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 comprehensible
 : 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 consider
 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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