### Biometrika Trust

The Distribution of `Student's t in Random Samples of any Size Drawn from Non-Normal Universes Author(s): A. K. Gayen Source: Biometrika, Vol. 36, No. 3/4 (Dec., 1949), pp. 353-369 Published by: Biometrika Trust 

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### [ 353 ]

THE DISTRIBUTION OF 'STUDENT'S' t IN RANDOM SAMPLES OF ANY SIZE DRAWN FROM NON-NORMAL UNIVERSES BY A. K. GAYEN, St Catharine's College, University

ofCambridge
1. INTRODUCTION 

### Theeffect

of universal non-normality

on 'Student's'thas been studied so far either byway of particular

numerical

examples

or by approximations

to itssampling

distribution

based on thelarge

sample assumption.

E. S. Pearson

(1928, 1929)has shown

in hisexperimental investigation

thatthe effect of universal

'excess' and of 'skewness' on 'Student's' ratioz (which is related

to t by t = z V(n

- 

### 1)) maybe considerable. He has also furnished

a small table, basedon experimental

results, showing some actualvalues

of theprobability

integral for the ratio insamples

of2, 5, 10and 20 from

a few universes with specified values of'skewness' and 'excess'. M. S. Bartlett (1935) obtained some theoretical

results for approximately representing

the distribution

of t, taking into account the universal

'skewness'

and 'excess'. The approximation

is basedon the

assumption

of large samples and is notvery

satisfactory, for inthat approach,

as he himself

hasalso pointed

out, the term 'approximation'

is perhaps misleading.

Stillfrom

the form ofthe expression

obtained,

it has been observed

thatthe effect of the 'skewness'

A3 in the original

distribution

is themore

serious and that of the ' excess' A4 is small. This, of course, confirms Pearson's

experimental

results, but the resulting expression

cannot clearly furnish a quantitative

measure of the corrections

to be applied tonormal

theory probabilities

of t, when the sample size is small.R. C. Geary

(1936) obtained anexpression

for the distribution

of t in samples of any size drawn from a slightly asymmetrical universe.

It consists

oftwo components,

one being

the 'normal

theory' t and the other

a term inA3, which hasbeen called the' corrective

function'

duetoit. He has,

alsogiven

a table of corrective

tail-area probabilities

for somerepresentative

values ofn. In a recentcommunication

(1947)he has given

from a different

approach

an approximate

formula for the frequency

of t, correct o n-2, 

### n being

the sizeof sample. From this and analogous

results on non-normal

Fisher's z, he has shown,

by a few

illustrative

xamples,

that the inferences drawn from the standard

tablesmay be seriously in error even in some cases where the parent is notconsiderably

non-normal.

He has accordingly

pointed outthe primaryimportance of testing

for normality

from the available samplesand has suggested that when universal

normality

cannot be assumed,

the standard

tablesshould

be corrected

by using thesample

estimates

ofA3 and A4, etc., in conjunction

with thetheoretical

results deduced therein. But so far

as thefrequency

oftis concerned,

satisfactory

measures

of probabilities for small samples cannot be obtained

from hisresults,

since they are mainly

based on large sample assumptions.

Alsothefirst

few terms ofhisresult,

as far as n'l, agree

with those of Bartlett, which, as has beenpointed

out, arenotvery

satisfactory. The purpose

of the present investigation

is to obtain

the form of thecorrective

terms due to population

cumulants

A3, A4, 

### etc.,in thefrequency

function oft for any size ofsample. For this the parent population

will be specified

bytheEdgeworth

series. It has been

proved rigorously

byCramer

( 1928) that this series gives a real asymptotic

expansion

of any universe f(x),in powers

ofv64, where voo is thenumber

ofsources

of 'elementary

errors'. Inclusion

of

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### 354 Distributionof 'Student's' t from non-normal universes terms of order vO, vO 

### i, l 1,..., P-Or

### gives

rise tothe first, second, third, ..., (r+ 1 )thapproximations to the law of error. We shall consider mainly the third approximation,

including

terms in A3, A4 and A2, 

### butshallalso deal with

the fourth

approximation

which takes in A5,

A3A4 and A3. Probabilities

of values of t, obtained from the derived formula, for theparent

specified by the third

approximation,

are in satisfactory

agreement

with the experimental

determinationsmade so far,

even in cases where

the sampled

populations

are represented

by the exponential

curve or a very

skew Type III curve

ofPearson.

Accordingly,

it appears

that thederived

expressions

for 'Student's't may perhaps provide quitesatisfactory

estimates of the probabilities,

for a fairly wide class of non-normal universes,

especially

whenthe sample sizeis nottoo small. The valuesofthe cumulants ofthepopulation

have beenassumed

to be known,

and the question oftheir estimation

has notbeenconsidered.

2. JOINT DISTRIBUTION OF THE SUM AND THE SUM OF SQUARES

OF DEVIATIONS OF n SAMPLE OBSERVATIONS 

### We shallfirst

ofall obtain

thedistribution

of 'Student's't,

with particular

eference to the parent population

beingspecified

by the third

approximation

to the law oferror.

This will giveus the corrective

terms due to A3,

A4and A2. 

### It will be shown afterwards

that similar corrective

terms due toA, A3A4and

A3 

### can be obtained

bythesamemethods

when the parent

population

is represented

by the Edgeworth

seriesup to the corresponding terms. Let us then

consider the parent

population

to be specified

by the third

approximation, correct o vi-', f(x) = 0(x)-3 0(3)(x) 

### + 4 0(4)(X) + 3 0(6)(X) 

### (2- 1) where A3 and A4 are themeasures

of universal

skewness and excess, 0(x) = 1 e-*x2 (2.2) V-(27r) 

### thenormal

function in standardized

form, and (p)(X)= 

### (d-) (x) = (- 

### 1) Hv(x) 

### q(x), (2.3) 

### where Hv(x) is thewell-known

Hermitian

polynomial

ofdegree v. Let xl,

### x2,

..., 

### xn be n independent

sample observations

drawn at random

from the universe (2.1),and letus adoptthenotations n n 

### S 

### E = X 

### d 2 2 (2-4) 1 1
n S2 

### and S2= (v-x2)2 = - (n-1)82 (25) 

### where hetmtva cn whereS2 is the estimatedvariance.

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### A. K. GAYEN 

### 355

By the use of Bartlett's method (1935) thejoint frequency density of (S1, S2) 

### (which

is also the jointfrequency

density of (S,,S'), since

theJacobian

oftransformation

is unity) maybe written

in theform g1(S1, S2) = W(n-1)-+3{Dl W(n+5)-6DiD2W(n+3)}
+ !4 {D4 W(n + 7)- 12D2D2W(n + 5) + 12D2W(n+ 3)}
+n2 {nD6W(n + 11)-6(2n + 3) AD 2W(n+9)+36(n+4)D2D2W(n+7)-12OD3W(n+5)}, (2.6) when terms

in Ai otherthan in A3,

A4 and A3 are neglected. In (2.6) W(n') = j(



### e n' 

### )J, 2 R&('2) e-iS2)

e-iSs (SI - S2/n) J(no-2) V/(2nrn) x 2in'r(In') and D as . D _-_ (2.8) 

# 2-esI& We now proceed to evaluate the derivatives involvedin the expression (2 6). It is possible to transform

theproduct

operators

of theform D1,D 2 (where

v1, v2= 0 1, 2, ...)into functions of the singleoperator

D1. For any typical function W(n4), defined in (2 7), where n' stands for (n-1), (n+ 1), 

### (n+ 3), 

## ...,it can be shown that
DvD2 2W(n') - ( 2)v2 

### [I -W() (V2) Dv1 W(n'-2) + 

# (2)Dv' W(n'-4)

+ (l (2 DvW(no-2r) +... + (-J)v2 Dv1W(nO-2V2) (2.9) By differentiating

W(n') successively

withrespectto S1 we get
D1W(nO) = (-1) W(nO-2) Si D2W(n') = 1)2 

### W(n'- 4) QS1)2 

### 1 S2 

### D3W(n')

= 1)3 

### W(n' 6)(!S1) 

## 3 

## (S1Q 

# ) ()S2} 1 W(nO) )= 1)' 

### W(n -8) -(- 8)() 

## () + (nO- 6)(n3 -8) and so on. It will be observedthat the numerical coefficients

are those ofthe Hermitianpolynomials 

### H = x 1

) X12-2 +.1(V-! ) (!-2) (V-.3) _

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### 356 Distribution of 'Student's'tfrom

non-normal

universes So we can write,

in general, 

### D =W(n') 1)vW(n-2v){((2) 1 ) 

# ()

(- 2)
+ v(v -1) 

### (v -2) (v 

### 3) {Si v-4S28 _22.0 22.2!(n'-2v+2)(n'-2v) n( ) 

### n 

# } (.0 For the present

case as we have specified the population by the thirdapproximation to the law of error

we requirethe values of Dv, forv = 0, 1, 2,

... up to 6. Now using results (2.9) and (2.10) in (2 6), we obtain, after some simplification, the following

expression

for thejoint frequency

function

of S, and S2: g1(S1,S 2) = 

### W(n-1 

### ) {+H3( 

### H ) + 3( ) H1Q!)}

+ ?A4 

### {HQI

S) + 6S2) H2(S) + 3( n ) (8)2) + n 

# {

[nQs)
- 3(2n + 3) (Si)4 4 (n) ] (n) (n 1) (n)2 ] ( l) n) x ((n +4) - 15

+- 6 -Q 3(n+3) (- 2+6 + n 1 ()2.

whereH1(Sl/n)

is the Hermitianpolynomial of degree v in (Sl/n), and W(n - 1) is given by (2 7). As a check it was foundthat, to the requiredapproximation,



# f g1(Sl, S2) dS2 gave the standard formula

for the distribution

of the sum of n sample observations.

3. THE FREQUENCY FUNCTION OF ESTIMATED VARIANCE Integrating

(2.11) for

S, between the limits - oo and oo, we find, for the frequency

of S2, aftersome simplification, 91(., S2) = V(S2, (n-1))[ + A4 (n- 1)2{( +2( 1) -2 (21) + I

+ 3 (n - 1) (n -_2__2_2 38

- (3.1) + (n-1)(n-2) ((+3) (n +21)(n-1 (n+1)(n-1) 

### + (n-1)1) where V(S2,(n - 1)) is the frequency

density of normal S2. Forthe distribution of 82 = S2/(n - 1), this becomes q1(, (n- 1)s2) (n- 1)d(s2). (3.2)

a Note that if we introduce the operator

D02 =[S2/(- 1)]' the frequency function

(3.1) can be put in the form V(n -1) + 4-D022 V(n +3) -
.

A(n 2)) D3V(n +5)) (3-3) 2n S 3n 3een 1i2 02 where S2 has been omitted

from the V's.

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All use subject to JSTOR Terms and ConditionsA. K. GAYEN 357 4. THE DISTRIBUTION OF 'STUDENT'S' t Put S1 = (n S2) t

in (2. 11)and integrate

for S2 from 0 to oo, when we find that a typicalterm W(n - 1) Srl Sr2 dS1dS2 becomes
nirn. 21(ri+2r2)

r[j(n + r, + 2r2)] tr' dt (4-1) 7T (n-1)*(r1+1) F[j(n-1)] [1 

### + t2/(n-1 )]J(n+rl+2r2) It will be noticed that (4.1) gives the normal theory t forr, = r2= 0. We shall denote the distribution

of t by p(t). Thus we get by using (4. 1) and after simplification 

### p(t)= F( n 1 + 

### A {3(n-1) t-(2n-1) t3} V1T[(n -1)] IF[ '(n - 1)]'[1 + t2/(n - 1)]Jn 3

### 6(n-1) V

(2nnr) [1I + t2/ (n 

### - 1)] j(n+3) -F[j(n + 2)] {3(n-1 )-6(n 

### + 1)t2 

### + (n + 1) t4} A424nV[r(n 

### -1)] r[j(n + 

### 3)] [I + t2/ (n -1)] 1(n+4) 

### rn2 J3(n

- 1)2 (2n+ 11)-9(n -1) (n+3) (2n- 1) t2 (2 -i( + -3(n + 1) (n+3) (2n+ 13) t4+(n+ 1) (n+3)(2n+5)t 

### 3 ~~~144n(n

-1)1 V71r

[I(n2 + 5)] [I + t2/(n

- 1)]J(X+6) p0(t)+ 

### ~ ~~~~~ +A~p'k(t). 42 = POMt) + A3PpA3(t)- A4pA4(t) 3 Obviously po(t) is the normal theory t for a sample of size n, and pA,(t), pA4(t) and pAi(t) are the corrective

terms due to universalA3,

A,,an A3 As a check on the above expression for t, momentsmay be calculated directly, about t = 0 and to n-2; theyare foundto be

/l~ -2nil +4n+ 82t I=1+ 2 (1I+ A2)

+ 2 (3 - A4)+... n (t) A ~(4.3) 2ni( )l 21 11t4(t) =3 + - (9 -A4 + 14A2)

+ 2(102 - 3A4 + 120A2) +
.. n 3n2 These are in agreement with the asymptotic

formulae

for the moments

of t forany universe as obtained by Geary (1936), up to first power of A4and second power of A3. All these corrective termswhen integrated between the limits - o and oo contribute nothing towards the integral,

as may be easily deduced from the fact that 

## f (t) dt = 1. -00

5. THE DISTRIBUTION OF 'STUDENT'S' t FOR THE PARENT SPECIFIED BY THE FOURTH APPROXIMATION TO THE LAW OF ERROR Let us specify

the parent universeby the fourth approximation

to the law of error,

correct up to vP-, given by f() = 0(X)--A30(3)(X) +A40(4)(X) 10A A5X 0(5)(X) 35A3A4 280A

3 ! 4 ! 

### ~ 6!/ 7! 9 S9() 51

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### 358 Distributionof 'Student's' t fromnon-normal universes It is of interest o know thedistribution

of82 and 'Student's'tin random samples from this universe. We canarrive

at these results byfollowing

a similar procedure,

and the expressions corresponding

to (2.6), (2.11)and (4.2) will haveadditional

terms involving

A AA a 3 

### We shall omit

these intermediate

stepsand give belowonlythe extraterms in thejoint distribution

of81and S2 

### andin thedistribution oft. Thus

corresponding

to (2.1 1), the joint distribution 2(Sl , '2), of 8 and S2 will be given by

92(S1 S2) =1(S1, 82) + 

### W(n - 1) 

### [n 5

(H5QSi) + io($) H (Si)tl5

)(') H1(SI)} 

### nA3/ S,4 

### 7S( S,3\S + 14i n + 3(3n

+ 4) () +21(n+4) () + 3(3n + 32)Qn)] 144 nnnn

- (I) 

## (I2) [3n(3n + 11) ( 5+ 6(7n2 + 43n + 20) ( )3 + 9(3n2 + 39n + 28) 

### (S')]
+n 1) n '822 (n+1) (-i) 2[3n(7n2+58n+75)( S +9(3n3t+46n2+71n+20)( S1) 

### ( n. + 3) (n - l) n}

- 1n [(n)

+ -129f; ([n2(J + 9'n(n + 3) (3n)7 + 27(n + 1) (n + 6) 

### 1 +9(n+3) (n+32) (!n 

### )+ l3(n+6) 

### ( n)] (n 

### -1) ( n) [n2(n 

### +7) +

3n(2nJ + 21n +33)

+ 3(3)3 + k51n + 166n+ 60) (Si) 

### + 3(23n2 + ldin + 

### 96)

+ (n 1)(n -Ij( 2)2 [n2fl2 + 14fl+ 4)(1) + n(3n3 +61n2 +271n + 225)()

+ 3(13n3+ i06n2+ 131n+ 30) (S)]
( + 3)(n+1)(n-1 (S

7 ) S[( i5(n)

+ 3(29n3 + 273n2 + 533n + 285) (in)] +(n +5)(n+ 3) (n + 1) (-1 ) (3,2 )4 [( 4163

+ 43n2+ I t8+i15)Q1)]}] 

### (5.2)
Since the expressions

appealinlg

withA5, A3A4 andl A3 in (5.2) are odldfunctions in S1, the integralof Y2(81' 22) with respect to SS will be independent of these cumulants, so that the distribution

of 82 will be the same as that obtained in (3n2) and (3.3).

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All use subject to JSTOR Terms and ConditionsA. K. GAYEN 359

For the distribution

of t we proceed as before using (4-1), and obtain after simplification (p(t) being given by (4.2)) (t) p(t) 

### + A{[n2

+(n-1)2] t5

+10(n l) t3-15(n-1)2 t} 40[V(2nn)]n(n - 1)2 El + t2/(n- 1)]4(i+5) + (4On3+ 258n2

+182n + 51) t7- 3(150n3+ 896n2+ 869n+ 297) t5) l+ 

### 15(n

- 1) (68n2 + 31 In + 

### 237)t3



### - 15(n - 1)2 (25n 

### + 59)tJ +A3A4--- 144[V(2Tn)] 

### n(n - 1)3[1 

### + t2/(n

- 1 )]4(l+7) (64n4 +952n3+ 3578n2+ 301In + 996) t9 - 9(144n4

+1918n3+ 701 n2 

# f + 7550n

+ 

### 2913)t7+

27(n - 1) (252n3+ 

### 2821n2

+8554n+7077) t5 13 t 

### - 45(n- 1

)2(541n2+ 2086n

### + 3049) t3+ 945(n- 1)3(3n + 13) 

### t 

### + A3 

### -
A -' 

### 1296[4(27n)] n(n - 

### 1)4 

### LI+ t2/(n

- 1)](n+9)

= pO(t) + A3pA3(t)

- A4Pk,(t) 

### + Alpkk(t)

+ A5pA5(t)

+ A3A4PA3A4(t)

+ A3pA3(t)* (5.3) Polynomials in t appearing in the numerator of the corrective

functions

are analogous in form to the Hermitianpolynomials of the same degree.The form of the corrective

term due to A5 is simple but those due to A3A4 and A33 are rathercomplicated.
6. TAIL-AREA PROBABILITIES We are interested

onlyin the tail area ofthese corrective terms and so consider

the integrals j-to X f p(t) dt or J'p(t) dt. For the first

ofthe four

right-hand

members

of (4.2) the integral

has been tabulated. For the next member

the two tail areas will be equal in magnitudebut opposite in sign. In each of the other two memberswe shall have equal areas at both tails. These integralsmay be tabulated for

givenvalues ofn and to, either directly from their algebraicexpressions

or more conveniently

by using the Incomplete Beta functiontable of Pearson. For the sake of completeness

we shall give here the expressions fortail areas of all the components,introducing Po(to),

PA3(to), PA4(to) and Pk2(to) to represent them. Thus, for one tail, the corrected probability

integral P(to), will be given by P(to) = PO(to)

+ A3PA3(to)

- A4PA4(to)

+ A3PA(t0), (6.1) in which P0(t0) = 

## fro(t)dt = fpo(t) dt = 2U,(-Lj- 2 (6 2) 

### PA (to) = -JPA 

### (t)dt 

### 12 

## (6) 30?) ('o

/(n 2

= 1)-(3(o2nl+) 

### [ ?(n2- 1)]. (63b)

We also have 

### 3 3- PA4(to)

_ - k

J A (t) dt 1)V irn- (t) dpttt (6__

_ __ ___ _3_(n 1

J-4 4 t12(m 

### (nT1)1 I[2(n-1)] 

### [1 +t-/(n-1)]i(n+2) (6.4)

__ (n-1)I {n-i !()n-I)(n+ 2) I nl -)+YIt4)(n-l)1 I 

### (6 4bis)

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### 360 Distributionof 'Student's' t fromnon-normal universes and lastly, PA2(to) = f-to = 

### fPAi(t) dt
- 

# ( (2n+5) (4 )2 2(2n-7) (-1)t3- 3(2n+ II) (n-1 )2to (n+ 1)(2n+.;) (n+l) (n+3) (2n+5) ? 6 5

[1 + t2/(n-1 )]k(n4) (65)
[( 72 )U( 2 ~2) 24n ) (2 2)

24n 2

2 72n 2 2 (6.5 bis) wheretois any typical value of t,and the transformed variate uo is given by 1 U0 = 1+tg/(n- 1) and 

### Iko (I vj, 2)is the Incomplete B-function as defined

by Pearson. Obviously

### PA4(to) and PA2(to) are of order n'1. For to= 0(0 5) 4 and n = 2, 3, 4, 5, 6, 7, 9, 13, 25 and so, values of Po(to), Pa3(to), PA4(to) and PAI(to)

have been tabulated (Table 1) with the help of Pearson's Incomplete B-function table. The sample sizes are chosen to correspond to the degrees of freedom1, 2, 3, 4, 5, 6, 8, 12, 24 and oo, as in Fisher's table of z. It willbe noticedthat the effect ofA4 is very small,as has been inferred by various authors. The corrective

tail area beyond I to



### J = 3 (which is nearly the 4 and 3 

### % points in normal samples of 5 and 6 respectively) is at most 1 

### % when the size of sample does not exceed 5, about 0'3 

### % for

samples of 10 and about 041

### % forsamples of 20. The actual probability is obtained by multiplying these tail areas by the given value of A4, and they are additive to that of the normaltheory t. As Geary pointsout, the effect of A3 is rather

serious, but so is that of A2. For a two-sided test the positive and negative tail areas will,of course,balance each other, but those due to Al will not. For a comparativestudy of the form of the frequency

curves of the normal theory t and those of the corrective functions

for A3,A and A2, diagrams have been constructedfor n = 3, 6, 13 and 25. They are shownin Figs. 1-4. As is to be expected, the curves ofthe corrective functions

tend to those of the third,fourth and sixth derivatives of the normal function

as n tends to infinity. An example to illustrate the use of the tables has been taken from Neyman & Pearson's (1928) paper 'On the use and interpretation of certaintest criteria'. The given values of A. and A4 are not large and more or less satisfy A. L. Bowley's (1928) criterion formoderately abnormal curves. Example (Neyman & Pearson, 1928, p. 203). Records of weightin a large population of mice for males between 120 and 140 days of age show a mean value of 23^823 g., and for

the frequency

distribution



### fl1 = 0-086, 

## f = 2-687. Can the group of six mice withthe weights 22*5, 26-0, 20-5, 24 0, 18-0and 24-5be considered a random sample from this population? Here A3= + 0-293 (inferred as positive),A2= 0-086,A4= -0.313, and we find t =-l*038 -to, say) for n' = 5 degrees of freedom. Considering both tails, the normal theory probability

is P0(j t 

### I > to) = 0'3468,

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### Table 1. Comparative valuzes 

### of PO (to), P,,,(to),

P,,,(to) 

### and PA2(to)

for different degrees of freedom n' ( 

### - n1, 

### n being

the size ofsarnple). 

### P (to) 

### PO P(to) + A3PA.,(to) -A4 PA, (to) A3PA(

| t O ) PA3(tO) PA,(tO) |PA(to) |PO(tO) |P'AyO) PA4(tO) PA (tO)

to 3
0-0 0.5000 0 0470 0.0000 0.0000 0.5000 0 0384 0 0000 0 0000 0.5 0.3524 0-0589 - 0 0064 - 0.0066 0 3333 0.0495 - 0 0069 -0 0066 1.0 0.2500 0-0665 0 0000 0-0044 0.2113 0 0597 - 0.0027 0.0009 1.5 0-1872 0 0622 0.0047 0-0147 0.1362 () 0563 0.0025 0.011]8 2-0 0-1476 0.0547 0 0064 0.0188 0.0918 0 0469 0.0047 0-0172 2-5 0-1211 0 0476 0 0066 0-0195 0-0648 0-0375 0-0051 0-0179 3-0 0.1024 0-0416 0 0064 0.0188 0-0477 0-0298 0 0047 0.0165 3-5 0-0886 0.0368 0.0059 0.0176 0.0364 0.0239 0.0041 0.0145 4 0 0-0780 0-0329 0 0055 0 0163 0 028,6 0-0194 0 0035 0-0125 n' 3 n' 4 0.0 0 5000 0.0332 0-0000 0 0000 0 50001 0 0297 0.0000 0.0000 0.5 0.3257 0.0431 - 0 0062 - 0 0056 0.3217 0 0387 - 0.0055 -0.0047 1.0 0.1955 0 0540 - 0 0034 - 0.0002 0.1870 0-0495 - 0.0036 -0.0005 1-5 0 1153 0.0513 0.0013 0.0098 0.1040 0 0473 0-0006 0-0084 2.0 0 0697 0.0413 0.0035 0.0152 0-0581 0.0372 0-0028 0.0135 2-5 0 0439 0.0310 0 0039 0-0157 0 0334 0.0266 0.0031 0.0139 3 0 0-0288 0 0229 0 0034 0.0139 0.0200 0.0184 0.0027 0-0119 3.5 0-0197 0.0169 0.0028 0.0114 0.0124 0.0127 0-0021 0.0095 4.0 0.0137 0.0126 0-0023 0.0093 0.0081 O OO88 0.0016 0-0072 n' =5 TV_ 0.0 0 5000 0.0271 0 0000 0 0000 0 5000 0-0251 0 0000 0-0000 0.5 0.3192 0-0355 - 0 0049 -0.0041 0.3174 0-0329 -0.0044 -0-0035 1.0 0-1816 0 0397 - 0.0035 -0.0005 0.1780 0.0430 -0 0033 -0-0005 1.5 0.0970 0.0440 0.0002 0 0074 0-0921N 0.0413 0-0000 0-0066 2.0 0.0510 0.0340 0.0022 0.0122 0-0462 0-0315 0.0019 0-0111 2-5 0-0272 0-0234 0.0025 0-0125 0-0233 0-0210 0-0021 0-0113 3 0 0.0150 0.0154 0.0021 0-0104 0-0120 0-0132 0 0017 0-0092 3.5 0-0086 0 0099 0.0016 0.0079 0-0064 0-0081 0-0012 0.0067 4 0 0 0052 0-0065 0.0011 0-0057 0 0036 0-0050 0-0008 0-0047 n' 8 w'=12 0.0 0 5000 0-0222 0.0000 0.0000 0-5000 0-0184 0-0000 0-0000 0.5 0-3153 0-0291 - 0.0037 - 0-0028 0-3131 0-0243 - ()-0027 - 0-00] 9 1.0 0-1733 0 0384 - 0.0030 - 0-0005 0-1685 0-0325 - 0)-0023 - 0-0002 1-5- 0-0860 0-0371 - 0-0002 - 0-0055 0-0797 0-0315 - 0-0004 0-0042 2-0 0 0403 0-0277 0-0014 0-0094 0.0343 0-0230 !0-0009 0 0073 2- 5 0.0185 0-0177 0-0016 0-0095 0-0140 0-0137 0-0011 0-0072 3 0 0 0085 0.0103 0.0013 0-0074 0-0055 0 0072 0-0008 0-0053 3-5 0 0040 0-0058 0-0008 0-0051 0-002!2 0 0036 0-0005 0-00331 4 0 0-0020 0-0032 0-0005 0-0033 0-0009 0-0017 0-0003 0-(0019 ?&' 24 n' = oo| -0.0- 0 500( 0-0133 0-0000 0-0000 0 5000 0 5 0-3101 10-0176 1 -0-0015 1 0.0010 0 30851 1.0 0 1636 10-0238 - 0-00014 -0-0001 0-1587I I 1.05 0 0733 10-0232 - 0-00003 10-0025 0-06681 2-0 0-0285 10-0164 10 0004 10 0043 0 02281 2-5 0.0098 |0.0090 |0.0005 |0-0041 0-0062I I 3-0 0-0031 10-0041 10-0003 10-0028 0-00131 3-5 0 0009 10-0016 10-0002 10-0015 0 0002 14 0 0 0003 10 0006 10 0001 10 0007 0 00001II

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### 362 Distribution of 'Student's' t 

### from

non-normal

universes whereasthe estimateof the actual probability 

### P(I tI > 

### to)

= 

### 03447. If we consider

the negative

tail only,then

### P(t < - to)= 04 1832. So that whichever tail is used the conclusion

that there

is no reason to doubt the origin of the sample is obvious.

I 004 - " I 0-03

-0-0 -4-0 -3-5 -3-0 -2T5 -1:5 -10 0 1*0 1-5 2*5 3.0 3.5 4.0

Fig. 1. Showing the terms in the distribution

p(t) pO(t) + A3PA3(t) -A4pA4(t) + A3pAI(t) forn' = n - 1 = 2 degreesof freedom.

Explanation of symbols for all figures: S If * po3p); Ok-- O -- O7A4(t);

0-04

//\ 0 03
-002

4.0 -35S -3.0 -25 5 -0 -5 0 0. 10 1525 310 35 410

Fig. 2. Showing the terms in the distribution

p(t) = Pot) + A3k)A3(t) - A4PA4(t) + Ap) for i' = -1 = 5 degrees of freedom.

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### A. K. GAYEN 

### 363

, ~~~~~~~~0-04 - j 

### ~~~~~~~~~0-03 -

-4-0 -35 -30 -25 5 0 2 3 3
002

-0-03-
-0-04 Fig. 3. Showing the terms in the distribution

p(t) = pO(t) + A3PA3(t)- A4PA4(t) + A MpA2(t) for n' = n-1 = 12 degrees of freedom.

004 / 

### ~~~~~~~~0-03 -

-4-0 -3-5 -3-0 -2. .5 025 304(0

-0-02 -0-03-
-0-04

. Fig. 4. Showingthe termsin the distribution

p(t) = pO(t) + A3PA3(t)-A4PA4(t) + APAI(t) for n' = n - I 24 degrees of freedom.

7. ASYMPTOTIC

CHARACTER

OF THE SERIES FOR p(t) 

### Pointing out some of the asymptotic

properties

up to order

n-1 possessed by the expression pO(t) + A3PA3(t) (being 

### his earlier approximation

for'Student's' t), Geary(1936) observed that for samples of moderatesize, the probability {pO(t) + A3pA3(t)}



### dt mighthave quite an extended range of applicability,provided that at least the lower frequency constants

A4, A. and A6

are small. He considered thisto be a matter

for experimental investigation.

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### 364 Distribution of 'Student's'tfrom

non-normal

universes The probability

function p(t) = {po(t) + A3PA3(t) PA4pA4(t) + A 2pA(t)} obtained in (4.2) possesses similarasymptotic properties

up to ordern-1. For it represents the frequency

density of 'Student's' t forsamples drawn from any parent population ifthe samples are so large that the higher ordertermsin n-r forr > 3 can be neglected. On the other hand, it gives the frequency density of 'Student's' tfor

samples ofany size drawnfrom the particular parent population (2.1), when the A termsother than in A3, A4and A3are negligible.From Geary's (1947) asymptotic formulae

for the cumulantsoft for

any universe (his results 2 18), it is apparent that these higherA termscan only occur in terms in n-iT for r > 3, so that it is not unlikely that for samples ofmoderatesize p(t) has a quite extended range of applicability, providedhigher

ordercumulantsA5,A6, etc., are small. The expression

forq(t) in (5.3), being correct to nu1, will provide a closer approximation to the actual distribution of t than the other

two expressions,

for samples from

any universe if the terms

in n-2 and higher

negative powers of n are negligibly small, as also forsamples of any size for

the parent population specified by (5.1) ifA6

... A4 are small. Table 2. Showing

the experimental

determinations

ofthe true probability

oftwith their standard errors for various sampled universes together withthecorresponding probability

obtained 

### from the frequency

function p(t) of(4.2)
Sampled Experimental Probability Size poIulations witli Normal determinations of as estimated of theory the probability from p(t) sample probability with their s.E.'s of (4.2)

2~fl A f-3

E. S. Pearson 5 0 00 - 050 0040 0 044 + 0 009 0 043 (1929) 5 000 1*12 0*040 0 038 + 0-009 0034

5 0*00 4*07 0*040 0*029 + 0*008 0*018 5 0-20 0*30 0*040 0 044 + 0 009 0 043 5 0 50 0 73 0 040 0-042 + 0009 0*048

H. L. Rietz 5 0 00 - 0-63 0-067 0-058 + 0-007 0.071 (1939) 5 0.03 - 0 55 0-067 0-070 + 0-008 0*071

A. N. K. Nair 5* 2.00 3-00 0-040 0-062+0 0011 0-072 (1941) 6t 4*00 6.00 0.030 0*088+0-010 0.088
* Type III curve. t Exponential curve. Approximate probabilitiescorrect to n-1- for a singletail will be obtained by applyingto the normaltheory

value necessarycorrections due to A3, A,

AP and A, A3A4,

As. But when bothtails are used, corrections for A4 and A2 only willfurnish

the same order

ofapproximation since the contributions due to odd-order

cumulants cancel each other,i.e. the use of p(t), which provides an approximation correctto n-1 only, will actually lead for a two-sided test to results

correct to n-A. This favoursthe application of our formula for p(t) to samples ofmoderate

size from

populations

whosedegree

of departure

from normality

is notnecessarily small.

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### A. K. GAYEN 

### 365



### Results ofinvestigations

made by various

writers support heconjecture

that p(t) or q(t) may have a quite extended range of applicability.

Table 2 shows the values of AM and A4 of the sampled populations,

the size of the samples considered,

the normal theory probability Po(to), and the experimental

determinations

with their standard errors, of the approximate probability

P(to) against heir values obtained from p(t). The agreement

is quite good, even in cases where samples of five or six only have been taken from markedly non-normal

populations. One of the two sampled populations

of Nair isa very skew Type III Pearson

curve, and theother is theexponential

curve. These populations,

as is wellknown,

can hardly

be represented

adequately

by the third or fourth approximation

tothe law of error. The agreement

between experimental

and theoretical

results, especially

for the two populations

considered

by Nair, may be partly due to the fact that the approximation

yielded by p(t) is, as we have noted, not only correct ton-1 but also toned, 

### since both tails have been used for these cases. For certain values of A3 and A4, estimates of P(to) at to = 

### 3,

for samples of 5and 6, obtained from formula (4 2),areshown

in Tables 3 and 4. Numerical

values of A2 

### andA4 have been selected to include

a fairly wide class of non-normal

populations

met with in practice.

In spite of the fact that there is a fair agreement

between the tabulated results and theexperimental determinations,

itis obviously

unwise toassume that they necessarily

furnish inall cases a satisfactory

estimate of the true probability

of tfor such small samples. The forms of the two approximations,

amely (4.2), correct ton-1, and(5.3),

### tona1,



### are different from those which may becorrespondingly

obtained from Geary's (1947) asymptotic expansion

to n-2 

### (his

result (2.24)). He has obtained

that expression

by using Charlier's '-Differential

Series' with the normal theory tas the generating

function, utilizing for the purpose his derived results for the first six cumulants

of non-normal

t. One of the advantages of Geary's expression

for tis that it takes into account some of the higher cumulants

and higher powers of A3 and A4. Accordingly

it may have an extended range of applicability. 

### But the fact that itagrees, toorder n-1, with M.S. Bartlett's

(1935) result suggests that the formula is not satisfactory,

since the asymptotic

character of the latter cannot beassumed. For Bartlett specified his parent population

by the first three terms of the Edgeworth

series which take no accountof the term in As, namely, 2OA20(6)(x)/6!. 

### The estimates of true 

### probabilities

shown inGeary's

Table 2 cannot as such beregarded

as satisfactory. 8. PROBABILITY CORRECTIONSDUE TO HIGHER CUMULANTSOF THE PARENT POPULATION 

### For the effects of A5, A3A4, A33 weconsider

as before the probability

integrals of the corrective functions anddenote

them by PA5(to), 

### PA3A4(to) and 

### PA3(tO)

respectively.

We then have

lbto 0 o PA(to) ft dt = fpAr(t)dt

(1+ (n-i))tP5 (n-i)2 ')t4} 24-3) (2n2- 2n+ (8.1 

### 1(n 

## 1)0(n + -1)2



### = (n-l)(n-4) I0 {n+3 )1 _J(n-l)(n-2) I 0n +l1 l 20 V(2T)n' 

# f

U? 

### 2' l 

### 10

V(27T) no UO\ 

### 2 

### + {(22_ 2T)

n+I}U(% 

### ,

1)2 (8-1

bis)

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### 366 Distributionof 'Student's' t fromnon-normal universes Table 3. Showing

comparative

values of to 0 = 1 - J 

### p(t)

dt= 2[PO(to)

- A4PA4(to)

+ A'PA2(to)]



### nearthe 4 

## % point

(approximately)

ofnormal

theory tforsamples

of 5 (n' = 

### degrees

of freedom = 4)

0.00 0.20 0*25 0*50 1.0 1.5 2.0 4.0 A4

- 2.0 0.0505 - 1.5 0*0479 0.0527 0*0539 0.0598 - 10 0*0452 0*0500 0*0512 0.0572 0.0691

- 05 0*0426 0*0474 0*0486 0*0545 0*0665 0.0784 0o0 0.0399 0.0447 0*0459 0*0519 0.0638 0.0758 0.0877 0 5 0*0373 0*0421 0.0433 0*0492 0*0612 0 0731 0.0851 10 0.0346 0*0394 0*0406 0*0466 0.0585 0.0705 0.0824 1.5 0*0320 0.0368 0*0380 0.0439 0*0559 0.0678 0.0798 2.0 0*0293 0.0341 0*0353 0*0413 0*0532 0.0652 0*0771 041249 3-0 0*0240 0*0288 0.0300 0.0360 0*0479 0.0599 0.0718 041196 4*0 0*0187 0*0235 0*0247 0*0307 0*0426 0*0546 0.0665 041143 6*0 0*0081 0-0129 0*0141 0*0201 0-0320 0-0440 0.0559 041037


### Table 4. Showing

comparative

values of
to 

### 0= 1- p(t) dt = 2[Po(to)-A4PA4(to)+ A2PA2(to)], to3 

### nearthe 3 

## % point

(approximately)

ofthe normal theory tfor samples of 6 (n' = 

### degrees of freedom = 5)

0.00 0*20 0-25 0*5 1.0 15 2.0 4-0 A4

- 2.0 0'0386 -15 0.0365 0*0406 0*0417 0*0468

- 10 0.0343 0*0385 0*0393 0.0447 0*0551

- 05 0*0322 0.0364 0*0374 0*0426 0*0530 0.0634 0o0 0*0301 0.0343 0*0353 0*0405 0*0509 0.0613 0.0717 0.5 0*0280 0*0321 0*0332 0*0384 0*0488 0*0592 0.0696 1*0 0*0259 0.0300 0*0311 0.0363 0.0467 0.0571 0.0674 1.5 0*0237 0.0279 0*0289 0.0341 0.0445 0.0549 0.0653 2.0 0*0216 0*0258 0*0268 0*0320 0.0424 0.0528 0.0632 0*1048 3.0 0.0174 0*0216 0*0226 0.0278 0*0382 0-0486 0*0590 0 1006 4-0 0'0132 0-0173 0-0184 0*0236 0-0340 0-0443 0 0547 0-0963 6*0 0*0047 0.0088 0o0099 0*0151 0*0255 0*0359 0.0463 0-0879

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### A. K. GAYEN 

### 367 r-to 00 PA3A4(tO) = 

# - PA3A4(t) dt = fPA3A4(t) 

### dt

-00t

= (n34) IuO( 4)-(n43) Iuo(2 3) + (n,52) u~O 2+ 2-(1)I (n~l +25u 1 (8.2) where(n34),

(n43), (n52) and (n61)

are given by

(n34) 

### = { 

### 2 )} (40n3+258n2+ 182n+ 51),



### Bn+' 

## 3f (n43) = 

### l ((22 

### T;2; (150n3+ 896n2+ 869n + 297), 96 V(2n)ni 

### ~~~~~~~~~~~~(8.3) 5SB n+3 , 

### 2~ (n52) 

### = 

## j9 7(z



## )t2 

## J (68n2+ 31In+ 237),

(n6l) = 

### { J96 V(2) n| (25n

+59),

P3(to) = pA3(t)t -PA3(t)dt



### n - I n + (n+5 In+ =(NO5 Uo( 2)5) + (n54) I( 2 

### 4 4- (n63) IU,( 2 3)
2n n , 2) n(8Duo 7 1) (8.4) +(72)'UO( 

### 2 ) )(8lIO 

### 2 1) (84 where 

### (n-1)B( 2 ' 5)

(n45) 

### = 2592 (2ir)n' 

# J (64n4+ 952n3+ 3578n2+ 

### 301 In+ 996), 

### ~B n+ 

## I4\f4 (n54) = 2

(2 ) 

### J

(144n4 +1918n3+ 7011n2+ 7550n + 2913), (/n+3 

### 3~ (n63) = t9(>/(2r) )|(252n3+ 2821n 2+ 8554n + 7077), (I n+52 2 (n72) 

### = 288 j(27T) 1 (541n2+ 2086n + 3049), 

# (5B n+7 \2 Il (n81)

### 96= (2)n ) (3n+ 

### 13), with the usual notationsof Completeand Incomplete B-functions. Biometrika

36 24

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### 368 Distribution of ' Student's' tfromnon-normal universes The additional formulae (8.1), (8.2) and (8.4) of this section enable us to deduce the probabilityof t when the parent population can be represented by the fourth

approximation. The corrected

probabilityP(to), fora singletail, may be obtained from the formula 

### P(t0) = 

### PO(to)

+ A3PA3(to)

A4PA4(to)

+A3PA3(to)

+AsPA5(to)

+ A3A4PA3A4(to)

+ A3 P(t) (8.6) It should be noted that for the negativetail ofthe distribution PA3(to) and PA3(to)

are positive, whereasPA5(to)

and PA3A4(to)

are negative,the reverse being the case with the othertail. For samples of 10 from such a population the probability at to= 2-2262 (here I to I is the 5 

### % point of the normaltheory t), will be given by P(- 2.262) = 0-0250

+ A3(0-0209) - A4(00015) + A2(0.0092)

- A5(-0.0017) + A3A4( - 00111) +3A(0@0221). If A3= 1, A4= 1, then for a Pearson curve, 

### A. = 0X25, and in that case the probability is found to be 0X0642,

whereas the estimate from termsas far as A2only is 0X0536. Thus for a singletail theydiffer considerably

from the normaltheory

probability

0f0250.

SUMMARY The theoreticaldistribution of 'Student's' t in non-normal samples of any size has been derived with reference to the parent population specifiedby a number of terms of the Edgeworth series. It contains, in addition to the normal theoryfrequency function

of t, corrective

termsdue to the cumulantsA3, A4, A3 and A5,

A3 A4, A3. It is assumed that the values of the population A's are known. If the population can be well represented by the thirdor the fourth approximation (in whichcases next higher A's and higher

powersofA3

and A4 are small), thenthe corresponding t-distributions

will be accurate for

any size of sample, as faras similarorder terms in A's. The two expressions

for t, namely, the one including

theterms

up to that in A3

and theother up to that in A3,

are asymptotic

to ordern-1 and nA- respectively, and as such they afford closer approximations to the actual distribution of t forlarge samples from any universe. The probabilities

calculated from

the former

willbe the same as those from the latter,

ifboth the tails of t are used. So it is not unlikely that for moderatesize of sample, the former may have an extendedrange ofapplicability. The satisfactory

agreement

betweenthe theoretical values and the experimental resultsof various writers appears to supportthis conjecture. Tail-area probabilities of the corrective terms for A3, A and A2 have been tabulated. An example is given to illustratethe use of the tables. Diagrams showingthe frequency curve of non-normal

t for certain values of n have been constructed. The probability

corrections

due to higher

cumulantsof the parent population have also been considered

for a representative

value of n. I wish to acknowledge my indebtedness to Dr H. E. Daniels for his kind advice and criticism

in the course of my investigations, also to Dr J. Wishart for suggesting

a number of improvementsto the paper. My thanks are due to Mr D. A. East for drawing the diagrams.

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### A. K. GAYEN 

### 369 REFERENCES BARTLETT, M. S. (1935). Proc. Camb. Phil. Soc. 31, 223. BOWLEY, A. L. (1928). F. Y. Edgeworth's Contributions

to Mathematical

Statistics. London: Royal Statistical Society. CRAMER,H. (1928). Skand. AktuarTidskr. 11, 13, 141. CRAMER, H. (1946). Mathematical Methods of Statistics. Princeton University Press. EDGEWORTH, F. Y. (1905). Trans. Camb. Phil. Soc. 20, 36, 113. EDGEWORTH, F. Y. (1906). J.R. Statist. Soc. 69, 497. GEARY, R. C. (1936). J.R. Statist. Soc. Suppl. 3, 178. GEARY, R. C. (1947). Biometrika, 34, 209. NAIR, A. N. K. (1941). Sankhya, 5, 393. NEYMAN, J. & PEARSON, E. S. (1928). Biometrika, 20A, 175, 263. PEARSON, E. S. (1928, 1929). Biometrika, 20A, 356 and 21, 259. RIETZ, H. L. (1939). Ann. Math. Statist.10, 265.

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