Karl Pearson and David Heron 
1!)3 
variate, when we consider the actual frequency of pigment granules, and it is 
only confusing the issue when eyes are divided into two classes, those with both 
posterior and anterior pigment, and those with only posterior pigment. We shall 
return to the question of eye-colour later, as it is a case out of which Mr Yule 
makes much capital, but which in his last paper is the only one he treats by his own 
association methods. If we take " prickliness " in fruit, there is no evidence yet 
that it has ever been properly measured, and that it would not prove to be a 
continuous variate, much as " hairiness " proved to be in the case of Lychnis when 
actually measured by Weldon*. To sum up, there are not among the attributes 
used by Mr Yule to illustrate association coefficients any but those of sex and 
nature of fertilisation which can reasonably be considered discrete quantities, and 
in using even these he always couples them with characters which in our opinion 
are distinctly continuous variates. 
We shall therefore start this criticism of Mr Yule's statistical investigations by 
indicating the fallacious nature of his coefficients of association and colligation as 
applied to continuous variates. We shall then deal with the question of discrete 
alternative variables, and show their absurdities in that case. Finally we shall 
show reason for questioning the details of the bulk of his memoir, which is not 
occupied with the discussion of his special coefficients at all, but in advocating a 
new empirical method which there is ample reason for considering equally fallacious. 
(7) On the Idleness of Mr Yules Coefficient of Association when applied 
to Continuous Variates. 
(a) The Need for either Knowledge or Hypothesis as to the Nature of the 
Frequency. 
Let us start from a very general case of two continuous variates. The first 
question that we require to answer is whether for a given value of one variate 
the mean value of the other variate changes. If we can find the mean values 
of the arrays of the second variate for given values of the first we obtain the 
regression line ; should this be a straight line the correlation coefficient r is a suit- 
able measure of the relationship, if it be not then the correlation ratio rj gives us 
a measure, and ^ 2 — r 2 marks the deviation of the regression from linearity. In 
cases where the regression lines are both straight and r = 0, it by no means 
follows that the two variates are absolutely independent. The next essential 
condition is the equal variability of the arrays, or what is nearly the same thing 
the probability of a combination of one variate x between x and x + Bsc with the 
other variate y lying between y and y + By is the product of the probabilities 
* Biometrika, Vol. n. pp. 47 — 55. The danger of this sort of classification has come home very 
emphatically to one of the present writers, who had tried to use the category of "short-muzzle" 
as against "long-muzzle" in breeding dogs. He was convinced that "short-muzzle" was a dominant 
character in the Mendelian sense, until he took actually to measuring the muzzles of the hybrids of 
first and later generations, when the idleness of treating such categories as discrete quantities became at 
once obvious. 
Biometrika ix 25 
