300 
On Theories of Association 
(a) that both characters proceed by discrete units, and are tabled as such. 
In this case the method of pseudo-ranks is identical with that of the product- 
moment, and there never has been doubt as to how the table is to be treated ; or : 
(b) that the frequency of the two characters is continuous and that this 
frequency follows or approximates to a definite theoretical system. There is only 
one such frequency system, which has up to the present been effectively discussed, 
i.e. that of Laplace, or as it is more frequently but less justly called, that of 
Gauss*. If the distribution be Gaussian or approximately Gaussian, there are 
many ways of dealing with an n x n-fold table. 
(ii) A majority of the cases which occur in statistical practice are so close 
to the Gaussian distribution that methods based upon Gaussian theory will give 
useful first approximations, i.e. correlations within + '05 say of the true values. 
Years ago one of the present writers insisted on the non-Gaussian character 
of many variables. But he also remarked on the large number of variables 
which can be described with sufficient practical accuracy by a Gaussian 
distribution. 
The present discussion demonstrates that even with distributions markedly 
skew the Gaussian theory, if applied to 2 x 2-fold tables — without extreme 
dichotomies — will give results not differing by more than '05 from the value of 
the true correlation and often differing by much less. Roughly, we may say 
that for reasonable divisions, the divergence between the true correlation and 
that obtained by Gaussian theory is hardly ever of practical importance and indeed 
in "populations" of the size usually dealt with rarely exceeds twice the probable 
error. 
(iii) The coefficient of correlation has such valuable and definite physical 
meanings that if it can be obtained for any material, even approximately, it is 
worth immensely more than any arbitrary coefficients of " association " and 
" colligation." 
Starting from these principles we ask ourselves to what data Mr Yule proposes 
to apply his three processes ; 
(a) The Boas-Yulean <f> for fourfold tables. 
(b) The coefficient of pseudo-ranks. 
(c) The coefficient of association or that of colligation. 
We have shown in this paper that for tables with a finite number of cells 
of the order 5 x 5-fold to 8 x 8-fold, the method of pseudo-rauks must lead to a 
value below and often 40 °/ o below the true correlation of variates. Mr Yule has 
stumbled into a statistical pitfall, for he has neglected the fact that correlation of 
ranks is not correlation of variates, and that his correlation of ranks would still 
have to be corrected for the class-index correlations, i.e. he has also neglected the 
''' Of course both these writers only dealt with the frequency of one variate ; Bravais extended it to 
two, but gives no admissible proof of his formula, which he practically gets by analogy. 
