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This section contains distribution tables for some of the more widely used probability distributions. The include the cumulative Normal distribution, the Student's t-distribution, the chi-square distribution and the F distribution. Note that distribution tables such as these can be produced directly using Excel or similar software via their various built-in functions. For example, the F distribution tables provided in this topic were generated using the Excel function FINV(0.05,df1,df2).
The following table provides the cumulative distribution function of the Normal distribution, from 0 to x. To obtain a one-sided value simply add 0.5 to the table entry, or for a two-sided value (the most commonly used case) double the value in the table. The highlighted cells identify the two-sided values for 90% (+/-5%) and 95% (+/2.5%) values.
x |
0 |
0.01 |
0.02 |
0.03 |
0.04 |
0.05 |
0.06 |
0.07 |
0.08 |
0.09 |
---|---|---|---|---|---|---|---|---|---|---|
— |
— |
0.00399 |
0.00798 |
0.01197 |
0.01595 |
0.01994 |
0.02392 |
0.02790 |
0.03188 |
0.03586 |
0.10 |
0.03983 |
0.04380 |
0.04776 |
0.05172 |
0.05567 |
0.05962 |
0.06356 |
0.06749 |
0.07142 |
0.07535 |
0.20 |
0.07926 |
0.08317 |
0.08706 |
0.09095 |
0.09483 |
0.09871 |
0.10257 |
0.10642 |
0.11026 |
0.11409 |
0.30 |
0.11791 |
0.12172 |
0.12552 |
0.12930 |
0.13307 |
0.13683 |
0.14058 |
0.14431 |
0.14803 |
0.15173 |
0.40 |
0.15542 |
0.15910 |
0.16276 |
0.16640 |
0.17003 |
0.17364 |
0.17724 |
0.18082 |
0.18439 |
0.18793 |
0.50 |
0.19146 |
0.19497 |
0.19847 |
0.20194 |
0.20540 |
0.20884 |
0.21226 |
0.21566 |
0.21904 |
0.22240 |
0.60 |
0.22575 |
0.22907 |
0.23237 |
0.23565 |
0.23891 |
0.24215 |
0.24537 |
0.24857 |
0.25175 |
0.25490 |
0.70 |
0.25804 |
0.26115 |
0.26424 |
0.26730 |
0.27035 |
0.27337 |
0.27637 |
0.27935 |
0.28230 |
0.28524 |
0.80 |
0.28814 |
0.29103 |
0.29389 |
0.29673 |
0.29955 |
0.30234 |
0.30511 |
0.30785 |
0.31057 |
0.31327 |
0.90 |
0.31594 |
0.31859 |
0.32121 |
0.32381 |
0.32639 |
0.32894 |
0.33147 |
0.33398 |
0.33646 |
0.33891 |
1.00 |
0.34134 |
0.34375 |
0.34614 |
0.34849 |
0.35083 |
0.35314 |
0.35543 |
0.35769 |
0.35993 |
0.36214 |
1.10 |
0.36433 |
0.36650 |
0.36864 |
0.37076 |
0.37286 |
0.37493 |
0.37698 |
0.37900 |
0.38100 |
0.38298 |
1.20 |
0.38493 |
0.38686 |
0.38877 |
0.39065 |
0.39251 |
0.39435 |
0.39617 |
0.39796 |
0.39973 |
0.40147 |
1.30 |
0.40320 |
0.40490 |
0.40658 |
0.40824 |
0.40988 |
0.41149 |
0.41308 |
0.41466 |
0.41621 |
0.41774 |
1.40 |
0.41924 |
0.42073 |
0.42220 |
0.42364 |
0.42507 |
0.42647 |
0.42785 |
0.42922 |
0.43056 |
0.43189 |
1.50 |
0.43319 |
0.43448 |
0.43574 |
0.43699 |
0.43822 |
0.43943 |
0.44062 |
0.44179 |
0.44295 |
0.44408 |
1.60 |
0.44520 |
0.44630 |
0.44738 |
0.44845 |
0.44950 |
0.45053 |
0.45154 |
0.45254 |
0.45352 |
0.45449 |
1.70 |
0.45543 |
0.45637 |
0.45728 |
0.45818 |
0.45907 |
0.45994 |
0.46080 |
0.46164 |
0.46246 |
0.46327 |
1.80 |
0.46407 |
0.46485 |
0.46562 |
0.46638 |
0.46712 |
0.46784 |
0.46856 |
0.46926 |
0.46995 |
0.47062 |
1.90 |
0.47128 |
0.47193 |
0.47257 |
0.47320 |
0.47381 |
0.47441 |
0.47500 |
0.47558 |
0.47615 |
0.47670 |
2.00 |
0.47725 |
0.47778 |
0.47831 |
0.47882 |
0.47932 |
0.47982 |
0.48030 |
0.48077 |
0.48124 |
0.48169 |
2.10 |
0.48214 |
0.48257 |
0.48300 |
0.48341 |
0.48382 |
0.48422 |
0.48461 |
0.48500 |
0.48537 |
0.48574 |
2.20 |
0.48610 |
0.48645 |
0.48679 |
0.48713 |
0.48745 |
0.48778 |
0.48809 |
0.48840 |
0.48870 |
0.48899 |
2.30 |
0.48928 |
0.48956 |
0.48983 |
0.49010 |
0.49036 |
0.49061 |
0.49086 |
0.49111 |
0.49134 |
0.49158 |
2.40 |
0.49180 |
0.49202 |
0.49224 |
0.49245 |
0.49266 |
0.49286 |
0.49305 |
0.49324 |
0.49343 |
0.49361 |
2.50 |
0.49379 |
0.49396 |
0.49413 |
0.49430 |
0.49446 |
0.49461 |
0.49477 |
0.49492 |
0.49506 |
0.49520 |
2.60 |
0.49534 |
0.49547 |
0.49560 |
0.49573 |
0.49585 |
0.49598 |
0.49609 |
0.49621 |
0.49632 |
0.49643 |
2.70 |
0.49653 |
0.49664 |
0.49674 |
0.49683 |
0.49693 |
0.49702 |
0.49711 |
0.49720 |
0.49728 |
0.49736 |
2.80 |
0.49744 |
0.49752 |
0.49760 |
0.49767 |
0.49774 |
0.49781 |
0.49788 |
0.49795 |
0.49801 |
0.49807 |
2.90 |
0.49813 |
0.49819 |
0.49825 |
0.49831 |
0.49836 |
0.49841 |
0.49846 |
0.49851 |
0.49856 |
0.49861 |
3.00 |
0.49865 |
0.49869 |
0.49874 |
0.49878 |
0.49882 |
0.49886 |
0.49889 |
0.49893 |
0.49896 |
0.49900 |
3.10 |
0.49903 |
0.49906 |
0.49910 |
0.49913 |
0.49916 |
0.49918 |
0.49921 |
0.49924 |
0.49926 |
0.49929 |
3.20 |
0.49931 |
0.49934 |
0.49936 |
0.49938 |
0.49940 |
0.49942 |
0.49944 |
0.49946 |
0.49948 |
0.49950 |
3.30 |
0.49952 |
0.49953 |
0.49955 |
0.49957 |
0.49958 |
0.49960 |
0.49961 |
0.49962 |
0.49964 |
0.49965 |
3.40 |
0.49966 |
0.49968 |
0.49969 |
0.49970 |
0.49971 |
0.49972 |
0.49973 |
0.49974 |
0.49975 |
0.49976 |
3.50 |
0.49977 |
0.49978 |
0.49978 |
0.49979 |
0.49980 |
0.49981 |
0.49981 |
0.49982 |
0.49983 |
0.49983 |
3.60 |
0.49984 |
0.49985 |
0.49985 |
0.49986 |
0.49986 |
0.49987 |
0.49987 |
0.49988 |
0.49988 |
0.49989 |
3.70 |
0.49989 |
0.49990 |
0.49990 |
0.49990 |
0.49991 |
0.49991 |
0.49992 |
0.49992 |
0.49992 |
0.49992 |
3.80 |
0.49993 |
0.49993 |
0.49993 |
0.49994 |
0.49994 |
0.49994 |
0.49994 |
0.49995 |
0.49995 |
0.49995 |
3.90 |
0.49995 |
0.49995 |
0.49996 |
0.49996 |
0.49996 |
0.49996 |
0.49996 |
0.49996 |
0.49997 |
0.49997 |
4.00 |
0.49997 |
0.49997 |
0.49997 |
0.49997 |
0.49997 |
0.49997 |
0.49998 |
0.49998 |
0.49998 |
0.49998 |
Note that specific critical values may be computed using standard inverse functions, such as NORMINV() in Excel. For the unit Normal the table below provides the x values associated with the critical probability levels indicated (upper tail).
p |
x |
p |
x |
0.9999 |
3.7190165 |
0.950 |
1.6438848 |
0.999 |
3.0902323 |
0.945 |
1.5972948 |
0.995 |
2.5689744 |
0.940 |
1.5539347 |
0.990 |
2.3226121 |
0.935 |
1.5133137 |
0.985 |
2.1674573 |
0.930 |
1.4750467 |
0.980 |
2.0516879 |
0.925 |
1.4388254 |
0.975 |
1.9582558 |
0.920 |
1.4043992 |
0.970 |
1.879326 |
0.915 |
1.3715615 |
0.965 |
1.810618 |
0.910 |
1.3401395 |
0.960 |
1.7495268 |
0.905 |
1.3099877 |
0.955 |
1.6943437 |
0.900 |
1.280982 |
Upper tail critical values for the Student's t-distribution, showing the degrees of freedom (df) and probability levels from 0.1 (10%) to 0.1%. Note that for df>30 the figures are close to those for the Normal distribution, and for the infinity line they match those for the Normal distribution. Only the upper critical values are required as the distribution is symmetric, as per the Normal.
df |
0.1 |
0.05 |
0.025 |
0.01 |
0.005 |
0.001 |
---|---|---|---|---|---|---|
1 |
3.078 |
6.314 |
12.706 |
31.821 |
63.657 |
318.313 |
2 |
1.886 |
2.920 |
4.303 |
6.965 |
9.925 |
22.327 |
3 |
1.638 |
2.353 |
3.182 |
4.541 |
5.841 |
10.215 |
4 |
1.533 |
2.132 |
2.776 |
3.747 |
4.604 |
7.173 |
5 |
1.476 |
2.015 |
2.571 |
3.365 |
4.032 |
5.893 |
6 |
1.440 |
1.943 |
2.447 |
3.143 |
3.707 |
5.208 |
7 |
1.415 |
1.895 |
2.365 |
2.998 |
3.499 |
4.782 |
8 |
1.397 |
1.860 |
2.306 |
2.896 |
3.355 |
4.499 |
9 |
1.383 |
1.833 |
2.262 |
2.821 |
3.250 |
4.296 |
10 |
1.372 |
1.812 |
2.228 |
2.764 |
3.169 |
4.143 |
11 |
1.363 |
1.796 |
2.201 |
2.718 |
3.106 |
4.024 |
12 |
1.356 |
1.782 |
2.179 |
2.681 |
3.055 |
3.929 |
13 |
1.350 |
1.771 |
2.160 |
2.650 |
3.012 |
3.852 |
14 |
1.345 |
1.761 |
2.145 |
2.624 |
2.977 |
3.787 |
15 |
1.341 |
1.753 |
2.131 |
2.602 |
2.947 |
3.733 |
16 |
1.337 |
1.746 |
2.120 |
2.583 |
2.921 |
3.686 |
17 |
1.333 |
1.740 |
2.110 |
2.567 |
2.898 |
3.646 |
18 |
1.330 |
1.734 |
2.101 |
2.552 |
2.878 |
3.610 |
19 |
1.328 |
1.729 |
2.093 |
2.539 |
2.861 |
3.579 |
20 |
1.325 |
1.725 |
2.086 |
2.528 |
2.845 |
3.552 |
21 |
1.323 |
1.721 |
2.080 |
2.518 |
2.831 |
3.527 |
22 |
1.321 |
1.717 |
2.074 |
2.508 |
2.819 |
3.505 |
23 |
1.319 |
1.714 |
2.069 |
2.500 |
2.807 |
3.485 |
24 |
1.318 |
1.711 |
2.064 |
2.492 |
2.797 |
3.467 |
25 |
1.316 |
1.708 |
2.060 |
2.485 |
2.787 |
3.450 |
26 |
1.315 |
1.706 |
2.056 |
2.479 |
2.779 |
3.435 |
27 |
1.314 |
1.703 |
2.052 |
2.473 |
2.771 |
3.421 |
28 |
1.313 |
1.701 |
2.048 |
2.467 |
2.763 |
3.408 |
29 |
1.311 |
1.699 |
2.045 |
2.462 |
2.756 |
3.396 |
30 |
1.310 |
1.697 |
2.042 |
2.457 |
2.750 |
3.385 |
31 |
1.309 |
1.696 |
2.040 |
2.453 |
2.744 |
3.375 |
32 |
1.309 |
1.694 |
2.037 |
2.449 |
2.738 |
3.365 |
33 |
1.308 |
1.692 |
2.035 |
2.445 |
2.733 |
3.356 |
34 |
1.307 |
1.691 |
2.032 |
2.441 |
2.728 |
3.348 |
35 |
1.306 |
1.690 |
2.030 |
2.438 |
2.724 |
3.340 |
36 |
1.306 |
1.688 |
2.028 |
2.434 |
2.719 |
3.333 |
37 |
1.305 |
1.687 |
2.026 |
2.431 |
2.715 |
3.326 |
38 |
1.304 |
1.686 |
2.024 |
2.429 |
2.712 |
3.319 |
39 |
1.304 |
1.685 |
2.023 |
2.426 |
2.708 |
3.313 |
40 |
1.303 |
1.684 |
2.021 |
2.423 |
2.704 |
3.307 |
41 |
1.303 |
1.683 |
2.020 |
2.421 |
2.701 |
3.301 |
42 |
1.302 |
1.682 |
2.018 |
2.418 |
2.698 |
3.296 |
43 |
1.302 |
1.681 |
2.017 |
2.416 |
2.695 |
3.291 |
44 |
1.301 |
1.680 |
2.015 |
2.414 |
2.692 |
3.286 |
45 |
1.301 |
1.679 |
2.014 |
2.412 |
2.690 |
3.281 |
46 |
1.300 |
1.679 |
2.013 |
2.410 |
2.687 |
3.277 |
47 |
1.300 |
1.678 |
2.012 |
2.408 |
2.685 |
3.273 |
48 |
1.299 |
1.677 |
2.011 |
2.407 |
2.682 |
3.269 |
49 |
1.299 |
1.677 |
2.010 |
2.405 |
2.680 |
3.265 |
50 |
1.299 |
1.676 |
2.009 |
2.403 |
2.678 |
3.261 |
51 |
1.298 |
1.675 |
2.008 |
2.402 |
2.676 |
3.258 |
52 |
1.298 |
1.675 |
2.007 |
2.400 |
2.674 |
3.255 |
53 |
1.298 |
1.674 |
2.006 |
2.399 |
2.672 |
3.251 |
54 |
1.297 |
1.674 |
2.005 |
2.397 |
2.670 |
3.248 |
55 |
1.297 |
1.673 |
2.004 |
2.396 |
2.668 |
3.245 |
56 |
1.297 |
1.673 |
2.003 |
2.395 |
2.667 |
3.242 |
57 |
1.297 |
1.672 |
2.002 |
2.394 |
2.665 |
3.239 |
58 |
1.296 |
1.672 |
2.002 |
2.392 |
2.663 |
3.237 |
59 |
1.296 |
1.671 |
2.001 |
2.391 |
2.662 |
3.234 |
60 |
1.296 |
1.671 |
2.000 |
2.390 |
2.660 |
3.232 |
61 |
1.296 |
1.670 |
2.000 |
2.389 |
2.659 |
3.229 |
62 |
1.295 |
1.670 |
1.999 |
2.388 |
2.657 |
3.227 |
63 |
1.295 |
1.669 |
1.998 |
2.387 |
2.656 |
3.225 |
64 |
1.295 |
1.669 |
1.998 |
2.386 |
2.655 |
3.223 |
65 |
1.295 |
1.669 |
1.997 |
2.385 |
2.654 |
3.220 |
66 |
1.295 |
1.668 |
1.997 |
2.384 |
2.652 |
3.218 |
67 |
1.294 |
1.668 |
1.996 |
2.383 |
2.651 |
3.216 |
68 |
1.294 |
1.668 |
1.995 |
2.382 |
2.650 |
3.214 |
69 |
1.294 |
1.667 |
1.995 |
2.382 |
2.649 |
3.213 |
70 |
1.294 |
1.667 |
1.994 |
2.381 |
2.648 |
3.211 |
71 |
1.294 |
1.667 |
1.994 |
2.380 |
2.647 |
3.209 |
72 |
1.293 |
1.666 |
1.993 |
2.379 |
2.646 |
3.207 |
73 |
1.293 |
1.666 |
1.993 |
2.379 |
2.645 |
3.206 |
74 |
1.293 |
1.666 |
1.993 |
2.378 |
2.644 |
3.204 |
75 |
1.293 |
1.665 |
1.992 |
2.377 |
2.643 |
3.202 |
76 |
1.293 |
1.665 |
1.992 |
2.376 |
2.642 |
3.201 |
77 |
1.293 |
1.665 |
1.991 |
2.376 |
2.641 |
3.199 |
78 |
1.292 |
1.665 |
1.991 |
2.375 |
2.640 |
3.198 |
79 |
1.292 |
1.664 |
1.990 |
2.374 |
2.640 |
3.197 |
80 |
1.292 |
1.664 |
1.990 |
2.374 |
2.639 |
3.195 |
81 |
1.292 |
1.664 |
1.990 |
2.373 |
2.638 |
3.194 |
82 |
1.292 |
1.664 |
1.989 |
2.373 |
2.637 |
3.193 |
83 |
1.292 |
1.663 |
1.989 |
2.372 |
2.636 |
3.191 |
84 |
1.292 |
1.663 |
1.989 |
2.372 |
2.636 |
3.190 |
85 |
1.292 |
1.663 |
1.988 |
2.371 |
2.635 |
3.189 |
86 |
1.291 |
1.663 |
1.988 |
2.370 |
2.634 |
3.188 |
87 |
1.291 |
1.663 |
1.988 |
2.370 |
2.634 |
3.187 |
88 |
1.291 |
1.662 |
1.987 |
2.369 |
2.633 |
3.185 |
89 |
1.291 |
1.662 |
1.987 |
2.369 |
2.632 |
3.184 |
90 |
1.291 |
1.662 |
1.987 |
2.368 |
2.632 |
3.183 |
91 |
1.291 |
1.662 |
1.986 |
2.368 |
2.631 |
3.182 |
92 |
1.291 |
1.662 |
1.986 |
2.368 |
2.630 |
3.181 |
93 |
1.291 |
1.661 |
1.986 |
2.367 |
2.630 |
3.180 |
94 |
1.291 |
1.661 |
1.986 |
2.367 |
2.629 |
3.179 |
95 |
1.291 |
1.661 |
1.985 |
2.366 |
2.629 |
3.178 |
96 |
1.290 |
1.661 |
1.985 |
2.366 |
2.628 |
3.177 |
97 |
1.290 |
1.661 |
1.985 |
2.365 |
2.627 |
3.176 |
98 |
1.290 |
1.661 |
1.984 |
2.365 |
2.627 |
3.175 |
99 |
1.290 |
1.660 |
1.984 |
2.365 |
2.626 |
3.175 |
100 |
1.290 |
1.660 |
1.984 |
2.364 |
2.626 |
3.174 |
infinity |
1.282 |
1.645 |
1.960 |
2.326 |
2.576 |
3.090 |
The following tables provide the upper and lower percentage points of the chi-square distribution with df degrees of freedom. Both are required as the distribution is asymmetric.
df |
0.100 |
0.050 |
0.025 |
0.010 |
0.001 |
---|---|---|---|---|---|
1 |
2.706 |
3.841 |
5.024 |
6.635 |
10.828 |
2 |
4.605 |
5.991 |
7.378 |
9.210 |
13.816 |
3 |
6.251 |
7.815 |
9.348 |
11.345 |
16.266 |
4 |
7.779 |
9.488 |
11.143 |
13.277 |
18.467 |
5 |
9.236 |
11.070 |
12.833 |
15.086 |
20.515 |
6 |
10.645 |
12.592 |
14.449 |
16.812 |
22.458 |
7 |
12.017 |
14.067 |
16.013 |
18.475 |
24.322 |
8 |
13.362 |
15.507 |
17.535 |
20.090 |
26.125 |
9 |
14.684 |
16.919 |
19.023 |
21.666 |
27.877 |
10 |
15.987 |
18.307 |
20.483 |
23.209 |
29.588 |
11 |
17.275 |
19.675 |
21.920 |
24.725 |
31.264 |
12 |
18.549 |
21.026 |
23.337 |
26.217 |
32.910 |
13 |
19.812 |
22.362 |
24.736 |
27.688 |
34.528 |
14 |
21.064 |
23.685 |
26.119 |
29.141 |
36.123 |
15 |
22.307 |
24.996 |
27.488 |
30.578 |
37.697 |
16 |
23.542 |
26.296 |
28.845 |
32.000 |
39.252 |
17 |
24.769 |
27.587 |
30.191 |
33.409 |
40.790 |
18 |
25.989 |
28.869 |
31.526 |
34.805 |
42.312 |
19 |
27.204 |
30.144 |
32.852 |
36.191 |
43.820 |
20 |
28.412 |
31.410 |
34.170 |
37.566 |
45.315 |
21 |
29.615 |
32.671 |
35.479 |
38.932 |
46.797 |
22 |
30.813 |
33.924 |
36.781 |
40.289 |
48.268 |
23 |
32.007 |
35.172 |
38.076 |
41.638 |
49.728 |
24 |
33.196 |
36.415 |
39.364 |
42.980 |
51.179 |
25 |
34.382 |
37.652 |
40.646 |
44.314 |
52.620 |
26 |
35.563 |
38.885 |
41.923 |
45.642 |
54.052 |
27 |
36.741 |
40.113 |
43.195 |
46.963 |
55.476 |
28 |
37.916 |
41.337 |
44.461 |
48.278 |
56.892 |
29 |
39.087 |
42.557 |
45.722 |
49.588 |
58.301 |
30 |
40.256 |
43.773 |
46.979 |
50.892 |
59.703 |
31 |
41.422 |
44.985 |
48.232 |
52.191 |
61.098 |
32 |
42.585 |
46.194 |
49.480 |
53.486 |
62.487 |
33 |
43.745 |
47.400 |
50.725 |
54.776 |
63.870 |
34 |
44.903 |
48.602 |
51.966 |
56.061 |
65.247 |
35 |
46.059 |
49.802 |
53.203 |
57.342 |
66.619 |
36 |
47.212 |
50.998 |
54.437 |
58.619 |
67.985 |
37 |
48.363 |
52.192 |
55.668 |
59.893 |
69.347 |
38 |
49.513 |
53.384 |
56.896 |
61.162 |
70.703 |
39 |
50.660 |
54.572 |
58.120 |
62.428 |
72.055 |
40 |
51.805 |
55.758 |
59.342 |
63.691 |
73.402 |
41 |
52.949 |
56.942 |
60.561 |
64.950 |
74.745 |
42 |
54.090 |
58.124 |
61.777 |
66.206 |
76.084 |
43 |
55.230 |
59.304 |
62.990 |
67.459 |
77.419 |
44 |
56.369 |
60.481 |
64.201 |
68.710 |
78.750 |
45 |
57.505 |
61.656 |
65.410 |
69.957 |
80.077 |
46 |
58.641 |
62.830 |
66.617 |
71.201 |
81.400 |
47 |
59.774 |
64.001 |
67.821 |
72.443 |
82.720 |
48 |
60.907 |
65.171 |
69.023 |
73.683 |
84.037 |
49 |
62.038 |
66.339 |
70.222 |
74.919 |
85.351 |
50 |
63.167 |
67.505 |
71.420 |
76.154 |
86.661 |
51 |
64.295 |
68.669 |
72.616 |
77.386 |
87.968 |
52 |
65.422 |
69.832 |
73.810 |
78.616 |
89.272 |
53 |
66.548 |
70.993 |
75.002 |
79.843 |
90.573 |
54 |
67.673 |
72.153 |
76.192 |
81.069 |
91.872 |
55 |
68.796 |
73.311 |
77.380 |
82.292 |
93.168 |
56 |
69.919 |
74.468 |
78.567 |
83.513 |
94.461 |
57 |
71.040 |
75.624 |
79.752 |
84.733 |
95.751 |
58 |
72.160 |
76.778 |
80.936 |
85.950 |
97.039 |
59 |
73.279 |
77.931 |
82.117 |
87.166 |
98.324 |
60 |
74.397 |
79.082 |
83.298 |
88.379 |
99.607 |
61 |
75.514 |
80.232 |
84.476 |
89.591 |
100.888 |
62 |
76.630 |
81.381 |
85.654 |
90.802 |
102.166 |
63 |
77.745 |
82.529 |
86.830 |
92.010 |
103.442 |
64 |
78.860 |
83.675 |
88.004 |
93.217 |
104.716 |
65 |
79.973 |
84.821 |
89.177 |
94.422 |
105.988 |
66 |
81.085 |
85.965 |
90.349 |
95.626 |
107.258 |
67 |
82.197 |
87.108 |
91.519 |
96.828 |
108.526 |
68 |
83.308 |
88.250 |
92.689 |
98.028 |
109.791 |
69 |
84.418 |
89.391 |
93.856 |
99.228 |
111.055 |
70 |
85.527 |
90.531 |
95.023 |
100.425 |
112.317 |
71 |
86.635 |
91.670 |
96.189 |
101.621 |
113.577 |
72 |
87.743 |
92.808 |
97.353 |
102.816 |
114.835 |
73 |
88.850 |
93.945 |
98.516 |
104.010 |
116.092 |
74 |
89.956 |
95.081 |
99.678 |
105.202 |
117.346 |
75 |
91.061 |
96.217 |
100.839 |
106.393 |
118.599 |
76 |
92.166 |
97.351 |
101.999 |
107.583 |
119.850 |
77 |
93.270 |
98.484 |
103.158 |
108.771 |
121.100 |
78 |
94.374 |
99.617 |
104.316 |
109.958 |
122.348 |
79 |
95.476 |
100.749 |
105.473 |
111.144 |
123.594 |
80 |
96.578 |
101.879 |
106.629 |
112.329 |
124.839 |
81 |
97.680 |
103.010 |
107.783 |
113.512 |
126.083 |
82 |
98.780 |
104.139 |
108.937 |
114.695 |
127.324 |
83 |
99.880 |
105.267 |
110.090 |
115.876 |
128.565 |
84 |
100.980 |
106.395 |
111.242 |
117.057 |
129.804 |
85 |
102.079 |
107.522 |
112.393 |
118.236 |
131.041 |
86 |
103.177 |
108.648 |
113.544 |
119.414 |
132.277 |
87 |
104.275 |
109.773 |
114.693 |
120.591 |
133.512 |
88 |
105.372 |
110.898 |
115.841 |
121.767 |
134.746 |
89 |
106.469 |
112.022 |
116.989 |
122.942 |
135.978 |
90 |
107.565 |
113.145 |
118.136 |
124.116 |
137.208 |
91 |
108.661 |
114.268 |
119.282 |
125.289 |
138.438 |
92 |
109.756 |
115.390 |
120.427 |
126.462 |
139.666 |
93 |
110.850 |
116.511 |
121.571 |
127.633 |
140.893 |
94 |
111.944 |
117.632 |
122.715 |
128.803 |
142.119 |
95 |
113.038 |
118.752 |
123.858 |
129.973 |
143.344 |
96 |
114.131 |
119.871 |
125.000 |
131.141 |
144.567 |
97 |
115.223 |
120.990 |
126.141 |
132.309 |
145.789 |
98 |
116.315 |
122.108 |
127.282 |
133.476 |
147.010 |
99 |
117.407 |
123.225 |
128.422 |
134.642 |
148.230 |
100 |
118.498 |
124.342 |
129.561 |
135.807 |
149.449 |
df |
0.9 |
0.95 |
0.975 |
0.99 |
0.999 |
---|---|---|---|---|---|
1 |
0.016 |
0.004 |
0.001 |
— |
— |
2 |
0.211 |
0.103 |
0.051 |
0.020 |
0.002 |
3 |
0.584 |
0.352 |
0.216 |
0.115 |
0.024 |
4 |
1.064 |
0.711 |
0.484 |
0.297 |
0.091 |
5 |
1.610 |
1.145 |
0.831 |
0.554 |
0.210 |
6 |
2.204 |
1.635 |
1.237 |
0.872 |
0.381 |
7 |
2.833 |
2.167 |
1.690 |
1.239 |
0.598 |
8 |
3.490 |
2.733 |
2.180 |
1.646 |
0.857 |
9 |
4.168 |
3.325 |
2.700 |
2.088 |
1.152 |
10 |
4.865 |
3.940 |
3.247 |
2.558 |
1.479 |
11 |
5.578 |
4.575 |
3.816 |
3.053 |
1.834 |
12 |
6.304 |
5.226 |
4.404 |
3.571 |
2.214 |
13 |
7.042 |
5.892 |
5.009 |
4.107 |
2.617 |
14 |
7.790 |
6.571 |
5.629 |
4.660 |
3.041 |
15 |
8.547 |
7.261 |
6.262 |
5.229 |
3.483 |
16 |
9.312 |
7.962 |
6.908 |
5.812 |
3.942 |
17 |
10.085 |
8.672 |
7.564 |
6.408 |
4.416 |
18 |
10.865 |
9.390 |
8.231 |
7.015 |
4.905 |
19 |
11.651 |
10.117 |
8.907 |
7.633 |
5.407 |
20 |
12.443 |
10.851 |
9.591 |
8.260 |
5.921 |
21 |
13.240 |
11.591 |
10.283 |
8.897 |
6.447 |
22 |
14.041 |
12.338 |
10.982 |
9.542 |
6.983 |
23 |
14.848 |
13.091 |
11.689 |
10.196 |
7.529 |
24 |
15.659 |
13.848 |
12.401 |
10.856 |
8.085 |
25 |
16.473 |
14.611 |
13.120 |
11.524 |
8.649 |
26 |
17.292 |
15.379 |
13.844 |
12.198 |
9.222 |
27 |
18.114 |
16.151 |
14.573 |
12.879 |
9.803 |
28 |
18.939 |
16.928 |
15.308 |
13.565 |
10.391 |
29 |
19.768 |
17.708 |
16.047 |
14.256 |
10.986 |
30 |
20.599 |
18.493 |
16.791 |
14.953 |
11.588 |
31 |
21.434 |
19.281 |
17.539 |
15.655 |
12.196 |
32 |
22.271 |
20.072 |
18.291 |
16.362 |
12.811 |
33 |
23.110 |
20.867 |
19.047 |
17.074 |
13.431 |
34 |
23.952 |
21.664 |
19.806 |
17.789 |
14.057 |
35 |
24.797 |
22.465 |
20.569 |
18.509 |
14.688 |
36 |
25.643 |
23.269 |
21.336 |
19.233 |
15.324 |
37 |
26.492 |
24.075 |
22.106 |
19.960 |
15.965 |
38 |
27.343 |
24.884 |
22.878 |
20.691 |
16.611 |
39 |
28.196 |
25.695 |
23.654 |
21.426 |
17.262 |
40 |
29.051 |
26.509 |
24.433 |
22.164 |
17.916 |
41 |
29.907 |
27.326 |
25.215 |
22.906 |
18.575 |
42 |
30.765 |
28.144 |
25.999 |
23.650 |
19.239 |
43 |
31.625 |
28.965 |
26.785 |
24.398 |
19.906 |
44 |
32.487 |
29.787 |
27.575 |
25.148 |
20.576 |
45 |
33.350 |
30.612 |
28.366 |
25.901 |
21.251 |
46 |
34.215 |
31.439 |
29.160 |
26.657 |
21.929 |
47 |
35.081 |
32.268 |
29.956 |
27.416 |
22.610 |
48 |
35.949 |
33.098 |
30.755 |
28.177 |
23.295 |
49 |
36.818 |
33.930 |
31.555 |
28.941 |
23.983 |
50 |
37.689 |
34.764 |
32.357 |
29.707 |
24.674 |
51 |
38.560 |
35.600 |
33.162 |
30.475 |
25.368 |
52 |
39.433 |
36.437 |
33.968 |
31.246 |
26.065 |
53 |
40.308 |
37.276 |
34.776 |
32.018 |
26.765 |
54 |
41.183 |
38.116 |
35.586 |
32.793 |
27.468 |
55 |
42.060 |
38.958 |
36.398 |
33.570 |
28.173 |
56 |
42.937 |
39.801 |
37.212 |
34.350 |
28.881 |
57 |
43.816 |
40.646 |
38.027 |
35.131 |
29.592 |
58 |
44.696 |
41.492 |
38.844 |
35.913 |
30.305 |
59 |
45.577 |
42.339 |
39.662 |
36.698 |
31.020 |
60 |
46.459 |
43.188 |
40.482 |
37.485 |
31.738 |
61 |
47.342 |
44.038 |
41.303 |
38.273 |
32.459 |
62 |
48.226 |
44.889 |
42.126 |
39.063 |
33.181 |
63 |
49.111 |
45.741 |
42.950 |
39.855 |
33.906 |
64 |
49.996 |
46.595 |
43.776 |
40.649 |
34.633 |
65 |
50.883 |
47.450 |
44.603 |
41.444 |
35.362 |
66 |
51.770 |
48.305 |
45.431 |
42.240 |
36.093 |
67 |
52.659 |
49.162 |
46.261 |
43.038 |
36.826 |
68 |
53.548 |
50.020 |
47.092 |
43.838 |
37.561 |
69 |
54.438 |
50.879 |
47.924 |
44.639 |
38.298 |
70 |
55.329 |
51.739 |
48.758 |
45.442 |
39.036 |
71 |
56.221 |
52.600 |
49.592 |
46.246 |
39.777 |
72 |
57.113 |
53.462 |
50.428 |
47.051 |
40.519 |
73 |
58.006 |
54.325 |
51.265 |
47.858 |
41.264 |
74 |
58.900 |
55.189 |
52.103 |
48.666 |
42.010 |
75 |
59.795 |
56.054 |
52.942 |
49.475 |
42.757 |
76 |
60.690 |
56.920 |
53.782 |
50.286 |
43.507 |
77 |
61.586 |
57.786 |
54.623 |
51.097 |
44.258 |
78 |
62.483 |
58.654 |
55.466 |
51.910 |
45.010 |
79 |
63.380 |
59.522 |
56.309 |
52.725 |
45.764 |
80 |
64.278 |
60.391 |
57.153 |
53.540 |
46.520 |
81 |
65.176 |
61.261 |
57.998 |
54.357 |
47.277 |
82 |
66.076 |
62.132 |
58.845 |
55.174 |
48.036 |
83 |
66.976 |
63.004 |
59.692 |
55.993 |
48.796 |
84 |
67.876 |
63.876 |
60.540 |
56.813 |
49.557 |
85 |
68.777 |
64.749 |
61.389 |
57.634 |
50.320 |
86 |
69.679 |
65.623 |
62.239 |
58.456 |
51.085 |
87 |
70.581 |
66.498 |
63.089 |
59.279 |
51.850 |
88 |
71.484 |
67.373 |
63.941 |
60.103 |
52.617 |
89 |
72.387 |
68.249 |
64.793 |
60.928 |
53.386 |
90 |
73.291 |
69.126 |
65.647 |
61.754 |
54.155 |
91 |
74.196 |
70.003 |
66.501 |
62.581 |
54.926 |
92 |
75.100 |
70.882 |
67.356 |
63.409 |
55.698 |
93 |
76.006 |
71.760 |
68.211 |
64.238 |
56.472 |
94 |
76.912 |
72.640 |
69.068 |
65.068 |
57.246 |
95 |
77.818 |
73.520 |
69.925 |
65.898 |
58.022 |
96 |
78.725 |
74.401 |
70.783 |
66.730 |
58.799 |
97 |
79.633 |
75.282 |
71.642 |
67.562 |
59.577 |
98 |
80.541 |
76.164 |
72.501 |
68.396 |
60.356 |
99 |
81.449 |
77.046 |
73.361 |
69.230 |
61.137 |
100 |
82.358 |
77.929 |
74.222 |
70.065 |
61.918 |
Upper critical values of the F distribution, 5% level. In the following table df1 (the row entries) provides the degrees of freedom for the numerator and df2 (the column entries) provides the degrees of freedom for the denominator in the F ratio.
df1, df2 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
---|---|---|---|---|---|---|---|---|---|---|
1 |
161.448 |
199.500 |
215.707 |
224.583 |
230.162 |
233.986 |
236.768 |
238.883 |
240.543 |
241.882 |
2 |
18.513 |
19.000 |
19.164 |
19.247 |
19.296 |
19.330 |
19.353 |
19.371 |
19.385 |
19.396 |
3 |
10.128 |
9.552 |
9.277 |
9.117 |
9.013 |
8.941 |
8.887 |
8.845 |
8.812 |
8.786 |
4 |
7.709 |
6.944 |
6.591 |
6.388 |
6.256 |
6.163 |
6.094 |
6.041 |
5.999 |
5.964 |
5 |
6.608 |
5.786 |
5.409 |
5.192 |
5.050 |
4.950 |
4.876 |
4.818 |
4.772 |
4.735 |
6 |
5.987 |
5.143 |
4.757 |
4.534 |
4.387 |
4.284 |
4.207 |
4.147 |
4.099 |
4.060 |
7 |
5.591 |
4.737 |
4.347 |
4.120 |
3.972 |
3.866 |
3.787 |
3.726 |
3.677 |
3.637 |
8 |
5.318 |
4.459 |
4.066 |
3.838 |
3.687 |
3.581 |
3.500 |
3.438 |
3.388 |
3.347 |
9 |
5.117 |
4.256 |
3.863 |
3.633 |
3.482 |
3.374 |
3.293 |
3.230 |
3.179 |
3.137 |
10 |
4.965 |
4.103 |
3.708 |
3.478 |
3.326 |
3.217 |
3.135 |
3.072 |
3.020 |
2.978 |
11 |
4.844 |
3.982 |
3.587 |
3.357 |
3.204 |
3.095 |
3.012 |
2.948 |
2.896 |
2.854 |
12 |
4.747 |
3.885 |
3.490 |
3.259 |
3.106 |
2.996 |
2.913 |
2.849 |
2.796 |
2.753 |
13 |
4.667 |
3.806 |
3.411 |
3.179 |
3.025 |
2.915 |
2.832 |
2.767 |
2.714 |
2.671 |
14 |
4.600 |
3.739 |
3.344 |
3.112 |
2.958 |
2.848 |
2.764 |
2.699 |
2.646 |
2.602 |
15 |
4.543 |
3.682 |
3.287 |
3.056 |
2.901 |
2.790 |
2.707 |
2.641 |
2.588 |
2.544 |
16 |
4.494 |
3.634 |
3.239 |
3.007 |
2.852 |
2.741 |
2.657 |
2.591 |
2.538 |
2.494 |
17 |
4.451 |
3.592 |
3.197 |
2.965 |
2.810 |
2.699 |
2.614 |
2.548 |
2.494 |
2.450 |
18 |
4.414 |
3.555 |
3.160 |
2.928 |
2.773 |
2.661 |
2.577 |
2.510 |
2.456 |
2.412 |
19 |
4.381 |
3.522 |
3.127 |
2.895 |
2.740 |
2.628 |
2.544 |
2.477 |
2.423 |
2.378 |
20 |
4.351 |
3.493 |
3.098 |
2.866 |
2.711 |
2.599 |
2.514 |
2.447 |
2.393 |
2.348 |
21 |
4.325 |
3.467 |
3.072 |
2.840 |
2.685 |
2.573 |
2.488 |
2.420 |
2.366 |
2.321 |
22 |
4.301 |
3.443 |
3.049 |
2.817 |
2.661 |
2.549 |
2.464 |
2.397 |
2.342 |
2.297 |
23 |
4.279 |
3.422 |
3.028 |
2.796 |
2.640 |
2.528 |
2.442 |
2.375 |
2.320 |
2.275 |
24 |
4.260 |
3.403 |
3.009 |
2.776 |
2.621 |
2.508 |
2.423 |
2.355 |
2.300 |
2.255 |
25 |
4.242 |
3.385 |
2.991 |
2.759 |
2.603 |
2.490 |
2.405 |
2.337 |
2.282 |
2.236 |
26 |
4.225 |
3.369 |
2.975 |
2.743 |
2.587 |
2.474 |
2.388 |
2.321 |
2.265 |
2.220 |
27 |
4.210 |
3.354 |
2.960 |
2.728 |
2.572 |
2.459 |
2.373 |
2.305 |
2.250 |
2.204 |
28 |
4.196 |
3.340 |
2.947 |
2.714 |
2.558 |
2.445 |
2.359 |
2.291 |
2.236 |
2.190 |
29 |
4.183 |
3.328 |
2.934 |
2.701 |
2.545 |
2.432 |
2.346 |
2.278 |
2.223 |
2.177 |
30 |
4.171 |
3.316 |
2.922 |
2.690 |
2.534 |
2.421 |
2.334 |
2.266 |
2.211 |
2.165 |
31 |
4.160 |
3.305 |
2.911 |
2.679 |
2.523 |
2.409 |
2.323 |
2.255 |
2.199 |
2.153 |
32 |
4.149 |
3.295 |
2.901 |
2.668 |
2.512 |
2.399 |
2.313 |
2.244 |
2.189 |
2.142 |
33 |
4.139 |
3.285 |
2.892 |
2.659 |
2.503 |
2.389 |
2.303 |
2.235 |
2.179 |
2.133 |
34 |
4.130 |
3.276 |
2.883 |
2.650 |
2.494 |
2.380 |
2.294 |
2.225 |
2.170 |
2.123 |
35 |
4.121 |
3.267 |
2.874 |
2.641 |
2.485 |
2.372 |
2.285 |
2.217 |
2.161 |
2.114 |
36 |
4.113 |
3.259 |
2.866 |
2.634 |
2.477 |
2.364 |
2.277 |
2.209 |
2.153 |
2.106 |
37 |
4.105 |
3.252 |
2.859 |
2.626 |
2.470 |
2.356 |
2.270 |
2.201 |
2.145 |
2.098 |
38 |
4.098 |
3.245 |
2.852 |
2.619 |
2.463 |
2.349 |
2.262 |
2.194 |
2.138 |
2.091 |
39 |
4.091 |
3.238 |
2.845 |
2.612 |
2.456 |
2.342 |
2.255 |
2.187 |
2.131 |
2.084 |
40 |
4.085 |
3.232 |
2.839 |
2.606 |
2.449 |
2.336 |
2.249 |
2.180 |
2.124 |
2.077 |
41 |
4.079 |
3.226 |
2.833 |
2.600 |
2.443 |
2.330 |
2.243 |
2.174 |
2.118 |
2.071 |
42 |
4.073 |
3.220 |
2.827 |
2.594 |
2.438 |
2.324 |
2.237 |
2.168 |
2.112 |
2.065 |
43 |
4.067 |
3.214 |
2.822 |
2.589 |
2.432 |
2.318 |
2.232 |
2.163 |
2.106 |
2.059 |
44 |
4.062 |
3.209 |
2.816 |
2.584 |
2.427 |
2.313 |
2.226 |
2.157 |
2.101 |
2.054 |
45 |
4.057 |
3.204 |
2.812 |
2.579 |
2.422 |
2.308 |
2.221 |
2.152 |
2.096 |
2.049 |
46 |
4.052 |
3.200 |
2.807 |
2.574 |
2.417 |
2.304 |
2.216 |
2.147 |
2.091 |
2.044 |
47 |
4.047 |
3.195 |
2.802 |
2.570 |
2.413 |
2.299 |
2.212 |
2.143 |
2.086 |
2.039 |
48 |
4.043 |
3.191 |
2.798 |
2.565 |
2.409 |
2.295 |
2.207 |
2.138 |
2.082 |
2.035 |
49 |
4.038 |
3.187 |
2.794 |
2.561 |
2.404 |
2.290 |
2.203 |
2.134 |
2.077 |
2.030 |
50 |
4.034 |
3.183 |
2.790 |
2.557 |
2.400 |
2.286 |
2.199 |
2.130 |
2.073 |
2.026 |
df1, df2 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
---|---|---|---|---|---|---|---|---|---|---|
1 |
242.983 |
243.906 |
244.690 |
245.364 |
245.950 |
246.464 |
246.918 |
247.323 |
247.686 |
248.013 |
2 |
19.405 |
19.413 |
19.419 |
19.424 |
19.429 |
19.433 |
19.437 |
19.440 |
19.443 |
19.446 |
3 |
8.763 |
8.745 |
8.729 |
8.715 |
8.703 |
8.692 |
8.683 |
8.675 |
8.667 |
8.660 |
4 |
5.936 |
5.912 |
5.891 |
5.873 |
5.858 |
5.844 |
5.832 |
5.821 |
5.811 |
5.803 |
5 |
4.704 |
4.678 |
4.655 |
4.636 |
4.619 |
4.604 |
4.590 |
4.579 |
4.568 |
4.558 |
6 |
4.027 |
4.000 |
3.976 |
3.956 |
3.938 |
3.922 |
3.908 |
3.896 |
3.884 |
3.874 |
7 |
3.603 |
3.575 |
3.550 |
3.529 |
3.511 |
3.494 |
3.480 |
3.467 |
3.455 |
3.445 |
8 |
3.313 |
3.284 |
3.259 |
3.237 |
3.218 |
3.202 |
3.187 |
3.173 |
3.161 |
3.150 |
9 |
3.102 |
3.073 |
3.048 |
3.025 |
3.006 |
2.989 |
2.974 |
2.960 |
2.948 |
2.936 |
10 |
2.943 |
2.913 |
2.887 |
2.865 |
2.845 |
2.828 |
2.812 |
2.798 |
2.785 |
2.774 |
11 |
2.818 |
2.788 |
2.761 |
2.739 |
2.719 |
2.701 |
2.685 |
2.671 |
2.658 |
2.646 |
12 |
2.717 |
2.687 |
2.660 |
2.637 |
2.617 |
2.599 |
2.583 |
2.568 |
2.555 |
2.544 |
13 |
2.635 |
2.604 |
2.577 |
2.554 |
2.533 |
2.515 |
2.499 |
2.484 |
2.471 |
2.459 |
14 |
2.565 |
2.534 |
2.507 |
2.484 |
2.463 |
2.445 |
2.428 |
2.413 |
2.400 |
2.388 |
15 |
2.507 |
2.475 |
2.448 |
2.424 |
2.403 |
2.385 |
2.368 |
2.353 |
2.340 |
2.328 |
16 |
2.456 |
2.425 |
2.397 |
2.373 |
2.352 |
2.333 |
2.317 |
2.302 |
2.288 |
2.276 |
17 |
2.413 |
2.381 |
2.353 |
2.329 |
2.308 |
2.289 |
2.272 |
2.257 |
2.243 |
2.230 |
18 |
2.374 |
2.342 |
2.314 |
2.290 |
2.269 |
2.250 |
2.233 |
2.217 |
2.203 |
2.191 |
19 |
2.340 |
2.308 |
2.280 |
2.256 |
2.234 |
2.215 |
2.198 |
2.182 |
2.168 |
2.155 |
20 |
2.310 |
2.278 |
2.250 |
2.225 |
2.203 |
2.184 |
2.167 |
2.151 |
2.137 |
2.124 |
21 |
2.283 |
2.250 |
2.222 |
2.197 |
2.176 |
2.156 |
2.139 |
2.123 |
2.109 |
2.096 |
22 |
2.259 |
2.226 |
2.198 |
2.173 |
2.151 |
2.131 |
2.114 |
2.098 |
2.084 |
2.071 |
23 |
2.236 |
2.204 |
2.175 |
2.150 |
2.128 |
2.109 |
2.091 |
2.075 |
2.061 |
2.048 |
24 |
2.216 |
2.183 |
2.155 |
2.130 |
2.108 |
2.088 |
2.070 |
2.054 |
2.040 |
2.027 |
25 |
2.198 |
2.165 |
2.136 |
2.111 |
2.089 |
2.069 |
2.051 |
2.035 |
2.021 |
2.007 |
26 |
2.181 |
2.148 |
2.119 |
2.094 |
2.072 |
2.052 |
2.034 |
2.018 |
2.003 |
1.990 |
27 |
2.166 |
2.132 |
2.103 |
2.078 |
2.056 |
2.036 |
2.018 |
2.002 |
1.987 |
1.974 |
28 |
2.151 |
2.118 |
2.089 |
2.064 |
2.041 |
2.021 |
2.003 |
1.987 |
1.972 |
1.959 |
29 |
2.138 |
2.104 |
2.075 |
2.050 |
2.027 |
2.007 |
1.989 |
1.973 |
1.958 |
1.945 |