Table 2 Recommended critical difference (CD) approximate tests for 1 × N and N × N comparisons of Friedman rank sums

1 × N

\( C{D}_N={z}_{\alpha /{c}_1}\sqrt{nk\left( k+1\right)/6},\kern0.75em {c}_1= k-1 \)

Demšar [2]

\( C{D}_M={m}_{\alpha, df= k-1,\rho ={\scriptscriptstyle \frac{1}{2}}}\sqrt{nk\left( k+1\right)/6} \)

Siegel and Castellan [18], Nemenyi [39], Miller [25], Hollander et al. [23], Zarr [20]

N × N

\( C{D}_N={z}_{{\scriptscriptstyle \frac{1}{2}}\alpha /{c}_2}\sqrt{nk\left( k+1\right)/6},\kern0.5em {c}_2= k\left( k-1\right)/2 \)

Siegel and Castellan [18], Gibbons and Chakraborti [21], Daniel [19], Hettmansperger [33], Sheskin [22]

\( \begin{array}{l} C{D}_Q = {q}_{\alpha, df= k,\infty}\sqrt{nk\left( k+1\right)/12}=\\ {}\kern3.25em \frac{q_{\alpha, df= k,\infty }}{\sqrt{2}}\sqrt{nk\left( k+1\right)/6}\end{array} \)

Nemenyi [39], Miller [25], Hollander et al. [23], Zarr [20], Desu and Raghavarao [40], Demšar [2]

\( C{D}_{\chi^2}=\sqrt{\chi_{\alpha, df= k-1}^2}\sqrt{nk\left( k+1\right)/6} \)

Miller [25], Bortz et al. [41], Wike [42]