VII.—MEASURES OF DISPERSION, ETC. 147
We pointed out in § 10 that in distributions of the simple forms
referred to, a range of six times the standard deviation contains
over 99 per cent. of all the observations. If the mean deviation
be employed as the measure of dispersion, we must substitute a
range of 71 times this measure.
20. The Quartile Deviation or Semi-interquartile Range.—1If a
value @, of the variable be determined of such magnitude that
one-quarter of all the values observed are less than ¢, and three-
quarters greater, then , is termed the lower quartile. Similarly,
if a value ; be determined such that three-quarters of all the
values observed are less than @, and one-quarter only greater,
then (J), is termed the upper quartile. The two quartiles and the
median divide the observed values of the variable into four
classes of equal frequency. If M7 be the value of the median, in
a symmetrical distribution
Me —- Q,=Q, - I,
and the difference may be taken as a measure of dispersion. But
as no distribution is rigidly symmetrical, it is usual to take as the
measure
0-924,
and @ is termed the quartile deviation, or better, the semi-
interquartile range—it is not a measure of the deviation from
any particular average: the old name probable error should be
confined to the theory of sampling (Chap. XV. § 17).
21. In the case of a short series of ungrouped observations
the quartiles are determined, like the median, by inspection.
In the wage statistics of Example i., for instance, there are
38 observations, and 38/4=9'5: What is the lower quartile ?
The student may be tempted to take it halfway between the
ninth and tenth observations from the bottom of the list;
but this would be wrong, for then there would be nine
observations only below the value chosen instead of 95. The
quartile must be taken as given by the tenth observation
itself, which may be regarded as divided by the quartile, and
falling half above it and half below. Therefore
Lower quartile @Q, = 14s. 10d.
Upper quartile Q,= 16s. 11d.
Q;-¢
and Q= Lgl = 12-54,
22. In the case of a grouped distribution, the quartiles, like
the median, are determined by simple arithmetical or by
-
