VIL—AVERAGES. 119 
involve the crude assumption that the frequency is uniformly 
distributed over the interval in which the median lies. 
17. A comparison of the calculations for the mean and 
for the median respectively will show that on the score of 
brevity of calculation the median has a distinct advantage. 
When, however, the ease of algebraical treatment of the two 
forms of average is compared, the superiority lies wholly on 
the side of the mean. As was shown in § 13, when several series 
of observations are combined into a single series, the mean of 
the resultant distribution can be simply expressed in terms 
of the means of the components. The expression of the 
median of the resultant distribution in terms of the medians 
of the components is, however, not merely complex and difficult, 
but impossible: the value of the resultant median depends on 
the forms of the component distributions, and not on their 
medians alone. If two symmetrical distributions of the same 
form and with the same numbers of observations, but with 
different medians, be combined, the resultant median must 
evidently (from symmetry) coincide with the resultant mean, z.e. 
lie halfway between the means of the components. But if the 
two components be asymmetrical, or (whatever their form) 
if the degrees of dispersion or numbers of observations in the 
two series be different, the resultant median will not coincide 
with the resultant mean, nor with any other simply assignable 
value. It is impossible, therefore, to give any theorem for 
medians analogous to equations (5) and (6) for means. It is 
equally impossible to give any theorem analogous to equations 
(8) and (9) of § 13. The median of the sum or difference of 
pairs of corresponding observations in two series is not, 
in general, equal to the sum or difference of the medians of 
the two series ; the median value of a measurement subject to 
error is not necessarily identical with the true median, even 
if the median error be zero, ¢.e. if positive and negative errors 
be equally frequent. 
18. These limitations render the applications of the median in 
any work in which theoretical considerations are necessary com- 
paratively circumscribed. On the other hand, the median may 
have an advantage over the mean for special reasons. (a) It is 
very readily calculated ; a factor to which, however, as already 
stated, too much weight ought not to be attached. (&) It is 
readily obtained, without the necessity of measuring all the 
objects to be observed, in any case in which they can be arranged 
by eye in order of magnitude. If, for instance, a number of men 
be ranked in order of stature, the stature of the middlemost is 
the median, and he alone need be measured. (On the other hand