In probability theory and statistics, skewness is a measure of the asymmetry of the probability
distribution of a real-valued random variable about its mean. The skewness value can be
positive or negative, or even undefined. The qualitative interpretation of the skew
is complicated. For a unimodal distribution, negative skew indicates that the tail on the
left side of the probability density function is longer or fatter than the right side – it
does not distinguish these shapes. Conversely, positive skew indicates that the tail on the
right side is longer or fatter than the left side. In cases where one tail is long but
the other tail is fat, skewness does not obey a simple rule. For example, a zero value indicates
that the tails on both sides of the mean balance out, which is the case both for a symmetric
distribution, and for asymmetric distributions where the asymmetries even out, such as one
tail being long but thin, and the other being short but fat. Further, in multimodal distributions
and discrete distributions, skewness is also difficult to interpret. Importantly, the skewness
does not determine the relationship of mean and median.
Introduction Consider the distribution in the figure. The
bars on the right side of the distribution taper differently than the bars on the left
side. These tapering sides are called tails, and they provide a visual means for determining
which of the two kinds of skewness a distribution has:
negative skew: The left tail is longer; the mass of the distribution is concentrated on
the right of the figure. The distribution is said to be left-skewed, left-tailed, or
skewed to the left. Example: 40, 49, 50, 51. positive skew: The right tail is longer; the
mass of the distribution is concentrated on the left of the figure. The distribution is
said to be right-skewed, right-tailed, or skewed to the right. Example: 49, 50, 51,
60. Relationship of mean and median
The skewness is not strictly connected with the relationship between the mean and median:
a distribution with negative skew can have the mean greater than or less than the median,
and likewise for positive skew. In the older notion of nonparametric skew,
defined as where µ is the mean, ν is the median, and σ is the standard deviation,
the skewness is defined in terms of this relationship: positive/right nonparametric skew means the
mean is greater than the median, while negative/left nonparametric skew means the mean is less
than the median. However, the modern definition of skewness and the traditional nonparametric
definition do not in general have the same sign: while they agree for some families of
distributions, they differ in general, and conflating them is misleading.
If the distribution is symmetric then the mean is equal to the median and the distribution
will have zero skewness. If, in addition, the distribution is 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 does not imply that the
mean is equal to the median. "Many textbooks," a 2005 article points out,
"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. [But] this rule fails with surprising frequency.
It can fail in multimodal distributions, or in distributions where one tail is long but
the other is fat. 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."
Definition Pearson's moment coefficient of skewness
The skewness of a random variable X is the third standardized moment, denoted γ1 and
defined as
where μ3 is the third central moment, μ is the mean, σ is the standard deviation,
and E is the expectation operator. The last equality expresses skewness in terms of the
ratio of the third cumulant κ3 and 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].
The formula expressing skewness in terms of the non-central moment E[X3] can be expressed
by expanding the previous formula,
Properties Skewness can be infinite, as when
or undefined, as when
In this latter example, the third cumulant is undefined. One can also have distributions
such as
where both the second and third cumulants are infinite, so the skewness is again undefined.
If Y is the sum of n independent and identically distributed random variables, all with the
distribution of X, then the third cumulant of Y is n times that of X and the second cumulant
of Y is n times that of X, so . This shows that the skewness of the sum is smaller, as
it approaches a Gaussian distribution in accordance with the central limit theorem.
Sample skewness For a sample of n values, a natural method
of moments estimator of the population skewness is
where is the sample mean, s is the sample standard deviation, and the numerator m3 is
the sample third central moment. Another common definition of the sample skewness
is
where is the unique symmetric unbiased estimator of the third cumulant and is the symmetric
unbiased estimator of the second cumulant. In general, the ratios and are both biased
estimators of the population skewness ; their expected values can even have the opposite
sign from the true skewness. Nevertheless, and each have obviously the correct expected
value of zero for any symmetric distribution with a finite third moment, including a normal
distribution. The variance of the skewness of a sample of
size n from a normal distribution is
An approximate alternative is 6/n but this is inaccurate for small samples.
In normal random samples, has the smaller variance of the two estimators, with
where m2 in the denominator is the sample second central moment.
The adjusted Fisher-Pearson standardized moment coefficient is the version found in Excel
and several statistical packages including Minitab, SAS and SPSS.
Applications Skewness has benefits in many areas. 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 Pearson's skewness coefficients
Karl Pearson suggested simpler calculations as a measure of skewness: the Pearson mode
or first skewness coefficient, defined by (mean − mode) / standard deviation,
as well as Pearson's median or second skewness coefficient, defined by
3 / standard deviation. The latter is a simple multiple of the nonparametric
skew. Starting from a standard cumulant expansion
around a Normal distribution, one can actually show that skewness = 6 / standard deviation
+ O(skewness2). One should keep in mind that above given equalities often don't hold even
approximately and these empirical formulas are abandoned nowadays. There is no guarantee
that these will be the same sign as each other or as the ordinary definition of skewness.
Quantile based measures A skewness function
can be defined, where F is the cumulative distribution function. This 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.
The function γ(u) satisfies -1 ≤ γ(u) ≤ 1 and is well defined without requiring
the existence of any moments of the distribution. Galton's measure of skewness is γ(u) evaluated
at u = 3 / 4. Other names for this same quantity are the Bowley Skewness, the Yule-Kendall
index and the quartile skewness. Kelley's measure of skewness uses u = 0.1.
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 which 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 which takes values in the d-dimensional Euclidean space,
X has finite expectation, X' is an independent identically distributed copy of X and denotes
the norm in the Euclidean space then a simple measure of asymmetry is
dSkew := 1 - E||X-X'|| / E||X + X'|| if X is not 0 with probability one,
and dSkew:= 0 for X = 0. Distance skewness is always between 0 and 1, equals 0 if and
only if X is diagonally symmetric and equals 1 if and only if X is a nonzero constant with
probability one. Thus there is a simple consistent statistical test of diagonal symmetry based
on the sample distance skewness: dSkewn(X):= 1- ∑i,j ||xi – xj|| / ∑i,j||xi
+ xj||. Groeneveld & Meeden’s coefficient
Groeneveld & Meeden have suggested, as an alternative measure of skewness,
where μ is the mean, ν is the median, |…| is the absolute value and E() is the expectation
operator. See also
Bragg peak Shape parameters
Skew normal distribution Skewness risk
Notes
References Johnson, NL, Kotz, S, Balakrishnan N Continuous
Univariate Distributions, Vol 1, 2nd Edition Wiley ISBN 0-471-58495-9
MacGillivray, HL. "Shape properties of the g- and h- and Johnson families". Comm. Statistics
— Theory and Methods 21: 1244–1250. External links
Hazewinkel, Michiel, ed., "Asymmetry coefficient", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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