[ 63 ] 



VI. A New Method of treating Correlated Averages. 

 By Professor F. Y. Edgeworth, M.A., D.C.L* 



THE following is a simpler method of solving one of the 

 problems treated in a former paper relating to correlated 

 averages f. Taking the case of three variables for conve- 

 nience of enunciation, let us put for the exponent (of the 

 expression for the probability that any particular values of 

 the variables will be associated) 



R== ax 2 + by 2 + cz 2 + 2fyz + 2gxz + 2hxy. 



And let it be required to determine the coefficients a, b, c, &c. 

 The most probable values of y and £ corresponding to any 

 assigned value of x 3 say x' ', are deduced from the equations 



fby+fz + hx f = 0, 



^cz+fy+gx' = 0. 



The values of y and z determined from these equations may 

 indeed diverge widely from the particular values corresponding 

 in any one specimen to x' . But if we take a number of 

 specimens, it becomes more true % that 



\cSz+fSy+gtx''=0. 



Dividing each equation by the sum of the assigned values 

 2#' ' , we have 



l^ 13 4-//?i2 + ^ = ; 



where p i2 , pn mean (as in the former paper) the coefficients 

 of correlation pertaining to each couple of organs (Mr. 

 Galton's r). 



By similarly assigning values of y and of z, and observing 

 the associated values of the other variables, we obtain six 

 (in general w(/i — 1)) equations for the sought coefficients 

 a, h, . . . g, h ; which, being rearranged, are as follows : — 



Pl2« + />23i/+ A = 0, \ 



p2^ + f+pzih=0,l 



Pi2b + Pdif+ h-0;S 



Ps\C + Puf+ tf = 0,\ 



P23C+ f+ pug = 0.1 



* Communicated by the Author. 

 t Phil. Mag. August 1892. 

 X Of. loc. cit. p. 191 et seq. 



