162 
On Theories of Association 
persons being ' dead ' or ' not dead ' and questioned whether Mr Yule was correct 
in treating the variate behind the class-index as discrete and not continuous. 
In the original paper of Pearson " strength to resist smallpox when incurred " 
was stated to be the variate, and all the evidence that has been produced since 
indicates its continuity ; in precisely the same way vaccination and non-vacci- 
nation represent degrees of immunity in a continuous variate of which area of 
vaccination as indicated by extent of cicatrix and period since vaccination are 
contributory quantitative factors. Again " employment or non-employment of the 
mother " are not taken by us as signifying the presence or absence of a mere 
discrete attribute — for example whether she works in a factory or not — but as a 
class-index indicating that employed women, who have not only their home work 
but factory labour also, have on the whole more physical exertion to endure than 
those who are simply housewives. We are really seeking how far the continuous 
variate physical exertion of women affects infant welfare, and this is not a discrete 
variate any more than survival or death of infant is a discrete variate, when you 
view them merely as class-indices of physical fitness to survive in the child. In 
other words, for the great bulk of attributes, to which Mr Yule without analysis of 
their nature applies association, we should assert continuous variation. We hold 
therefore that in the main we are applying fourfold or other class divisions to 
continuous variates. Mr Yule thinks he has freed himself from all consideration 
of what the nature of this continuity may be ; we consider his belief wholly 
fallacious. You cannot free yourself from some assumption as to the nature of the 
distribution when you are dealing with the association of attributes. And in 
ignorance of what the true distribution may be, what assumption will help you to 
the most probable result ? On the basis of a very large experience of frequency 
curves and surfaces we have no hesitation in saying that up to the present time no 
distribution has been proposed which roundly represents experience so effectively 
as the Gaussian frequency. One of the present writers has indicated over and 
over again how it fails, and he has measured the significance of its failure, but has 
always recognised that he must put against this the large percentage of cases in 
which it gives reasonable results, close enough for all practical purposes*. Mr Yule 
(J. R. S. S. Vol. lxxv. p. 612). Who the "we" are, Mr Yule does not tell us; but suppose "we" 
started to find the relation between age and place in an examination — say the mathematical tripos — 
should we learn more or less by treating the wranglers as a class-mark with no graduations and age 21 
as a fixed division, or by assuming that the fourfold table : wrangler — not wrangler, minor — not minor, 
really covered continuous variations of age and class-place? Vaccination means vaccinated a week 
ago or forty years ago, a graduated immunity ; the dead means a group who not only had no power 
to resist an attack of the given intensity, but in certain of the cases an attack of a far less intensity — it 
covers a class with graduated power of resistance. Mr Yule here as elsewhere is mastered by words, not 
seeking the realities behind classification. 
* See, for example, C. D. Fawcett, Biometrika, Vol. i. p. 443 ; W. E. Macdonell, Vol. i. p. 184, 
Vol. m. p. 227 ; Pearson and Lee, Vol. li. pp. 362 — 7; E. Pearl, Vol. iv. p. 40; etc. etc. Compare, 
however, J. F. Tocher, Vol. v. p. 300, who for long series finds a certain amount of deviation from 
normality, generally in the direction of " leptokurtosis " (/3 2 is large) not of much asymmetry (p x is small). 
Even in these cases we doubt whether any serious practical error would be introduced by the use of the 
Gaussian distribution, unless extreme dichotomies are made. 
