Skewness in terms of Non-Central Moments

Theorem

Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$.

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

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

$\ds \gamma_1$

$=$

$\ds \expect {\paren {\dfrac {X - \mu} \sigma}^3}$

Definition of Skewness

$\ds $

$=$

$\ds \frac {\expect {X^3 - 3 \mu X^2 + 3 \mu^2 X - \mu^3} } {\sigma^3}$

$\ds $

$=$

$\ds \frac {\expect {X^3} - 3 \mu \expect {X^2} + 3 \mu^2 \expect X - \mu^3} {\sigma^3}$

$\ds $

$=$

$\ds \frac {\expect {X^3} - 3 \mu \paren {\expect {X^2} - \mu \expect X} - \mu^3} {\sigma^3}$

$\ds $

$=$

$\ds \frac {\expect {X^3} - 3 \mu \paren {\expect {X^2} - \paren {\expect X}^2} - \mu^3} {\sigma^3}$

$\mu = \expect X$

$\ds $

$=$

$\ds \frac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$

$\blacksquare$