Editing Skewness (section)

===Quantile-based measures===

Bowley's measure of skewness (from 1901),<ref>Bowley, A. L. (1901). Elements of Statistics, P.S. King & Son, Laondon.  Or in a later edition: BOWLEY, AL. "Elements of Statistics, 4th Edn (New York, Charles Scribner)."(1920).</ref><ref name=Kenney1962>Kenney JF and Keeping ES (1962) ''Mathematics of Statistics, Pt. 1, 3rd ed.'', Van Nostrand, (page 102).</ref> also called '''Yule's coefficient''' (from 1912)<ref>Yule, George Udny. An introduction to the theory of statistics. C. Griffin, limited, 1912.</ref><ref>{{cite journal | last1 = Groeneveld | first1 = Richard A | year = 1991 | title = An influence function approach to describing the skewness of a distribution | journal = The American Statistician | volume = 45 | issue = 2| pages = 97–102 | doi = 10.2307/2684367 | jstor = 2684367 }}</ref> is defined as: 
<math display="block">\frac{\frac{Q(3/4) + Q(1/4)}{2} - Q(1/2)}{\frac{Q(3/4) - Q(1/4)}{2}}
= \frac{Q(3/4) + Q(1/4) - 2 Q(1/2)}{Q(3/4) - Q(1/4)},</math>
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]] <math>(Q(3/4)}-{Q(1/4))/2</math>, which for symmetric distributions is equal to the [[Average absolute deviation|MAD]] measure of [[statistical dispersion|dispersion]].{{citation needed|date=May 2024}}

Other names for this measure are Galton's measure of skewness,<ref name=Johnson1994>{{harvp|Johnson, NL|Kotz, S|Balakrishnan, N|1994}} p. 3 and p. 40</ref> the Yule–Kendall index<ref name=Wilks1995>Wilks DS (1995) ''Statistical Methods in the Atmospheric Sciences'', p 27. Academic Press. {{isbn|0-12-751965-3}}</ref> and the quartile skewness,<ref>{{Cite web|url=http://mathworld.wolfram.com/Skewness.html|title=Skewness|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-21}}</ref>

Similarly, Kelly's measure of skewness is defined as<ref>{{cite web |url=http://www.math.ruh.ac.lk/~pubudu/app5.pdf |title=Applied Statistics I: Chapter 5: Measures of skewness |author=A.W.L. Pubudu Thilan |website=University of Ruhuna |page=21}}</ref>
<math display="block">\frac{Q(9/10) + Q(1/10) - 2 Q(1/2)}{Q(9/10) - Q(1/10)}.</math>

A more general formulation of a skewness function was described by Groeneveld, R. A. and Meeden, G. (1984):<ref name=Groeneveld1984>{{Cite journal | doi = 10.2307/2987742 | last1 = Groeneveld | first1 = R.A. | last2 = Meeden | first2 = G. | year = 1984 | title = Measuring Skewness and Kurtosis | jstor = 2987742| journal = The Statistician | volume = 33 | issue = 4| pages = 391–399 }}</ref><ref name=MacGillivray1992>MacGillivray (1992)</ref><ref name=Hinkley1975>Hinkley DV (1975) "On power transformations to symmetry", ''[[Biometrika]]'', 62, 101–111</ref>
<math display="block"> \gamma( u ) = \frac{ Q( u ) +Q( 1 - u )-2Q( 1 / 2 ) }{Q( u ) -Q( 1 - u ) } </math>
The function {{math|''γ''(''u'')}} satisfies {{math|−1 ≤ ''γ''(''u'') ≤ 1}} and is well defined without requiring the existence of any moments of the distribution.<ref name=Groeneveld1984/> Bowley's measure of skewness is {{math|''γ''(''u'')}} evaluated at {{math|1=''u'' = 3/4}} while Kelly's measure of skewness is {{math|''γ''(''u'')}} evaluated at {{math|1=''u'' = 9/10}}. This definition leads to a corresponding overall measure of skewness<ref name=MacGillivray1992>MacGillivray (1992)</ref> defined as the [[supremum]] of this over the range {{math|1/2 ≤ ''u'' < 1}}. Another measure can be obtained by integrating the numerator and denominator of this expression.<ref name=Groeneveld1984/> 

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.