494 Prof. Karl Pearson. 



case in which the coefficients of variation are all the same, p = 0'5. 

 When the absolute sizes of organs are very feebly correlated, then 

 in most cases there will be a considerable correlation of indices. 



Example (a). Suppose three organs, x\, # 2 , and x 3 to have sensibly 

 equal coefficients of variation, and that the correlation of x\ and Xz = 

 r J2 = r and of #1 and X A , as well as of x 2 and x 3 = r. 



Then: 



= O'5 + O'o 



l-r 1 



This formula illustrates well in a specially simple case how the 

 correlation in the indices diverges from the spurious value 0*5, as we 

 alter r and r' from zero, i.e., as we introduce organic correlation. 

 According as r, the correlation of the numerators, is greater or less 

 than r', the correlation of the numerator with the denominator, the 

 actual index correlation can be greater or less than the spurious 

 value. 



Example (6). If e\, z 2 be the indices, then in the case of normal 

 correlation the contour lines of the correlation surface for the indices 

 are given by 



= constant - 



where 2j, Sa, and p are given by (iii) and (iv) above. 



The contour lines of a surface of spurious index correlation are 

 given by 



= constant, 



while the uncorrelated distribution of the numerators a? t and x z is 

 given by the contours, 



x\l<r\+x*l<r% = constant. 



We are thus able to mark the growth of the spurious correlation 

 as we increase v z from zero ; we see the axes of the ellipses diminishing 

 and their directions beginning to rotate. 



Example (c). To find the spurious correlation between the two chief 

 cephalic indices. 



I have calculated the following results from the measurements 

 made on 100 " Altbayerisch " -$ skulls, by Professor J. E/anke. See 

 his ' Anthropologie der Bayern,' Bd. i, Kapifcel v, S. 194. 



