1909-10.] The Significance of the Correlation Coefficient, etc. 475 
Dividing by the common factor x + y this becomes 
Parents. 
Offspring. 
(a, a). 
(a, b). 
(b, b). 
(a, a) 
a 3 
• x 2 y 
(a, b) . 
xhj 
xy(x+y) 
xy 2 
(b, b) . 
xy* 
y 3 
In this table the regression is linear, and therefore the correlation between 
parent and offspring may be determined by the product method and is 
given by 
r — m 5. 
This shows that in a stable population the correlation is independent of the 
relative proportions of purer races. Now in ascertaining the correlation 
when the hybrid can be distinguished from the dominant the process given 
above is correct, but when the hybrid has no points of special distinction 
and must therefore be included in the dominant, the table is condensed to 
the following : — 
Parents. 
Offspring. 
(a, a) + (a, b). 
(b, b). 
(a, a) + (a, b) . 
(b,b) . 
x 3 + 3x 2 y + xy 2 
xy 2 
xy 2 
if 
Here the regression is linear as shown by Professor Pearson, so that by the 
product method 
r= J 
x + 2y 
or, 
= '333 when oc = y. 
5. By repeating the above process the correlation of offspring with 
remoter ancestors can be easily evaluated. The first hypothesis, namely, 
that the hybrid is independent of the dominant, leads to correlation of '5, 
‘25, 125, etc., or, in other words, they are there given by Galton’s Law of 
Ancestral Inheritance.* On the second hypothesis, the one investigated by 
* Professor Pearson, Royal Soc. Trans ., vol. excv. p. 119, Table IX., “Exclusive 
Inheritance.” 
