Skewness

skewedskewskewed distributionright-skewednegative skewDistance skewnesspositively skewedright-hand tailright-skewed curveskewnesses
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.wikipedia
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Nonparametric skew

In the older notion of nonparametric skew, defined as where \mu is the mean, \nu is the median, and \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.
It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean.

Median

averagesample medianmedian-unbiased estimator
In the older notion of nonparametric skew, defined as where \mu is the mean, \nu is the median, and \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. If the distribution is both symmetric and unimodal, then the mean = median = mode.
The basic advantage of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a "typical" value.

Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
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.
Skewness: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution.

Mode (statistics)

modemodalmodes
If the distribution is both symmetric and unimodal, then the mean = median = mode.
The numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions.

Mean

mean valuepopulation meanaverage
In the older notion of nonparametric skew, defined as where \mu is the mean, \nu is the median, and \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. If the distribution is both symmetric and unimodal, then the mean = median = mode.
The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode).

Kurtosis

excess kurtosisleptokurticplatykurtic
The last equality expresses skewness in terms of the ratio of the third cumulant κ 3 to the 1.5th power of the second cumulant κ 2 . This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant.
In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population.

Symmetric probability distribution

symmetricsymmetric distributionSymmetry
If the distribution is symmetric, then the mean is equal to the median, and the distribution has zero skewness.
Every measure of skewness equals zero for a symmetric distribution.

Unimodality

unimodalunimodal distributionunimodal function
If the distribution is both symmetric and unimodal, then the mean = median = mode. For a unimodal distribution, negative skew commonly indicates that the tail is on the left side of the distribution, and positive skew indicates that the tail is on the right.
Rohatgi and Szekely have shown that the skewness and kurtosis of a unimodal distribution are related by the inequality:

Moment (mathematics)

momentsmomentraw moment
where is the sample mean, s is the sample standard deviation, and the numerator m 3 is the sample third central moment.
If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.

Coskewness

this example
Note that the assumption that the variables be independent for the above formula is very important because it is possible even for the sum of two Gaussian variables to have a skewed distribution (see this example).
Coskewness is the third standardized cross central moment, related to skewness as covariance is related to variance.

Multimodal distribution

bimodalbimodal distributionmultimodal
It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy.
where γ is the skewness and κ is the kurtosis.

Central moment

central momentsmoment about the meanmoments about the mean
where μ is the mean, σ is the standard deviation, E is the expectation operator, μ 3 is the third central moment, and κ t are the t th cumulants.
The third and fourth central moments are used to define the standardized moments which are used to define skewness and kurtosis, respectively.

Q–Q plot

plotting positionnormal quantile plotprobability plot correlation coefficient
Skewness is a descriptive statistic that can be used on conjunction with the histogram and the normal quantile plot to characterize the data or distribution.
A Q–Q plot is used to compare the shapes of distributions, providing a graphical view of how properties such as location, scale, and skewness are similar or different in the two distributions.

Standardized moment

standardized central moments
The skewness of a random variable X is the third standardized moment γ 1, defined as:

Medcouple

fast algorithm
The medcouple is a scale-invariant robust measure of skewness, with a breakdown point of 25%.
In statistics, the medcouple is a robust statistic that measures the skewness of a univariate distribution.

D'Agostino's K-squared test

D'Agostino's K-squared test is a goodness-of-fit normality test based on sample skewness and sample kurtosis.
The test is based on transformations of the sample kurtosis and skewness, and has power only against the alternatives that the distribution is skewed and/or kurtic.

L-moment

probability weighted moments
Use of L-moments in place of moments provides a measure of skewness known as the L-skewness.
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).

Histogram

histogramsbin sizebin
Skewness is a descriptive statistic that can be used on conjunction with the histogram and the normal quantile plot to characterize the data or distribution.
where g_1 is the estimated 3rd-moment-skewness of the distribution and

Normality test

Non-normality of errorsnormality testingstatistical test for the normality
D'Agostino's K-squared test is a goodness-of-fit normality test based on sample skewness and sample kurtosis.
Historically, the third and fourth standardized moments (skewness and kurtosis) were some of the earliest tests for normality.

Skewness risk

skewness
Skewness risk
Skewness risk in financial modeling is the risk that results when observations are not spread symmetrically around an average value, but instead have a skewed distribution.

Confidence interval

confidence intervalsconfidence levelconfidence
With pronounced skewness, standard statistical inference procedures such as a confidence interval for a mean will be not only incorrect, in the sense of having true coverage level unequal to the nominal (e.g., 95%) level, but also with unequal error probabilities on each side.
The approximation will be quite good with only a few dozen observations in the sample if the probability distribution of the random variable is not too different from the normal distribution (e.g. its cumulative distribution function does not have any discontinuities and its skewness is moderate).

Shape parameter

shape
Shape parameters
Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the skewness (3rd moment) or kurtosis (4th moment), if the higher moments are defined and finite.

Skew normal distribution

Skewskew normalskew-normal distribution
Skew normal distribution
In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

Cornish–Fisher expansion

Skewness can be used to obtain approximate probabilities and quantiles of distributions (such as the value at risk in finance) via the Cornish-Fisher expansion.
The values γ 1 and γ 2 are the random variable's skewness and (excess) kurtosis respectively.

Probability theory

theory of probabilityprobabilityprobability theorist
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.