# Skewness

skewedskewskewed distributionright-skewednegative skewDistance skewnessPearson's skewness coefficientspositively skewedright-hand tailright-skewed curve
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

Based on the formula of nonparametric skew, defined as the sequence is negative. 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.

### Outlier

We can transform this sequence into a negatively skewed distribution by adding a value far below the mean, which is probably a negative outlier, e.g. (40, 49, 50, 51).
In the former case one wishes to discard them or use statistics that are robust to outliers, while in the latter case they indicate that the distribution has high skewness and that one should be very cautious in using tools or intuitions that assume a normal distribution.

### 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 a small proportion of 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.

### 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 valueaveragepopulation mean
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
This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant.
Like skewness, kurtosis describes the shape of a probability distribution and, like 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.

### Multimodal distribution

bimodalbimodal distributionmultimodal
It can fail in multimodal distributions, or in distributions where one tail is long but the other is heavy.
and S i and K i are the skewness and kurtosis of the i th 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 0, 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.

### Q–Q plot

plotting positionQ-Q plotProbability plot correlation coefficient
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.
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, 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.

### L-moment

L-momentsL-kurtosisL-scale
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).

### 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.

### Histogram

histogramsbin sizebin
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.
where g_1 is the estimated 3rd-moment-skewness of the distribution and

### Normality test

Non-normality of errorsnormality testingNormality tests
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 in financial modeling is the risk that results when observations are not spread symmetrically around an average value, but instead have a skewed distribution.

### Shape parameter

shape
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.

### 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 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.
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).

### Skew normal distribution

Skewskew normalskew-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

Cornish-Fisher expansion
Skewness can be used to obtain approximate probabilities and quantiles of distributions (such as 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.