Nonparametric skew

In [[statistics]] and [[probability theory]], the '''nonparametric skew''' is a [[statistic]] occasionally used with [[random variable]]s that take [[real number|real]] values.<ref name="Arnold1995">Arnold BC, Groeneveld RA (1995) Measuring skewness with respect to the mode. The American Statistician 49 (1) 34–38 DOI:10.1080/00031305.1995.10476109</ref><ref name="Rubio2012">Rubio F.J.; Steel M.F.J. (2012) "On the Marshall–Olkin transformation as a skewing mechanism". ''Computational Statistics & Data Analysis'' [http://www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/steel/steel_homepage/techrep/mosrevcsda.pdf Preprint]</ref> It is a measure of the [[skewness]] of a random variable's [[Probability distribution|distribution]]—that is, the distribution's tendency to "lean" to one side or the other of the [[mean]]. Its calculation does not require any knowledge of the form of the underlying distribution—hence the name [[Non-parametric statistics|nonparametric]]. It has some desirable properties: it is zero for any [[symmetric distribution]]; it is unaffected by a [[Scale parameter|scale]] shift; and it reveals either left- or right-skewness equally well. In some [[Sample (statistics)|statistical sample]]s it has been shown to be less [[statistical power|powerful]]<ref name="Tabor2010">Tabor J (2010) Investigating the Investigative Task: Testing for skewness - An investigation of different test statistics and their power to detect skewness. J Stat Ed 18: 1–13</ref> than the usual measures of skewness in detecting departures of the [[Population (statistics)|population]] from [[Normal distribution|normality]].<ref name="amstat">{{cite journal |title=Measuring Skewness: A Forgotten Statistic? |first1=David P. |last1=Doane |first2=Lori E. |last2=Seward |journal=Journal of Statistics Education |volume=19 |issue=2 |year=2011 |url=http://www.amstat.org/publications/jse/v19n2/doane.pdf |archive-date=2016-03-04 |access-date=2012-01-26 |archive-url=https://web.archive.org/web/20160304054803/http://www.amstat.org/publications/jse/v19n2/doane.pdf |url-status=dead }}</ref>

==Properties==

===Definition===

The nonparametric skew is defined as

: <math> S = \frac{ \mu - \nu } { \sigma } </math>

where the [[mean]] (''μ''), [[median]] (''ν'') and [[standard deviation]] (''σ'') of the population have their usual meanings.

===Properties===

The nonparametric skew is one third of the [[Skewness#Pearson's skewness coefficients|Pearson 2 skewness coefficient]] and lies between −1 and +1 for any distribution.<ref name=Hotelling1932>Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114</ref><ref name=Garver1932>Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142</ref> This range is implied by the fact that the mean lies within one standard deviation of any median.<ref name="O’Cinneide1990">O’Cinneide CA (1990) The mean is within one standard deviation of any median. Amer Statist 44, 292–293</ref>

Under an [[affine transformation]] of the variable (''X''), the value of ''S'' does not change except for a possible change in sign. In symbols

: <math> S( aX + b ) = \operatorname{sign} (a)\, S(X) </math>

where ''a'' ≠ 0 and ''b'' are constants and ''S''( ''X'' ) is the nonparametric skew of the variable ''X''.

==Sharper bounds==

The bounds of this statistic ( ±1 ) were sharpened by Majindar<ref name=Majindar1962>Majindar KN (1962) "Improved bounds on a measure of skewness". ''Annals of Mathematical Statistics'', 33, 1192–1194 {{doi|10.1214/aoms/1177704482}}</ref> who showed that its [[absolute value]] is bounded by

: <math> \frac{ 2 ( p q )^{ 1 / 2 } } { ( p + q )^{ 1 / 2 } } </math>

with

: <math> p = \Pr( X > \operatorname{ E }( X ) ) </math>

and

: <math> q = \Pr( X < \operatorname{ E }( X ) ) ,</math>

where ''X'' is a random variable with finite [[variance]], ''E''() is the expectation operator and ''Pr''() is the probability of the event occurring.

When ''p'' = ''q'' = 0.5 the absolute value of this statistic is bounded by 1. With ''p'' = 0.1 and ''p'' = 0.01, the statistic's absolute value is bounded by 0.6 and 0.199 respectively.

==Extensions==

It is also known that<ref name=Mallows1969>Mallows CCC, Richter D (1969) "Inequalities of Chebyschev type involving conditional expectations". ''Annals of Mathematical Statistics'', 40:1922–1932</ref>

: <math> | \mu - \nu_0 | \le \operatorname{E} ( | X - \nu_0 | ) \le \operatorname{ E } ( | X - \mu | ) \le \sigma ,</math>

where ''ν''0 is any median and ''E''(.) is the [[Expected value|expectation operator]].

It has been shown that : <math> \frac{ | \mu - x_q | }{ \sigma } \le \max\left( \sqrt{\frac{( 1 - q ) } { q }}, \sqrt{\frac{ q } { ( 1 - q ) } } \right)</math> where ''x''''q'' is the ''q''th [[Quantile function|quantile]].<ref name="O’Cinneide1990"/> Quantiles lie between 0 and 1: the median (the 0.5 quantile) has ''q'' = 0.5. This inequality has also been used to define a measure of skewness.<ref name=Dziubinska1996>Dziubinska R, Szynal D (1996) On functional measures of skewness. Applicationes Mathematicae 23(4) 395–403</ref>

This latter inequality has been sharpened further.<ref name=Dharmadhikari1991>Dharmadhikari SS (1991) Bounds on quantiles: a comment on O'Cinneide. The Am Statist 45: 257-58</ref>

: <math> \mu -\sigma \sqrt{ \frac{ 1-q }{ q } } \le x_q \le \mu + \sigma \sqrt{ \frac{ q }{1- q } }</math> Another extension for a distribution with a finite mean has been published:<ref name=Gilat1993>Gilat D, Hill TP(1993) Quantile-locating functions and the distance between the mean and quantiles. Statistica Neerlandica 47 (4) 279–283 DOI: 10.1111/j.1467-9574.1993.tb01424.x [http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1037&context=rgp_rsr]</ref>

: <math> \mu - \frac{ 1 }{ 2q } \operatorname{ E }| X - \mu | \le x_q \le \mu + \frac{ 1 }{ ( 2 - 2q ) } \operatorname{ E }| X - \mu |</math>

The bounds in this last pair of inequalities are attained when <math>\Pr (X=a) = q</math> and <math>\Pr (X=b) = 1-q</math> for fixed numbers ''a'' < ''b''.

===Finite samples===

For a finite sample with sample size ''n'' ≥ 2 with ''x''r is the ''r''th [[order statistic]], ''m'' the sample mean and ''s'' the [[sample standard deviation]] corrected for degrees of freedom,<ref name=David1991>David HA (1991) Mean minus median: A comment on O'Cinneide. The Am Statist 45: 257</ref>

<math> \frac{ | m - x_r | }{ s } \le \text{max}\left[ \sqrt{ \frac{ ( n - 1 )( r - 1 ) } { n ( n - r + 1 ) } } , \sqrt{ \frac{ ( n - 1 )( n - r ) }{ nr } } \right] </math>

Replacing ''r'' with ''n'' / 2 gives the result appropriate for the sample median:<ref name=Joarder2004>Joarder AH, Laradji A (2004) Some inequalities in descriptive statistics. Technical Report Series TR 321</ref>

where ''a'' is the sample median.

==Statistical tests==

Hotelling and Solomons considered the distribution of the test statistic<ref name="Hotelling1932"/>

: <math> D = \frac{ n ( m - a ) }{ s } </math>

where ''n'' is the sample size, ''m'' is the sample mean, ''a'' is the sample median and ''s'' is the sample's standard deviation.

Statistical tests of ''D'' have assumed that the null hypothesis being tested is that the distribution is symmetric .

Gastwirth estimated the asymptotic [[variance]] of ''n''−1/2''D''.<ref name=Gastwirth1971>Gastwirth JL (1971) "On the sign test for symmetry". ''[[Journal of the American Statistical Association]]'' 66:821–823</ref> If the distribution is unimodal and symmetric about 0, the asymptotic variance lies between 1/4 and 1. Assuming a conservative estimate (putting the variance equal to 1) can lead to a true level of significance well below the nominal level.

Assuming that the underlying distribution is symmetric Cabilio and Masaro have shown that the distribution of ''S'' is asymptotically normal.<ref name=Cabilio1996>Cabilio P, Masaro J (1996) "A simple test of symmetry about an unknown median". ''Canadian Journal of Statistics-Revue Canadienne De Statistique'', 24:349–361</ref> The asymptotic variance depends on the underlying distribution: for the normal distribution, the asymptotic variance of ''S''{{radic|''n''}} is 0.5708...

Assuming that the underlying distribution is symmetric, by considering the distribution of values above and below the median Zheng and Gastwirth have argued that<ref name=Zheng2010>Zheng T, Gastwirth J (2010) "On bootstrap tests of symmetry about an unknown median". ''Journal of Data Science'', 8(3): 413–427</ref>

: <math> \sqrt{ 2n } \left( \frac{ m - a }{ s } \right) </math>

where ''n'' is the sample size, is distributed as a [[Student's t-distribution|t distribution]].

==Related statistics==

[[Antonietta Mira]] studied the distribution of the difference between the mean and the median.<ref name=Mira1999>[[Antonietta Mira|Mira A]] (1999) "Distribution-free test for symmetry based on Bonferroni’s measure", ''Journal of Applied Statistics'', 26:959–972</ref>

: <math> \gamma_1 = 2 ( m - a ) ,</math>

where ''m'' is the sample mean and ''a'' is the median. If the underlying distribution is symmetrical ''γ''1 itself is asymptotically normal. This statistic had been earlier suggested by Bonferroni.<ref name=Bonferroni1999>Bonferroni CE (1930) ''Elementi di statistica generale''. Seeber, Firenze</ref>

Assuming a symmetric underlying distribution, a modification of ''S'' was studied by Miao, [[Yulia Gel|Gel]] and Gastwirth who modified the standard deviation to create their statistic.<ref name=Miao2006>Miao W, [[Yulia Gel|Gel YR]], Gastwirth JL (2006) "A new test of symmetry about an unknown median". In: Hsiung A, Zhang C-H, Ying Z, eds. ''Random Walk, Sequential Analysis and Related Topics — A Festschrift in honor of Yuan-Shih Chow''. World Scientific; Singapore</ref>

: <math> J = \frac{ 1 }{ n } \sqrt { \frac{ \pi }{ 2 } } \sum{ | X_i -a | } </math>

where ''X''i are the sample values, || is the [[absolute value]] and the sum is taken over all ''n'' sample values.

The test statistic was

: <math> T = \frac{ m - a } { J } .</math>

The scaled statistic ''T''{{radic|''n''}} is asymptotically normal with a mean of zero for a symmetric distribution. Its asymptotic variance depends on the underlying distribution: the limiting values are, for the normal distribution {{nowrap|1=var(''T''{{radic|''n''}})}} = 0.5708... and, for the [[Student's t-distribution|t distribution]] with three [[degrees of freedom]], {{nowrap|1=var(''T''{{radic|''n''}})}} = 0.9689...<ref name="Miao2006"/>

==Values for individual distributions==

===Symmetric distributions===

For [[symmetric probability distribution]]s the value of the nonparametric skew is 0.

===Asymmetric distributions===

It is positive for right skewed distributions and negative for left skewed distributions. Absolute values ≥ 0.2 indicate marked skewness.

It may be difficult to determine ''S'' for some distributions. This is usually because a closed form for the median is not known: examples of such distributions include the [[gamma distribution]], [[inverse-chi-squared distribution]], the [[inverse-gamma distribution]] and the [[scaled inverse chi-squared distribution]].

The following values for ''S'' are known:

*[[Beta distribution]]: 1 < ''α'' < ''β'' where ''α'' and ''β'' are the parameters of the distribution, then to a good approximation<ref name=Kerman2011>Kerman J (2011) "A closed-form approximation for the median of the beta distribution". {{arxiv|1111.0433v1}}</ref>

:: <math> S = \frac{ 1 }{ 3 }\frac{ ( \alpha - 2 \beta ) ( \alpha + \beta + 1 )^{ 1 / 2 } }{ ( \alpha + \beta - 2 / 3 ) ( \alpha \beta )^{ 1 / 2 } } </math>

: If 1 < ''β'' < ''α'' then the positions of ''α'' and ''β'' are reversed in the formula. ''S'' is always < 0. *[[Binomial distribution]]: varies. If the mean is an [[integer]] then ''S'' = 0. If the mean is not an integer ''S'' may have either sign or be zero.<ref name=Kaas1980>Kaas R, Buhrman JM (1980) Mean, median and mode in binomial distributions. Statistica Neerlandica 34 (1) 13–18</ref> It is bounded by ±min{ max{ ''p'', 1 − ''p'' }, loge2 } / ''σ'' where ''σ'' is the standard deviation of the binomial distribution.<ref name=Hamza1995>Hamza K (1995) "The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions". ''Statistics and Probability Letters'', 23 (1) 21–25</ref> *[[Burr distribution]]: *[[Birnbaum–Saunders distribution]]:

:: <math> S = \frac{ 2 }{ \beta^2 ( 4 + 5 \alpha^2 ) }</math>

: where ''α'' is the shape parameter and ''β'' is the location parameter.

*[[Cantor distribution]]: despite the distribution being symmetric about its mean of <math>\tfrac12</math>, the median can be any value in <math>\left[\tfrac13,\tfrac23\right]</math> as this central interval has zero probability

:: <math> \frac{ -4 }{ 3 } \le S \le \frac{ 4 }{ 3 } </math>

*[[Chi square distribution]]: Although ''S'' ≥ 0 its value depends on the numbers of [[degrees of freedom]] (''k'').

:: <math> S \approx \frac{ 1 - ( 1 - \frac{ 2 }{ k } )^3 }{ 2 } </math>

*[[Dagum distribution]]: *[[Exponential distribution]]: :: <math> S = 1 - \log_e( 2 ) \approx 0.31 </math>

*[[Exponential distribution]] with two parameters:<ref name=Caltech00>{{Cite web |url=http://web.ipac.caltech.edu/staff/fmasci/home/statistics_refs/UsefulDistributions.pdf |title=Archived copy |access-date=2012-09-30 |archive-date=2008-04-19 |archive-url=https://web.archive.org/web/20080419002641/http://web.ipac.caltech.edu/staff/fmasci/home/statistics_refs/UsefulDistributions.pdf |url-status=dead }}</ref> :: <math> S = 1 - \log_e( 2 ) \approx 0.31 </math>

*[[Exponential-logarithmic distribution]] :: <math> S = - \frac{ polylog( 2, 1 - p ) + \ln( 1 + \sqrt{ p } ) \ln p }{ \sqrt{ -[2 polylog( 3, 1 - p ) + polylog^2( 2, 1 -p ) ] } }</math>

: Here ''S'' is always > 0. *[[Exponentially modified Gaussian distribution]]:

:: <math> 0 \le S \le 1 - \log_e( 2 ) </math>

*[[F distribution]] with ''n'' and ''n'' [[degrees of freedom]] ( ''n'' > 4 ):<ref name=Terrell1986>Terrell GR (1986) "Pearson's rule for sample medians". Technical Report 86-2{{full citation needed|date=November 2012}}</ref>

:: <math> S = n^{ -3 / 2 } \sqrt{ \frac{ n - 4 }{ n - 2 } } + O( n^{ -5 / 2 } ) </math>

*[[Fréchet distribution]]: The variance of this distribution is defined only for ''α'' > 2.

:: <math> S = \frac{ \Gamma \left( 1 - \frac{ 1 }{ \alpha } \right) - \frac{ 1 }{ \sqrt{\alpha} \log_e( 2 ) } } \sqrt { \Gamma \left( 1 - \frac{ 2 }{ \alpha } \right ) - \left( \Gamma \left( 1 - \frac{ 1 }{ \alpha } \right ) \right )^2 } </math>

*[[Gamma distribution]]: The median can only be determined approximately for this distribution.<ref name=Banneheka2009>Banneheka BMSG, Ekanayake GEMUPD (2009) A new point estimator for the median of Gamma distribution. Viyodaya J Science 14:95–103</ref> If the shape parameter ''α'' is ≥ 1 then

:: <math> S \approx \frac{ \beta }{ 3 \alpha + 0.2 } </math>

: where ''β'' > 0 is the rate parameter. Here ''S'' is always > 0.

*[[Generalized normal distribution]] version 2

:: <math> S = - \frac{ \exp( \frac {-k^2 } { 2 } ) - 1 }{ \sqrt{ \exp( \frac{ k^2 }{ 2 } ) - 1 } } </math>

: ''S'' is always < 0.

*[[Generalized Pareto distribution]]: ''S'' is defined only when the shape parameter ( ''k'' ) is < 1/2. ''S'' is < 0 for this distribution.

:: <math> S = \left( \frac{ 2^k - 1 }{ k } - 2^k \right)( 1 - 2k )^{ 0.5 } </math>

*[[Gumbel distribution]]:

:: <math> \frac{ \sqrt{ 6 } [ \gamma + \log_e( \log_e( 2 ) ) ] }{ \pi } \approx 0.1643 </math>

: where ''γ'' is [[Euler's constant]].<ref name=Ferguson>Ferguson T. [https://www.math.ucla.edu/~tom/papers/unpublished/meanmed.pdf "Asymptotic Joint Distribution of Sample Mean and a Sample Quantile"], Unpublished</ref>

*[[Half-normal distribution]]:<ref name="Caltech00"/>

:: <math> S \approx \frac{ \sqrt{ 2 } - 0.6745 \sqrt{ \pi } }{ \sqrt{ \pi - 2 } } \approx 0.36279 </math>

*[[Kumaraswamy distribution]] *[[Log-logistic distribution]] (Fisk distribution): Let ''β'' be the shape parameter. The variance and mean of this distribution are only defined when ''β'' > 2. To simplify the notation let ''b'' = ''β'' / {{pi}}.

:: <math> S = \frac{ b - \sin ( b ) }{ \sqrt{ b \tan ( b ) - b^2 } } </math>

: The standard deviation does not exist for values of ''b'' > 4.932 (approximately). For values for which the standard deviation is defined, ''S'' is > 0.

*[[Log-normal distribution]]: With mean ( ''μ'' ) and variance ( ''σ''2 )

:: <math> S = \frac{ 1 }{ ( e^{ \frac{ \sigma^2 }{ 2 } } + 1 ) ( e^{ \mu + \sigma^2 } ) } </math>

*[[Log-Weibull distribution]]:<ref name="Caltech00"/>

:: <math> S \approx \frac{ [ \log_e( \log_e( 2 ) ) - 0.5772 ] \sqrt{ 6 } }{ \pi } \approx -0.1643 </math>

*[[Lomax distribution]]: ''S'' is defined only for ''α'' > 2

:: <math> S = \frac{ ( \alpha - 1 )( \alpha - 2 )( 1 - ( \alpha - 1 )( 2^{ 1 / \alpha } - 1 ) ) }{ \alpha^{ 1/2 } } </math>

*[[Maxwell–Boltzmann distribution]]:<ref name="Caltech00"/>

:: <math> S \approx \frac{ \sqrt{ 2 } - 1.5382 \Gamma( \frac{ 3 }{ 2 } ) }{ \sqrt{ 2 ( \Gamma( \frac{ 5 }{ 2 } ) - \Gamma( \frac{ 3 }{ 2 } ) ) } } \approx 0.0854 </math>

*[[Nakagami distribution]] :: <math> S = -1 </math>

*[[Pareto distribution]]: for ''α'' > 2 where ''α'' is the shape parameter of the distribution,

:: <math> S = ( \alpha - 2^{ 1 / \alpha }[ \alpha - 1 ] ) ( \frac{ \alpha - 2 }{ \alpha } )^{ 1 / 2 } ,</math>

:and ''S'' is always > 0.

*[[Poisson distribution]]:

:: <math> \frac{ -\log_e(2)} { \lambda^\frac{ 1 }{ 2 } } \le S \le \frac{ 1 }{ 3 \lambda^\frac{ 1 }{ 2 } }</math>

: where ''λ'' is the parameter of the distribution.<ref name=Choi1994>Choi KP (1994) "On the medians of Gamma distributions and an equation of Ramanujan". ''Proc Amer Math Soc'' 121 (1) 245–251</ref>

*[[Rayleigh distribution]]:

:: <math> S = \sqrt { \frac{ 2 }{ 4 - \pi } } [ ( \frac{ \pi }{ 2 } )^{ 0.5 } - \log_e( 4 ) ] \approx 0.1251 </math>

*[[Weibull distribution]]:

:: <math> S = \frac{ \Gamma( 1 + 1 / k ) - \log_e( 2 )^{ 1 / k } }{ ( \Gamma( 1 + 2 / k ) - \Gamma( 1 + 1 / k ) )^{ 1 / 2 } }, </math>

: where ''k'' is the shape parameter of the distribution. Here ''S'' is always > 0.

==History==

In 1895 [[Karl Pearson|Pearson]] first suggested measuring skewness by standardizing the difference between the mean and the [[Mode (statistics)|mode]],<ref name=Pearson1895>Pearson K (1895) Contributions to the Mathematical Theory of Evolution–II. Skew variation in homogenous material. Phil Trans Roy Soc A. 186: 343–414</ref> giving

: <math> \frac{ \mu - \theta } { \sigma } ,</math>

where ''μ'', ''θ'' and ''σ'' is the mean, mode and standard deviation of the distribution respectively. Estimates of the population mode from the sample data may be difficult but the difference between the mean and the mode for many distributions is approximately three times the difference between the mean and the median<ref name=Stuart1994>Stuart A, Ord JK (1994) ''Kendall’s advanced theory of statistics. Vol 1. Distribution theory''. 6th Edition. Edward Arnold, London</ref> which suggested to Pearson a second skewness coefficient:

: <math> \frac{ 3 ( \mu - \nu ) } { \sigma } ,</math>

where ''ν'' is the median of the distribution. [[Arthur Lyon Bowley|Bowley]] dropped the factor 3 from this formula in 1901 leading to the nonparametric skew statistic.

The relationship between the median, the mean and the mode was first noted by Pearson when he was investigating his type III distributions.

==Relationships between the mean, median and mode==

For an arbitrary distribution the mode, median and mean may appear in any order.<ref>[http://www.se16.info/hgb/median.htm Relationship between the mean, median, mode, and standard deviation in a unimodal distribution]</ref><ref>von Hippel, Paul T. (2005) [http://www.amstat.org/publications/jse/v13n2/vonhippel.html "Mean, Median, and Skew: Correcting a Textbook Rule"] {{Webarchive|url=https://web.archive.org/web/20160220181456/http://www.amstat.org/publications/jse/v13n2/vonhippel.html |date=2016-02-20 }}, ''Journal of Statistics Education'', 13(2)</ref><ref name=Dharmadhikari1983>Dharmadhikari SW, Joag-dev K (1983) Mean, Median, Mode III. Statistica Neerlandica, 33: 165–168</ref>

Analyses have been made of some of the relationships between the mean, median, mode and standard deviation.<ref>Bottomly, H.(2002,2006) [http://www.se16.info/hgb/median.htm "Relationship between the mean, median, mode, and standard deviation in a unimodal distribution"] Personal webpage</ref> and these relationships place some restrictions on the sign and magnitude of the nonparametric skew.

A simple example illustrating these relationships is the [[binomial distribution]] with ''n'' = 10 and ''p'' = 0.09.<ref name=Lesser2005>Lesser LM (2005).[http://www.amstat.org/publications/jse/v13n3/lesser_letter.html "Letter to the editor" ], [comment on von Hippel (2005)]. ''Journal of Statistics Education'' 13(2).</ref> This distribution when plotted has a long right tail. The mean (0.9) is to the left of the median (1) but the skew (0.906) as defined by the third standardized moment is positive. In contrast the nonparametric skew is -0.110.

===Pearson's rule===

The rule that for some distributions the difference between the mean and the mode is three times that between the mean and the median is due to Pearson who discovered it while investigating his Type 3 distributions. It is often applied to slightly asymmetric distributions that resemble a normal distribution but it is not always true.

In 1895 Pearson noted that for what is now known as the [[gamma distribution]] that the relation<ref name=Pearson1895/>

: <math> \nu - \theta = 2( \mu - \nu ) </math>

where ''θ'', ''ν'' and ''μ'' are the mode, median and mean of the distribution respectively was approximately true for distributions with a large shape parameter.

Doodson in 1917 proved that the median lies between the mode and the mean for moderately skewed distributions with finite fourth moments.<ref name=Doodson1917>Doodson AT (1917) "Relation of the mode, median and mean in frequency functions". ''[[Biometrika]]'', 11 (4) 425–429 {{doi|10.1093/biomet/11.4.425}}</ref> This relationship holds for all the [[Pearson distribution]]s and all of these distributions have a positive nonparametric skew.

Doodson also noted that for this family of distributions to a good approximation,

: <math> \theta = 3 \nu - 2 \mu ,</math>

where ''θ'', ''ν'' and ''μ'' are the mode, median and mean of the distribution respectively. Doodson's approximation was further investigated and confirmed by [[J. B. S. Haldane|Haldane]].<ref name=Haldane1942>Haldane JBS (1942) "The mode and median of a nearly normal distribution with given cumulants". ''[[Biometrika]]'', 32: 294–299</ref> Haldane noted that samples with identical and independent variates with a third [[cumulant]] had sample means that obeyed Pearson's relationship for large sample sizes. Haldane required a number of conditions for this relationship to hold including the existence of an [[Edgeworth expansion]] and the uniqueness of both the median and the mode. Under these conditions he found that mode and the median converged to 1/2 and 1/6 of the third moment respectively. This result was confirmed by Hall under weaker conditions using [[characteristic function (probability theory)|characteristic function]]s.<ref name=Hall1980>Hall P (1980) "On the limiting behaviour of the mode and median of a sum of independent random variables". ''Annals of Probability'' 8: 419–430</ref>

Doodson's relationship was studied by Kendall and Stuart in the [[log-normal distribution]] for which they found an exact relationship close to it.<ref name=Kendall1958>Kendall M.G., Stuart A. (1958) ''The advanced theory of statistics''. p53 Vol 1. Griffin. London</ref>

Hall also showed that for a distribution with regularly varying tails and exponent ''α'' that{{clarify|reason=explain this terminology properly|date=July 2012}}<ref name=Hall1980/>

: <math> \mu - \theta = \alpha ( \mu - \nu ) </math>

===Unimodal distributions===

Gauss showed in 1823 that for a [[unimodal distribution]]<ref name=Gauss1823>Gauss C.F. Theoria Combinationis Observationum Erroribus Minimis Obnoxiae. Pars Prior. Pars Posterior. Supplementum. Theory of the Combination of Observations Least Subject to Errors. Part One. Part Two. Supplement. 1995. Translated by G.W. Stewart. Classics in Applied Mathematics Series, Society for Industrial and Applied Mathematics, Philadelphia</ref>

: <math> \sigma \le \omega \le 2 \sigma </math>

and

: <math> | \nu - \mu | \le \sqrt{ \frac{ 3 }{ 4 } } \omega ,</math>

where ''ω'' is the root mean square deviation from the mode.

For a large class of unimodal distributions that are positively skewed the mode, median and mean fall in that order.<ref name=MacGillivray1981>MacGillivray HL (1981) The mean, median, mode inequality and skewness for a class of densities. Aust J Stat 23(2) 247–250</ref> Conversely for a large class of unimodal distributions that are negatively skewed the mean is less than the median which in turn is less than the mode. In symbols for these positively skewed unimodal distributions

: <math> \theta \le \nu \le \mu </math>

and for these negatively skewed unimodal distributions

: <math> \mu \le \nu \le \theta </math>

This class includes the important F, beta and gamma distributions.

This rule does not hold for the unimodal Weibull distribution.<ref name=Groeneveld1986>Groeneveld RA (1986) Skewness for the Weibull family. Statistica Neerlandica 40: 135–140</ref>

For a unimodal distribution the following bounds are known and are sharp:<ref name=Johnson1951>Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". ''Annals of Mathematical Statistics'', 22 (3) 433–439</ref>

: <math> \frac{| \theta - \mu |}{ \sigma } \le \sqrt{ 3 } ,</math>

: <math> \frac{| \nu - \mu |}{ \sigma } \le \sqrt{ 0.6 } ,</math>

: <math> \frac{| \theta - \nu |}{ \sigma } \le \sqrt{ 3 } ,</math>

where ''μ'',''ν'' and ''θ'' are the mean, median and mode respectively.

The middle bound limits the nonparametric skew of a unimodal distribution to approximately ±0.775.

===van Zwet condition===

The following inequality,

: <math> \theta \le \nu \le \mu ,</math>

where ''θ'', ''ν'' and ''μ'' is the mode, median and mean of the distribution respectively, holds if

: <math> F( \nu - x ) + F( \nu + x ) \ge 1 \text{ for all } x, </math>

where ''F'' is the [[cumulative distribution function]] of the distribution.<ref name=vanZwet1979>van Zwet W.R. (1979) "Mean, median, mode II". ''Statistica Neerlandica'' 33(1) 1–5</ref> These conditions have since been generalised<ref name=Dharmadhikari1983/> and extended to discrete distributions.<ref name=Abdous1998>Abdous B, Theodorescu R (1998) Mean, median, mode IV. Statistica Neerlandica. 52 (3) 356–359</ref> Any distribution for which this holds has either a zero or a positive nonparametric skew.

==Notes==

===Ordering of skewness===

In 1964 van Zwet proposed a series of axioms for ordering measures of skewness.<ref>van Zwet, W.R. (1964) "Convex transformations of random variables". ''Mathematics Centre Tract'', 7, Mathematisch Centrum, Amsterdam</ref> The nonparametric skew does not satisfy these axioms.

===Benford's law===

[[Benford's law]] is an empirical law concerning the distribution of digits in a list of numbers. It has been suggested that random variates from distributions with a positive nonparametric skew will obey this law.<ref name=Durtschi2004>Durtschi C, Hillison W, Pacini C (2004) The effective use of Benford’s Law to assist in detecting fraud in accounting data. J Forensic Accounting 5: 17–34</ref>

=== Relation to Bowley's coefficient===

This statistic is very similar to [[Skewness#Quantile-based measures|Bowley's coefficient of skewness]]<ref name=Bowley1920>Bowley AL (1920) Elements of statistics. New York: Charles Scribner's Sons</ref>

: <math> SK_2 = \frac{ Q_3 + Q_1 - 2 Q_2 }{ Q_3 - Q_1 } </math>

where Qi is the ith quartile of the distribution.

Hinkley generalised this<ref name=Hinkley1975>Hinkley DV (1975) On power transformations to symmetry. Biometrika 62: 101–111</ref>

: <math> SK = \frac{ F^{-1}( 1 - \alpha ) + F^{-1}( \alpha ) - 2 Q_2 }{ Q_3 - Q_1 } </math>

where <math> \alpha </math> lies between 0 and 0.5. Bowley's coefficient is a special case with <math> \alpha </math> equal to 0.25.

Groeneveld and Meeden<ref name=Groeneveld1984>Groeneveld RA, Meeden G (1984) Measuring skewness and kurtosis. The Statistician, 33: 391–399</ref> removed the dependence on <math> \alpha </math> by integrating over it.

: <math> SK_3 = \frac{ \mu - Q_2 }{ E | y - Q_2 | } </math>

The denominator is a measure of dispersion. Replacing the denominator with the standard deviation we obtain the nonparametric skew.

==References== {{reflist|2}}

[[Category:Summary statistics]]