10 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF TTS CHARACTERISTJCS. 



CO 



%(*/) = - \dyF{x p ,y){y-y x /:F l (x p )ol p (y)2, 



— 00 

 00 



E Xp (y)=JdyF(x p ,y)(y- y Xp Y: ^(a^oy-y) 8-|, 



r s (yh p = JdyF(x p ,y)(y - y Xp )' : F x (x p ) . 



Plotting ont against x p the value of 



y Xp we obtain the curve of regression of y on x 



ol p {y):o\ » » > scedasticity 



Sx p (y) » clisy 



.EVj, (y) » » synagic » » » » 



The last curve has not been dealt with by Pearson. For the others the ter- 

 minology is his. 



The curve obtained by plotting out any characteristic or moment of the arrays 

 against x p as argument we shall shortly call the curve of regression of that charac- 

 teristic. 



Of course we have — muiatis mutandis — the same expressions for the cha- 

 racteristics and moments in the ?/-arrays; i. e. 



% g ; <%(*); 8 Vg (x); E v q i x ); M x )v g - 



There is, however, one more characteristic of the arrays that is of importance. 

 That is the position of the mode, or the most probable value of y or x, associated icitlt 

 any x p or y q respeciively. 



As the mode is approximately at the distance 



S 

 a-— 





1 + 5E 

 from the mean we have for its position in the .r-array of y:s 



and for its position in the y-array of x:s 



., S Vq {x) 



% •% + %W, +5 e v (x) 



