# 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 [itex]\gamma_1[itex] and defined as

[itex]\gamma_1 = \frac{\mu_3}{\sigma^3}, \![itex]

where [itex]\mu_3[itex] is the third moment about the mean and [itex]\sigma[itex] is the standard deviation. Equivalently, skewness can be defined as the ratio of the third cumulant [itex]\kappa_3[itex] and the third power of the square root of the second cumulant [itex]\kappa_2[itex]:

[itex]\gamma_1 = \frac{\kappa_3}{\kappa_2^{3/2}}. \![itex]

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

[itex]g_1 = \frac{m_3}{m_2^{3/2}} = \frac{\sqrt{n\,}\sum_{i=1}^N (x_i-\bar{x})^3}{\left(\sum_{i=1}^N (x_i-\bar{x})^2\right)^{3/2}}, \![itex]

where [itex]x_i[itex] is the ith value, [itex]\bar{x}[itex] is the sample mean, [itex]m_3[itex] is the sample third central moment, and [itex]m_2[itex] is the sample variance.

Given samples from a population, the equation for the sample skewness [itex]g_1[itex] above is a biased estimator of the population skewness. An unbiased estimator of skewness is

[itex]G_1 = \frac{k_3}{k_2^{3/2}}

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

where [itex]k_3[itex] is the unique symmetric unbiased estimator of the third cumulant and [itex]k_2[itex] 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:

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

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