In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Roughly speaking, a distribution has positive skew (right-skewed) if the right (higher value) tail is longer and negative skew (left-skewed) if the left (lower value) tail is longer (confusing the two is a common error).

Skewness, the third standardized moment, is written as $\backslash gamma\_1$ and defined as

$\backslash gamma\_1\; =\; \backslash frac\{\backslash mu\_3\}\{\backslash sigma^3\},\; \backslash !$

where $\backslash mu\_3$ is the third moment about the mean and $\backslash sigma$ is the standard deviation. Equivalently, skewness can be defined as the ratio of the third cumulant $\backslash kappa\_3$ and the third power of the square root of the second cumulant $\backslash kappa\_2$:

$\backslash gamma\_1\; =\; \backslash frac\{\backslash kappa\_3\}\{\backslash kappa\_2^\{3/2\}\}.\; \backslash !$

This is analogous to the definition of kurtosis, which is expressed as the fourth cumulant divided by the fourth power of the square root of the second cumulant.

For a sample of N values the sample skewness is

$g\_1\; =\; \backslash frac\{m\_3\}\{m\_2^\{3/2\}\}\; =\; \backslash frac\{\backslash sqrt\{n\backslash ,\}\backslash sum\_\{i=1\}^N\; (x\_i-\backslash bar\{x\})^3\}\{\backslash left(\backslash sum\_\{i=1\}^N\; (x\_i-\backslash bar\{x\})^2\backslash right)^\{3/2\}\},\; \backslash !$

where $x\_i$ is the i^th value, $\backslash bar\{x\}$ is the sample mean, $m\_3$ is the sample third central moment, and $m\_2$ is the sample variance.

Given samples from a population, the equation for the sample skewness $g\_1$ above is a biased estimator of the population skewness. The usual estimator of skewness is

$G\_1\; =\; \backslash frac\{k\_3\}\{k\_2^\{3/2\}\}$

= \frac{\sqrt{n\,(n-1)}}{n-2}\; g_1, \!

where $k\_3$ is the unique symmetric unbiased estimator of the third cumulant and $k\_2$ is the symmetric unbiased estimator of the second cumulant. Unfortunately $G\_1$ is, nevertheless, generally biased. Its expected value can even have the opposite sign from the true skewness.

The skewness of a random variable X is sometimes denoted SkewIf Y is the sum of n independent random variables, all with the same distribution as X, then it can be shown that Skew^* / √n.

Skewness has benefits in many areas. Many simplistic models assume normal distribution i.e. data is symmetric about the mean. But in reality, data points are not perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative.

Sample skewness

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Tha sample skewness can be calculated using formula

$D\; =\; \{1\; \backslash over\; n\}\; \backslash sum\_\{i=1\}^n\{\; (x\_i\; -\; \backslash bar\{x\})^2\}$,

$A=\backslash frac\{1\}\{n\; D^\{\backslash frac\{3\}\{2\}\}\}\; \backslash sum\_\{i=1\}^n\; (x\_i\; -\; \backslash overline\{x\})^3$

where n - the sample size, D - the pre-computed variance, x_i - the value of the x'th measurement and $\backslash bar\{x\}$ - the pre-computed arithmetic mean.

Pearson skewness coefficients

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Karl Pearson suggested two simpler calculations as a measure of skewness:

• 3 (mean - mode) / standard deviation
• 3 (mean - median) / standard deviation

There is no guarantee that these will be the same sign as each other or as the ordinary definition of skewness.

See also

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• kurtosis

External links

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• Free Online Software (Calculator) computes various types of Skewness and Kurtosis statistics for any dataset (includes small and large sample tests).

Probability theory | Statistics

Schiefe (Statistik) | Asimetrijas koeficients | 歪度 | Obliquidade | Коэффициент асимметрии | Skewness