% <html><head><title>Upper Critical Values for Kruskal-Wallis (k Samples)</title>

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  {\Large\textbf{Upper Critical Values for the Kruskal-Wallis Test}} \\
  \textbf{($k$ samples)}
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  \begin{tabular}{rl}
    Notes: 1. & In the table below, the critical values give significance
                levels as close \\
              & as possible to, but not exceeding the nominal $\alpha$. 
                The actual levels of \\
              & significance are in brackets. \\
    \ \\
           2. & When the table below is not applicable and $k = 3$ with 
                the three group \\
              & sizes above 5 or $k > 3$ with all group sizes above 4, 
                use the following \\
              & approximation: \\
              & \multicolumn{1}{c}
                {$h_{\alpha}=\chi_{\alpha}^2,\quad\text{df}=k-1.$}
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\footnotesize

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\begin{tabular}{|ccc|c|c|c|c|} \hline 
&&& \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{4-7} \multicolumn{3}{|c|}{Group} &&&& \\
\multicolumn{3}{|c|}{Sizes} & 0.10 & 0.05 & 0.025 & 0.01 \\ \hline 
2 & 2 & 2 & 4.571 (.06667) & - - & - - & - -\\
3 & 2 & 1 & 4.286 (.10000) & - - & - - & - -\\
3 & 2 & 2 & 4.500 (.06667) & 4.714 (.04762) & - - & - -\\
3 & 3 & 1 & 4.571 (.10000) & 5.143 (.04286) & - - & - -\\
3 & 3 & 2 & 4.556 (.10000) & 5.361 (.03214) & 5.556 (.02500) & - -\\
3 & 3 & 3 & 4.622 (.10000) & 5.600 (.05000) & 5.956 (.02500) & 7.200 (.00357)\\
4 & 2 & 1 & 4.500 (.07619) & - - & - - & - -\\
4 & 2 & 2 & 4.458 (.10000) & 5.333 (.03333) & 5.500 (.02381) & - -\\
4 & 3 & 1 & 4.056 (.09286) & 5.208 (.05000) & 5.833 (.02143) & - -\\
4 & 3 & 2 & 4.511 (.09841) & 5.444 (.04603) & 6.000 (.02381) & 6.444 (.00794)\\
4 & 3 & 3 & 4.709 (.09238) & 5.791 (.04571) & 6.155 (.02476) & 6.745 (.01000)\\
4 & 4 & 1 & 4.167 (.08254) & 4.967 (.04762) & 6.167 (.02222) & 6.667 (.00952)\\
4 & 4 & 2 & 4.555 (.09778) & 5.455 (.04571) & 6.327 (.02413) & 7.036 (.00571)\\
4 & 4 & 3 & 4.545 (.09905) & 5.598 (.04866) & 6.394 (.02476) & 7.144 (.00970)\\
4 & 4 & 4 & 4.654 (.09662) & 5.692 (.04866) & 6.615 (.02424) & 7.654 (.00762)\\
5 & 2 & 1 & 4.200 (.09524) & 5.000 (.04762) & - - & - -\\
5 & 2 & 2 & 4.373 (.08995) & 5.160 (.03439) & 6.000 (.01852) & 6.533 (.00794)\\
5 & 3 & 1 & 4.018 (.09524) & 4.960 (.04762) & 6.044 (.01984) & - -\\
5 & 3 & 2 & 4.651 (.09127) & 5.251 (.04921) & 6.004 (.02460) & 6.909 (.00873)\\
5 & 3 & 3 & 4.533 (.09697) & 5.648 (.04892) & 6.315 (.02121) & 7.079 (.00866)\\
5 & 4 & 1 & 3.987 (.09841) & 4.985 (.04444) & 5.858 (.02381) & 6.955 (.00794)\\
5 & 4 & 2 & 4.541 (.09841) & 5.273 (.04877) & 6.068 (.02482) & 7.205 (.00895)\\
5 & 4 & 3 & 4.549 (.09892) & 5.656 (.04863) & 6.410 (.02496) & 7.445 (.00974)\\
5 & 4 & 4 & 4.668 (.09817) & 5.657 (.04906) & 6.673 (.02429) & 7.760 (.00946)\\
5 & 5 & 1 & 4.109 (.08586) & 5.127 (.04618) & 6.000 (.02165) & 7.309 (.00938)\\
5 & 5 & 2 & 4.623 (.09704) & 5.338 (.04726) & 6.346 (.02489) & 7.338 (.00962)\\
5 & 5 & 3 & 4.545 (.09965) & 5.705 (.04612) & 6.549 (.02436) & 7.578 (.00968)\\
5 & 5 & 4 & 4.523 (.09935) & 5.666 (.04931) & 6.760 (.02490) & 7.823 (.00978)\\
5 & 5 & 5 & 4.560 (.09952) & 5.780 (.04878) & 6.740 (.02475) & 8.000 (.00946)\\
\hline
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\begin{tabular}{|ccc|c|c|c|c|} \hline 
&&& \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{4-7} \multicolumn{3}{|c|}{Group} &&&& \\
\multicolumn{3}{|c|}{Sizes} & 0.10 & 0.05 & 0.025 & 0.01 \\ \hline 
6 & 1 & 1 & - - & - - & - - & - -\\
6 & 2 & 1 & 4.200 (.09524) & 4.822 (.04762) & 5.600 (.02381) & - -\\
6 & 2 & 2 & 4.545 (.08889) & 5.345 (.03810) & 5.745 (.02063) & 6.655 (.00794)\\
6 & 3 & 1 & 3.909 (.09524) & 4.855 (.05000) & 5.945 (.02143) & 6.873 (.00714)\\
6 & 3 & 2 & 4.682 (.08528) & 5.348 (.04632) & 6.136 (.02294) & 6.970 (.00909)\\
6 & 3 & 3 & 4.590 (.09773) & 5.615 (.04968) & 6.436 (.02229) & 7.410 (.00779)\\
6 & 4 & 1 & 4.038 (.09437) & 4.947 (.04675) & 5.856 (.02424) & 7.106 (.00866)\\
6 & 4 & 2 & 4.494 (.09986) & 5.340 (.04906) & 6.186 (.02453) & 7.340 (.00967)\\
6 & 4 & 3 & 4.604 (.09997) & 5.610 (.04862) & 6.538 (.02498) & 7.500 (.00966)\\
6 & 4 & 4 & 4.595 (.09847) & 5.681 (.04881) & 6.667 (.02495) & 7.795 (.00990)\\
6 & 5 & 1 & 4.128 (.09271) & 4.990 (.04726) & 5.951 (.02453) & 7.182 (.00974)\\
6 & 5 & 2 & 4.596 (.09807) & 5.338 (.04729) & 6.196 (.02481) & 7.376 (.00982)\\
6 & 5 & 3 & 4.535 (.09932) & 5.602 (.04956) & 6.667 (.02452) & 7.590 (.00999)\\
6 & 5 & 4 & 4.522 (.09974) & 5.661 (.04991) & 6.750 (.02473) & 7.936 (.00998)\\
6 & 5 & 5 & 4.547 (.09835) & 5.729 (.04973) & 6.788 (.02484) & 8.028 (.00988)\\
6 & 6 & 1 & 4.000 (.09774) & 4.945 (.04779) & 5.923 (.02381) & 7.121 (.00932)\\
6 & 6 & 2 & 4.438 (.09824) & 5.410 (.04993) & 6.210 (.02443) & 7.467 (.00982)\\
6 & 6 & 3 & 4.558 (.09948) & 5.625 (.04999) & 6.725 (.02462) & 7.725 (.00985)\\
6 & 6 & 4 & 4.548 (.09982) & 5.724 (.04950) & 6.812 (.02458) & 8.000 (.00998)\\
6 & 6 & 5 & 4.542 (.09987) & 5.765 (.04993) & 6.848 (.02489) & 8.124 (.00990)\\
6 & 6 & 6 & 4.643 (.09874) & 5.801 (.04905) & 6.889 (.02493) & 8.222 (.00994)\\
7 & 7 & 7 & 4.594 (.09933) & 5.819 (.04911) & 6.954 (.02446) & 8.378 (.00992)\\
8 & 8 & 8 & 4.595 (.09933) & 5.805 (.04973) & 6.995 (.02485) & 8.465 (.00991)\\
\hline
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