Skewness of Chi-Squared Distribution

Theorem
Let $n$ be a strictly positive integer.

Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom.

Then the skewness $\gamma_1$ of $X$ is given by:


 * $\gamma_1 = \sqrt{\dfrac 8 n}$

Proof
From Skewness in terms of Non-Central Moments, we have:


 * $\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$

where:
 * $\mu$ is the expectation of $X$.
 * $\sigma$ is the standard deviation of $X$.

By Expectation of Chi-Squared Distribution we have:


 * $\mu = n$

By Variance of Chi-Squared Distribution we have:


 * $\sigma = \sqrt {2 n}$

We also have:

So: