THEORY OF STATISTICS. 
4. (a) In the first place, it almost goes without saying that an 
average should be rigidly defined, and not left to the mere estimation 
of the observer. An average that was merely estimated would 
depend too largely on the observer as well as the data. (b) An 
average should be based on all the observations made. If not, 
it is not really a characteristic of the whole distribution. (c) It 
is desirable that the average should possess some simple and 
obvious properties to render its general nature readily compre- 
hensible : an average should not be of too abstract a mathematical 
character. (d) It is, of course, desirable that an average should 
be calculated with reasonable ease and rapidity. Other things 
being equal, the easier calculated is the better of two forms of 
average. At the same time too great weight must not be attached 
to mere ease of calculation, to the neglect of other factors. (e) 
It is desirable that the average should be as little affected as 
may be possible by what we have termed fluctuations of sampling. 
If different samples be drawn from the same material, however 
carefully they may be taken, the averages of the different samples 
will rarely be quite the same, but one form of average may show 
much greater differences than another. Of the two forms, the 
more stable is the better. The full discussion of this condition 
must, however, be postponed ‘to a later section of this work 
(Chap. XVIL). (f) Finally, by far the most important desideratum 
is this, that the measure chosen shall lend itself readily to 
algebraical treatment. If, e.g., two or more series of observations 
on similar material are given, the average of the combined series 
should be readily expressed in terms of the averages of the 
component series : if a variable may be ‘expressed as the sum of 
two or more others, the average of the whole should be readily 
expressed in terms of the averages of its parts. A measure for 
which simple relations of this kind cannot be readily determined 
is likely to prove of somewhat limited application. 
5. There are three forms of average in common use, the 
arithmetic mean, the median, and the mode, the first named being 
by far the most widely used in general statistical work. To 
these may be added the geometric mean and the harmonic mean, 
more rarely used, but of service in special cases. We will con- 
sider these in the order named. 
6. The arithmetic mean.—The arithmetic mean of a series of 
values of a variable X;, X,, X;, . .. X,, & in number, is the 
quotient of the sum of the values by their number. That is to 
say, if M be the arithmetic mean, 
M= 3 (X; + Xt Xt itis: Fn) 
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