D'Agostino's K-squared test

Skewness and kurtosis

In the following, { x_i } denotes a sample of n observations, g₁ and g₂ are the sample skewness and kurtosis, m_j’s are the j-th sample central moments, and $\bar{x}$ is the sample mean. Frequently in the literature related to normality testing, the skewness and kurtosis are denoted as √β₁ and β₂ respectively. Such notation can be inconvenient since, for example, √β₁ can be a negative quantity.

The sample skewness and kurtosis are defined as

\begin{matrix} g_{1} = \frac{m_{3}}{m_{2}^{3 / 2}} = \frac{\frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{3}}{{(\frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2})}^{3 / 2}}, \\ g_{2} = \frac{m_{4}}{m_{2}^{2}} - 3 = \frac{\frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{4}}{{(\frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2})}^{2}} - 3 . \end{matrix}

These quantities consistently estimate the theoretical skewness and kurtosis of the distribution, respectively. Moreover, if the sample indeed comes from a normal population, then the exact finite sample distributions of the skewness and kurtosis can themselves be analysed in terms of their means μ₁, variances μ₂, skewnesses γ₁, and kurtosis γ₂. This has been done by Pearson (1931), who derived the following expressions:

\begin{matrix} μ_{1} (g_{1}) = 0, \\ μ_{2} (g_{1}) = \frac{6 (n - 2)}{(n + 1) (n + 3)}, \\ γ_{1} (g_{1}) \equiv \frac{μ_{3} (g_{1})}{μ_{2} (g_{1})^{3 / 2}} = 0, \\ γ_{2} (g_{1}) \equiv \frac{μ_{4} (g_{1})}{μ_{2} (g_{1})^{2}} - 3 = \frac{36 (n - 7) (n^{2} + 2 n - 5)}{(n - 2) (n + 5) (n + 7) (n + 9)} . \end{matrix}

and

\begin{matrix} μ_{1} (g_{2}) = - \frac{6}{n + 1}, \\ μ_{2} (g_{2}) = \frac{24 n (n - 2) (n - 3)}{(n + 1)^{2} (n + 3) (n + 5)}, \\ γ_{1} (g_{2}) \equiv \frac{μ_{3} (g_{2})}{μ_{2} (g_{2})^{3 / 2}} = \frac{6 (n^{2} - 5 n + 2)}{(n + 7) (n + 9)} \sqrt{\frac{6 (n + 3) (n + 5)}{n (n - 2) (n - 3)}}, \\ γ_{2} (g_{2}) \equiv \frac{μ_{4} (g_{2})}{μ_{2} (g_{2})^{2}} - 3 = \frac{36 (15 n^{6} - 36 n^{5} - 628 n^{4} + 982 n^{3} + 5777 n^{2} - 6402 n + 900)}{n (n - 3) (n - 2) (n + 7) (n + 9) (n + 11) (n + 13)} . \end{matrix}

For example, a sample with size n = 1000 drawn from a normally distributed population can be expected to have a skewness of 0, SD 0.08 and a kurtosis of 0, SD 0.15, where SD indicates the standard deviation.

2 more chapters are free with a Lably account.

D'Agostino's K-squared test

Skewness and kurtosis

Transformed sample skewness and kurtosis

Omnibus K2 statistic

More in Statistics