278 



MR. O. UDNY VULE ON THE .\ssoCT\TION 



There is one case, and one only, where Q is independent of the axes chosen, and that 

 is where the variables are strictly independent. Let /, // be the elementary 



frequencies corresponding to values x m , ?/ of the variables, and let F MM be the 

 frequency of the pair (.r y n ). Then, if the variables are strictly independent, we 

 must have in every case 



K .,.= f m x/ H ' 



N being the total number of observations. Therefore, summing over any one 

 quadrant, whatever the position of the axes, 



or 



N(AB) = (A)(B) 



and so on, so that Q is zero for all axes. It is impossible to create an artificial 

 association, out of real independence, by mere choice of special axes. This is a most 

 important limitation. At the same time it must be borne in mind that where the 

 variables are not independent, as in the table on p. 277, Q may lie changed in 

 sign or rendered vanishingly small by the choice of special (possibly exceptional) 



,-l.XfS. 



The whole subject of the connection between correlation and association demands 

 further investigation, as it bristles with difficulties and possibilities of fallacy. In 

 some practical cases there seems no doubt that the signs of Q and r would be different, 

 and, indeed, the physical meaning attached to their interpretation. In the present 

 paper, however, I do not deal further with the subject. 



* Vf. also the example of assortative mating according to stature from Mr. GALTON'S 'Natural 

 Selection,' p. 82. 



