2.340 2.329 2.320 2.312 2.305 2.298
10 2.302 2.284 2.269 2.255
2.244 2.233 2.224 2.215 2.208 2.201
11 2.227 2.209 2.193 2.179
2.167 2.156 2.147 2.138 2.130 2.123
12 2.166 2.147 2.131 2.117
2.105 2.094 2.084 2.075 2.067 2.060
13 2.116 2.097 2.080 2.066
2.053 2.042 2.032 2.023 2.014 2.007
14 2.073 2.054 2.037 2.022
2.010 1.998 1.988 1.978 1.970 1.962
15 2.037 2.017 2.000 1.985
1.972 1.961 1.950 1.941 1.932 1.924
16 2.005 1.985 1.968 1.953
1.940 1.928 1.917 1.908 1.899 1.891
17 1.978 1.958 1.940 1.925
1.912 1.900 1.889 1.879 1.870 1.862
18 1.954 1.933 1.916 1.900
1.887 1.875 1.864 1.854 1.845 1.837
19 1.932 1.912 1.894 1.878
1.865 1.852 1.841 1.831 1.822 1.814
20 1.913 1.892 1.875 1.859
1.845 1.833 1.821 1.811 1.802 1.794
21 1.896 1.875 1.857 1.841
1.827 1.815 1.803 1.793 1.784 1.776
22 1.880 1.859 1.841 1.825
1.811 1.798 1.787 1.777 1.768 1.759
23 1.866 1.845 1.827 1.811
1.796 1.784 1.772 1.762 1.753 1.744
24 1.853 1.832 1.814 1.797

1.783 1.770 1.759 1.748 1.739 1.730
25 1.841 1.820 1.802 1.785
1.771 1.758 1.746 1.736 1.726 1.718
26 1.830 1.809 1.790 1.774
1.760 1.747 1.735 1.724 1.715 1.706
27 1.820 1.799 1.780 1.764
1.3.6.7.3. Upper Critical Values of the F Distribution
(21 of 38) [5/1/2006 9:58:27 AM]
1.749 1.736 1.724 1.714 1.704 1.695
28 1.811 1.790 1.771 1.754
1.740 1.726 1.715 1.704 1.694 1.685
29 1.802 1.781 1.762 1.745
1.731 1.717 1.705 1.695 1.685 1.676
30 1.794 1.773 1.754 1.737
1.722 1.709 1.697 1.686 1.676 1.667
31 1.787 1.765 1.746 1.729
1.714 1.701 1.689 1.678 1.668 1.659
32 1.780 1.758 1.739 1.722
1.707 1.694 1.682 1.671 1.661 1.652
33 1.773 1.751 1.732 1.715
1.700 1.687 1.675 1.664 1.654 1.645
34 1.767 1.745 1.726 1.709
1.694 1.680 1.668 1.657 1.647 1.638
35 1.761 1.739 1.720 1.703
1.688 1.674 1.662 1.651 1.641 1.632
36 1.756 1.734 1.715 1.697
1.682 1.669 1.656 1.645 1.635 1.626
37 1.751 1.729 1.709 1.692
1.677 1.663 1.651 1.640 1.630 1.620
38 1.746 1.724 1.704 1.687

1.672 1.658 1.646 1.635 1.624 1.615
39 1.741 1.719 1.700 1.682
1.667 1.653 1.641 1.630 1.619 1.610
40 1.737 1.715 1.695 1.678
1.662 1.649 1.636 1.625 1.615 1.605
41 1.733 1.710 1.691 1.673
1.658 1.644 1.632 1.620 1.610 1.601
42 1.729 1.706 1.687 1.669
1.654 1.640 1.628 1.616 1.606 1.596
43 1.725 1.703 1.683 1.665
1.650 1.636 1.624 1.612 1.602 1.592
44 1.721 1.699 1.679 1.662
1.646 1.632 1.620 1.608 1.598 1.588
45 1.718 1.695 1.676 1.658
1.3.6.7.3. Upper Critical Values of the F Distribution
(22 of 38) [5/1/2006 9:58:27 AM]
1.643 1.629 1.616 1.605 1.594 1.585
46 1.715 1.692 1.672 1.655
1.639 1.625 1.613 1.601 1.591 1.581
47 1.712 1.689 1.669 1.652
1.636 1.622 1.609 1.598 1.587 1.578
48 1.709 1.686 1.666 1.648
1.633 1.619 1.606 1.594 1.584 1.574
49 1.706 1.683 1.663 1.645
1.630 1.616 1.603 1.591 1.581 1.571
50 1.703 1.680 1.660 1.643
1.627 1.613 1.600 1.588 1.578 1.568
51 1.700 1.677 1.658 1.640
1.624 1.610 1.597 1.586 1.575 1.565
52 1.698 1.675 1.655 1.637

1.621 1.607 1.594 1.583 1.572 1.562
53 1.695 1.672 1.652 1.635
1.619 1.605 1.592 1.580 1.570 1.560
54 1.693 1.670 1.650 1.632
1.616 1.602 1.589 1.578 1.567 1.557
55 1.691 1.668 1.648 1.630
1.614 1.600 1.587 1.575 1.564 1.555
56 1.688 1.666 1.645 1.628
1.612 1.597 1.585 1.573 1.562 1.552
57 1.686 1.663 1.643 1.625
1.610 1.595 1.582 1.571 1.560 1.550
58 1.684 1.661 1.641 1.623
1.607 1.593 1.580 1.568 1.558 1.548
59 1.682 1.659 1.639 1.621
1.605 1.591 1.578 1.566 1.555 1.546
60 1.680 1.657 1.637 1.619
1.603 1.589 1.576 1.564 1.553 1.543
61 1.679 1.656 1.635 1.617
1.601 1.587 1.574 1.562 1.551 1.541
62 1.677 1.654 1.634 1.616
1.600 1.585 1.572 1.560 1.549 1.540
63 1.675 1.652 1.632 1.614
1.3.6.7.3. Upper Critical Values of the F Distribution
(23 of 38) [5/1/2006 9:58:27 AM]
1.598 1.583 1.570 1.558 1.548 1.538
64 1.673 1.650 1.630 1.612
1.596 1.582 1.569 1.557 1.546 1.536
65 1.672 1.649 1.628 1.610
1.594 1.580 1.567 1.555 1.544 1.534
66 1.670 1.647 1.627 1.609

1.593 1.578 1.565 1.553 1.542 1.532
67 1.669 1.646 1.625 1.607
1.591 1.577 1.564 1.552 1.541 1.531
68 1.667 1.644 1.624 1.606
1.590 1.575 1.562 1.550 1.539 1.529
69 1.666 1.643 1.622 1.604
1.588 1.574 1.560 1.548 1.538 1.527
70 1.665 1.641 1.621 1.603
1.587 1.572 1.559 1.547 1.536 1.526
71 1.663 1.640 1.619 1.601
1.585 1.571 1.557 1.545 1.535 1.524
72 1.662 1.639 1.618 1.600
1.584 1.569 1.556 1.544 1.533 1.523
73 1.661 1.637 1.617 1.599
1.583 1.568 1.555 1.543 1.532 1.522
74 1.659 1.636 1.616 1.597
1.581 1.567 1.553 1.541 1.530 1.520
75 1.658 1.635 1.614 1.596
1.580 1.565 1.552 1.540 1.529 1.519
76 1.657 1.634 1.613 1.595
1.579 1.564 1.551 1.539 1.528 1.518
77 1.656 1.632 1.612 1.594
1.578 1.563 1.550 1.538 1.527 1.516
78 1.655 1.631 1.611 1.593
1.576 1.562 1.548 1.536 1.525 1.515
79 1.654 1.630 1.610 1.592
1.575 1.561 1.547 1.535 1.524 1.514
80 1.653 1.629 1.609 1.590
1.574 1.559 1.546 1.534 1.523 1.513
81 1.652 1.628 1.608 1.589

1.3.6.7.3. Upper Critical Values of the F Distribution
(24 of 38) [5/1/2006 9:58:27 AM]
1.573 1.558 1.545 1.533 1.522 1.512
82 1.651 1.627 1.607 1.588
1.572 1.557 1.544 1.532 1.521 1.511
83 1.650 1.626 1.606 1.587
1.571 1.556 1.543 1.531 1.520 1.509
84 1.649 1.625 1.605 1.586
1.570 1.555 1.542 1.530 1.519 1.508
85 1.648 1.624 1.604 1.585
1.569 1.554 1.541 1.529 1.518 1.507
86 1.647 1.623 1.603 1.584
1.568 1.553 1.540 1.528 1.517 1.506
87 1.646 1.622 1.602 1.583
1.567 1.552 1.539 1.527 1.516 1.505
88 1.645 1.622 1.601 1.583
1.566 1.551 1.538 1.526 1.515 1.504
89 1.644 1.621 1.600 1.582
1.565 1.550 1.537 1.525 1.514 1.503
90 1.643 1.620 1.599 1.581
1.564 1.550 1.536 1.524 1.513 1.503
91 1.643 1.619 1.598 1.580
1.564 1.549 1.535 1.523 1.512 1.502
92 1.642 1.618 1.598 1.579
1.563 1.548 1.534 1.522 1.511 1.501
93 1.641 1.617 1.597 1.578
1.562 1.547 1.534 1.521 1.510 1.500
94 1.640 1.617 1.596 1.578
1.561 1.546 1.533 1.521 1.509 1.499
95 1.640 1.616 1.595 1.577

1.560 1.545 1.532 1.520 1.509 1.498
96 1.639 1.615 1.594 1.576
1.560 1.545 1.531 1.519 1.508 1.497
97 1.638 1.614 1.594 1.575
1.559 1.544 1.530 1.518 1.507 1.497
98 1.637 1.614 1.593 1.575
1.558 1.543 1.530 1.517 1.506 1.496
99 1.637 1.613 1.592 1.574
1.3.6.7.3. Upper Critical Values of the F Distribution
(25 of 38) [5/1/2006 9:58:27 AM]
1.557 1.542 1.529 1.517 1.505 1.495
100 1.636 1.612 1.592 1.573
1.557 1.542 1.528 1.516 1.505 1.494
Upper critical values of the F distribution
for
numerator degrees of freedom and denominator
degrees of freedom
1% significance level
\ 1 2 3 4 5
6 7 8 9 10

1 4052.19 4999.52 5403.34 5624.62
5763.65 5858.97 5928.33 5981.10 6022.50
6055.85
2 98.502 99.000 99.166 99.249
99.300 99.333 99.356 99.374 99.388 99.399
3 34.116 30.816 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
1.3.6.7.3. Upper Critical Values of the F Distribution
(26 of 38) [5/1/2006 9:58:27 AM]
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
1.3.6.7.3. Upper Critical Values of the F Distribution
(27 of 38) [5/1/2006 9:58:27 AM]
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 3.305 3.173 3.067 2.979
31 7.530 5.362 4.484 3.993
3.675 3.449 3.281 3.149 3.043 2.955
32 7.499 5.336 4.459 3.969
3.652 3.427 3.258 3.127 3.021 2.934

33 7.471 5.312 4.437 3.948
3.630 3.406 3.238 3.106 3.000 2.913
34 7.444 5.289 4.416 3.927
3.611 3.386 3.218 3.087 2.981 2.894
35 7.419 5.268 4.396 3.908
3.592 3.368 3.200 3.069 2.963 2.876
36 7.396 5.248 4.377 3.890
3.574 3.351 3.183 3.052 2.946 2.859
37 7.373 5.229 4.360 3.873
3.558 3.334 3.167 3.036 2.930 2.843
38 7.353 5.211 4.343 3.858
3.542 3.319 3.152 3.021 2.915 2.828
39 7.333 5.194 4.327 3.843
3.528 3.305 3.137 3.006 2.901 2.814
40 7.314 5.179 4.313 3.828
3.514 3.291 3.124 2.993 2.888 2.801
41 7.296 5.163 4.299 3.815
3.501 3.278 3.111 2.980 2.875 2.788
42 7.280 5.149 4.285 3.802
1.3.6.7.3. Upper Critical Values of the F Distribution
(28 of 38) [5/1/2006 9:58:27 AM]
3.488 3.266 3.099 2.968 2.863 2.776
43 7.264 5.136 4.273 3.790
3.476 3.254 3.087 2.957 2.851 2.764
44 7.248 5.123 4.261 3.778
3.465 3.243 3.076 2.946 2.840 2.754
45 7.234 5.110 4.249 3.767
3.454 3.232 3.066 2.935 2.830 2.743
46 7.220 5.099 4.238 3.757
3.444 3.222 3.056 2.925 2.820 2.733

47 7.207 5.087 4.228 3.747
3.434 3.213 3.046 2.916 2.811 2.724
48 7.194 5.077 4.218 3.737
3.425 3.204 3.037 2.907 2.802 2.715
49 7.182 5.066 4.208 3.728
3.416 3.195 3.028 2.898 2.793 2.706
50 7.171 5.057 4.199 3.720
3.408 3.186 3.020 2.890 2.785 2.698
51 7.159 5.047 4.191 3.711
3.400 3.178 3.012 2.882 2.777 2.690
52 7.149 5.038 4.182 3.703
3.392 3.171 3.005 2.874 2.769 2.683
53 7.139 5.030 4.174 3.695
3.384 3.163 2.997 2.867 2.762 2.675
54 7.129 5.021 4.167 3.688
3.377 3.156 2.990 2.860 2.755 2.668
55 7.119 5.013 4.159 3.681
3.370 3.149 2.983 2.853 2.748 2.662
56 7.110 5.006 4.152 3.674
3.363 3.143 2.977 2.847 2.742 2.655
57 7.102 4.998 4.145 3.667
3.357 3.136 2.971 2.841 2.736 2.649
58 7.093 4.991 4.138 3.661
3.351 3.130 2.965 2.835 2.730 2.643
59 7.085 4.984 4.132 3.655
3.345 3.124 2.959 2.829 2.724 2.637
60 7.077 4.977 4.126 3.649
1.3.6.7.3. Upper Critical Values of the F Distribution
(29 of 38) [5/1/2006 9:58:27 AM]
3.339 3.119 2.953 2.823 2.718 2.632

61 7.070 4.971 4.120 3.643
3.333 3.113 2.948 2.818 2.713 2.626
62 7.062 4.965 4.114 3.638
3.328 3.108 2.942 2.813 2.708 2.621
63 7.055 4.959 4.109 3.632
3.323 3.103 2.937 2.808 2.703 2.616
64 7.048 4.953 4.103 3.627
3.318 3.098 2.932 2.803 2.698 2.611
65 7.042 4.947 4.098 3.622
3.313 3.093 2.928 2.798 2.693 2.607
66 7.035 4.942 4.093 3.618
3.308 3.088 2.923 2.793 2.689 2.602
67 7.029 4.937 4.088 3.613
3.304 3.084 2.919 2.789 2.684 2.598
68 7.023 4.932 4.083 3.608
3.299 3.080 2.914 2.785 2.680 2.593
69 7.017 4.927 4.079 3.604
3.295 3.075 2.910 2.781 2.676 2.589
70 7.011 4.922 4.074 3.600
3.291 3.071 2.906 2.777 2.672 2.585
71 7.006 4.917 4.070 3.596
3.287 3.067 2.902 2.773 2.668 2.581
72 7.001 4.913 4.066 3.591
3.283 3.063 2.898 2.769 2.664 2.578
73 6.995 4.908 4.062 3.588
3.279 3.060 2.895 2.765 2.660 2.574
74 6.990 4.904 4.058 3.584
3.275 3.056 2.891 2.762 2.657 2.570
75 6.985 4.900 4.054 3.580
3.272 3.052 2.887 2.758 2.653 2.567

76 6.981 4.896 4.050 3.577
3.268 3.049 2.884 2.755 2.650 2.563
77 6.976 4.892 4.047 3.573
3.265 3.046 2.881 2.751 2.647 2.560
78 6.971 4.888 4.043 3.570
1.3.6.7.3. Upper Critical Values of the F Distribution
(30 of 38) [5/1/2006 9:58:27 AM]
3.261 3.042 2.877 2.748 2.644 2.557
79 6.967 4.884 4.040 3.566
3.258 3.039 2.874 2.745 2.640 2.554
80 6.963 4.881 4.036 3.563
3.255 3.036 2.871 2.742 2.637 2.551
81 6.958 4.877 4.033 3.560
3.252 3.033 2.868 2.739 2.634 2.548
82 6.954 4.874 4.030 3.557
3.249 3.030 2.865 2.736 2.632 2.545
83 6.950 4.870 4.027 3.554
3.246 3.027 2.863 2.733 2.629 2.542
84 6.947 4.867 4.024 3.551
3.243 3.025 2.860 2.731 2.626 2.539
85 6.943 4.864 4.021 3.548
3.240 3.022 2.857 2.728 2.623 2.537
86 6.939 4.861 4.018 3.545
3.238 3.019 2.854 2.725 2.621 2.534
87 6.935 4.858 4.015 3.543
3.235 3.017 2.852 2.723 2.618 2.532
88 6.932 4.855 4.012 3.540
3.233 3.014 2.849 2.720 2.616 2.529
89 6.928 4.852 4.010 3.538
3.230 3.012 2.847 2.718 2.613 2.527

90 6.925 4.849 4.007 3.535
3.228 3.009 2.845 2.715 2.611 2.524
91 6.922 4.846 4.004 3.533
3.225 3.007 2.842 2.713 2.609 2.522
92 6.919 4.844 4.002 3.530
3.223 3.004 2.840 2.711 2.606 2.520
93 6.915 4.841 3.999 3.528
3.221 3.002 2.838 2.709 2.604 2.518
94 6.912 4.838 3.997 3.525
3.218 3.000 2.835 2.706 2.602 2.515
95 6.909 4.836 3.995 3.523
3.216 2.998 2.833 2.704 2.600 2.513
96 6.906 4.833 3.992 3.521
1.3.6.7.3. Upper Critical Values of the F Distribution
(31 of 38) [5/1/2006 9:58:27 AM]
3.214 2.996 2.831 2.702 2.598 2.511
97 6.904 4.831 3.990 3.519
3.212 2.994 2.829 2.700 2.596 2.509
98 6.901 4.829 3.988 3.517
3.210 2.992 2.827 2.698 2.594 2.507
99 6.898 4.826 3.986 3.515
3.208 2.990 2.825 2.696 2.592 2.505
100 6.895 4.824 3.984 3.513
3.206 2.988 2.823 2.694 2.590 2.503
\
11 12 13 14 15
16 17 18 19 20

1. 6083.35 6106.35 6125.86 6142.70
6157.28 6170.12 6181.42 6191.52 6200.58

6208.74
2. 99.408 99.416 99.422 99.428
99.432 99.437 99.440 99.444 99.447 99.449
3. 27.133 27.052 26.983 26.924
26.872 26.827 26.787 26.751 26.719 26.690
4. 14.452 14.374 14.307 14.249
14.198 14.154 14.115 14.080 14.048 14.020
5. 9.963 9.888 9.825 9.770
9.722 9.680 9.643 9.610 9.580 9.553
6. 7.790 7.718 7.657 7.605
7.559 7.519 7.483 7.451 7.422 7.396
7. 6.538 6.469 6.410 6.359
6.314 6.275 6.240 6.209 6.181 6.155
8. 5.734 5.667 5.609 5.559
5.515 5.477 5.442 5.412 5.384 5.359
9. 5.178 5.111 5.055 5.005
4.962 4.924 4.890 4.860 4.833 4.808
1.3.6.7.3. Upper Critical Values of the F Distribution
(32 of 38) [5/1/2006 9:58:27 AM]
10. 4.772 4.706 4.650 4.601
4.558 4.520 4.487 4.457 4.430 4.405
11. 4.462 4.397 4.342 4.293
4.251 4.213 4.180 4.150 4.123 4.099
12. 4.220 4.155 4.100 4.052
4.010 3.972 3.939 3.909 3.883 3.858
13. 4.025 3.960 3.905 3.857
3.815 3.778 3.745 3.716 3.689 3.665
14. 3.864 3.800 3.745 3.698
3.656 3.619 3.586 3.556 3.529 3.505
15. 3.730 3.666 3.612 3.564

3.522 3.485 3.452 3.423 3.396 3.372
16. 3.616 3.553 3.498 3.451
3.409 3.372 3.339 3.310 3.283 3.259
17. 3.519 3.455 3.401 3.353
3.312 3.275 3.242 3.212 3.186 3.162
18. 3.434 3.371 3.316 3.269
3.227 3.190 3.158 3.128 3.101 3.077
19. 3.360 3.297 3.242 3.195
3.153 3.116 3.084 3.054 3.027 3.003
20. 3.294 3.231 3.177 3.130
3.088 3.051 3.018 2.989 2.962 2.938
21. 3.236 3.173 3.119 3.072
3.030 2.993 2.960 2.931 2.904 2.880
22. 3.184 3.121 3.067 3.019
2.978 2.941 2.908 2.879 2.852 2.827
23. 3.137 3.074 3.020 2.973
2.931 2.894 2.861 2.832 2.805 2.781
24. 3.094 3.032 2.977 2.930
2.889 2.852 2.819 2.789 2.762 2.738
25. 3.056 2.993 2.939 2.892
2.850 2.813 2.780 2.751 2.724 2.699
26. 3.021 2.958 2.904 2.857
2.815 2.778 2.745 2.715 2.688 2.664
27. 2.988 2.926 2.871 2.824
2.783 2.746 2.713 2.683 2.656 2.632
1.3.6.7.3. Upper Critical Values of the F Distribution
(33 of 38) [5/1/2006 9:58:27 AM]
28. 2.959 2.896 2.842 2.795
2.753 2.716 2.683 2.653 2.626 2.602
29. 2.931 2.868 2.814 2.767

2.726 2.689 2.656 2.626 2.599 2.574
30. 2.906 2.843 2.789 2.742
2.700 2.663 2.630 2.600 2.573 2.549
31. 2.882 2.820 2.765 2.718
2.677 2.640 2.606 2.577 2.550 2.525
32. 2.860 2.798 2.744 2.696
2.655 2.618 2.584 2.555 2.527 2.503
33. 2.840 2.777 2.723 2.676
2.634 2.597 2.564 2.534 2.507 2.482
34. 2.821 2.758 2.704 2.657
2.615 2.578 2.545 2.515 2.488 2.463
35. 2.803 2.740 2.686 2.639
2.597 2.560 2.527 2.497 2.470 2.445
36. 2.786 2.723 2.669 2.622
2.580 2.543 2.510 2.480 2.453 2.428
37. 2.770 2.707 2.653 2.606
2.564 2.527 2.494 2.464 2.437 2.412
38. 2.755 2.692 2.638 2.591
2.549 2.512 2.479 2.449 2.421 2.397
39. 2.741 2.678 2.624 2.577
2.535 2.498 2.465 2.434 2.407 2.382
40. 2.727 2.665 2.611 2.563
2.522 2.484 2.451 2.421 2.394 2.369
41. 2.715 2.652 2.598 2.551
2.509 2.472 2.438 2.408 2.381 2.356
42. 2.703 2.640 2.586 2.539
2.497 2.460 2.426 2.396 2.369 2.344
43. 2.691 2.629 2.575 2.527
2.485 2.448 2.415 2.385 2.357 2.332
44. 2.680 2.618 2.564 2.516

2.475 2.437 2.404 2.374 2.346 2.321
45. 2.670 2.608 2.553 2.506
2.464 2.427 2.393 2.363 2.336 2.311
1.3.6.7.3. Upper Critical Values of the F Distribution
(34 of 38) [5/1/2006 9:58:27 AM]
46. 2.660 2.598 2.544 2.496
2.454 2.417 2.384 2.353 2.326 2.301
47. 2.651 2.588 2.534 2.487
2.445 2.408 2.374 2.344 2.316 2.291
48. 2.642 2.579 2.525 2.478
2.436 2.399 2.365 2.335 2.307 2.282
49. 2.633 2.571 2.517 2.469
2.427 2.390 2.356 2.326 2.299 2.274
50. 2.625 2.562 2.508 2.461
2.419 2.382 2.348 2.318 2.290 2.265
51. 2.617 2.555 2.500 2.453
2.411 2.374 2.340 2.310 2.282 2.257
52. 2.610 2.547 2.493 2.445
2.403 2.366 2.333 2.302 2.275 2.250
53. 2.602 2.540 2.486 2.438
2.396 2.359 2.325 2.295 2.267 2.242
54. 2.595 2.533 2.479 2.431
2.389 2.352 2.318 2.288 2.260 2.235
55. 2.589 2.526 2.472 2.424
2.382 2.345 2.311 2.281 2.253 2.228
56. 2.582 2.520 2.465 2.418
2.376 2.339 2.305 2.275 2.247 2.222
57. 2.576 2.513 2.459 2.412
2.370 2.332 2.299 2.268 2.241 2.215
58. 2.570 2.507 2.453 2.406

2.364 2.326 2.293 2.262 2.235 2.209
59. 2.564 2.502 2.447 2.400
2.358 2.320 2.287 2.256 2.229 2.203
60. 2.559 2.496 2.442 2.394
2.352 2.315 2.281 2.251 2.223 2.198
61. 2.553 2.491 2.436 2.389
2.347 2.309 2.276 2.245 2.218 2.192
62. 2.548 2.486 2.431 2.384
2.342 2.304 2.270 2.240 2.212 2.187
63. 2.543 2.481 2.426 2.379
2.337 2.299 2.265 2.235 2.207 2.182
1.3.6.7.3. Upper Critical Values of the F Distribution
(35 of 38) [5/1/2006 9:58:27 AM]
64. 2.538 2.476 2.421 2.374
2.332 2.294 2.260 2.230 2.202 2.177
65. 2.534 2.471 2.417 2.369
2.327 2.289 2.256 2.225 2.198 2.172
66. 2.529 2.466 2.412 2.365
2.322 2.285 2.251 2.221 2.193 2.168
67. 2.525 2.462 2.408 2.360
2.318 2.280 2.247 2.216 2.188 2.163
68. 2.520 2.458 2.403 2.356
2.314 2.276 2.242 2.212 2.184 2.159
69. 2.516 2.454 2.399 2.352
2.310 2.272 2.238 2.208 2.180 2.155
70. 2.512 2.450 2.395 2.348
2.306 2.268 2.234 2.204 2.176 2.150
71. 2.508 2.446 2.391 2.344
2.302 2.264 2.230 2.200 2.172 2.146
72. 2.504 2.442 2.388 2.340

2.298 2.260 2.226 2.196 2.168 2.143
73. 2.501 2.438 2.384 2.336
2.294 2.256 2.223 2.192 2.164 2.139
74. 2.497 2.435 2.380 2.333
2.290 2.253 2.219 2.188 2.161 2.135
75. 2.494 2.431 2.377 2.329
2.287 2.249 2.215 2.185 2.157 2.132
76. 2.490 2.428 2.373 2.326
2.284 2.246 2.212 2.181 2.154 2.128
77. 2.487 2.424 2.370 2.322
2.280 2.243 2.209 2.178 2.150 2.125
78. 2.484 2.421 2.367 2.319
2.277 2.239 2.206 2.175 2.147 2.122
79. 2.481 2.418 2.364 2.316
2.274 2.236 2.202 2.172 2.144 2.118
80. 2.478 2.415 2.361 2.313
2.271 2.233 2.199 2.169 2.141 2.115
81. 2.475 2.412 2.358 2.310
2.268 2.230 2.196 2.166 2.138 2.112
1.3.6.7.3. Upper Critical Values of the F Distribution
(36 of 38) [5/1/2006 9:58:27 AM]
82. 2.472 2.409 2.355 2.307
2.265 2.227 2.193 2.163 2.135 2.109
83. 2.469 2.406 2.352 2.304
2.262 2.224 2.191 2.160 2.132 2.106
84. 2.466 2.404 2.349 2.302
2.259 2.222 2.188 2.157 2.129 2.104
85. 2.464 2.401 2.347 2.299
2.257 2.219 2.185 2.154 2.126 2.101
86. 2.461 2.398 2.344 2.296

2.254 2.216 2.182 2.152 2.124 2.098
87. 2.459 2.396 2.342 2.294
2.252 2.214 2.180 2.149 2.121 2.096
88. 2.456 2.393 2.339 2.291
2.249 2.211 2.177 2.147 2.119 2.093
89. 2.454 2.391 2.337 2.289
2.247 2.209 2.175 2.144 2.116 2.091
90. 2.451 2.389 2.334 2.286
2.244 2.206 2.172 2.142 2.114 2.088
91. 2.449 2.386 2.332 2.284
2.242 2.204 2.170 2.139 2.111 2.086
92. 2.447 2.384 2.330 2.282
2.240 2.202 2.168 2.137 2.109 2.083
93. 2.444 2.382 2.327 2.280
2.237 2.200 2.166 2.135 2.107 2.081
94. 2.442 2.380 2.325 2.277
2.235 2.197 2.163 2.133 2.105 2.079
95. 2.440 2.378 2.323 2.275
2.233 2.195 2.161 2.130 2.102 2.077
96. 2.438 2.375 2.321 2.273
2.231 2.193 2.159 2.128 2.100 2.075
97. 2.436 2.373 2.319 2.271
2.229 2.191 2.157 2.126 2.098 2.073
98. 2.434 2.371 2.317 2.269
2.227 2.189 2.155 2.124 2.096 2.071
99. 2.432 2.369 2.315 2.267
2.225 2.187 2.153 2.122 2.094 2.069
1.3.6.7.3. Upper Critical Values of the F Distribution
(37 of 38) [5/1/2006 9:58:27 AM]
100. 2.430 2.368 2.313 2.265

2.223 2.185 2.151 2.120 2.092 2.067
1.3.6.7.3. Upper Critical Values of the F Distribution
(38 of 38) [5/1/2006 9:58:27 AM]
Given a specified value for :
For a two-sided test, find the column corresponding to
/2 in the
table for upper critical values and reject the null hypothesis if the
test statistic is greater than the tabled value. Similarly, find the
column corresponding to 1 -
/2 in the table for lower critical
values and reject the null hypothesis if the test statistic is less than
the tabled value.
1.
For an upper one-sided test, find the column corresponding to
in the upper critical values table and reject the null hypothesis if
the test statistic is greater than the tabled value.
2.
For a lower one-sided test, find the column corresponding to 1 -
in the lower critical values table and reject the null hypothesis
if the computed test statistic is less than the tabled value.
3.
1.3.6.7.4. Critical Values of the Chi-Square Distribution
(2 of 15) [5/1/2006 9:58:28 AM]
Upper critical values of chi-square distribution with degrees of
freedom
Probability of exceeding the
critical value

0.10 0.05 0.025
0.01 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
1.3.6.7.4. Critical Values of the Chi-Square Distribution
(3 of 15) [5/1/2006 9:58:28 AM]
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
1.3.6.7.4. Critical Values of the Chi-Square Distribution

(4 of 15) [5/1/2006 9:58:28 AM]
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
1.3.6.7.4. Critical Values of the Chi-Square Distribution
(5 of 15) [5/1/2006 9:58:28 AM]
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
1.3.6.7.4. Critical Values of the Chi-Square Distribution
(6 of 15) [5/1/2006 9:58:28 AM]