Karl Pearson and David Heron 
105 
attribute associated with a given value of a second changes continuously as we 
change that given value. It was only his wide experience of anthropometric data 
which led him to believe that in most cases the function that the mean of one 
organ is for a given value of the second may be adequately represented by a 
straight line. There was nothing either in his own treatment or in the work of 
his followers of the Biometric School, which pinned them down to the use of the 
word correlation for a particular constant found by the product-moment method. 
The "correlation ratio " has in the work of that school just as much significance as 
the " correlation coefficient," and it is only Mr Yule who proposes to confine the 
use of the word correlation to the narrow sense of straight line regression deter- 
mined by product-moment methods. To the biometrician correlation when it 
ceases to be linear is not determined by the product-moment value of r at all, 
and the grade of correlation may be far higher than the value as determined by 
the coefficient of correlation. In the same way other constants may be found 
defining the relationship of two continuous characters, or measuring their degree of 
dependence. These are equally measures of correlation in our sense of the word. 
Mr Yule's coefficient of association is not in our sense of the word a measure of 
correlation at all, it shows in no manner how the mean of one attribute for a given 
value of a second attribute varies as we modify this value. It is, as we shall show 
below, impossible to give it in the case of continuous variates any rational 
significance whatever. Where there is no true correlation at all, the size of 
Mr Yule's Q may be produced solely by a lack of homoscedasticity — of equal varia- 
tion — in the arrays of one variable associated with constant values of the second, 
but in what manner it measures this heteroscedastic property is quite beyond 
interpretation. Mr Yule claims that the nature of the frequency is of no conse- 
quence, he states that the coefficient of association may be applied without any 
general theory of frequency. For us this is not a correct attitude; we admit wide 
deviations from Gaussian distributions, but such cases are not the rule. Mr Yule 
can pick out special instances which are far from Gaussian, such as age distributions, 
barometric heights, or heterogeneous mixtures of growing organs like ivy leaf 
lengths. Even for such cases he has not examined in any adequate manner how 
far methods based on Gaussian or other allied considerations do give practical 
results, nor how far even the Gaussian fourfold r — tetrachoric r or r t , we will call it 
for the purposes of this paper — is a more stable and reliable coefficient than those 
suggested by himself. 
He has indeed criticised the application of a tetrachoric r t to eye-colour data — 
his discussion of the subject will be considered in a separate section of our reply- — ■ 
but he does not inform his readers that the one of the present writers concerned in 
the eye-colour treatment fully admitted in June 1907* "the unsatisfactory approach 
to the Gaussian distribution found in pigmentation tables" and stated that his 
very knowledge of this point had led him to develop the method of contingency for 
such problems. Yet Mr Yule raises five years later the applicability of the 
* Diometrika, Vol. v. p. 472. 
