Karl Pearson and David Heron 
225 
What has Mr Yule to place against this method for tables 3 x 3-fold up to 
6 x 6-fold in classification ? He writes: "From several trials — more than are here 
given — I have come to the tentative conclusion that the best guide to the cor- 
relation that would be found for given data, if the grouping were other than that 
which in fact it is, is the correlation for the existing grouping, provided that you 
are given at least some five or six arrays" (loc. cit. p. 618). He does not venture 
to inform us what he would do for a table of 3 x 3, or 4x4 cells ! As a matter 
of fact such tables can give even by Mr Yule's method of pseudo-ranks better 
results than the 5x5 or 6x6 groupings. Mr Yule is, however, not content with 
his statement that his method is tentative; before he has done with it* he has 
assumed that by increasing his classes he will approach a limit which is the true 
product-moment correlation. As a matter of fact there is no such approach at 
all ; the Yulean method of pseudo-ranks may give a better result for a lower than 
a higher number of cells, and if it did go far enough to reach a limit, it would 
Order of 
0*5 Gaussian Correlation 
Barometer Results 
Age of Husband and Wife 
arranged in Father and Son 
Table. 
Number 
Eye-Colour Groups 
of 
Classes 
Contingency 
Pseudo-Ranks 
Contingency 
Pseudo-Ranks 
Contingency 
Pseudo-Ranks 
7x7 
•75 
•73 
•49 
•46 
6x6 
•74 
•73 
•88 
■89 
•48 
•45 
5x5 
•73 
•72 
•87 
•88 
•48 
•44 
1st 4x4 
•73 
•67 
•86 
•86 
•49 
•42 
2nd4x4 
•72 
■71 
•48 
•43 
1st 3x3 
•73 
•66 
■91 
•83 
•48 
•41 
2nd3x3 
•79 
■65 
•84 
•81 
■51 
•40 
3rd 3 x 3 
•75 
•67 
4th 3 x 3 
•76 
■52f 
•87 
•79 
Mean 
•744 
•674 
■890 
•845 
•488 
•431 
True r 
•780 
•780 
•925 
•925 
•500 
•500 
It will be seen that the method of contingency, especially with few classes, is markedly better than 
that of pseudo-ranks. We bave purposely introduced the Age of Husband and Wife, because the 
divisions there have ranges not very diverse in magnitude. In such cases the method of pseudo-ranks 
becomes almost exactly that of the true correlation of variates, if the proper Sheppard's correction be 
made. In the case of Ages of Husband and Wife, if this correction be included, we are practically 
finding the true correlation by the method of pseudo-ranks. It is remarkable that Mr Yule has not 
drawn attention to this, because it at once indicates how fallacious the method is, if the subranges 
be unequal. 
* On the basis of what he starts by calling a tentative method, he then proceeds to assert that all 
the biometric pigmentation work is "wholly untrustworthy" (loc. cit. p. 622) — a characteristic illustra- 
tion of how Mr Yule's mind rapidly grows obsessed by a theory which he has not properly investigated. 
t This case is of remarkable interest as indicating the futility of Mr Yule's method of pseudo-ranks. 
The table although 3 x 3-fold has sensibly equal ranges. Therefore the correction to pass from ranks to 
variates is closely Sheppard's. The Yulean pseudo-ranks coefficient thus corrected is raised from -522 
to -775, close to the true correlation ! 
Biometrika ix 29 
