for Regression, fyc., in the case of Skew Correlation. 483 



^ = ^ 2 = ^=, . ;=f- 1 (8)v 



2/1 2/3 2/3 "2 



the sigh of the last term depending on the sign of r. Hence the 

 statement that two variables are "perfectly correlated "implies that 

 relation (8) holds good, or that all pairs of deviations bear the same 

 ratio to one another. It follows that in correlation, where the means 

 of arrays are not collinear, or the deviation of the mean of the array 

 is not a linear function of the deviation of the type, r can never be 

 unity, though we know from experience that it can approach pretty 

 closely to that value. If the regression be very far from linear, some 

 caution must evidently be used in employing r to compare two diffe- 

 rent distributions. 



In the case of normal correlation, o-^l r 2 is the standard devia- 

 tion of any array of the x variables, corresponding to a single type of 

 2/'s. <r a ^T r 8 is similarly the standard deviation of any array of 

 the y variables, corresponding to a single type of o>'s. In the general 

 case, the first expression may be interpreted as the mean standard 

 deviation of the ^-arrays from the line of regression, and the second 

 expression as the mean standard deviation of the y-arrays from the 

 line of regression. Otherwise we may regard 



r 2 

 as the standard error made in estimating x from the relation 



x = %, 

 and 



as the standard error made in estimating y from the relation 



y = M, 



these interpretations being independent of the form of the correla- 

 tion. 



(2.) Case of Three Variables. 



Let the three correlated variables be Xj, X 2 , X 3 , and let # l5 a? 2 , #3 

 denote deviations of these variables from their respective means. Let 

 us write, for brevity, 



NV, S(> 2 a ) = 



2 o 2 



