Skewness

Relationship of mean and median

The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, and likewise for positive skew.

In the older notion of nonparametric skew, defined as ${\displaystyle (\mu -\nu )/\sigma ,}$ where ${\displaystyle \mu }$ is the mean, ${\displaystyle \nu }$ is the median, and ${\displaystyle \sigma }$ 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:

Many textbooks teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. This rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy. Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median.

For example, in the distribution of adult residents across US households, the skew is to the right. However, since the majority of cases is less than or equal to the mode, which is also the median, the mean sits in the heavier left tail. As a result, the rule of thumb that the mean is right of the median under right skew failed.

Definition

Fisher's moment coefficient of skewness

The skewness ${\displaystyle {\gamma }_1}$ of a random variable X is the third standardized moment ${\displaystyle {\tilde{\mu }}_3}$, defined as:

$${\displaystyle \begin{array}{cc}{\gamma }_1& :={\tilde{\mu }}_3\equiv \frac{{\mu }_3}{{\sigma }^3}\\ & =\mathrm{E}\left[{\left(\frac{X-\mu }{\sigma }\right)}^3\right]=\frac{\mathrm{E}\left[(X-\mu {)}^3\right]}{{\left(\mathrm{E}\left[(X-\mu {)}^2\right]\right)}^{3/2}}\\ & =\frac{{\kappa }_3}{{\kappa }_2^{3/2}}\end{array}}$$ 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, 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 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 and μ is finite too, then skewness can be expressed in terms of the non-central moment E[X³] by expanding the previous formula: $${\displaystyle \begin{array}{cc}{\tilde{\mu }}_3& =\mathrm{E}\left[{\left(\frac{X-\mu }{\sigma }\right)}^3\right]\\ & =\frac{\mathrm{E}[X^3]-3\mu \mathrm{E}[X^2]+3{\mu }^2\mathrm{E}[X]-{\mu }^3}{{\sigma }^3}\\ & =\frac{\mathrm{E}[X^3]-3\mu \mathrm{E}[X^2]+3{\mu }^2\mu -{\mu }^3}{{\sigma }^3}\\ & =\frac{\mathrm{E}[X^3]-3\mu \mathrm{E}[X^2]+2{\mu }^3}{{\sigma }^3}.\end{array}}$$

Examples

Skewness can be infinite, as when $${\displaystyle Pr(X=x)=\{\begin{array}{ll}{x}^{-2}& \text{ for }x>1,\\ 0& \text{ for }x<1,\end{array}}$$ where the third cumulants are infinite, or as when $${\displaystyle Pr[X<x]=\{\begin{array}{ll}\frac{1}{2}(1-x{)}^{-3}& \text{ for }x<0,\\ \frac{1}{2}(1+x{)}^{-3}& \text{ for }x>0.\end{array}}$$ 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

$${\displaystyle b_1=\frac{m_3}{s^3}=\frac{\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^3}{{\left[\frac{1}{n-1}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2\right]}^{3/2}}}$$

and

$${\displaystyle g_1=\frac{m_3}{m_2^{3/2}}=\frac{\frac{1}{n}\sum _{i=1}^n(x_i-\overline{x}{)}^3}{{\left[\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2\right]}^{3/2}},}$$

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 (biased) sample third central moment. ${\displaystyle g_1}$ is a method of moments estimator.

Another common definition of the sample skewness is

$${\displaystyle \begin{array}{cc}{G}_1& =\frac{k_3}{k_2^{3/2}}=\frac{n^2}{(n-1)(n-2)}{b}_1=\frac{\sqrt{n(n-1)}}{n-2}{g}_1,\end{array}}$$

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.

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 ${\displaystyle \sqrt{n}{b}_1\stackrel{d}{\to }N(0,6)}$, i.e., their distributions converge to a normal distribution with mean 0 and variance 6 (Fisher, 1930). 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,

$${\displaystyle \mathrm{var}(G_1)=\frac{6n(n-1)}{(n-2)(n+1)(n+3)}.}$$

In normal samples, ${\displaystyle b_1}$ has the smaller variance of the three estimators, with

$${\displaystyle \mathrm{var}(b_1)<\mathrm{var}(g_1)<\mathrm{var}(G_1).}$$

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.

Applications

Skewness is a descriptive statistic that can be used in conjunction with the histogram and the normal quantile plot to characterize the data or distribution.

Skewness indicates the direction and relative magnitude of a distribution's deviation from the normal distribution.

With pronounced skewness, standard statistical inference procedures such as a confidence interval for a mean will be not only incorrect, in the sense that the true coverage level will differ from the nominal (e.g., 95%) level, but they will also result in unequal error probabilities on each side.

Skewness can be used to obtain approximate probabilities and quantiles of distributions (such as value at risk in finance) via the Cornish–Fisher expansion.

Many models assume normal distribution; i.e., data are symmetric about the mean. The normal distribution has a skewness of zero. But in reality, data points may not be perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative.

D'Agostino's K-squared test is a goodness-of-fit normality test based on sample skewness and sample kurtosis.

Other measures of skewness

Other measures of skewness have been used, including simpler calculations suggested by Karl Pearson (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

⁠mean − mode/standard deviation⁠.

Pearson's second skewness coefficient (median skewness)

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

⁠3 (mean − median)/standard deviation⁠.

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: $${\displaystyle \frac{\frac{Q(3/4)+Q(1/4)}{2}-Q(1/2)}{\frac{Q(3/4)-Q(1/4)}{2}}=\frac{Q(3/4)+Q(1/4)-2Q(1/2)}{Q(3/4)-Q(1/4)},}$$ 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 ${\displaystyle \frac{Q(3/4)-Q(1/4)}{2}}$, 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 $${\displaystyle \frac{Q(9/10)+Q(1/10)-2Q(1/2)}{Q(9/10)-Q(1/10)}.}$$

A more general formulation of a skewness function was described by Groeneveld, R. A. and Meeden, G. (1984): $${\displaystyle \gamma (u)=\frac{Q(u)+Q(1-u)-2Q(1/2)}{Q(u)-Q(1-u)}}$$ The function γ(u) satisfies −1 ≤ γ(u) ≤ 1 and is well defined without requiring the existence of any moments of the distribution. Bowley's measure of skewness is γ(u) evaluated at u = 3/4 while Kelly's measure of skewness is γ(u) evaluated at u = 9/10. This definition leads to a corresponding overall measure of skewness 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.

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.

Groeneveld and Meeden's coefficient

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

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

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.

L-moments

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

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 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 ${\displaystyle \Vert \cdot \Vert }$ denotes the norm in the Euclidean space, then a simple measure of asymmetry with respect to location parameter θ is $${\displaystyle \mathrm{dSkew}(X):=1-\frac{\mathrm{E}\Vert X-X^{\prime }\Vert }{\mathrm{E}\Vert X+X^{\prime }-2\theta \Vert }\text{ if }Pr(X=\theta )\ne 1}$$ 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 (${\displaystyle c\ne \theta }$) with probability one. Thus there is a simple consistent statistical test of diagonal symmetry based on the sample distance skewness: $${\displaystyle {\mathrm{dSkew}}_n(X):=1-\frac{\sum _{i,j}\Vert x_i-x_j\Vert }{\sum _{i,j}\Vert x_i+x_j-2\theta \Vert }.}$$

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 $${\displaystyle h(x_i,x_j)=\frac{(x_i-x_m)-(x_m-x_j)}{x_i-x_j}}$$ taken over all couples ${\displaystyle (x_i,x_j)}$ such that ${\displaystyle x_i\ge x_m\ge x_j}$, where ${\displaystyle x_m}$ is the median of the sample ${\displaystyle \{x_1,x_2,\dots ,x_n\}}$. It can be seen as the median of all possible quantile skewness measures.