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XXII. Correlated Averages. 

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



THE " correlation " f between the members of a system 

 such as the limbs or other measurable attributes of an 

 organism may in general be expressed by the formula 



11 = Je~ n dx 1 , dx 2 , dx 3 , &c. ; 



where 



B.=p Y (xi — ,1'J 2 + p 2 (x 2 — x 2 ) 2 + &c, 



+ 2q 12 (x 1 — x l )(x 2 —x 2 ) + 2q n (x 1 —x 1 )(x s -x 3 ) + &c; 



x 1? x 2 , x 3 &c. are the average values of the respective organs ; 

 irj, x 2 , &c. are particular values of the same ; p iv p 2 . . -. <7i 2 , <]\z 

 are constants to be obtained from observation ; J is a constant 

 deduced from the condition that the integral of II between 

 extreme limits should be unity. The expression II represents 

 the probability that any particular values of x^ x 2 , &c. should 

 concur. It enables us to answer the questions : What is the 

 most probable value of one deviation x r corresponding to 

 assigned values a?/, x 2 &c. of the other variables ? and What 

 is the dispersion of the values of x r about its mean (the other 

 variables being assigned) ? 



This general formula for the concurrence of particular 

 values of several organs is deducible from the proposition, 

 proved by theory and observation, that each organ considered 

 by itself assumes different values according to the exponential 

 law of error. In a subsequent paper I hope to justify this 

 principle ; at present, assuming the propriety of the above- 

 written formula, I propose to show how the constants 

 />,, p 2 . . . q l2t q n . . . are calculated. This problem has been 

 solved by Mr. Galton for the case of two variables. The 

 happy device of measuring each deviation by the corre- 

 sponding quartile taken as unit enables him to express the 

 sought quadratic in terms of a single parameter ; as thus : — 



tj &\ 2px v r 2 x 2 



^— 1 _ «2 "" 1 _ -2 "*" 



2 s 



1-P 2 1-p' 1-P 



where our p is Mr. Galton' s r, and the x ]5 x 2 of our general 

 formula are zero. The parameter is found by observing the 



* Communicated by the Author. 



f See Galton, Proc. Roy. Soc. 1888, " Co-relations and their Measure- 

 ment;" and Weldon, Proc. Roy. Soc. 1892, "Certain Correlated Varia- 

 tions in Crangon vulgaris" 



