G. U. YüLK 127 



This poiiit is t'requently t'orgottcn. In an iuvestigation as to Uiu iiiheritance 

 of dcaf-mutism in America*, for instance, only the offspring of deaf-mutes were 

 obscrvcd, and the argument consequently brcaks down on page after page into 

 conjectural statcmcnts as to points on which the editor has no inforniation — 

 e.g. the Proportion of deaf-mutes amongst the children of normals. 



The difference of iAB)/{A) frcia (yi)/.Y and of {AB)!{B) from {A)!N are of 

 coursc not, as a ruie, the same, and it would be useful and convenient to measure 

 the " association " by some more symmetrical method — a " coefficient of association " 

 ranging between + 1 like the coefficient of correlation. In the first memoir 

 referred to in note §, p. 121, such a coefficient, of empirical form, was suggested, 

 but that portion of the memoir should now lio read in connection with a later 

 memoir by Professor Pearson f. 



4. On the theory of complete iiidependence of a series vf Attrilmtes. 



The tests for independence are by no means simple when the inimber of 

 attributes is more than two. Under what cirenmstances should we say that a 

 series of attributes ABCD... were completely independent ? I believe not a few 

 statisticians would reply at once " if the chance of finding them together were 

 equal to the product of the chances of finding them separately," yd such a reply 

 would be in error. The mere result 



{ABCD...) JA) (B) (C) (D) 



N N ' N ' N ' JSr ^■' 



does not in general give auy informatiou as to the independence or otherwise 

 of the attributes concerned. If the attributes are knoiun to he completely inde- 

 pendent then certainly the relation (0) holds good, but the converse is not true. 

 " Equations of independence " of the form (9) must be shewn to hold for more 

 than one class of any aggregate, of an order higher than the second, before the 

 complete independence of the attributes can be inferred. 



From the physical point of view complete independence can only be said to 

 subsist for a series of attributes ABCD... within a given universe, when every 

 pair of such attributes exhibits independence not only within the universe at large 

 but also in every sub-universe specified by one or more of the remaining attributes 

 of the series, or their cnntraries. Thus three attributes A, B, C are completely 

 independent within a given universe if AB, ^t'and BC are independent within 

 that universe and also 



AB independent within the universes C and 7, 

 -^C' „ „ „ B „ ß, 



^G „ „ „ A „ a. 



* Mnrridflfs of the Deaf iti Amerirn, ed. by E. A. Fay. Volta Bureau, Washington, 1898. 

 t Plul. Trans. Vol. 195, p. 16. 



