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

Stable URL: http://www.jstor.org/stable/2332673 .

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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 of Cambridge 

1. INTRODUCTION 

The effect of universal non-normality on 'Student's' t has been studied so far either by way 

of particular numerical examples or by approximations to its sampling distribution based 

on the large sample assumption. E. S. Pearson (1928, 1929) has shown in his experimental 

investigation that the effect of universal 'excess' and of 'skewness' on 'Student's' ratio z 

(which is related to t by t = z V(n - 1)) may be considerable. He has also furnished a small 

table, based on experimental results, showing some actual values of the probability integral 

for the ratio in samples of 2, 5, 10 and 20 from afew 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 isbased on the assumption of large samples and is not very satisfactory, 

for in that approach, as he himself has also pointed out, the term 'approximation' is perhaps 

misleading. Still from the form of the expression obtained, it has been observed that the 

effect of the 'skewness' A3 in the original distribution isthe more 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 

to normal theory probabilities of t, when the sample size is small. R. C. Geary (1936) obtained 

an expression for the distribution of t in samples of any size drawn from aslightly asymmetrical 

universe. It consists of two components, one being the 'normal theory' t and the other a 

term in A3, which has been called the ' corrective function' due to it. He has, also given a table 

of corrective tail-area probabilities for some representative values of n. In a recent communication (1947) he has given from a different approach an approximate formula for the 

frequency of t, correcto n-2, n being the size of sample. From this and analogous results on 

non-normal Fisher's z, he has shown, by a few illustrativexamples, that the inferences 

drawn from the standard tables may be seriously in error even in some cases where the 

parent is not considerably non-normal. He has accordingly pointed out the primary importance of testing for normality from the available samples and has suggested that when 

universal normality cannot be assumed, the standard tables should be corrected by using 

the sample estimates of A3 and A4, etc., in conjunction with the theoretical results deduced 

therein. But so far as the frequency of t is concerned, satisfactory measures of probabilities 

for small samples cannot be obtained from his results, since they are mainly based on large 

sample assumptions. Also the first few terms of his result, as far as n'l, agree with those of 

Bartlett, which, as has been pointed out, are not very satisfactory. 

The purpose of the present investigation isto obtain the form of the corrective terms due 

to population cumulants A3, A4, etc., in the frequency function of t for any size of sample. 

For this the parent population will be specified by the Edgeworth series. It has been proved 

rigorously by Cramer ( 1928) that this series gives a real asymptotic expansion of any universe 

f(x), in powers of v64, where voo is the number of sources of 'elementary errors'. Inclusion of 
354 Distribution of 'Student's' t from non-normal universes 

terms of order vO, vO i, l 1, ..., P- Or gives rise to the first, second, third, ..., (r + 1 )th approximations to the law of error. We shall consider mainly the third approximation, including terms 

in A3, A4 and A2, but shall also deal with the fourth approximation which takes in A5, A3A4 

and A3. 

Probabilities of values of t, obtained from the derived formula, for the parent specified by 

the third approximation, are in satisfactory agreement with the experimental determinations made so far, even in cases where the sampled populations are represented by the 

exponential curve or a very skew Type III curve of Pearson. Accordingly, itappears that 

the derived expressions for 'Student's' t may perhaps provide quite satisfactory estimates 

of the probabilities, for a fairly wide class of non-normal universes, especially when the 

sample size is not too small. 

The values of the cumulants of the population have been assumed to be known, and the 

question of their estimation has not been considered. 

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

OF DEVIATIONS OF n SAMPLE OBSERVATIONS 

We shall first of all obtain the distribution of 'Student's' t, with particulareference to the 

parent population being specified by the third approximation to the law of error. This 

will give us the corrective terms due to A3, A4 and A2. It will be shown afterwards that 

similar corrective terms due toA, A3A4and A3 can be obtained by the same methods when 

the parent population is represented by the Edgeworth series up to the corresponding 

terms. 

Let us then consider the parent population to be specified by the third approximation, 

correcto vi-', 

f(x) = 0(x)-3 0(3)(x) + 4 0(4)(X) + 3 0(6)(X) (2- 1) 

where A3 and A4 are the measures of universal skewness and excess, 

0(x) = 1 e-*x2 (2.2) V-(27r) 

the normal function in standardized form, and 

(p)(X)= (d-) (x) = (- 1) Hv(x) q(x), (2.3) 

where Hv(x) is the well-known Hermitian polynomial of degree v. 

Let xl, x2, ..., xn be n independent sample observations drawn at random from the universe 

(2.1), and let us adopt the notations 

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 

where S2 is the estimated variance. 
A. K. GAYEN 355 

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

also the joint frequency density of (S,, S'), since the Jacobian of transformation is unity) 

may be written in the form 

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 + AD 3) 2W(n+9)+36(n+4)D2D2W(n+7)-12OD3W(n+5)}, 

(2.6) 

when terms in Ai other than 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 involved in the expression (2 6). It is possible 

to transform the product operators of the form D1,D 2 (where v1, v2= 0 1, 2, ...)into functions 

of the single operator 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 with respect to 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 observed that the numerical coefficients are those of the Hermitian polynomials 

H = x 1 ) X12-2 +.1(V-! ) (!-2) (V-.3) _ 
356 Distribution of 'Student's' t from non-normal universes 

So we can write, in general, 

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

+ 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 third approximation to the 

law of error we require the values of Dv, for v = 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 the joint frequency function of S, and S2: 

g1(S1,S 2) = W(n-1 ) {+H3( H ) + 3( ) H1Q ! )} 

+ ?A4 {HQI S) + 6S2) H2(S) + 3( ) (8)2) + n { [nQs) n - 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. 

where H1(Sl/n) is the Hermitian polynomial of degree v in (Sl/n), and W(n - 1) is given by 

(2 7). As a check it was found that, to the required approximation, 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, 

after some 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) S 3een 1i2 2n 3n 02 

where S2 has been omitted from the V's. 
A. 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 typical term 

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 for r, = 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} 

A4 24n V[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 universal A3, A,,an A3 

As a check on the above expression for t, moments may be calculated directly, about 

t = 0 and to n-2; they are found to 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 for any universe 

as obtained by Geary (1936), up to first power of A4 and second power of A3. All these corrective terms when 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 universe by the fourth approximation to the law of error, correct 

up to vP-, given by 

f() = 0(X)--A30(3)(X) +A4 0(4)(X) 10A A5X 0(5)(X) 35A3A4 280A 

3 !4 ! ~ 6 !/ 7 ! 9 S9() 51 
358 Distribution of 'Student's' t from non-normal universes 

It is of interesto know the distribution of 82 and 'Student's' t in random samples from this 

universe. We can arrive at these results by following a similar procedure, and the expressions 

corresponding to (2.6), (2.11) and (4.2) will have additional terms involving A AA a 3 

We shall omit these intermediate steps and give below only the extra terms in the joint 

distribution of 81 and S2 and in the distribution of t. 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 7 (S 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 with A5, A3A4 andl A3 in (5.2) are odld functions in S1, the 

integral of 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). 
A. 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+ 301 In + 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 Hermitian polynomials of the same degree. The form of the corrective term due 

to A5 is simple but those due to A3A4 and A33 are rather complicated. 

6. TAIL-AREA PROBABILITIES 

We are interested only in the tail area of these corrective terms and so consider the integrals 

j-to X 

f p(t) dt or J'p(t) dt. 

For the first of the four right-hand members of (4.2) the integral has been tabulated. For the 

next member the two tail areas will be equal in magnitude but opposite in sign. In each of 

the other two members we shall have equal areas at both tails. These integrals may be 

tabulated for given values of n and to, either directly from their algebraic expressions or more 

conveniently by using the Incomplete Beta function table of Pearson. For the sake of 

completeness we shall give here the expressions for tail 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) _ dpttt - J A k (t) dt (6__ (t) 1)V _ __ irn- ___ _3_(n 1 

J-4 4 t12(m (nT1)1 I[2(n-1)] +t-/(n-1)]i(n+2) [1 

(6.4) 

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

(6 4bis) 
360 Distribution of 'Student's' t from non-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) 

where to is 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 

freedom 1, 2, 3, 4, 5, 6, 8, 12, 24 and oo, as in Fisher's table of z. 

It will be noticed that the effect of A4 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 % for samples 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 normal theory t. 

As Geary points out, 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 comparative study 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 constructed for 

n = 3, 6, 13 and 25. They are shown in Figs. 1-4. As is to be expected, the curves of the 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 certain test criteria'. The given values of A. 

and A4 are not large and more or less satisfy A. L. Bowley's (1928) criterion for moderately 

abnormal curves. 

Example (Neyman & Pearson, 1928, p. 203). Records of weight in 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 with the weights 

22*5, 26-0, 20-5, 24 0, 18-0 and 24-5 be 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, 
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 of sarnple). P (to) PO P(to) + A3 PA.,(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 
362 Distribution of 'Student's' t from non-normal universes 

whereas the estimate of the actual probability 

P(I t I > 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) for n' = n - 1 = 2 degrees of 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. 1 0 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. 
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. Showing the terms in 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 moderate size, the probability 

{pO(t) + A3pA3(t)} dt 

might have quite an extended range of applicability, provided that at least the lower frequency constants A4, A. and A6 are small. He considered this to be a matter for experimental 

investigation. 
364 Distribution of 'Student's' t from non-normal universes 

The probability function 

p(t) = {po(t) + A3PA3(t) PA4pA4(t) + A 2pA(t)} 

obtained in (4.2) possesses similar asymptotic properties up to order n-1. For it represents 

the frequency density of 'Student's' t for samples drawn from any parent population if the 

samples are so large that the higher order terms in n-r for r > 3 can be neglected. On the 

other hand, it gives the frequency density of 'Student's' t for samples of any size drawn from 

the particular parent population (2.1), when the A terms other than in A3, A4 and A3 are 

negligible. From Geary's (1947) asymptotic formulae for the cumulants of t for any universe 

(his results 2 18), it is apparent that these higher A terms can only occur in terms in n-iT 

for r > 3, so that it is not unlikely that for samples of moderate size p(t) has a quite extended 

range of applicability, provided higher order cumulants A5, A6, etc., are small. 

The expression for q(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 for samples 

of any size for the parent population specified by (5.1) if A6 ... A4 are small. 

Table 2. Showing the experimental determinations of the true probability of t with their standard 

errors for various sampled universes together with the corresponding 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 probabilities correct to n-1- for a single tail will be obtained by applying to 

the normal theory value necessary corrections due to A3, A, AP and A, A3A4, As. But when 

both tails are used, corrections for A4 and A2 only will furnish the same order of approximation 

since the contributions due to odd-order cumulants cancel each other, i.e. the use of p(t), 

which provides an approximation correct to n-1 only, will actually lead for a two-sided 

test to results correct to n-A. This favours the application of our formula for p(t) to samples 

of moderate size from populations whose degree of departure from normality is not necessarily 

small. 
A. K. GAYEN 365 

Results of investigations made by various writers supporthe conjecture 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) againstheir 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 is a very skew Type III Pearson curve, and 

the other is the exponential curve. These populations, asis well known, can hardly be 

represented adequately by the third or fourth approximation to the 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 to n-1 but also to ned, 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 5 and 6, obtained 

from formula (4 2), are shown in Tables 3 and 4. Numerical values of A2 and A4 have been 

selected to include afairly 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 the experimental determinations, it isobviously unwise to assume that they necessarily furnish in all 

cases a satisfactory estimate of the true probability of t for such small samples. 

The forms of the two approximations,amely (4.2), correct to n-1, and (5.3), to na1, are 

different from those which may be correspondingly 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 t as 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 t is 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 it agrees, to order n-1, with M. S. Bartlett's (1935) result suggests that the 

formula isnot satisfactory, since the asymptotic character of the latter cannot be assumed. 

For Bartlett specified his parent population by the first three terms of the Edgeworth series 

which take no account of the term in As, 2OA20(6)(x)/6!. namely, The estimates of true 

probabilities shown in Geary's Table 2 cannot as such be regarded as satisfactory. 

8. PROBABILITY CORRECTIONS DUE TO HIGHER CUMULANTS OF THE 

PARENT POPULATION 

For the effects of A5, A3A4, A33 we consider as before the probability integrals of the corrective 

functions and denote 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) 
366 Distribution of 'Student's' t from non-normal universes 

Table 3. Showing comparative values of 

to 

0 = 1 - J p(t) dt = 2[PO(to) - A4PA4(to) + A'PA2(to)] 

near the 4 % point (approximately) of normal theory t for samples 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 

near the 3 % point (approximately) of the normal theory t for 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 
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 notations of Complete and Incomplete B-functions. 

Biometrika 36 24 
368 Distribution of ' Student's' t from non-normal universes 

The additional formulae (8.1), (8.2) and (8.4) of this section enable us to deduce the probability of t when the parent population can be represented by the fourth approximation. 

The corrected probability P(to), for a single tail, 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 negative tail of the distribution PA3(to) and PA3(to) are positive, 

whereas PA5(to) and PA3A4(to) are negative, the reverse being the case with the other tail. 

For samples of 10 from such a population the probability at to = 2-2262 (here I to I is the 

5 % point of the normal theory 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 terms as far as A2 only is 0X0536. Thus for 

a single tail they differ considerably from the normal theory probability 0f0250. 

SUMMARY 

The theoretical distribution of 'Student's' t in non-normal samples of any size has been 

derived with reference to the parent population specified by a number of terms of the 

Edgeworth series. It contains, in addition to the normal theory frequency function of t, 

corrective terms due to the cumulants A3, 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 third or the fourth approximation (in 

which cases next higher A's and higher powers of A3 and A4 are small), then the corresponding 

t-distributions will be accurate for any size of sample, as far as similar order terms in A's. 

The two expressions for t, namely, the one including the terms up to that in A3 and the other 

up to that in A3, are asymptotic to order n-1 and nA- respectively, and as such they afford 

closer approximations to the actual distribution of t for large samples from any universe. 

The probabilities calculated from the former will be the same as those from the latter, if both 

the tails of t are used. So it is not unlikely that for moderate size of sample, the former may 

have an extended range of applicability. The satisfactory agreement between the theoretical 

values and the experimental results of various writers appears to support this conjecture. 

Tail-area probabilities of the corrective terms for A3, A and A2 have been tabulated. 

An example is given to illustrate the use of the tables. Diagrams showing the frequency 

curve of non-normal t for certain values of n have been constructed. 

The probability corrections due to higher cumulants of 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 improvements to the paper. My thanks are due to Mr D. A. East for drawing the 

diagrams. 
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.