Skewness

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 if the positive tail is longer and negative skew if the negative tail is longer.

Skewness, the third standardized moment, is defined as μ3 / σ3, where μ3 is the third moment about the mean and σ is the standard deviation. The skewness of a random variable X is sometimes denoted Skew[X].

For a sample of N values the sample skewness is Σi(xi − μ)3 / Nσ3, where xi is the ith value and μ is the mean.

If Y is the sum of n independent random variables, all with the same distribution as X, then it can be shown that Skew[Y] = Skew[X] / √n.

Given samples from a population, the equation for population skewness above is a biased estimator of the population skewness. An unbiased estimator of skewness is

\mbox{Skew} = \frac{n}{(n-1)(n-2)} \sum_{i=1}^N \left( \frac{x_i - \mu}{\sigma} \right)^3

where σ is the sample standard deviation and μ is the sample mean.

See also: mean, variance, kurtosis, cumulant.

de:Schiefe

External links

  • Free Online Software (Calculator) (http://www.wessa.net/skewkurt.wasp) computes various types of Skewness and Kurtosis statistics for any dataset (includes small and large sample tests).

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