22 On the Association of Attribute s in Statistics, dr. 



" On the Association of Attributes in Statistics, with Examples 

 from the Material of the Childhood Society, &c." By G. 

 Udny Yule, formerly Assistant Professor of Ap]3lied Mathe- 

 matics, University College, London. Communicated by Karl 

 Pearson, F.E.S. Eeceived October 20, — Eead December 7, 

 1899. 



(Abstract.) 



The paper deals with the theory of association of attributes, i.e., 

 invariable attributes, as opposed to the " correlation " of variables. 

 Two attributes A and B are independent or unassociated if 



(AB) = (A)(B)/N, 



(A) being the frequency of the attribute A, (B) that of B, and (AB) the 

 frequency of the pair AB ; N being the total number of observations. 

 If this relation do not hold, they are " associated." 



Section (I) of the paper is introductory, describing the subject-matter 

 and notation, which is essentially that of Jevons."^ Calling a group 

 defined by n attributes ABGD......N an ?ith-order group. Section (II) 



deals with the fundamental problem of the number of independent ?ith 

 order frequencies that can be formed from m attributes ; i.e., the number 

 of such frequencies that must be given before the remaining frequencies 

 of the same order can be calculated. Certain extremely curious 

 relations are shown to hold in the special case of " equality of con- 

 traries," where all pairs of contrary frequencies (A) (a), (AB) (a^), 

 (ABC) (a^y) are equal, a being the contrary of A — i.e. not A — and so 

 on. 



Section (III) proceeds to the theory of association proper. The 

 function 



Q _ (AB)(a^)- (A^)(aB) 

 ^~(AB)(a^)+"(A^)(aB) 



is proposed as a " coefficient of association." It is zero when the 

 attributes are independent, + 1 when all A's are B or all B's are A, 

 and - 1 when all A's are /3 or all /3's are A, and thus measures the 

 approach towards " perfect association " in the same sort of way as 

 the correlation coefficient measures the approach towards perfect 

 correlation. The connection between correlation and association is 

 touched upon, and it is pointed out that one may form " partial " 

 coefficients of association (by limiting the extent of the universe of 



* On a G-eneral System of Numerically Definite Eeasoning," ' Manchester Lit. 

 and Pliil. Soc.,' 1870, and " Pure Logic and other minor works " p. 173. 



