OF ATTlMliriT.s |\ sTATKI'KS 



271 



Therefore 



(AB) {(,3) - (A) + (AB)} = {(A) - (AB)J {(B) - (AB)} 

 (AB) [(ft) - (A)} = (A) (B) - (AB) {(A) + (B)} 



(AB)(U) = 



20.* Now it seems to me that one of the chief needs in handling statistics of the 

 kind we are considering is some sort of " coefficient of association," which should 

 take the place of the " coefficient of correlation " for continuous variables, and be a 

 measure of the approach of association towards complete independence on the one 

 hand and complete association on the other. Such a coefficient should 



(1) Be zero when the variables or attributes A, B, are independent, and only when 

 they are independent. 



(2) It should be +1 when, and only when, A and B are completely associated, i.e., 

 when either 



all A's are B 1 

 all ft's are a J 



or all B's are A 1 



all a's are ft 1 



or when both of these statements are true together, which can only be when 



(A) = (B), () = (/*). 

 The three diagrams below illustrate the three cases which correspond to 



(A/8) = , (aB) = , (A0) = (aB) = 0. 



(3) It should be 1 when, and only when, A and ft or B aud a are completely 



associated, i.e., when either 



all A's are ft } 



all B's are a I 

 all yffs are A 1 

 all a's are B J 



* .\ote added 19/1/00. It has several times occurred to me us quite possible that I have limited myself 

 too much in this section by defining the case of " complete association " us equivalent simply to the logical 

 case. An association coefficient of greater analytical convenience might have been obtained by defining 

 attributes A and B as completely associated only when all A's were B ami all B's were A. The distinction 

 of the logical case by a definite value of the association has, however, obvious convenience*. 



