144 Co-relations and their Measurement. [Dec. 20, 



found to be two-thirds of the deviation of the mid-parents, while the 

 mean deviation in inches of the mid-parent was one-third of the devia- 

 tion of the sons. Here the regression, when calculated in Q units, is in 



1 2 J- 



the first case from ,-7^ to ~Xl'7 = 1 to O47, and in the second case 

 .1 ' o 



from r^ to ~ X ,-^ = 1 to 0'44, which is practically the same. 

 J.' / o L' u 



The rationale of all this will be found discussed in the paper on 

 " Hereditary Stature," to which reference has already been made, and 

 in the appendix to it by Mr. J. D. Hamilton Dickson. The entries in 

 any table, such as Table II, may be looked upon as the values of 

 the vertical ordinates to a surface of frequency, whose mathematical 

 properties were discussed in the above-mentioned appendix, there- 

 fore I need not repeat them here. But there is always room for 

 legitimate doubt whether conclusions based on the strict properties of 

 the ideal law of error would be sufficiently correct to be serviceable in 

 actual cases of co-relation between variables that conform only 

 approximately to that law. It is therefore exceedingly desirable to 

 put the theoretical conclusions to frequent test, as has been done with 

 these anthropometric data. The result is that anthropologists may 

 now have much less hesitation than before, in availing themselves of 

 the properties of the law of frequency of error. 



I have given in Table V a column headed v/(l r 3 ) = /. The 

 meaning of / is explained in the paper on " Hereditary Stature." It is 

 the Q value of the distribution of any system of x values, as x lt x%, x$, 

 &c., round the mean of all of them, which we may call X. The 

 knowledge of / enables dotted lines to be drawn, as in the figure above, 

 parallel to the line of M values, between which one half of the tc 

 observations, for each value of T/, will be included. This value of / 

 has much anthropological interest of its own, especially in connexion 

 with M. Bertillon's system of anthropometric identification, to which 

 I will not call attention now. 



It is not necessary to extend the list of examples to show how to 

 measure the degree in which one variable may be co-related with the 

 combined effect of n other variables, whether these be themselves 

 co-related or not. To do so, we begin by reducing each measure into 

 others, each having the Q of its own system for a unit. We thus 

 obtain a set of values that can be treated exactly in the same way 

 as the measures of a single variable were treated in Tables II and 

 onwards. Neither is it necessary to give examples of a method 

 by which the degree may be measured, in which the variables in a 

 series each member of which is the summed effect of n variables, 

 may be modified by their partial co-relation. After transmuting the 

 separate measures as above, and then summing them, we should find 

 the probable error of any one of them to be </n if the variables were 



