NOV. 19, 1921 REED : CORRELATION BETWEEN FUNCTIONS 451 



Equations (1), (2) and (3) furnish a complete set of formulae for 

 the determination of the means, standard deviations, and coefficient 

 of correlation of any two functions of two sets of variables in terms 

 of the means, standard deviations, and coefficients of correlation 

 of the variables themselves. 



It should be noted that, in the derivation of these formulae, all 



r 1 • Ci (To 



terms of the third or higher orders in the variables — .-^' etc., were 



nil mi 



disregarded. Therefore in a practical problem when the standard 

 deviation is large as compared with the mean the formula might not 

 give a sufficiently close approximation to the true value of the cor- 

 relation coefficient. In the majority of cases, however, the ratio 

 of the standard deviation to the mean is a sufficiently small decimal 

 so that disregarding its powers higher than the second can have no 

 appreciable effect on the result. 



To illustrate the use of equation (1) we may first apply it to the 

 case of the correlation between ratios. Let the variables be Xi, %i 



and x[, x'l and the ratios be >- = — and 2 = — . 



X<i X-i 



1 



Then 



Substituting these values in (1) and replacing "=— by ^i, — by i^i, etc., 



Wi W2 



we have 



ry:.= (4) 



This is Pearson's^ well known formula for the coefficient of cor- 

 relation of two indices or ratios. Formulae for the means and standard 

 deviations of y and z can be obtained from equations (2) and (3). 



They are m = — [1 -f 1-2^ - rriX2^i^2] 

 nu 



and o-y = — V i'r + i'2- - 2r x^x^x 112 

 mi 



' Pearson, K. On a form of spurious correlation which may arise when indices are used 

 in the measurement of organs. Proc. Roy. Soc, London 60. 1896. 



