OF ATTRIBUTES IN STATISTICS. 



mth order frequencies must be given (nothing else being given) in order that the 

 remaining frequencies of the same order maybe calculated? These questions are 

 considered in the next section (II.). In Section III. correlation or association audits 

 measurement are treated ; Section I V. deals with jiroUible errors; and in Section V. 

 some arithmetical examples are given of the methods and results previously discussed. 



II. GI:M:I:\I- RELATIONS. 

 Number of Independent Frequencies. 



7. Before proceeding to the problem above described, we will first prove the 

 theorem 



" The frequency of any group whatever can always be expressed entirely in terms 

 of the frequencies of the positive groups of its own and lower orders, and the total 

 frequency (U)." 



Tli is theorem may most simply be proved by the method of multiplying operators 

 as described in the introduction, replacing any negative operator like a by A and 

 multiplying out. We may, however, effect the reduction by step-by-step substitution. 

 Thus 



(Afr) = (A0) - (A/?C) 



= (A) - (AB) - (AC) + (ABC). 



To take terms of the fourth order, for instance 



(ABCD) = (ABCD) 



(ABC8) = (ABC) - (ABCD) 



(AByS) = (AB) - (ABC) - (ABD) + (ABCD) 



(Ay8) = (A) - (AB) - (AC) - (AD) + (ABC) + (ABD) + (ACD) - (ABCD) 



(a/JyS) = (U) - (A) - (B) - (C) - (D) + (AB) + (AC) + (AD) + (BC) 



+ (BD) + (CD) - (ABC) - (ABD) - (ACD) - (BCD) + (ABCD) 



.... (1). 



Evidently from the form of the last equation all the positive groups are required 

 to express the frequency of an entirely negative group. 



8. Now to the problem 



" To find the number of independent frequencies of mth order groups, the number 

 of variables being n." 



The number of positive groups of order m is (number of combinations of n things m 



together) 



n(n 1). . . . (n m + 1) 

 ml 



But by the theorem of 7 the frequency of any group of the mth order can be 

 expressed entirely in terms of the frequencies of positive groups. Therefore the 

 number of independent mth order frequencies must be equal simply to the total 



