MEASUREMENTS OF TYPE AND VARIABILITY 9 



4. The Relation Between the Amount of a Central Tendency and 



the Amount of the Variability of the Group about the 



Central Tendency 



In comparing groups with respect to variability allowance must 

 be made for the fact that, in certain cases at least, the amounts of 

 the central tendency influence the amounts of the variabilities. Thus 

 the A.D. of men in weight is hundreds of times that of butterflies, 

 yet the former are of course not really a hundred times as variable. 

 Thus the A.D. of a group in a test of addition was, for trials of 40 

 seconds, 2.18 ; for trials of 80 seconds, 3.41 ; and for trials of 120 

 seconds, 5.18. It would obviously be silly if we had tested men with 

 trials of 80 seconds and women with trials of 40 seconds, and ob- 

 tained these results, to infer that men are 50 per cent, more variable 

 in ability to add than are women. 



In using the so-called coefficient of variation (proposed by Pear- 

 son) onemakes allowance for the possible influence of the central 

 tendencies' amounts by dividing through the gross variabilities each 

 by the amount of its corresponding central tendency. I have else- 

 where shown that for mental and social measurements no one such 

 rule can be always or even often right and suggested that in any 

 case a division through by the square root of the corresponding cen- 

 tral tendency is more in accord with both theory and facts. 1 



In this section enough data will be presented to practically dem- 

 onstrate both of these assertions. It is not important to investigate 

 the matter exhaustively for the very reason that no one general rule 

 for comparing groups with respect to variability can be found. All 

 that is needed is a clear enough proof of the inadequacy of the prac- 

 tice of comparing groups after dividing through the gross variabili- 

 ties by the corresponding means clear enough to stop the spread of 

 the practice and to warn readers against conclusions based on such 

 comparisons. 



If we take the arrays of y in a case where y is positively cor- 

 related with x we have a series of groups with central tendencies 

 varying from lower to higher which are selected at random so far 

 as concerns any influence on the variability except the influence of 

 the amount of the mean. The differences in variability found for 

 these arrays give, then, in connection with the differences in the 

 amounts of their central tendencies, the answer to our problem for 

 the case of comparisons of groups with respect to their variability 

 in the same trait. If we find that even in such cases there is no 

 constant relation of difference in central tendency to difference in 



1 Mental gnd Social Measurements, pp. 102-103. 



