682 APPENDIX 



all ? (3) What are the objects of having different kinds of averages ? 

 Such questions as these are apt to be overlooked by those who have formed 

 the habit of averaging all kinds of results without careful thought. 



In popular language, we speak of the average daily temperature, the 

 average length of ears of corn, the average student, the average citizen ; 

 and we should know the exact meaning to be conveyed by these and simi- 

 lar expressions before using them in scientific discourse. 



The taking of an average presupposes a population whose variates have 

 a certain measurable character about which we are concerned, and that 

 the measurement of this character differs in different individuals. We 

 attempt to describe this population by putting aside the measurements of 

 individuals and constructing a single intermediate number which shall be 

 descriptive of the total population, in so far as one number can describe 

 a population. 



The single intermediate number which answers this purpose is, in the 

 general sense, some kind of an average. We thus use averages for descrip- 

 tive purposes in the interest of brevity ; but, taken alone, an average can- 

 not completely describe a population any more than the motion of the 

 center of gravity of a system of material particles can completely describe 

 the motion of the separate particles. 



In stating what an average is we have also stated its function ; but, as 

 just indicated, it must not be assumed that an average gives all the infor- 

 mation which is to be secured from the measurement of a population. It 

 can only take the place of the mass of figures for certain special purposes. 

 In fact, there has been a tendency, by somewhat careless workers with 

 statistical data, to attach too much importance to averages and not enough 

 to deviations from the average, a point that will be dealt with in a later 

 section. 



There are five different kinds of averages in common use for different 

 purposes. These are (i) the arithmetic mean, (2) the weighted arithmetic 

 mean, (3) the geometric mean, (4) the mode, (5) the median. 



While some of these averages have been defined, and used freely in the 

 text, it seems well to restate these definitions together with the others, the 

 better to discuss their respective advantages and disadvantages, and some 

 of the purposes to which each is adapted. 



The arithmetic mean. The arithmetic mean of a population of n variates 

 may be defined as follows : 



sum of measurement of n variates 



arithmetic mean = 



n 



That is, to find the arithmetic mean of n variates, we divide the sum of the 

 measurements of these variates by the number of variates. 



Thus, in the case of a thousand ears of corn, the arithmetic mean of the 

 lengths of the ears is the sum of the lengths of 1000 ears divided by 1000. 

 The use of this kind of an average has always been taken by observers as the 

 best method of combining direct measurements of the same quantity. This is 



