David Heron 
119 
have just seen that the further away from the mean the dividing line is taken, 
i.e. the higher the values of h and k and the fewer the number of defectives, then 
the higher the value of the coefficient of association ; so that the " apparent law " 
arises solely from his faulty method. Had Mr Yule used his " theoretical value 
of r" instead of the coefficient of association, it would have been suggested that 
the associations were on the whole lower where populations were healthier or less 
defective. Neither coefficient enables any light at all to be thrown on the question 
at issue. 
For precisely similar reasons his discuss, m of the change of association with 
age must be dismissed as entirely fallacious. There may be and probably is, some 
decrease of association with advancing age, but the enunciation of such a law on 
the basis of a number of coefficients of association is purely idle. In the example 
that has already been given, Mr Yule has stated that the association between 
blindness and mental derangement decreased steadily from "921 at the age group 
5 — 15 to —'126 at the age group 75 and upwards. But the proportions of the 
blind increase steadily from 26 per 100,000 to 1051 per 100,000, while the pro- 
portions of the mentally deranged increase steadily from 85 per 100,000 to 079 
per 100,000, and this increase in the number of defectives will in itself produce 
a large decrease in the value of the coefficients of association. Until the amount 
of association has been expressed in terms of coefficients of correlation, no con- 
clusions can be drawn from the data. 
In the same way Mr Yule's statements that " the differences exhibited by the 
sexes as regards association are so marked that they can hardly have failed to have 
struck the reader of the foregoing tables," and that " in an immense majority of 
cases, the associations are greater for females than for males" must be rejected. 
The apparent difference arises from the fact that "besides being more highly 
associated, women are also in general less defective." 
That so much labour should have been spent on the analysis of data of very 
doubtful value by methods which can throw no light on any problem whatsoever 
is deeply to be regretted. As a warning of the clanger of basing definite and far- 
reachiug conclusions on results obtained by methods which have not been 
adequately tested, Mr Yule's memoir may in fact be commended, but it must be 
stated emphatically that not a single one of the conclusions reached by Mr Yule 
can be justified by the data or the methods he has used. 
One further point is of some interest. The process of finding the actual value 
of the coefficient of correlation from a four-fold table involves somewhat lengthy 
arithmetic and hence various formulae have been suggested by Professor Pearson 
by which an approximate value of r might be obtained without excessive labour. 
Since the publication of Everitt's Tables, however, these approximations are of 
considerably less importance owing to the very great reduction in the labour 
involved. I have, however, tested the comparative accuracy of the various 
approximations suggested for the special case where r is actually 5. 
