450 JOURNAL OP THE WASHINGTON ACADEMY OF SCIENCES VOL. 11, NO. 19 



The same notation will be used for the means, standard deviations 

 and coefficients of correlation of the primed variables. 

 Now lety = f {x^, x.^, x„) 



and z = f{x[,x'„ x'k) 



represent any two analytic functions of these two sets of variables. 

 Then the correlation between these two functions is measured by 



X^j ^j fa Fp rxaXp(Ta<r0 

 a=l ^=1 



I'yz — 



r n n 



n r /^ 



(1) 





1 





J 



in which 



/« = 



d f(if2i,m2, mn) 



d m 



a 



and 



F^ = 



c) F (ni{, mj, m'k) 



The derivation of equation (1) yielded as a by-product the ex- 

 pressions for the mean and standard deviation of any function of 

 a set of variables in terms of the means, standard deviations and first 

 order coefficients of correlation of the variables themselves.- These 

 formulae are 



(a) for the mean value of y = f{xi, x^, Xn), 



n n 



niy = / + 3^ ^ ^ /a/3 rxaX^(^a(^0 (2) 



a=l /3=1 



where / = / (wi, 7no, ntn) 



^^/(wi, mo, m„) 



and 



f al3 = 



d nia ^fn ft 

 (b) for the standard deviation of y — f{xi, Xo, Xn), 



n n 



crv = 



V X'i Xl f'^Jt^ *'XaXfi (Ta^-ft 

 « = ! /3=1 



(3) 



2 For another derivation of equations (2) and (.3) see E- Czuber, Uber Funktionen von 

 Variablen zzvischen welchen Korrelationen bestehen. Metron 1, No. 1, July, 1920. 



