Theorem 1.7.4. Alternate Formulas for Skewness and Kurtosis.
Skewness =
\begin{equation*}
\gamma_1 = \frac{1}{\sigma^3}
\left [ \frac{\sum_{k=1}^n x_k^3}{n} - 3 v \mu - \mu^3 \right ]
\\ = \frac{1}{\sigma^3}
\left [ \frac{\sum_{k=1}^n x_k^3}{n} - 3 \mu \sum_{k=1}^n \frac{x_k^2}{n} + 2 \mu^3 \right ]
\end{equation*}
and Kurtosis =
\begin{equation*}
\gamma_2 = \frac{1}{\sigma^4} \left [ \frac{\sum_{k=1}^n x_k^4}{n} - 4 \mu \frac{\sum_{k=1}^n x_k^3 }{n} + 6 \mu^2 v + 3 \mu^4
\right ]
\\ = \frac{1}{\sigma^4}
\left [ \frac{\sum_{k=1}^n x_k^4}{n} - 4 \mu \frac{\sum_{k=1}^n x_k^3 }{n} + 6 \mu^2 \frac{\sum_{k=1}^n x_k^2}{n} - 3 \mu^4 \right ]
\end{equation*}
Similar formulas can be used for sample data with \(\mu\) replaced with \(\overline{x}\) and then by adjusting at the end with the factor \(\frac{n}{n-1}\) in the same way that was used when computing the sample variance.