lute lengths ; these would now exhibit no organic correlation, but the indices calculated from this random distribution would have a, correlation nearly as high, if not in some cases higher than before. The biologist would be not unlikely to argue that the index correlation of the imp-assorted, but probably, from the vital standpoint, impossible beings was “ organic.”
As a lq,st illustration, suppose 1000 skeletons obtained by distributing component bones at random. Between none of their bones will these individuals exhibit correlation. Wire the spurious skeletons together and photograph them all, so that their stature in the photographs is the same; the series of photographs, if measured, will show correlation between their parts. It seems to me that the biologist who reduces the parts of an animal to fractions of some one length measured upon it is dealing with a series very much like these photographs. A part of the correlation he discovers between organs is undoubtedly organic, but another part is solely due to the nature of his arithmetic, and as a measure of organic relationship is spurious.
Returning to our problem of the randomly distributed bones, let us suppose the indices femur/humerus and tibia/humerus to have a correlation of 0*45. Now suppose successively 1, 2, 3, 4, &c., per cent, of the bones are assorted in their true groupings, then begins the true organic correlating of the bones. It starts from 0’45, and will alter gradually until 100 per cent, of the bones are truly grouped. The final value may be greater or less than 0‘45, but it would seem that 0*45 is a more correct point to measure the organic correlation from than zero. At any rate it appears fairly certain that if a biologist recognised that a perfectly random selection of organs would still lead to a correlation of organ-indices, he would be unlikely to accept index-correlation as a fair measure of the relative intensity of correlation between organs. I shall accordingly define spurious organic correlation as the correlation which will be found between indices, when the absolute values of the organs have been selected purely at random. In estimating relative correlation by the hitherto usual measurement of indices, it seems to me that a statement of the amount of spurious correlation ought always to be made. 2
(2) Proposition I.—To find the mean of an index in terms of the means, coefficients of variton, and coefficient of correlation of the two absolute measurements.*
Let X\, x?, 0%, a?4 be the absolute sizes of any four correlated organs ; Wj, m2, m3, m4 their mean values; <r1} <r2, <x3, <r4 their standard deviations ;
- In all that follows, unless otherwise stated, the correlation may be of any kind
whatever, i.e., the frequencies are not supposed to follow the G-aussian or normal law of error.
