: G. U. YULE 125 



These conditions give liinits to aiiy ouc of thr t-hree frequencies (j4ß), (/16') aiid 

 {BC) in terms of the other two and Ihc fnciui'iicii's of tlu; first order, i.e. enablc us 

 to infer liniits to the one class-frcinu'ncy in terms of the othors. It will very 

 usually happen in practica! Statistical oases that the limits so obtaincd are vahie- 

 less, lying ontside those given by tlie simpler conditions (4), but that is merely 

 because in practice the values of the assigned fretpiencies, e.g. (AB) and (AC), 

 seldom approach sutticiently closely to their limiting values to render inference 

 possible. 



3. Association. 



Two attributes, A and B, are usually dcfinod to be indepcndent, witliin any 

 given field of Observation or " universe," when the chance of finding theni together 

 is the product of the chances of finding either of them separately. The physical 

 meaning of the definition seems rather clearer in a different form of Statement, 

 viz. if we define A and B to be independent ivhen the proportion of A's amongst 

 the B's of the yiven universe is the sarne as in that universe at (arge. If for 

 instance the (piestion were put " Wliat is the test for independence of sniall-pox 

 attack and vaccination ?", the natural reply would be " The percentage of 

 vaccinatetl amongst the attacked should he the same as in the general popu- 

 lation " or " The percentage of attacked amongst the vaccinated should be the 

 same as in the general population." The two definitions are of course identical 

 in efFect, and permit of the same simple symbolical expression in our notation ; 

 the criterion of independence of A and B is in fixet 



(AByJ'^P ,7). 



In this etpuition the attributes spocifying the universe are understood, not 

 expressed. If all objects or individuals in the universe are to possess an attribute 

 or series of attributes K it may be written 



(AK)(BK) 



(ABA) = ^j^y-. 



An equation of such form must be recognised as the criterion of independence 

 for A and B within the universe K. As I have shewn in the finst memoir referred 

 to in note §, p. 121, if the relation (7) hold good, the three similar relations for the 

 reniaining fretpiencies of the " aggregate " — i.e. the set of frequencies obtained by 

 substituting their contraries a, ß for A or B or both — must also hold, viz. 



(A)(ß)\ 



•(8). 



