Mathematical Contributions to the Theory of Evolution. 345 



is thus impossible to lay much stress on any individual relative 

 difference ; we are bound to consider only the general trend of our 

 results. In order, however, to make the probable errors as small as 

 possible it is desirable in the case of correlation to make the coeffi- 

 cients as large as possible. Now, for measurements on the skull, 

 the correlation is small, but, for the long bones, it is large. Hence, 

 the latter will give better results for comparative pnrposes, if the 

 series be smaller than the former. They have accordingly been 

 selected for the present investigation. In all cases the standard 

 deviations, coefficients of variation, and coefficients of correlation 

 have been calculated zuithout grouping by means of the formula? given 

 in a memoir on " Regression, Heredity, and Panmixia."* The 

 standard deviations and coefficients of correlation had their errors 

 determined by the formulae of the same memoir (pp. 266 and 275) ; 

 the coefficients of variation have for probable error, if n be the 

 number of measurements in the series, and V the coefficient, 



or sensibly, 0*6745 , since v/100 is of the order 0*05, for our pre- 

 sent investigations. It will thus be seen that the probable error of 

 a coefficient of variation can, like that of a standard deviation, 

 amount to a considerable percentage of its value if n be small. For 

 example, if n — 23 (case of the Aino women) it can amount to about 

 10 per cent. ; while for n = 50 (case of the French men or women) it 

 will amount to 6*7 per cent. The probable errors of the standard 

 deviations will also in the two cases amount to 10 and 6" 7 per cent, 

 respectively. On the other hand, if n be as small as 23, the pro- 

 bable error of the coefficent of correlation will only be 2*2 per cent, 

 of its value if r be as large as 0*9, and only 6*56 per cent, of its 

 value if r = 0*75. On the other hand for n — 50 the probable errors 

 of r for the two cases of r — 0'9 and 0*75 are respectively 1*5 and 4*45 

 per cent. It thus appears that if correlation be large, or even fairly 

 large, it can be determined with far less probable error than varia- 

 tion. The results for correlation are thus likely in the present case 

 to be more significant than those for variation. 



3. As to the material itself the French measurements are taken 

 from a work by Dr. E. Rollet: ' De la Mensuration des Os longs 

 des Membres,' published in the ' Bibliotheque d'Anthropologie 

 Criminelle,' Lyons, 1889. The bodies from which the bones were 

 taken having passed through the post-mortem room, the sex was 

 known without question. The right member was always taken 



v 



* ' Phil. Trans.,' A, vol. 189 pp. 265 and 277. 

 VOL. LXI. 2 B 



