OF ATTRIBUTES IN STATISTICS. 283 



35. I do not profess to have given in the foregoing pages more than an outline of 

 the theory of the case with which the statistician lias to deal ; in stronger hands it 

 could prolmbly l>e carried much further. The method 1 have suggested ha* the 

 advantage of bringing the case of association somewhat into line with that of 

 con-elation ; assimilating the method and conceptions of the case of association to 

 those of the better known field. 



The statistician has to handle problems of peculiar difficulty, where the association 

 may have any value. The logician demands Q = 1 before he will consent to 

 infer, and limits himself to this special and elementary case. At the opposite pole 

 to that of the logician we may imagine a " logic of independence," where Q is always 

 zero a case hardly less artificial and quite as interesting as the converse, but one 

 where inference is frequently im{x>ssible. 



IV. PROBABLE ERRORS. 



36. Let f be the frequency of any one group of any order, and let N be the total 

 frequency observed. Also let <f> = //N. Then the standard deviation of <f or <r+ is 

 at once given by 



,,= ^/^SES <. 



The S.D. of the frequency/ is N times this. 



37. Now consider the frequencies of two groups and let us find the correlation 

 between errors in their frequencies. We must here consider two different cases, 

 (1) where we are dealing with two ultimate groups, e.g., (AB) (A/J), or (ABC) 

 (aBC), or (ABC) (a/Jy) ; (2) where we are dealing with the two non-ultimate groups, 

 e.g. t (A) (B), or (AB) (AC), or (AB) (CD). 



CASE 1. Ultimate Groups. Let/i, /j be the two frequencies, fa, fa their ratios to N. 

 Suppose fa to undergo an increment A^ ; there is then a total decrement A^, to 

 be spread over the remaining groups in proportion to their frequencies, the sum of 

 the <f's being constant and equal to unity. Therefore 



. . (2). 



Let R^ t be the correlation coefficient between errors in fa and fa. Then the 

 above equation gives us 



or for ultimate groups 



Hence T <PIP 



N 



;in expression that we shall frequently require. 



2 o 2 



