| \(df \backslash p\) | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 15 | 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 |
| 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 |
| 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 | 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 |
| 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 |
| 6 | 3.46 | 3.586 | 3.649 | 3.68 | 3.694 | 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 |
| 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 |
| 9 | 3.199 | 3.339 | 3.42 | 3.47 | 3.502 | 3.523 | 3.536 | 3.544 | 3.547 | 3.547 | 3.547 | 3.547 | 3.547 |
| 10 | 3.151 | 3.293 | 3.376 | 3.43 | 3.465 | 3.489 | 3.505 | 3.516 | 3.522 | 3.525 | 3.525 | 3.525 | 3.525 |
| 11 | 3.113 | 3.256 | 3.341 | 3.397 | 3.435 | 3.462 | 3.48 | 3.493 | 3.501 | 3.506 | 3.509 | 3.51 | 3.51 |
| 12 | 3.081 | 3.225 | 3.312 | 3.37 | 3.41 | 3.439 | 3.459 | 3.474 | 3.484 | 3.491 | 3.495 | 3.498 | 3.498 |
| 13 | 3.055 | 3.2 | 3.288 | 3.348 | 3.389 | 3.419 | 3.441 | 3.458 | 3.47 | 3.478 | 3.484 | 3.49 | 3.49 |
| 14 | 3.033 | 3.178 | 3.268 | 3.328 | 3.371 | 3.403 | 3.426 | 3.444 | 3.457 | 3.467 | 3.474 | 3.484 | 3.484 |
| 15 | 3.014 | 3.16 | 3.25 | 3.312 | 3.356 | 3.389 | 3.413 | 3.432 | 3.446 | 3.457 | 3.465 | 3.478 | 3.48 |
| 16 | 2.998 | 3.144 | 3.235 | 3.297 | 3.343 | 3.376 | 3.402 | 3.422 | 3.437 | 3.449 | 3.458 | 3.473 | 3.477 |
| 17 | 2.984 | 3.13 | 3.222 | 3.285 | 3.331 | 3.365 | 3.392 | 3.412 | 3.429 | 3.441 | 3.451 | 3.469 | 3.475 |
| 18 | 2.971 | 3.117 | 3.21 | 3.274 | 3.32 | 3.356 | 3.383 | 3.404 | 3.421 | 3.435 | 3.445 | 3.465 | 3.474 |
| 19 | 2.96 | 3.106 | 3.199 | 3.264 | 3.311 | 3.347 | 3.375 | 3.397 | 3.415 | 3.429 | 3.44 | 3.462 | 3.474 |
| 20 | 2.95 | 3.097 | 3.19 | 3.255 | 3.303 | 3.339 | 3.368 | 3.39 | 3.409 | 3.423 | 3.435 | 3.459 | 3.473 |
| 21 | 2.941 | 3.088 | 3.181 | 3.247 | 3.295 | 3.332 | 3.361 | 3.385 | 3.403 | 3.418 | 3.431 | 3.456 | 3.473 |
| 22 | 2.933 | 3.08 | 3.173 | 3.239 | 3.288 | 3.326 | 3.355 | 3.379 | 3.398 | 3.414 | 3.427 | 3.453 | 3.472 |
| 23 | 2.926 | 3.072 | 3.166 | 3.233 | 3.282 | 3.32 | 3.35 | 3.374 | 3.394 | 3.41 | 3.423 | 3.451 | 3.472 |
| 24 | 2.919 | 3.066 | 3.16 | 3.226 | 3.276 | 3.315 | 3.345 | 3.37 | 3.39 | 3.406 | 3.42 | 3.449 | 3.472 |
| 25 | 2.913 | 3.059 | 3.154 | 3.221 | 3.271 | 3.31 | 3.341 | 3.366 | 3.386 | 3.403 | 3.417 | 3.447 | 3.471 |
| 26 | 2.907 | 3.054 | 3.149 | 3.216 | 3.266 | 3.305 | 3.336 | 3.362 | 3.382 | 3.4 | 3.414 | 3.445 | 3.471 |
| 27 | 2.902 | 3.049 | 3.144 | 3.211 | 3.262 | 3.301 | 3.332 | 3.358 | 3.379 | 3.397 | 3.412 | 3.443 | 3.471 |
| 28 | 2.897 | 3.044 | 3.139 | 3.206 | 3.257 | 3.297 | 3.329 | 3.355 | 3.376 | 3.394 | 3.409 | 3.442 | 3.47 |
| 29 | 2.892 | 3.039 | 3.135 | 3.202 | 3.253 | 3.293 | 3.326 | 3.352 | 3.373 | 3.392 | 3.407 | 3.44 | 3.47 |
| 30 | 2.888 | 3.035 | 3.131 | 3.199 | 3.25 | 3.29 | 3.322 | 3.349 | 3.371 | 3.389 | 3.405 | 3.439 | 3.47 |
| 31 | 2.884 | 3.031 | 3.127 | 3.195 | 3.246 | 3.287 | 3.319 | 3.346 | 3.368 | 3.387 | 3.403 | 3.438 | 3.47 |
| 32 | 2.881 | 3.028 | 3.123 | 3.192 | 3.243 | 3.284 | 3.317 | 3.344 | 3.366 | 3.385 | 3.401 | 3.436 | 3.47 |
| 33 | 2.877 | 3.024 | 3.12 | 3.188 | 3.24 | 3.281 | 3.314 | 3.341 | 3.364 | 3.383 | 3.399 | 3.435 | 3.47 |
| 34 | 2.874 | 3.021 | 3.117 | 3.185 | 3.238 | 3.279 | 3.312 | 3.339 | 3.362 | 3.381 | 3.398 | 3.434 | 3.469 |
| 35 | 2.871 | 3.018 | 3.114 | 3.183 | 3.235 | 3.276 | 3.309 | 3.337 | 3.36 | 3.379 | 3.396 | 3.433 | 3.469 |
| 36 | 2.868 | 3.015 | 3.111 | 3.18 | 3.232 | 3.274 | 3.307 | 3.335 | 3.358 | 3.378 | 3.395 | 3.432 | 3.469 |
| 37 | 2.865 | 3.013 | 3.109 | 3.178 | 3.23 | 3.272 | 3.305 | 3.333 | 3.356 | 3.376 | 3.393 | 3.431 | 3.469 |
| 38 | 2.863 | 3.01 | 3.106 | 3.175 | 3.228 | 3.27 | 3.303 | 3.331 | 3.355 | 3.375 | 3.392 | 3.431 | 3.469 |
| 39 | 2.861 | 3.008 | 3.104 | 3.173 | 3.226 | 3.268 | 3.301 | 3.33 | 3.353 | 3.373 | 3.391 | 3.43 | 3.469 |
| 40 | 2.858 | 3.005 | 3.102 | 3.171 | 3.224 | 3.266 | 3.3 | 3.328 | 3.352 | 3.372 | 3.389 | 3.429 | 3.469 |
| 48 | 2.843 | 2.991 | 3.087 | 3.157 | 3.211 | 3.253 | 3.288 | 3.318 | 3.342 | 3.363 | 3.382 | 3.424 | 3.468 |
| 60 | 2.829 | 2.976 | 3.073 | 3.143 | 3.198 | 3.241 | 3.277 | 3.307 | 3.333 | 3.355 | 3.374 | 3.419 | 3.468 |
| 80 | 2.814 | 2.961 | 3.059 | 3.13 | 3.185 | 3.229 | 3.266 | 3.297 | 3.323 | 3.346 | 3.366 | 3.414 | 3.467 |
| 120 | 2.8 | 2.947 | 3.045 | 3.116 | 3.172 | 3.217 | 3.254 | 3.286 | 3.313 | 3.337 | 3.358 | 3.409 | 3.466 |
| 240 | 2.786 | 2.933 | 3.031 | 3.103 | 3.159 | 3.205 | 3.243 | 3.276 | 3.304 | 3.329 | 3.35 | 3.404 | 3.466 |
| \(\infty\) | 2.772 | 2.918 | 3.017 | 3.089 | 3.146 | 3.193 | 3.232 | 3.265 | 3.294 | 3.32 | 3.343 | 3.399 | 3.466 |
26 Appendix 2
Duncan’s multiple range test is one of the most widely used post-hoc procedures for comparing treatment means once an ANOVA has shown a significant difference among them. Table 26.1 gives the critical values \(r_p (p, df, \alpha = 0.05)\) needed to carry out the test, tabulated against the error degrees of freedom (rows) and the number of means being compared, \(p\) (columns).