478 Mr. G. U. Yule. On the Significance of Bravais Formula? 



subjecting tlie distances of the means from the line to some minimal 

 condition. If the slope of RR is positive we may say that large 

 values of x are on the whole associated with large values of t/, if it is 

 negative large values of x are associated with small values of y. 

 Further, if the slope of RR to the vertical be given we shall have a 

 measure of a rough practical kind of the shift of the mean of an 

 #-array when its type y is altered. The equation to RR conse- 

 quently gives a concise and definite answer to two most important 

 statistical questions. It is also evident that if the means of the 

 arrays actually lie in a straight line (as in normal correlation), the 

 equation to RR must be the equation to the line of regression. 



Let n be the number of observations in any a?-array, and let d be 

 the horizontal distance of the mean of this array from the line RR. 

 I propose to subject the line to the condition that the sum of all 

 quantities like nd~ shall be a minimum, i.e., I shall use the condition 

 of least squares. I do this solely for convenience of analysis ; I do- 

 not claim for the method adopted any peculiar advantage as regards 

 the probability of its results. It would, in fact, be absurd to do so y 

 for I am postulating at the very outset that the curve of regression is 

 only exceptionally a straight line ; there can consequently be no 

 meaning in seeking for the most probable straight line to represent 

 the regression. 



Let x, y be a pair of associated deviations, let a be the standard 

 deviation of any array about its own mean, and let 



