VIL.—AVERAGES. 117
such that the frequencies in excess and defect are equal.
In the case of the buttercups of Table XIV. (Chap. VI. § 15)
there is no number of petals that even remotely fulfils the
required condition. An analogous difficulty may arise, it may
be remarked, even in the case of an odd number of observations
of a continuous variable if the number of observations be small
and several of the observed values identical. The median is
therefore a form of average of most uncertain meaning in cases
of strictly discontinuous variation, for it may be exceeded by
5, 10, 15, or 20 per cent. only of the observed values, instead of
by 50 per cent.: its use in such cases is to be deprecated, and
is perhaps best avoided in any case, whether the variation be
continuous or discontinuous, in which small series of observations
have to be dealt with.
15. When a table showing the frequency-distribution for a
long series of observations of a continuous variable is given, no
difficulty arises, as a sufficiently approximate value of the median
can be readily determined by simple interpolation on the hypo-
thesis that the values in each class are uniformly distributed
throughout the interval. Thus, taking the figures in our first
illustration of the method of calculating the mean, the total
number of observations (registration districts) is 632, of which
the half is 316. Looking down the table, we see that there are
227 districts with not more than 2-75 per cent. of the population
in receipt of relief, and 100 more with between 2-75 and 3-25
per cent. But only 89 are required to make up the total of 316 :
bence the value of the median is taken as
276 + on. §=2T5 + 0-445
+100 2= +
=3'195 per cent.
The mean being 3:29, the median is slightly less ; its position
is indicated by M7 in fig. 21.
The value of the median stature of males may be similarly
calculated from the data of the second illustration. The work
may be indicated thus: —
Half the total number of observations (8585) =4292'5
Total frequency under 661% inches . . =3589
Difference . = 7035
Frequency in next interval . =1329
a Miguniass
Therefore median = 6615 + 1399
= 6747 inches.
