THE STATISTICAL STUDY OF EVOLUTION. 449 



])lu'ii()iiu'iion must make use of it. But in measurements made on a 

 series of distinct individuals the average does not signify the value 

 nearest tlu^ tnith, and we cannot infer that it is the most significant 

 single representative of the series. The average has this disadvantage, 

 moreover, that the introduction of a few very extraordinary individuals 

 has undue influence on the result. Thus in calculating the average 

 income of American colleges, one institution with an income of $1,200,- 

 000 increases the average by an amount ($2,500) equal to the total 

 income of about 5 per cent, of the 'colleges and universities.' The 

 average has indeed been over-rated and over-used, as though it were 

 always the best single representative of a series; whereas there are 

 other representatives wdiich are sometimes superior. Among these is 

 the middle value, Avhich is usually got without much calculation. It 



Fig. 1. File of Forty University of Chicago Students arranged (APPnoxiMATELV) in ur- 



DEK OF Height. 



is the value above and below which fifty per cent, of the cases lie. In 

 the case of income of American colleges the middle value is not far 

 from $15,000, while the average is $43,000; the former amount un- 

 fortunately gives the truer idea of the usual American college; for 

 about 80 per cent, of the colleges have an income of less than $43,000 

 Still another rei^resentative value is the geometric mean which is 

 especially important in many biologic and economic statistics. The 

 geometric mean is the number corresponding to the average of the 

 logarithms of the individual quantities. 



Finally, if there is one representative of a biological series that is 

 more apt to be significant than any other, it is tlie value that occurs 

 with the greatest frequency or, in other words, the commonest value. 

 Since this value may be said figuratively to be the most fashionable one, 



VOL. LIX. 31 



