H. E. Soper 
107 
Hence by subtraction, 
mean {drf = ^ (1 - pj + ^ (1 - r)' (1 + 5^ 2 ). 
Or taking the square root 
Vn \ 2w 
which may be expressed in like manner with r in the form 
1 - 
ov = -= 
VM 
4(w- 1) 
.(xliii), 
to the same degree of approximation. 
It appears then from the above results that if the coefficient of correlation 
existing between two measured characters in a large aggregate of individuals be 
computed from the product moment values in small samples, these values are 
subject to errors from a mean value, the standard deviation of which errors may 
be very approximately represented by the formula 
I-P" 
V?z — 1 ' 
and with greater degree of accuracy by the formula 
W / iv 
V» - 1 V 4>n 
p being the coefficient of correlation between the characters in the material 
sampled and n being the number in the sample. 
Moreover the mean value of the correlation coefficients obtained from such 
small samples will be less than the true coefficient of the aggregate and will be 
approximately represented by the formula 
'('"^ 
the defect being very small when p is large, and when p is small being of the order 
5°/ 0 in samples of 10 and 5 °j o in samples of 100. 
On the other hand the modal value of the correlation coefficients, or the most 
likely value in a single sample, will be greater than the true correlation coefficient 
(that is to say numerically greater: the correlation being supposed to be measured 
positively). 
We have, by definition l/\ = 
and so from equations (xli), (xliii) putting n — 1 
ti \ 4n' 
n V 4n' 
using second approximations. 
