Wilcoxon-Mann-Whitney-test for two independent groups

$$\begin{equation*} z_{r_1} = \frac{r_1-n_1(n+1)/2 \pm \mathbf{0.5}}{\sqrt{n_1n_2(n+1)/12}}, \end{equation*}$$ where $r_1$ is the sum of the rank numbers in the smallest group ($n_1\le n_2$) and $n=n_1+n_2$ and the continuity correction $\pm \mathbf{0.5}$ equals

• $+0.5$ for a left-sided test
• $-0.5$ for a right-sided test
• $+0.5$ for a two-sided test if $r_1\le n_1(n+1)/2$, and
• $-0.5$ for a two-sided test if $r_1> n_1(n+1)/2$.

Kruskal-Wallis-test for more than two independent groups

$$\begin{equation*} \chi^2 = \Bigg(\frac{12}{n(n+1)}\Bigg)\sum_{j=1}^g\limits n_j(\overline{r}_j-\overline{r})^2, \end{equation*}$$ where $\overline{r}_j$ is the mean of rank numbers in group $j$ of size $n_j$, with $g$ groups in total and $df=g-1$ degrees of freedom.

$Z$-score sign-test for two dependent groups

$$\begin{equation*} z_{p}=\frac{p-0.5}{\sqrt{0.25/n}}, \end{equation*}$$ where $n$ is the number of pairs of observations.

Wilcoxon's signed rank-test for two dependent groups

$$\begin{equation*} W_{+} = \sum{r_{d_i}}, \end{equation*}$$ where $r_{d_{i}}$ denotes the rank score of the positive difference score $i$.