Karl Pearson and David Heron 
289 
Pearson did not give a name to his coefficient fa above, but he carefully 
distinguished it from the partial coefficient {r w and stated that the former 
generalised form which did not select at a given value but round it was more 
important for both natural and artificial selection (loc. cit. p. 31). Mr Yule has 
apparently just awoke to the importance of fa, but that is no reason why he 
should confuse it in the minds of his readers with or' lead his readers to believe 
that we do not know the difference between the two. To avoid confusion of this 
kind in future we shall henceforth speak of jr a as a singular partial correlation and 
as a plural partial correlation. For, the former expresses the relation between 
A and B for a single value of C and the latter for a plurality or universe of values 
of C. In actual practice there is little difficulty in determining ^ if there is 
enough material, for all we have to do is to take the given universe out of C and 
correlate the resulting A's and B's. On the other hand, when we speak of the 
relation of health of child to health of mother for constant employment of mother 
or constant habits of mother, we do not look upon the universe of employed 
mothers as a whole or the universe of mothers with bad habits as a whole. We 
are thinking of employment of mother as a graduated character and parental habit 
also as a graduated character, and we properly use ^ to measure the relationship 
of health in mother and child for a constant grade of employment or constant 
grade of bad habit in the mother. In this case the use of fa has precisely the 
same justification, if r ls , r 2S and r 12 are found by tetrachoric tables, as if they had 
been found by product-moments, provided the assumption of a Gaussian distribution 
be reasonably justified for the material in question. There is no other source of 
error in the use of ,r B as Mr Yule obscurely seems to indicate. It did not need 
Mr Yule's numerical illustration (loc. cit. p. 629) to prove that fa for the two 
sections of an unequally divided normal curve — 'defectives' and ' undefectives ' 
(sic !) — is in neither case equal to fa. The two coefficients have different values 
and different significance whether the frequency be Gaussian or non-Gaussian. 
(b) Mr Yide's Failure to distinguish between Criticism of Method and 
Criticism of Conclusion. 
We have seen in the course of this paper that Mr Yule's coefficient of 
association automatically rises in all cases examined when our dichotomy is very 
one-sided. This is very obvious even in skew distributions ; compare Diagrams 
XVIII, XIX, XXI and XXII where the rapid increase of Q for small per- 
centages is obvious. Heron working on the Gaussian surface had demonstrated 
that this was an absolute necessity which flowed from the theory and that there- 
fore it must follow, even for surfaces only approximately Gaussian, that two or 
more values of Q were quite incomparable if the dichotomic lines were at 
different percentages of marginal frequencies. He argued that no valid proof 
could therefore be based on the relative sizes of Q in a series of tables for which 
Biometrika ix 37 
