272 MR. G. UDNY YULE ON THE ASSOCIATION 



or when both of these statements are true, which again r.m only be it' 



A = (a) (B) = (ft). 



The three diagrams below illustrate these cases of negative association which corre- 



(AB)=0 , (ofl = (AB) = (aft) = 0. 



spond to 



21. The theorems just given show that 



__ 



) + (Ay9)(B) 



(i) 



will serve as such a coefficient of association for 



(1) When A and B are independent the numerator is zero and therefore Q zero ; 

 and conversely when Q is zero the variables are independent. 



(2) When (Aft) = or (aB) = 0, or both, Q = + 1 ; and conversely when Q = + 1 

 (Aft) = 0, (aB) = 0, or both. 



(3) When (A/8) = or (aB) = 0, or both, Q = 1 ; and conversely when Q = 1 

 (AB) = or (aft) = 0, or both. 



It is perfectly possible that other simple functions of the frequencies might be 

 devised which should have the same properties, but Q at any rate will serve ; I do 

 not wish to attach too great importance to the identical function employed. If we 

 choose Q for such a purpose, however, its properties must lie investigated. 



22. The numerator, or difference of the cross-products, has, as Professor KAEL 

 PEARSON has pointed out to me, a very simple and important physical meaning. It 

 follows immediately from the equations used in 19 for showing that when the cross- 

 products were equal A and B were independent ; namely 



(AB) (aft) - (Aft) (aB) = (AB) (U) - (A) (B) ; 

 or if (AB) be the value (AB) would have if Q were zero 

 (AB)(a/3) - 



- (AB)J 

 = (U) {(aft) - (aft), } 

 = (U) {(Aft) - (Aft)} 

 = (U) {(aB) - (aB) } 



