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 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 set of N values the skewness can be calculated as Σ_i(x_i - μ)³ / Nσ³, where x_i is the i^th 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

<math> \mbox{Skew} = \frac{n}{(n-1)(n-2)}

\sum_{i=1}^N \left( \frac{x_i - \bar{x}}{\sigma} \right)^3 </math>

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

See also: mean, variance, kurtosis.