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16.5: Critical Values for F-Test

  • Page ID
    220802
  • 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.

    Table \(\PageIndex{1}\): Critical Values of F for a One-Tailed F-Test
    \(\frac {\nu_\text{num}\ce{->} }{\nu_{denom} \ce{ v }}\) 1 2 3 4 5 6 7 8 9 10 15 20 \(\infty\)
    1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 245.9 248.0 254.3
    2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 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 8.660 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 5.050 4.950 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 4.060 3.938 3.874 3.669
    7 5.591 4.737 4.347 4.120 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 3.500 3.438 3.388 3.347 3.218 3.150 2.928
    9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 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 3.020 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 3.490 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 2.790 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 2.810 2.699 2.614 2.548 2.494 2.450 2.308 2.230 1.960
    18 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.456 2.412 2.269 2.191 1.917
    19 4.381 3.552 3.127 2.895 2.740 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 2.010 1.938 1.880 1.831 1.666 1.570 1.000

    Table \(\PageIndex{2}\): Critical Values of F for a Two-Tailed F-Test
    \(\frac {\nu_\text{num}\ce{->} }{\nu_{denom} \ce{ v }}\) 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 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.43 39.45 39.50
    3 17.44 16.04 15.44 15.10 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 8.980 8.905 8.444 8.657 8.560 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 7.260 6.599 6.227 5.988 5.820 5.695 5.600 5.523 5.461 5.269 5.168 4.894
    7 8.073 6.542 5.890 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 4.320 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 3.950 3.855 3.779 3.717 3.522 3.419 3.080
    11 6.724 5.256 4.630 4.275 4.044 3.881 3.759 3.644 3.588 3.526 3.330 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 3.250 3.053 2.948 2.596
    14 6.298 4.857 4.242 3.892 3.663 3.501 3.380 3.285 3.209 3.147 2.949 2.844 2.487
    15 6.200 4.765 4.153 3.804 3.576 3.415 3.293 3.199 3.123 3.060 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 4.560 3.954 3.608 3.382 3.221 3.100 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 2.880 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
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