27G 



MR. O. UDNY YULE ON THK ASSOCIATION 



, A R)^ ,(M>. 



(10) 



= COS 



=- IT 



fhere 



1 - Q 

 /r = as before. 



The figures llow give corresponding values of Q and / : 



Q is always slightly in excess of r, the greatest difference being rather more than 

 1 for Q = 7. 



27. In the general case the value of Q is necessarily a function of the position of 

 the origin, or of the arbitrary axes which are chosen for dividing A from and B 

 from /8. The evaluation of Q for any pair of axes in the case of normal correlation, 

 depends on that of certain definite integrals which have not yet been tabulated. To 

 get some idea of the general character of the dependence I have calculated the value 

 of Q for every possible pair of axes in the annexed (observed) frequency table ; the 

 frequencies* being the small figures, and the values of Q those entered in heavy type 

 at each origin. An inspection of the table will show that Q is a minimum for axes 

 near the mean of the whole table, and a maximum for origins near the limits. At the 

 exti'eme boundary the values vary suddenly and erratically, owing to the necessary 

 discontinuity of the observed frequencies, and here we may get values i 1 for the 

 association. In other parts of the table, however, negative values only occur in most 

 exceptional positions, and appear to l>e due to accidental irregularities. The sign of Q 

 agrees with that of the correlation coefficient r over almost the whole table. 



28. It does not seem possible to obtain for Q a function that shall not vary with 

 the position of the axes in the general case, so long, at all events, as we adhere to 

 certain conditions of symmetry for the function Q that seem to me almost necessary. 

 It may perhaps be possible for a strictly normal frequency distribution. 



* There is some slight error, possibly due to copying, in the frequencies of the table, as the totals of 

 rows and columns occasionally contain odd quarters, whereas they should only contain odd halves. I do 

 not think this is of any practical consequence. 



