thumb|100px|Example distribution with positive skewness. The data presented is from experiments on wheat grass growth.

Skewness in probability theory and statistics is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. Similarly to kurtosis, it provides insights into shape-related characteristics of a distribution. The skewness value can be positive, zero, negative, or undefined.

For a unimodal distribution (a distribution with a single peak), negative skew commonly indicates that the is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is thick, skewness does not obey a simple rule. For example, a zero value in skewness means that the tails on both sides of the mean balance out overall; this is the case for a symmetric distribution but can also be true for an asymmetric distribution where one tail is long and thin, and the other is short but thick. Thus, the symmetry of a distribution cannot be inferred using only its skewness; the distribution shape must be taken into account.

Introduction

Consider the two distributions in the figure. Within each graph, the values on the right side of the distribution taper differently from the values on the left side. These tapering sides are called and they provide a visual means to determine which of the two kinds of skewness a distribution has:

  1. : The left tail is longer; the mass of the distribution is concentrated on the right of the figure. The distribution is said to be , or , despite the fact that the curve itself appears to be skewed or leaning to the right; "left" refers instead to the left tail being drawn out and, often, the mean being skewed to the left of a typical center of the data. A left-skewed distribution usually appears as a right-leaning curve.
  2. : The right tail is longer; the mass of the distribution is concentrated on the left of the figure. The distribution is said to be , or , despite the fact that the curve itself appears to be skewed or leaning to the left; "right" refers instead to the right tail being drawn out and, often, the mean being skewed to the right of a typical center of the data. A right-skewed distribution usually appears as a left-leaning curve.

thumb|434x434px|A general relationship of mean and median under differently skewed unimodal distribution.

In the older notion of nonparametric skew, defined as <math>(\mu - \nu)/\sigma,</math> where <math>\mu</math> is the mean, <math>\nu</math> is the median, and <math>\sigma</math> is the standard deviation, the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than (to the right of) the median, while negative/left nonparametric skew means the mean is less than (to the left of) the median. However, the modern definition of skewness and the traditional nonparametric definition do not always have the same sign: while they agree for some families of distributions, they differ in some of the cases, and conflating them is misleading.

If the distribution is symmetric, then the mean is equal to the median, and the distribution has zero skewness. If the distribution is both symmetric and unimodal, then the mean = median = mode. This is the case of a coin toss or the series 1,2,3,4,... Note, however, that the converse is not true in general, i.e. zero skewness (defined below) does not imply that the mean is equal to the median.

A 2005 journal article points out: or simply the moment coefficient of skewness, 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 to the 1.5th power of the second cumulant . 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 .

If is finite and is finite too, then skewness can be expressed in terms of the non-central moment by expanding the previous formula:

<math display="block">

\begin{align}

\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{align}

</math>

<!-- EDITORS BEWARE: DO NOT CHANGE THIS INTO E[X^3] - 3\mu E[X^2] + 2\mu^3 /// SEE TALK PAGE -->

Examples

Skewness can be infinite, as when

<math display="block">\Pr(X=x) = \begin{cases}

x^{-2} & \text{ for } x >1, \\[2pt]

0 & \text{ for } x < 1,

\end{cases}</math>

where the third cumulants are infinite, or as when

<math display="block">\Pr[X<x] = \begin{cases}

\frac{1}{2} (1-x)^{-3} & \text{ for } x < 0, \\[2pt]

\frac{1}{2} (1+x)^{-3} & \text{ for } x > 0.

\end{cases}</math>

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

Sample skewness

For a sample of n values, two natural estimators of the population skewness are

<math display="block">

b_1 = \frac{m_3}{s^3}

= \frac{\tfrac{1}{n} \sum_{i=1}^n \left(x_i-\bar{x}\right)^3}{\left[\tfrac{1}{n-1} \sum_{i=1}^n \left(x_i-\bar{x}\right)^2 \right]^{3/2

</math>

and

<math display="block">

g_1 = \frac{m_3}{m_2^{3/2

= \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i-\bar{x})^3}{\left[\tfrac{1}{n} \sum_{i=1}^n \left(x_i-\bar{x}\right)^2 \right]^{3/2,

</math>

where <math>\bar{x}</math> is the sample mean, is the sample standard deviation, is the (biased) sample second central moment, and is the (biased) sample third central moment.

<math display="block">\begin{align}

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{align}</math>

where <math>k_3</math> is the unique symmetric unbiased estimator of the third cumulant and <math>k_2 = s^2</math> is the symmetric unbiased estimator of the second cumulant (i.e. the sample variance). This adjusted Fisher–Pearson standardized moment coefficient <math> G_1 </math> is the version found in Excel and several statistical packages including Minitab, SAS and SPSS.

<math display="block"> \operatorname{var}(G_1)= \frac{6n ( n - 1 )}{ ( n - 2 )( n + 1 )( n + 3 ) } .</math>

In normal samples, <math>b_1</math> has the smaller variance of the three estimators, with (not to be confused with Pearson's moment coefficient of skewness, see above). These other measures are:

Pearson's first skewness coefficient (mode skewness)

The Pearson mode skewness, or first skewness coefficient, is defined as

Pearson's second skewness coefficient (median skewness)

The Pearson median skewness, or second skewness coefficient, is defined as

Which is a simple multiple of the nonparametric skew.

Quantile-based measures

Bowley's measure of skewness (from 1901), also called Yule's coefficient (from 1912) 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>\frac{Q(3/4)-Q(1/4)}{2}</math>, 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

<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):

<math display="block"> \gamma( u ) = \frac{ Q( u ) +Q( 1 - u )-2Q( 1 / 2 ) }{Q( u ) -Q( 1 - u ) } </math>

The function satisfies and is well defined without requiring the existence of any moments of the distribution. defined as the supremum of this over the range . Another measure can be obtained by integrating the numerator and denominator of this expression.

Distance skewness

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. It is called distance skewness and denoted by . If X is a random variable taking values in the -dimensional Euclidean space, has finite expectation, is an independent identically distributed copy of , and <math>\|\cdot\|</math> denotes the norm in the Euclidean space, then a simple measure of asymmetry with respect to location parameter is

<math display="block">

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

</math>

and for (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only if X is diagonally symmetric with respect to ( and have the same probability distribution) and equals 1 if and only if X is a constant c (<math>c \neq \theta</math>) with probability one. Thus there is a simple consistent statistical test of diagonal symmetry based on the sample distance skewness:

<math display="block">

\operatorname{dSkew}_n(X) := 1 - \frac{\sum_{i,j} \|x_i-x_j\| }{\sum_{i,j} \|x_i+x_j-2\theta \|}.

</math>

Medcouple

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

<math display="block"> h(x_i, x_j) = \frac{ (x_i - x_m) - (x_m - x_j)}{x_i - x_j} </math>

taken over all couples <math>(x_i, x_j)</math> such that <math>x_i \geq x_m \geq x_j</math>, where <math>x_m</math> is the median of the sample <math>\{x_1, x_2, \ldots, x_n\}</math>. It can be seen as the median of all possible quantile skewness measures.

See also

  • Bragg peak
  • Coskewness
  • Shape parameters
  • Skew normal distribution
  • Skewness risk

References

Citations

Sources

  • Premaratne, G., Bera, A. K. (2001). Adjusting the Tests for Skewness and Kurtosis for Distributional Misspecifications. Working Paper Number 01-0116, University of Illinois. Forthcoming in Comm in Statistics, Simulation and Computation. 2016 1–15
  • Premaratne, G., Bera, A. K. (2000). Modeling Asymmetry and Excess Kurtosis in Stock Return Data. Office of Research Working Paper Number 00-0123, University of Illinois.
  • Skewness Measures for the Weibull Distribution
  • An Asymmetry Coefficient for Multivariate Distributions by Michel Petitjean
  • On More Robust Estimation of Skewness and Kurtosis Comparison of skew estimators by Kim and White.
  • Closed-skew Distributions — Simulation, Inversion and Parameter Estimation