Skewness
Quantile-based measures

Bowley's measure of skewness (from 1901), also called Yule's coefficient (from 1912) is defined as:

where Q is the quantile function (i.e., the inverse of the cumulative distribution function). The numerator is difference between the average of the upper and lower quartiles (a measure of location) and the median (another measure of location), while the denominator is the semi-interquartile range (Q (3 / 4) − Q (1 / 4)) / 2 {\displaystyle ({Q}(3/4)}-{{Q}(1/4))/2}, which for symmetric distributions is equal to the MAD measure of dispersion.

Other names for this measure are Galton's measure of skewness, the Yule–Kendall index and the quartile skewness,

Similarly, Kelly's measure of skewness is defined as

A more general formulation of a skewness function was described by Groeneveld, R. A. and Meeden, G. (1984):

The function γ (u) satisfies −1 ≤ γ (u) ≤ 1 and is well defined without requiring the existence of any moments of the distribution. Bowley's measure of skewness is γ(u) evaluated at u = 3/4 while Kelly's measure of skewness is γ(u) evaluated at u = 9/10. This definition leads to a corresponding overall measure of skewness defined as the supremum of this over the range 1/2 ≤ u < 1. Another measure can be obtained by integrating the numerator and denominator of this expression.

Quantile-based skewness measures are at first glance easy to interpret, but they often show significantly larger sample variations than moment-based methods. This means that often samples from a symmetric distribution (like the uniform distribution) have a large quantile-based skewness, just by chance.
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