Skewed distribution

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 $$\gamma_1$$ and defined as


 * $$\gamma_1 = \frac{\mu_3}{\sigma^3}, \!$$

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


 * $$\gamma_1 = \frac{\kappa_3}{\kappa_2^{3/2}}. \!$$

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


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

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

Given samples from a population, the equation for the sample skewness $$g_1$$ above is a biased estimator of the population skewness. The usual estimator of skewness is


 * $$G_1 = \frac{k_3}{k_2^{3/2}}

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

where $$k_3$$ is the unique symmetric unbiased estimator of the third cumulant and $$k_2$$ is the symmetric unbiased estimator of the second cumulant. Unfortunately $$G_1$$ is, nevertheless, generally biased. Its expected value can even have the opposite sign from the true skewness.

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] / &radic;n.

Section to develop: Why should we care about skew? what difference does it make!

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.
 * 3 (mean minus mode) / standard deviation
 * 3 (mean minus median) / standard deviation