Multiple Comparisons and Outliers
Duncan's Multiple Range Test Table: Significant Studentized Ranges
Critical ranges r_p for Duncan's new multiple range test, which compares ordered means after ANOVA and sets a different critical value for means that are p steps apart.
- p = 2 to 20 means
- df = 1 to ∞
- α = 0.05 and 0.01
- Shortest significant range
Duncan's Multiple Range Table
Jump to a significance level below. Rows: error df · Columns: number of means p in the range being tested · Cells: critical value rp. Multiply by √(MSE / n) to get the shortest significant range.
α = 0.05
| Means in the range (p) → Error df ↓ | p = 2 | p = 3 | p = 4 | p = 5 | p = 6 | p = 7 | p = 8 | p = 9 | p = 10 | p = 11 | p = 12 | p = 13 | p = 14 | p = 15 | p = 16 | p = 17 | p = 18 | p = 19 | p = 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 | 17.969 |
| 2 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 | 6.085 |
| 3 | 4.501 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 | 4.516 |
| 4 | 3.926 | 4.013 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 | 4.033 |
| 5 | 3.635 | 3.749 | 3.796 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 | 3.814 |
| 6 | 3.460 | 3.586 | 3.649 | 3.680 | 3.694 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 | 3.697 |
| 7 | 3.344 | 3.477 | 3.548 | 3.588 | 3.611 | 3.622 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 | 3.625 |
| 8 | 3.261 | 3.398 | 3.475 | 3.521 | 3.549 | 3.566 | 3.575 | 3.579 | 3.579 | 3.579 | 3.579 | 3.579 | 3.579 | 3.579 | 3.579 | 3.579 | 3.579 | 3.579 | 3.579 |
| 9 | 3.199 | 3.339 | 3.420 | 3.470 | 3.502 | 3.523 | 3.536 | 3.544 | 3.547 | 3.547 | 3.547 | 3.547 | 3.547 | 3.547 | 3.547 | 3.547 | 3.547 | 3.547 | 3.547 |
| 10 | 3.151 | 3.293 | 3.376 | 3.430 | 3.465 | 3.489 | 3.505 | 3.516 | 3.522 | 3.525 | 3.525 | 3.525 | 3.525 | 3.525 | 3.525 | 3.525 | 3.525 | 3.525 | 3.525 |
| 11 | 3.113 | 3.256 | 3.341 | 3.397 | 3.435 | 3.462 | 3.480 | 3.493 | 3.501 | 3.506 | 3.509 | 3.510 | 3.510 | 3.510 | 3.510 | 3.510 | 3.510 | 3.510 | 3.510 |
| 12 | 3.081 | 3.225 | 3.312 | 3.370 | 3.410 | 3.439 | 3.459 | 3.474 | 3.484 | 3.491 | 3.495 | 3.498 | 3.498 | 3.498 | 3.498 | 3.498 | 3.498 | 3.498 | 3.498 |
| 13 | 3.055 | 3.200 | 3.288 | 3.348 | 3.389 | 3.419 | 3.441 | 3.458 | 3.470 | 3.478 | 3.484 | 3.488 | 3.490 | 3.490 | 3.490 | 3.490 | 3.490 | 3.490 | 3.490 |
| 14 | 3.033 | 3.178 | 3.268 | 3.328 | 3.371 | 3.403 | 3.426 | 3.444 | 3.457 | 3.467 | 3.474 | 3.479 | 3.482 | 3.484 | 3.484 | 3.484 | 3.484 | 3.484 | 3.484 |
| 15 | 3.014 | 3.160 | 3.250 | 3.312 | 3.356 | 3.389 | 3.413 | 3.432 | 3.446 | 3.457 | 3.465 | 3.471 | 3.476 | 3.478 | 3.480 | 3.480 | 3.480 | 3.480 | 3.480 |
| 16 | 2.998 | 3.144 | 3.235 | 3.297 | 3.343 | 3.376 | 3.402 | 3.422 | 3.437 | 3.449 | 3.458 | 3.465 | 3.470 | 3.473 | 3.476 | 3.477 | 3.477 | 3.477 | 3.477 |
| 17 | 2.984 | 3.130 | 3.222 | 3.285 | 3.331 | 3.365 | 3.392 | 3.412 | 3.429 | 3.441 | 3.451 | 3.459 | 3.465 | 3.469 | 3.472 | 3.474 | 3.475 | 3.475 | 3.475 |
| 18 | 2.971 | 3.117 | 3.210 | 3.274 | 3.320 | 3.356 | 3.383 | 3.404 | 3.421 | 3.435 | 3.445 | 3.454 | 3.460 | 3.465 | 3.469 | 3.472 | 3.473 | 3.474 | 3.474 |
| 19 | 2.960 | 3.106 | 3.199 | 3.264 | 3.311 | 3.347 | 3.375 | 3.397 | 3.415 | 3.429 | 3.440 | 3.449 | 3.456 | 3.462 | 3.466 | 3.469 | 3.472 | 3.473 | 3.474 |
| 20 | 2.950 | 3.097 | 3.190 | 3.255 | 3.303 | 3.339 | 3.368 | 3.390 | 3.409 | 3.423 | 3.435 | 3.445 | 3.452 | 3.459 | 3.463 | 3.467 | 3.470 | 3.472 | 3.473 |
| 24 | 2.919 | 3.066 | 3.160 | 3.226 | 3.276 | 3.315 | 3.345 | 3.370 | 3.390 | 3.406 | 3.420 | 3.431 | 3.441 | 3.449 | 3.455 | 3.461 | 3.465 | 3.469 | 3.472 |
| 30 | 2.888 | 3.035 | 3.131 | 3.199 | 3.250 | 3.290 | 3.322 | 3.349 | 3.371 | 3.389 | 3.405 | 3.418 | 3.429 | 3.439 | 3.447 | 3.454 | 3.460 | 3.466 | 3.470 |
| 40 | 2.858 | 3.005 | 3.102 | 3.171 | 3.224 | 3.266 | 3.300 | 3.328 | 3.352 | 3.372 | 3.389 | 3.404 | 3.418 | 3.429 | 3.439 | 3.448 | 3.456 | 3.463 | 3.469 |
| 60 | 2.829 | 2.976 | 3.073 | 3.143 | 3.198 | 3.241 | 3.277 | 3.307 | 3.333 | 3.355 | 3.374 | 3.391 | 3.406 | 3.419 | 3.431 | 3.441 | 3.451 | 3.460 | 3.468 |
| 120 | 2.800 | 2.947 | 3.045 | 3.116 | 3.172 | 3.217 | 3.254 | 3.286 | 3.313 | 3.337 | 3.358 | 3.377 | 3.394 | 3.409 | 3.423 | 3.435 | 3.446 | 3.457 | 3.466 |
| ∞ | 2.772 | 2.918 | 3.017 | 3.089 | 3.146 | 3.193 | 3.232 | 3.265 | 3.294 | 3.320 | 3.343 | 3.363 | 3.382 | 3.399 | 3.414 | 3.428 | 3.442 | 3.454 | 3.466 |
α = 0.01
| Means in the range (p) → Error df ↓ | p = 2 | p = 3 | p = 4 | p = 5 | p = 6 | p = 7 | p = 8 | p = 9 | p = 10 | p = 11 | p = 12 | p = 13 | p = 14 | p = 15 | p = 16 | p = 17 | p = 18 | p = 19 | p = 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 | 90.024 |
| 2 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 | 14.036 |
| 3 | 8.260 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 | 8.321 |
| 4 | 6.511 | 6.677 | 6.740 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 | 6.755 |
| 5 | 5.702 | 5.893 | 5.989 | 6.040 | 6.065 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 | 6.074 |
| 6 | 5.243 | 5.439 | 5.549 | 5.614 | 5.655 | 5.680 | 5.694 | 5.701 | 5.703 | 5.703 | 5.703 | 5.703 | 5.703 | 5.703 | 5.703 | 5.703 | 5.703 | 5.703 | 5.703 |
| 7 | 4.949 | 5.145 | 5.260 | 5.333 | 5.383 | 5.416 | 5.439 | 5.454 | 5.464 | 5.470 | 5.472 | 5.472 | 5.472 | 5.472 | 5.472 | 5.472 | 5.472 | 5.472 | 5.472 |
| 8 | 4.745 | 4.939 | 5.056 | 5.134 | 5.189 | 5.227 | 5.256 | 5.276 | 5.291 | 5.302 | 5.309 | 5.313 | 5.316 | 5.317 | 5.317 | 5.317 | 5.317 | 5.317 | 5.317 |
| 9 | 4.596 | 4.787 | 4.906 | 4.986 | 5.043 | 5.086 | 5.117 | 5.142 | 5.160 | 5.174 | 5.185 | 5.193 | 5.199 | 5.202 | 5.205 | 5.206 | 5.206 | 5.206 | 5.206 |
| 10 | 4.482 | 4.671 | 4.789 | 4.871 | 4.931 | 4.975 | 5.010 | 5.036 | 5.058 | 5.074 | 5.087 | 5.098 | 5.106 | 5.112 | 5.117 | 5.120 | 5.122 | 5.123 | 5.124 |
| 11 | 4.392 | 4.579 | 4.697 | 4.780 | 4.841 | 4.887 | 4.923 | 4.952 | 4.975 | 4.994 | 5.009 | 5.021 | 5.031 | 5.039 | 5.045 | 5.050 | 5.054 | 5.057 | 5.059 |
| 12 | 4.320 | 4.504 | 4.622 | 4.705 | 4.767 | 4.815 | 4.852 | 4.882 | 4.907 | 4.927 | 4.944 | 4.957 | 4.969 | 4.978 | 4.986 | 4.993 | 4.998 | 5.002 | 5.005 |
| 13 | 4.260 | 4.442 | 4.560 | 4.643 | 4.706 | 4.754 | 4.793 | 4.824 | 4.850 | 4.871 | 4.889 | 4.904 | 4.917 | 4.927 | 4.936 | 4.944 | 4.950 | 4.955 | 4.960 |
| 14 | 4.210 | 4.391 | 4.508 | 4.591 | 4.654 | 4.703 | 4.743 | 4.775 | 4.802 | 4.824 | 4.843 | 4.859 | 4.872 | 4.884 | 4.894 | 4.902 | 4.909 | 4.916 | 4.921 |
| 15 | 4.167 | 4.346 | 4.463 | 4.547 | 4.610 | 4.660 | 4.700 | 4.733 | 4.760 | 4.783 | 4.803 | 4.820 | 4.834 | 4.846 | 4.857 | 4.866 | 4.874 | 4.881 | 4.887 |
| 16 | 4.131 | 4.308 | 4.425 | 4.508 | 4.572 | 4.622 | 4.662 | 4.696 | 4.724 | 4.748 | 4.768 | 4.785 | 4.800 | 4.813 | 4.825 | 4.835 | 4.843 | 4.851 | 4.858 |
| 17 | 4.099 | 4.275 | 4.391 | 4.474 | 4.538 | 4.589 | 4.630 | 4.664 | 4.692 | 4.717 | 4.737 | 4.755 | 4.771 | 4.785 | 4.797 | 4.807 | 4.816 | 4.824 | 4.832 |
| 18 | 4.071 | 4.246 | 4.361 | 4.445 | 4.509 | 4.559 | 4.601 | 4.635 | 4.664 | 4.689 | 4.710 | 4.729 | 4.745 | 4.759 | 4.771 | 4.782 | 4.792 | 4.801 | 4.808 |
| 19 | 4.046 | 4.220 | 4.335 | 4.418 | 4.483 | 4.533 | 4.575 | 4.610 | 4.639 | 4.664 | 4.686 | 4.705 | 4.722 | 4.736 | 4.749 | 4.760 | 4.771 | 4.780 | 4.788 |
| 20 | 4.024 | 4.197 | 4.312 | 4.395 | 4.459 | 4.510 | 4.552 | 4.587 | 4.617 | 4.642 | 4.664 | 4.684 | 4.701 | 4.716 | 4.729 | 4.741 | 4.751 | 4.761 | 4.769 |
| 24 | 3.955 | 4.126 | 4.239 | 4.322 | 4.386 | 4.437 | 4.480 | 4.516 | 4.546 | 4.573 | 4.596 | 4.616 | 4.634 | 4.651 | 4.665 | 4.678 | 4.690 | 4.700 | 4.710 |
| 30 | 3.889 | 4.056 | 4.168 | 4.250 | 4.314 | 4.366 | 4.409 | 4.445 | 4.477 | 4.504 | 4.528 | 4.550 | 4.569 | 4.586 | 4.601 | 4.615 | 4.628 | 4.640 | 4.650 |
| 40 | 3.825 | 3.988 | 4.098 | 4.180 | 4.243 | 4.295 | 4.339 | 4.376 | 4.408 | 4.436 | 4.461 | 4.483 | 4.503 | 4.521 | 4.537 | 4.552 | 4.566 | 4.579 | 4.591 |
| 60 | 3.762 | 3.922 | 4.030 | 4.111 | 4.174 | 4.226 | 4.270 | 4.307 | 4.340 | 4.368 | 4.394 | 4.417 | 4.437 | 4.456 | 4.474 | 4.489 | 4.504 | 4.518 | 4.530 |
| 120 | 3.702 | 3.858 | 3.964 | 4.044 | 4.107 | 4.158 | 4.202 | 4.239 | 4.272 | 4.301 | 4.327 | 4.351 | 4.372 | 4.392 | 4.410 | 4.426 | 4.442 | 4.456 | 4.469 |
| ∞ | 3.643 | 3.796 | 3.900 | 3.978 | 4.040 | 4.091 | 4.135 | 4.172 | 4.205 | 4.235 | 4.261 | 4.285 | 4.307 | 4.327 | 4.345 | 4.363 | 4.379 | 4.394 | 4.408 |
Tip: click any value to highlight its row and column.
How to read this table
- Run a one-way ANOVA and note the MSE, its error degrees of freedom and the number of observations n per group.
- Order the group means from smallest to largest.
- For each pair, let p be the number of steps between the two means in the ordered list (adjacent means have p = 2). Read rp from the row for the error df.
- The pair differs significantly if its difference exceeds rp × √(MSE / n).
Worked example
Five treatment means come from a one-way ANOVA with MSE = 4.2, n = 6 per group and 20 error df. The standard error of a mean is √(4.2 / 6) = 0.8367. In the df = 20 row of the α = 0.05 table, p = 2, p = 3, p = 4, p = 5 give r2 = 2.950, r3 = 3.097, r4 = 3.190 and r5 = 3.255.
The shortest significant ranges are rp × SE: 2.468 (p = 2), 2.591 (p = 3), 2.669 (p = 4) and 2.723 (p = 5). Two means that are p steps apart in the ordered list differ significantly if their difference exceeds the range for that p. For example, the largest and smallest means (p = 5) differ by 6.0, which exceeds 2.723.
Click the underlined links to highlight the value in the table.
Calculator
Enter p, error df, MSE and n. Values between the tabulated rows are interpolated and are accurate to about ±0.01.
Formula and how the values were computed
Duncan's protection level for p means is (1 − α)^(p − 1), so the critical value is the upper αp = 1 − (1 − α)^(p − 1) point of the studentized range with p means and ν error df. As in Duncan's published table, values are held non-decreasing across p, which produces equal values across p at very small df (for example 17.969 for every p at df = 1, α = 0.05). Duncan's test is liberal: its familywise error rate is above α, so Tukey's HSD is preferred when strong control of Type I error is required.
Frequently asked questions
How is Duncan's test different from Tukey's HSD?
Tukey's HSD uses one critical value for every pair and controls the familywise error rate at α. Duncan's test uses a smaller critical value for means that are closer together in the ordering, so it finds more differences but its familywise error rate is higher.
Why are some values in the small-df rows equal?
Duncan's critical values never decrease as p grows. At very small df the calculated value would fall slightly, so the value from the smaller p is carried across, as in Duncan's published table.
When should I use Duncan's test?
It is common in agricultural and some laboratory literature, and available in SAS and SPSS. For clinical trial analyses, Tukey, Dunnett or a pre-specified multiplicity procedure is the usual choice.
How do I handle unequal group sizes?
Replace n with the harmonic mean of the group sizes, which is an approximation. For clearly unbalanced designs prefer the Tukey-Kramer procedure.