JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. 11 November 19, 1921 No. 19 



MATHEMATICS.- — On the correlation between any two functions and 

 its application to the general case of spurious correlation.'^ TowELiv 

 J. Reed, Johns Hopkins University. (Communicated by Ray- 

 mond Pearl.) 

 In problems where correlation methods are employed it is often 

 necessary to determine the coefficient of correlation between one of 

 the measured variables and some function of the others. In certain 

 cases we need to go still further and determine the correlation between 

 two different functions of the measured variables. Such cases arise 

 where two different index numbers are correlated with each other 

 or when an index number is correlated with one of the measured 

 variables. In all problems of this type the coefficient of correlation 

 may be found by computing the value of the function in question 

 at each position for which the values of the variables themselves 

 are known and then finding the correlation coefficient in the usual 

 way. In a great many cases however there would be a considerable 

 saving of labor if this coefficient could be determined directly from 

 the means, standard deviations, and first order correlation coefficients 

 of the variables themselves. The following general equation has 

 been derived to accomplish this and it should prove to be of use in 

 problems of the type outlined above. The proof of the formula 

 is too long to be given in the present paper, but will be published 

 later, together w4th additional illustrations of its application. 



Let Xi, X2, Xn be a set of n variables, 



and Xi, 00^, x^ be a second set of k variables. 



Let Wi, Wo, m„ be the means of the variables of the first 



set, 



0-1, 0-2, (Tn be their standard deviations, 



and rxix,, fxiXz >'xn-ixn be the coefficients of correlation between 



these variables taken in pairs. 



1 Papers from the Department of Biometry and Vital Statistics, School of Hygiene, 

 Johns Hopkins University, No. 38. Received October 18, 1921. 



449 



