Multiple Comparisons and Outliers
Bonferroni-Adjusted Critical Values of t
Critical t-values that keep the familywise error rate at α when you run m tests, each at level α/m.
- m = 2 to 30 comparisons
- df 1 to ∞
- Holm adjustment too
Bonferroni t Table
Rows: degrees of freedom · Columns: number of comparisons m · Cells: two-sided critical t at level α/m per comparison
Familywise α = 0.05 (two-sided)
| Number of comparisons (m) → df ↓ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 25.452 | 38.188 | 50.923 | 63.657 | 76.390 | 89.123 | 101.9 | 114.6 | 127.3 | 152.8 | 191.0 | 254.6 | 318.3 | 382.0 |
| 2 | 6.205 | 7.649 | 8.860 | 9.925 | 10.886 | 11.769 | 12.590 | 13.360 | 14.089 | 15.443 | 17.277 | 19.962 | 22.327 | 24.464 |
| 3 | 4.177 | 4.857 | 5.392 | 5.841 | 6.232 | 6.580 | 6.895 | 7.185 | 7.453 | 7.940 | 8.575 | 9.465 | 10.215 | 10.869 |
| 4 | 3.495 | 3.961 | 4.315 | 4.604 | 4.851 | 5.068 | 5.261 | 5.437 | 5.598 | 5.885 | 6.254 | 6.758 | 7.173 | 7.529 |
| 5 | 3.163 | 3.534 | 3.810 | 4.032 | 4.219 | 4.382 | 4.526 | 4.655 | 4.773 | 4.983 | 5.247 | 5.604 | 5.893 | 6.138 |
| 6 | 2.969 | 3.287 | 3.521 | 3.707 | 3.863 | 3.997 | 4.115 | 4.221 | 4.317 | 4.486 | 4.698 | 4.981 | 5.208 | 5.398 |
| 7 | 2.841 | 3.128 | 3.335 | 3.499 | 3.636 | 3.753 | 3.855 | 3.947 | 4.029 | 4.174 | 4.355 | 4.595 | 4.785 | 4.944 |
| 8 | 2.752 | 3.016 | 3.206 | 3.355 | 3.479 | 3.584 | 3.677 | 3.759 | 3.833 | 3.962 | 4.122 | 4.334 | 4.501 | 4.640 |
| 9 | 2.685 | 2.933 | 3.111 | 3.250 | 3.364 | 3.462 | 3.547 | 3.622 | 3.690 | 3.808 | 3.954 | 4.146 | 4.297 | 4.422 |
| 10 | 2.634 | 2.870 | 3.038 | 3.169 | 3.277 | 3.368 | 3.448 | 3.518 | 3.581 | 3.691 | 3.827 | 4.005 | 4.144 | 4.259 |
| 11 | 2.593 | 2.820 | 2.981 | 3.106 | 3.208 | 3.295 | 3.370 | 3.437 | 3.497 | 3.600 | 3.728 | 3.895 | 4.025 | 4.132 |
| 12 | 2.560 | 2.779 | 2.934 | 3.055 | 3.153 | 3.236 | 3.308 | 3.371 | 3.428 | 3.527 | 3.649 | 3.807 | 3.930 | 4.031 |
| 13 | 2.533 | 2.746 | 2.896 | 3.012 | 3.107 | 3.187 | 3.256 | 3.318 | 3.372 | 3.467 | 3.584 | 3.735 | 3.852 | 3.948 |
| 14 | 2.510 | 2.718 | 2.864 | 2.977 | 3.069 | 3.146 | 3.214 | 3.273 | 3.326 | 3.417 | 3.530 | 3.675 | 3.787 | 3.880 |
| 15 | 2.490 | 2.694 | 2.837 | 2.947 | 3.036 | 3.112 | 3.177 | 3.235 | 3.286 | 3.375 | 3.484 | 3.624 | 3.733 | 3.822 |
| 16 | 2.473 | 2.673 | 2.813 | 2.921 | 3.008 | 3.082 | 3.146 | 3.202 | 3.252 | 3.339 | 3.444 | 3.581 | 3.686 | 3.773 |
| 17 | 2.458 | 2.655 | 2.793 | 2.898 | 2.984 | 3.056 | 3.119 | 3.173 | 3.222 | 3.307 | 3.410 | 3.543 | 3.646 | 3.730 |
| 18 | 2.445 | 2.639 | 2.775 | 2.878 | 2.963 | 3.034 | 3.095 | 3.149 | 3.197 | 3.279 | 3.380 | 3.510 | 3.610 | 3.692 |
| 19 | 2.433 | 2.625 | 2.759 | 2.861 | 2.944 | 3.014 | 3.074 | 3.127 | 3.174 | 3.255 | 3.354 | 3.481 | 3.579 | 3.660 |
| 20 | 2.423 | 2.613 | 2.744 | 2.845 | 2.927 | 2.996 | 3.055 | 3.107 | 3.153 | 3.233 | 3.331 | 3.455 | 3.552 | 3.630 |
| 21 | 2.414 | 2.601 | 2.732 | 2.831 | 2.912 | 2.980 | 3.038 | 3.090 | 3.135 | 3.214 | 3.310 | 3.432 | 3.527 | 3.604 |
| 22 | 2.405 | 2.591 | 2.720 | 2.819 | 2.899 | 2.965 | 3.023 | 3.074 | 3.119 | 3.196 | 3.291 | 3.412 | 3.505 | 3.581 |
| 23 | 2.398 | 2.582 | 2.710 | 2.807 | 2.886 | 2.952 | 3.009 | 3.059 | 3.104 | 3.181 | 3.274 | 3.393 | 3.485 | 3.560 |
| 24 | 2.391 | 2.574 | 2.700 | 2.797 | 2.875 | 2.941 | 2.997 | 3.046 | 3.091 | 3.166 | 3.258 | 3.376 | 3.467 | 3.540 |
| 25 | 2.385 | 2.566 | 2.692 | 2.787 | 2.865 | 2.930 | 2.986 | 3.035 | 3.078 | 3.153 | 3.244 | 3.361 | 3.450 | 3.523 |
| 26 | 2.379 | 2.559 | 2.684 | 2.779 | 2.856 | 2.920 | 2.975 | 3.024 | 3.067 | 3.141 | 3.231 | 3.346 | 3.435 | 3.507 |
| 27 | 2.373 | 2.552 | 2.676 | 2.771 | 2.847 | 2.911 | 2.966 | 3.014 | 3.057 | 3.130 | 3.219 | 3.333 | 3.421 | 3.492 |
| 28 | 2.368 | 2.546 | 2.669 | 2.763 | 2.839 | 2.902 | 2.957 | 3.004 | 3.047 | 3.120 | 3.208 | 3.321 | 3.408 | 3.479 |
| 29 | 2.364 | 2.541 | 2.663 | 2.756 | 2.832 | 2.894 | 2.949 | 2.996 | 3.038 | 3.110 | 3.198 | 3.310 | 3.396 | 3.466 |
| 30 | 2.360 | 2.536 | 2.657 | 2.750 | 2.825 | 2.887 | 2.941 | 2.988 | 3.030 | 3.102 | 3.189 | 3.300 | 3.385 | 3.454 |
| 35 | 2.342 | 2.515 | 2.633 | 2.724 | 2.797 | 2.857 | 2.910 | 2.955 | 2.996 | 3.066 | 3.150 | 3.258 | 3.340 | 3.407 |
| 40 | 2.329 | 2.499 | 2.616 | 2.704 | 2.776 | 2.836 | 2.887 | 2.931 | 2.971 | 3.039 | 3.122 | 3.227 | 3.307 | 3.372 |
| 45 | 2.319 | 2.487 | 2.602 | 2.690 | 2.760 | 2.819 | 2.869 | 2.913 | 2.952 | 3.019 | 3.100 | 3.203 | 3.281 | 3.345 |
| 50 | 2.311 | 2.477 | 2.591 | 2.678 | 2.747 | 2.805 | 2.855 | 2.898 | 2.937 | 3.003 | 3.083 | 3.184 | 3.261 | 3.324 |
| 60 | 2.299 | 2.463 | 2.575 | 2.660 | 2.729 | 2.785 | 2.834 | 2.877 | 2.915 | 2.979 | 3.057 | 3.156 | 3.232 | 3.293 |
| 80 | 2.284 | 2.445 | 2.555 | 2.639 | 2.705 | 2.761 | 2.809 | 2.850 | 2.887 | 2.950 | 3.026 | 3.122 | 3.195 | 3.254 |
| 100 | 2.276 | 2.435 | 2.544 | 2.626 | 2.692 | 2.747 | 2.793 | 2.834 | 2.871 | 2.933 | 3.007 | 3.102 | 3.174 | 3.232 |
| 120 | 2.270 | 2.428 | 2.536 | 2.617 | 2.683 | 2.737 | 2.783 | 2.824 | 2.860 | 2.921 | 2.995 | 3.088 | 3.160 | 3.217 |
| ∞ | 2.241 | 2.394 | 2.498 | 2.576 | 2.638 | 2.690 | 2.734 | 2.773 | 2.807 | 2.865 | 2.935 | 3.023 | 3.090 | 3.144 |
Familywise α = 0.01 (two-sided)
| Number of comparisons (m) → df ↓ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 127.3 | 191.0 | 254.6 | 318.3 | 382.0 | 445.6 | 509.3 | 573.0 | 636.6 | 763.9 | 954.9 | 1273.2 | 1591.5 | 1909.9 |
| 2 | 14.089 | 17.277 | 19.962 | 22.327 | 24.464 | 26.429 | 28.258 | 29.975 | 31.599 | 34.619 | 38.710 | 44.705 | 49.985 | 54.759 |
| 3 | 7.453 | 8.575 | 9.465 | 10.215 | 10.869 | 11.453 | 11.984 | 12.471 | 12.924 | 13.745 | 14.819 | 16.326 | 17.598 | 18.709 |
| 4 | 5.598 | 6.254 | 6.758 | 7.173 | 7.529 | 7.841 | 8.122 | 8.376 | 8.610 | 9.029 | 9.568 | 10.306 | 10.915 | 11.438 |
| 5 | 4.773 | 5.247 | 5.604 | 5.893 | 6.138 | 6.352 | 6.541 | 6.713 | 6.869 | 7.146 | 7.499 | 7.976 | 8.363 | 8.693 |
| 6 | 4.317 | 4.698 | 4.981 | 5.208 | 5.398 | 5.563 | 5.709 | 5.840 | 5.959 | 6.169 | 6.434 | 6.788 | 7.074 | 7.314 |
| 7 | 4.029 | 4.355 | 4.595 | 4.785 | 4.944 | 5.082 | 5.202 | 5.310 | 5.408 | 5.580 | 5.795 | 6.082 | 6.311 | 6.503 |
| 8 | 3.833 | 4.122 | 4.334 | 4.501 | 4.640 | 4.759 | 4.864 | 4.957 | 5.041 | 5.189 | 5.374 | 5.617 | 5.811 | 5.973 |
| 9 | 3.690 | 3.954 | 4.146 | 4.297 | 4.422 | 4.529 | 4.622 | 4.706 | 4.781 | 4.912 | 5.076 | 5.291 | 5.461 | 5.602 |
| 10 | 3.581 | 3.827 | 4.005 | 4.144 | 4.259 | 4.357 | 4.442 | 4.518 | 4.587 | 4.706 | 4.855 | 5.049 | 5.202 | 5.329 |
| 11 | 3.497 | 3.728 | 3.895 | 4.025 | 4.132 | 4.223 | 4.303 | 4.373 | 4.437 | 4.548 | 4.685 | 4.863 | 5.004 | 5.120 |
| 12 | 3.428 | 3.649 | 3.807 | 3.930 | 4.031 | 4.117 | 4.192 | 4.258 | 4.318 | 4.422 | 4.550 | 4.716 | 4.847 | 4.955 |
| 13 | 3.372 | 3.584 | 3.735 | 3.852 | 3.948 | 4.030 | 4.101 | 4.164 | 4.221 | 4.319 | 4.440 | 4.597 | 4.721 | 4.822 |
| 14 | 3.326 | 3.530 | 3.675 | 3.787 | 3.880 | 3.958 | 4.026 | 4.086 | 4.140 | 4.234 | 4.349 | 4.499 | 4.616 | 4.712 |
| 15 | 3.286 | 3.484 | 3.624 | 3.733 | 3.822 | 3.897 | 3.963 | 4.021 | 4.073 | 4.163 | 4.273 | 4.417 | 4.528 | 4.620 |
| 16 | 3.252 | 3.444 | 3.581 | 3.686 | 3.773 | 3.846 | 3.909 | 3.965 | 4.015 | 4.102 | 4.208 | 4.346 | 4.454 | 4.542 |
| 17 | 3.222 | 3.410 | 3.543 | 3.646 | 3.730 | 3.801 | 3.862 | 3.917 | 3.965 | 4.049 | 4.152 | 4.286 | 4.390 | 4.475 |
| 18 | 3.197 | 3.380 | 3.510 | 3.610 | 3.692 | 3.762 | 3.822 | 3.874 | 3.922 | 4.003 | 4.104 | 4.233 | 4.334 | 4.416 |
| 19 | 3.174 | 3.354 | 3.481 | 3.579 | 3.660 | 3.727 | 3.786 | 3.837 | 3.883 | 3.963 | 4.061 | 4.187 | 4.285 | 4.365 |
| 20 | 3.153 | 3.331 | 3.455 | 3.552 | 3.630 | 3.697 | 3.754 | 3.804 | 3.850 | 3.928 | 4.023 | 4.146 | 4.241 | 4.319 |
| 21 | 3.135 | 3.310 | 3.432 | 3.527 | 3.604 | 3.669 | 3.726 | 3.775 | 3.819 | 3.896 | 3.989 | 4.110 | 4.203 | 4.279 |
| 22 | 3.119 | 3.291 | 3.412 | 3.505 | 3.581 | 3.645 | 3.700 | 3.749 | 3.792 | 3.867 | 3.959 | 4.077 | 4.168 | 4.243 |
| 23 | 3.104 | 3.274 | 3.393 | 3.485 | 3.560 | 3.623 | 3.677 | 3.725 | 3.768 | 3.841 | 3.932 | 4.047 | 4.137 | 4.210 |
| 24 | 3.091 | 3.258 | 3.376 | 3.467 | 3.540 | 3.603 | 3.656 | 3.703 | 3.745 | 3.818 | 3.907 | 4.021 | 4.109 | 4.181 |
| 25 | 3.078 | 3.244 | 3.361 | 3.450 | 3.523 | 3.584 | 3.637 | 3.684 | 3.725 | 3.797 | 3.884 | 3.996 | 4.083 | 4.154 |
| 26 | 3.067 | 3.231 | 3.346 | 3.435 | 3.507 | 3.567 | 3.620 | 3.666 | 3.707 | 3.777 | 3.864 | 3.974 | 4.060 | 4.129 |
| 27 | 3.057 | 3.219 | 3.333 | 3.421 | 3.492 | 3.552 | 3.604 | 3.649 | 3.690 | 3.759 | 3.845 | 3.954 | 4.038 | 4.107 |
| 28 | 3.047 | 3.208 | 3.321 | 3.408 | 3.479 | 3.538 | 3.589 | 3.634 | 3.674 | 3.743 | 3.827 | 3.935 | 4.018 | 4.086 |
| 29 | 3.038 | 3.198 | 3.310 | 3.396 | 3.466 | 3.525 | 3.575 | 3.620 | 3.659 | 3.728 | 3.811 | 3.918 | 4.000 | 4.067 |
| 30 | 3.030 | 3.189 | 3.300 | 3.385 | 3.454 | 3.513 | 3.563 | 3.607 | 3.646 | 3.714 | 3.796 | 3.902 | 3.983 | 4.049 |
| 35 | 2.996 | 3.150 | 3.258 | 3.340 | 3.407 | 3.463 | 3.511 | 3.553 | 3.591 | 3.656 | 3.735 | 3.836 | 3.914 | 3.977 |
| 40 | 2.971 | 3.122 | 3.227 | 3.307 | 3.372 | 3.426 | 3.473 | 3.514 | 3.551 | 3.614 | 3.691 | 3.788 | 3.864 | 3.925 |
| 45 | 2.952 | 3.100 | 3.203 | 3.281 | 3.345 | 3.398 | 3.444 | 3.484 | 3.520 | 3.582 | 3.657 | 3.752 | 3.825 | 3.885 |
| 50 | 2.937 | 3.083 | 3.184 | 3.261 | 3.324 | 3.376 | 3.421 | 3.461 | 3.496 | 3.556 | 3.630 | 3.723 | 3.795 | 3.853 |
| 60 | 2.915 | 3.057 | 3.156 | 3.232 | 3.293 | 3.344 | 3.388 | 3.426 | 3.460 | 3.519 | 3.590 | 3.681 | 3.750 | 3.807 |
| 80 | 2.887 | 3.026 | 3.122 | 3.195 | 3.254 | 3.304 | 3.346 | 3.383 | 3.416 | 3.473 | 3.542 | 3.629 | 3.696 | 3.750 |
| 100 | 2.871 | 3.007 | 3.102 | 3.174 | 3.232 | 3.280 | 3.322 | 3.358 | 3.390 | 3.446 | 3.513 | 3.598 | 3.664 | 3.716 |
| 120 | 2.860 | 2.995 | 3.088 | 3.160 | 3.217 | 3.265 | 3.306 | 3.342 | 3.373 | 3.428 | 3.494 | 3.578 | 3.642 | 3.694 |
| ∞ | 2.807 | 2.935 | 3.023 | 3.090 | 3.144 | 3.189 | 3.227 | 3.261 | 3.291 | 3.341 | 3.403 | 3.481 | 3.540 | 3.588 |
Tip: click any value to highlight its row and column.
How to read this table
- Count m, the number of tests in the family (for all pairs of g groups, m = g(g − 1)/2).
- Find the row for the df of each test (the error df from the ANOVA for pairwise comparisons).
- A comparison is significant if |t| is larger than the table value.
Worked example
A trial compares four doses with placebo using separate t-tests, each on 20 df. With m = 4 comparisons and familywise α = 0.05, row 20, column m = 4 gives 2.744. A dose with t = 2.60 would be significant on its own (2.60 > 2.086), but it is not significant after the Bonferroni correction.
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
tcrit = t1 − α/(2m), df · adjusted p = min(1, m·p)
Each cell is the two-sided critical value of Student's t at level α/m, that is the 1 − α/(2m) quantile of t with the given df, computed in double precision.
Frequently asked questions
Is Bonferroni too conservative?
It controls the familywise error rate under any dependence between tests, so it can be conservative when tests are correlated. Holm's step-down procedure is uniformly more powerful and equally valid; the calculator gives both.
Bonferroni or Dunnett?
When every test compares a treatment with the same control, Dunnett's test uses the correlation between comparisons and has more power.