302 
On Theories of Association 
Mr Yule does not really appear to know which of his ducklings to prefer, 
even for what in his estimation — although not in ours — are discrete attributes. 
We are quite clear that none of them are appropriate. Still less are they 
appropriate for the definitely continuous data to which Mr Yule and his disciples 
apply them. 
The controversy between us is much more important than an idle reader will 
at once comprehend. It is the old controversy of nominalism against realism. 
Mr Yule is juggling with class-names as if they represented real entities, and his 
statistics are only a form of symbolic logic. No knowledge of a practical kind 
ever came out of these logical theories. As exercises for students of logic they 
may be of educational value, but great harm will arise to modern statistical 
practice, if Mr Yule's methods of treating all individuals under a class-index as 
identities become widespread, and there is grave danger of such a result, for his 
path is easy to follow and most men shirk the arduous. 
The very large amount of arithmetic involved in this paper would have been 
impossible without friendly help from a number of our colleagues ; we have 
especially to thank Miss Julia Bell and Mr Herbert E. Soper; the former for 
much work in calculating and the latter for diagram draughtsmanship as well as 
calculation. Miss Ethel M. Elderton also most kindly undertook one or two 
pieces of heavy arithmetic. We can hardly hope to have escaped numerical slips, 
but we feel confident that such slips, if they occur, will as frequently tell against 
us as for us ; and we have not, knowing the fallibility of the best calculators, 
done more than draw attention to points where our arithmetic differs from that 
of Mr Yule. We have laid sole stress on errors of interpretation and on fallacious 
theory. 
APPENDIX I. 
On the Fallacy of asserting Perfect Association when one 
Quadrant in a Fourfold Table is Vacant. 
In all the values of the probable errors hitherto determined the constants 
of the distribution found in the formula are truly constants of the actual 
distribution of which the population under discussion is a sample. Because we 
do not know the actual distribution, we replace its constants by those of the 
observed sample. This method will as a rule not lead us astray, but it may do 
so grievously in cases where the observed frequencies take limiting values. For 
example, consider the population described in the following fourfold table : 
TABLE I. 
A 
Totals 
D 
Not-/?... 
971,138 
5,862 
22,862 
138 
994,000 
6,000 
Totals 
977,000 
23,000 
1000,000 
