26G PROF. K. PEARSON" OR THE MATHEMATICAL THEORY OF EVOLUTION". 
of correlation r is r + p instead of r. Hence the above is the law of distribution of 
variation in the coefficient of correlation. If the second term be negligible as com¬ 
pared with the first, we see that p follows the normal law of distribution. Thus we 
may say that with sufficient accuracy for most cases the standard deviation of a 
coefficient of correlation is 
1 - r 2 
vAmT + ffi> ’ 
1 — r 3 
or its probable error = ‘674506 —=rr . 
1 Sn (1 + r 2 ) 
The ratio of the first term neglected to the term retained 
4 ^ W + 3) 
3 (?’ 2 + 1) (1 - r 2 ) P ’ 
or to determine the order, giving p its probable value on a first approximation, we 
have 
1 r ( r - _L 3 ) 
ratio = | /. , ' X ‘674506. 
15 V n (r 2 + l)' 1 
This may be shown to be a maximum for r 2 = 1, and the ratio then takes the value 
1-272 
- y— , or the second term in this most unfavourable case will only be about 4 per 
cent, of the first when n = 1000. For r = ‘5, the ratio takes the value l‘046/\/nor 
for n = 1000 is about 3‘3 per cent. 
It will be sufficient, therefore, for most practical purposes to assume that the 
probable error of a coefficient of correlation 
= ‘674506 
1 - r 2 
(d .) Constancy of Correlation Coefficients for Local Races .—This result is not only 
of importance in dealing with the problem of heredity, it is crucial for determining 
whether constancy of correlation is characteristic of all races of the same species. 
Mr. Galton has suggested that the coefficient of correlation might be found to be 
constant for any pair of organs in different families of the same race. Professor 
Weldon has determined a series of coefficients of correlation for shrimps and crabs, 
which he thinks justify him in assuming “ as at least an empirical working rule that 
Galton’s function has the same value in all local races. The question whether the 
empirical rule is rigidly true will have to be determined by fuller investigation, 
based on larger samples.”* 
* ‘ Roy. Soc. Proc.,’ vol. 54, p. 329, 1S93. 
