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 (confusing the two is a common error).

Skewness, the third standardized moment, is written as and defined as

where is the third moment about the mean and is the standard deviation. Equivalently, skewness can be defined as the ratio of the third cumulant and the third power of the square root of the second cumulant :

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

where is the i^th value, is the sample mean, is the sample third central moment, and is the sample variance.

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

where is the unique symmetric unbiased estimator of the third cumulant and is the symmetric unbiased estimator of the second cumulant.

The skewness of a random variable X is sometimes denoted Skew[X]. 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.

Pearson skewness coefficients

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

• 3(mean minus mode)/standard deviation
• (mean minus 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.

External links

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

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