250 Oh a Neiv Method of Determining Correlation 
and from this it follows that 
.(iii). 
This is the value of the correlation ratio, based, so far, solely on the assumption 
that the arrays are sufficiently homoscedastic to permit of our replacing the ratio 
y<^xl<^x by its mean value. There has been no hypothesis of linear regression, still 
less of a Gaussian distribution. 
It now remains to consider on what hypothesis we shall determine Xy/yO-x and 
xjax, which are not directly determinable from the data. We shall assume that 
the mean of each array of sc's as well as the mean of the whole population may 
be found in terms of their respective standard deviations by tables of the 
probability integral. In other words we assume the distribution of the a;-variable 
to be approximately Gaussian. Thus we find uniquely the correlation ratio of a 
double row contingency table by the following assumptions: 
(i) The alternative variate is sufficiently Gaussian to admit of the means 
being found by tables of the probability integral. 
(ii) The arrays are approximately homoscedastic. This latter condition is 
unnecessary, if the alternative variable be replaced by one grouped in thi-ee classes, 
for then the actual value of yO-x/a-x can be found for each array and we need not 
give it the mean value Vl-??-. With regard to the multiple categoric variable 
no assumption as to the nature of its distribution has been, or need be, made. 
Illustrations. 
(3) I will take first some illustrations from a forthcoming paper dealing with 
the relation of pigmentation to psychical characters. 
(A) Conscientiousness in Boys and Hair Colour. The conscientiousness was 
classified as "keen " and "dull" with an intermediate class, which was found to be 
relatively very small and distributed accordingly between the two main groups. 
We have the following table : 
Hair Colour. 
Red 
Fair 
Brown 
Dark 
Jet 
Black 
Totals 
Keen 
Dull 
59-5 
24-5 
441-5 
194-5 
433 
188 
354-5 
162-5 
20-5 
16-5 
1309 
586 
Totals 
84 
636 
621 
517 
37 
1895 
