Skewness

Skewness is a statistical measure of the asymmetry of a probability distribution around its mean. It describes whether data tend to have tails that extend more to one side than the other.

Population and sample definitions: The population skewness, often denoted gamma1, is defined as the third standardized moment: E[(X − μ)³] / σ³, where μ is the mean and σ is the standard deviation. A positive gamma1 indicates a distribution with a longer or fatter tail on the right (right-skewed); a negative gamma1 indicates a left tail. A gamma1 of zero suggests symmetry of the distribution around its center.

A common way to estimate skewness from data is to compute the sample skewness. One widely used estimator is g1 = [n / ((n−1)(n−2))] × ∑[(xi − x̄)³] / s³, where xi are data values, x̄ is the sample mean, s is the sample standard deviation, and n is the sample size. Other formulas exist and may be biased for small samples, but bias-adjusted versions are available.

Related measures include Pearson’s moment coefficient of skewness, which equals the standardized third moment, and location-based or quartile-based measures such as Pearson’s simple skewness (3(mean − median)/sd) and Bowley’s skewness (Q3 − 2Q2 + Q1) / (Q3 − Q1), where Q1, Q2, Q3 are the first, second, and third quartiles.

Interpretation and cautions: Zero skewness implies symmetry in many standard distributions, but zero does not guarantee symmetry in all cases. Skewness is sensitive to scale and outliers, and it can be unreliable with small samples. It is often complemented by visual tools (histograms, Q–Q plots) and by considering transformations or alternative distributions when modeling skewed data.