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#.