54

•90000

•016 7080

54

•01500

•0002618

55

•9699311

115

2-007 1286

175

3-054 3262

55

•91667

•015 9989

66

•01528

•000 26(i6

56

•9773844

116

2-024 5819

176

3-071 7796

56

•93333

•016 2897

56

•01556

•0002715

57

•994 8377

117

2-042 0352

177

3-089 2328

57

•95IKX)

•016 5806

57

•01583

•000 2763

58

1O122910

118

2-059 4885

178

3-1066861

58

•96667

•0168715

58

•01611

•0002812

59

1-0297443

119

2-0769418

179

3-1241394

59

•98333

•017 1624

59

•01639

•000 2800

60

1-0471976

1X0

2-094 3951

ISO

3-141 5927

60

1-00000

•017 4533

60

•01667

•0002909

I -JO

Table* for Statisticians and

TABLE L1V. The 0(r, v)-Integral*.

'

r-1

r-2

^<,,>

A

A«

*FCr..>

logH(r,i.)

A

A«

0

0-301 0300

o/wi

0-196 1199

0-196 1199

1

•3010609

ouy

619

•196 2052

•liiii 13IM

383

I

•301 1538

928

621

•1964614

•196 l!H;ii

675

383

3

•3013087

1550

O 1 ~ ~

62r>

•1968890

•196 2924

958

382

4
6

•301 5268
•301 8067

1170

2- '.-,

C3o
636

•197 4890
•198 2627

•196 4264
•106 6985

1340
1722

381
380

6
7
8
9
10

•3021508
•302 5594
•303 0335
•3035741
•304 1825

3441
4086
4740
5406
6085

645
654
666
679
693

•1992118
•2003385
•201 6452
•203 1349
•2048110

•1968087
•187 0567
•197 3424
•197 6655
•1980260

2102
21*0
2857
3231
3604

379
377
376
373
370

11

1 ;

•304 8603
•3056091

6778

Tiss

710
728

•206 6774
•208 7382

•198 4234
•198 8574

3974
4341

367
364

I',

•306 4307
•307 3-271
•308 3006

8964
9735

749
771
796

•210 9985
•213 4631
•216 1383

•199 3280
•199 8344
•200 3764

4705
BOM

6420

360
366
352

1 :
17
18
19

to

•3093538
•310 4892
•311 7098
•313 0189
•314 4200

10532
11353
12206
13091
14011

822
853
885
919
958

•219 0303
•222 1462
•225 4936
•22!) 0807
•232 9167

•200 9537
•2ol 5657
•202 2120
•202 8921
•203 6054

6773
6120
6463
6801
7133

347
343
338
332
327

tl
U

•315 9169
317 5137
•319 2160
•321 0266
•322 9507

14969
15968
17013
18106
19252

999
1045
1093
1146
1204

•2370114
•241 3755
•246 0203
•250 9584
•256 2034

•204 3514
•205 1294
•205 9387
•206 7787
•207 6487

7460
7780
8093
8400
8700

320
313
307
•00

291

17
.'8

SO

•324 9963
•327 1685
•329 4740
•331 9202
•3345150

20456
21722
23066
24462
25948

1266
1334
1407
1486
1573

•261 7697
•267 6733
•2739311
•280 5618
•287 5852

•208 5478
•209 4753
•210 4302
•211 4118
•212 4190

8991
9275
9549
9816
1 0072

284
275
266

2M!
247

31

.:
33
34
35

•337 2672
•340 1860
•343 2818
•346 5656
•3500496

27521
29188
30958
32838
34840

1667
1769
1881
2002
2134

•295 0232
•302 8992
•311 2388
•320 0695
•329 4214

•213 4609
•214 6064
•2156846
•2166842
•217 8041

10319
1 0556
1 0782
10996
1 1199

236

ne

213
203
191

36
37
38
39
40

•353 7469
•357 6722
•361 8410
•306 2708
•370 9805

36974
39252
41689
44298
47096

2279
2436
2609
2799
3007

•339 3271
•349 8221
•360 9461
•372 7382
•385 2475

•218 9431
•22(1 1000
•221 2734
•222 4621
•223 6644

1 1390
1 1569
1 1734
1 1886
12023

178
165
152
138
125

41

43
44
45

•375 9908
•381 3246
•387 0070
•393 0658
•399 6316

60103
53338
56824
60588
64658

3235
MM

3764
4069

•398 6232
•412 6205
•427 5995
•443 5266
•460 4746

•224 8791
•226 1048
•227 3397

•22* 5.H24
•229 8313

1 2148
1 2256
1 2350
1 2427
12489

109
94
77
62

Tables of the (7 (r, v)-Integrals
TABLE LIV— (continued).

127

,

r=3

r=4

r=5

lo3F(r,»)

logH(r.r)

A

A>

.<**-(,-„>

logH(r,i.)

A

A»

log^,,.

log H (r, F)

A

A2

0

0-124 9387

0-275 4537

0-071 1811

0-309 7418

105

0-028 0289

0-329 0589

QJ

1

•125 0847

•275 4674

41 A

273

•071 3902

•309 7523

316

211

•028 3019

•329 0673

o4

173

2

•125 5230

•275 5084

*xlU
CQ*3

273

•072 0177

•309 7840

527

211

•029 1221

•329 0930

257

171

3

•126 2545

•275 5767

272

•073 0650

•309 8366

707

210

•030 4908

•329 1357

428

171

4

•127 2807

•275 6721

1 9->ft •* •

•074 5342

•309 9103

191

Q4R

209

•0324110

•329 1956

598

fT£»O

170

5

•128 6039

•275 7948

I2-2" 270

•076 4285

•310 0049

J7tU

;208

•034 8865

•329 2723

Too

169

6

•130 2270

•2759444 !-^! 269

•078 7517

•310 1204

1154
lifil

207

•037 9224

•329 3661

937

1 1 AK

168

7
8
9
10

•132 1533
•134 3870
•136 9331
•139 7969

•276 1209
•276 3242
•276 5540
•276 8101

1 /OO

2032
U98

2561

267
266
263
261

•081 5088
•084 7055
•088 3486
•092 4458

•310 2565
•310 4132
•310 5904
•310 7877

1OD1

1567
1771
1973

206
204
202
201

•041 5250
•045 7016
•050 4609
•055 8130

•329 4765
•329 6037
•329 7474
•329 9075

1105

1272
1437
1601

167
165
164
162

11

•142 9850

•277 0923

2822

258

•0970060

•3110051

2174
2372

198

•0617690

•330 0838

1763

1 QOO.

160

12
IS

•146 5043
•150 3626

•277 4003
•277 7338

,,,,, 25fi
333f) .,..,

•102 0399
•107 5555

•311 2423
•311 4990

2567
2760

196
193

•0683415
•075 5446

•330 2761
•330 4842

lyz.3

2081

158
156

15

•154 5688
•159 1322

•278 0926
•278 4762

3837

249
245

•1135680
•120 0895

•311 7751
•312 0701

2950

190

187

•083 3937
•0919061

•330 7079
•330 9469

2390

153
151

16
17
18
19
SO

•164 0636
•169 3743
•175 0768
•181 1848
187 7130

•278 8844
•279 3167
•279 7726
•280 2519
•280 7539

4082
4323
4560
4793
5020

241
237
•J33

22H
223

•127 1349
•134 7199
•142 8621
•151 5802
•160 8948

•312 3838
•312 7159
•313 0660
•313 4337
•313 8186

3137
3321

3501
3677
3849

184
180
176
172
168

•101 1002
•110 9967
•121 6176
•132 9872
•145 1317

•331 2010
•331 4700
•331 7532
•332 0508
•332 3621

2541
2689
2833
2975
3114

148
144
142
138
135

tl

S3

•194 6774
•202 0955

•281 2782
•281 8243

5243
5460

217
212

•170 8281
•181 4042

•314 2204
•314 6385

4018
4181

164
160

•158 0795
•171 8614

•332 6870
•333 0250

3249
3380

131
127

S.I

S4
SS

•209 9858
•218 3688
•227 2664

•282 3915
•282 9794
•283 5873

5879
6079

206
200
194

•192 6491
•204 5907
•217 2596

•315 0726
•315 5222
•315 9866

4495
4645

154
150
144

•186 5105
•202 0627
•218 5568

333 3757
•333 7387
334 1137

3507
3630
3750

123
119
115

26

27

;.'X

SO

•236 7023
•246 7020
•257 2933
•268 5060
•280 3725

•284 2145
•284 8604
•285 5244
•280 2057
•286 9035

6272
6460
6640
6813
6978

188
180
173
165
158

•230 6885
•244 9127
•259 9707
•275 9034
•292 7555

•316 4655
•316 9583
•317 4644
•317 9833
•318 5143

4789
4927
5062
5189
5310

138
134
127
121
116

•236 0346
•254 5413
•274 1255
•294 8399
•316 7413

•334 5001
•334 8976
•335 3057
•335 7239
•336 1516

3864
3975
4081
4182
4277

110

106
101
96
91

SI
32
S3
34
SS

•292 9278
•3062096
•320 2589
•33.-> 1201
•350 8413

•287 6170
•288 3456
•289 0883
•289 8443
•290 6127

7136
7286
7427
7560
7684

150
141
133
124
115

•310 5754
•329 4149
•3493304
•370 3832
•392 6390

•319 0569
•319 6104
•320 1741
•320 7474
•321 3297

5426
5535
5637
5733
5822

109
102
96
89
82

•339 8909
•364 3553
•390 2059
•417 5203
•446 3827

•336 5884
•337 0339
•337 4874
•337 9485
•338 4164

4368
4455
4535
4610
4680

87
80
75
70
64

36
37
38
39
40

•367 4747
•385 0770
•403 7099
•423 4403
•444 3416

•291 3927
•2d2 1832
•292 9834
•293 7923
•294 6089

7800
7905
8002
8068

8166

106

97

87

77
67

•416 1697
•441 0529
•467 3733
•495 2227
•624 7011

•321 9201
•322 5181
•323 1228
•323 7335
•324 3495

5904
5980
6047
6107
6160

75

68
60
53
45

•476 8841
•509 1232
•543 2072
•579 2529
•617 3872

•338 8908
•339 3709
•339 8563
•340 3463
•340 8404

4744
4802
4854
4900
4940

58
52
46
40
34

41g

•466 4933

•2954321 \

57

•555 9177

•324 9700

6205

fi941

38

•657 7483

•341 3378

4975

•ifift'-l

28

•480 MM

•296 2610 "™ 46

•588 9916

•325 5943

v&to
00*70

30

•700 4872

•341 8381

OUU«5

22

A5

•514 9055

•2970946 oo-'i' 38

•624 0530

•326 2216

OZ7Z

, • _M i "

22

•745 7688

•342 3405

5024

16

•541 3<i58

•297 9316 -iAi

23

•061 2446

•326 8510

bzyo
Rim

12

•793 7739

•342 8446

5040

KO.4Q

9

'i '

•:.<;:> 4783

•2987710 c

•700 7225

•327 4817

VOV/ 1

•844 6999

•343 3495

J\.'4:i/

1

r.'s

I': il, leg for Statisticians and ni<nnrtri<-itiH9
TABLE LlV—(cvntiHued).

'

,-.

r-7

r-8

logf(r.r)

logH(r.r)

A

A»

Iog7>(r.r)

logH(r, r)

A

A>

logF(r.r)

logW(r,,)

A

A*

0

1-991 9999

0-341 4849

70

1-961 0819

0-350 1576

1-9340080

0-356 6570

J

•9923379

•341 4921

f 9
91 A

144

•9614851

•350 1638

1 ftft

124

•934 4765

•356 6624

1 fi't

109

s

•9933626

•341 5137

21O

144

•9626953

•350 1824

loo

124

•935 8831

•356 6788

HM

979

109

•995 0459

•341 5496

i.

143

•964 7161

•350 2134

i*

123

•938 2304

•356 6059

mtm

10S

'.

•9974213

•341 6999

fUfi

143

•967 5483

•360 2567

r,r c

123

•941 5232

•356 6440

AQQ

108

.

0-0004836

•3416644

142

•9712008

•3503123

OOD

122

•945 7679

•356 6928

*OO

107

I

•0042390

•341 7432

787

QOQ

141

•975 6796

•350 3801

878

122

•950 9729

•356 7524

596

107

7

•0086960

•341 8360

yzo

140

•980 9940

•350 4601

QOl

121

•957 1486

•356 8226

OAU

KM;

8

•013 8607

•311 iU'.'s

1 9A7

139

•987 1544

•350 5522

HU

120

•964 3073

•366 9034

DUO

105

9
10

•019 7468
•026 3653

•:i 1-2 0635
•342 1980

1 £\J4

1345

137
136

•994 1735
0-0020658

•350 6662
•350 7720

1158

118
117

•972 4634
•981 6333

•366 9947
•357 0961

1017

104
103

77

•0337300

•342 3461

1481

134

•010 8465

•350 8995

1275

i om

116

•991 8357

•357 2083

1120

1 991

101

75
75

•0418562
•0507609
•0604633
•070 9839

•342 5076
•3426823
•342 8701
•343 0707

1747
1878
2006

133
131

128
127

•020 5347
•031 1503
•042 7157
•056 2551

•351 0386
•351 1891
•351 3508
•351 6235

UVJ

1505
1617
1727

114
112
110
109

0003 0914
•015 4237
•028 8583
•043 4233

•357 3304
•357 462.-.
•357 6044

•357 7560

1 — _ i

1321
1419
1516

100
98
97
95

76
77
18

•082 3457
•094 5734
•107 6941

•343 2839
•343 5095
•343 7472

2133

2256
2377

123
121
119

•0687956
•083 3665
•0989997

•351 7071
•351 9012
•352 1058

1836
1942
2046

•» 1 .IS

106
104
102

•059 1498
•076 0714
•094 2250

•357 9170
•358 0874
•358 2669

1611
1704
1795

1884

93
91
89

.'"

•121 7375
•136 7352

•343 9968
•3442579

2611

115
112

•1157298
•133 5946

•3.12 3206
•352 5452

— 1**O

2247

99
97

•1136505
•1343912

•35S 4553
•358 6524

1971

87
84

tl

:.'
S3
S4
S5

•152 7219
•169 7350
•187 8149
•207 0053
•227 3532

•344 5302
•3448135
•345 1074
•3454116
•345 7256

2723
2833
2939
3041
3141

109
106
103
99
95

•152 6347
•172 8941
•194 4206
•217 2653
•241 4839

•352 7795
•353 0232
•353 2760
•353 5375
•353 8075

2343
2437
2528
2615
2700

94
91

ss
85
82

•156 4939
•180 0093
•204 9923
•231 5019
•259 6019

.358 8579
•359 0716
•359 2933
•359 5226
•359 7594

2055
2137
2217
2293
2368

82
80
77
74
71

26

in

•2489095
•271 7291

•346 0492
•346 3819

3236
3328

92

88

•267 1362
•294 2868

•354 0857
•354 3717

2782
2860

OOQH

79
75

•2893613
•320 8543

•360 0033
•360 2540

2439
2507

68
65

28
29
90

•295 8713
•321 3998
•348 3836

•346 7235
•347 0734
•347 4311

3415
3499
3677

84
78
76

•323 0053
•353 3670
•385 4529

•354 6652
•354 9659
•355 2732

WBO

3007
3074

72
67
64

•354 1610
•389 3678
•426 5682

•3605112
•360 7747
•361 0441

2635
2694

62
59
56

•376 8974

•347 7965

3653

71

•419 3506

•355 5871

3138

Ql Of.

60

•465 8626

•361 3191

-750

52

.

•407 0214

•348 1689

3724

3""Q1

66

•465 1549

•355 9070

J19!'

M

•507 3601

•361 5993

OQR1

49

33

- ;

35

•4388428
•472 4666
•5079618

•3485480
•348 9332
•349 3242

Ivl

3852
3910

62
67
52

•492 9680
•5329005
•575 0721

•356 2324
•356 5632
•356 8988

3308
3356

53
49
44

•651 1780
•597 4436
•646 2944

•361 8844
•362 1741
•362 4681

-. • ' 1

2897
2939

'46
42
39

36
37
38

sa

40

•545 4718
•585 1052
•626 9922
•671 2739
•718 1040

•3497204
•3501213
•350 5265
•350 9354
•351 3477

3962
4009
4052
4090
4122

47
43
38
33
28

•6196127
•666 6R29
•716 3753
•768 9159
•824 4653

•357 2388
•357 5829
•357 9306
•358 2814
•358 6350

3400
3441

3477
3509
3536

40
30
32
28
23

•697 8795
•752 3605
•809 9127
•870 7267
•935 0097

•362 7659
•363 0671
•363 3715
•363 6787
•363 9883

2978
3013
3044
3072
3096

35
31
28
M
20

*i
43
44
46

•767 6502
•820 0951
•8766383
•934 4983
•9969145

•351 7627
•352 1799
•352 6989
•353 0191
353 4399

4150
4172
4190
4202
4209

22
17
12

7

•883 2199
•!M.r) 3944
1-011 2229
•0809618
•154 8920

•358 9910
•359 3488
•359 7080
•3600682
•360 4290

3559
3578
3592
3602
3608

19
14
10
6

1002 9876
•074 9063
•151 0352
•231 6680
•317 1271

•3642998
•364 6130
•:(»i4 9274
•31 !5 2427
•365 5584

3116
3132
3144
3153
3157

16
12
9
6

Tables of the G (r, v}-Tntegrals
TABLE LIV— (continued).

129

,

r=9

r=10

r=ll

logF(r, r)

log H (r, »)

A

A*

loSF(r, v)

loglTtr,,,

A

A*

log*C>,,)

logH(r, »)

A

A2

0

1-909 9294

0-361 4744

1-888 2505

0-365 3717

id.

T-868 5367

0-368 5367

1

•910 4635

•361 4793

J

97

•888 8502

•365 3761

***±

87

•869 2023

•368 5407

4U

80

•912 0669

•361 4938

146

97

•890 6508

•365 3892

131

87

•871 2002

•368 5527

120

1 QQ

79

s

•914 7427

•361 5180

™T

97

•893 6556

•3654111

8JLH

87

•874 5345

•368 5725

um

79

4

•918 4961

•361 5520

339

96

•897 8705

•365 4416

306

?Q9

87

•879 2114

•368 6004

278

79

5

•923 3346

•361 5955

96

•903 3037

•365 4808

OM

86

•885 2402

•368 6361

78

ill

<T7Q

J.'}ei

6

•929 2675

•361 6485

OOl

95

•909 9658

•365 5287

"*/ a

564

86

•892 6324

•368 6796

**Oc.

514

78

7

•936 3067

•361 7111

7OA

94

•917 8700

•365 5851

85

•901 4027

•368 7310

Kni

77

8

•944 4661

•361 7831

I - '
Ol O

94

•927 0317

•365 6500

•70*1

84

•911 5681

•368 7900

Ot7l
ffCO

77

9

•953 7619

•361 8644

ol.)

92

•937 4692

•365 7233

Mo

Qi 7

83

•923 1487

•368 8568

DDo

76

10

•964 2128

•361 9550

906

92

•949 2033

•365 8050

Ol/

82

•936 1674

•3689311

75

QQ7

AQO

Q1Q

11

•975 8398

•362 0547

• <<> t

lilSS

90

•962 2575

•365 8949

O*7»7

82

•950 6505

•369 0129

O1O

892

74

IS

•9886667

•362 1635

1 VOO

89

•976 6581

•365 9929

HOI

80

•966 6268

•369 1022

73

IS

0-002 7197

•362 2312

1177

•992 4345

•366 0990

1061

1 10Q

79

•984 1288

•369 1987

1037

72

14
15

•018 0278
•034 6230

•362 4075
•362 5426

1350

W
84

0-009 6193
•028 2479

•3662129
•366 3346

1 1 • >• '

1217

78
76

0-003 1923
•023 8567

•369 3024
•369 4132

1108

71
69

16

•i i.->2 5403

•3626860

1435

1 PL] Q

83

•048 3595

•366 4639

1293

1 Q«Q

75

•046 1651

•369 5308

1177
1245

68

17

•071 8179

•362 8378

1 • ' 1O

81

•069 9966

•3666007 \"™

73

•070 1646

•369 6553

l *ii i

66

18
19

•092 4974
•114 6239

•362 9976
•363 1654

1678

79

77

•093 2058
•1180374

•366 7447
•366 8960

i****l

1512

72
69

•095 9063
•123 4460

•369 7864
•369 9240

1O11

1376

1 J.1Q

65
63

20

•138 2465

•3633409

75

•144 5461

•367 0541

68

•152 8439

•370 0679

IrtO*/

62

21

23
24

u

•1634180
•190 1960
•SU64U

•248 sj.'iT
•280 8124

•363 5239
•:i*;s 7142

•303 9115
•364 1157
•364 3264

1830
1903
1973
2042
2107

73
71
68
66
63

•172 7910
•202 8360
•2:! 47504
•268 6087
•304 4913

•367 2190
•367 3904
•367 5682
•367 7522
•367 9420

1649
1714
1778
1839
1899

66
64
61
59
57

•184 1654
•217 4809
•252 8669
•290 4057
•330 1857

•370 2179
•370 3739
•370 5357
•370 7030
•370 8757

1500
1560
1618
1673
1727

60
58
56
54
52

26

•314 6864

•3645435 *J'J

61

•342 48.11

•368 1376

IftM

55

•372 3033

•371 0536

1779

50

27

sa

•350 5295

•388 i:iL.':i

•364 7666
•364 9955

2290

58
55

•3826838
•-12.} 1882

•368 3386
•368 5448

2062

52
50

•4168615
•463 9718

•371 2365
•371 4241

1876

47
45

2'J
SO

•I2S 4925
•470 8156

•365 2300
•365 4697

2397

50

•470 1076
•517 5593

•3687559
•368 9718

2112
2159

47
44

•513 7544
•566 3390

•371 6162
•371 8125

1921
1964

43

40

31
S2
33

•615 5153
-.'.'; 2 7146
•6125463

•365 7144
•365 9636
•366 2173

2447
2493
2537

OR*"-

47
44

40

•567 6703
•620 5776
•676 4293

•369 1922
•369 4167
•369 6451

2203
U4fi

2284

910 1

12
39
30

•621 8657
•680 488 1
•7423617

•372 0129
•372 2171
•372 4249

2004
2042
2078
91 in

38
36
33

34
35

I Ml

•7S069M

•3664750
•366 7365

mOi t

2615

37

M

•735 3855
•797 6194

•369 8772
•3701126

Mm

2354

31

•807 6710
•876 6045

•372 6359
•372 8500

£iL 1<J

2141

30
28

36
37
38

•779 3322
•841 2534
•9066548

•367 0013
•••','••- 2693
•367 5400

2649
2680
2707

97Q1

n

28

24

•863 3188
•932 6869
1-005 9444

•370 3510

•370 5!»23
•370 8360

2385
2412
2437

28
25
22

•949 3690
1-026 1888
•107 3073

•373 0669
•373 2862
•373 5078

2169
2194
2216
223G

25
23

20

39

1)7.'. 7519

•367 8131

XI 04
07 to

21

•083 3313

•371 OM'.i

O/I7Q

19

•192 9888

•373 7314

17

•i"

1-048 7784

•3i;8 0884

2i ->J,

18

•165 1080

•371 3297

z47o

16

•283 5209

•373 9567

''

14

41
4*
43
44

•125 (txii!
•207 W2 J
•2!I4 1013
•38.-) 6379
•482 •

•368 3654
•368 6438
•368 9:i.T(
•3<J!) 2036
•369 4842

2770
2784
2795
2803
2806

14

11

7
4

•251 5588
•342 9931
•139 7491
•:.12 1966
•6607407

•371 5790
•371 8296
•372(1*13
•372 3335
•372 5861

2494
2506
2516
2522
2526

13
10
6
4

•379 2165
•480 4171
•587 4953
•700 8587
•820 9540

•374 1834
•374 4113
•374 6400
•374 8693
•375 0990

2267
2279
2287
2293
2296

12
9
6
3

17

130

Tables for Shitixticiam an<l

TABLE LIV— (continued).

<•

r-12

r-13

r=14

log *•(•-. »)

logH(r.r)

A

A-

logF(r.r)

logH(r,»)

A

A*

log *•(!-, ,)

logff(r.r)

A

A.

0

1-850 4619

0-371 1582

T-833 7746

0-373 3653

T-818 2772

0-375 2489

5tl

1

•851 1933

•371 1619

73

•834 5719

•373 3686

101

68

•819 1404

•376 2620

01

63

t

•8533889

•371 1729

1 UQ

73

•836 9652

•373 3787

1 VI

IfiO

67

•821 7316

•375 2614

63

3

•8570528

•371 1912

100

73

•840 9592

•373 3956

LOT
O-lf'

• 17

•826 0558

•375 2771

21ft

62

4

•8621923

•371 2166

72

•846 5616

•37341:12

B0U
•>AO

67

•832 1212

•375 2990

QQl

62

6

•868 8171

•371 2494

72

•853 7829

•373 4494

OvO

67

•839 9395

•3753271

•01

62

ACH\

OOQ

0^0

6

•876 9402

•371 2893

t±\AJ

471

71

•862 6374

•373 4864

OO<7

66

•849 6258

•375 3614

O4O

Al)A

61

7

•886 5774

•371 3364

** 1 i

71

•873 1421

•373 6299

rm

66

•860 8985

•375 4019

^V"«

61

X

•897 7474

•371 3907

i ; 1 0

70

•885 3174

•373 5800

r.i'i'

65

•874 0798

•376 4484

BM

60

9

•910 4722

•371 4519

ulo

69

•899 1873

•373 6365

ilDO

65

•889 0954

•375 5010

KQK

60

10

•924 7770

•371 5201

69

•914 7789

•373 6995

630

64

•905 9746

•375 5595

HOP

59

11

•9406901

•371 5952

751

Q1 O

68

•932 1233

•373 7689

694

63

•924 7510

•375 6239

644

58

It

•958 2436

•371 6771

olU

ss1.;

67

•951 2549

•373 8445

01 Q

62

•945 4618

•375 6942

"60

57

•977 4727

•371 7656

OOD

T, 1

66

•972 2124

•373 9263

Olo

O»TQ

61

•968 1485

•376 7702

QIC

56

14

•998 4167

•371 8608

yui

65

•995 0382

•374 0142

O/J7

60

•992 8571

•375 8518

O1O

s7->

55

15

OO21 1187

•371 9624

64

0019 7792

•374 1081

938

59

0-019 6382

•375 9390

Oli

54

16

•045 6258

•372 0703

1080

1 149

62

•046 4865

•374 2078

997

58

•048 5469

•376 0317

QS<1

53

17

•0719896

•372 1846

1 i i —

61

•075 2161

•374 3132

10OJ

56

•079 6435

'376 I^-'T

OO\J
1 {Vift

62

18
10
SO

•100 2660
•130 5159
•162 8054

•3723048
•372 4310
•372 5630

1262
1320

59
58
56

•106 0288
•138 9908
•174 1737

•374 4243
•374 5409
•374 6628

1111
1166
1219

55
53
52

•1129939
•148 6693
•186 7470

•376 2329
•3763412
•376 4544

lOoZ
1083
1133

61
50
48

.7

•197 2059

•372 7006

1376

1 JQ1

65

•211 6551

•374 7899

1271

51

•227 3107

•376 5725

1181

1 99ft

47

_• • >

•233 7945

•372 8437

l*sOl

53

•251 5187

•374 9221

1322

49

•270 4509

•376 6952

i 440

1 O"'*!

45

',;

•272 6547
•313 8765

•372 9921
•373 1456

1535

51
49

•293 8552
•338 7623

•375 0591
•375 2008

1370
1417

47
46

•316 2653
•364 8594

•376 8225
•376 9542

1X79

1317

44
42

.'•'>

•357 5571

•373 3040

4

47

•386 3455

•375 3472

1464

43

•416 3469

•377 0901

40

„;

•403 8013

•373 4672

1632

45

•436 7186

•375 4978

1506

42

•470 8506

•377 2301

1399

1 A "' i

39

.'X

•452 7221
•504 4412

•373 6349
•373 8069

1720

43

41

•490 0042
•546 3346

•375 6527
•375 8115

154!)
1589

40
38

•528 5029
•589 4465

•377 3739
•377 5215

14.sU
1476
I'll l

37
35

.'.'I

•559 0903

•373 9831

1 DAI

39

•605 8526

•375 9742

1D1/

36

•653 8352

•377 6726

iiji i

33

.;n

•616 8111

•374 1631

1801

37

•668 7120

•376 1405

1663

34

•721 8353

•377 8270

1544

32

31

•677 7567

•374 3469

1838

35

•735 0790

•376 3102

1697

1 7OQ

32

•793 6258

•377 9846

1576
ifinfi

30

S2

•742 0923

•374 5341

on

33

•805 1331

•376 4831

1 i „.'

30

•869 4003

•378 1452

iov/\>

28

S3

34
35

•8099966
•881 6625
•957 2989

•374 7246
•374 9182
•375 1145

1935
1963

30
28
25

•879 0680
•957 0932
1-039 4354

•376 6589
•376 8376
•377 0188

1759
1787
1812

28
26
23

9483978

1-033 7545
•122 8047

•378 3085
•378 4745
•378 6428

1034
1659
1683

26
24
22

36

1-037 1322

•375 3133

1988

23

•126 3402

•377 2024

1836

21

•216 7831

•378 8133

1705

1 *7O/i

19

37
38

•121 4073
•210 3905
•304 3704
•403 6615

•375 5144
•375 7176
•375 9226
•376 1291

2011
2032
MM
M66

21
18
16
13

•218 0735
•314 9240
•417 2053
•525 2853

•377 3881
•377 5757
•377 7649
•377 9556

1857
1876
1893
1907

19
17
14
12

•315 9768
•420 6970
•531 2818
•648 0989

•378 9857
•379 I.V.MI
•379 3356
•379 5127

17X4

1742
1757
1771

18
15
13
11

M

•5086058

•376 3370

2078

11

•639 4542

•378 1475

1919

10

•771 5487

•379 6909

1782

8

•619 5764

•376 5458

::' i^ii

8

•760 1977

•378 3403

1928

7

•902 0674

•379 8699

1791

7

:'

;;

•••

•7369806
•861 2638
•9929140

•376 7555
•376 9657
•377 1762

2102
2105

5
3

•887 9308
2-023 1367
•166 3448

•378 5338
•378 7279
•378 9222

1936
1941
1943

5

2

2-040 1318
•186 2627
•341 0309

•380 0497
•380 2299
•380 4103

1802
1804

6
2

Tables of the G (r, v)-Tntegrah
TABLE LIV— (continued).

131

*

,.15

,= 16

r=17

log /•' (r, »)

log*(r,,>

A

AS

log F (r, v)

logH(r,»)

A

A*

logJF(r,F)

logH(r.r)

A

A2

0

1-803 8114

0-376 8754

1-790 2485

0-378 2941

9ft

1-777 4825

0-379 5425

1

•804 7405

•376 8783

ftft

59

•791 2436

•378 2969

20
89

65

•778 5436

•379 5450

7ft

52

2

•807 5297

•376 8871

oo

58

•794 2308

•378 3051

QV

56

•781 7289

•379 6528

i O
1 OO

52

3

•812 1842

•376 9018

2<y

58

•799 2158

•378 3188

1 ')•'

66

•787 0445

•379 5657

LSV

181

51

4

•818 7130

•376 9222

u)J './

58

•806 2081

•378 3380

ftV9

64

•794 5004

•379 5838

9QO

51

5

•827 1285

•376 9485

58

•815 2211

•378 3026

54

•804 1110

•379 6070

£iO£t

51

C

•837 4469

•376 9805

321

37ft

57

•826 2719

•378 3927

300

54

•815 8945

•379 6352

283

•JO')

51

7

•849 6881

•377 0183

i O

57

•839 3819

•378 4281

fn«

53

•829 8736

•379 6686

OOO

384

50

8

•863 8758

•377 0617

4Q1

56

•854 5764

•378 4688

4fiO

53

•846 0751

•379 7070

433

50

9

•880 0376

•377 1108

*Xl7l
E ,47

56

•871 8848

•378 5149

El O

52

•864 5306

•379 7503

49

10

•898 2051

•377 1655

M3

55

•891 3410

•378 5661

ulo

52

•885 2768

•379 7986

49

11

•918 4141

•377 2256

602

54

•912 9831

•378 6226

564

fil *\

61

•908 3516

•379 8517

531

F17Q

48

1 :

•940 7047

•377 2912

71 A

54

•936 8541

•378 6841

DID

fifift

50

•933 8035

•379 9096

Of 0

fi97

47

IS

•965 1215

•377 3622

/ l\J

53

•963 0016

•378 7507

OOD

49

•961 6821

•379 9723

\JA I

47

Ht

•991 7138

•377 4384

762
QIA

52

•991 4782

•378 8222

715

49

•992 0436

•380 0396

71Q

46

15

0-020 5357

•377 5199

Oi*x

51

0-022 3418

•378 8985

48

0-024 9495

•380 1115

1 1»7

45

16

•051 6467

•377 6064

860

50

•056 6559

•378 9796

811

QRO

47

•0604672

•380 1878

764

QC\Q

44

17

•085 1115

•377 6979

QfU

49

•091 4895

•379 0654

oOo

46

•098 6704

•380 2686

ouo
811

43

18

•121 0005

•377 7942

ini i

48

•129 9181

•379 1558

Q4ft

45

•139 6392

•380 3537

Ou 1

893

42

19

•159 3904

•377 8953

i ' 1 1

46

•171 0234

•379 2506

PV0

43

•183 4606

•380 4430

41

ao

•200 3641

•378 0011

1057

45

•214 8939

•379 3498

992

42

•230 2288

•380 5363

933

40

si

22
23

25

•2440113
•290 4293
•339 7229
•392 0053
•447 3985

•378 1113
•378 2259
•378 3448
•378 4677
•378 5946

1102
1146
1189

1229
1269

44
43
41
39
38

•261 6258
•311 3226
•304 0964
•420 0681
•479 3681

•379 4532
•379 6606
•379 6721
•379 7874
•379 9063

1075
1114
1153
1190

41
40
38
37
35

•280 0460
•333 0225
•389 2774
•448 9393
•512 1471

•380 6336
•380 7348
•380 8397
•380 9482
•381 0602

973

1012
1049
1085
1120

39
37
36
35
33

27
,-x
29
SO

•506 0343
•068 0547
•633 6129
•702 8740
•776 0162

•378 7252
•378 8595
•378 9973
•379 1383
•379 2825

1307
1343
1377
1411
1442

36
34
33
31
30

•642 1372
•608 5269
•678 7007
•762 8356
•831 1213

•380 0289
•380 1548
•380 2839
•380 4162
•380 5514

1225
1259
1292
1323

1352

34
33
31

29

28

•579 0504
•649 8104
•724 6013
•803 6106
•887 0408

•381 1756
•381 2941
•381 4157
•381 5402
•381 6674

1153

1185
1216
1245
1273

32
31
29
28
26

SI
32
SS

•853 2318
•934 7286
1-020 7304

•379 4296
•379 5795
•379 7320

1471
1499
1525

28
26
24

•913 7633
1-000 9834
O930211

•380 6893
•380 8299
•380 9729

1380
1405
1430

26
24
23

•975 1103
1-0680549
•166 1294

•381 7973
•381 9296
•382 0642

1299
1323
1346
l^fi?

24
23
21

94

So

•111 4801
•207 2399

•379 8869
•3800440

1571

22
20

•190 1352
•2926060

•381 1181
•381 2654

1473

21
19

•269 6092
•378 7922

•382 2009
•382 3395

i'j<> t
1387

20
18

36
S7
38
S9
40

•308 2938
•414 9495

•610 4313
•772 0144

•380 2031
•380 3641
•380 5267
•380 6907
•380 8660

1691

1610
1626
1640
1653

18
17
14
12
11

•400 7368
•514 8561
•635 3206
•762 5175
•896 8681

•381 4146
•381 5655
•381 7180
•381 8718
•382 0267

1492
1509
1525
1538
1549

17
15
13
12
10

•494 0009
•615 5849
•743 9235
•879 4285
2-022 5477

•382 4800
•382 6220
•382 7655
•382 9102
•383 0561

1404
1420
1435
1448
1458

16
14
13
11
9

41

42
43
44
45

•9047199
2-045 0157
•193 4131
•350 4709
•5168011

•381 0223
•381 1894
•381 3572
•381 5254
•381 6938

1663

1671
1678
1682
1684

8
6
4

2

2-038 8307
•188 9051
•347 6370
•515 6232
•693 6170

•382 1826
•382 3393
•382 4966
•382 6543
•382 8122

1559
1567
1573
1577
1579

8
6
4
2

•173 7686
•333 6229
•502 6906
•681 6063
•871 0649

•383 2028
•383 3503
•383 4984
•383 6468
•383 7953

1468
1475
1480
1484
1480

7
6
4
2

17-2

132

fnr Stitfixfii-nin* and fft
TABLE L1V— (continued).

'

r-18

r-19

r-20

W(r.*

logH(r.r)

A

A

.o^<r.,,

logH(^)

A

A*

lug*'(r, r)

log 11 (r. r)

A

A»

0

i24y

0-380 6494

94

1-754 0014

0-381 6376

23

1-743 1485

0-382 5253

i

•766 6520

•3806518

£4
70

49

•755 1945

•381 6399

2>O

46

•744 4077

•382 5275

44

*

•769 9355

•3806691

10

199

49

•758 7762

•381 6469

4i ;

•748 1876

•382 5341

i in

44

3

•775 6818

•380 6713

LB

49

•764 7532

•381 6584

1 <

M

•754 4955

•382 5451

i r '

1 fi A

41

4

•783 6015

•3806884

48

•773 1369

•381 6746

ftvfl

41;

•763 3431

•3826605

1 • ' t

1Q7

II

6

•793 7099

•3807103

48

•783 9431

•381 6954

46

•774 7474

•382 5802

ftvl

-i:i

967

9r»?

941

e

•8062268

•380 7370

•OJ

48

•797 1925

•381 7207

•wfl

46

•788 7299

•3826042

£*t 1

"S 1

4::

7

•821 0746

•380 7685

•M

47

•8129104

•381 7605

040

46

•805 3174

•382 6326

2O*x

^97

4:1

8

•8382835

•3808048

A K~

47

•831 1269

•381 7849

O4O

45

•824 5417

•3826652

O£ f

42

9

•8678863

•3808467

*rX

47

•851 8772

•381 8237

T?"

'

44

•846 4398

•382 7021

41 1

42

10

•879 9809

•380 8913

46

•875 2015

•381 8669

44

•871 0541

•382 7432

111

41

11

•9044307

•3809415

502

45

•901 1455

•381 9144

476

- 1 .,

43

•898 4326

•382 7883

452

4Q1

41

I :

•931 4638

•3809962

roo

45

•929

•381 9663

919

42

•928 6293

•382 8376

tyo
111

40

IS

•961 0741

•381 0554

filfi

44

•961 1020

•382 0224

fiO't

42

•961 7039

•3828909

«JOO

673

40

14

•993 3209

•381 1190

OOV

43

•995 2351

•382 0826

wo

41

•997 722 1

•382 9482

All

M

15

0-028 2696

•381 1869

42

0032 2269

•382 1470

40

0-036 7674

•383 0093

vl 1

39

*799

ftQj

fiin

16

•0659915

•381 2591

i - _

42

•072 1535

•382 2154

\>O*1

7OQ

39

•078 8894

•383 0743

QUU

f:s~

37

17

•1065648

•381 3354

ftfu

41

•1150974

•382 2877

1 20

7fil

38

•124 2043

•383 M:M

uo /
724

37

;x

•150 0744

•381 4157

ou<*

40

•161 1483

•382 3638

1 v l

38

•172 7969

•383 2153

7">O

36

19

•196 6125

•381 5001

SS9

39

•210 4036

•382 4437

25!

37

•224 7699

•3832912

1 •'.'

35

SO

•246 2790

•381 5882

38

•262 9690

•382 5273

36

•280 2346

•383 3706

34

SI

•299 1823

•381 6802

919

36

•318 9588

•382 6144

871

QOfi

35

•3393115

•383 4534

828

M;D

33

S3

•355 4391

•381 7757

QQl

35

•378 4966

•382 7049

S7VU

qoq

33

•402 1306

•383 5394

\j\j\j

ftQ9

32

S3

•415 1758

•381 8749

in?*

34

•441 7157

•382 7988

«7O>J

Q71

32

•468 8327

•383 6286

OM

31

•',

•478 5288

•381 9774

inis

33

•508 7603

•382 8960

is 1 1

lllll-1

31

•539 5695

•383 7209

QfjO

30

«5

•545 6451

•382 0832

31

•579 7858

•382 9962

1 \f\Jiu

30

•614 5047

•383 8162

28

?G

•616 6833
•691 8145

•382 1921
•382 3041

1120

1 I A Q

30
29

•654 9597
•734 4627

•383 0994
•383 2055

1061

1AQQ

29

27

•693 8148
•777 6902

•383 9142
•384 0150

981

1008

27
26

S8

•771 2230

•382 4189

114O
1 1 7ft

27

•818 4896

•383 3143

1V/OO

26

•866 3362

•3841184

1 (),->!)

25

,':'
50

•855 1078
•943 6834

•382 5365
•382 6567

11 1 ' '

1202

26
25

•907 2506
1-000 9723

•383 4257
•383 5396

1139

25
23

•95U U74U
1-058 8424

•384 2243
•384 3325

1082

23

22

51

St

1-037 1813
•135 8513

•382 7794
•382 9043

1227
1249

23

22

•0998993
•204 2956

•383 6558
•383 7742

1162
1184

22
20

•163 1992
•273 3224

•384 4429
•384 5554

1104
1126

1111

21

19

S3
34
35

L-:f:> >,::»;
•3498099
•465 7060

•383 0314
•383 1605
•383 2915

1271
1291
1310

20
19

17

•314 4464
•430 6601
•553 2702

•383 8946
•384 0170
•384 1411

1204
1223
1241

19
17
16

' 5124
•6UOMJ
•641 4188

•3846698
•384 7860
•384 9039

1 144
1162
1179

18
17
15

3G

•587 9937

•383 4241

1326

15

•682 6377

•384 2667

1 971

14

•777 8668

•385 0233

1194

1208

14

37
38

•717 0435
•863 2571

•383 5583
•383 6938

1355

14
12

•819 1539
•963 2435

•384 3938
•384 .IJL'L'

1 at L

1284

1 '"I".

13
11

•921 8503
2-073 8165

•385 1440
•385 2660

1220

191?!

12
11

39

•9970711

•3838305

1 ^77

10

2-115 3673

•384 6518

1 _ • '« »

10

•234 2509

•385 3891

&XOJ

1240

9

40

2-148 9600

•3839683

10 1 t

9

•276 0267

•384 7823

J

8

•403 6815

•385 5130

8

','

•309 4403
•479 0754
•658 4799

•384 1069
•384 2462
•3843860

1386
1393
1398

7
5
4

•445 7673
•625 1841
•814 9263

•384 9136
•385 0455
•385 1780

1313
1320

1321

1 '{•' S

7
5
3

•582 6831.
•771 8822
•971 9629

•385 6378

• •Ju~ "flQO

•JoO 1 ' l-Jw

•385 8HIK)

1247
1254
1258

i-'i;i

6
5
3

'•'•
4S

•848 3263
3-049 3507

•384 6262
•3846666

1403

2

3 015 7041
•228 2952

•385 3108
•385 4437

luZO

1330

2

3-1836730
•407 '

•386 0151
•386 1414

i —'ii
1263

2

Tables of the G (r, ^-Integrals
TABLE LIV— (continued).

133

,

r=21

r = 22

, = 23

logP(r,»)

logH(r,»)

A

*

log F(r,,)

logH(r,r)

A

A*

logF(r, f)

logH(r,»)

A

A2

0

1-732 8121

0-383 3271

91

1-722 9451

0-384 0548

1-713 5069

0-384 7182

1

734 1352

•383 3292

21

42

•724 3364

•384 0568

fiO

40

•714 9643

•384 7202

R7

38

2

•738 1155

•383 3354

1(\r,

42

•728 5130

•384 0628

w

40

•719 3391

•384 7259

Ul

38

S

•744 7542

•383 3459

IIW

42

•735 4826

•384 0728

14T)

40

•726 6397

•384 7355

38

4

•754 0660

•383 3606

1 QQ

41

•745 2584

•384 0867

f±\j

40

•736 8798

•384 7488

38

5

•766 0683

•383 3793

IOO

41

•757 8590

•384 1047

39

•750 0786

•384 7660

38

99Q

219

209

6

•780 7641

•383 4022

tLfifj
9.7O

41

•773 3082

•384 1266

258

39

•766 2613

•384 7869

247

37

7

•798 2414

•383 4293

£ti\j

41

•791 6354

•384 1523

2Q7

39

•785 4585

•384 8116

284

37

8

•818 4736

•383 4604

Vi\

40

•812 8757

•384 1820

fttj t
335

38

•807 7070

•384 8400

321

37

9

•841 5196

•383 4955

40

•837 0698

•384 2155

^71

38

•833 0493

•384 8721

36

10

•867 4241

•383 5346

39

•864 2646

•384 2529

o t o

38

•861 5346

•384 9078

36

AW)

411

w*

11

•896 2374

•383 5776

"*«>U

39

•894 5129

•384 2940

*±L L

448

37

•893 2180

•384 9471

oi/o
49Q

36

IS

•928 0162

•383 6245

"508

38

•927 8740

•384 3388

485

37

•928 1617

•384 9899

lit/

464

35

IS

•962 8233

•383 6753

38

•964 4139

•384 3872

521

36

•966 4346

•385 0363

34

14

0-000 7283

•383 7299

*

37

0-004 2054

•384 4393

556

35

0-008 1129

•385 0861

632

34

15

•041 8074

•383 7881

36

•047 3287

•384 4949

35

•053 2804

•385 1393

33

16

•086 1444

•3838500

654

36

•093 8713

•384 5540

591

625

34

•102 0289

•385 1958

565
598

33

17

•133 8306

•383 9154

CVI

35

•143 9291

•334 6164

658

33

•154 4586

•385 2556

32

18

•184 9653

•383 9843

UOJJ

700

34

•197 6061

•384 6822

r- ti i

32

•210 6783

•385 3185

ftfiA

31

29

•239 Go63

•384 0566

m

33

•255 0156

•384 7513

OJAJ
799

31

•270 8064

•385 3845

DDU

(I'll

30

20

•2980208

•384 1322

32

•316 2801

•384 8235

t ~ —

31

•334 9713

•385 4536

\JiJl

29

XI

•360 1851

•384 2111

788

wOi i

31

•381 5322

•384 8987

789

30

•403 3116

•385 5256

720

74ft

28

gs

•426 2861

•384 2930

ozu

30

•450 9155

•384 9770

iOA
Oil

29

•475 9774

•385 6004

I*±O

28

S3

•496 4716

•384 3780

Q7Q

29

•524 5848

•385 0581

oil

QOQ

28

•553 1308

•385 6780

776

ono

27

24

•570 9011

•384 4659

Ola
QO7

28

•602 7073

•385 1420

OOo

>•»'' ,

27

•634 9466

•385 7583

ouo

QOQ

26

25

•649 7464

•384 5566

•A//

27

•685 4632

•385 2286

OOO

26

•721 6136

•385 8411

OBO

25

S6

•733 1933

•384 6500

960

26

•773 0472

•385 3178

'lit;

25

•813 3351

•385 9264

077

24

27

•821 4416

•384 7460

985

26

•8656689

•385 4094

01 U

Q4O

24

•910 3304

•386 0141

Oil

23

S8

•914 7070

•384 8445

23

•963 5543

•385 5034

tt*t\j

22

1-012 8362

•386 1040

22

29

1-0132222

•384 9453

1031

22

1-0669472

•385 5996

flS4

21

•121 1074

•386 1961

921
Q41

20

SO

•117 2380

•385 0484

21

•176 1108

•385 6980

20

•235 4191

•386 2902

{741

19

31

•227 0250

•385 1535

1052

1 1 1 ~ i

20

•291 3286

•385 7984

1004

19

•356 0681

•386 3862

960

18

SS
S3
S4

55

•342 8756
•465 1055
•594 0558
•730 0957

•385 2607
•385 3696
•385 4803
•385 5926

1071
1090
1107
1123

18
17
16
14

•412 9072
•541 1773
•670 4967
•819 2523

•385 9007
•386 0047
•386 1104
•386 2176

1023
1040
1057
1072

18
16
15
14

•483 3760
•617 6859
•759 3748
•908 8466

•386 4840
•386 5836
•386 6846
•386 7871

978
995
1011
1025

17
16
14
13

36

n
u

39
40

•873 6248
2-025 0761
•184 9195
•353 6651
•531 8676

•385 7063
•385 8213
•385 9375
•386 0547
•386 1728

1137
1150
1162
1172
1181

13
12
10
9
7

•9698630
2-128 7827
•296 5039
•473 6612
•6605361

•386 3261
•386 4359
•386 5468
•386 6586
•386 7713

1085
1098
1109
1119
1127

12
11
10
8
7

2-066 5394
•232 9279
•408 5272
•593 8968
•789 6446

•386 8910
•386 9960
•387 1021
•387 2091
•387 3169

1038
1050
1061
1070
1078

12
11
10
8

7

41

4-

•720 1308
•919 1130

•386 2916
•3864110

1188
1194

11 'IS

6

4

•858 0615
3-066 8272

•386 8848
•386 9987

1134
1140

1 144

6
4

•996 4325
3-214 9823

•387 4264
•387 5344

1085
1090

1OQ4

5

4

43
44
45

3-129 5327
•352 1756
•687 9021

•386 5308
•386 6510
•386 7712

i lyo
1201
1203

3

2

•287 6865
•521 1629
•768 4580

•387 1131
•387 2278
•387 3426

1 lit

1147
1148

3
1

•446 0817
•690 5919
•949 4561

•387 6438
•387 7535
•387 8633

l' '• ' i

1097
1098

3
1

134

Tables for Station- hi us ((ml />'/<>///< trichina
TABLE LJV— (continual).

,

r-24

r-26

r-26

•**>,»

log//(r,r)

A

-i-

W.,>

log//(r,r)

A

A»

log*'(r,r)

*H(r.,,

A

A»

0

1 7ol 4618

0-385 3266

1 Q

1-695 7781

0-385 8838

1 O

T'687 4284

0-386 3984

1

•7059862

•386 3276

1s

37

•697 3569

•385 8855

lo

35

•6890840

•386 4001

61

34

t

•7105684

•385 3330

M

37

•702 1393

•3868908

KV

35

•C:il 0540

•386 4052

86

'•', 1

3

•718 1899

•385 3421

Wm

1 • ' s

36

•710 1019

•386 S:I:H;

DO

I 00

36

•7<i2 317(i

•3864136

118

31

4

•7288942

•385 3549

1ZO

IfU

30

•721 2704

•3859119

J - • '

1 no

35

•713 9805

•386 4264

152

34

6

•742 6914

•385 3714

lU*i

36

•7356660

•385 9277

100

36

•728 9746

•386 4406

33

6

•769 6077

•385 3915

201

36

•753 3158

•385 9470

193

OO"7

35

•747 3681

•386 4591

186

•'Is

33

7

•779 6760

•385 4151

272

36

•774 2534

•3*:, 9697

IB?

.p.; 1

M

•769 1658

•386 Mo

£1O

251

33

8

•8029317

•385 4423

QA**

35

•798 5185

•385 9958

•Dl

34

•7!U 4395

•3865061

9A4

33

9

•829 4225

•385 4730

00 4

35

•826 1578

•386 0253

295

34

-.^23 2272

•38C .r>:il5

•Bfl
lift

32

•859 1983

•385 5073

34

•857 2243

•386 0582

329

33

•855 5847

vAU

32

11

•892 3170

•385 5460

377
<ii i

34

•891 7784

•386 0943

362

33

•891 6743

•3866009

348
379

31

I '

9 8434

•385 5860

** 1 1

34

•929 8876

•386 1338

!io-

32

•931 2665

•386 6888

411

31

rj

•968 8495

•385 6305

*177

33

•971 6270

•386 1764

*~'

32

•974 7393

•386 t',7:iH

440

30

14

0-0124148

•385 6782

**i t

Mil

32

0-017 0795

•386 2223

AOa

31

0-022 0791

•386 72.-J9

471

30

15

•0696268

•385 7292

UiU

32

•066 3362

•386 2712

31

•073 3806

•386 7710

29

16

•1105814

•3857834

642

It^Q

31

•1194971

•386 3232

520

""ii

30

•128 7480

•386 8210

500

29

ir

•]<;:> 3831

•3858406

078

f\t k«>

80

•1766711

•386 3782

550

29

•188 2!u:i

•386 8738

557

28

lit

•221 1458

•3859009

DUo

!'•>'>

30

•237 9768

•386 4361

29

•252 1435

•386 9295

27

19

•^-i; xn!s

•385 9642

two

29

•303 5430

•386 4969

608

i") "

28

•320 4290

•386 9880

fill

27

90

•3540583

•3860304

662

28

•373 5093

•386 5604

635

27

•393 2963

•387 0491

26

f Nil .

<*('•>

637

SI

•425 4870

•386 0994

OlA/
*71 1

27

•448 0267

•386 6267

DDa

26

•470 9025

•387 1128

662

25

•501 4356

•386 1711

717

— i <

26

•527 2584

•386 6955

"it

25

•553 4176

•387 1790

(iM7

24

23

•582 0734

•386 2455

/44

26

•611 3809

•386 7669

714

*7*>O

25

•641 0250

•387 2477

\J\J i

710

24

~4

•667 5830

•386 3225

25

•700 5843

•386 8408

7*>y

— . •_)

24

•733 1I22(;

•387 3187

23

2.7

•768 1612

•386 4018

24

•795 0741

•386 9170

762

23

•832 3242

•387 3920

22

01 0

•TQK

765

/<;

•8540205

•3864836

OiO

H 11 t

23

•895 0716

•386 9955

/oo

22

•9364600

•387 4674

776

21

.•;

•9553899

•386 5676

OVJ

W( '-)

22

1-000 8153

•387 0762

QO**

21

1-046 6783

•387 5450

796

20

28

1-062 6163

•386 6538

on

OQO

21

•112 5627

•387 1589

Dal

QjC7

20

•162 9470

•387 6246

815

19

:'•

•175 6661

•386 7420

Boa

19

•230 5913

•387 2436

H47

19

•285 8547

•3877060

18

90

•295 1263

•386 8322

902

18

•355 2004

•387 3302

866

18

•415 6129

•387 7893

17

31

•4212069

•386 9242

920

17

•486 7129

•387 4185

883

17

•552 5576

•387 8742

860
865

16

n

•654 2426

•387 0180

ii" i

16

•625 4776

•387 6085

900

16

•697 0516

•3879608

ouu

880

15

33

•094 5945

•387 1134

BM

15

•771 8709

•387 6001

910

14

•849 4867

•388 0488

894

14

•i

•842 6534

•387 2102

tK>a

14

•9263001

•387 6931

930

13

2-010 2864

•388 1382

907

13

• -.

•9988417

•387 3085

982

13

2-089 2053

•387 7874

943

12

•179 9088

•388 2289

12

36

2-163 6169

•3874080

995

1 1 M it;

11

•261 0633

•387 8829

955

11

•358 8499

•388 3208

919
929

11

37

•337 4747

•387 5086

IUUO
1 A1 T

10

•442 3906

•387 9795

~~

10

517 6471

•388 4137

938

9

38

•520 9526

•387 6103

1017

9

•633 7476

•388 0771

MA

9

•746 8833

•388 6075

947

8

39
40

•7146347
•919 1558

•387 7128
•387 8162

1033

8
6

•836 7426
3049 0373

•388 1756
•388 2748

992

7
6

-IC.7 1916
3-179 2602

•388 6022
•388 6976

954

7
6

41

3-135 8068
•3635411
•6049809

•387 9201
•388 0246
•388 1295

1045
1048

6
3
3

•274 3517
•612 4708
•764 2515

•388 3746
•388 4749
•388 5766

998
1003
1007

6
4

2

•413 8384
•661 7426
•923 8647

•388 7935
•388 8900
•388 9868

964

B68

((-i i

6
4

2

M

45

•8604254
4-1308591

•388 2346
•3883398

1051
1052

1

4-030 6307
•312 6342

•388 6765
•388 7776

1010

1

4-201 1786
•494 7624

•389 0838
•389 1810

*7I \J

972

1

Tables of the G (r, v)-Integrals
TABLE LIV— (continued).

135

,

r=27

r=28

, = 29

logF(r, »)

logtffr.r)

A

A"

logf(r, r)

logH(r,r)

A

A"

logF(r, »)

logH(r, r)

A

A3

0

1-679 3877

0-386 8744

1ft

1-671 6341

0-387 3160

16

T'664 1478

0-387 7268

in

1

•681 1094

•386 8760

i\>

33

•673 4219

•387 3176

47

31

•666 0017

•387 7283

xu

30

.?

•686 2778

•386 8809

Ql

33

•678 7887

•387 3223

^ t

79

31

•671 5669

•387 7328

76

30

3

•694 9025

•386 8891

Ol

32

•687 7445

•387 3301

110

31

•680 8538

•387 7404

lOft

30

4

•706 9998

•386 9005

1 4fi

32

•700 3062

•387 3411

141

31

•693 8799

•387 7510

i\j\j
i.ifi

30

5

•722 5923

•386 9151

LtlJ

32

•716 4973

•387 3552

31

•710 6695

•387 7647

K0Q

30

6

•741 7096

•386 9329

178

91 A

32

•736 3483

•387 3724

172

31

•731 2544

•387 7813

166

I'M;

30

7

•764 3877

•386 9539

£.l\J o.-j
9.19

•759 8968

•387 3927

233

30

•755 6734

•387 8008

X{7O

225

30

t

•790 6698

•386 9781

ni

•787 1876

•387 4160

264

30

•783 9728

•387 8234

254

29

9

•820 6063

•387 0055

•*n!i ^

•818 272H

•387 4424

293

30

•816 2068

•387 8488

29

10

•854 2546

•387 0359

31

•853 2121

•387 4717

30

•852 4372

•387 8772

29

11

•891 6799

•387 0694

335

30

•892 0731

•387 5041

352

29

•892 7340

•387 9084

340

28

12

•932 9552

•387 1059

one 30

•934 9315

•387 5393

ift l

29

•937 1757

•387 9424

28

IS

•978 1615

•387 1454

*2

29

•981 8715

•387 5774

001

409

28

•985 8495

•387 9792

395

27

14

0-027 3887

•387 1879

~~*

29

0-032 9863

•387 6183

437

28

0-038 8519

•388 0187

422

27

15

•080 7353

•387 2332

28

•0883780

•387 6620

27

•0962888

•388 0609

26

16

•1383092

•387 2814

"tii' »

28

•148 1586

•387 7084

464

4Q1

27

•158 2763

•388 1057

448

26

17

•200 2283

•387 3323

ouy

27

•212 4505

•387 7576

ftt7i

517

26

•224 9410

•388 1531

499

25

18

•266 6208

•387 3859

o«>o

26

•281 3865

•387 8093

25

•296 4208

•388 2031

25

19

•337 6258

•387 4422

563

K.QQ

26

•355 1112

•387 8635

567

25

•372 8653

•388 2555

548

24

20

•413 3944

•387 5010

JOO

25

•4337811

•387 9203

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24

•454 4367

•388 3103

23

21

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•387 5624

coo

24

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•387 9794

fit fi

23

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•388 3674

671

fiQJ.

23

n

•579 8882

•387 6261

24

•606 6479

•3880409

VlU
COO

23

•633 6767

•388 4268

U«^r

(; 1 1;

22

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•670 9806

•387 6923 ™A

23

•701 2256

•388 1047

ooo
fifirt

22

•731 7398

•388 4883

O1Q

COY

21

*4

•767 5727

•387 7607

wv

22

•801 5122

•388 1706

\J\1\J

fiHl

21

•835 7212

•388 5520

QO /

fiV7

20

25

•869 8863

•387 8312

21

•9077380

•388 2387

UO1

20

•945 8594

•388 6177

•JO 4

20

797

7f»1

fl77

26

•978 1616

•387 9039

t fit

20

1-020 1612

•3883088

* ' ' 1

19

1-0624115

•388 6854

D i I

19

27

1-0926538

•387 9786

"fifi

19

•139 0194

•3883808

719

18

•185 6549

•388 7550

71 ^

17

28

•213 6439

•388 0552

18

•264 6312

•388 4547

7^K

18

•315 8886

•388 8263

t i '*

17

29

•341 4310

•388 1337

Q/\Q

17

•397 2979

•388 5303

771

17

•453 4351

•388 8993

7^7

16

SO

•476 3386

•388 2138

BOB

16

•537 3550

•388 6077

773

16

•598 6420

•388 9740

15

31

•618 7157

•388 2956

818

QOO

15

•685 1648

•3886865

789

15

•751 8846

•389 0501

762

77ft

14

32

•768 9393

•3*H 3790 j

14

•841 1182

•388 7669

Ql Q

14

•913 5680

•389 1277

/ i O

13

S3

•927 4164

•388 4638

13

2-005 6375

•3888487

O1O

QOfi

13

2-084 1297

•389 2067

ano

12

34

2-094 5869

•388 5499 ^i

12

•179 1790

•388 9317

OuU

. 1 Q

12

•264 0426

•389 2868

Ql t

11

SB

•270 9268

•388 6372

o/o

11

•362 2366

•389 0159

MS

11

•453 8181

•389 3682

81«>

10

36

•456 9512

•388 7257

885

10

•555 3447

•389 1012

853

10

•654 0100

•389 4505

824

9

37

•653 2186

•388 8161

895

9

•759 0825

•389 1875

863

QTO

9

•865 2184

•389 5338

833

,ii

8

38
39

•8603344
3-078 9562

•388 9055
•3889967

912

8

7

•974 0781
3-201 0137

•389 2746
•389 3625

o7z
879

QQO

7

7

3-0880940
•323 3436

•389 6179
•389 7028

841
849

oe e

7
6

40

•309 7991

•389 0885

918

6

•440 6310

•389 4511

Bob

5

•571 7357

•389 7883

855

5

41
42
43

•553 6412
•811 3309
4-083 7948

•389 1809
•389 2738
•389 3670

924
929
932

5
3
2

•693 7374
•961 2127
4-2440178

•389 5402
•389 6298
•389 7197

891

896
899

lull

4
3

2

•834 1065
4-111 3678
•404 5148

•389 8744
•389 9608
•390 0476

860
865

868

4
3
2

44

•372 0436

•389 4604

a

1

•543 2028

•3898098

901

1

•714 6355

•390 1346

870

1

45

•677 1878

•389 5540

936

•859 9176

•3899000

902

5-042 9213

•390 2217

871

I

Tables for Stal!*li<'i<imt <n»l

TABLE LIV— (continn,

„

r-30

r-31

,-»

lo^r,,)

logH(r. r)

A

*

logF(r, ,)

IogH(r,,)

A

A-

log F(r, »)

bi*fc4

A

A'

0

T-6669109

0-3881099

1-649 !H173

0-388 IC71I

1-CI3 1226

H-3SS8034

/

•<:-,8 8309

•3881113

16

29

•651 8936

•3**. 4(i'J4

1 1

28

1115 1748

•3-vS 8048

14

28

.

M 1 5;i|5

•388 1157

44

29

•6678666

•388 4736

13

28

•I-.51 3354

•3883089

41

i'ii

27

' : ; UM

•388 1231

73

29

•6678048

•3884808

71

28

•661 6158

•388 *|5v

69

27

;

•687 7031

•388 1333

1(13

29

•681 7598

•3vs 4906

99

•i;7(i 0352

:»;

1 )•>

a?

-

•705 0914

•388 1466

132

29

•699 7466

•388 5034

12<

2*

•694 6208

•3888378

123

27

6

•726 4101

•388 1625

161

29

•721 7993

•388 5189

155

28

•717 4073

•3*8 8528

151

1— ••

(V

7

•751 •

•388 1*15

189

29

•717 9592

•388 5372

183

28

•744 4379

•388 8706

, i

27

-

•781 0077

•388

218

28

•778 2762

•388 5583

211

27

•775 7637

•388 8910

204

26

9

•8143906

•388 2278

246

28

•812 8080

•388 6822

238

27

•811 1111

•388 9140

231

• - —

2(5

10

•851 9121

•388 2562

274

28

•851 6207

•388 6086

265

27

•851 5483

•388 9397

257

26

11

:<>447

•388 2854

302

27

•894 7893

•388 6378

292

26

•896 1530

•38,*

283

«}/\0

26

12

•939 6098

•388 3183

329

27

•942 3977

•388 6696

318

26

•945 3119

•388 9888

.ills
ooo

25

•990 0775

•388 3538

356

26

•994 5391

•388 7041

344

•»— . ,

•999 2205

•389 0322

888
'i " k.

25

14

0-044 9676

•388 3920

382

26

0-051 3173

•388 7410

•J7U

QO!",

25

0-057 !-vi;i

•3890680

806

oyO-

24

IS

•104 4498

•388 4328

408

25

•1128450

•388 7805

3!l.>

25

•121 4595

•389 Ki(i2

tKSJ

24

16

•168 6443

•388 4762

433

25

•179 2464

•388 8225

419

24

•1900681

•389 I ItiH

23

17

•237 6820

•388 5220

458

24

•250 6573

•388 8668

444

23

•2(i3 8522

•389 1*9! i

1 - •}

23

18

•311 7056

•388 5703

483

24

•327 2249

•388 9136

467

23

312 9638

•389 2361

tot

A1H

22

19

•390 8701

•3886209

23

•409 1093

468 9626

490

22

•427 5684

"389 2s''*'

1 I •)

22

JO

•475 3432

•3886739

530

22

•4964842

•389 0138

513

22

•517 8452

•3*9 3323

r.i7

21

SI

•5653065

•388 7291

552

22

•589 5372

•389 0673

53 1

21

•til 3

•389 3840

518

20

•660 9566

•3** 7*'i5

574

21

•688 4714

•389 1228

5.)6

20

•7 Hi 2dti3

•3*9 4379

20

Us

•762 5053

•388 8460

595

20

•793 6068

•389 1804

576

20

•824 72<i<i

•3*9 1937

.1. .S

5 MM

19

•870 181(i

•3889076

616

20

: 8771

•389 2400

596

c i -

19

•939 79211

•389 551 I

< «

-1 11 •

19

•984 2323

•3889711

635

19

1 -022 8405

•389 3015

DID

18

1 -061 6692

•389 UK >9

Ol/D

18

:>

1-104 !I235

•389 0366

654

18

•1176710

•389 3C, is

633

18

•1906I01

•389 6723

(i!3

17

.>;

•232 5423

•389 1038

672

17

•279 6653

•389 4299

650

17

•327 0091

•3-9 7353

i- \(\

16

/,-

•367 39*1

•389 1727

(i: in
— , ,, •

16

•419 1433

•3*9 4966

£UO

Hi

•171 1094

•3-9 8000

l> 4T>
<•<;.>

15

•609 *2 15

•389 2433

70(5

16

•566 419-'

•;;-,]! ;,i| |i)

DOtJ

15

•(123 2962

•389 8661

' »')_

15

SO

•660 1814

•389 3155

722

15

•721 9568

•3-9 6348

(,9V

14

•783 953 1

•3*9 9338

14

31

•818-8570

•389 3891

736

14

•8860657

•389 7060

713

13

•953 11151;

•3! (0002*

— , ,'j

13

1707

•389 4642

750

7C1

13

2-059 2096

•389 7786

786

-OQ

13

2-132 3701

•391 1 0732

i (M

71 ">

12

8749

•3*:» 5405

i It.}

12

•241 8567

•3*9 *525

(OO

12

•321 (Mini

•390 1 117

i i •*

*~ >*"

11

34

•3111 1593

775

11

•434 5127

•3*9 9275

750

"O 1

11

•5900679

•31HI 2171

i -i

10

35

•545 6529

•M i;:..;.;

786

10

•037 7246

•390 (H135

(61

10

•7300183

•31HI 2911

""

9

S6

752 9288

•389 7762

796

9

•852 0847

•39(10806

770

••"•A

9

•951 4628

•390 3657

746

9

37

•9716080

•389 85(i7

805

8

3-078 2349

•31X1 15*5

(79

— .; —

8

3-1850841

•3904412

IP

8

08

1 3639

•389 9380

813

7

•316 8712

•390 2372

78 (

7

•431 (5009

•3!H) 5171

- !

7

•115 9277

•390 0201

820

6

•568 7494

•390 3166

791

6

•691 7ii3*

•390 51 HI

i-'.'

6

40

•703 0947

•390 1028

827

5

•834 6915

•390 3966

900

6

•9665111

•390 6719

6

41

974 7302

•390 1859

832

4

1-1155919

•390 4771

805

4

1-3066766

•390 7498

780

4

42

4 261 7776

•39"

3

•31HI 55*d

«?9

3

•5ii3 2967

•3!tO 8282

--,-

3

4$

•390 3534

2

•72fi 257 1

•31KIC392

H12

2

•**7 4707

•3909069

(H|

2

44

•886 3234

•390 1375

841

O4O

1

•390 7206

01 -

1

5-2303999

•3! Mi

TAB

1

-V-

6 l*d|

•3!Mi 5JI7

* M

• B7M

•390 8021

BAH

•593 3997

•391 0646

, .^.t

Tables of the G (r, v}- Integrals
TABLE LIV— (continued).

137

00

r=33

r=34

,=35

log -/•'('"> »)

logff(r,i.)

A

A*

logF(r, i.)

log if (r, ,)

A

A2

logF (r, x)

*«M

A

A2

0

1-636 5434

0-389 1183

13

1-630 1576

0-389 4146

i "^

1-623 9542

0-389 6937

1 °.

1

•638 6617

•389 1197

40

27

•632 3421

•389 4159

10
1Q

26

•626 2048

•389 6949

lo

00

25

%

•645 0208

•389 1237

67

27

•638 8996

•389 4198

Ot7

26

•632 9608

•389 6987

oo

00

25

s

•li.V. 6323

•389 1304

uq

26

•649 8424

•389 4262

Ol

26

•644 2348

•389 7050

Do

00

25

4

•670 5163

•389 1397

«/«>
120

26

•665 1908

•389 4353

26

•660 0478

•389 7138

00

25

5

•689 7005

•389 1516

26

•684 9738

•389 4469

26

•680 4295

•389 7251

25

MI;

1 49

1 °.M

6

•713 2210

•389 1662

1 1U

172

26

•709 2283

•389 4611

&U

25

•705 4180

•389 7389

loo

25

7

•741 1222

•389 1835

26

•738 0001

•389 4778

25

•735 0604

•389 7551

IB?

24

8

•773 4568

•389 2033

994

26

•771 3436

•389 4970

91 7

25

•769 4129

•389 7738

24

9

•810 2865

•389 2256

ZZ4
94Q

25

•809 3225

•389 5187

•At
949

25

•808 5407

•389 7948

.

00 K

24

10

•851 6818

•389 2505

tito

25

•8520090

•389 5429

Z4Z

24

•852 5188

•389 8183

20O

24

11

•897 7225

•389 2780

274

25

•899 4858

•389 5695

266

24

•901 4317

•389 8442

259

900

23

12

•948 4980

•389 3078

323

24

•951 8449

•389 5985

MA

24

•955 3744

•389 8724

•ox

23

IS

0-004 1077

•389 3402

24

0-009 1887

•389 6299

007

23

0-014 4525

•389 9029

•}•>«

23

14

•064 6615

•389 3749

371

24

•071 6306

•389 6636

Oo t

23

•078 7824

•389 9356

aso

22

15

•130 2802

•389 4120

23

•139 2949

•389 6996

23

•148 4925

•389 9706

a

22

16

•201 0960

•389 4514

417

23

•2123180

•389 7379

382

Af)f\

22

•223 7229

•390 0078

372

21

17

•277 2533

•389 4931

11 1

22

•290 8487

•389 7783

t*\J*J
A9fi

21

•304 6270

•3900471

41 -1

21

18

•358 9091

•389 5370

~™

22

•3750487

•389 8209

•BQ

21

•391 3713

•390 0884

*±l ~±

20

19

•446 2340

•389 5830

461

21

•465 0938

•389 8656

447

20

•484 1369

•390 1319

d<-i

20

to

•539 4127

•389 6312

482

20

•561 1746

•389 9123

467

20

•583 1198

•390 1773

d

19

..;

•638 6453

•389 6814

602

-,.'-•

20

•663 4971

•3899611

487

19

•688 5323

•390 2246

JQO

19

22

•744 1480

•389 7336

*)£i£i

19

•772 2842

•3900117

V '.

19

•800 6039

•390 2738

K.in

18

'.':

•856 1542

•3*9 7M77

O41

19

•887 7765

•390 0643

mo

18

•919 5823

•390 3248

OIU

17

,

•974 9160

•389 8437

560

f 7Q

18

1-010 2336

•390 1186

543

17

1-045 7350

•390 3776

528

r 4 f

17

25

1-100 7050

•389 9014

078

17

•139 9367

•390 1746

561

17

•179 3502

•390 4321

040

16

,,;

•233 8145

•389 9609

695

rtl i

17

•277 1848

•390 2324

577

16

•320 7390

•SIX) 4881

561

16

!8

•374 5602
•523 2830

•390 0220
•390 0847

Oil

627

16
15

•422 3065
•575 6518

•390 2917
•390 3525

593
609

15
14

•470 2366
•628 2047

•390 5458
•390 6049

591

f*f\[L

15
14

so

•6*0 3501
•846 1578

•390 1489
•390 2145

656

14
13

•737 5995
•908 5576

•390 4148
•390 4785

623
637

14
13

•795 0329
•971 1417

•390 6654
•390 7273

bUO
619

13
13

81

2021 1335
•2U5 7387

•390 2815
•390 3497

669
682

13
12

2-088 9669
•279 3029

•390 5435
•3906097

650
662

12
11

2-156 9847
•353 0516

•390 7904
•390 8547

631

643

i>K •

12
11

S3

•400 4717

•390 4190

oJ3

11

•4HO 0791

•390 6770

673

11

•5598711

•390 9201

bo4

10

34
86

•608 H714
•822 5204

•390 4895
•390 5610

715

10
9

•691 8509
•915 2186

•390 7454
•390 8148

684
694

10
9

•778 0150
3-008 1016

•390 9865
•391 0539

664

674

9
9

SH

3-051 0494

•390 6333

724

7Oil

8

3-150 8322

•390 8850

703

8

•250 8000

•391 1222

682

i "1 H 1

8

31

•292 1420

•3: HI 7065

7.i2

7

•3! lit 3963

•390 9561

711

7

•506 8356

•391 1912

byo

7

38

•'< 16 5396

•31)0 7805

798

6

•661 6747

•391 0278

718

6

•776 9950

•391 2609

o97

6

S'J

•815 0471

•390 8551

/46

6

•1I3M 4971

•391 UK 12

724

5

4-062 1324

•391 3312

^03

5

40

4-098 5399

•390 9302

751

5

4-230 7i M!

•391 1732

729

4

•363 1764

•391 4021

709

4

4'

•397 !>7"l

•391 0058

756

4

•539 4613

•391 2466

734

4

•681 1377

•391 4734

713

71 fi

4

4*

•714 3773

•391 0818

fOU

3

•865 6549

•391 3203

738

3

5-017 1183

•391 5450

71O

71 O

3

4-1

5O48 3939

•391 1581

763

*"ftt*

2

5-2105144

•391 3943

7 in

2

•372 3207

•391 6169

719

791

2

44

•402 759';

•391 2345

/DO

1

•575 3166

•391 4686

742

1

•748 0596

•391 6890

1ml

700

1

46

•777 3311

•391 3111

•961 4600

•391 5429

743

6-145 7749

•391 7612

Y22

B.

18

Tables for StittixticiaHs amf li'unm frici
TABLE LIV— (continued).

'

r-36

r-37

r-38

h.Ffr.1

log H(r. r)

A

A*

logF(r, ,)

logHfc,)

A

A>

logF(r.r)

logH(r, r)

A

A>

0

T-617 9231

0-389 9572

1 O

1-612 0660

•03902063

1 O

1 -606 341 3

0.390 4421

'

1

•6208399

•389 9584

11

24

•614 4378

•3! ID 2074

12

24

•608 7902

•390 4433

12

ne

23

9

•627 1943

•3899620

i ' \

24

•621 6908

•3902110

.i .

24

•616 1417

•3904468

OO

23

3

BM TWO

•389 9682

ol

ftfl

24

•633 5272

•390 2170

00

UQ

24

•628 l<>:':t

•3904526

68

A1

23

4

•6550771

•389 9767

BB

21

•650 2694

•390 22.r>3

O»J

in?

M

•646 6161

•3904607

Ol

23

5

•676 0576

•389 9877

24

•671 8485

•390 2300

IVf

23

•667 7940

•390 4711

23

114

1 *Vl

6

•701 7800

•3900011

IBBj

24

•C98 3051

•390 2490

LBV

23

•694 9847

•390 4837

23

7

•732 2932

•390 0168

1 A9

24

•729 6890

•390 2643

23

7:.' 7 2393

•390 4987

1 TO

23

8

•767 6546

•3900350

1OZ

24

•766 0594

•390 2*211

.-, X i

23

•764 6187

•3905159 !,'~

22

9

•807 9316

•390 0555

99X

23

•807 4855

•3903020

•uu

999

23

•807 1940

•390 5353 i,o

--

10

•853 2010

•390 0783

£2O

23

•854 0464

•390 3242

BBB

22

•855 0464

•390 55G9

22

11

•903 5502

•390 1035

251

97.1

23

•9058318

•390 3486

245

9fi7

22

•9082680

•390 5808

22

IS

•959 0766

•390 1309

•fv

22

•962 9420

•390 3753

ZO i
OQft

22

•966 9620

•uv

OU1

21

IS

, .tllll *v<!l

•390 1605

318

22

IHI254886

•3904041

•DO

21

0-031 2429

•390 <;:i •!*

•O 1
"V)9

21

'4

•086 1070

•390 1924

idn

22

•093 5948

•390 4351

J~~

21

•101 2371

•390 6650

0UB

20

15

•157 8628

•390 2264

•w

21

•167 3965

•390 4682

21

•177 0849

•390 6972

20

361

0*0

1d9

16

•235 3007

•390 2625

ooo

21

•247 0418

•3905034

BBB
070

20

•258 9378

•390 7314

Ot£

'•1C-*

20

17

•318 6782

•3903007

•OB

402

20

•332 6929

•390 5405

971
tftl

20

•346 9624

•3; in 7676

BvB

19

18

•4078669

•390 3409

20

•424 5260

•390 5797

09 1
41 1

19

•441 3400

•3908057

JAA

19

11

•5033529

•390 3832

"

19

•522 7325

•3906208

*1 1

19

•642 2671

•390 8457

BJUV

18

M

•6052380

•390 4273

19

•627 5199

•3906637

18

•649 9569

•390 8876

18

.-:

•713 7407

•390 4733

460

47ft

18

•7391128

•390 7085

ICC

18

•764 6400

•3!K) 9312

A***)

17

.'.•

•8290968

•390 6212

%l O

18

•857 7536

•390 7551

4OO

17

•886 5655

•3H09765

4*)o

47O

17

•951 6615

•390 6708

__-

17

•983 7045

•390 8033

4QQ

16

1-0160028

•391 023.-)

*t i \j

16

.•;

1-081 4096

•390 6221

*i2fl

16

1-117 2483

•390 8532

r i e

16

•153 2423

•391 0721

•.,,->

16

S5

•218 9381

•390 6760

16

•258 6900

•390 9048

O10

15

•298 5974

•391 1223

DUB

15

S4S

Ron

M7

26

•364 4667

•390 7296

BBB

560

15

•408 3585

•390 9578

UOU

15

•452 4059

•391 1739

01 *

14

37

•518 3405

•390 7856

14

•566 6085

•391 0123

KK.fl

14

•615 0321

•391 227t)

f i -"

14

,'8

•6809313

•390 8431

675

KQQ

14

•733 8222

•391 0682

559

13

•786 8688

•3! II :;*!:.

13

.'£>

•852 6402

•390 9019

Ooo

fiol

13

•9104119

•391 1256

573

fift*

13

•968 3394

•391 3372

557

12

30

2-033 8997

•3909681

vlv/i

12

2 096 8223

•391 1840

BOB

12

2-159 9006

•391 3'.M-

12

31

•225 1766

•391 0234

614

12

•293 5330

•391 2437

£AU

11

•362 0454

•391 4523

581
MM

11

SS

•426 9744

•391 0859

6^fi

11

•501 0620

•391 3046

vUO

t; 1 O

10

•676 3055

•391 5115

Wm

10

S3

•639 8374

•391 1495

646

10

•719 9685

•391 3664

vl>/

9

•800 2556

•3! il 5718

fi!2

9

34

•864 3536

•391 2141

9

•950 8571

•391 4293

"

9

3-037 5167

•391 6330

fi9l

9

S5

3 101 1591

•391 2796

8

3-194 3816

•391 4930

8

•287 7604

•391 6950

8

S6

•350 9424

•391 3460

671

8

•451 2499

•31)1 5576

646

7

•651 7138

•391 7">7!i

o»

I»M*{

7

37

•614 4497

•391 4131

\fl L

C~K

7

•72-2

•391 6229

.--.I

6

•830 1647

•3!»1 8215

VO»J

C 1 •'

6

38

•892 4902

•391 4809

o/o

6

4O08 1507

•391 6888

ODv

6

II -J3 9677

•3111 8857

M * -
i : i ^

6

S'J

4-185 9427

•391 5492

• >•!

6

•309 9183

•391 7553

«7rt

6

•434 0507

•31)1 9606

6

40

•495 7624

•391 6181

BBB

4

•628 5140

•391 8224

4

•761 4223

•392 0157

ww

4

41

•822 9894

•391 6874

693

3

•965006H

•391 8898

674

3

5-107 1807

•392 0814

657

fttiO

3

:-•

5-168 7569

•3! 11 7571

B90

3

5-320 5613

•391 9676

flOn

3

•472 5225

•392 1474

\f\fj

m

2

J .

•534 3023

•3!il 8270

*~i 1 1

2

•696 4499

•392 0256

" ' • '

2

•858 7544

•3!I2 213ti

* " '•,

2

::

•920 <»7»2
6-330 2'i.v.

•391 NI71
•391 9673

(01

702

1

6-094 0627
•511

•392

•:V.K Ki21

682
683

1

6-267 3013
•699 7361

•3!)2 2MH)
•392

86B

1

Tables of the G (r, v)-fntegrals
TABLE LIV— (continued).

139

,

r=39

r = 40

r=41

logF(r, K)

logH(r, r)

A

A*

logf(r, •>)

logH(r,»)

A

A2

log*(,,)

log H (r, v)

A

A2

0

1-600 7740

0 -OJO 6658

1-595 3459

0-390 8782

T-590 0501

0-391 0801

1

603 2891

•390 6669

23

•597 9271

•390 8793

19

22

•592 6975

•391 0812

21

s

•610 8391

•390 6703

M

23

•605 6756

•390 8826

oo
55

22

•600 6445

•391 0844

21

5

•623 4380

•390 6760

ou
7Q

23

•618 6058

•390 8881

77

22

•613 9059

•391 0898

21

4

•641 1093

•390 6838

I o

22

•636 7416

•390 8958

t II 1

22

•632 5064

•391 0973

'

21

S

•663 8860

•390 6940

101

22

•660 1171

•390 9057

99

22

•656 4807

•391 1069

96

21

124

120

118

6

•691 8107

•390 7063

1 dfi

22

•688 7760

•390 9177

142

22

•685 8737

•391 1187

21

7

•724 9361

•390 7209

14U

1 f'Q

22

•722 7721

•390 9319

IftO

21

•720 7406

•391 1325

1 r 1 1

21

8

•7«3 324C

1 7377

lOo

22

•762 1697

•390 9483

lOo

185

21

•761 1472

•391 1485

1W9

180

21

9

•807 0490

•:>,'.» 7566

911

22

•807 0434

•390 9667

206

21

•807 1702

•391 1665

200

20

10

•856 1930

•390 7777

-. 1 1

21

•857 4789

•390 9873

21

•858 8973

•391 1865

20

9T->

226

221

11

•910 8510

•3908009

BOfl

,»KO

21

•913 5732

•391 0099

0/17

20

•916 4280

•391 2086

OyM

20

1 .'

•971 1287

•390 8262

zoo

21

•975 4348

•391 0346

- < /

20

•979 8734

•391 2327

MM

20

13

0-037 1440

•390 8535

2<U

20

0-043 1846

•391 0612

9R7

20

0-049 3576

•391 2587

280

19

14

•109 0268

•390 8829

114

20

•1169556

•391 0899

iO 1

19

•1250171

•391 2867

299

19

15

•186 9202

•390 9143

Ol**

20

•196 8949

•391 1205

19

•207 0023

•391 3165

19

Hi

•270 9806

•390 9477

ORQ

19

•283 1630

•391 1530

344

19

•295 4780

•391 3483

317
335

18

17

•361 3789

•390 9829

•KM
071

19

•375 9350

•391 1874

18

•390 6238

•391 3818

353

18

18

•458 3010

•391 0201

'it 1

QQA

18

•475 4017

•391 2236

'l80

18

•492 6351

•391 4172

371

17

19

•561 9488

•391 0591

0VU

18

•581 7701

•391 2616

1Q7

17

•601 7243

•391 4542

388

17

;»

•672 5410

•391 0998

17

•695 2648

•391 3014

17

•718 1215

•391 4930

16

21

•790 3143

•391 1423

425

17

•816 1285

•391 3428

414

16

•842 0755

•391 5334

404
420

16

tt

•915 5247

•391 1865

A ^ft

16

•944 6237

•391 3859

1 IT

16

•973 8556

•391 5754

400

16

1-048 4484

•391 2323

4*>o
474

16

1-081 0339

•391 4305

****/

15

1-113 7524

•391 6190

too
451

15

~4

•189 3837

•391 279fi

*4f1

16

-i-i:> 6650

•391 4767

477

15

•2C2 0794

•391 6640

465

14

M

•338 6522

•391 3285

14

•378 8470

•391 5243

**< I

14

•419 1750

•391 7105

14

26

•4966007

•391 3788

M *7

14

•640 9357

•391 5734 -'!!!

14

•585 4038

•391 7684

479
4QO

13

:7

•663 6033

•391 4306

ol (

KQA

13

•712 3146

•391 6238

mn
M7

13

•761 1693

•391 8076

wv
505

13

28

•8400880

•391 4836

DVU

13

•893 3974

•391 6766

«1 i

12

•946 8652

•391 8581

si 7

12

29

2-026 4145

•391 5379

r.f.r.

12

2-084 6300

•391 7285

rV?

12

2-142 9789

•391 9097

vft 1

528

11

SO

•223 1268

•391 5935

BwD

11

•286 4933

•391 7827

J

11

•349 9933

•391 9626

11

SI

•430 7056

•391 6601

566

11

•499 5062

•391 8379

Mfl

UM

10

•568 4405

•392 0164

549

10

M

•649 6970

•391

KQ7

10

•724 2290

•391 8942

«M>.5
'-.)

10

•7! is 8947

•392 0713

558

9

S3

•880 6908

•391 7665

~z

9

•961 2w;

•391 9514

Oif

'i- 1

9

3-041 9761

•392 1272

567

9

34

3-1243*11

•391 8261

CAR

8

3-211 2729

•392 0095

• ' I
r,iu»

8

•I'iiX 3551

•392 1839

8

88

•381 2874

•391 8866

vllv

8

•474 9551

•392 0686

UW

8

•568 7567

•392 2414

8

36

•652 3259

•391 9479

612

C 1 '1

7

•763 0789

•392 1282

597

7

•853 9658

•392 2997

583

7

31

•938 2488

•392 0098

01V

r.-T

6

4O46 4738

•392 1886

fi?O

6

4-154 8329

•392 3686

695

6

38

4-239 9332

•392 0724

VZO

ftOl

6

•356 0397

•392 2496

*• 1 ".

6

•472 2803

•392 4181

(tf\f)

6

• ','.<

•558 3315

•392 1355

')•> I

6

•682 753:>

•392 3112

D1O

*

5

•807 3096

•392 4782

DvJV
fiOfi

6

40

•894 47!«

•392 1991

*

4

5-027 6774

•392 3732

4

5-161 0098

•392 5380

\J\JU

4

i CtAf\

f •-> 1

ROQ

41

5-249 5035

•392 2031

3

•391 9676

•392 4355

DM

fi97

3

•534 5660

•392 5995

UwI7

612

3

4%

•624 6328

•392 3274

«jr

2

•776 8842

•392 4982

OA I

' i ' " t

2

•929 2701

•392 6607

614

2

43

6-021 2079

•3923919

R4.7

2

6-183 8028

•392 5612

1

6-346 6322

•392 7221

616

1

44

•440 6950

•392 4 :.»;<;

D* '

1

•614 2272

•392 6242

fjOl

0

•787 8938

•392 7836

i;i i;

1

45

•884 6991

•3925213

7-069 8037

•392 6874

Ool

7-2.")5 0429

•392 8452

01V

18—2

140

/'<//-/..</,./• Sfittfttftciana an<? Bionntrii-ians
TABLE LIV— (continued).

i

r-4i

,.43

r = 4l

logF(r, r)

lo«H(r, ,)

A

*

log », »)

log H (r, ••)

A

tf

.o«^,,,

,o,*(r>,,

A

^

u

; 8804

0-391 2724

1-579 8310

0-391 4656

in

1-674 8962

o-3:i:

in

I

•5876939

•391 2734

01

21

•582 6106

31)1 4566

Hf

01

21

•577 7420

31)1 6316

•V

20

s

•6857394

•391 2766

OI

KO

21

•590 9546

•391 15H7

• , I

20

•6862845

•391 6345

20

s

•6093321

•3!)1 2818

'-

21

•6048786

31)1 4648

51

— ,

BO

•600 5397

31)1 63»5

.',11

BO

4

•6283972

•391 2891

'11

21

•6244083

•391 4720

72

20

•6205341

•391

7<>

BO

s

•6529704

•391 2986

J-i

21

•649 6803

•391 4812

20

•6463049

•391 6664

BO

6

• notn

•391 3100

115

1 or.

21

•680 4416

•391 4924

112

20

•6779004

•391

i -ii.

2"

7

•718

31)1 3235

I .i.i
1 *\fl

20

7170601

•391 5056

1 *\Q

20

•715 3798

•31)1 (i7!l3

i -•'
i 1*4

2'.

-

•760 2510

•3'Jl 33!) 1

1 iIU

20

•759 4750

•391 6208

1 ...

BO

•768 8138

'3!)1 (i!) 12

i *. •

19

:•

•807 4232

•31U 35U7

/o

1 Mt'

20

•807 7965

•391 6380

172

19

•8082846

•:',: H

1 s~

19

JO

•860 4419

•391 3763

lif\t

20

•862 1068

•391 5571

19

•863 8860

•391 72!«;

1O J

19

n

•9194089

•391 3978

216

19

1P22 5103

•391 5781

210

19

•925 7265

•391 7502

206

•»•> i

1!)

•9844383

•391 4213

->-. 1

19

•989 1236

•391 6011

230

a A Q

19

•993 9238

•31H 772(1

__ i

•M"4

18

IS

MM 8MB

•391 4467

_. > 1
2-0

19

0-062 0767

•391 6259

•vo

OP*"

19

0-0686113

•3!)1 77119

_ *o

18

n

•1332048

•391 4740

i •'

19

•141 5130

•391 6526

mK

18

•149 9361

•3!)1 822!)

",-

18

IS

•217 2361

•391 6032

292

18

•227 5903

•391 6810

IBB

18

•238 0595

•31U 8508

17

•••

•307 9195

•391 6341

•i OH

18

•320 4814

•391 7113

303

17

•333 1584

•3!)1 8803

MM

;,•

•405 4390

•391 6669

020

17

•420 3748

•391 7 133

!> -

17

•435 4256

•31)1 in H;

•!•»!

17

18

•5099950

•391 6014

MO

17

•527 4755

•391 7770

OOl

17

•545 0710

•3!)l 1)115

•I"'-

16

19

SO

•621 8060
•741 1047

•391 6376
•391 6754

378

16
16

•642 0063
•764 2086

•391 8123
•391 8493

370

16
16

•662 3227
•787 4277

31)1 9791
•392 0152

38]

16
16

tl

•868 1492

•391 7149

395
j.in

16

•894 3438

•391 8878

385

15

•920 6532

•392 0628

377

QQl

IS

:

1-003 21 13

•391 7569

41U
49R

16

1-032 6937

•391 9279

—•

16

1-062 2883

•31)2 01120

O.7 1

406

14

•14'-

•391 7984

4 — •*

15

•179 5637

•391 9694

0

14

•212 6450

•392 132-;

14

. ;

BH BBOfl

•391 8424

A " 1

14

•336 2826

•392 0124

4.10

14

•372 0600

•392 1746

j«»*i

13

•4696888

•391 8878

454

13

•600 2054

•392 0567

443

13

•5108966

•392 2179

4.J.5

13

---

•6299989

•391 9345

467

13

•674 7149

•392 1024

457

13

•7196463

•3!)2 2625

44<;

12

t7

•810 1308

•391 9826

-4QO

12

•859 2234

•392 1493

469

12

•908 4315

•3923084

168

12

.:>

2-0004600

•392 0318

CM

Kf\A

12

2-054 1759

•392 1974

481

12

2-108 0073

•31)2 355 I

4(U

11

SO

•2014548
•413 6204

•392 0823
•392 1338

5O4
516

11
10

•260 0519
•477 3687

•392 2467
•392 2970

493
504

11
10

•318 7(i 15
•511 2327

•3112 103..
•392

)H2

11
10

SI

•637 5019

•392 1864

526

p; no

10

•7066845

•392 3 is |

514

10

•77.'.

•3H2 5030

- 1 -•

9

•873 6876

•392 2400

O.TO
E je

9

•948 6018

•31)2 4007

523

!)

3-023 (S3 17

•31)2 6641

O 1 .

9

S,1

1 LSJ BUB

•392 2945

M0

9

3-203 7711

•392 4510

.).J.)

8

•284 8450

•3!)2 6062

- .1C

8

34

•3856647

•392 3499

554

- , . .

8

•472 8957

•3H2 5081

541

8

•5603l2r,

•392 6590

OZo
nor-

8

S.5

•6626867

•3924060

on

7

7 ,<! 7363

•392 5629

548

•7
i

•850 9027

•392 7126

OoO

7

S6

•9649802

•392 4629

- —

7

4-056 1162

•392 6185

556

6

4-157 3682

•392 7i;<;it

543

6

S7

4-263 3195

31)2 5204

075

6

•371 9277

31(2 6747

66B

6

•4*o i;.-.2o

3112 8218

v,~,

6

S8

•588 6485

•31)2 5786

ool

5

•705 1384

•392 7314

667

5

•821 7 t ! 1

-3112 8773

.).».!
~ - i i

6

-•'•

1)31 Ii:i3;,

31)2 6371

M?

4

5-056 7991

•31)2 7886

572

4

5-181 720*

•3!)2 11332

.!.».»

.v ; i

4

40

5-294 4700

392 6962

4

•428 0520

•392 8463

677

3

•561 7502

•31)2 'JH96

4

41

1177 2923

31.2 net

694

ItfVT

3

•820 1404

•392 9044

580

e QO.

3

•9631049

•393 0463

;,7o

3

'

6-081 7H3!)

•392 8153

• ' ' i

2

6-234 4197

•392 9627

MM

2

6-387 1711*

•31)3 1033

2

•$••'

•;,•,:. :;-...,

till

1

•672 3690

•3930212

J2

2

•S35 4i;i!)

•31)3 ico;,

57*

1

44

•9616887

•392 9353

DU1

,-, , i

1

7-1366065

•3113 07!t!l

KQ"

1

7-3011 •;.•{*'.)

•3!)3 2178

574

1

4*

7-440 4103

•3929964

WJI

•ii25 8999

3!)3 1386

l)O*

•811 5060

•3i)3 2752

Tables of the G (r, v)-Integrals
TABLE LIV— (continued).

141

,

r=4o

r=46

r-47

^

lo^' H (r, r)

A

*

log F(r, »)

W(r,»

A

A2

log F (r, v)

logtf(r, r)

A

A"

II

T -570 0711

0-391 7975

in

1-565 3509

0-391 9572

10

1-5607311

0-392 1100

9

1

•572 9830

•391 7984

BU
'" 1

20

•568 3289

•391 9581

19

•563 7753

•392 1110

28

19

s

•581 7241

•391 8014

20

•577 2685

•391 9610

48

19

•572 9134

•392 1138

47

19

3

•5963100

•391 8063

t'~

19

•592 1863

•391 9658

19

•588 1625

•392 1185

66

19

4

•616 7090

•391 8131

Oo

QQ

19

•613 1100

•391 9725

86

19

•609 5508

•392 1250

84

19

6

•043 1393

•391 8219

OO

19

•640 0785

•391 9811

19

•637 1182

•392 1334

18

1H7

105

103

,;

••;75 40>9

•391 8320

1U1

19

•673 1423

•391 9915

124

19

•670 9162

•392 1437

121

18

7

•7 13 H| 92

•391 8452

1 IT

19

•712 3034

•392 0039

142

19

•711 0082

•392 1557

139

18

8

•75* 2i -.23

•391 8598

1 -4 J

19

•757 8157

•39201*1

11:1

18

•757 4696

•392 1697

157

18

y

•808 8*24

•391 8762

1 U'J

19

•8095852

•392 0342

1 VI

171

18

•810 3885

•392 1854

175

18

10

•N',5 7761

•391 8944

1OO

18

•867 7706

•392 0520

L t *7

18

•869 8656

•392 2029

18

11

•0524

•391 9145

•M

18

•932 4834

•392 0717

197
215

18

•936 0149

•392 2221

193

210

17

I!

•998 M337

•391 9365

OQ-

18

OO038487

•392 0932

232

18

0-0089642

•392 2431

227

17

IS

0-075 2558

•391 !)002

~V-

18

•082 0054

•392 1104

249

17

•088 8555

•392 2658

244

17

14

•1584691

•391 9857

2-p

17

•167 1071

•392 1413

17

•175 8458

•392 2902

261

17

IB

•248 6386

•3920129

. Z

17

•259 322"

•392 1679

17

•270 1076

•392 3163

16

OQQ

OQO

277

1G

•345 9453

•392 0418

* 17

•358 8372

•392 1962

200

•''!')

16

•371 8299

•392 3440

293

16

17

•4.r)0 5863

•392 0724

•\->->

16

•465

•392 2261

_.T. "

315

16

•481 2188

•392 3732

308

16

18

502 7765

•392 1040

ozz

QQQ

10

. " w| I ' v""'J
• 1 ' 1 t)O 1 * 1

•392 2576

331

10

•598 4987

•392 4041

323

15

19

.; 7492

•392 1383

ooo

15

•7< 13 2808

•392 2906

346

15

•723 9132

•392 4364

338

15

.'n

•810 7568

•392 1737

15

•834 1911

•392 3252

15

•857 7263

•392 4702

14

OftU

360

353

21

•947 0729

•392 2105

«>DO
OQQ

15

•973 5979

•392 3612

375

14

1-000 2236

•392 5055

367

14

an

1-091 9931

•3922488

ooo

QUT

14

1-121 8031

•392 3987

388

14

•151 7141

•392 5421

380

13

;.:

•245 -*305

•392 2*85

• ' - ' t
A\ 1

14

•279 1333

•392 4375

402

13

•312 5311

•392 5801

393

13

*4

M08 9476

•392 3295

-411

13

•4 15 9406

•392 4776

414

13

•483 0345

•392 6194

406

13

0

•581 6979

•392 3719

13

•022 6047

•392 5191

12

•663 6124

•392 6600

12

26

•7644880

•392 4155

448

12

•8095352

•392 5618

427

438

12

•854 6834

•392 7018

418

429

12

ft

7 7499

•392 4603

460

12

2O07 1738

•392 6056

450

11

2-056 6988

•392 7447

440

11

28

2-161 9490

•.•592 5063

471

11

•215 9964

•3926506

461

11

•270 1448

•392 7887

451

11

10

•377 5876

•392 5534

1 1 i

JUl

10

•4365163

•392 6967

10

•495 5462

•392 8338

461

10

50

•oo.-, 2071

•392 6015

toi

10

•6692873

•392 7437

10

•733 4685

•392 8799

9

31

-15 3918

•392 6506

-,,, |

9

•9149064

•392 7918

480
MO

0

•984 5222

•392 9269

470

479

9

32

3-O98 772:;

•392 7006

9

3-1740186

•392 8407

498

8

3-249 3662

•392 9748

487

8

33

•3660297

392 7515

117

8

•447 3201

•392 8905

505

8

•528 7119

•393 0235

495

8

34

•6479002

•392 8032

9AM

7

•735 5636

•392 9410

513

7

•823 3284

•393 0729

502

7

S5

•945 1800

•392 8556

7

4-039 5631

•392 9923

6

4-134 0476

•393 1231

6

30

4 258 7310

•31.2 :><i-7

U9

6

•3601998

•393 0442

519

525

6

•461 7700

•393 1740

508

5 1 1

6

37

•589 4872

•392 9624

• >r» i

5

•6984283

•393 0967

530

6

•807 4710

•393 2254

519

5

38

•938 101 \

•393 0166

547

5

5O55 2844

•393 1498

6

5-172 2090

•393 277:5

524

5

39

5-30<i 7534

•3930713

t**» i
551

4

•431 8925

•393 2033

539

4

•557 1330

•393 3297

528

4

40

095 5595

•393 1264

3

•829 4749

•393 2572

3

•963 4920

•393 3824

3

r.rt

"ill

631

41

6-106 1805

•393 1818

•Mis

3

6-249 3623

•3933115

• • i.j

r, i -.

3

6-392 6458

•393 4355

534

3

4-

•540 0352

•393 2370

-,-.li

2

093 0048

•393 3660

\J**O

' • 1 t

2

•846 07G1

•393 4889

536

2

',.:

ffJK

•393 2935

• i. >.t

561

1

7-161 9854

•393 4207

548

1

7-325 4006

•393 5424

537

1

44

•483 7836

•393 3496

1

•658 0347

•393 4755

549

1

•832 3876

•393 5961

537

0

V>

•997 2231

•393 4057

8-183 0474

•393 5304

8-308 9732

•393 6498

14-J

or Statist it-inn* «n<l /iioiin tr
TABLK

'

r-48

r-49

r-50

.o.*<r,,)

log H (r. r)

A

A«

!<*,>,„

logH(r, r)

A

A<

logFfr, ,)

IOR // (r, ,)

A

A<

0

T-666 2075

•:: <J IBM

1-551 7763

0-392 3969

1 •:. 17 4336

0-392 5316

1

•659 3178

•392 2574

oo

18

•564 !i:.27

•if: 12 3978

97

18

•660 6762

•392 5325

18

*

•6686645

•3!)2 2601

ZO

18

•564 4879

•392 4005

•4

18

•6604099

•392 5352

44

18

a

•6842348

•392 2(147

18

•680 3995

•392 4050

18

•576 6629

•392 5396

18

4

•6060878

•31(2 2711

•

18

•602 7172

•392 4113

Rl

18

•599 4362

•932 5457

79

17

r,

•i, ;i L-. U

•392 2794

18

•631 4824

•392 4193

01

18

•6287993

•392 6536

4 o

17

u

•6687863

•392 2H94

100

18

•6667488

•392 4291

98

18

•6647999

•392 5633

96
114

17

7

•709 7492

•3923012

i • v

18

•708 5826

•392 4407

17

•707 5046

•392 6746

131

17

8

•757 21U-

•392 3149

loO

1 *\±

18

•757 0624

•392 4541

17

•756 9936

•392 6877

148

17

:•

•811 2881

•392 3302

!«>•*

17

•812 2801

•392 4692

iftft

17

•813 3607

•392 6025

164

17

10

•8720669

•392 3474

17

•874 3406

•392 4859

1W5

17

•876 7129

•392 6189

17

1 ftQ,

IftK

181

11

•939 r. 127

•3:'2 3662

lOtf

17

•943 3629

•392 5044

1OU

•'ill

17

•947 1718

•392 6370

197

16

: .

OO14 1760

•3!i2 3868

000

17

00194803

•392 5245

ZU1
91ft

17

0-024 8733

•31)2 6568

lot

213

16

1,1

•0958020

•392 4090

ZZZ

17

•102 8409

•392 5463

Z1O

16

•1099685

•3! 12 6781

229

16

14

•1846808

•392 4329

-P- r

16

•193 6083

•3! 12 5697

2JVO

16

•202 6245

•3:i2 7010

245

16

15

•2809888

•392 4584

•BW

16

•291 9625

•392 5947

16

•303 0249

•392 7255

16

971

9fifi

260

16

•384 9189

•392 4856

Bf 1
007

16

•398 1006

•3926213

•0U
•'SI

15

•411 3709

•392 7516

275

16

i:

486 '>l-

•392 5142

ZOi

15

•512 2374

•392 6493

•01

•AH

15

•527 8818

•392 7791

15

18

•6166066

•392 5444

O17

16

•6346070

•392 6789

HPv

U

•6627964

3!)2 8080

304

14

19

•744 6421

•3:i2 .".7iio

•> 1 *
Oil

16

•765 4636

•392 7099

^91

14

•786 3739

•392 8384

118

14

.•"

•881 3579

•3926002

pl9J

14

•9060822

•392 7424

14

•928 8954

•392 8702

o 10

14

.:

10269460

•392 6437

345

14

1-0537610

•392 7762

338

OKO

13

1-080 6649

•392 9034

331

345

14

.:

•181 7216

•392 6796

JJJ

13

•211 8218

•392 8114

9Om

13

•2420110

•392 9378

357

13

•34U 1 12.1 1

•392 7168

JYz

13

•379 6125

•3H2 8478

0-7>7

12

•413 2886

•392 9736

369

12

*4

•520 2260

•392 7553

385

12

•557 8084

•392 8855

•t«Q

12

•594 8807

•393 0105

381

12

K

7o| 7169

•392 7950

397

12

•746 9141

•392 9244

12

•787 2004

•393 0486

11

K

•8999284

•392 8359

409

tnf\

11

•945 2662

•392 9645

401

11

•9906930

•393 0879

393

11

S7

2-106 3205

•392 8779

42O

I'M

11

2-1560351

•393 0057

0

11

2-205 8389

•393 1282

414

10

as

324 3901

•392 9210

4ol

10

•37* 7283

•393 0479

4/J

10

•433 1566

•393 1696

tit

10

S9

•554 6729

•392 9652

441

i " i

10

•613 8925

•393 0911

432

10

•673 2014

•393 2120

413

9

so

•797 7467

•393 0103

4ol

9

•862 1178

•393 1353

442

9

•926 5783

•393 2653

too

9

SI

30642360

•393 0563

460 j 9

3-124 0408

•393 1804

461

8

3-193 9359

•933 2995

442
450

8

SS

•324 8107

•393 1032

IS 8

•400 3485

•393 2263

'.-

8

•475 9753

:5!>3 3445

458

8

S3

•6102007

•393 1509 !„[

8

•691 7826

•393 2731

1?I

7

•773 l.»3K

•393 3903

465

t

34

•911 1903

393 1993

Oil

7

•999 1454

•393 3205

ifll

7

4-087 1898

•393 4368

479

7

SB

; HI .;L-.-3

•3932486

491

6

4-323 3042

•3933686

6

•418 0685

•3934840

^1 i

6

36

•6634374

•3932982

498

6

•666 1980

•3934174

188

ITf

6

•767 0481

•393 6318

478

.)>;{

6

S7

•9166109

3'.i3 3486

6O3

6

5O25 8441

•3!13 4<>i;7

WM

5

5-135 1668

•393 5801

6

S8

5-289 230!)

•393 3994

.'i' IV
M •'

4

•406 3461

• :. iii.'i

wv>

4

•523 5508

•393 6289

492

4

SS

•6824708

•3934507

O I*}

KIT

4

•807 9020

•3!i3 5667

OTJB

4

•933 422"

•393 6781

4

40

6-0976055

•3935024

.817

3

6-231 8144

•393 6174

3

6-3661119

•393 7277

3

41

•'<•

•53i;

•99! <

•393 5543
3936066

no

622

8

•679 5011
7-162 5073

•3936683
•393 7195

PJ08

512

3
2

•823 0651
7-305 8593

•393 7776
•393 8278

499
601

2

2

^

43
44
4*

7-488 9134
6-0068381
•6649966

•3937116
•393 7'ilJ

."'2 i
626
526

1
1

•662 5197
8-181 3822
•741 1137

•393 7708
•393 8223
•393 S73'.i

515
616

1

1

•816 2167
8-356 0160
•9273206

•393 8781
•393 9286
•393 9791

606
606

1
1

Miscellaneous Constants in Frequent Use 143

TABLE LV.

Miscellaneous Coitsbints.

ir 3-141 592fi 54
l«.gir -4971499
log 2* -798 1799

log 1-2018201

log -. _ 1-600 9100

« 2-718 2818 28
•367 8794 41

log •• -434 2944 82
log«A -036 1912 07
log log 9 -63777U9 16

1 centimetre = -393 70432 ins.

1 inch = 2-539 9772 cm.

1 square cm. = -155 00309 sq. ins.

1 square inch= 6'451 4842 sq. < m>-

1 cubic cm. = O61 025386 cub. ins.

1 cubic inch = 16-386 623 cub. cm.

1 kilogram = 2-204 6212 Ibs. avoir.

1 Ib. avoir. = -453 59266 kg.

1 radian =57'295 7795 degrees.

1 degree = -017 4532 925 radians.

CAMBRIDGE : PB1NTKD BT JOHN CLAY. M.A. AT THE UHIVEBSITT PIIE8S.

Pearson, Karl

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A Preliminary Study of Extreme Alcoholism in Adults

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The Treasury of Human Inheritance. Prefatory Matter

I ices to Vol. I With
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A Second Study of Extreme Alcoholism in Adults
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I. The Scope and Importance to the State of the Science
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II. The Groundwork of Eugenics. By

III. The Relative Strength of Nurture and Nature. By

IV. On the Marriage of First Cousins.

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IX. Darwinism Medical Progress, and Eugenics: The

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I. The Influence o* Parental Alcoholism on the Physique

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•J Defect, MaJ-Nutrition, and the Teacher's ap-

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An Attempt to correct some of the Misstatements

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IV. The Fight against Tuberculosis and the Death-rate

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V. Social Problems: Their treatment, Past, Present and

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VI. Eugenics and Public Health: Lecture to the York Con-

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VII. Mendelism and the Problem of Mental Defect. I. A

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VIII. Mendelism and the Problem of Mental Defect II. The

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