Skewness

From Academic Kids

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 <math>\gamma_1<math> and defined as

<math>\gamma_1 = \frac{\mu_3}{\sigma^3}, \!<math>

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

<math>\gamma_1 = \frac{\kappa_3}{\kappa_2^{3/2}}. \!<math>

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

<math>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}}, \!<math>

where <math>x_i<math> is the ith value, <math>\bar{x}<math> is the sample mean, <math>m_3<math> is the sample third central moment, and <math>m_2<math> is the sample variance.

Given samples from a population, the equation for the sample skewness <math>g_1<math> above is a biased estimator of the population skewness. An unbiased estimator of skewness is

<math>G_1 = \frac{k_3}{k_2^{3/2}}

= \frac{\sqrt{n\,(n-1)}}{n-2}\; g_1, \!<math>

where <math>k_3<math> is the unique symmetric unbiased estimator of the third cumulant and <math>k_2<math> 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.

External links

pt:Obliquidade

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