Multiple Comparisons and Outliers
Tukey HSD q Table, α = 0.10: Studentized Range Critical Values
Critical values of the studentized range q at the 10% level for Tukey's honestly significant difference (HSD) test, for 2 to 20 groups and error df from 1 to infinity.
- k = 2 to 20 groups
- df = 1 to ∞
- α = 0.10
- HSD calculator
Tukey q Table, α = 0.10
Rows: error degrees of freedom · Columns: number of groups k · Cells: critical value q at α = 0.10. Tukey's HSD = q × √(MSE / n).
| Number of groups (k) → Error df ↓ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 8.929 | 13.437 | 16.358 | 18.488 | 20.150 | 21.504 | 22.642 | 23.621 | 24.477 | 25.237 | 25.918 | 26.536 | 27.100 | 27.618 | 28.097 | 28.542 | 28.958 | 29.347 | 29.713 |
| 2 | 4.129 | 5.733 | 6.772 | 7.538 | 8.139 | 8.633 | 9.049 | 9.409 | 9.725 | 10.006 | 10.259 | 10.488 | 10.698 | 10.891 | 11.070 | 11.237 | 11.392 | 11.538 | 11.676 |
| 3 | 3.328 | 4.467 | 5.199 | 5.738 | 6.162 | 6.511 | 6.806 | 7.062 | 7.287 | 7.487 | 7.667 | 7.831 | 7.982 | 8.120 | 8.248 | 8.368 | 8.479 | 8.584 | 8.683 |
| 4 | 3.015 | 3.976 | 4.586 | 5.035 | 5.388 | 5.679 | 5.926 | 6.139 | 6.327 | 6.494 | 6.645 | 6.783 | 6.909 | 7.025 | 7.132 | 7.233 | 7.326 | 7.414 | 7.497 |
| 5 | 2.850 | 3.717 | 4.264 | 4.664 | 4.979 | 5.238 | 5.458 | 5.648 | 5.816 | 5.965 | 6.100 | 6.223 | 6.336 | 6.439 | 6.536 | 6.626 | 6.710 | 6.788 | 6.863 |
| 6 | 2.748 | 3.558 | 4.065 | 4.435 | 4.726 | 4.966 | 5.168 | 5.344 | 5.499 | 5.637 | 5.762 | 5.875 | 5.979 | 6.075 | 6.164 | 6.247 | 6.325 | 6.398 | 6.466 |
| 7 | 2.679 | 3.451 | 3.931 | 4.280 | 4.555 | 4.780 | 4.971 | 5.137 | 5.283 | 5.413 | 5.530 | 5.637 | 5.735 | 5.826 | 5.910 | 5.988 | 6.061 | 6.130 | 6.195 |
| 8 | 2.630 | 3.374 | 3.834 | 4.169 | 4.431 | 4.646 | 4.829 | 4.987 | 5.126 | 5.250 | 5.362 | 5.464 | 5.558 | 5.644 | 5.724 | 5.799 | 5.869 | 5.935 | 5.997 |
| 9 | 2.592 | 3.316 | 3.761 | 4.084 | 4.337 | 4.545 | 4.721 | 4.873 | 5.007 | 5.126 | 5.234 | 5.333 | 5.423 | 5.506 | 5.583 | 5.655 | 5.722 | 5.786 | 5.845 |
| 10 | 2.563 | 3.270 | 3.704 | 4.018 | 4.264 | 4.465 | 4.636 | 4.783 | 4.913 | 5.029 | 5.134 | 5.229 | 5.316 | 5.397 | 5.472 | 5.542 | 5.607 | 5.668 | 5.726 |
| 11 | 2.540 | 3.234 | 3.658 | 3.965 | 4.205 | 4.401 | 4.567 | 4.711 | 4.838 | 4.951 | 5.053 | 5.145 | 5.231 | 5.309 | 5.382 | 5.450 | 5.514 | 5.573 | 5.630 |
| 12 | 2.521 | 3.204 | 3.621 | 3.921 | 4.156 | 4.349 | 4.511 | 4.652 | 4.776 | 4.886 | 4.986 | 5.076 | 5.160 | 5.236 | 5.308 | 5.374 | 5.436 | 5.495 | 5.550 |
| 13 | 2.504 | 3.179 | 3.589 | 3.885 | 4.116 | 4.304 | 4.464 | 4.602 | 4.724 | 4.832 | 4.930 | 5.019 | 5.100 | 5.175 | 5.245 | 5.310 | 5.371 | 5.429 | 5.483 |
| 14 | 2.491 | 3.158 | 3.563 | 3.854 | 4.081 | 4.267 | 4.424 | 4.560 | 4.679 | 4.786 | 4.882 | 4.969 | 5.050 | 5.124 | 5.192 | 5.256 | 5.316 | 5.372 | 5.426 |
| 15 | 2.479 | 3.140 | 3.540 | 3.828 | 4.052 | 4.235 | 4.390 | 4.524 | 4.641 | 4.746 | 4.841 | 4.927 | 5.006 | 5.079 | 5.146 | 5.209 | 5.268 | 5.324 | 5.376 |
| 16 | 2.469 | 3.124 | 3.520 | 3.804 | 4.026 | 4.207 | 4.360 | 4.492 | 4.608 | 4.712 | 4.805 | 4.890 | 4.968 | 5.040 | 5.106 | 5.169 | 5.227 | 5.282 | 5.333 |
| 17 | 2.460 | 3.110 | 3.503 | 3.784 | 4.003 | 4.182 | 4.334 | 4.464 | 4.579 | 4.681 | 4.774 | 4.857 | 4.934 | 5.005 | 5.071 | 5.133 | 5.190 | 5.244 | 5.295 |
| 18 | 2.452 | 3.098 | 3.487 | 3.766 | 3.984 | 4.161 | 4.310 | 4.440 | 4.553 | 4.654 | 4.746 | 4.829 | 4.905 | 4.975 | 5.040 | 5.101 | 5.158 | 5.211 | 5.262 |
| 19 | 2.445 | 3.087 | 3.474 | 3.751 | 3.966 | 4.142 | 4.290 | 4.418 | 4.530 | 4.630 | 4.721 | 4.803 | 4.878 | 4.948 | 5.012 | 5.072 | 5.129 | 5.182 | 5.232 |
| 20 | 2.439 | 3.077 | 3.462 | 3.736 | 3.950 | 4.124 | 4.271 | 4.398 | 4.510 | 4.609 | 4.699 | 4.780 | 4.855 | 4.923 | 4.987 | 5.047 | 5.103 | 5.155 | 5.205 |
| 24 | 2.420 | 3.047 | 3.423 | 3.692 | 3.900 | 4.070 | 4.213 | 4.336 | 4.445 | 4.541 | 4.628 | 4.707 | 4.780 | 4.847 | 4.909 | 4.966 | 5.020 | 5.071 | 5.119 |
| 30 | 2.400 | 3.017 | 3.386 | 3.648 | 3.851 | 4.016 | 4.155 | 4.275 | 4.381 | 4.474 | 4.559 | 4.635 | 4.706 | 4.770 | 4.830 | 4.886 | 4.939 | 4.988 | 5.034 |
| 40 | 2.381 | 2.988 | 3.348 | 3.605 | 3.802 | 3.963 | 4.099 | 4.215 | 4.317 | 4.408 | 4.490 | 4.564 | 4.632 | 4.694 | 4.752 | 4.806 | 4.857 | 4.904 | 4.949 |
| 60 | 2.363 | 2.959 | 3.312 | 3.562 | 3.755 | 3.911 | 4.042 | 4.155 | 4.254 | 4.342 | 4.421 | 4.493 | 4.558 | 4.619 | 4.675 | 4.727 | 4.775 | 4.821 | 4.864 |
| 120 | 2.344 | 2.930 | 3.276 | 3.520 | 3.707 | 3.859 | 3.986 | 4.096 | 4.191 | 4.276 | 4.353 | 4.422 | 4.485 | 4.543 | 4.597 | 4.647 | 4.694 | 4.738 | 4.779 |
| ∞ | 2.326 | 2.902 | 3.240 | 3.478 | 3.661 | 3.808 | 3.931 | 4.037 | 4.129 | 4.211 | 4.285 | 4.351 | 4.412 | 4.468 | 4.519 | 4.568 | 4.612 | 4.654 | 4.694 |
Tip: click any value to highlight its row and column.
How to read this table
- Run a one-way ANOVA and note the mean square error (MSE) and its error degrees of freedom (N − k).
- Find the row for the error df and the column for the number of groups k.
- Compute HSD = q × √(MSE / n), where n is the number of observations per group.
- Two group means that differ by more than HSD are significantly different at the 10% familywise level.
Worked example
A trial compares k = 4 groups with 11 patients per group; the ANOVA gives MSE = 12 with 20 error df. Row df = 20, k = 4 gives 3.462.
The honestly significant difference is HSD = 3.462 × √(12 / 11) = 3.615. At α = 0.10, two group means that differ by more than this are significantly different, with the familywise error rate controlled across all 6 pairs.
Click the underlined links to highlight the value in the table.
Calculator
Choose k and error df from the lists, then enter MSE and n.
Formula and how the values were computed
q is the upper α = 0.10 point of the studentized range distribution with k groups and ν error degrees of freedom, computed by numerical integration of the studentized range distribution. For ν = ∞ and k = 2 the value is 2.326 = 1.645 √2, which is the normal result.
Frequently asked questions
When should I use the α = 0.10 Tukey table?
When a 10% familywise error rate is acceptable, for example in exploratory or early-phase comparisons. For confirmatory work, use the α = 0.05 or α = 0.01 tables.
What if my group sizes are unequal?
Use the Tukey-Kramer modification: replace √(MSE / n) with √[(MSE / 2)(1/n_i + 1/n_j)] for each pair.
Why not just use t-tests?
Each pairwise t-test has its own error rate, so the chance of at least one false positive across all pairs grows quickly with k. Tukey's q controls the error rate over all pairs together.
Where do rows like df = 24 come from?
Rows follow the usual printed layout: every df from 1 to 20, then 24, 30, 40, 60, 120 and ∞. The calculator uses the listed rows.