Karl Pearson and David Heron 
241 
the eye-colour, (ii) by Mr Yule's method of unit groupings to indicate that even 
that method shows the same result, the inheritance of the intensity of pigment 
inside the two main groups*. In an inset figure we have given, by the same 
methods, the increasing pigmentation of the second brother as the pigmentation 
of the first brother increases, when we remove all the blue-grey group from con- 
sideration. The general weakening of the correlation produced by using the 
method of pseudo-ranks will be obvious if Diagrams VI and VII are compared. 
The later diagrams, Nos. VIII and IX, show precisely the same point for the 
parental eye-colour data. It will be clear that if i were the limit to the corre- 
lation of these eye-colour tables, then all the correlation within the blue-grey and 
hazel-brown groups ought to have disappeared. Instead of this we find that the 
£ is only the correlation on the Mendelian hypothesis that there is a unit difference 
between individuals in the one group ami individuals in the other; whatever 
this " unit difference " may refer to, it does not refer to a quantitative difference 
in pigmentation, because there is correlation within the groups. 
We have then the following results for fourfold parental eye-colour tables, the 
division being made between marked and less marked anterior pigment : 
Classes 
Variates supposed 
to be discrete. 
Yulean <j> 
Variates supposed to be 
continuous and Gaussian. 
Tetrachoric r t 
Father and Son ... 
Father and Daughter 
Mother and Son ... 
Mother and Daughter ... 
■37 
•28 
■32 
•34 
■55 
•44 
■48 
•51 
Mean 
■33 
•49 
Mr Yule's statement that the correlation is " something like 4-, not J " amounts 
to the denial that the variate is continuous ; and the slightest inspection of our 
diagrams shows that the variate is continuous ; the question therefore turns on 
whether the Gaussian assumption gives a l'easonable approximation to the influence 
of this continuity in increasing the correlation. We have shown that the fourfold 
table result (</>) obtained by treating the variate as discrete requires, whether the 
distribution be skew or Gaussian, to be increased by amounts ranging from 37 °/ 0 to 
80°/ o , when the value of $ is '3 or upwards. We have accordingly little hesitation 
in asserting that the true correlation exceeds + by at least 40%. Another way 
of approaching the problem is the one adopted by Pearson, when he found the 
want of stability in the tetrachoric r t as applied to these eye-colour tables, the 
method of mean square contingency. 
* Note how the slope of this regression line is *5 for the main portion of Diagram VI. 
Biometiika ix 31 
