Co r related Averages. 



199 



given the coefficients in this expression, we could deduce the 

 coefficient of correlation between any two of the variables, 

 x and y. The sought coefficient, say r, is such that to any 

 assigned value of x, e. g. x', there corresponds rx as the most 

 probable value of y : that is to say, the y which in the long- 

 run — the long run whose stages are different values of z — most 

 frequently occurs. The direct method is to substitute x' for 

 x in w, integrate between extreme limits with regard to z, 

 differentiate with regard to y, and equate to zero. Geometri- 

 cally we may imagine a surface in the fourth dimension of 

 space at a distance iv± from the plane x = x' ; where w 1 is what 

 w becomes when x f is substituted for x. The annexed diagram 

 is intended to assist the imagination by representing the 

 curves of probability projected on the plane yz. The point 

 Oi is the centre of that system of ellipses which is formed by 





***** "*""*» 





S 





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s 



s^ \ ^ 



s 



jT \ \ 



/ 



/ \ 



/ , 





/ / 



J \ 



/ / 



\ \ 



/ / 



I J 



/ / 

 / / 



/ 1 



f / 



1 1 



f / 



/ 1 



/ / 



I 1 



/ / 



/ / 



f / 



/ / 

 / / 



r / rt 





/ / Oi 



/ / 



/ / 



/ / 



/ / 



/ / 



/ / 



/ / 



/ / 

 / / 



/ / 

 f / 



/ / / 



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1 / S 



/ 



' / s 



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1 / / 



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1 1 / 



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1 V <s 



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I \ jT / 





\ v ^y^ / 





\ ^-^ y 





\ yS 





the section of the plane x = x f with the ellipsoid 



ax* + hf + cz* + 2fyz + 2gxz f 2hxy = const. 



The surface iv 1 is evidently symmetrical on either side of 

 planes parallel to the axes y and z through the point O x . 

 Thus the sought y to which corresponds the greatest number 

 of instances in the long run, the y for which the strip 



w 1 dz is greatest, is the ordinate of 1 ; which is also 



j: 



