Nonparametric Tests
Kruskal-Wallis Table: Exact Critical Values of H
Exact critical values for the Kruskal-Wallis test with small samples, where the chi-square approximation is unreliable.
- k = 3 (nᵢ ≤ 7) and k = 4 (nᵢ ≤ 4)
- Exact values
- Exact calculator
Kruskal-Wallis Table
Rows: group sizes, largest first · Columns: α · Cells: critical H. Reject H₀ if H ≥ the cell.
k = 3 groups
| α → Sample sizes ↓ | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 |
|---|---|---|---|---|---|---|
| 2, 2, 2 | 4.5715 | — | — | — | — | — |
| 3, 2, 1 | 4.2858 | — | — | — | — | — |
| 3, 2, 2 | 4.5000 | 4.7143 | — | — | — | — |
| 3, 3, 1 | 4.5715 | 5.1429 | — | — | — | — |
| 4, 2, 1 | 4.5000 | — | — | — | — | — |
| 3, 3, 2 | 4.5556 | 5.3612 | 5.5556 | — | — | — |
| 4, 2, 2 | 4.4584 | 5.3334 | 5.5000 | — | — | — |
| 4, 3, 1 | 4.0556 | 5.2084 | 5.8334 | — | — | — |
| 5, 2, 1 | 4.2000 | 5.0000 | — | — | — | — |
| 3, 3, 3 | 4.6223 | 5.6000 | 5.9556 | 7.2000 | 7.2000 | — |
| 4, 3, 2 | 4.5112 | 5.4445 | 6.0000 | 6.4445 | 7.0000 | — |
| 4, 4, 1 | 4.1667 | 4.9667 | 6.1667 | 6.6667 | — | — |
| 5, 2, 2 | 4.3734 | 5.1600 | 6.0000 | 6.5334 | — | — |
| 5, 3, 1 | 4.0178 | 4.9600 | 6.0445 | — | — | — |
| 6, 2, 1 | 4.2000 | 4.8223 | 5.6000 | — | — | — |
| 7, 1, 1 | 4.2667 | — | — | — | — | — |
| 4, 3, 3 | 4.7091 | 5.7910 | 6.1546 | 6.7455 | 7.3182 | — |
| 4, 4, 2 | 4.5546 | 5.4546 | 6.3273 | 7.0364 | 7.2819 | — |
| 5, 3, 2 | 4.6510 | 5.2510 | 6.0037 | 6.9091 | 7.1819 | — |
| 5, 4, 1 | 3.9873 | 4.9855 | 5.8582 | 6.9546 | 7.3637 | — |
| 6, 2, 2 | 4.5455 | 5.3455 | 5.7455 | 6.6546 | 6.9819 | — |
| 6, 3, 1 | 3.9091 | 4.8546 | 5.9455 | 6.8728 | — | — |
| 7, 2, 1 | 4.2000 | 4.7065 | 5.7273 | — | — | — |
| 4, 4, 3 | 4.5455 | 5.5985 | 6.3940 | 7.1440 | 7.5985 | 8.9091 |
| 5, 3, 3 | 4.5334 | 5.6485 | 6.3152 | 7.0788 | 7.6364 | 8.7273 |
| 5, 4, 2 | 4.5410 | 5.2728 | 6.0682 | 7.2046 | 7.5728 | 8.5910 |
| 5, 5, 1 | 4.1091 | 5.1273 | 6.0000 | 7.3091 | 8.1819 | — |
| 6, 3, 2 | 4.6819 | 5.3485 | 6.1364 | 6.9697 | 7.5152 | — |
| 6, 4, 1 | 4.0379 | 4.9470 | 5.8561 | 7.1061 | 7.6137 | — |
| 7, 2, 2 | 4.5260 | 5.1429 | 5.8182 | 7.0000 | 7.3637 | — |
| 7, 3, 1 | 4.1732 | 4.9524 | 5.7576 | 7.0304 | 7.2728 | — |
| 4, 4, 4 | 4.6539 | 5.6924 | 6.6154 | 7.6539 | 8.0000 | 9.2693 |
| 5, 4, 3 | 4.5488 | 5.6565 | 6.4103 | 7.4449 | 7.9270 | 8.7949 |
| 5, 5, 2 | 4.6231 | 5.3385 | 6.3462 | 7.3385 | 8.1308 | 8.9385 |
| 6, 3, 3 | 4.5898 | 5.6154 | 6.4359 | 7.4103 | 7.8718 | 8.6924 |
| 6, 4, 2 | 4.4936 | 5.3398 | 6.1859 | 7.3398 | 7.8462 | 8.8270 |
| 6, 5, 1 | 4.1283 | 4.9898 | 5.9513 | 7.1821 | 8.0770 | — |
| 7, 3, 2 | 4.5825 | 5.3572 | 6.2015 | 6.8389 | 7.5495 | 8.6539 |
| 7, 4, 1 | 4.1209 | 4.9863 | 5.7913 | 6.9863 | 7.9341 | — |
| 5, 4, 4 | 4.6682 | 5.6572 | 6.6726 | 7.7605 | 8.1891 | 9.1682 |
| 5, 5, 3 | 4.5451 | 5.7055 | 6.5495 | 7.5781 | 8.3165 | 9.2836 |
| 6, 4, 3 | 4.6044 | 5.6099 | 6.5385 | 7.5000 | 8.0330 | 9.1704 |
| 6, 5, 2 | 4.5957 | 5.3385 | 6.1957 | 7.3759 | 8.1957 | 9.1891 |
| 6, 6, 1 | 4.0001 | 4.9451 | 5.9231 | 7.1209 | 7.9341 | 9.6924 |
| 7, 3, 3 | 4.6029 | 5.6201 | 6.4490 | 7.2277 | 7.9059 | 9.2622 |
| 7, 4, 2 | 4.5495 | 5.3760 | 6.1837 | 7.3211 | 7.9168 | 9.1979 |
| 7, 5, 1 | 4.0352 | 5.0638 | 5.9530 | 7.0606 | 8.0026 | 9.1781 |
| 5, 5, 4 | 4.5229 | 5.6658 | 6.7600 | 7.8229 | 8.5229 | 9.6058 |
| 6, 4, 4 | 4.5953 | 5.6810 | 6.6667 | 7.7953 | 8.3810 | 9.6810 |
| 6, 5, 3 | 4.5353 | 5.6020 | 6.6667 | 7.5905 | 8.3143 | 9.6686 |
| 6, 6, 2 | 4.4381 | 5.4096 | 6.2096 | 7.4667 | 8.2096 | 9.7524 |
| 7, 4, 3 | 4.5273 | 5.6232 | 6.5783 | 7.5497 | 8.1844 | 9.6701 |
| 7, 5, 2 | 4.4849 | 5.3927 | 6.2213 | 7.4498 | 8.2400 | 9.6400 |
| 7, 6, 1 | 4.0327 | 5.0667 | 6.0667 | 7.2545 | 7.9878 | 9.7470 |
| 5, 5, 5 | 4.5600 | 5.7800 | 6.7400 | 8.0000 | 8.7800 | 9.9200 |
| 6, 5, 4 | 4.5225 | 5.6609 | 6.7500 | 7.9359 | 8.6434 | 9.9609 |
| 6, 6, 3 | 4.5584 | 5.6250 | 6.7250 | 7.7250 | 8.4584 | 10.1500 |
| 7, 4, 4 | 4.5625 | 5.6500 | 6.7072 | 7.8143 | 8.5072 | 9.8411 |
| 7, 5, 3 | 4.5353 | 5.6067 | 6.6267 | 7.6972 | 8.4210 | 9.8743 |
| 7, 6, 2 | 4.5000 | 5.3572 | 6.2227 | 7.4905 | 8.3727 | 10.0620 |
| 7, 7, 1 | 3.9858 | 4.9858 | 6.0572 | 7.1572 | 8.2286 | 9.8715 |
| 6, 5, 5 | 4.5471 | 5.7295 | 6.7883 | 8.0280 | 8.8589 | 10.2868 |
| 6, 6, 4 | 4.5478 | 5.7243 | 6.8126 | 8.0000 | 8.7537 | 10.3420 |
| 7, 5, 4 | 4.5416 | 5.7328 | 6.7384 | 7.9311 | 8.6670 | 10.1619 |
| 7, 6, 3 | 4.5505 | 5.6891 | 6.6944 | 7.7564 | 8.5841 | 10.2648 |
| 7, 7, 2 | 4.4906 | 5.3982 | 6.3278 | 7.4906 | 8.4738 | 10.2385 |
| 6, 6, 5 | 4.5425 | 5.7648 | 6.8484 | 8.1242 | 8.9870 | 10.5242 |
| 7, 5, 5 | 4.5715 | 5.7076 | 6.8348 | 8.1076 | 8.9009 | 10.4538 |
| 7, 6, 4 | 4.5617 | 5.7059 | 6.7867 | 8.0388 | 8.8488 | 10.4622 |
| 7, 7, 3 | 4.6126 | 5.6882 | 6.7078 | 7.8096 | 8.6462 | 10.4538 |
| 6, 6, 6 | 4.6433 | 5.8012 | 6.8889 | 8.2223 | 9.1696 | 10.8889 |
| 7, 6, 5 | 4.5598 | 5.7700 | 6.8575 | 8.1568 | 9.0515 | 10.7478 |
| 7, 7, 4 | 4.5627 | 5.7657 | 6.7883 | 8.1417 | 8.9236 | 10.6943 |
| 7, 6, 6 | 4.5301 | 5.7301 | 6.8970 | 8.2572 | 9.1707 | 10.9993 |
| 7, 7, 5 | 4.5456 | 5.7456 | 6.8861 | 8.2575 | 9.1489 | 10.9191 |
| 7, 7, 6 | 4.5681 | 5.7926 | 6.9266 | 8.3449 | 9.2776 | 11.1313 |
| 7, 7, 7 | 4.5937 | 5.8182 | 6.9537 | 8.3785 | 9.3730 | 11.3173 |
k = 4 groups
| α → Sample sizes ↓ | 0.1 | 0.05 | 0.025 | 0.01 | 0.005 | 0.001 |
|---|---|---|---|---|---|---|
| 2, 2, 2, 1 | 5.3572 | 5.6786 | — | — | — | — |
| 3, 2, 1, 1 | 5.1429 | — | — | — | — | — |
| 2, 2, 2, 2 | 5.6667 | 6.1667 | 6.6667 | 6.6667 | — | — |
| 3, 2, 2, 1 | 5.5556 | 5.8334 | 6.2500 | — | — | — |
| 3, 3, 1, 1 | 5.3334 | 6.3334 | 6.3334 | — | — | — |
| 4, 2, 1, 1 | 5.2500 | 5.8334 | — | — | — | — |
| 3, 2, 2, 2 | 5.6445 | 6.3334 | 6.9778 | 7.1334 | 7.5334 | — |
| 3, 3, 2, 1 | 5.6889 | 6.2445 | 6.6889 | 7.2000 | 7.4000 | — |
| 4, 2, 2, 1 | 5.5334 | 6.1334 | 6.5334 | 7.0000 | — | — |
| 4, 3, 1, 1 | 5.0667 | 6.1778 | 6.7112 | 7.0667 | — | — |
| 3, 3, 2, 2 | 5.7455 | 6.5273 | 7.0546 | 7.6364 | 7.8728 | 8.4546 |
| 3, 3, 3, 1 | 5.6546 | 6.6000 | 7.0364 | 7.4000 | 8.0546 | — |
| 4, 2, 2, 2 | 5.7546 | 6.5455 | 7.0637 | 7.3910 | 7.9637 | — |
| 4, 3, 2, 1 | 5.5910 | 6.3091 | 6.9546 | 7.4546 | 7.7728 | — |
| 4, 4, 1, 1 | 5.1819 | 5.9455 | 6.9546 | 7.9091 | 7.9091 | — |
| 3, 3, 3, 2 | 5.8788 | 6.7273 | 7.5152 | 8.0152 | 8.3788 | 9.0304 |
| 4, 3, 2, 2 | 5.7500 | 6.6213 | 7.3258 | 7.8713 | 8.2728 | 8.9091 |
| 4, 3, 3, 1 | 5.6894 | 6.5455 | 7.3258 | 7.7576 | 8.2122 | 9.1819 |
| 4, 4, 2, 1 | 5.5682 | 6.3864 | 7.1591 | 7.9091 | 8.3410 | 8.9091 |
| 3, 3, 3, 3 | 6.0257 | 7.0000 | 7.6667 | 8.5385 | 8.8975 | 9.5129 |
| 4, 3, 3, 2 | 5.8718 | 6.7949 | 7.5642 | 8.3334 | 8.7180 | 9.4552 |
| 4, 4, 2, 2 | 5.8077 | 6.7308 | 7.5385 | 8.3462 | 8.6924 | 9.4616 |
| 4, 4, 3, 1 | 5.6924 | 6.6347 | 7.5000 | 8.2308 | 8.5834 | 9.3270 |
| 4, 3, 3, 3 | 6.0165 | 6.9836 | 7.7748 | 8.6594 | 9.2528 | 10.0165 |
| 4, 4, 3, 2 | 5.9011 | 6.8737 | 7.7473 | 8.6209 | 9.1649 | 9.9451 |
| 4, 4, 4, 1 | 5.6539 | 6.7253 | 7.6484 | 8.5880 | 9.0000 | 9.7583 |
| 4, 4, 3, 3 | 6.0191 | 7.0381 | 7.9286 | 8.8762 | 9.4953 | 10.4667 |
| 4, 4, 4, 2 | 5.9143 | 6.9572 | 7.9143 | 8.8715 | 9.4858 | 10.4286 |
| 4, 4, 4, 3 | 6.0417 | 7.1417 | 8.0792 | 9.0750 | 9.7417 | 10.9292 |
| 4, 4, 4, 4 | 6.0883 | 7.2353 | 8.2280 | 9.2868 | 9.9706 | 11.3603 |
Tip: click any value to highlight its row and column.
How to read this table
- Rank all N observations together and add up the ranks Rᵢ in each group.
- Compute H = 12/(N(N + 1)) Σ Rᵢ²/nᵢ − 3(N + 1).
- Find the row for your group sizes (in any order; the table lists the largest first).
- Reject H₀ if H is greater than or equal to the table value. For larger samples, compare H with chi-square on k − 1 df.
Worked example
Pain scores are compared across three analgesics with 5, 4 and 3 patients, and H = 5.80. Row 5, 4, 3, α = 0.05 gives 5.6565, so the groups differ at the 5% level. The chi-square approximation would give a critical value of 5.991 and miss this result, which is why exact values matter for small samples. At α = 0.01 the exact value is 7.4449.
Click the underlined links to highlight the value in the table.
Calculator
For values between the rows and columns of the table. Results update as you type.
Formula and how the values were computed
Values come from the exact permutation distribution of H, computed by dynamic programming over all assignments of ranks to groups. Each entry is the smallest attainable H with P(H ≥ h) ≤ α, rounded up. The table assumes no ties.
Frequently asked questions
When can I use the chi-square approximation?
A common rule is when every group has at least 5 observations. With smaller groups, use the exact values here or the exact calculator.
What do I do after a significant H?
Compare pairs of groups with Dunn's test (with a multiplicity adjustment) or pairwise Mann-Whitney U tests with a Bonferroni or Holm correction.