Mathematical Contributions to the Theory of Evolution. 491 



lute lengths j 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 correla- 

 tion of the imp-assorted, but probably, from the vital standpoint, 

 impossible beings was " organic." 



As a last illustration, suppose 1000' skeletons obtained by distribut- 

 ing 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 photo- 

 graphs 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 pho- 

 tographs. 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 f emur/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 rela- 

 tive 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; Proposition L To find the mean of an index in terms of the 

 means, coefficients of variation, and coefficient of correlation of the two 

 absolute measurements.* 



Let a?!, a^, a? 3 , a? 4 be the absolute sizes of any four correlated organs ; 

 m lt Wz, Wa, m 4 their mean values ; <?i, <r 2 , <r 3 , <r 4 their standard deviations ; 



* In all that follows, unless otherwise stated, the correlation may be of any kind 

 whatever, Le., the frequencies are not supposed to follow , the Gaussian or normal 

 law of error. 



