270 MR. G. ITPNY YULE ON 7 THE ASSOCIATION 



III. ASSOCIATION. 



16. Two qualities or attributes, A and B, are defined to be independent if the 

 chance of finding them together is the product of the chances of finding either of 

 them separately, i.e., if 



(AB)_(A) (B) 

 (U) - 



or (AB)(U) = (A)(B). - ,.' 



This is, I think, the only legitimate test of dependence or independence association 

 or non-association in the general case. 

 17. Theorem. To show that if 



(AB)(U) = 

 then 



(aB)(U)=()(B) 



(aft)(V)=,(a)(ft). 



Take the first equation of these three 



(A)08) = (A) {(U) - (B)} = (U) (A) - (AB) (U) 



and so on for the others. So that if the chance of finding the two qualities together 

 js the product of the chances of finding either of them separately, the chance of 

 finding the one without the other is the chance of finding the one multiplied by the 

 chance of not finding the other, and so on. Any one of the relations implies all the 

 others. 



18. It follows at once from the above that if two attributes (A) and (B) are inde- 

 pendent, the products of the contrary second order frequencies are equal, i.e., 



for each is equal to (A) (B) (a) (ft) divided by (U) 2 . Not only so, but the converse is 

 also true 



19. If the cross-products are equal the variables are independent. Thus let 



(AB)(o0)=(A/8)(aB). 

 Now 



(Aft) = (A) - (AB) 

 (aB) = (B) - (AB) 

 (aft) = (ft) - (A/8) 



= (ft) - (A) + (AB). 



