THEORY OF STATISTICS. 
difference to some quantity of the same dimensions, e.g. the semi- 
interquartile range. Our measure would then be, taking the 
skewness to be positive if the longer tail of the distribution runs 
in the direction of high values of JX, 
skownoss = (Gam 20 = (hi ~@)_Q+Q,-208 
Q Q 
This would not be a bad measure if we were using the quartile 
deviation as a measure of dispersion : its lowest value is zero, 
when the distribution is symmetrical ; and while its highest possible 
value is 2, it would rarely in practice attain higher numerical 
values than +1. A similar measure might be based on the mean 
deviations in excess and in defect of the mean. There is, however, 
only one generally recognised measure of skewness, and that is 
Pearson’s measure (ref. 9)— 
mean — mode 
Hick standard deviation 4 (17) 
This is evidently zero for a symmetrical distribution, in which 
mode and mean coincide. No upper limit to the ratio is apparent 
from the formula, but, as a fact, the value does not exceed unity for 
frequency-distributions resembling generally the ideal distributions 
of fig. 9. As the mode is a difficult form of average to determine 
by elementary methods, it may be noted that the numerator of the 
above fraction may, in the case of frequency-distributions of the 
forms referred to, be replaced approximately by 3(mean — median), 
(¢f. Chap. VIL §20). The measure (12) is much more sensitive 
than (11) for moderate degrees of asymmetry. 
27. The Method of Percentiles.—We may conclude this chapter 
by describing briefly a method that has been largely used in the 
past in lieu of the methods dealt with in Chapters VI. and VII, 
and the preceding paragraphs of this chapter, for summarising 
such statistics as we have been considering. If the values of the 
variable (variates, as they are sometimes termed) be ranged in 
order of magnitude, and a value P of the variable be determined 
such that a percentage p of the total frequency lies below it and 
100 - p above, then P is termed a percentile. If a series of per- 
centiles be determined for short intervals, e.g. 5 per cent. or 10 
per cent., they suffice by themselves to show the general form 
of the distribution. This is Sir Francis Galton’s method of 
percentiles. The deciles, or values of the variable which divide 
the total frequency into ten equal parts, form a natural and 
convenient series of percentiles to use. The fifth decile, or value 
of the variable which has 50 per cent. of the observed values 
150