% <html><head><title>Upper Critical Values for Kruskal-Wallis</title>

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  {\Large\textbf{Upper Critical Values for the Kruskal-Wallis Test}}
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\footnotesize

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\begin{tabular}{|ccccc|c|c|c|c|} \hline 
&&&&& \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{6-9} \multicolumn{5}{|c|}{Group} &&&& \\
\multicolumn{5}{|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}{|ccccc|c|c|c|c|} \hline 
&&&&& \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{6-9} \multicolumn{5}{|c|}{Group} &&&& \\
\multicolumn{5}{|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
\end{tabular}
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\begin{tabular}{|ccccc|c|c|c|c|} \hline 
&&&&& \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{6-9} \multicolumn{5}{|c|}{Group} &&&& \\
\multicolumn{5}{|c|}{Sizes} & 0.10 & 0.05 & 0.025 & 0.01 \\ \hline 
2&2&1&1&  & - - & - - & - - & - -\\
2&2&2&1&  & 5.357 (.06667) & 5.679 (.03810) & - - & - -\\
2&2&2&2&  & 5.667 (.07619) & 6.167 (.03810) & 6.667 (.00952) & 6.667 (.00952)\\
3&1&1&1&  & - - & - - & - - & - -\\
3&2&1&1&  & 5.143 (.08571) & - - & - - & - -\\
3&2&2&1&  & 5.556 (.07143) & 5.833 (.04286) & 6.250 (.02143) & - -\\
3&2&2&2&  & 5.644 (.10000) & 6.333 (.04762) & 6.978 (.01746) & 7.133 (.00794)\\
3&3&1&1&  & 5.333 (.09643) & 6.333 (.02143) & 6.333 (.02143) & -\\
3&3&2&1&  & 5.689 (.08571) & 6.244 (.04246) & 6.689 (.01786) & 7.200 (.00595)\\
3&3&2&2&  & 5.745 (.09921) & 6.527 (.04921) & 7.055 (.02317) & 7.636 (.01000)\\
3&3&3&1&  & 5.655 (.09786) & 6.600 (.04929) & 7.036 (.02429) & 7.400 (.00857)\\
3&3&3&2&  & 5.879 (.09974) & 6.727 (.04948) & 7.515 (.02390) & 8.015 (.00961)\\
3&3&3&3&  & 6.026 (.09779) & 7.000 (.04351) & 7.667 (.02338) & 8.538 (.00838)\\
4&1&1&1&  & - - & - - & - - & - -\\
4&2&1&1&  & 5.250 (.09048) & 5.833 (.04286) & - - & - -\\
4&2&2&1&  & 5.533 (.09788) & 6.133 (.04180) & 6.533 (.02063) & 7.000 (.00952)\\
4&2&2&2&  & 5.755 (.09302) & 6.545 (.04921) & 7.064 (.02222) & 7.391 (.00889)\\
4&3&1&1&  & 5.067 (.09524) & 6.178 (.04921) & 6.711 (.01905) & 7.067 (.00952)\\
4&3&2&1&  & 5.591 (.09857) & 6.309 (.04937) & 6.955 (.02317) & 7.455 (.00984)\\
4&3&2&2&  & 5.750 (.09980) & 6.621 (.04949) & 7.326 (.02496) & 7.871 (.00999)\\
4&3&3&1&  & 5.689 (.09602) & 6.545 (.04952) & 7.326 (.02329) & 7.758 (.00974)\\
4&3&3&2&  & 5.872 (.09929) & 6.795 (.04925) & 7.564 (.02494) & 8.333 (.00985)\\
4&3&3&3&  & 6.016 (.09779) & 6.984 (.04897) & 7.775 (.02437) & 8.659 (.00990)\\
4&4&1&1&  & 5.182 (.09968) & 5.945 (.04952) & 6.955 (.02349) & 7.909 (.00381)\\
4&4&2&1&  & 5.568 (.09980) & 6.386 (.04981) & 7.159 (.02459) & 7.909 (.00906)\\
4&4&2&2&  & 5.808 (.09882) & 6.731 (.04872) & 7.538 (.02453) & 8.346 (.00941)\\
4&4&3&1&  & 5.692 (.09853) & 6.635 (.04978) & 7.500 (.02462) & 8.231 (.00955)\\
4&4&3&2&  & 5.901 (.09950) & 6.874 (.04983) & 7.747 (.02500) & 8.621 (.00999)\\
4&4&3&3&  & 6.019 (.09948) & 7.038 (.04990) & 7.929 (.02487) & 8.876 (.00974)\\
4&4&4&1&  & 5.654 (.09801) & 6.725 (.04979) & 7.648 (.02470) & 8.588 (.00986)\\
4&4&4&2&  & 5.914 (.09940) & 6.957 (.04960) & 7.914 (.02499) & 8.871 (.00987)\\
4&4&4&3&  & 6.042 (.09980) & 7.142 (.04954) & 8.079 (.02494) & 9.075 (.01000)\\
4&4&4&4&  & 6.088 (.09900) & 7.235 (.04922) & 8.228 (.02476) & 9.287 (.00999)\\
\hline
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\begin{tabular}{|ccccc|c|c|c|c|} \hline 
&&&&& \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{6-9} \multicolumn{5}{|c|}{Group} &&&& \\
\multicolumn{5}{|c|}{Sizes} & 0.10 & 0.05 & 0.025 & 0.01 \\ \hline 
2&2&1&1&1 & 5.786 (.09524) & - - & - - & - -\\
2&2&2&1&1 & 6.250 (.08810) & 6.750 (.02381) & 6.750 (.02381) & - -\\
2&2&2&2&1 & 6.600 (.08889) & 7.133 (.04127) & 7.333 (.02222) & 7.533 (.00952)\\
2&2&2&2&2 & 6.982 (.09101) & 7.418 (.04868) & 7.964 (.02222) & 8.291 (.00952)\\
3&1&1&1&1 & - - & - - & - - & - -\\
3&2&1&1&1 & 6.139 (.10000) & 6.583 (.03571) & - - & - -\\
3&2&2&1&1 & 6.511 (.10000) & 6.800 (.04921) & 7.200 (.02460) & 7.600 (.00794)\\
3&2&2&2&1 & 6.709 (.09873) & 7.309 (.04889) & 7.745 (.02317) & 8.127 (.00937)\\
3&2&2&2&2 & 6.955 (.09922) & 7.682 (.04745) & 8.182 (.02384) & 8.682 (.00958)\\
3&3&1&1&1 & 6.311 (.09286) & 7.111 (.04048) & 7.467 (.01190) & - -\\
3&3&2&1&1 & 6.600 (.09929) & 7.200 (.05000) & 7.618 (.02452) & 8.073 (.00738)\\
3&3&2&2&1 & 6.788 (.09892) & 7.591 (.04919) & 8.121 (.02437) & 8.576 (.00984)\\
3&3&2&2&2 & 7.026 (.09897) & 7.910 (.04934) & 8.538 (.02408) & 9.115 (.00996)\\
3&3&3&1&1 & 6.788 (.09779) & 7.576 (.04545) & 8.061 (.02325) & 8.424 (.00909)\\
3&3&3&2&1 & 6.910 (.09916) & 7.769 (.04885) & 8.449 (.02471) & 9.051 (.00976)\\
3&3&3&2&2 & 7.121 (.09979) & 8.044 (.04915) & 8.813 (.02472) & 9.505 (.00999)\\
3&3&3&3&1 & 7.077 (.09836) & 8.000 (.04792) & 8.703 (.02396) & 9.451 (.00997)\\
3&3&3&3&2 & 7.210 (.09965) & 8.200 (.04940) & 9.038 (.02452) & 9.876 (.00966)\\
3&3&3&3&3 & 7.333 (.09922) & 8.333 (.04955) & 9.200 (.02500) & 10.200 (.00986)\\
\hline
\end{tabular}
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