# Nonparametric skew

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Statistical quantity

In [statistics](/source/Statistics) and [probability theory](/source/Probability_theory), the **nonparametric skew** is a [statistic](/source/Statistic) occasionally used with [random variables](/source/Random_variable) that take [real](/source/Real_number) values.[1][2] It is a measure of the [skewness](/source/Skewness) of a random variable's [distribution](/source/Probability_distribution)—that is, the distribution's tendency to "lean" to one side or the other of the [mean](/source/Mean). Its calculation does not require any knowledge of the form of the underlying distribution—hence the name [nonparametric](/source/Non-parametric_statistics). It has some desirable properties: it is zero for any [symmetric distribution](/source/Symmetric_distribution); it is unaffected by a [scale](/source/Scale_parameter) shift; and it reveals either left- or right-skewness equally well. In some [statistical samples](/source/Sample_(statistics)) it has been shown to be less [powerful](/source/Statistical_power)[3] than the usual measures of skewness in detecting departures of the [population](/source/Population_(statistics)) from [normality](/source/Normal_distribution).[4]

## Properties

### Definition

The nonparametric skew is defined as

- S = μ − ν σ {\displaystyle S={\frac {\mu -\nu }{\sigma }}}

where the [mean](/source/Mean) (*μ*), [median](/source/Median) (*ν*) and [standard deviation](/source/Standard_deviation) (*σ*) of the population have their usual meanings.

### Properties

The nonparametric skew is one third of the [Pearson 2 skewness coefficient](/source/Skewness#Pearson's_skewness_coefficients) and lies between −1 and +1 for any distribution.[5][6] This range is implied by the fact that the mean lies within one standard deviation of any median.[7]

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

- S ( a X + b ) = sign ⁡ ( a ) S ( X ) {\displaystyle S(aX+b)=\operatorname {sign} (a)\,S(X)}

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[8] who showed that its [absolute value](/source/Absolute_value) is bounded by

- 2 ( p q ) 1 / 2 ( p + q ) 1 / 2 {\displaystyle {\frac {2(pq)^{1/2}}{(p+q)^{1/2}}}}

with

- p = Pr ( X > E ⁡ ( X ) ) {\displaystyle p=\Pr(X>\operatorname {E} (X))}

and

- q = Pr ( X < E ⁡ ( X ) ) , {\displaystyle q=\Pr(X<\operatorname {E} (X)),}

where *X* is a random variable with finite [variance](/source/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[9]

- | μ − ν 0 | ≤ E ⁡ ( | X − ν 0 | ) ≤ E ⁡ ( | X − μ | ) ≤ σ , {\displaystyle |\mu -\nu _{0}|\leq \operatorname {E} (|X-\nu _{0}|)\leq \operatorname {E} (|X-\mu |)\leq \sigma ,}

where *ν*0 is any median and *E*(.) is the [expectation operator](/source/Expected_value).

It has been shown that

- | μ − x q | σ ≤ max ( ( 1 − q ) q , q ( 1 − q ) ) {\displaystyle {\frac {|\mu -x_{q}|}{\sigma }}\leq \max \left({\sqrt {\frac {(1-q)}{q}}},{\sqrt {\frac {q}{(1-q)}}}\right)}

where *x**q* is the *q*th [quantile](/source/Quantile_function).[7] 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.[10]

This latter inequality has been sharpened further.[11]

- μ − σ 1 − q q ≤ x q ≤ μ + σ q 1 − q {\displaystyle \mu -\sigma {\sqrt {\frac {1-q}{q}}}\leq x_{q}\leq \mu +\sigma {\sqrt {\frac {q}{1-q}}}}

Another extension for a distribution with a finite mean has been published:[12]

- μ − 1 2 q E ⁡ | X − μ | ≤ x q ≤ μ + 1 ( 2 − 2 q ) E ⁡ | X − μ | {\displaystyle \mu -{\frac {1}{2q}}\operatorname {E} |X-\mu |\leq x_{q}\leq \mu +{\frac {1}{(2-2q)}}\operatorname {E} |X-\mu |}

The bounds in this last pair of inequalities are attained when Pr ( X = a ) = q {\displaystyle \Pr(X=a)=q} and Pr ( X = b ) = 1 − q {\displaystyle \Pr(X=b)=1-q} 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](/source/Order_statistic), *m* the sample mean and *s* the [sample standard deviation](/source/Sample_standard_deviation) corrected for degrees of freedom,[13]

| m − x r | s ≤ max [ ( n − 1 ) ( r − 1 ) n ( n − r + 1 ) , ( n − 1 ) ( n − r ) n r ] {\displaystyle {\frac {|m-x_{r}|}{s}}\leq {\text{max}}\left[{\sqrt {\frac {(n-1)(r-1)}{n(n-r+1)}}},{\sqrt {\frac {(n-1)(n-r)}{nr}}}\right]}

Replacing *r* with *n* / 2 gives the result appropriate for the sample median:[14]

| m − a | s ≤ n 2 − n n 2 = n − 1 n {\displaystyle {\frac {|m-a|}{s}}\leq {\sqrt {\frac {n^{2}-n}{n^{2}}}}={\sqrt {\frac {n-1}{n}}}}

where *a* is the sample median.

## Statistical tests

Hotelling and Solomons considered the distribution of the test statistic[5]

- D = n ( m − a ) s {\displaystyle D={\frac {n(m-a)}{s}}}

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](/source/Variance) of *n*−1/2*D*.[15] 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.[16] The asymptotic variance depends on the underlying distribution: for the normal distribution, the asymptotic variance of *S*√*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[17]

- 2 n ( m − a s ) {\displaystyle {\sqrt {2n}}\left({\frac {m-a}{s}}\right)}

where *n* is the sample size, is distributed as a [t distribution](/source/Student's_t-distribution).

## Related statistics

[Antonietta Mira](/source/Antonietta_Mira) studied the distribution of the difference between the mean and the median.[18]

- γ 1 = 2 ( m − a ) , {\displaystyle \gamma _{1}=2(m-a),}

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.[19]

Assuming a symmetric underlying distribution, a modification of *S* was studied by Miao, [Gel](/source/Yulia_Gel) and Gastwirth who modified the standard deviation to create their statistic.[20]

- J = 1 n π 2 ∑ | X i − a | {\displaystyle J={\frac {1}{n}}{\sqrt {\frac {\pi }{2}}}\sum {|X_{i}-a|}}

where *X*i are the sample values, || is the [absolute value](/source/Absolute_value) and the sum is taken over all *n* sample values.

The test statistic was

- T = m − a J . {\displaystyle T={\frac {m-a}{J}}.}

The scaled statistic *T*√*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 var(*T*√*n*) = 0.5708... and, for the [t distribution](/source/Student's_t-distribution) with three [degrees of freedom](/source/Degrees_of_freedom), var(*T*√*n*) = 0.9689...[20]

## Values for individual distributions

### Symmetric distributions

For [symmetric probability distributions](/source/Symmetric_probability_distribution) 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](/source/Gamma_distribution), [inverse-chi-squared distribution](/source/Inverse-chi-squared_distribution), the [inverse-gamma distribution](/source/Inverse-gamma_distribution) and the [scaled inverse chi-squared distribution](/source/Scaled_inverse_chi-squared_distribution).

The following values for *S* are known:

- [Beta distribution](/source/Beta_distribution): 1 < *α* < *β* where *α* and *β* are the parameters of the distribution, then to a good approximation[21]

- - S = 1 3 ( α − 2 β ) ( α + β + 1 ) 1 / 2 ( α + β − 2 / 3 ) ( α β ) 1 / 2 {\displaystyle S={\frac {1}{3}}{\frac {(\alpha -2\beta )(\alpha +\beta +1)^{1/2}}{(\alpha +\beta -2/3)(\alpha \beta )^{1/2}}}}

- If 1 < *β* < *α* then the positions of *α* and *β* are reversed in the formula. *S* is always < 0.

- [Binomial distribution](/source/Binomial_distribution): varies. If the mean is an [integer](/source/Integer) then *S* = 0. If the mean is not an integer *S* may have either sign or be zero.[22] It is bounded by ±min{ max{ *p*, 1 − *p* }, loge2 } / *σ* where *σ* is the standard deviation of the binomial distribution.[23]

- [Burr distribution](/source/Burr_distribution):

- [Birnbaum–Saunders distribution](/source/Birnbaum%E2%80%93Saunders_distribution):

- - S = 2 β 2 ( 4 + 5 α 2 ) {\displaystyle S={\frac {2}{\beta ^{2}(4+5\alpha ^{2})}}}

- where *α* is the shape parameter and *β* is the location parameter.

- [Cantor distribution](/source/Cantor_distribution): despite the distribution being symmetric about its mean of 1 2 {\displaystyle {\tfrac {1}{2}}} , the median can be any value in [ 1 3 , 2 3 ] {\displaystyle \left[{\tfrac {1}{3}},{\tfrac {2}{3}}\right]} as this central interval has zero probability

- - − 4 3 ≤ S ≤ 4 3 {\displaystyle {\frac {-4}{3}}\leq S\leq {\frac {4}{3}}}

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

- - S ≈ 1 − ( 1 − 2 k ) 3 2 {\displaystyle S\approx {\frac {1-(1-{\frac {2}{k}})^{3}}{2}}}

- [Dagum distribution](/source/Dagum_distribution):

- [Exponential distribution](/source/Exponential_distribution):

- - S = 1 − log e ⁡ ( 2 ) ≈ 0.31 {\displaystyle S=1-\log _{e}(2)\approx 0.31}

- [Exponential distribution](/source/Exponential_distribution) with two parameters:[24]

- - S = 1 − log e ⁡ ( 2 ) ≈ 0.31 {\displaystyle S=1-\log _{e}(2)\approx 0.31}

- [Exponential-logarithmic distribution](/source/Exponential-logarithmic_distribution)

- - S = − p o l y l o g ( 2 , 1 − p ) + ln ⁡ ( 1 + p ) ln ⁡ p − [ 2 p o l y l o g ( 3 , 1 − p ) + p o l y l o g 2 ( 2 , 1 − p ) ] {\displaystyle S=-{\frac {polylog(2,1-p)+\ln(1+{\sqrt {p}})\ln p}{\sqrt {-[2polylog(3,1-p)+polylog^{2}(2,1-p)]}}}}

- Here *S* is always > 0.

- [Exponentially modified Gaussian distribution](/source/Exponentially_modified_Gaussian_distribution):

- - 0 ≤ S ≤ 1 − log e ⁡ ( 2 ) {\displaystyle 0\leq S\leq 1-\log _{e}(2)}

- [F distribution](/source/F_distribution) with *n* and *n* [degrees of freedom](/source/Degrees_of_freedom) ( *n* > 4 ):[25]

- - S = n − 3 / 2 n − 4 n − 2 + O ( n − 5 / 2 ) {\displaystyle S=n^{-3/2}{\sqrt {\frac {n-4}{n-2}}}+O(n^{-5/2})}

- [Fréchet distribution](/source/Fr%C3%A9chet_distribution): The variance of this distribution is defined only for *α* > 2.

- - S = Γ ( 1 − 1 α ) − 1 α log e ⁡ ( 2 ) Γ ( 1 − 2 α ) − ( Γ ( 1 − 1 α ) ) 2 {\displaystyle 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}}}}}

- [Gamma distribution](/source/Gamma_distribution): The median can only be determined approximately for this distribution.[26] If the shape parameter *α* is ≥ 1 then

- - S ≈ β 3 α + 0.2 {\displaystyle S\approx {\frac {\beta }{3\alpha +0.2}}}

- where *β* > 0 is the rate parameter. Here *S* is always > 0.

- [Generalized normal distribution](/source/Generalized_normal_distribution) version 2

- - S = − exp ⁡ ( − k 2 2 ) − 1 exp ⁡ ( k 2 2 ) − 1 {\displaystyle S=-{\frac {\exp({\frac {-k^{2}}{2}})-1}{\sqrt {\exp({\frac {k^{2}}{2}})-1}}}}

- *S* is always < 0.

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

- - S = ( 2 k − 1 k − 2 k ) ( 1 − 2 k ) 0.5 {\displaystyle S=\left({\frac {2^{k}-1}{k}}-2^{k}\right)(1-2k)^{0.5}}

- [Gumbel distribution](/source/Gumbel_distribution):

- - 6 [ γ + log e ⁡ ( log e ⁡ ( 2 ) ) ] π ≈ 0.1643 {\displaystyle {\frac {{\sqrt {6}}[\gamma +\log _{e}(\log _{e}(2))]}{\pi }}\approx 0.1643}

- where *γ* is [Euler's constant](/source/Euler's_constant).[27]

- [Half-normal distribution](/source/Half-normal_distribution):[24]

- - S ≈ 2 − 0.6745 π π − 2 ≈ 0.36279 {\displaystyle S\approx {\frac {{\sqrt {2}}-0.6745{\sqrt {\pi }}}{\sqrt {\pi -2}}}\approx 0.36279}

- [Kumaraswamy distribution](/source/Kumaraswamy_distribution)

- [Log-logistic distribution](/source/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* / π.

- - S = b − sin ⁡ ( b ) b tan ⁡ ( b ) − b 2 {\displaystyle S={\frac {b-\sin(b)}{\sqrt {b\tan(b)-b^{2}}}}}

- 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](/source/Log-normal_distribution): With mean ( *μ* ) and variance ( *σ*2 )

- - S = 1 ( e σ 2 2 + 1 ) ( e μ + σ 2 ) {\displaystyle S={\frac {1}{(e^{\frac {\sigma ^{2}}{2}}+1)(e^{\mu +\sigma ^{2}})}}}

- [Log-Weibull distribution](/source/Log-Weibull_distribution):[24]

- - S ≈ [ log e ⁡ ( log e ⁡ ( 2 ) ) − 0.5772 ] 6 π ≈ − 0.1643 {\displaystyle S\approx {\frac {[\log _{e}(\log _{e}(2))-0.5772]{\sqrt {6}}}{\pi }}\approx -0.1643}

- [Lomax distribution](/source/Lomax_distribution): *S* is defined only for *α* > 2

- - S = ( α − 1 ) ( α − 2 ) ( 1 − ( α − 1 ) ( 2 1 / α − 1 ) ) α 1 / 2 {\displaystyle S={\frac {(\alpha -1)(\alpha -2)(1-(\alpha -1)(2^{1/\alpha }-1))}{\alpha ^{1/2}}}}

- [Maxwell–Boltzmann distribution](/source/Maxwell%E2%80%93Boltzmann_distribution):[24]

- - S ≈ 2 − 1.5382 Γ ( 3 2 ) 2 ( Γ ( 5 2 ) − Γ ( 3 2 ) ) ≈ 0.0854 {\displaystyle S\approx {\frac {{\sqrt {2}}-1.5382\Gamma ({\frac {3}{2}})}{\sqrt {2(\Gamma ({\frac {5}{2}})-\Gamma ({\frac {3}{2}}))}}}\approx 0.0854}

- [Nakagami distribution](/source/Nakagami_distribution)

- - S = − 1 {\displaystyle S=-1}

- [Pareto distribution](/source/Pareto_distribution): for *α* > 2 where *α* is the shape parameter of the distribution,

- - S = ( α − 2 1 / α [ α − 1 ] ) ( α − 2 α ) 1 / 2 , {\displaystyle S=(\alpha -2^{1/\alpha }[\alpha -1])({\frac {\alpha -2}{\alpha }})^{1/2},}

- and *S* is always > 0.

- [Poisson distribution](/source/Poisson_distribution):

- - − log e ⁡ ( 2 ) λ 1 2 ≤ S ≤ 1 3 λ 1 2 {\displaystyle {\frac {-\log _{e}(2)}{\lambda ^{\frac {1}{2}}}}\leq S\leq {\frac {1}{3\lambda ^{\frac {1}{2}}}}}

- where *λ* is the parameter of the distribution.[28]

- [Rayleigh distribution](/source/Rayleigh_distribution):

- - S = 2 4 − π [ ( π 2 ) 0.5 − log e ⁡ ( 4 ) ] ≈ 0.1251 {\displaystyle S={\sqrt {\frac {2}{4-\pi }}}[({\frac {\pi }{2}})^{0.5}-\log _{e}(4)]\approx 0.1251}

- [Weibull distribution](/source/Weibull_distribution):

- - S = Γ ( 1 + 1 / k ) − log e ⁡ ( 2 ) 1 / k ( Γ ( 1 + 2 / k ) − Γ ( 1 + 1 / k ) ) 1 / 2 , {\displaystyle S={\frac {\Gamma (1+1/k)-\log _{e}(2)^{1/k}}{(\Gamma (1+2/k)-\Gamma (1+1/k))^{1/2}}},}

- where *k* is the shape parameter of the distribution. Here *S* is always > 0.

## History

In 1895 [Pearson](/source/Karl_Pearson) first suggested measuring skewness by standardizing the difference between the mean and the [mode](/source/Mode_(statistics)),[29] giving

- μ − θ σ , {\displaystyle {\frac {\mu -\theta }{\sigma }},}

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[30] which suggested to Pearson a second skewness coefficient:

- 3 ( μ − ν ) σ , {\displaystyle {\frac {3(\mu -\nu )}{\sigma }},}

where *ν* is the median of the distribution. [Bowley](/source/Arthur_Lyon_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.[31][32][33]

Analyses have been made of some of the relationships between the mean, median, mode and standard deviation.[34] 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](/source/Binomial_distribution) with *n* = 10 and *p* = 0.09.[35] 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](/source/Gamma_distribution) that the relation[29]

- ν − θ = 2 ( μ − ν ) {\displaystyle \nu -\theta =2(\mu -\nu )}

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.[36] This relationship holds for all the [Pearson distributions](/source/Pearson_distribution) and all of these distributions have a positive nonparametric skew.

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

- θ = 3 ν − 2 μ , {\displaystyle \theta =3\nu -2\mu ,}

where *θ*, *ν* and *μ* are the mode, median and mean of the distribution respectively. Doodson's approximation was further investigated and confirmed by [Haldane](/source/J._B._S._Haldane).[37] Haldane noted that samples with identical and independent variates with a third [cumulant](/source/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](/source/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 functions](/source/Characteristic_function_(probability_theory)).[38]

Doodson's relationship was studied by Kendall and Stuart in the [log-normal distribution](/source/Log-normal_distribution) for which they found an exact relationship close to it.[39]

Hall also showed that for a distribution with regularly varying tails and exponent *α* that[*[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify)*][38]

- μ − θ = α ( μ − ν ) {\displaystyle \mu -\theta =\alpha (\mu -\nu )}

### Unimodal distributions

Gauss showed in 1823 that for a [unimodal distribution](/source/Unimodal_distribution)[40]

- σ ≤ ω ≤ 2 σ {\displaystyle \sigma \leq \omega \leq 2\sigma }

and

- | ν − μ | ≤ 3 4 ω , {\displaystyle |\nu -\mu |\leq {\sqrt {\frac {3}{4}}}\omega ,}

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.[41] 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

- θ ≤ ν ≤ μ {\displaystyle \theta \leq \nu \leq \mu }

and for these negatively skewed unimodal distributions

- μ ≤ ν ≤ θ {\displaystyle \mu \leq \nu \leq \theta }

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

This rule does not hold for the unimodal Weibull distribution.[42]

For a unimodal distribution the following bounds are known and are sharp:[43]

- | θ − μ | σ ≤ 3 , {\displaystyle {\frac {|\theta -\mu |}{\sigma }}\leq {\sqrt {3}},}

- | ν − μ | σ ≤ 0.6 , {\displaystyle {\frac {|\nu -\mu |}{\sigma }}\leq {\sqrt {0.6}},}

- | θ − ν | σ ≤ 3 , {\displaystyle {\frac {|\theta -\nu |}{\sigma }}\leq {\sqrt {3}},}

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,

- θ ≤ ν ≤ μ , {\displaystyle \theta \leq \nu \leq \mu ,}

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

- F ( ν − x ) + F ( ν + x ) ≥ 1 for all x , {\displaystyle F(\nu -x)+F(\nu +x)\geq 1{\text{ for all }}x,}

where *F* is the [cumulative distribution function](/source/Cumulative_distribution_function) of the distribution.[44] These conditions have since been generalised[33] and extended to discrete distributions.[45] 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.[46] The nonparametric skew does not satisfy these axioms.

### Benford's law

[Benford's law](/source/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.[47]

### Relation to Bowley's coefficient

This statistic is very similar to [Bowley's coefficient of skewness](/source/Skewness#Quantile-based_measures)[48]

- S K 2 = Q 3 + Q 1 − 2 Q 2 Q 3 − Q 1 {\displaystyle SK_{2}={\frac {Q_{3}+Q_{1}-2Q_{2}}{Q_{3}-Q_{1}}}}

where Qi is the ith quartile of the distribution.

Hinkley generalised this[49]

- S K = F − 1 ( 1 − α ) + F − 1 ( α ) − 2 Q 2 Q 3 − Q 1 {\displaystyle SK={\frac {F^{-1}(1-\alpha )+F^{-1}(\alpha )-2Q_{2}}{Q_{3}-Q_{1}}}}

where α {\displaystyle \alpha } lies between 0 and 0.5. Bowley's coefficient is a special case with α {\displaystyle \alpha } equal to 0.25.

Groeneveld and Meeden[50] removed the dependence on α {\displaystyle \alpha } by integrating over it.

- S K 3 = μ − Q 2 E | y − Q 2 | {\displaystyle SK_{3}={\frac {\mu -Q_{2}}{E|y-Q_{2}|}}}

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

## References

1. **[^](#cite_ref-Arnold1995_1-0)** 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

1. **[^](#cite_ref-Rubio2012_2-0)** Rubio F.J.; Steel M.F.J. (2012) "On the Marshall–Olkin transformation as a skewing mechanism". *Computational Statistics & Data Analysis* [Preprint](http://www2.warwick.ac.uk/fac/sci/statistics/staff/academic-research/steel/steel_homepage/techrep/mosrevcsda.pdf)

1. **[^](#cite_ref-Tabor2010_3-0)** 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

1. **[^](#cite_ref-amstat_4-0)** Doane, David P.; Seward, Lori E. (2011). ["Measuring Skewness: A Forgotten Statistic?"](https://web.archive.org/web/20160304054803/http://www.amstat.org/publications/jse/v19n2/doane.pdf) (PDF). *Journal of Statistics Education*. **19** (2). Archived from [the original](http://www.amstat.org/publications/jse/v19n2/doane.pdf) (PDF) on 2016-03-04. Retrieved 2012-01-26.

1. ^ [***a***](#cite_ref-Hotelling1932_5-0) [***b***](#cite_ref-Hotelling1932_5-1) Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114

1. **[^](#cite_ref-Garver1932_6-0)** Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142

1. ^ [***a***](#cite_ref-O’Cinneide1990_7-0) [***b***](#cite_ref-O’Cinneide1990_7-1) O’Cinneide CA (1990) The mean is within one standard deviation of any median. Amer Statist 44, 292–293

1. **[^](#cite_ref-Majindar1962_8-0)** Majindar KN (1962) "Improved bounds on a measure of skewness". *Annals of Mathematical Statistics*, 33, 1192–1194 [doi](/source/Doi_(identifier)):[10.1214/aoms/1177704482](https://doi.org/10.1214%2Faoms%2F1177704482)

1. **[^](#cite_ref-Mallows1969_9-0)** Mallows CCC, Richter D (1969) "Inequalities of Chebyschev type involving conditional expectations". *Annals of Mathematical Statistics*, 40:1922–1932

1. **[^](#cite_ref-Dziubinska1996_10-0)** Dziubinska R, Szynal D (1996) On functional measures of skewness. Applicationes Mathematicae 23(4) 395–403

1. **[^](#cite_ref-Dharmadhikari1991_11-0)** Dharmadhikari SS (1991) Bounds on quantiles: a comment on O'Cinneide. The Am Statist 45: 257-58

1. **[^](#cite_ref-Gilat1993_12-0)** 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 [\[1\]](http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1037&context=rgp_rsr)

1. **[^](#cite_ref-David1991_13-0)** David HA (1991) Mean minus median: A comment on O'Cinneide. The Am Statist 45: 257

1. **[^](#cite_ref-Joarder2004_14-0)** Joarder AH, Laradji A (2004) Some inequalities in descriptive statistics. Technical Report Series TR 321

1. **[^](#cite_ref-Gastwirth1971_15-0)** Gastwirth JL (1971) "On the sign test for symmetry". *[Journal of the American Statistical Association](/source/Journal_of_the_American_Statistical_Association)* 66:821–823

1. **[^](#cite_ref-Cabilio1996_16-0)** 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

1. **[^](#cite_ref-Zheng2010_17-0)** Zheng T, Gastwirth J (2010) "On bootstrap tests of symmetry about an unknown median". *Journal of Data Science*, 8(3): 413–427

1. **[^](#cite_ref-Mira1999_18-0)** [Mira A](/source/Antonietta_Mira) (1999) "Distribution-free test for symmetry based on Bonferroni’s measure", *Journal of Applied Statistics*, 26:959–972

1. **[^](#cite_ref-Bonferroni1999_19-0)** Bonferroni CE (1930) *Elementi di statistica generale*. Seeber, Firenze

1. ^ [***a***](#cite_ref-Miao2006_20-0) [***b***](#cite_ref-Miao2006_20-1) Miao W, [Gel YR](/source/Yulia_Gel), 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

1. **[^](#cite_ref-Kerman2011_21-0)** Kerman J (2011) "A closed-form approximation for the median of the beta distribution". [arXiv](/source/ArXiv_(identifier)):[1111.0433v1](https://arxiv.org/abs/1111.0433v1)

1. **[^](#cite_ref-Kaas1980_22-0)** Kaas R, Buhrman JM (1980) Mean, median and mode in binomial distributions. Statistica Neerlandica 34 (1) 13–18

1. **[^](#cite_ref-Hamza1995_23-0)** 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

1. ^ [***a***](#cite_ref-Caltech00_24-0) [***b***](#cite_ref-Caltech00_24-1) [***c***](#cite_ref-Caltech00_24-2) [***d***](#cite_ref-Caltech00_24-3) ["Archived copy"](https://web.archive.org/web/20080419002641/http://web.ipac.caltech.edu/staff/fmasci/home/statistics_refs/UsefulDistributions.pdf) (PDF). Archived from [the original](http://web.ipac.caltech.edu/staff/fmasci/home/statistics_refs/UsefulDistributions.pdf) (PDF) on 2008-04-19. Retrieved 2012-09-30.{{[cite web](https://en.wikipedia.org/wiki/Template:Cite_web)}}: CS1 maint: archived copy as title ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_archived_copy_as_title))

1. **[^](#cite_ref-Terrell1986_25-0)** Terrell GR (1986) "Pearson's rule for sample medians". Technical Report 86-2[*[full citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citing_sources#What_information_to_include)*]

1. **[^](#cite_ref-Banneheka2009_26-0)** Banneheka BMSG, Ekanayake GEMUPD (2009) A new point estimator for the median of Gamma distribution. Viyodaya J Science 14:95–103

1. **[^](#cite_ref-Ferguson_27-0)** Ferguson T. ["Asymptotic Joint Distribution of Sample Mean and a Sample Quantile"](https://www.math.ucla.edu/~tom/papers/unpublished/meanmed.pdf), Unpublished

1. **[^](#cite_ref-Choi1994_28-0)** Choi KP (1994) "On the medians of Gamma distributions and an equation of Ramanujan". *Proc Amer Math Soc* 121 (1) 245–251

1. ^ [***a***](#cite_ref-Pearson1895_29-0) [***b***](#cite_ref-Pearson1895_29-1) Pearson K (1895) Contributions to the Mathematical Theory of Evolution–II. Skew variation in homogenous material. Phil Trans Roy Soc A. 186: 343–414

1. **[^](#cite_ref-Stuart1994_30-0)** Stuart A, Ord JK (1994) *Kendall’s advanced theory of statistics. Vol 1. Distribution theory*. 6th Edition. Edward Arnold, London

1. **[^](#cite_ref-31)** [Relationship between the mean, median, mode, and standard deviation in a unimodal distribution](http://www.se16.info/hgb/median.htm)

1. **[^](#cite_ref-32)** von Hippel, Paul T. (2005) ["Mean, Median, and Skew: Correcting a Textbook Rule"](http://www.amstat.org/publications/jse/v13n2/vonhippel.html) [Archived](https://web.archive.org/web/20160220181456/http://www.amstat.org/publications/jse/v13n2/vonhippel.html) 2016-02-20 at the [Wayback Machine](/source/Wayback_Machine), *Journal of Statistics Education*, 13(2)

1. ^ [***a***](#cite_ref-Dharmadhikari1983_33-0) [***b***](#cite_ref-Dharmadhikari1983_33-1) Dharmadhikari SW, Joag-dev K (1983) Mean, Median, Mode III. Statistica Neerlandica, 33: 165–168

1. **[^](#cite_ref-34)** Bottomly, H.(2002,2006) ["Relationship between the mean, median, mode, and standard deviation in a unimodal distribution"](http://www.se16.info/hgb/median.htm) Personal webpage

1. **[^](#cite_ref-Lesser2005_35-0)** Lesser LM (2005).["Letter to the editor"](http://www.amstat.org/publications/jse/v13n3/lesser_letter.html), [comment on von Hippel (2005)]. *Journal of Statistics Education* 13(2).

1. **[^](#cite_ref-Doodson1917_36-0)** Doodson AT (1917) "Relation of the mode, median and mean in frequency functions". *[Biometrika](/source/Biometrika)*, 11 (4) 425–429 [doi](/source/Doi_(identifier)):[10.1093/biomet/11.4.425](https://doi.org/10.1093%2Fbiomet%2F11.4.425)

1. **[^](#cite_ref-Haldane1942_37-0)** Haldane JBS (1942) "The mode and median of a nearly normal distribution with given cumulants". *[Biometrika](/source/Biometrika)*, 32: 294–299

1. ^ [***a***](#cite_ref-Hall1980_38-0) [***b***](#cite_ref-Hall1980_38-1) 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

1. **[^](#cite_ref-Kendall1958_39-0)** Kendall M.G., Stuart A. (1958) *The advanced theory of statistics*. p53 Vol 1. Griffin. London

1. **[^](#cite_ref-Gauss1823_40-0)** 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

1. **[^](#cite_ref-MacGillivray1981_41-0)** MacGillivray HL (1981) The mean, median, mode inequality and skewness for a class of densities. Aust J Stat 23(2) 247–250

1. **[^](#cite_ref-Groeneveld1986_42-0)** Groeneveld RA (1986) Skewness for the Weibull family. Statistica Neerlandica 40: 135–140

1. **[^](#cite_ref-Johnson1951_43-0)** Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". *Annals of Mathematical Statistics*, 22 (3) 433–439

1. **[^](#cite_ref-vanZwet1979_44-0)** van Zwet W.R. (1979) "Mean, median, mode II". *Statistica Neerlandica* 33(1) 1–5

1. **[^](#cite_ref-Abdous1998_45-0)** Abdous B, Theodorescu R (1998) Mean, median, mode IV. Statistica Neerlandica. 52 (3) 356–359

1. **[^](#cite_ref-46)** van Zwet, W.R. (1964) "Convex transformations of random variables". *Mathematics Centre Tract*, 7, Mathematisch Centrum, Amsterdam

1. **[^](#cite_ref-Durtschi2004_47-0)** 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

1. **[^](#cite_ref-Bowley1920_48-0)** Bowley AL (1920) Elements of statistics. New York: Charles Scribner's Sons

1. **[^](#cite_ref-Hinkley1975_49-0)** Hinkley DV (1975) On power transformations to symmetry. Biometrika 62: 101–111

1. **[^](#cite_ref-Groeneveld1984_50-0)** Groeneveld RA, Meeden G (1984) Measuring skewness and kurtosis. The Statistician, 33: 391–399

v t e Statistics Outline Index Descriptive statistics Continuous data Center Mean Arithmetic Arithmetic-Geometric Contraharmonic Cubic Generalized/power Geometric Harmonic Heronian Heinz Lehmer Median Mode Dispersion Average absolute deviation Coefficient of variation Interquartile range Percentile Range Standard deviation Variance Shape Central limit theorem Moments Kurtosis L-moments Skewness Count data Index of dispersion Summary tables Contingency table Frequency distribution Grouped data Dependence Partial correlation Pearson product-moment correlation Rank correlation Kendall's τ Spearman's ρ Scatter plot Graphics Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q–Q plot Radar chart Run chart Scatter plot Stem-and-leaf display Violin plot Heatmap Scatter Plot Matrix ECDF plot Line chart Statistical data processing Transformations Data transformation Log transformation Power transform Box–Cox transformation Yeo–Johnson transformation Variance-stabilizing transformation Anscombe transform Fisher transformation Scaling and normalization Feature scaling Normalization Standardization (z-score) Min–max normalization Unit vector normalization Data cleaning Data cleaning Outlier Winsorizing Truncation Missing data Data reduction Dimensionality reduction Principal component analysis Factor analysis Time-series preprocessing Differencing Detrending Seasonal adjustment Stationarity transformation Data collection Study design Effect size Missing data Optimal design Population Replication Sample size determination Statistic Statistical power Survey methodology Sampling Cluster Stratified Opinion poll Questionnaire Standard error Controlled experiments Blocking Factorial experiment Interaction Random assignment Randomized controlled trial Randomized experiment Scientific control Adaptive designs Adaptive clinical trial Stochastic approximation Up-and-down designs Observational studies Cohort study Cross-sectional study Natural 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Adapted from the Wikipedia article [Nonparametric skew](https://en.wikipedia.org/wiki/Nonparametric_skew) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Nonparametric_skew?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.