Skewness

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Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790)

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 right (higher value) tail is longer or fatter and negative skew (left-skewed) if the left (lower value) tail is longer or fatter. The two are often confused, since most of the mass of a right (or left) skewed distribution is to the left (or right) of its respective tail.

Skewness, the third standardized moment, is written as $γ 1$ and defined as

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

where $μ 3$ is the third moment about the mean and $σ$ is the standard deviation. Equivalently, skewness can be defined as the ratio of the third cumulant $κ 3$ and the third power of the square root of the second cumulant $κ 2$ :

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

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 i^th 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] / √n.

Skewness has benefits in many areas. Many simplistic models assume normal distribution i.e. data is symmetric about the mean. But in reality, data points are not perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be positive or negative.