Page:Proceedings of the Royal Society of London Vol 60.djvu/529

494 case in which the coefficients of variation are all the same, = 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, xu and x3 to have sensibly equal coefficients of variation, and that the correlation of and x2 = rJ2 = r and of xx and x3,as well as of x2 and = r.

Then: P 0-5 X l i l — 1— — 0*5-f0*5 r —r' 1^7'*

This formula illustrates well in a specially simple case how the correlation in the indices diverges from the spurious value 05, as we alter ? and / from zero, i.e.t 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 (b). If zh z2 be the indices, then in the case of normal correlation the contour lines of the correlation surface for the indices are given by _fL____ 2PZi** z? 2^(1 — />2) + “ consfcant>

where Sj, S2, and p are given by (iii) and (iv) above. The contour lines of a surface of spurious index correlation are given by £l

while the uncorrelated distribution of the numerators xx and x2 is given by the contours, %ifai + x2\ <j2 = constant.

We are thus able to mark the growth of the spurious correlation as we increase v3 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 ” 3 skulls, by Professor J. Ranke. See his ‘ Anthropologie der Bayern,’ Bd. i, Kapitel v, S. 194.