Skewness

Topics: Measure of Shape - Statistics

---

(definition)

Given a variable $X$, its skewness is a measure of how asymmetrical its data is with respect to its mean.

Let $x_1,\dots ,x_n$ be observations of the variable $X$. The skewness of $X$ is defined as:

$$\mathrm{Sk}(X)=\frac{1}{s^3}\left[\frac{1}{n}\sum _{i=1}^n{\left(x_i-\overline{X}\right)}^3\right]$$

…where $s$ is $X$’s standard deviation.

Moments

Notice that the second and third moment of $X$, which we’ll denote by $m_2$ and $m_3$ respectively, are involved in this definition:

$$\begin{array}{rl}\mathrm{sk}(X)& =\frac{1}{s^3}(m_3)\\ & =\frac{m_3}{(m_2{)}^{3/2}}\end{array}$$

When skewness is negative, mass tends to be concentrated to the right of the mean (with a tail on the left); when it’s positive, it tends to be concentrated to its left (with a tail on the right):

(courtesy of Wikimedia Commons)

Negative and positive skew are sometimes referred to as left-skewed and right-skewed, respectively.