Karl Pearson and David Heron 
1 95 
taken in the quadrants EOD and FOG, Q is negative and varies from 0 to — 1. 
Hence for two continuous variates, whose actual correlation is zero but which 
are not homoscedastic, we find Mr Yule's coefficient will run through the whole 
range of possible values from —1 to + 1, and what its observed value will be 
depends solely on where we take our dividing planes. 
It is perfectly true that Mr Yule's Q vanishes if the variables are theoretically 
independent, but no variables are practically independent, and in actual statistics 
of two continuous variates when the grouping is fairly small we do not get DEFG 
a rectangle, the sole bounding contour for which Mr Yule's coefficient of association 
is zero all round. The fact that Mr Yule gets + 1 or — 1 for his coefficient round 
his boundary contour would be of small importance were it not that he appears 
to hold that, when Q= ± 1, then its probable error is zero. Round the bounding 
contour of a distribution of this kind Pearson's normal coefficient has usually a big 
probable error, and the investigator is thus warned that its vagaries are of no 
account *. When the investigator comes, however, to a fourfold classification leading 
to Q = ± 1 by the zero of one of its classes, he would, if he were to follow Mr Yule, 
assume his result absolutely reliable, and to be so independently of the total 
population used. As a matter of fact, his dividing lines may have given Q = ± 1 
solely because they chanced to be taken near the bounding contour of a frequency 
distribution, of which the investigator knows nothing ! The real fact of the case 
is that Mr Yule's investigation of the probable error of Q, while correct as long 
as the frequency of any of his four classes is substantial, fails entirely when one 
of his four classes is zero, and is correspondingly in error when Q is very large 
owing to one of the four classes being very small. Even if Mr Yule determines 
the probable error of Q for such cases by higher approximations, it will be 
meaningless without consideration of the frequency distribution of Q, which is 
an exceedingly skew curve for high numerical values of Q. 
Before we leave these cases of zero correlation it is worth while to indicate 
how Q works for various artificial frequency surfaces. 
(1) A rectangular block. Q is zero all round the boundary and for all 
possible divisions. 
(2) A square prism ; diagonal planes parallel to the variate axes. Q is positive 
in two quadrants and negative in two, and takes all values between + 1 and - 1 
according to the point of division. It is essential to note that Q does not, as 
* The matter is discussed more at length in Appendix I. 
