ON A NOVEL METHOD OF KEGAEDING THE ASSOCIATION OF TWO 

 VARIATES CLASSED SOLELY IN ALTERNATIVE CATEGORIES. By 

 Karl Pea.iison, F.R.S. 



In a memoir published twelve years ago in the Phil. Trans. I have shewn 

 that in the case of the fourfold table for the correlation of two variates, i.e. 



the correlation between the means of the two variates, when each is measured 

 in terms of its standard deviation, is* 



_ ad — he ... 



'^'^~-J{b + d){a + c){c + d) (a +T) ^'^" 



This correlation naturally vanishes with the transfer, i.e. e = (ad — be)/N, or if the 

 two variates are absolutely independent. Further if r^ be the correlation between 

 the two variates x and y concerned, r^ must of course vanish with r^, but it is 

 very far from equal to it, or proportional to it, as has been apparently assumed by 

 certain recent writers on correlation, the multiplying factor varying with the values 

 of ^ and Jc, i.e. with the positions where the dividing classifications are made. 



The above statements depend upon the assumption that the distribution of 

 frequency is normal or Gaussian in character. 



Quite apart from any assumption as to the nature of the distribution, I have 

 shewn that the mean square contingency t of a fourfold table is 



(ab — cdy 



<!>'=-, 



= r. 



hk 



•(ii). 



'{a + d){c + b){a + c){d + b) 

 and that, if we take 



we can from the general theory of the deviations from the probable in a correlated 

 system of variables^ reach a quantity P giving the probabihty that the system is 



* Phil. Trans. Vol. 195, A, p. 12, 1900. It is well to take as our standard arrangement of the table 

 one in which a + b>c + d and a + ob + d. 



f " On the Theory of Contingency and its relation to Association and Normal Correlation," Drapers' 

 Company Besea/rch Memoirs, i. p. 21. 



I Phil. Mag. July 1900, pp. 157—75. 



1—2 



