482 Mr. G. U. Yule. On the Significance of Bravais 9 Formula 1 



where &i& 2 , &'i&' 2 are the regressions in the two cases. To which 

 distribution are we, in such a case, to attribute the greater corre- 

 lation ? Bravais' coefficient solves the difficulty, we may say, in 

 one way, by taking the* geometrical mean of the two regressions as 

 the measure of correlation. It will still remain valid for non-normal 

 correlation. But there are other and less arbitrary interpretations 

 even in the general case. 



Suppose that instead of measuring x and y in arbitrary units we 

 measure each in terms of its own standard deviation, Then let us 

 write 



X - = f y ~ ......... ............. (5), 



and solve for p by the method of least squares. We have omitted a 

 constant on the right-hand side, since it would vanish as before. We 

 have, at once, 



That is to say, if we measure x and y each in terms of its own 

 standard deviation, r becomes at once the regression of x on y, and 

 the regression of y on x. The regressions being, in fact,, the funda- 

 mental physical quantities, r is a coefficient of correlation because it 

 is a coefficient of regression.* , 



Again, let us form the sums of the squares of residuals in equations 

 (1) and (5). Inserting the values of 6 l5 6 2 , and />, we have 



(7). 



Any one of these quantities, 'being the sum of a series of squares, 

 must be positive. Hence r cannot be greater than unity. If r be 

 equal to unity, or if the correlation be perfect, all the above three 

 sums become zero. But 



can only vanish if 



x y 

 -- = 



<T 2 



in every case, or if the relation hold good, 



* That the regression becomes the coefficient of correlation when each deviation 

 is measured in terms of its standard-deviation in the case of normal correlation has 

 been pointed out by Mr. Francis G-alton. Vide Pearson ' Phil. Trans.,' A, vol. 187, 

 p. 307, note. 



