H. E. Soper 
109 
p, is zero and A = 3, r is indeterminate. This case is a little singular and (ix) 
shows that m Y and m. 2 are zero and that the frequency is therefore constant or 
that all values of r are equally likely, from — 1 to +1. Remembering that 
1 — r 3 
X = — it will happen when a r * = ±. But when p = 0 the value we have found 
for a 3 is which will be \ when n = 4. If therefore from material possessing 
zero correlation samples of four are drawn, the values of the correlation coefficients 
in these samples should, if the above formulae and assumed type of frequency 
distribution are correct, be equally distributed in value throughout the whole 
range. 
Complete experimental confirmation of the distributions found in this paper 
for the product moment expression of the coefficient of correlation in small samples 
is difficult to obtain. The second order differences in the values of the mean and 
standard deviation of the distribution are necessarily small in large samples and 
comparable with the errors of sampling of such samples unless a very large number 
of samples is taken. On the other hand if small samples are taken the theory is 
not so exact and in addition the distribution tends to concentrate in certain 
grades if the original material is coarsely grouped, whilst fine grouping adds very 
much to the labour of computation. 
In a paper published in Biometrika, Vol. VI., p. 302, " Student " gives the 
results of a very painstaking investigation to determine experimentally these 
distributions. Material based on W. R. Macdonell's measures of finger length and 
stature in 3000 criminals [Biometrika, Vol. I., p. 219] was formed having correla- 
tion with coefficient* *66 and samples of 4, 8 and 30 were drawn. Material with 
correlation zero was also constructed and samples of 4 and 8 drawn. 
The comparison of the actual means and standard deviations found in these 
experiments with those computed from the formulae in this paper is shown in the 
table that follows. 
1 
2 
3 
4 
5 
Number in sample, n ... 
4 
8 
4 
8 
30 
Correlation coefficient in material, p 
0 
0 
•66 
■66 
■66 
Mean correlation coefficient in samples, observed 
•5609 
■6139 
•661 
„ „ ,, „ calculated, r 
0 
0 
■5933 
•6317 
•6542 
Standard deviation of coefficient in samples, observed ... 
•5512 
■3731 
■4680 
•2684 
•1001 
,, ,, ,, „ calculated, cr r 
■5773 
•3780 
•4234 
■2453 
•1088 
Number of samples taken 
745 
750 
745 
750 
100 
Probable error of mean 
•0143 
•0093 
•0073 
•0105 
•0060 
„ „ standard deviation 
■0101 
•0066 
■0052 
•0074 
•0043 
* The exact figure was -G608. 
