An F-table, also known as an F-distribution table, is a statistical reference table used in hypothesis testing and analysis of variance (ANOVA) to determine critical values of the F-statistic. The F-distribution is a probability distribution that arises in various statistical tests, particularly when comparing the variances of two or more groups.
Here’s a brief explanation of the F-table and its significance:
What Is the F-Table?
The F-table provides critical values for the F-statistic, which is calculated as the ratio of two variances. In hypothesis testing, the F-statistic is used to compare the variances of two or more groups. It helps determine whether there are statistically significant differences between group variances.
How It Works:
- Calculate the F-statistic for your data by dividing the variance of one group by the variance of another group. This F-statistic represents the ratio of the variances you are comparing.
- Determine the degrees of freedom for the numerator and denominator. The numerator degrees of freedom are associated with the first variance, and the denominator degrees of freedom are associated with the second variance.
- Choose a significance level (alpha), which represents the probability of making a Type I error (rejecting a true null hypothesis).
- Use the F-table to find the critical value corresponding to your degrees of freedom and significance level. If the calculated F-statistic is greater than the critical value from the table, you can reject the null hypothesis.
Significance:
The F-table is essential in ANOVA and regression analysis, where it helps researchers determine if there are significant differences or relationships between groups or variables. By comparing the calculated F-statistic to the critical value from the table, researchers can make informed decisions about their hypotheses.
Interpretation:
- If the calculated F-statistic is greater than the critical value from the table, it suggests that there are statistically significant differences or relationships between the groups or variables being compared.
- If the calculated F-statistic is less than the critical value, it indicates that there are no statistically significant differences or relationships, and the null hypothesis cannot be rejected.
Note: F-tables can vary based on the degrees of freedom and the chosen significance level. Different tables may be used for one-tailed or two-tailed tests. Researchers need to select the appropriate table corresponding to their specific analysis and significance level.
In summary, the F-table is a valuable tool in statistical analysis, helping researchers determine the significance of differences or relationships between groups or variables. It allows for data-driven decision-making by providing critical values for the F-statistic, which aids in hypothesis testing and ANOVA.
F Table for alpha=.10
DF1 (see next table below for 12 to ∞)
| DF2 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
| 1 |
39.86346 |
49.50000 |
53.59324 |
55.83296 |
57.24008 |
58.20442 |
58.90595 |
59.43898 |
59.85759 |
60.19498 |
| 2 |
8.52632 |
9.00000 |
9.16179 |
9.24342 |
9.29263 |
9.32553 |
9.34908 |
9.36677 |
9.38054 |
9.39157 |
| 3 |
5.53832 |
5.46238 |
5.39077 |
5.34264 |
5.30916 |
5.28473 |
5.26619 |
5.25167 |
5.24000 |
5.23041 |
| 4 |
4.54477 |
4.32456 |
4.19086 |
4.10725 |
4.05058 |
4.00975 |
3.97897 |
3.95494 |
3.93567 |
3.91988 |
| 5 |
4.06042 |
3.77972 |
3.61948 |
3.52020 |
3.45298 |
3.40451 |
3.36790 |
3.33928 |
3.31628 |
3.29740 |
| 6 |
3.77595 |
3.46330 |
3.28876 |
3.18076 |
3.10751 |
3.05455 |
3.01446 |
2.98304 |
2.95774 |
2.93693 |
| 7 |
3.58943 |
3.25744 |
3.07407 |
2.96053 |
2.88334 |
2.82739 |
2.78493 |
2.75158 |
2.72468 |
2.70251 |
| 8 |
3.45792 |
3.11312 |
2.92380 |
2.80643 |
2.72645 |
2.66833 |
2.62413 |
2.58935 |
2.56124 |
2.53804 |
| 9 |
3.36030 |
3.00645 |
2.81286 |
2.69268 |
2.61061 |
2.55086 |
2.50531 |
2.46941 |
2.44034 |
2.41632 |
| 10 |
3.28502 |
2.92447 |
2.72767 |
2.60534 |
2.52164 |
2.46058 |
2.41397 |
2.37715 |
2.34731 |
2.32260 |
| 11 |
3.22520 |
2.85951 |
2.66023 |
2.53619 |
2.45118 |
2.38907 |
2.34157 |
2.30400 |
2.27350 |
2.24823 |
| 12 |
3.17655 |
2.80680 |
2.60552 |
2.48010 |
2.39402 |
2.33102 |
2.28278 |
2.24457 |
2.21352 |
2.18776 |
| 13 |
3.13621 |
2.76317 |
2.56027 |
2.43371 |
2.34672 |
2.28298 |
2.23410 |
2.19535 |
2.16382 |
2.13763 |
| 14 |
3.10221 |
2.72647 |
2.52222 |
2.39469 |
2.30694 |
2.24256 |
2.19313 |
2.15390 |
2.12195 |
2.09540 |
| 15 |
3.07319 |
2.69517 |
2.48979 |
2.36143 |
2.27302 |
2.20808 |
2.15818 |
2.11853 |
2.08621 |
2.05932 |
| 16 |
3.04811 |
2.66817 |
2.46181 |
2.33274 |
2.24376 |
2.17833 |
2.12800 |
2.08798 |
2.05533 |
2.02815 |
| 17 |
3.02623 |
2.64464 |
2.43743 |
2.30775 |
2.21825 |
2.15239 |
2.10169 |
2.06134 |
2.02839 |
2.00094 |
| 18 |
3.00698 |
2.62395 |
2.41601 |
2.28577 |
2.19583 |
2.12958 |
2.07854 |
2.03789 |
2.00467 |
1.97698 |
| 19 |
2.98990 |
2.60561 |
2.39702 |
2.26630 |
2.17596 |
2.10936 |
2.05802 |
2.01710 |
1.98364 |
1.95573 |
| 20 |
2.97465 |
2.58925 |
2.38009 |
2.24893 |
2.15823 |
2.09132 |
2.03970 |
1.99853 |
1.96485 |
1.93674 |
| 21 |
2.96096 |
2.57457 |
2.36489 |
2.23334 |
2.14231 |
2.07512 |
2.02325 |
1.98186 |
1.94797 |
1.91967 |
| 22 |
2.94858 |
2.56131 |
2.35117 |
2.21927 |
2.12794 |
2.06050 |
2.00840 |
1.96680 |
1.93273 |
1.90425 |
| 23 |
2.93736 |
2.54929 |
2.33873 |
2.20651 |
2.11491 |
2.04723 |
1.99492 |
1.95312 |
1.91888 |
1.89025 |
| 24 |
2.92712 |
2.53833 |
2.32739 |
2.19488 |
2.10303 |
2.03513 |
1.98263 |
1.94066 |
1.90625 |
1.87748 |
| 25 |
2.91774 |
2.52831 |
2.31702 |
2.18424 |
2.09216 |
2.02406 |
1.97138 |
1.92925 |
1.89469 |
1.86578 |
| 26 |
2.90913 |
2.51910 |
2.30749 |
2.17447 |
2.08218 |
2.01389 |
1.96104 |
1.91876 |
1.88407 |
1.85503 |
| 27 |
2.90119 |
2.51061 |
2.29871 |
2.16546 |
2.07298 |
2.00452 |
1.95151 |
1.90909 |
1.87427 |
1.84511 |
| 28 |
2.89385 |
2.50276 |
2.29060 |
2.15714 |
2.06447 |
1.99585 |
1.94270 |
1.90014 |
1.86520 |
1.83593 |
| 29 |
2.88703 |
2.49548 |
2.28307 |
2.14941 |
2.05658 |
1.98781 |
1.93452 |
1.89184 |
1.85679 |
1.82741 |
| 30 |
2.88069 |
2.48872 |
2.27607 |
2.14223 |
2.04925 |
1.98033 |
1.92692 |
1.88412 |
1.84896 |
1.81949 |
| 40 |
2.83535 |
2.44037 |
2.22609 |
2.09095 |
1.99682 |
1.92688 |
1.87252 |
1.82886 |
1.79290 |
1.76269 |
| 60 |
2.79107 |
2.39325 |
2.17741 |
2.04099 |
1.94571 |
1.87472 |
1.81939 |
1.77483 |
1.73802 |
1.70701 |
| 120 |
2.74781 |
2.34734 |
2.12999 |
1.99230 |
1.89587 |
1.82381 |
1.76748 |
1.72196 |
1.68425 |
1.65238 |
| inf |
2.70554 |
2.30259 |
2.08380 |
1.94486 |
1.84727 |
1.77411 |
1.71672 |
1.67020 |
1.63152 |
1.59872 |
DF 1
| DF2 |
12 |
15 |
20 |
24 |
30 |
40 |
60 |
120 |
inf |
| 1 |
60.70521 |
61.22034 |
61.74029 |
62.00205 |
62.26497 |
62.52905 |
62.79428 |
63.06064 |
63.32812 |
| 2 |
9.40813 |
9.42471 |
9.44131 |
9.44962 |
9.45793 |
9.46624 |
9.47456 |
9.48289 |
9.49122 |
| 3 |
5.21562 |
5.20031 |
5.18448 |
5.17636 |
5.16811 |
5.15972 |
5.15119 |
5.14251 |
5.13370 |
| 4 |
3.89553 |
3.87036 |
3.84434 |
3.83099 |
3.81742 |
3.80361 |
3.78957 |
3.77527 |
3.76073 |
| 5 |
3.26824 |
3.23801 |
3.20665 |
3.19052 |
3.17408 |
3.15732 |
3.14023 |
3.12279 |
3.10500 |
| 6 |
2.90472 |
2.87122 |
2.83634 |
2.81834 |
2.79996 |
2.78117 |
2.76195 |
2.74229 |
2.72216 |
| 7 |
2.66811 |
2.63223 |
2.59473 |
2.57533 |
2.55546 |
2.53510 |
2.51422 |
2.49279 |
2.47079 |
| 8 |
2.50196 |
2.46422 |
2.42464 |
2.40410 |
2.38302 |
2.36136 |
2.33910 |
2.31618 |
2.29257 |
| 9 |
2.37888 |
2.33962 |
2.29832 |
2.27683 |
2.25472 |
2.23196 |
2.20849 |
2.18427 |
2.15923 |
| 10 |
2.28405 |
2.24351 |
2.20074 |
2.17843 |
2.15543 |
2.13169 |
2.10716 |
2.08176 |
2.05542 |
| 11 |
2.20873 |
2.16709 |
2.12305 |
2.10001 |
2.07621 |
2.05161 |
2.02612 |
1.99965 |
1.97211 |
| 12 |
2.14744 |
2.10485 |
2.05968 |
2.03599 |
2.01149 |
1.98610 |
1.95973 |
1.93228 |
1.90361 |
| 13 |
2.09659 |
2.05316 |
2.00698 |
1.98272 |
1.95757 |
1.93147 |
1.90429 |
1.87591 |
1.84620 |
| 14 |
2.05371 |
2.00953 |
1.96245 |
1.93766 |
1.91193 |
1.88516 |
1.85723 |
1.82800 |
1.79728 |
| 15 |
2.01707 |
1.97222 |
1.92431 |
1.89904 |
1.87277 |
1.84539 |
1.81676 |
1.78672 |
1.75505 |
| 16 |
1.98539 |
1.93992 |
1.89127 |
1.86556 |
1.83879 |
1.81084 |
1.78156 |
1.75075 |
1.71817 |
| 17 |
1.95772 |
1.91169 |
1.86236 |
1.83624 |
1.80901 |
1.78053 |
1.75063 |
1.71909 |
1.68564 |
| 18 |
1.93334 |
1.88681 |
1.83685 |
1.81035 |
1.78269 |
1.75371 |
1.72322 |
1.69099 |
1.65671 |
| 19 |
1.91170 |
1.86471 |
1.81416 |
1.78731 |
1.75924 |
1.72979 |
1.69876 |
1.66587 |
1.63077 |
| 20 |
1.89236 |
1.84494 |
1.79384 |
1.76667 |
1.73822 |
1.70833 |
1.67678 |
1.64326 |
1.60738 |
| 21 |
1.87497 |
1.82715 |
1.77555 |
1.74807 |
1.71927 |
1.68896 |
1.65691 |
1.62278 |
1.58615 |
| 22 |
1.85925 |
1.81106 |
1.75899 |
1.73122 |
1.70208 |
1.67138 |
1.63885 |
1.60415 |
1.56678 |
| 23 |
1.84497 |
1.79643 |
1.74392 |
1.71588 |
1.68643 |
1.65535 |
1.62237 |
1.58711 |
1.54903 |
| 24 |
1.83194 |
1.78308 |
1.73015 |
1.70185 |
1.67210 |
1.64067 |
1.60726 |
1.57146 |
1.53270 |
| 25 |
1.82000 |
1.77083 |
1.71752 |
1.68898 |
1.65895 |
1.62718 |
1.59335 |
1.55703 |
1.51760 |
| 26 |
1.80902 |
1.75957 |
1.70589 |
1.67712 |
1.64682 |
1.61472 |
1.58050 |
1.54368 |
1.50360 |
| 27 |
1.79889 |
1.74917 |
1.69514 |
1.66616 |
1.63560 |
1.60320 |
1.56859 |
1.53129 |
1.49057 |
| 28 |
1.78951 |
1.73954 |
1.68519 |
1.65600 |
1.62519 |
1.59250 |
1.55753 |
1.51976 |
1.47841 |
| 29 |
1.78081 |
1.73060 |
1.67593 |
1.64655 |
1.61551 |
1.58253 |
1.54721 |
1.50899 |
1.46704 |
| 30 |
1.77270 |
1.72227 |
1.66731 |
1.63774 |
1.60648 |
1.57323 |
1.53757 |
1.49891 |
1.45636 |
| 40 |
1.71456 |
1.66241 |
1.60515 |
1.57411 |
1.54108 |
1.50562 |
1.46716 |
1.42476 |
1.37691 |
| 60 |
1.65743 |
1.60337 |
1.54349 |
1.51072 |
1.47554 |
1.43734 |
1.39520 |
1.34757 |
1.29146 |
| 120 |
1.60120 |
1.54500 |
1.48207 |
1.44723 |
1.40938 |
1.36760 |
1.32034 |
1.26457 |
1.19256 |
| inf |
1.54578 |
1.48714 |
1.42060 |
1.38318 |
1.34187 |
1.29513 |
1.23995 |
1.16860 |
1.00000 |
F Table for alpha=.05
DF1 (see next table below for 12 to ∞)
| df2 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
| 1 |
161.4476 |
199.5000 |
215.7073 |
224.5832 |
230.1619 |
233.9860 |
236.7684 |
238.8827 |
240.5433 |
241.8817 |
| 2 |
18.5128 |
19.0000 |
19.1643 |
19.2468 |
19.2964 |
19.3295 |
19.3532 |
19.3710 |
19.3848 |
19.3959 |
| 3 |
10.1280 |
9.5521 |
9.2766 |
9.1172 |
9.0135 |
8.9406 |
8.8867 |
8.8452 |
8.8123 |
8.7855 |
| 4 |
7.7086 |
6.9443 |
6.5914 |
6.3882 |
6.2561 |
6.1631 |
6.0942 |
6.0410 |
5.9988 |
5.9644 |
| 5 |
6.6079 |
5.7861 |
5.4095 |
5.1922 |
5.0503 |
4.9503 |
4.8759 |
4.8183 |
4.7725 |
4.7351 |
| 6 |
5.9874 |
5.1433 |
4.7571 |
4.5337 |
4.3874 |
4.2839 |
4.2067 |
4.1468 |
4.0990 |
4.0600 |
| 7 |
5.5914 |
4.7374 |
4.3468 |
4.1203 |
3.9715 |
3.8660 |
3.7870 |
3.7257 |
3.6767 |
3.6365 |
| 8 |
5.3177 |
4.4590 |
4.0662 |
3.8379 |
3.6875 |
3.5806 |
3.5005 |
3.4381 |
3.3881 |
3.3472 |
| 9 |
5.1174 |
4.2565 |
3.8625 |
3.6331 |
3.4817 |
3.3738 |
3.2927 |
3.2296 |
3.1789 |
3.1373 |
| 10 |
4.9646 |
4.1028 |
3.7083 |
3.4780 |
3.3258 |
3.2172 |
3.1355 |
3.0717 |
3.0204 |
2.9782 |
| 11 |
4.8443 |
3.9823 |
3.5874 |
3.3567 |
3.2039 |
3.0946 |
3.0123 |
2.9480 |
2.8962 |
2.8536 |
| 12 |
4.7472 |
3.8853 |
3.4903 |
3.2592 |
3.1059 |
2.9961 |
2.9134 |
2.8486 |
2.7964 |
2.7534 |
| 13 |
4.6672 |
3.8056 |
3.4105 |
3.1791 |
3.0254 |
2.9153 |
2.8321 |
2.7669 |
2.7144 |
2.6710 |
| 14 |
4.6001 |
3.7389 |
3.3439 |
3.1122 |
2.9582 |
2.8477 |
2.7642 |
2.6987 |
2.6458 |
2.6022 |
| 15 |
4.5431 |
3.6823 |
3.2874 |
3.0556 |
2.9013 |
2.7905 |
2.7066 |
2.6408 |
2.5876 |
2.5437 |
| 16 |
4.4940 |
3.6337 |
3.2389 |
3.0069 |
2.8524 |
2.7413 |
2.6572 |
2.5911 |
2.5377 |
2.4935 |
| 17 |
4.4513 |
3.5915 |
3.1968 |
2.9647 |
2.8100 |
2.6987 |
2.6143 |
2.5480 |
2.4943 |
2.4499 |
| 18 |
4.4139 |
3.5546 |
3.1599 |
2.9277 |
2.7729 |
2.6613 |
2.5767 |
2.5102 |
2.4563 |
2.4117 |
| 19 |
4.3807 |
3.5219 |
3.1274 |
2.8951 |
2.7401 |
2.6283 |
2.5435 |
2.4768 |
2.4227 |
2.3779 |
| 20 |
4.3512 |
3.4928 |
3.0984 |
2.8661 |
2.7109 |
2.5990 |
2.5140 |
2.4471 |
2.3928 |
2.3479 |
| 21 |
4.3248 |
3.4668 |
3.0725 |
2.8401 |
2.6848 |
2.5727 |
2.4876 |
2.4205 |
2.3660 |
2.3210 |
| 22 |
4.3009 |
3.4434 |
3.0491 |
2.8167 |
2.6613 |
2.5491 |
2.4638 |
2.3965 |
2.3419 |
2.2967 |
| 23 |
4.2793 |
3.4221 |
3.0280 |
2.7955 |
2.6400 |
2.5277 |
2.4422 |
2.3748 |
2.3201 |
2.2747 |
| 24 |
4.2597 |
3.4028 |
3.0088 |
2.7763 |
2.6207 |
2.5082 |
2.4226 |
2.3551 |
2.3002 |
2.2547 |
| 25 |
4.2417 |
3.3852 |
2.9912 |
2.7587 |
2.6030 |
2.4904 |
2.4047 |
2.3371 |
2.2821 |
2.2365 |
| 26 |
4.2252 |
3.3690 |
2.9752 |
2.7426 |
2.5868 |
2.4741 |
2.3883 |
2.3205 |
2.2655 |
2.2197 |
| 27 |
4.2100 |
3.3541 |
2.9604 |
2.7278 |
2.5719 |
2.4591 |
2.3732 |
2.3053 |
1.7306 |
1.6717 |
| 28 |
4.1960 |
3.3404 |
2.9467 |
2.7141 |
2.5581 |
2.4453 |
2.3593 |
2.2913 |
2.2360 |
2.1900 |
| 29 |
4.1830 |
3.3277 |
2.9340 |
2.7014 |
2.5454 |
2.4324 |
2.3463 |
2.2783 |
2.2229 |
2.1768 |
| 30 |
4.1709 |
3.3158 |
2.9223 |
2.6896 |
2.5336 |
2.4205 |
2.3343 |
2.2662 |
2.2107 |
2.1646 |
| 40 |
4.0847 |
3.2317 |
2.8387 |
2.6060 |
2.4495 |
2.3359 |
2.2490 |
2.1802 |
2.1240 |
2.0772 |
| 60 |
4.0012 |
3.1504 |
2.7581 |
2.5252 |
2.3683 |
2.2541 |
2.1665 |
2.0970 |
2.0401 |
1.9926 |
| 120 |
3.9201 |
3.0718 |
2.6802 |
2.4472 |
2.2899 |
2.1750 |
2.0868 |
2.0164 |
1.9588 |
1.9105 |
| inf |
3.8415 |
2.9957 |
2.6049 |
2.3719 |
2.2141 |
2.0986 |
2.0096 |
1.9384 |
1.8799 |
1.8307 |
| df2 |
12 |
15 |
20 |
24 |
30 |
40 |
60 |
120 |
inf |
| 1 |
243.9060 |
245.9499 |
248.0131 |
249.0518 |
250.0951 |
251.1432 |
252.1957 |
253.2529 |
254.3144 |
| 2 |
19.4125 |
19.4291 |
19.4458 |
19.4541 |
19.4624 |
19.4707 |
19.4791 |
19.4874 |
19.4957 |
| 3 |
8.7446 |
8.7029 |
8.6602 |
8.6385 |
8.6166 |
8.5944 |
8.5720 |
8.5494 |
8.5264 |
| 4 |
5.9117 |
5.8578 |
5.8025 |
5.7744 |
5.7459 |
5.7170 |
5.6877 |
5.6581 |
5.6281 |
| 5 |
4.6777 |
4.6188 |
4.5581 |
4.5272 |
4.4957 |
4.4638 |
4.4314 |
4.3985 |
4.3650 |
| 6 |
3.9999 |
3.9381 |
3.8742 |
3.8415 |
3.8082 |
3.7743 |
3.7398 |
3.7047 |
3.6689 |
| 7 |
3.5747 |
3.5107 |
3.4445 |
3.4105 |
3.3758 |
3.3404 |
3.3043 |
3.2674 |
3.2298 |
| 8 |
3.2839 |
3.2184 |
3.1503 |
3.1152 |
3.0794 |
3.0428 |
3.0053 |
2.9669 |
2.9276 |
| 9 |
3.0729 |
3.0061 |
2.9365 |
2.9005 |
2.8637 |
2.8259 |
2.7872 |
2.7475 |
2.7067 |
| 10 |
2.9130 |
2.8450 |
2.7740 |
2.7372 |
2.6996 |
2.6609 |
2.6211 |
2.5801 |
2.5379 |
| 11 |
2.7876 |
2.7186 |
2.6464 |
2.6090 |
2.5705 |
2.5309 |
2.4901 |
2.4480 |
2.4045 |
| 12 |
2.6866 |
2.6169 |
2.5436 |
2.5055 |
2.4663 |
2.4259 |
2.3842 |
2.3410 |
2.2962 |
| 13 |
2.6037 |
2.5331 |
2.4589 |
2.4202 |
2.3803 |
2.3392 |
2.2966 |
2.2524 |
2.2064 |
| 14 |
2.5342 |
2.4630 |
2.3879 |
2.3487 |
2.3082 |
2.2664 |
2.2229 |
2.1778 |
2.1307 |
| 15 |
2.4753 |
2.4034 |
2.3275 |
2.2878 |
2.2468 |
2.2043 |
2.1601 |
2.1141 |
2.0658 |
| 16 |
2.4247 |
2.3522 |
2.2756 |
2.2354 |
2.1938 |
2.1507 |
2.1058 |
2.0589 |
2.0096 |
| 17 |
2.3807 |
2.3077 |
2.2304 |
2.1898 |
2.1477 |
2.1040 |
2.0584 |
2.0107 |
1.9604 |
| 18 |
2.3421 |
2.2686 |
2.1906 |
2.1497 |
2.1071 |
2.0629 |
2.0166 |
1.9681 |
1.9168 |
| 19 |
2.3080 |
2.2341 |
2.1555 |
2.1141 |
2.0712 |
2.0264 |
1.9795 |
1.9302 |
1.8780 |
| 20 |
2.2776 |
2.2033 |
2.1242 |
2.0825 |
2.0391 |
1.9938 |
1.9464 |
1.8963 |
1.8432 |
| 21 |
2.2504 |
2.1757 |
2.0960 |
2.0540 |
2.0102 |
1.9645 |
1.9165 |
1.8657 |
1.8117 |
| 22 |
2.2258 |
2.1508 |
2.0707 |
2.0283 |
1.9842 |
1.9380 |
1.8894 |
1.8380 |
1.7831 |
| 23 |
2.2036 |
2.1282 |
2.0476 |
2.0050 |
1.9605 |
1.9139 |
1.8648 |
1.8128 |
1.7570 |
| 24 |
2.1834 |
2.1077 |
2.0267 |
1.9838 |
1.9390 |
1.8920 |
1.8424 |
1.7896 |
1.7330 |
| 25 |
2.1649 |
2.0889 |
2.0075 |
1.9643 |
1.9192 |
1.8718 |
1.8217 |
1.7684 |
1.7110 |
| 26 |
2.1479 |
2.0716 |
1.9898 |
1.9464 |
1.9010 |
1.8533 |
1.8027 |
1.7488 |
1.6906 |
| 27 |
2.1323 |
2.0558 |
1.9736 |
1.9299 |
1.8842 |
1.8361 |
1.7851 |
1.7306 |
1.6717 |
| 28 |
2.1179 |
2.0411 |
1.9586 |
1.9147 |
1.8687 |
1.8203 |
1.7689 |
1.7138 |
1.6541 |
| 29 |
2.1045 |
2.0275 |
1.9446 |
1.9005 |
1.8543 |
1.8055 |
1.7537 |
1.6981 |
1.6376 |
| 30 |
2.0921 |
2.0148 |
1.9317 |
1.8874 |
1.8409 |
1.7918 |
1.7396 |
1.6835 |
1.6223 |
| 40 |
2.0035 |
1.9245 |
1.8389 |
1.7929 |
1.7444 |
1.6928 |
1.6373 |
1.5766 |
1.5089 |
| 60 |
1.9174 |
1.8364 |
1.7480 |
1.7001 |
1.6491 |
1.5943 |
1.5343 |
1.4673 |
1.3893 |
| 120 |
1.8337 |
1.7505 |
1.6587 |
1.6084 |
1.5543 |
1.4952 |
1.4290 |
1.3519 |
1.2539 |
| inf |
1.7522 |
1.6664 |
1.5705 |
1.5173 |
1.4591 |
1.3940 |
1.3180 |
1.2214 |
1.0000 |
F Table for alpha=.025
DF1 (see next table below for 12 to ∞)
| df2 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
| 1 |
647.7890 |
799.5000 |
864.1630 |
899.5833 |
921.8479 |
937.1111 |
948.2169 |
956.6562 |
963.2846 |
968.6274 |
| 2 |
38.5063 |
39.0000 |
39.1655 |
39.2484 |
39.2982 |
39.3315 |
39.3552 |
39.3730 |
39.3869 |
39.3980 |
| 3 |
17.4434 |
16.0441 |
15.4392 |
15.1010 |
14.8848 |
14.7347 |
14.6244 |
14.5399 |
14.4731 |
14.4189 |
| 4 |
12.2179 |
10.6491 |
9.9792 |
9.6045 |
9.3645 |
9.1973 |
9.0741 |
8.9796 |
8.9047 |
8.8439 |
| 5 |
10.0070 |
8.4336 |
7.7636 |
7.3879 |
7.1464 |
6.9777 |
6.8531 |
6.7572 |
6.6811 |
6.6192 |
| 6 |
8.8131 |
7.2599 |
6.5988 |
6.2272 |
5.9876 |
5.8198 |
5.6955 |
5.5996 |
5.5234 |
5.4613 |
| 7 |
8.0727 |
6.5415 |
5.8898 |
5.5226 |
5.2852 |
5.1186 |
4.9949 |
4.8993 |
4.8232 |
4.7611 |
| 8 |
7.5709 |
6.0595 |
5.4160 |
5.0526 |
4.8173 |
4.6517 |
4.5286 |
4.4333 |
4.3572 |
4.2951 |
| 9 |
7.2093 |
5.7147 |
5.0781 |
4.7181 |
4.4844 |
4.3197 |
4.1970 |
4.1020 |
4.0260 |
3.9639 |
| 10 |
6.9367 |
5.4564 |
4.8256 |
4.4683 |
4.2361 |
4.0721 |
3.9498 |
3.8549 |
3.7790 |
3.7168 |
| 11 |
6.7241 |
5.2559 |
4.6300 |
4.2751 |
4.0440 |
3.8807 |
3.7586 |
3.6638 |
3.5879 |
3.5257 |
| 12 |
6.5538 |
5.0959 |
4.4742 |
4.1212 |
3.8911 |
3.7283 |
3.6065 |
3.5118 |
3.4358 |
3.3736 |
| 13 |
6.4143 |
4.9653 |
4.3472 |
3.9959 |
3.7667 |
3.6043 |
3.4827 |
3.3880 |
3.3120 |
3.2497 |
| 14 |
6.2979 |
4.8567 |
4.2417 |
3.8919 |
3.6634 |
3.5014 |
3.3799 |
3.2853 |
3.2093 |
3.1469 |
| 15 |
6.1995 |
4.7650 |
4.1528 |
3.8043 |
3.5764 |
3.4147 |
3.2934 |
3.1987 |
3.1227 |
3.0602 |
| 16 |
6.1151 |
4.6867 |
4.0768 |
3.7294 |
3.5021 |
3.3406 |
3.2194 |
3.1248 |
3.0488 |
2.9862 |
| 17 |
6.0420 |
4.6189 |
4.0112 |
3.6648 |
3.4379 |
3.2767 |
3.1556 |
3.0610 |
2.9849 |
2.9222 |
| 18 |
5.9781 |
4.5597 |
3.9539 |
3.6083 |
3.3820 |
3.2209 |
3.0999 |
3.0053 |
2.9291 |
2.8664 |
| 19 |
5.9216 |
4.5075 |
3.9034 |
3.5587 |
3.3327 |
3.1718 |
3.0509 |
2.9563 |
2.8801 |
2.8172 |
| 20 |
5.8715 |
4.4613 |
3.8587 |
3.5147 |
3.2891 |
3.1283 |
3.0074 |
2.9128 |
2.8365 |
2.7737 |
| 21 |
5.8266 |
4.4199 |
3.8188 |
3.4754 |
3.2501 |
3.0895 |
2.9686 |
2.8740 |
2.7977 |
2.7348 |
| 22 |
5.7863 |
4.3828 |
3.7829 |
3.4401 |
3.2151 |
3.0546 |
2.9338 |
2.8392 |
2.7628 |
2.6998 |
| 23 |
5.7498 |
4.3492 |
3.7505 |
3.4083 |
3.1835 |
3.0232 |
2.9023 |
2.8077 |
2.7313 |
2.6682 |
| 24 |
5.7166 |
4.3187 |
3.7211 |
3.3794 |
3.1548 |
2.9946 |
2.8738 |
2.7791 |
2.7027 |
2.6396 |
| 25 |
5.6864 |
4.2909 |
3.6943 |
3.3530 |
3.1287 |
2.9685 |
2.8478 |
2.7531 |
2.6766 |
2.6135 |
| 26 |
5.6586 |
4.2655 |
3.6697 |
3.3289 |
3.1048 |
2.9447 |
2.8240 |
2.7293 |
2.6528 |
1.878 |
| 27 |
5.6331 |
4.2421 |
3.6472 |
3.3067 |
3.0828 |
2.9228 |
2.8021 |
2.7074 |
2.6309 |
2.5676 |
| 28 |
5.6096 |
4.2205 |
3.6264 |
3.2863 |
3.0626 |
2.9027 |
2.7820 |
2.6872 |
2.6106 |
2.5473 |
| 29 |
5.5878 |
4.2006 |
3.6072 |
3.2674 |
3.0438 |
2.8840 |
2.7633 |
2.6686 |
2.5919 |
2.5286 |
| 30 |
5.5675 |
4.1821 |
3.5894 |
3.2499 |
3.0265 |
2.8667 |
2.7460 |
2.6513 |
2.5746 |
2.5112 |
| 40 |
5.4239 |
4.0510 |
3.4633 |
3.1261 |
2.9037 |
2.7444 |
2.6238 |
2.5289 |
2.4519 |
2.3882 |
| 60 |
5.2856 |
3.9253 |
3.3425 |
3.0077 |
2.7863 |
2.6274 |
2.5068 |
2.4117 |
2.3344 |
2.2702 |
| 120 |
5.1523 |
3.8046 |
3.2269 |
2.8943 |
2.6740 |
2.5154 |
2.3948 |
2.2994 |
2.2217 |
2.1570 |
| inf |
5.0239 |
3.6889 |
3.1161 |
2.7858 |
2.5665 |
2.4082 |
2.2875 |
2.1918 |
2.1136 |
2.0483 |
| df2 |
12 |
15 |
20 |
24 |
30 |
40 |
60 |
120 |
inf |
| 1 |
976.7079 |
984.8668 |
993.1028 |
997.2492 |
1001.414 |
1005.598 |
1009.800 |
1014.020 |
1018.258 |
| 2 |
39.4146 |
39.4313 |
39.4479 |
39.4562 |
39.465 |
39.473 |
39.481 |
39.490 |
39.498 |
| 3 |
14.3366 |
14.2527 |
14.1674 |
14.1241 |
14.081 |
14.037 |
13.992 |
13.947 |
13.902 |
| 4 |
8.7512 |
8.6565 |
8.5599 |
8.5109 |
8.461 |
8.411 |
8.360 |
8.309 |
8.257 |
| 5 |
6.5245 |
6.4277 |
6.3286 |
6.2780 |
6.227 |
6.175 |
6.123 |
6.069 |
6.015 |
| 6 |
5.3662 |
5.2687 |
5.1684 |
5.1172 |
5.065 |
5.012 |
4.959 |
4.904 |
4.849 |
| 7 |
4.6658 |
4.5678 |
4.4667 |
4.4150 |
4.362 |
4.309 |
4.254 |
4.199 |
4.142 |
| 8 |
4.1997 |
4.1012 |
3.9995 |
3.9472 |
3.894 |
3.840 |
3.784 |
3.728 |
3.670 |
| 9 |
3.8682 |
3.7694 |
3.6669 |
3.6142 |
3.560 |
3.505 |
3.449 |
3.392 |
3.333 |
| 10 |
3.6209 |
3.5217 |
3.4185 |
3.3654 |
3.311 |
3.255 |
3.198 |
3.140 |
3.080 |
| 11 |
3.4296 |
3.3299 |
3.2261 |
3.1725 |
3.118 |
3.061 |
3.004 |
2.944 |
2.883 |
| 12 |
3.2773 |
3.1772 |
3.0728 |
3.0187 |
2.963 |
2.906 |
2.848 |
2.787 |
2.725 |
| 13 |
3.1532 |
3.0527 |
2.9477 |
2.8932 |
2.837 |
2.780 |
2.720 |
2.659 |
2.595 |
| 14 |
3.0502 |
2.9493 |
2.8437 |
2.7888 |
2.732 |
2.674 |
2.614 |
2.552 |
2.487 |
| 15 |
2.9633 |
2.8621 |
2.7559 |
2.7006 |
2.644 |
2.585 |
2.524 |
2.461 |
2.395 |
| 16 |
2.8890 |
2.7875 |
2.6808 |
2.6252 |
2.568 |
2.509 |
2.447 |
2.383 |
2.316 |
| 17 |
2.8249 |
2.7230 |
2.6158 |
2.5598 |
2.502 |
2.442 |
2.380 |
2.315 |
2.247 |
| 18 |
2.7689 |
2.6667 |
2.5590 |
2.5027 |
2.445 |
2.384 |
2.321 |
2.256 |
2.187 |
| 19 |
2.7196 |
2.6171 |
2.5089 |
2.4523 |
2.394 |
2.333 |
2.270 |
2.203 |
2.133 |
| 20 |
2.6758 |
2.5731 |
2.4645 |
2.4076 |
2.349 |
2.287 |
2.223 |
2.156 |
2.085 |
| 21 |
2.6368 |
2.5338 |
2.4247 |
2.3675 |
2.308 |
2.246 |
2.182 |
2.114 |
2.042 |
| 22 |
2.6017 |
2.4984 |
2.3890 |
2.3315 |
2.272 |
2.210 |
2.145 |
2.076 |
2.003 |
| 23 |
2.5699 |
2.4665 |
2.3567 |
2.2989 |
2.239 |
2.176 |
2.111 |
2.041 |
1.968 |
| 24 |
2.5411 |
2.4374 |
2.3273 |
2.2693 |
2.209 |
2.146 |
2.080 |
2.010 |
1.935 |
| 25 |
2.5149 |
2.4110 |
2.3005 |
2.2422 |
2.182 |
2.118 |
2.052 |
1.981 |
1.906 |
| 26 |
2.4908 |
2.3867 |
2.2759 |
2.2174 |
2.157 |
2.093 |
2.026 |
1.954 |
1.878 |
| 27 |
2.4688 |
2.3644 |
2.2533 |
2.1946 |
2.133 |
2.069 |
2.002 |
1.930 |
1.853 |
| 28 |
2.4484 |
2.3438 |
2.2324 |
2.1735 |
2.112 |
2.048 |
1.980 |
1.907 |
1.829 |
| 29 |
2.4295 |
2.3248 |
2.2131 |
2.1540 |
2.092 |
2.028 |
1.959 |
1.886 |
1.807 |
| 30 |
2.4120 |
2.3072 |
2.1952 |
2.1359 |
2.074 |
2.009 |
1.940 |
1.866 |
1.787 |
| 40 |
2.2882 |
2.1819 |
2.0677 |
2.0069 |
1.943 |
1.875 |
1.803 |
1.724 |
1.637 |
| 60 |
2.1692 |
2.0613 |
1.9445 |
1.8817 |
1.815 |
1.744 |
1.667 |
1.581 |
1.482 |
| 120 |
2.0548 |
1.9450 |
1.8249 |
1.7597 |
1.690 |
1.614 |
1.530 |
1.433 |
1.310 |
| inf |
1.9447 |
1.8326 |
1.7085 |
1.6402 |
1.566 |
1.484 |
1.388 |
1.268 |
1.000 |
F Table for alpha=.01
DF1 (see table below for 12 to ∞)
| df2/df1 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
| 1 |
4052.181 |
4999.500 |
5403.352 |
5624.583 |
5763.650 |
5858.986 |
5928.356 |
5981.070 |
6022.473 |
6055.847 |
| 2 |
98.503 |
99.000 |
99.166 |
99.249 |
99.299 |
99.333 |
99.356 |
99.374 |
99.388 |
99.399 |
| 3 |
34.116 |
30.817 |
29.457 |
28.710 |
28.237 |
27.911 |
27.672 |
27.489 |
27.345 |
27.229 |
| 4 |
21.198 |
18.000 |
16.694 |
15.977 |
15.522 |
15.207 |
14.976 |
14.799 |
14.659 |
14.546 |
| 5 |
16.258 |
13.274 |
12.060 |
11.392 |
10.967 |
10.672 |
10.456 |
10.289 |
10.158 |
10.051 |
| 6 |
13.745 |
10.925 |
9.780 |
9.148 |
8.746 |
8.466 |
8.260 |
8.102 |
7.976 |
7.874 |
| 7 |
12.246 |
9.547 |
8.451 |
7.847 |
7.460 |
7.191 |
6.993 |
6.840 |
6.719 |
6.620 |
| 8 |
11.259 |
8.649 |
7.591 |
7.006 |
6.632 |
6.371 |
6.178 |
6.029 |
5.911 |
5.814 |
| 9 |
10.561 |
8.022 |
6.992 |
6.422 |
6.057 |
5.802 |
5.613 |
5.467 |
5.351 |
5.257 |
| 10 |
10.044 |
7.559 |
6.552 |
5.994 |
5.636 |
5.386 |
5.200 |
5.057 |
4.942 |
4.849 |
| 11 |
9.646 |
7.206 |
6.217 |
5.668 |
5.316 |
5.069 |
4.886 |
4.744 |
4.632 |
4.539 |
| 12 |
9.330 |
6.927 |
5.953 |
5.412 |
5.064 |
4.821 |
4.640 |
4.499 |
4.388 |
4.296 |
| 13 |
9.074 |
6.701 |
5.739 |
5.205 |
4.862 |
4.620 |
4.441 |
4.302 |
4.191 |
4.100 |
| 14 |
8.862 |
6.515 |
5.564 |
5.035 |
4.695 |
4.456 |
4.278 |
4.140 |
4.030 |
3.939 |
| 15 |
8.683 |
6.359 |
5.417 |
4.893 |
4.556 |
4.318 |
4.142 |
4.004 |
3.895 |
3.805 |
| 16 |
8.531 |
6.226 |
5.292 |
4.773 |
4.437 |
4.202 |
4.026 |
3.890 |
3.780 |
3.691 |
| 17 |
8.400 |
6.112 |
5.185 |
4.669 |
4.336 |
4.102 |
3.927 |
3.791 |
3.682 |
3.593 |
| 18 |
8.285 |
6.013 |
5.092 |
4.579 |
4.248 |
4.015 |
3.841 |
3.705 |
3.597 |
3.508 |
| 19 |
8.185 |
5.926 |
5.010 |
4.500 |
4.171 |
3.939 |
3.765 |
3.631 |
3.523 |
3.434 |
| 20 |
8.096 |
5.849 |
4.938 |
4.431 |
4.103 |
3.871 |
3.699 |
3.564 |
3.457 |
3.368 |
| 21 |
8.017 |
5.780 |
4.874 |
4.369 |
4.042 |
3.812 |
3.640 |
3.506 |
3.398 |
3.310 |
| 22 |
7.945 |
5.719 |
4.817 |
4.313 |
3.988 |
3.758 |
3.587 |
3.453 |
3.346 |
3.258 |
| 23 |
7.881 |
5.664 |
4.765 |
4.264 |
3.939 |
3.710 |
3.539 |
3.406 |
3.299 |
3.211 |
| 24 |
7.823 |
5.614 |
4.718 |
4.218 |
3.895 |
3.667 |
3.496 |
3.363 |
3.256 |
3.168 |
| 25 |
7.770 |
5.568 |
4.675 |
4.177 |
3.855 |
3.627 |
3.457 |
3.324 |
3.217 |
3.129 |
| 26 |
7.721 |
5.526 |
4.637 |
4.140 |
3.818 |
3.591 |
3.421 |
3.288 |
3.182 |
3.094 |
| 27 |
7.677 |
5.488 |
4.601 |
4.106 |
3.785 |
3.558 |
3.388 |
3.256 |
3.149 |
3.062 |
| 28 |
7.636 |
5.453 |
4.568 |
4.074 |
3.754 |
3.528 |
3.358 |
3.226 |
3.120 |
3.032 |
| 29 |
7.598 |
5.420 |
4.538 |
4.045 |
3.725 |
3.499 |
3.330 |
3.198 |
3.092 |
3.005 |
| 30 |
7.562 |
5.390 |
4.510 |
4.018 |
3.699 |
3.473 |
3.304 |
3.173 |
3.067 |
2.979 |
| 40 |
7.314 |
5.179 |
4.313 |
3.828 |
3.514 |
3.291 |
3.124 |
2.993 |
2.888 |
2.801 |
| 60 |
7.077 |
4.977 |
4.126 |
3.649 |
3.339 |
3.119 |
2.953 |
2.823 |
2.718 |
2.632 |
| 120 |
6.851 |
4.787 |
3.949 |
3.480 |
3.174 |
2.956 |
2.792 |
2.663 |
2.559 |
2.472 |
| inf |
6.635 |
4.605 |
3.782 |
3.319 |
3.017 |
2.802 |
2.639 |
2.511 |
2.407 |
2.321 |
| df2 |
12 |
15 |
20 |
24 |
30 |
40 |
60 |
120 |
inf |
| 1 |
6106.321 |
6157.285 |
6208.730 |
6234.631 |
6260.649 |
6286.782 |
6313.030 |
6339.391 |
6365.864 |
| 2 |
99.416 |
99.433 |
99.449 |
99.458 |
99.466 |
99.474 |
99.482 |
99.491 |
99.499 |
| 3 |
27.052 |
26.872 |
26.690 |
26.598 |
26.505 |
26.411 |
26.316 |
26.221 |
26.125 |
| 4 |
14.374 |
14.198 |
14.020 |
13.929 |
13.838 |
13.745 |
13.652 |
13.558 |
13.463 |
| 5 |
9.888 |
9.722 |
9.553 |
9.466 |
9.379 |
9.291 |
9.202 |
9.112 |
9.020 |
| 6 |
7.718 |
7.559 |
7.396 |
7.313 |
7.229 |
7.143 |
7.057 |
6.969 |
6.880 |
| 7 |
6.469 |
6.314 |
6.155 |
6.074 |
5.992 |
5.908 |
5.824 |
5.737 |
5.650 |
| 8 |
5.667 |
5.515 |
5.359 |
5.279 |
5.198 |
5.116 |
5.032 |
4.946 |
4.859 |
| 9 |
5.111 |
4.962 |
4.808 |
4.729 |
4.649 |
4.567 |
4.483 |
4.398 |
4.311 |
| 10 |
4.706 |
4.558 |
4.405 |
4.327 |
4.247 |
4.165 |
4.082 |
3.996 |
3.909 |
| 11 |
4.397 |
4.251 |
4.099 |
4.021 |
3.941 |
3.860 |
3.776 |
3.690 |
3.602 |
| 12 |
4.155 |
4.010 |
3.858 |
3.780 |
3.701 |
3.619 |
3.535 |
3.449 |
3.361 |
| 13 |
3.815 |
3.665 |
3.587 |
3.507 |
3.425 |
3.341 |
3.255 |
3.165 |
| 14 |
3.800 |
3.656 |
3.505 |
3.427 |
3.348 |
3.266 |
3.181 |
3.094 |
3.004 |
| 15 |
3.666 |
3.522 |
3.372 |
3.294 |
3.214 |
3.132 |
3.047 |
2.959 |
2.868 |
| 16 |
3.553 |
3.409 |
3.259 |
3.181 |
3.101 |
3.018 |
2.933 |
2.845 |
2.753 |
| 17 |
3.455 |
3.312 |
3.162 |
3.084 |
3.003 |
2.920 |
2.835 |
2.746 |
2.653 |
| 18 |
3.371 |
3.227 |
3.077 |
2.999 |
2.919 |
2.835 |
2.749 |
2.660 |
2.566 |
| 19 |
3.297 |
3.153 |
3.003 |
2.925 |
2.844 |
2.761 |
2.674 |
2.584 |
2.489 |
| 20 |
3.231 |
3.088 |
2.938 |
2.859 |
2.778 |
2.695 |
2.608 |
2.517 |
2.421 |
| 21 |
3.173 |
3.030 |
2.880 |
2.801 |
2.720 |
2.636 |
2.548 |
2.457 |
2.360 |
| 22 |
3.121 |
2.978 |
2.827 |
2.749 |
2.667 |
2.583 |
2.495 |
2.403 |
2.305 |
| 23 |
3.074 |
2.931 |
2.781 |
2.702 |
2.620 |
2.535 |
2.447 |
2.354 |
2.256 |
| 24 |
3.032 |
2.889 |
2.738 |
2.659 |
2.577 |
2.492 |
2.403 |
2.310 |
2.211 |
| 25 |
2.993 |
2.850 |
2.699 |
2.620 |
2.538 |
2.453 |
2.364 |
2.270 |
2.169 |
| 26 |
2.958 |
2.815 |
2.664 |
2.585 |
2.503 |
2.417 |
2.327 |
2.233 |
2.131 |
| 27 |
2.926 |
2.783 |
2.632 |
2.552 |
2.470 |
2.384 |
2.294 |
2.198 |
2.097 |
| 28 |
2.896 |
2.753 |
2.602 |
2.522 |
2.440 |
2.354 |
2.263 |
2.167 |
2.064 |
| 29 |
2.868 |
2.726 |
2.574 |
2.495 |
2.412 |
2.325 |
2.234 |
2.138 |
2.034 |
| 30 |
2.843 |
2.700 |
2.549 |
2.469 |
2.386 |
2.299 |
2.208 |
2.111 |
2.006 |
| 40 |
2.665 |
2.522 |
2.369 |
2.288 |
2.203 |
2.114 |
2.019 |
1.917 |
1.805 |
| 60 |
2.496 |
2.352 |
2.198 |
2.115 |
2.028 |
1.936 |
1.836 |
1.726 |
1.601 |
| 120 |
2.336 |
2.192 |
2.035 |
1.950 |
1.860 |
1.763 |
1.656 |
1.533 |
1.381 |
| inf |
2.185 |
2.039 |
1.878 |
1.791 |
1.696 |
1.592 |
1.473 |
1.325 |
1.000 |
