172 



SCIENCE 



[N. S. Vol. XXXI. No. 788 



and this is one of the most important facts 

 to be borne in mind in understanding any 

 collection of individuals. It is specially 

 important in comparing different groups of 

 men, since the range of variation within 

 either group is usually much greater than 

 the difference between the averages of the 

 groups. The groups overlap to such an 

 extent that the majority of the individuals 

 composing either group might perfectly 

 well belong to the other. 



No doubt statements like this will be 

 readily accepted as far as concerns the dif- 

 ferent nations belonging to the same race. 

 One could not seriously doubt that the 

 nations of Europe, though they might dif- 

 fer slightly on the average, would so much 

 overlap one another that, except for lan- 

 guage and superficial mannerisms, the 

 great majority of the members of one 

 nation might be exchanged with a majority 

 from another nation without altering the 

 characteristics of either. But when we 

 extend our view to all the peoples of the 

 earth, the case would at first appear quite 

 changed. Certainly whites and negroes do 

 not overlap, to any extent, in color of skin, 

 nor negroes and Chinamen in kinkiness of 

 hair, nor Indians and Pygmies in stature. 

 Such specialization of traits is, however, 

 the exception. Whites and negroes, though 

 differing markedly in complexion and hair, 

 overlap very extensively in almost every 

 other trait, as, for example, in stature. 

 Even in brain weight, which would seem a 

 trait of great importance in relation to in- 

 telligence and civilization, the overlapping 

 is much more impressive than the differ- 

 ence; since while the brain of negroes 

 averages perhaps two ounces lighter than 

 the brain of Europeans, the range of varia- 

 tion within either race amounts to 25 

 ounces. 



Our inveterate love for types and sharp 

 distinctions is apt to stay with us even after 



we have become scientific, and vitiate our 

 use of statistics to such an extent that the 

 average becomes a stumbling-block rather 

 than an aid to knowledge. We desire, for 

 example, to compare the brain weights of 

 whites and of negroes. We weigh the 

 brains of a sufficient number of each race 

 —or let us at least assume the number to 

 be sufficient. When our measurements are 

 all obtained and spread before us, they 

 convey to the unaided eye no clear idea of 

 a racial difference, so much do they over- 

 lap. If they should become jumbled to- 

 gether, we should never be able to separate 

 the negroes from the whites by aid of brain 

 weight. But now we east up the average 

 of each group, and find them to differ ; and 

 though the difference is small, we straight- 

 way seize on it as the important result, and 

 announce that the negro has a smaller brain 

 than the white. We go a step further, and 

 class the white as a large-brained race, the 

 negro as a small-brained. Such transform- 

 ing of differences of degree into differences 

 of kind, and making antitheses between 

 overlapping groups, partakes not a little of 

 the ludicrous. 



We seem to be confronted by a dilemma ; 

 for the group as a whole is too unwieldy 

 to grasp, while the average, though con- 

 venient, is treacherous. What we should 

 like is some picture or measure of the dis- 

 tribution of a given trait throughout the 

 members of a group ; and, fortunately, such 

 measures and pictures can be had. Con- 

 venient and compact measures of variabil- 

 ity are afforded by the science of statistics, 

 and are of no less importance than the 

 average. But still better, because closer 

 to the actual facts, are graphic or tabular 

 pictures of the distribution of the trait, 

 showing the frequency with which it occurs 

 in each degree. The distribution of a trait 

 is for some purposes more important than 

 the average. Let us suppose, for instance, 



