# 12.3: Critical Values for F-Test

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

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

12.3: Critical Values for F-Test is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.