<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 (k Samples)&lt;/title&gt;

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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. &amp; In the table below, the critical values give significance
                levels as close \\
              &amp; as possible to, but not exceeding the nominal $\alpha$. 
                The actual levels of \\
              &amp; significance are in brackets. \\
    \ \\
           2. &amp; When the table below is not applicable and $k = 3$ with 
                the three group \\
              &amp; sizes above 5 or $k &gt; 3$ with all group sizes above 4, 
                use the following \\
              &amp; approximation: \\
              &amp; \multicolumn{1}{c}
                {$h_{\alpha}=\chi_{\alpha}^2,\quad\text{df}=k-1.$}
  \end{tabular}
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\footnotesize

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