Skewness

The skewness of a random variable X is the third standardized moment ${\displaystyle {\tilde {\mu }}_{3}}$, defined as:[4][5]

${\displaystyle {\tilde {\mu }}_{3}=\operatorname {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{3}\right]={\frac {\mu _{3}}{\sigma ^{3}}}={\frac {\operatorname {E} \left[(X-\mu )^{3}\right]}{(\operatorname {E} \left[(X-\mu )^{2}\right])^{3/2}}}={\frac {\kappa _{3}}{\kappa _{2}^{3/2}}}}$

where μ is the mean, σ is the standard deviation, E is the expectation operator, μ₃ is the third central moment, and κ_t are the t-th cumulants. It is sometimes referred to as Pearson's moment coefficient of skewness,[5] or simply the moment coefficient of skewness,[4] but should not be confused with Pearson's other skewness statistics (see below). The last equality expresses skewness in terms of the ratio of the third cumulant κ₃ to the 1.5th power of the second cumulant κ₂. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant. The skewness is also sometimes denoted Skew[X].

If σ is finite, μ is finite too and skewness can be expressed in terms of the non-central moment E[X³] by expanding the previous formula,

${\displaystyle {\begin{aligned}{\tilde {\mu }}_{3}&=\operatorname {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{3}\right]\\&={\frac {\operatorname {E} [X^{3}]-3\mu \operatorname {E} [X^{2}]+3\mu ^{2}\operatorname {E} [X]-\mu ^{3}}{\sigma ^{3}}}\\&={\frac {\operatorname {E} [X^{3}]-3\mu (\operatorname {E} [X^{2}]-\mu \operatorname {E} [X])-\mu ^{3}}{\sigma ^{3}}}\\&={\frac {\operatorname {E} [X^{3}]-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}.\end{aligned}}}$

Skewness can be infinite, as when

${\displaystyle \Pr \left[X>x\right]=x^{-2}{\mbox{ for }}x>1,\ \Pr[X<1]=0}$

where the third cumulants are infinite, or as when

${\displaystyle \Pr[X<x]=(1-x)^{-3}/2{\mbox{ for negative }}x{\mbox{ and }}\Pr[X>x]=(1+x)^{-3}/2{\mbox{ for positive }}x.}$

where the third cumulant is undefined.

Examples of distributions with finite skewness include the following.

• A normal distribution and any other symmetric distribution with finite third moment has a skewness of 0
• A half-normal distribution has a skewness just below 1
• An exponential distribution has a skewness of 2
• A lognormal distribution can have a skewness of any positive value, depending on its parameters

For a sample of n values, two natural method of moments estimators of the population skewness are[6]

${\displaystyle b_{1}={\frac {m_{3}}{s^{3}}}={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{3}}{\left[{\tfrac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}\right]^{3/2}}}}$

and

${\displaystyle g_{1}={\frac {m_{3}}{m_{2}^{3/2}}}={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{3}}{\left[{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}\right]^{3/2}}},}$

where ${\displaystyle {\overline {x}}}$ is the sample mean, s is the sample standard deviation, m₂ is the (biased) sample second central moment, and m₃ is the sample third central moment.[6]

Another common definition of the sample skewness is[6][7]

${\displaystyle {\begin{aligned}G_{1}&={\frac {k_{3}}{k_{2}^{3/2}}}={\frac {n^{2}}{(n-1)(n-2)}}\;b_{1}={\frac {\sqrt {n(n-1)}}{n-2}}\;g_{1},\\\end{aligned}}}$

where ${\displaystyle k_{3}}$ is the unique symmetric unbiased estimator of the third cumulant and ${\displaystyle k_{2}=s^{2}}$ is the symmetric unbiased estimator of the second cumulant (i.e. the sample variance). This adjusted Fisher–Pearson standardized moment coefficient ${\displaystyle G_{1}}$ is the version found in Excel and several statistical packages including Minitab, SAS and SPSS.[8]

Under the assumption that the underlying random variable ${\displaystyle X}$ is normally distributed, it can be shown that all three ratios ${\displaystyle b_{1}}$, ${\displaystyle g_{1}}$ and ${\displaystyle G_{1}}$ are unbiased and consistent estimators of the population skewness ${\displaystyle \gamma _{1}=0}$, with ${\displaystyle {\sqrt {n}}b_{1}{\xrightarrow {d}}N(0,6)}$, i.e., their distributions converge to a normal distribution with mean 0 and variance 6 (Fisher, 1930).[6] The variance of the sample skewness is thus approximately ${\displaystyle 6/n}$ for sufficiently large samples. More precisely, in a random sample of size n from a normal distribution,[9][10]

${\displaystyle \operatorname {var} (G_{1})={\frac {6n(n-1)}{(n-2)(n+1)(n+3)}}.}$

In normal samples, ${\displaystyle b_{1}}$ has the smaller variance of the three estimators, with[6]

${\displaystyle \operatorname {var} (b_{1})<\operatorname {var} (g_{1})<\operatorname {var} (G_{1}).}$

For non-normal distributions, ${\displaystyle b_{1}}$, ${\displaystyle g_{1}}$ and ${\displaystyle G_{1}}$ are generally biased estimators of the population skewness ${\displaystyle \gamma _{1}}$; their expected values can even have the opposite sign from the true skewness. For instance, a mixed distribution consisting of very thin Gaussians centred at −99, 0.5, and 2 with weights 0.01, 0.66, and 0.33 has a skewness ${\displaystyle \gamma _{1}}$ of about −9.77, but in a sample of 3 ${\displaystyle G_{1}}$ has an expected value of about 0.32, since usually all three samples are in the positive-valued part of the distribution, which is skewed the other way.

The Pearson mode skewness,[12] or first skewness coefficient, is defined as

mean − mode/standard deviation.

The Pearson median skewness, or second skewness coefficient,[13][14] is defined as

3 (mean − median)/standard deviation.

Which is a simple multiple of the nonparametric skew.

Bowley's measure of skewness (from 1901),[15][16] also called Yule's coefficient (from 1912)[17][18] is defined as:

${\displaystyle B_{1}={\frac {{\frac {{{Q}_{3}}+{{Q}_{1}}}{2}}-{{Q}_{2}}}{\frac {{{Q}_{3}}-{{Q}_{1}}}{2}}}={\frac {{{Q}_{3}}-2{{Q}_{2}+{{Q}_{1}}}}{{{Q}_{3}}-{{Q}_{1}}}}.}$

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 ${\displaystyle ({Q}_{3}}-{{Q}_{1})/2}$, which for symmetric distributions is the MAD measure of dispersion.

Other names for this measure are Galton's measure of skewness,[19] the Yule–Kendall index[20] and the quartile skewness,[21]

A more general formulation of a skewness function was described by Groeneveld, R. A. and Meeden, G. (1984):[22][23][24]

${\displaystyle \gamma (u)={\frac {F^{-1}(u)+F^{-1}(1-u)-2F^{-1}(1/2)}{F^{-1}(u)-F^{-1}(1-u)}}}$

where F is the cumulative distribution function. This leads to a corresponding overall measure of skewness[23] 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.[22] The function γ(u) satisfies −1 ≤ γ(u) ≤ 1 and is well defined without requiring the existence of any moments of the distribution.[22] 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.

Bowley's measure of skewness is γ(u) evaluated at u = 3/4. Kelley's measure of skewness uses u = 0.1.[25]

Groeneveld and Meeden have suggested, as an alternative measure of skewness,[22]

${\displaystyle B_{3}=\mathrm {skew} (X)={\frac {(\mu -\nu )}{E(|X-\nu |)}},}$

where μ is the mean, ν is the median, |...| is the absolute value, and E() is the expectation operator. This is closely related in form to Pearson's second skewness coefficient.

Use of L-moments in place of moments provides a measure of skewness known as the L-skewness.[26]

A value of skewness equal to zero does not imply that the probability distribution is symmetric. Thus there is a need for another measure of asymmetry that has this property: such a measure was introduced in 2000.[27] It is called distance skewness and denoted by dSkew. If X is a random variable taking values in the d-dimensional Euclidean space, X has finite expectation, X' is an independent identically distributed copy of X, and ${\displaystyle \|\cdot \|}$ denotes the norm in the Euclidean space, then a simple measure of asymmetry with respect to location parameter θ is

${\displaystyle \operatorname {dSkew} (X):=1-{\frac {\operatorname {E} \|X-X'\|}{\operatorname {E} \|X+X'-2\theta \|}}{\text{ if }}\Pr(X=\theta )\neq 1}$

and dSkew(X) := 0 for X = θ (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only if X is diagonally symmetric with respect to θ (X and 2θ−X have the same probability distribution) and equals 1 if and only if X is a constant c (${\displaystyle c\neq \theta }$) with probability one.[28] Thus there is a simple consistent statistical test of diagonal symmetry based on the sample distance skewness:

${\displaystyle \operatorname {dSkew} _{n}(X):=1-{\frac {\sum _{i,j}\|x_{i}-x_{j}\|}{\sum _{i,j}\|x_{i}+x_{j}-2\theta \|}}.}$

The medcouple is a scale-invariant robust measure of skewness, with a breakdown point of 25%.[29] It is the median of the values of the kernel function

${\displaystyle h(x_{i},x_{j})={\frac {(x_{i}-x_{m})-(x_{m}-x_{j})}{x_{i}-x_{j}}}}$

taken over all couples ${\displaystyle (x_{i},x_{j})}$ such that ${\displaystyle x_{i}\geq x_{m}\geq x_{j}}$, where ${\displaystyle x_{m}}$ is the median of the sample ${\displaystyle \{x_{1},x_{2},\ldots ,x_{n}\}}$. It can be seen as the median of all possible quantile skewness measures.