<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% &lt;html&gt;&lt;head&gt;&lt;title&gt;Upper Critical Values for Kruskal-Wallis&lt;/title&gt;

\documentclass{article}

\begin{document}

\thispagestyle{empty}

\begin{center}
  {\Large\textbf{Upper Critical Values for the Kruskal-Wallis Test}}
\end{center}

\footnotesize

\begin{center}
\begin{tabular}{|ccccc|c|c|c|c|} \hline 
&amp;&amp;&amp;&amp;&amp; \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{6-9} \multicolumn{5}{|c|}{Group} &amp;&amp;&amp;&amp; \\
\multicolumn{5}{|c|}{Sizes} &amp; 0.10 &amp; 0.05 &amp; 0.025 &amp; 0.01 \\ \hline 
2&amp;2&amp;2&amp;&amp; &amp; 4.571 (.06667) &amp; - - &amp; - - &amp; - - \\
3&amp;2&amp;1&amp;&amp; &amp; 4.286 (.10000) &amp; - - &amp; - - &amp; - - \\
3&amp;2&amp;2&amp;&amp; &amp; 4.500 (.06667) &amp; 4.714 (.04762) &amp; - - &amp; - - \\
3&amp;3&amp;1&amp;&amp; &amp; 4.571 (.10000) &amp; 5.143 (.04286) &amp; - - &amp; - - \\
3&amp;3&amp;2&amp;&amp; &amp; 4.556 (.10000) &amp; 5.361 (.03214) &amp; 5.556 (.02500) &amp; - - \\
3&amp;3&amp;3&amp;&amp; &amp; 4.622 (.10000) &amp; 5.600 (.05000) &amp; 5.956 (.02500) &amp; 7.200 (.00357) \\
4&amp;2&amp;1&amp;&amp; &amp; 4.500 (.07619) &amp; - - &amp; - - &amp; - - \\
4&amp;2&amp;2&amp;&amp; &amp; 4.458 (.10000) &amp; 5.333 (.03333) &amp; 5.500 (.02381) &amp; - - \\
4&amp;3&amp;1&amp;&amp; &amp; 4.056 (.09286) &amp; 5.208 (.05000) &amp; 5.833 (.02143) &amp; - - \\
4&amp;3&amp;2&amp;&amp; &amp; 4.511 (.09841) &amp; 5.444 (.04603) &amp; 6.000 (.02381) &amp; 6.444 (.00794) \\
4&amp;3&amp;3&amp;&amp; &amp; 4.709 (.09238) &amp; 5.791 (.04571) &amp; 6.155 (.02476) &amp; 6.745 (.01000) \\
4&amp;4&amp;1&amp;&amp; &amp; 4.167 (.08254) &amp; 4.967 (.04762) &amp; 6.167 (.02222) &amp; 6.667 (.00952) \\
4&amp;4&amp;2&amp;&amp; &amp; 4.555 (.09778) &amp; 5.455 (.04571) &amp; 6.327 (.02413) &amp; 7.036 (.00571) \\
4&amp;4&amp;3&amp;&amp; &amp; 4.545 (.09905) &amp; 5.598 (.04866) &amp; 6.394 (.02476) &amp; 7.144 (.00970) \\
4&amp;4&amp;4&amp;&amp; &amp; 4.654 (.09662) &amp; 5.692 (.04866) &amp; 6.615 (.02424) &amp; 7.654 (.00762) \\
5&amp;2&amp;1&amp;&amp; &amp; 4.200 (.09524) &amp; 5.000 (.04762) &amp; - - &amp; - - \\
5&amp;2&amp;2&amp;&amp; &amp; 4.373 (.08995) &amp; 5.160 (.03439) &amp; 6.000 (.01852) &amp; 6.533 (.00794) \\
5&amp;3&amp;1&amp;&amp; &amp; 4.018 (.09524) &amp; 4.960 (.04762) &amp; 6.044 (.01984) &amp; - - \\
5&amp;3&amp;2&amp;&amp; &amp; 4.651 (.09127) &amp; 5.251 (.04921) &amp; 6.004 (.02460) &amp; 6.909 (.00873) \\
5&amp;3&amp;3&amp;&amp; &amp; 4.533 (.09697) &amp; 5.648 (.04892) &amp; 6.315 (.02121) &amp; 7.079 (.00866) \\
5&amp;4&amp;1&amp;&amp; &amp; 3.987 (.09841) &amp; 4.985 (.04444) &amp; 5.858 (.02381) &amp; 6.955 (.00794) \\
5&amp;4&amp;2&amp;&amp; &amp; 4.541 (.09841) &amp; 5.273 (.04877) &amp; 6.068 (.02482) &amp; 7.205 (.00895) \\
5&amp;4&amp;3&amp;&amp; &amp; 4.549 (.09892) &amp; 5.656 (.04863) &amp; 6.410 (.02496) &amp; 7.445 (.00974) \\
5&amp;4&amp;4&amp;&amp; &amp; 4.668 (.09817) &amp; 5.657 (.04906) &amp; 6.673 (.02429) &amp; 7.760 (.00946) \\
5&amp;5&amp;1&amp;&amp; &amp; 4.109 (.08586) &amp; 5.127 (.04618) &amp; 6.000 (.02165) &amp; 7.309 (.00938) \\
5&amp;5&amp;2&amp;&amp; &amp; 4.623 (.09704) &amp; 5.338 (.04726) &amp; 6.346 (.02489) &amp; 7.338 (.00962) \\
5&amp;5&amp;3&amp;&amp; &amp; 4.545 (.09965) &amp; 5.705 (.04612) &amp; 6.549 (.02436) &amp; 7.578 (.00968) \\
5&amp;5&amp;4&amp;&amp; &amp; 4.523 (.09935) &amp; 5.666 (.04931) &amp; 6.760 (.02490) &amp; 7.823 (.00978) \\
5&amp;5&amp;5&amp;&amp; &amp; 4.560 (.09952) &amp; 5.780 (.04878) &amp; 6.740 (.02475) &amp; 8.000 (.00946) \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|ccccc|c|c|c|c|} \hline 
&amp;&amp;&amp;&amp;&amp; \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{6-9} \multicolumn{5}{|c|}{Group} &amp;&amp;&amp;&amp; \\
\multicolumn{5}{|c|}{Sizes} &amp; 0.10 &amp; 0.05 &amp; 0.025 &amp; 0.01 \\ \hline 
6&amp;1&amp;1&amp;&amp; &amp; - - &amp; - - &amp; - - &amp; - - \\
6&amp;2&amp;1&amp;&amp; &amp; 4.200 (.09524) &amp; 4.822 (.04762) &amp; 5.600 (.02381) &amp; - - \\
6&amp;2&amp;2&amp;&amp; &amp; 4.545 (.08889) &amp; 5.345 (.03810) &amp; 5.745 (.02063) &amp; 6.655 (.00794) \\
6&amp;3&amp;1&amp;&amp; &amp; 3.909 (.09524) &amp; 4.855 (.05000) &amp; 5.945 (.02143) &amp; 6.873 (.00714) \\
6&amp;3&amp;2&amp;&amp; &amp; 4.682 (.08528) &amp; 5.348 (.04632) &amp; 6.136 (.02294) &amp; 6.970 (.00909) \\
6&amp;3&amp;3&amp;&amp; &amp; 4.590 (.09773) &amp; 5.615 (.04968) &amp; 6.436 (.02229) &amp; 7.410 (.00779) \\
6&amp;4&amp;1&amp;&amp; &amp; 4.038 (.09437) &amp; 4.947 (.04675) &amp; 5.856 (.02424) &amp; 7.106 (.00866) \\
6&amp;4&amp;2&amp;&amp; &amp; 4.494 (.09986) &amp; 5.340 (.04906) &amp; 6.186 (.02453) &amp; 7.340 (.00967) \\
6&amp;4&amp;3&amp;&amp; &amp; 4.604 (.09997) &amp; 5.610 (.04862) &amp; 6.538 (.02498) &amp; 7.500 (.00966) \\
6&amp;4&amp;4&amp;&amp; &amp; 4.595 (.09847) &amp; 5.681 (.04881) &amp; 6.667 (.02495) &amp; 7.795 (.00990) \\
6&amp;5&amp;1&amp;&amp; &amp; 4.128 (.09271) &amp; 4.990 (.04726) &amp; 5.951 (.02453) &amp; 7.182 (.00974) \\
6&amp;5&amp;2&amp;&amp; &amp; 4.596 (.09807) &amp; 5.338 (.04729) &amp; 6.196 (.02481) &amp; 7.376 (.00982) \\
6&amp;5&amp;3&amp;&amp; &amp; 4.535 (.09932) &amp; 5.602 (.04956) &amp; 6.667 (.02452) &amp; 7.590 (.00999) \\
6&amp;5&amp;4&amp;&amp; &amp; 4.522 (.09974) &amp; 5.661 (.04991) &amp; 6.750 (.02473) &amp; 7.936 (.00998) \\
6&amp;5&amp;5&amp;&amp; &amp; 4.547 (.09835) &amp; 5.729 (.04973) &amp; 6.788 (.02484) &amp; 8.028 (.00988) \\
6&amp;6&amp;1&amp;&amp; &amp; 4.000 (.09774) &amp; 4.945 (.04779) &amp; 5.923 (.02381) &amp; 7.121 (.00932) \\
6&amp;6&amp;2&amp;&amp; &amp; 4.438 (.09824) &amp; 5.410 (.04993) &amp; 6.210 (.02443) &amp; 7.467 (.00982) \\
6&amp;6&amp;3&amp;&amp; &amp; 4.558 (.09948) &amp; 5.625 (.04999) &amp; 6.725 (.02462) &amp; 7.725 (.00985) \\
6&amp;6&amp;4&amp;&amp; &amp; 4.548 (.09982) &amp; 5.724 (.04950) &amp; 6.812 (.02458) &amp; 8.000 (.00998) \\
6&amp;6&amp;5&amp;&amp; &amp; 4.542 (.09987) &amp; 5.765 (.04993) &amp; 6.848 (.02489) &amp; 8.124 (.00990) \\
6&amp;6&amp;6&amp;&amp; &amp; 4.643 (.09874) &amp; 5.801 (.04905) &amp; 6.889 (.02493) &amp; 8.222 (.00994) \\
7&amp;7&amp;7&amp;&amp; &amp; 4.594 (.09933) &amp; 5.819 (.04911) &amp; 6.954 (.02446) &amp; 8.378 (.00992) \\
8&amp;8&amp;8&amp;&amp; &amp; 4.595 (.09933) &amp; 5.805 (.04973) &amp; 6.995 (.02485) &amp; 8.465 (.00991) \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|ccccc|c|c|c|c|} \hline 
&amp;&amp;&amp;&amp;&amp; \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{6-9} \multicolumn{5}{|c|}{Group} &amp;&amp;&amp;&amp; \\
\multicolumn{5}{|c|}{Sizes} &amp; 0.10 &amp; 0.05 &amp; 0.025 &amp; 0.01 \\ \hline 
2&amp;2&amp;1&amp;1&amp;  &amp; - - &amp; - - &amp; - - &amp; - -\\
2&amp;2&amp;2&amp;1&amp;  &amp; 5.357 (.06667) &amp; 5.679 (.03810) &amp; - - &amp; - -\\
2&amp;2&amp;2&amp;2&amp;  &amp; 5.667 (.07619) &amp; 6.167 (.03810) &amp; 6.667 (.00952) &amp; 6.667 (.00952)\\
3&amp;1&amp;1&amp;1&amp;  &amp; - - &amp; - - &amp; - - &amp; - -\\
3&amp;2&amp;1&amp;1&amp;  &amp; 5.143 (.08571) &amp; - - &amp; - - &amp; - -\\
3&amp;2&amp;2&amp;1&amp;  &amp; 5.556 (.07143) &amp; 5.833 (.04286) &amp; 6.250 (.02143) &amp; - -\\
3&amp;2&amp;2&amp;2&amp;  &amp; 5.644 (.10000) &amp; 6.333 (.04762) &amp; 6.978 (.01746) &amp; 7.133 (.00794)\\
3&amp;3&amp;1&amp;1&amp;  &amp; 5.333 (.09643) &amp; 6.333 (.02143) &amp; 6.333 (.02143) &amp; -\\
3&amp;3&amp;2&amp;1&amp;  &amp; 5.689 (.08571) &amp; 6.244 (.04246) &amp; 6.689 (.01786) &amp; 7.200 (.00595)\\
3&amp;3&amp;2&amp;2&amp;  &amp; 5.745 (.09921) &amp; 6.527 (.04921) &amp; 7.055 (.02317) &amp; 7.636 (.01000)\\
3&amp;3&amp;3&amp;1&amp;  &amp; 5.655 (.09786) &amp; 6.600 (.04929) &amp; 7.036 (.02429) &amp; 7.400 (.00857)\\
3&amp;3&amp;3&amp;2&amp;  &amp; 5.879 (.09974) &amp; 6.727 (.04948) &amp; 7.515 (.02390) &amp; 8.015 (.00961)\\
3&amp;3&amp;3&amp;3&amp;  &amp; 6.026 (.09779) &amp; 7.000 (.04351) &amp; 7.667 (.02338) &amp; 8.538 (.00838)\\
4&amp;1&amp;1&amp;1&amp;  &amp; - - &amp; - - &amp; - - &amp; - -\\
4&amp;2&amp;1&amp;1&amp;  &amp; 5.250 (.09048) &amp; 5.833 (.04286) &amp; - - &amp; - -\\
4&amp;2&amp;2&amp;1&amp;  &amp; 5.533 (.09788) &amp; 6.133 (.04180) &amp; 6.533 (.02063) &amp; 7.000 (.00952)\\
4&amp;2&amp;2&amp;2&amp;  &amp; 5.755 (.09302) &amp; 6.545 (.04921) &amp; 7.064 (.02222) &amp; 7.391 (.00889)\\
4&amp;3&amp;1&amp;1&amp;  &amp; 5.067 (.09524) &amp; 6.178 (.04921) &amp; 6.711 (.01905) &amp; 7.067 (.00952)\\
4&amp;3&amp;2&amp;1&amp;  &amp; 5.591 (.09857) &amp; 6.309 (.04937) &amp; 6.955 (.02317) &amp; 7.455 (.00984)\\
4&amp;3&amp;2&amp;2&amp;  &amp; 5.750 (.09980) &amp; 6.621 (.04949) &amp; 7.326 (.02496) &amp; 7.871 (.00999)\\
4&amp;3&amp;3&amp;1&amp;  &amp; 5.689 (.09602) &amp; 6.545 (.04952) &amp; 7.326 (.02329) &amp; 7.758 (.00974)\\
4&amp;3&amp;3&amp;2&amp;  &amp; 5.872 (.09929) &amp; 6.795 (.04925) &amp; 7.564 (.02494) &amp; 8.333 (.00985)\\
4&amp;3&amp;3&amp;3&amp;  &amp; 6.016 (.09779) &amp; 6.984 (.04897) &amp; 7.775 (.02437) &amp; 8.659 (.00990)\\
4&amp;4&amp;1&amp;1&amp;  &amp; 5.182 (.09968) &amp; 5.945 (.04952) &amp; 6.955 (.02349) &amp; 7.909 (.00381)\\
4&amp;4&amp;2&amp;1&amp;  &amp; 5.568 (.09980) &amp; 6.386 (.04981) &amp; 7.159 (.02459) &amp; 7.909 (.00906)\\
4&amp;4&amp;2&amp;2&amp;  &amp; 5.808 (.09882) &amp; 6.731 (.04872) &amp; 7.538 (.02453) &amp; 8.346 (.00941)\\
4&amp;4&amp;3&amp;1&amp;  &amp; 5.692 (.09853) &amp; 6.635 (.04978) &amp; 7.500 (.02462) &amp; 8.231 (.00955)\\
4&amp;4&amp;3&amp;2&amp;  &amp; 5.901 (.09950) &amp; 6.874 (.04983) &amp; 7.747 (.02500) &amp; 8.621 (.00999)\\
4&amp;4&amp;3&amp;3&amp;  &amp; 6.019 (.09948) &amp; 7.038 (.04990) &amp; 7.929 (.02487) &amp; 8.876 (.00974)\\
4&amp;4&amp;4&amp;1&amp;  &amp; 5.654 (.09801) &amp; 6.725 (.04979) &amp; 7.648 (.02470) &amp; 8.588 (.00986)\\
4&amp;4&amp;4&amp;2&amp;  &amp; 5.914 (.09940) &amp; 6.957 (.04960) &amp; 7.914 (.02499) &amp; 8.871 (.00987)\\
4&amp;4&amp;4&amp;3&amp;  &amp; 6.042 (.09980) &amp; 7.142 (.04954) &amp; 8.079 (.02494) &amp; 9.075 (.01000)\\
4&amp;4&amp;4&amp;4&amp;  &amp; 6.088 (.09900) &amp; 7.235 (.04922) &amp; 8.228 (.02476) &amp; 9.287 (.00999)\\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|ccccc|c|c|c|c|} \hline 
&amp;&amp;&amp;&amp;&amp; \multicolumn{4}{|c|}{Nominal size $\alpha$} \\
\cline{6-9} \multicolumn{5}{|c|}{Group} &amp;&amp;&amp;&amp; \\
\multicolumn{5}{|c|}{Sizes} &amp; 0.10 &amp; 0.05 &amp; 0.025 &amp; 0.01 \\ \hline 
2&amp;2&amp;1&amp;1&amp;1 &amp; 5.786 (.09524) &amp; - - &amp; - - &amp; - -\\
2&amp;2&amp;2&amp;1&amp;1 &amp; 6.250 (.08810) &amp; 6.750 (.02381) &amp; 6.750 (.02381) &amp; - -\\
2&amp;2&amp;2&amp;2&amp;1 &amp; 6.600 (.08889) &amp; 7.133 (.04127) &amp; 7.333 (.02222) &amp; 7.533 (.00952)\\
2&amp;2&amp;2&amp;2&amp;2 &amp; 6.982 (.09101) &amp; 7.418 (.04868) &amp; 7.964 (.02222) &amp; 8.291 (.00952)\\
3&amp;1&amp;1&amp;1&amp;1 &amp; - - &amp; - - &amp; - - &amp; - -\\
3&amp;2&amp;1&amp;1&amp;1 &amp; 6.139 (.10000) &amp; 6.583 (.03571) &amp; - - &amp; - -\\
3&amp;2&amp;2&amp;1&amp;1 &amp; 6.511 (.10000) &amp; 6.800 (.04921) &amp; 7.200 (.02460) &amp; 7.600 (.00794)\\
3&amp;2&amp;2&amp;2&amp;1 &amp; 6.709 (.09873) &amp; 7.309 (.04889) &amp; 7.745 (.02317) &amp; 8.127 (.00937)\\
3&amp;2&amp;2&amp;2&amp;2 &amp; 6.955 (.09922) &amp; 7.682 (.04745) &amp; 8.182 (.02384) &amp; 8.682 (.00958)\\
3&amp;3&amp;1&amp;1&amp;1 &amp; 6.311 (.09286) &amp; 7.111 (.04048) &amp; 7.467 (.01190) &amp; - -\\
3&amp;3&amp;2&amp;1&amp;1 &amp; 6.600 (.09929) &amp; 7.200 (.05000) &amp; 7.618 (.02452) &amp; 8.073 (.00738)\\
3&amp;3&amp;2&amp;2&amp;1 &amp; 6.788 (.09892) &amp; 7.591 (.04919) &amp; 8.121 (.02437) &amp; 8.576 (.00984)\\
3&amp;3&amp;2&amp;2&amp;2 &amp; 7.026 (.09897) &amp; 7.910 (.04934) &amp; 8.538 (.02408) &amp; 9.115 (.00996)\\
3&amp;3&amp;3&amp;1&amp;1 &amp; 6.788 (.09779) &amp; 7.576 (.04545) &amp; 8.061 (.02325) &amp; 8.424 (.00909)\\
3&amp;3&amp;3&amp;2&amp;1 &amp; 6.910 (.09916) &amp; 7.769 (.04885) &amp; 8.449 (.02471) &amp; 9.051 (.00976)\\
3&amp;3&amp;3&amp;2&amp;2 &amp; 7.121 (.09979) &amp; 8.044 (.04915) &amp; 8.813 (.02472) &amp; 9.505 (.00999)\\
3&amp;3&amp;3&amp;3&amp;1 &amp; 7.077 (.09836) &amp; 8.000 (.04792) &amp; 8.703 (.02396) &amp; 9.451 (.00997)\\
3&amp;3&amp;3&amp;3&amp;2 &amp; 7.210 (.09965) &amp; 8.200 (.04940) &amp; 9.038 (.02452) &amp; 9.876 (.00966)\\
3&amp;3&amp;3&amp;3&amp;3 &amp; 7.333 (.09922) &amp; 8.333 (.04955) &amp; 9.200 (.02500) &amp; 10.200 (.00986)\\
\hline
\end{tabular}
\end{center}

\end{document}

% &lt;/pre&gt;&lt;/body&gt;&lt;/html&gt;
</pre></body></html>