24 BELL SYSTEM TECHNICAL JOURNAL 



to be controlled by the manufacturing process, and N is the number of 

 observations. Now, if errors of observation are made in determining 

 X and y, the observed correlation coefficient r^oyo is known to be given 

 by the expression 



r«„„ = ^^-. (10) 



where o-xo= v o-a + <^^^. and (jy^-=\/ Oy-^-ay^j 



Cxe and cTvE being the root mean square errors of observation of x and 

 y respectively. 



Attention is directed to Equation (10) which shows that the observed 

 correlation coefficient r^oyo is always less than the true correlation co- 

 efficient rxy irrespective of the number of observations made. Ob- 

 \iously, this point is of considerable commercial importance as we 

 shall now see. 



If the observed correlation is small, we customarily assume that 

 there is little need of trying to control the quality X by controlling the 

 manufacturing factor Y, whereas this conclusion cannot be justified 

 unless it can be shown that the true correlation has not been masked 

 by the errors of measurement. 



This point has had to be taken into account in the development 

 of machine methods for testing transmitters and receivers, because 

 the calibration curves of the machines in terms of ear-voice tests 

 depend upon the correlation coefficient. 



Appendix I 



It may be of some interest to certain readers to note that the results 

 given in Equations (4) and (7) can also be obtained in the following 

 way by the method of moments so often used in statistical investi- 

 gations. 



Assuming that/r(A'+.\-) is expansible in terms of a Taylor's series, 



we get 



MX) =/r(X)+^/'/(A) + | j- (^) V'(A') + 



l(^)VrW+.... (11) 



If we substitute a normal form for /r(A) in Equation (11) and 

 solve for the moments of/o(A), we find that the odd moments are zero 



