684 APPENDIX 



Let us assume that during a decade the attendance at a university in- 

 creased 100 per cent, and let us propose the problem of rinding the average 

 annual rate of increase. Will it do to resort to the arithmetic mean in 

 this case and say that the average rate of increase is 10 per cent ? No ; an 

 increase of 10 per cent annually would give an attendance (i.io) 10 = 2.59 

 times the attendance at the beginning of the decade. What we really want 

 is ^2~= 1.07 + ; that is, an increase of a little more than 7 per cent each 

 year will double the population in a decade. 



The geometrical average is but little used in our work, but it is brought 

 forward here to remind us that an average can, in general, be depended 

 upon only to serve a definite purpose ; and, when the purpose is known, we 

 are sometimes confined to one kind of average, or at least able to see the 

 advantage of one kind of average over another. Suppose that we know the 

 protein content of corn to have been increased 50 per cent in ten years' 

 breeding. Can we say that the average annual rate of increase was 5 per 

 cent ? Clearly we cannot. What we should do is to take 



^1.50 i .00 = 0.041 



and say that the average annual rate of increase is approximately 4 per cent. 



The mode. When we speak of the average college student or the aver- 

 age citizen we certainly do not have reference to the arithmetic or geo- 

 metric average of anything. When we say a man is an average citizen 

 we mean that he represents a type which is met oftener than any other. 



If a community has ten millionaires, but all the other citizens are in pov- 

 erty, an arithmetical average might give the impression that the people of 

 the community are in good financial condition, while really the " average 

 citizen " is in poverty. The averages thus far discussed are in no way 

 suited to describe this population, but the average called the " mode " is 

 useful for this purpose. 



If a population be arranged in seriate order with respect to some char- 

 acter, a mode is a value to which there corresponds a greater frequency 

 than to values just preceding and immediately following it in the arrange- 

 ment. A population may have more than one mode, but the populations 

 with which we shall deal have, in general, only one. 



This kind of average seems to be about the same as that of the news- 

 papers when they speak of the average citizen. In a democracy we often 

 hear the cry of " the greatest good for the greatest number," and insist that 

 legislation shall benefit the average man, the man at the mode. 



Reverting again to the thousand ears of corn arranged in half-inch groups, 

 it should be noted that the frequency increases up to the class of mark 

 9 inches and then decreases. We might conclude that 9 inches is exactly 

 the mode for this population. It must be remembered that all measure- 

 ments from 8.75 to 9.25 inches were placed in the 9-inch group, and that 

 a different grouping might change the frequencies somewhat. Hence 9 is 

 said to be the empirical mode, and the theoretical mode is defined as a 



