16.5: Critical Values for F-Test

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The following tables provide values for $$F(0.05, \nu_\text{num}, \nu_\text{denom})$$ for one-tailed and for two-tailed F-tests. To use these tables, we first decide whether the situation calls for a one-tailed or a two-tailed analysis and calculate Fexp

$F_\text{exp} = \frac {s_A^2} {s_B^2} \nonumber$

where $$S_A^2$$ is greater than $$s_B^2$$. Next, we compare Fexp to $$F(0.05, \nu_\text{num}, \nu_\text{denom})$$ and reject the null hypothesis if $$F_\text{exp} > F(0.05, \nu_\text{num}, \nu_\text{denom})$$. You may replace s with $$\sigma$$ if you know the population’s standard deviation.

 $$\frac {\nu_\text{num}\ce{->} }{\nu_{denom} \ce{ v }}$$ 1 2 3 4 5 6 7 8 9 10 15 20 $$\infty$$ 1 161.4 199.5 215.7 224.6 230.2 234 236.8 238.9 240.5 241.9 245.9 248 254.3 2 18.51 19 19.16 19.25 19.3 19.33 19.35 19.37 19.38 19.4 19.43 19.45 19.50 3 10.13 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786 8.703 8.66 8.526 4 7.709 6.994 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964 5.858 5.803 5.628 5 6.608 5.786 5.409 5.192 5.05 4.95 4.876 4.818 4.722 4.753 4.619 4.558 4.365 6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.06 3.938 3.874 3.669 7 5.591 4.737 4.347 4.12 3.972 3.866 3.787 3.726 3.677 3.637 3.511 3.445 3.230 8 5.318 4.459 4.066 3.838 3.687 3.581 3.5 3.438 3.388 3.347 3.218 3.15 2.928 9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.23 3.179 3.137 3.006 2.936 2.707 10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.02 2.978 2.845 2.774 2.538 11 4.844 3.982 3.587 3.257 3.204 3.095 3.012 2.948 2.896 2.854 2.719 2.646 2.404 12 4.747 3.885 3.49 3.259 3.106 2.996 2.913 2.849 2.796 2.753 2.617 2.544 2.296 13 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671 2.533 2.459 2.206 14 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602 2.463 2.388 2.131 15 4.534 3.682 3.287 3.056 2.901 2.79 2.707 2.641 2.588 2.544 2.403 2.328 2.066 16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.538 2.494 2.352 2.276 2.010 17 4.451 3.592 3.197 2.965 2.81 2.699 2.614 2.548 2.494 2.45 2.308 2.23 1.960 18 4.414 3.555 3.16 2.928 2.773 2.661 2.577 2.51 2.456 2.412 2.269 2.191 1.917 19 4.381 3.552 3.127 2.895 2.74 2.628 2.544 2.477 2.423 2.378 2.234 2.155 1.878 20 4,351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348 2.203 2.124 1.843 $$\infty$$ 3.842 2.996 2.605 2.372 2.214 2.099 2.01 1.938 1.88 1.831 1.666 1.57 1.000

 $$\frac {\nu_\text{num}\ce{->} }{\nu_{denom} \ce{ v }}$$ 1 2 3 4 5 6 7 8 9 10 15 20 $$\infty$$ 1 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3 968.6 984.9 993.1 1018 2 38.51 39 39.17 39.25 39.3 39.33 39.36 39.37 39.39 39.4 39.43 39.45 39.50 3 17.44 16.04 15.44 15.1 14.88 14.73 14.62 14.54 14.47 14.42 14.25 14.17 13.90 4 12.22 10.65 9.979 9.605 9.364 9.197 9.074 8.98 8.905 8.444 8.657 8.56 8.257 5 10.01 8.434 7.764 7.388 7.146 6.978 6.853 6.757 6.681 6.619 6.428 6.329 6.015 6 8.813 7.26 6.599 6.227 5.988 5.82 5.695 5.6 5.523 5.461 5.269 5.168 4.894 7 8.073 6.542 5.89 5.523 5.285 5.119 4.995 4.899 4.823 4.761 4.568 4.467 4.142 8 7.571 6.059 5.416 5.053 4.817 4.652 4.529 4.433 4.357 4.259 4.101 3.999 3.670 9 7.209 5.715 5.078 4.718 4.484 4.32 4.197 4.102 4.026 3.964 3.769 3.667 3.333 10 6.937 5.456 4.826 4.468 4.236 4.072 3.95 3.855 3.779 3.717 3.522 3.419 3.080 11 6.724 5.256 4.63 4.275 4.044 3.881 3.759 3.644 3.588 3.526 3.33 3.226 2.883 12 6.544 5.096 4.474 4.121 3.891 3.728 3.607 3.512 3.436 3.374 3.177 3.073 2.725 13 6.414 4.965 4.347 3.996 3.767 3.604 3.483 3.388 3.312 3.25 3.053 2.948 2.596 14 6.298 4.857 4.242 3.892 3.663 3.501 3.38 3.285 3.209 3.147 2.949 2.844 2.487 15 6.2 4.765 4.153 3.804 3.576 3.415 3.293 3.199 3.123 3.06 2.862 2.756 2.395 16 6.115 4.687 4.077 3.729 3.502 3.341 3.219 3.125 3.049 2.986 2.788 2.681 2.316 17 6.042 4.619 4.011 3.665 3.438 3.277 3.156 3.061 2.985 2.922 2.723 2.616 2.247 18 5.978 4.56 3.954 3.608 3.382 3.221 3.1 3.005 2.929 2.866 2.667 2.559 2.187 19 5.922 4.508 3.903 3.559 3.333 3.172 3.051 2.956 2.88 2.817 2.617 2.509 2.133 20 5.871 4.461 3.859 3.515 3.289 3.128 3.007 2.913 2.837 2.774 2.573 2.464 2.085 $$\infty$$ 5.024 3.689 3.116 2.786 2.567 2.408 2.288 2.192 2.114 2.048 1.833 1.708 1.000

This page titled 16.5: Critical Values for F-Test is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by David Harvey.