Skewness

Definition

Let $X$ be a real random variable with $E[|X|^3] < \infty$, mean $\mu$ and standard deviation $\sigma > 0$. The skewness of $X$ is $$\gamma_1 = E\left[\left(\frac{X - \mu}{\sigma}\right)^3\right] = \frac{\mu_3}{\sigma^3},$$ where $\mu_3$ is the third central moment.

Remarks

A distribution that is symmetric about its mean has skewness $0$. The converse is false. Positive skewness usually means a longer tail on the right. Negative skewness usually means a longer tail on the left. The skewness does not change under $X \mapsto aX + b$ with $a > 0$. For $a < 0$, it changes sign. The exponential distribution has skewness $2$.

Explanations

L1 human

Skewness tells to which side a distribution leans. Incomes have positive skewness: most values are moderate, and a few are very large.