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 (right-skewed) if the higher tail is longer and negative skew (left-skewed) if the lower tail is longer; getting these the wrong way round is a common error.

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 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{\sqrt{n(n-1)}}{(n-2)}

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

where <math>\sigma<math> is the sample standard deviation and <math>\bar{x}<math> is the sample mean.

Pearson Skewness Coefficients

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

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

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

See also

• cumulant
• kurtosis

External links

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