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Pr( )
!
Y y
i
e
i
i
y
≤ =
=
−∑ λ λ
0
Table A2. Cumulative Poisson(λ) probabilities
The body of the table represents Pr(Y ≤ y) where Y~Poisson(λ)
Example: The probability that a Poisson(4) random variable is at most 5 is 0.7851
y
λ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
0.25 .7788 .9735 .9978 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.50 .6065 .9098 .9856 .9982 .9998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.75 .4724 .8266 .9595 .9927 .9989 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.00 .3679 .7358 .9197 .9810 .9963 .9994 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.25 .2865 .6446 .8685 .9617 .9909 .9982 .9997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.50 .2231 .5578 .8088 .9344 .9814 .9955 .9991 .9998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.75 .1738 .4779 .7440 .8992 .9671 .9909 .9978 .9995 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2.00 .1353 .4060 .6767 .8571 .9473 .9834 .9955 .9989 .9998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2.25 .1054 .3425 .6093 .8094 .9220 .9726 .9916 .9977 .9994 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2.50 .0821 .2873 .5438 .7576 .8912 .9580 .9858 .9958 .9989 .9997 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2.75 .0639 .2397 .4815 .7030 .8554 .9392 .9776 .9927 .9978 .9994 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3.00 .0498 .1991 .4232 .6472 .8153 .9161 .9665 .9881 .9962 .9989 .9997 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3.25 .0388 .1648 .3696 .5914 .7717 .8888 .9523 .9817 .9937 .9980 .9994 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3.50 .0302 .1359 .3208 .5366 .7254 .8576 .9347 .9733 .9901 .9967 .9990 .9997 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000
3.75 .0235 .1117 .2771 .4838 .6775 .8229 .9137 .9624 .9852 .9947 .9983 .9995 .9999 1.000 1.000 1.000 1.000 1.000 1.000 1.000
4.00 .0183 .0916 .2381 .4335 .6288 .7851 .8893 .9489 .9786 .9919 .9972 .9991 .9997 .9999 1.000 1.000 1.000 1.000 1.000 1.000
4.25 .0143 .0749 .2037 .3862 .5801 .7449 .8617 .9326 .9702 .9880 .9956 .9985 .9995 .9999 1.000 1.000 1.000 1.000 1.000 1.000
4.50 .0111 .0611 .1736 .3423 .5321 .7029 .8311 .9134 .9597 .9829 .9933 .9976 .9992 .9997 .9999 1.000 1.000 1.000 1.000 1.000
4.75 .0087 .0497 .1473 .3019 .4854 .6597 .7978 .8914 .9470 .9764 .9903 .9963 .9987 .9996 .9999 1.000 1.000 1.000 1.000 1.000
5.00 .0067 .0404 .1247 .2650 .4405 .6160 .7622 .8666 .9319 .9682 .9863 .9945 .9980 .9993 .9998 .9999 1.000 1.000 1.000 1.000
5.25 .0052 .0328 .1051 .2317 .3978 .5722 .7248 .8392 .9144 .9582 .9812 .9922 .9970 .9989 .9996 .9999 1.000 1.000 1.000 1.000
5.50 .0041 .0266 .0884 .2017 .3575 .5289 .6860 .8095 .8944 .9462 .9747 .9890 .9955 .9983 .9994 .9998 .9999 1.000 1.000 1.000
5.75 .0032 .0215 .0741 .1749 .3199 .4866 .6464 .7776 .8719 .9322 .9669 .9850 .9937 .9975 .9991 .9997 .9999 1.000 1.000 1.000
6.00 .0025 .0174 .0620 .1512 .2851 .4457 .6063 .7440 .8472 .9161 .9574 .9799 .9912 .9964 .9986 .9995 .9998 .9999 1.000 1.000
6.25 .0019 .0140 .0517 .1303 .2530 .4064 .5662 .7089 .8204 .8978 .9462 .9737 .9880 .9949 .9979 .9992 .9997 .9999 1.000 1.000
6.50 .0015 .0113 .0430 .1118 .2237 .3690 .5265 .6728 .7916 .8774 .9332 .9661 .9840 .9929 .9970 .9988 .9996 .9998 .9999 1.000
6.75 .0012 .0091 .0357 .0958 .1970 .3338 .4876 .6359 .7611 .8549 .9183 .9571 .9790 .9904 .9958 .9983 .9994 .9998 .9999 1.000
Calculated with the Poisson function of The SAS System, Vs. 6.12.
Table A2. Cumulative Poisson(λ) probabilities (Continued)
y
λ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
7.00 .0009 .0073 .0296 .0818 .1730 .3007 .4497 .5987 .7291 .8305 .9015 .9467 .9730 .9872 .9943 .9976 .9990 .9996 .9999 1.000
7.25 .0007 .0059 .0245 .0696 .1514 .2699 .4132 .5615 .6960 .8043 .8828 .9345 .9658 .9832 .9923 .9966 .9986 .9995 .9998 .9999
7.50 .0006 .0047 .0203 .0591 .1321 .2414 .3782 .5246 .6620 .7764 .8622 .9208 .9573 .9784 .9897 .9954 .9980 .9992 .9997 .9999
7.75 .0004 .0038 .0167 .0501 .1149 .2152 .3449 .4884 .6274 .7471 .8399 .9053 .9475 .9727 .9866 .9938 .9973 .9989 .9996 .9998
8.00 .0003 .0030 .0138 .0424 .0996 .1912 .3134 .4530 .5925 .7166 .8159 .8881 .9362 .9658 .9827 .9918 .9963 .9984 .9993 .9997
8.25 .0003 .0024 .0113 .0358 .0862 .1694 .2838 .4186 .5577 .6852 .7903 .8692 .9234 .9578 .9781 .9893 .9950 .9978 .9991 .9996
8.50 .0002 .0019 .0093 .0301 .0744 .1496 .2562 .3856 .5231 .6530 .7634 .8487 .9091 .9486 .9726 .9862 .9934 .9970 .9987 .9995
8.75 .0002 .0015 .0076 .0253 .0640 .1317 .2305 .3540 .4890 .6203 .7352 .8266 .8932 .9380 .9661 .9824 .9914 .9960 .9982 .9992
9.00 .0001 .0012 .0062 .0212 .0550 .1157 .2068 .3239 .4557 .5874 .7060 .8030 .8758 .9261 .9585 .9780 .9889 .9947 .9976 .9989
9.25 .0001 .0010 .0051 .0178 .0471 .1013 .1849 .2954 .4232 .5545 .6760 .7781 .8568 .9129 .9499 .9727 .9859 .9931 .9968 .9986
9.50 .0001 .0008 .0042 .0149 .0403 .0885 .1649 .2687 .3918 .5218 .6453 .7520 .8364 .8981 .9400 .9665 .9823 .9911 .9957 .9980
9.75 .0001 .0006 .0034 .0124 .0344 .0772 .1467 .2436 .3617 .4896 .6143 .7248 .8146 .8820 .9289 .9594 .9780 .9886 .9944 .9974
10.0 .0000 .0005 .0028 .0103 .0293 .0671 .1301 .2202 .3328 .4579 .5830 .6968 .7916 .8645 .9165 .9513 .9730 .9857 .9928 .9965
10.3 .0000 .0004 .0023 .0086 .0249 .0582 .1151 .1985 .3054 .4271 .5518 .6680 .7673 .8456 .9029 .9420 .9671 .9822 .9909 .9955
10.5 .0000 .0003 .0018 .0071 .0211 .0504 .1016 .1785 .2794 .3971 .5207 .6387 .7420 .8253 .8879 .9317 .9604 .9781 .9885 .9942
10.8 .0000 .0003 .0015 .0059 .0179 .0435 .0895 .1601 .2549 .3682 .4900 .6091 .7157 .8039 .8716 .9201 .9527 .9733 .9857 .9926
11.0 .0000 .0002 .0012 .0049 .0151 .0375 .0786 .1432 .2320 .3405 .4599 .5793 .6887 .7813 .8540 .9074 .9441 .9678 .9823 .9907
11.3 .0000 .0002 .0010 .0041 .0128 .0323 .0689 .1278 .2105 .3140 .4304 .5495 .6611 .7576 .8352 .8935 .9344 .9615 .9784 .9884
11.5 .0000 .0001 .0008 .0034 .0107 .0277 .0603 .1137 .1906 .2888 .4017 .5198 .6329 .7330 .8153 .8783 .9236 .9542 .9738 .9857
11.8 .0000 .0001 .0006 .0028 .0090 .0238 .0526 .1010 .1721 .2649 .3740 .4905 .6045 .7076 .7942 .8619 .9117 .9461 .9686 .9825
12.0 .0000 .0001 .0005 .0023 .0076 .0203 .0458 .0895 .1550 .2424 .3472 .4616 .5760 .6815 .7720 .8444 .8987 .9370 .9626 .9787
Table A3. Areas under the standard normal probability density function 
21
21
2
z
z
P(Z z ) . e dz
∞
−
≥ =
pi ∫ 
Example: The probability that a standard Gaussian r.v. Z exceeds 1.5 is P(Z > 1.5) = 0.06681 
 
21
2
1 5
11 5 0 06681
2
z
.
P(Z . ) e dz .
∞
−
≥ = =
pi ∫ 
 
Table A5. Areas under the t-student probability density function 
 
 - t{n-1, ����/2} 0 t{n-1, ����/2} (Simétrica) 
Example: The probability that a t-student r.v. t exceeds 1.5 is P(t{8} > 2.3060) = 0.025 
 
 
Table A6. Percentiles of the χ2 distributions
The body of the table represents Pr(χ2ν ≥ x) where χ2ν is a Chi-square random variable with ν degrees of freedom.
Example: The probability that a χ21 r.v. exceeds 3.8415 is 0.05; Pr(χ21 > 3.8415) =0.05
One-sided, right-tail probability
v 0.001 0.005 0.010 0.020 0.025 0.045 0.050 0.100 0.900 0.950 0.955 0.975 0.980 0.990 0.995 0.999 v
 1 10.828 7.8794 6.6349 5.4119 5.0239 4.0186 3.8415 2.7055 .01579 .00393 .00318 .00098 .00063 .00016 .00004 .00000 1
 2 13.816 10.597 9.2103 7.8240 7.3778 6.2022 5.9915 4.6052 .21072 .10259 .09209 .05064 .04041 .02010 .01003 .00200 2
 3 16.266 12.838 11.345 9.8374 9.3484 8.0495 7.8147 6.2514 .58437 .35185 .32634 .21580 .18483 .11483 .07172 .024303
 4 18.467 14.860 13.277 11.668 11.143 9.7423 9.4877 7.7794 1.0636 .71072 .66980 .48442 .42940 .29711 .20699 .09080 4
 5 20.515 16.750 15.086 13.388 12.833 11.342 11.070 9.2364 1.6103 1.1455 1.0898 .83121 .75189 .55430 .41174 .21021 5
 6 22.458 18.548 16.812 15.033 14.449 12.879 12.592 10.645 2.2041 1.6354 1.5659 1.2373 1.1344 .87209 .67573 .38107 6
 7 24.322 20.278 18.475 16.622 16.013 14.369 14.067 12.017 2.8331 2.1673 2.0848 1.6899 1.5643 1.2390 .98926 .59849 7
 8 26.124 21.955 20.090 18.168 17.535 15.822 15.507 13.362 3.4895 2.7326 2.6377 2.1797 2.0325 1.6465 1.3444 .85710 8
 9 27.877 23.589 21.666 19.679 19.023 17.246 16.919 14.684 4.1682 3.3251 3.2185 2.7004 2.5324 2.0879 1.7349 1.1519 9
 10 29.588 25.188 23.209 21.161 20.483 18.646 18.307 15.987 4.8652 3.9403 3.8225 3.2470 3.0591 2.5582 2.1559 1.4787 10
 11 31.264 26.757 24.725 22.618 21.920 20.025 19.675 17.275 5.5778 4.5748 4.4463 3.8157 3.6087 3.0535 2.6032 1.8339 11
 12 32.909 28.300 26.217 24.054 23.337 21.386 21.026 18.549 6.3038 5.2260 5.0873 4.4038 4.1783 3.5706 3.0738 2.2142 12
 13 34.528 29.819 27.688 25.472 24.736 22.733 22.362 19.812 7.0415 5.8919 5.7432 5.0088 4.7654 4.1069 3.5650 2.6172 13
 14 36.123 31.319 29.141 26.873 26.119 24.065 23.685 21.064 7.7895 6.5706 6.4125 5.6287 5.3682 4.6604 4.0747 3.0407 14
 15 37.697 32.801 30.578 28.259 27.488 25.385 24.996 22.307 8.5468 7.2609 7.0936 6.2621 5.9849 5.2293 4.6009 3.4827 15
 16 39.252 34.267 32.000 29.633 28.845 26.695 26.296 23.542 9.3122 7.9616 7.7854 6.9077 6.6142 5.8122 5.1422 3.9416 16
 17 40.790 35.718 33.409 30.995 30.191 27.995 27.587 24.769 10.085 8.6718 8.4868 7.5642 7.2550 6.4078 5.6972 4.4161 17
 18 42.312 37.156 34.805 32.346 31.526 29.285 28.869 25.989 10.865 9.3905 9.1971 8.2307 7.9062 7.0149 6.2648 4.9048 18
 19 43.820 38.582 36.191 33.687 32.852 30.568 30.144 27.204 11.651 10.117 9.9155 8.9065 8.5670 7.6327 6.8440 5.4068 19
 20 45.315 39.997 37.566 35.020 34.170 31.843 31.410 28.412 12.443 10.851 10.641 9.5908 9.2367 8.2604 7.4338 5.9210 20
 21 46.797 41.401 38.932 36.343 35.479 33.111 32.671 29.615 13.240 11.591 11.374 10.283 9.9146 8.8972 8.0337 6.4467 21
 22 48.268 42.796 40.289 37.659 36.781 34.373 33.924 30.813 14.041 12.338 12.113 10.982 10.600 9.5425 8.6427 6.9830 22
 23 49.728 44.181 41.638 38.968 38.076 35.628 35.172 32.007 14.848 13.091 12.858 11.689 11.293 10.196 9.2604 7.5292 23
 24 51.179 45.559 42.980 40.270 39.364 36.878 36.415 33.196 15.659 13.848 13.609 12.401 11.992 10.856 9.8862 8.0849 24
 25 52.620 46.928 44.314 41.566 40.646 38.123 37.652 34.382 16.473 14.611 14.365 13.120 12.697 11.524 10.520 8.6493 25
 Calculated with the CInv function of The SAS System, Vs. 6.12.
Table A6. Percentiles of the χ2 distributions (Continued)
One-sided, right-tail probability
v 0.001 0.005 0.010 0.020 0.025 0.045 0.050 0.100 0.900 0.950 0.955 0.975 0.980 0.990 0.995 0.999 v
 26 54.052 48.290 45.642 42.856 41.923 39.363 38.885 35.563 17.292 15.379 15.125 13.844 13.409 12.198 11.160 9.2221 26
 27 55.476 49.645 46.963 44.140 43.195 40.598 40.113 36.741 18.114 16.151 15.891 14.573 14.125 12.879 11.808 9.8028 27
 28 56.892 50.993 48.278 45.419 44.461 41.828 41.337 37.916 18.939 16.928 16.660 15.308 14.847 13.565 12.461 10.391 28
 29 58.301 52.336 49.588 46.693 45.722 43.055 42.557 39.087 19.768 17.708 17.434 16.047 15.574 14.256 13.121 10.986 29
 30 59.703 53.672 50.892 47.962 46.979 44.277 43.773 40.256 20.599 18.493 18.212 16.791 16.306 14.953 13.787 11.588 30
 31 61.098 55.003 52.191 49.226 48.232 45.496 44.985 41.422 21.434 19.281 18.993 17.539 17.042 15.655 14.458 12.196 31
 32 62.487 56.328 53.486 50.487 49.480 46.712 46.194 42.585 22.271 20.072 19.778 18.291 17.783 16.362 15.134 12.811 32
 33 63.870 57.648 54.776 51.743 50.725 47.923 47.400 43.745 23.110 20.867 20.567 19.047 18.527 17.074 15.815 13.431 33
 34 65.247 58.964 56.061 52.995 51.966 49.132 48.602 44.903 23.952 21.664 21.358 19.806 19.275 17.789 16.501 14.057 34
 35 66.619 60.275 57.342 54.244 53.203 50.338 49.802 46.059 24.797 22.465 22.153 20.569 20.027 18.509 17.192 14.688 35
 36 67.985 61.581 58.619 55.489 54.437 51.540 50.998 47.212 25.643 23.269 22.951 21.336 20.783 19.233 17.887 15.324 36
 37 69.346 62.883 59.893 56.730 55.668 52.740 52.192 48.363 26.492 24.075 23.751 22.106 21.542 19.960 18.586 15.965 37
 38 70.703 64.181 61.162 57.969 56.896 53.937 53.384 49.513 27.343 24.884 24.554 22.878 22.304 20.691 19.289 16.611 38
 39 72.055 65.476 62.428 59.204 58.120 55.132 54.572 50.660 28.196 25.695 25.360 23.654 23.069 21.426 19.996 17.262 39
 40 73.402 66.766 63.691 60.436 59.342 56.324 55.758 51.805 29.051 26.509 26.168 24.433 23.838 22.164 20.707 17.916 40
 41 74.745 68.053 64.950 61.665 60.561 57.513 56.942 52.949 29.907 27.326 26.979 25.215 24.609 22.906 21.421 18.575 41
 42 76.084 69.336 66.206 62.892 61.777 58.700 58.124 54.090 30.765 28.144 27.792 25.999 25.383 23.650 22.138 19.239 42
 43 77.419 70.616 67.459 64.116 62.990 59.885 59.304 55.230 31.625 28.965 28.607 26.785 26.159 24.398 22.859 19.906 43
 44 78.750 71.893 68.710 65.337 64.201 61.068 60.481 56.369 32.487 29.787 29.425 27.575 26.939 25.148 23.584 20.576 44
 45 80.077 73.166 69.957 66.555 65.410 62.249 61.656 57.505 33.350 30.612 30.244 28.366 27.720 25.901 24.311 21.251 45
 46 81.400 74.437 71.201 67.771 66.617 63.427 62.830 58.641 34.215 31.439 31.065 29.160 28.505 26.657 25.041 21.929 46
 47 82.720 75.704 72.443 68.985 67.821 64.604 64.001 59.774 35.081 32.268 31.889 29.956 29.291 27.416 25.775 22.610 47
 48 84.037 76.969 73.683 70.197 69.023 65.779 65.171 60.907 35.949 33.098 32.714 30.755 30.080 28.177 26.511 23.295 48
 49 85.351 78.231 74.919 71.406 70.222 66.952 66.339 62.038 36.818 33.930 33.541 31.555 30.871 28.941 27.249 23.983 49
 50 86.661 79.490 76.154 72.613 71.420 68.123 67.505 63.167 37.689 34.764 34.370 32.357 31.664 29.707 27.991 24.674 50
 60 99.607 91.952 88.379 84.580 83.298 79.749 79.082 74.397 46.459 43.188 42.746 40.482 39.699 37.485 35.534 31.738 60
Table A7. Upper one-sided percentiles of the F distributions
The body of the table represents Pr(Fν1,ν2 ≥ f ) where Fν1,ν2 is a F random variable with ν1 numerator and ν2 denominator degrees of freedom.
Example: The probability that a F3,4 random variable. exceeds 4.191 is 0.1; Pr(F3,4 > 4.191)
Numerator degrees of freedom ν1
ν2 p 1 2 3 4 5 6 7 8 9 10 15 20 25 30 50 p ν2
1 .005 16211 19999 21615 22500 23056 23437 23715 23925 24091 24224 24630 24836 24960 25044 25211 .005 1
1 .010 4052 5000 5403 5625 5764 5859 5928 5981 6022 6056 6157 6209 6240 6261 6303 .010 1
1 .025 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3 968.6 984.9 993.1 998.1 1001 1008 .025 1
1 .050 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 245.9 248.0 249.3 250.1 251.8 .050 1
1 .100 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86 60.19 61.22 61.74 62.05 62.26 62.69 .100 1
2 .005 198.5 199.0 199.2 199.2 199.3 199.3 199.4 199.4 199.4 199.4 199.4 199.4 199.5 199.5 199.5 .005 2
2 .010 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.43 99.45 99.46 99.47 99.48 .010 2
2 .025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.43 39.45 39.46 39.46 39.48 .025 2
2 .050 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.43 19.45 19.46 19.46 19.48 .050 2
2 .100 8.526 9.000 9.162 9.243 9.293 9.326 9.349 9.367 9.381 9.392 9.425 9.441 9.451 9.458 9.471 .100 2
3 .005 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.88 43.69 43.08 42.78 42.59 42.47 42.21 .005 3
3 .010 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 26.87 26.69 26.58 26.50 26.35 .010 3
3 .025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.25 14.17 14.12 14.08 14.01 .025 3
3 .050 10.13 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786 8.703 8.660 8.634 8.617 8.581 .050 3
3 .100 5.538 5.462 5.391 5.343 5.309 5.285 5.266 5.252 5.240 5.230 5.200 5.184 5.175 5.168 5.155 .100 3
4 .005 31.3326.28 24.26 23.15 22.46 21.97 21.62 21.35 21.14 20.97 20.44 20.17 20.00 19.89 19.67 .005 4
4 .010 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.20 14.02 13.91 13.84 13.69 .010 4
4 .025 12.22 10.65 9.979 9.605 9.364 9.197 9.074 8.980 8.905 8.844 8.657 8.560 8.501 8.461 8.381 .025 4
4 .050 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964 5.858 5.803 5.769 5.746 5.699 .050 4
4 .100 4.545 4.325 4.191 4.107 4.051 4.010 3.979 3.955 3.936 3.920 3.870 3.844 3.828 3.817 3.795 .100 4
5 .005 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.77 13.62 13.15 12.90 12.76 12.66 12.45 .005 5
5 .010 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.722 9.553 9.449 9.379 9.238 .010 5
5 .025 10.01 8.434 7.764 7.388 7.146 6.978 6.853 6.757 6.681 6.619 6.428 6.329 6.268 6.227 6.144 .025 5
5 .050 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772 4.735 4.619 4.558 4.521 4.496 4.444 .050 5
5 .100 4.060 3.780 3.619 3.520 3.453 3.405 3.368 3.339 3.316 3.297 3.238 3.207 3.187 3.174 3.147 .100 5
6 .005 18.63 14.54 12.92 12.03 11.46 11.07 10.79 10.57 10.39 10.25 9.814 9.589 9.451 9.358 9.170 .005 6
6 .010 13.75 10.92 9.780 9.148 8.746 8.466 8.260 8.102 7.976 7.874 7.559 7.396 7.296 7.229 7.091 .010 6
6 .025 8.813 7.260 6.599 6.227 5.988 5.820 5.695 5.600 5.523 5.461 5.269 5.168 5.107 5.065 4.980 .025 6
6 .050 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.060 3.938 3.874 3.835 3.808 3.754 .050 6
6 .100 3.776 3.463 3.289 3.181 3.108 3.055 3.014 2.983 2.958 2.937 2.871 2.836 2.815 2.800 2.770 .100 6
7 .005 16.24 12.40 10.88 10.05 9.522 9.155 8.885 8.678 8.514 8.380 7.968 7.754 7.623 7.534 7.354 .005 7
7 .010 12.25 9.547 8.451 7.847 7.460 7.191 6.993 6.840 6.719 6.620 6.314 6.155 6.058 5.992 5.858 .010 7
7 .025 8.073 6.542 5.890 5.523 5.285 5.119 4.995 4.899 4.823 4.761 4.568 4.467 4.405 4.362 4.276 .025 7
7 .050 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637 3.511 3.445 3.404 3.376 3.319 .050 7
7 .100 3.589 3.257 3.074 2.961 2.883 2.827 2.785 2.752 2.725 2.703 2.632 2.595 2.571 2.555 2.523 .100 7
Calculated with the FInv function of The SAS System, Vs. 6.12.
Table A7. Upper one-sided percentiles of the F distributions (Continued …)
Numerator degrees of freedom �1
ν2 p 1 2 3 4 5 6 7 8 9 10 15 20 25 30 50 p ν2
8 .005 14.69 11.04 9.596 8.805 8.302 7.952 7.694 7.496 7.339 7.211 6.814 6.608 6.482 6.396 6.222 .005 8
8 .010 11.26 8.649 7.591 7.006 6.632 6.371 6.178 6.029 5.911 5.814 5.515 5.359 5.263 5.198 5.065 .010 8
8 .025 7.571 6.059 5.416 5.053 4.817 4.652 4.529 4.433 4.357 4.295 4.101 3.999 3.937 3.894 3.807 .025 8
8 .050 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347 3.218 3.150 3.108 3.079 3.020 .050 8
8 .100 3.458 3.113 2.924 2.806 2.726 2.668 2.624 2.589 2.561 2.538 2.464 2.425 2.400 2.383 2.348 .100 8
9 .005 13.61 10.11 8.717 7.956 7.471 7.134 6.885 6.693 6.541 6.417 6.032 5.832 5.708 5.625 5.454 .005 9
9 .010 10.56 8.022 6.992 6.422 6.057 5.802 5.613 5.467 5.351 5.257 4.962 4.808 4.713 4.649 4.517 .010 9
9 .025 7.209 5.715 5.078 4.718 4.484 4.320 4.197 4.102 4.026 3.964 3.769 3.667 3.604 3.560 3.472 .025 9
9 .050 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137 3.006 2.936 2.893 2.864 2.803 .050 9
9 .100 3.360 3.006 2.813 2.693 2.611 2.551 2.505 2.469 2.440 2.416 2.340 2.298 2.272 2.255 2.218 .100 9
10 .005 12.83 9.427 8.081 7.343 6.872 6.545 6.302 6.116 5.968 5.847 5.471 5.274 5.153 5.071 4.902 .005 10
10 .010 10.04 7.559 6.552 5.994 5.636 5.386 5.200 5.057 4.942 4.849 4.558 4.405 4.311 4.247 4.115 .010 10
10 .025 6.937 5.456 4.826 4.468 4.236 4.072 3.950 3.855 3.779 3.717 3.522 3.419 3.355 3.311 3.221 .025 10
10 .050 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978 2.845 2.774 2.730 2.700 2.637 .050 10
10 .100 3.285 2.924 2.728 2.605 2.522 2.461 2.414 2.377 2.347 2.323 2.244 2.201 2.174 2.155 2.117 .100 10
11 .005 12.23 8.912 7.600 6.881 6.422 6.102 5.865 5.682 5.537 5.418 5.049 4.855 4.736 4.654 4.488 .005 11
11 .010 9.646 7.206 6.217 5.668 5.316 5.069 4.886 4.744 4.632 4.539 4.251 4.099 4.005 3.941 3.810 .010 11
11 .025 6.724 5.256 4.630 4.275 4.044 3.881 3.759 3.664 3.588 3.526 3.330 3.226 3.162 3.118 3.027 .025 11
11 .050 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.896 2.854 2.719 2.646 2.601 2.570 2.507 .050 11
11 .100 3.225 2.860 2.660 2.536 2.451 2.389 2.342 2.304 2.274 2.248 2.167 2.123 2.095 2.076 2.036 .100 11
12 .005 11.75 8.510 7.226 6.521 6.071 5.757 5.525 5.345 5.202 5.085 4.721 4.530 4.412 4.331 4.165 .005 12
12 .010 9.330 6.927 5.953 5.412 5.064 4.821 4.640 4.499 4.388 4.296 4.010 3.858 3.765 3.701 3.569 .010 12
12 .025 6.554 5.096 4.474 4.121 3.891 3.728 3.607 3.512 3.436 3.374 3.177 3.073 3.008 2.963 2.871 .025 12
12 .050 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753 2.617 2.544 2.498 2.466 2.401 .050 12
12 .100 3.177 2.807 2.606 2.480 2.394 2.331 2.283 2.245 2.214 2.188 2.105 2.060 2.031 2.011 1.970 .100 12
13 .005 11.37 8.186 6.926 6.233 5.791 5.482 5.253 5.076 4.935 4.820 4.460 4.270 4.153 4.073 3.908 .005 13
13 .010 9.074 6.701 5.739 5.205 4.862 4.620 4.441 4.302 4.191 4.100 3.815 3.665 3.571 3.507 3.375 .010 13
13 .025 6.414 4.965 4.347 3.996 3.767 3.604 3.483 3.388 3.312 3.250 3.053 2.948 2.882 2.837 2.744 .025 13
13 .050 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671 2.533 2.459 2.412 2.380 2.314 .050 13
13 .100 3.136 2.763 2.560 2.434 2.347 2.283 2.234 2.195 2.164 2.138 2.053 2.007 1.978 1.958 1.915 .100 13
14 .005 11.06 7.922 6.680 5.998 5.562 5.257 5.031 4.857 4.717 4.603 4.247 4.059 3.942 3.862 3.698 .005 14
14 .010 8.862 6.515 5.564 5.035 4.695 4.456 4.278 4.140 4.030 3.939 3.656 3.505 3.412 3.348 3.215 .010 14
14 .025 6.298 4.857 4.242 3.892 3.663 3.501 3.380 3.285 3.209 3.147 2.949 2.844 2.778 2.732 2.638 .025 14
14 .050 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602 2.463 2.388 2.341 2.308 2.241 .050 14
14 .100 3.102 2.726 2.522 2.395 2.307 2.243 2.193 2.154 2.122 2.095 2.010 1.962 1.933 1.912 1.869 .100 14
15 .005 10.80 7.701 6.476 5.803 5.372 5.071 4.847 4.674 4.536 4.424 4.070 3.883 3.766 3.687 3.523 .005 15
15 .010 8.683 6.359 5.417 4.893 4.556 4.318 4.142 4.004 3.895 3.805 3.522 3.372 3.278 3.214 3.081 .010 15
15 .025 6.200 4.765 4.153 3.804 3.576 3.415 3.293 3.199 3.123 3.060 2.862 2.756 2.689 2.644 2.549 .025 15
15 .050 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544 2.403 2.328 2.280 2.247 2.178 .050 15
15 .100 3.073 2.695 2.490 2.361 2.273 2.208 2.158 2.119 2.086 2.059 1.972 1.924 1.894 1.873 1.828 .100 15
Table A7. Upper one-sided percentiles of the F distributions (Continued …)
Numerator degrees of freedom ν1
ν2 p 1 2 3 4 5 6 7 8 9 10 15 20 25 30 50 p ν2
16 .005 10.58 7.514 6.303 5.638 5.212 4.913 4.692 4.521 4.384 4.272 3.920 3.734 3.618 3.539 3.375 .005 16
16 .010 8.531 6.226 5.292 4.773 4.437 4.202 4.026 3.890 3.780 3.691 3.409 3.259 3.165 3.101 2.967 .010 16
16 .025 6.115 4.687 4.077 3.729 3.502 3.341 3.219 3.125 3.049 2.986 2.788 2.681 2.614 2.568 2.472 .025 16
16 .050 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.538 2.494 2.352 2.276 2.227 2.194 2.124 .050 16
16 .100 3.048 2.668 2.462 2.333 2.244 2.178 2.128 2.088 2.055 2.028 1.940 1.891 1.860 1.839 1.793 .100 16
17 .005 10.38 7.354 6.156 5.497 5.075 4.779 4.559 4.389 4.254 4.142 3.793 3.607 3.492 3.412 3.248 .005 17
17 .010 8.400 6.112 5.185 4.669 4.336 4.102 3.927 3.791 3.682 3.593 3.312 3.162 3.068 3.003 2.869 .010 17
17 .025 6.042 4.619 4.011 3.665 3.438 3.277 3.156 3.061 2.985 2.922 2.723 2.616 2.548 2.502 2.405 .025 17
17 .050 4.451 3.592 3.197 2.965 2.810 2.699 2.614 2.548 2.494 2.450 2.308 2.230 2.181 2.148 2.077 .050 17
17 .100 3.026 2.645 2.437 2.308 2.218 2.152 2.102 2.061 2.028 2.001 1.912 1.862 1.831 1.809 1.763 .100 17
18 .005 10.22 7.215 6.028 5.375 4.956 4.663 4.445 4.276 4.141 4.030 3.683 3.498 3.382 3.303 3.139 .005 18
18 .010 8.285 6.013 5.092 4.579 4.248 4.015 3.841 3.705 3.597 3.508 3.227 3.077 2.983 2.919 2.784 .010 18
18 .025 5.978 4.5603.954 3.608 3.382 3.221 3.100 3.005 2.929 2.866 2.667 2.559 2.491 2.445 2.347 .025 18
18 .050 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.456 2.412 2.269 2.191 2.141 2.107 2.035 .050 18
18 .100 3.007 2.624 2.416 2.286 2.196 2.130 2.079 2.038 2.005 1.977 1.887 1.837 1.805 1.783 1.736 .100 18
19 .005 10.07 7.093 5.916 5.268 4.853 4.561 4.345 4.177 4.043 3.933 3.587 3.402 3.287 3.208 3.043 .005 19
19 .010 8.185 5.926 5.010 4.500 4.171 3.939 3.765 3.631 3.523 3.434 3.153 3.003 2.909 2.844 2.709 .010 19
19 .025 5.922 4.508 3.903 3.559 3.333 3.172 3.051 2.956 2.880 2.817 2.617 2.509 2.441 2.394 2.295 .025 19
19 .050 4.381 3.522 3.127 2.895 2.740 2.628 2.544 2.477 2.423 2.378 2.234 2.155 2.106 2.071 1.999 .050 19
19 .100 2.990 2.606 2.397 2.266 2.176 2.109 2.058 2.017 1.984 1.956 1.865 1.814 1.782 1.759 1.711 .100 19
20 .005 9.944 6.986 5.818 5.174 4.762 4.472 4.257 4.090 3.956 3.847 3.502 3.318 3.203 3.123 2.959 .005 20
20 .010 8.096 5.849 4.938 4.431 4.103 3.871 3.699 3.564 3.457 3.368 3.088 2.938 2.843 2.778 2.643 .010 20
20 .025 5.871 4.461 3.859 3.515 3.289 3.128 3.007 2.913 2.837 2.774 2.573 2.464 2.396 2.349 2.249 .025 20
20 .050 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348 2.203 2.124 2.074 2.039 1.966 .050 20
20 .100 2.975 2.589 2.380 2.249 2.158 2.091 2.040 1.999 1.965 1.937 1.845 1.794 1.761 1.738 1.690 .100 20
21 .005 9.830 6.891 5.730 5.091 4.681 4.393 4.179 4.013 3.880 3.771 3.427 3.243 3.128 3.049 2.884 .005 21
21 .010 8.017 5.780 4.874 4.369 4.042 3.812 3.640 3.506 3.398 3.310 3.030 2.880 2.785 2.720 2.584 .010 21
21 .025 5.827 4.420 3.819 3.475 3.250 3.090 2.969 2.874 2.798 2.735 2.534 2.425 2.356 2.308 2.208 .025 21
21 .050 4.325 3.467 3.072 2.840 2.685 2.573 2.488 2.420 2.366 2.321 2.176 2.096 2.045 2.010 1.936 .050 21
21 .100 2.961 2.575 2.365 2.233 2.142 2.075 2.023 1.982 1.948 1.920 1.827 1.776 1.742 1.719 1.670 .100 21
22 .005 9.727 6.806 5.652 5.017 4.609 4.322 4.109 3.944 3.812 3.703 3.360 3.176 3.061 2.982 2.817 .005 22
22 .010 7.945 5.719 4.817 4.313 3.988 3.758 3.587 3.453 3.346 3.258 2.978 2.827 2.733 2.667 2.531 .010 22
22 .025 5.786 4.383 3.783 3.440 3.215 3.055 2.934 2.839 2.763 2.700 2.498 2.389 2.320 2.272 2.171 .025 22
22 .050 4.301 3.443 3.049 2.817 2.661 2.549 2.464 2.397 2.342 2.297 2.151 2.071 2.020 1.984 1.909 .050 22
22 .100 2.949 2.561 2.351 2.219 2.128 2.060 2.008 1.967 1.933 1.904 1.811 1.759 1.726 1.702 1.652 .100 22
23 .005 9.635 6.730 5.582 4.950 4.544 4.259 4.047 3.882 3.750 3.642 3.300 3.116 3.001 2.922 2.756 .005 23
23 .010 7.881 5.664 4.765 4.264 3.939 3.710 3.539 3.406 3.299 3.211 2.931 2.781 2.686 2.620 2.483 .010 23
23 .025 5.750 4.349 3.750 3.408 3.183 3.023 2.902 2.808 2.731 2.668 2.466 2.357 2.287 2.239 2.137 .025 23
23 .050 4.279 3.422 3.028 2.796 2.640 2.528 2.442 2.375 2.320 2.275 2.128 2.048 1.996 1.961 1.885 .050 23
23 .100 2.937 2.549 2.339 2.207 2.115 2.047 1.995 1.953 1.919 1.890 1.796 1.744 1.710 1.686 1.636 .100 23
Table A7. Upper one-sided percentiles of the F distributions (Continued …)
Numerator degrees of freedom ν1
ν2 p 1 2 3 4 5 6 7 8 9 10 15 20 25 30 50 p ν2
24 .005 9.551 6.661 5.519 4.890 4.486 4.202 3.991 3.826 3.695 3.587 3.246 3.062 2.947 2.868 2.702 .005 24
24 .010 7.823 5.614 4.718 4.218 3.895 3.667 3.496 3.363 3.256 3.168 2.889 2.738 2.643 2.577 2.440 .010 24
24 .025 5.717 4.319 3.721 3.379 3.155 2.995 2.874 2.779 2.703 2.640 2.437 2.327 2.257 2.209 2.107 .025 24
24 .050 4.260 3.403 3.009 2.776 2.621 2.508 2.423 2.355 2.300 2.255 2.108 2.027 1.975 1.939 1.863 .050 24
24 .100 2.927 2.538 2.327 2.195 2.103 2.035 1.983 1.941 1.906 1.877 1.783 1.730 1.696 1.672 1.621 .100 24
25 .005 9.475 6.598 5.462 4.835 4.433 4.150 3.939 3.776 3.645 3.537 3.196 3.013 2.898 2.819 2.652 .005 25
25 .010 7.770 5.568 4.675 4.177 3.855 3.627 3.457 3.324 3.217 3.129 2.850 2.699 2.604 2.538 2.400 .010 25
25 .025 5.686 4.291 3.694 3.353 3.129 2.969 2.848 2.753 2.677 2.613 2.411 2.300 2.230 2.182 2.079 .025 25
25 .050 4.242 3.385 2.991 2.759 2.603 2.490 2.405 2.337 2.282 2.236 2.089 2.007 1.955 1.919 1.842 .050 25
25 .100 2.918 2.528 2.317 2.184 2.092 2.024 1.971 1.929 1.895 1.866 1.771 1.718 1.683 1.659 1.607 .100 25
26 .005 9.406 6.541 5.409 4.785 4.384 4.103 3.893 3.730 3.599 3.492 3.151 2.968 2.853 2.774 2.607 .005 26
26 .010 7.721 5.526 4.637 4.140 3.818 3.591 3.421 3.288 3.182 3.094 2.815 2.664 2.569 2.503 2.364 .010 26
26 .025 5.659 4.265 3.670 3.329 3.105 2.945 2.824 2.729 2.653 2.590 2.387 2.276 2.205 2.157 2.053 .025 26
26 .050 4.225 3.369 2.975 2.743 2.587 2.474 2.388 2.321 2.265 2.220 2.072 1.990 1.938 1.901 1.823 .050 26
26 .100 2.909 2.519 2.307 2.174 2.082 2.014 1.961 1.919 1.884 1.855 1.760 1.706 1.671 1.647 1.594 .100 26
27 .005 9.342 6.489 5.361 4.740 4.340 4.059 3.850 3.687 3.557 3.450 3.110 2.928 2.812 2.733 2.565 .005 27
27 .010 7.677 5.488 4.601 4.106 3.785 3.558 3.388 3.256 3.149 3.062 2.783 2.632 2.536 2.470 2.330 .010 27
27 .025 5.633 4.242 3.647 3.307 3.083 2.923 2.802 2.707 2.631 2.568 2.364 2.253 2.183 2.133 2.029 .025 27
27 .050 4.210 3.354 2.960 2.728 2.572 2.459 2.373 2.305 2.250 2.204 2.056 1.974 1.921 1.884 1.806 .050 27
27 .100 2.901 2.511 2.299 2.165 2.073 2.005 1.952 1.909 1.874 1.845 1.749 1.695 1.660 1.636 1.583 .100 27
28 .005 9.284 6.440 5.317 4.698 4.300 4.020 3.811 3.649 3.519 3.412 3.073 2.890 2.775 2.695 2.527 .005 28
28 .010 7.636 5.453 4.568 4.074 3.754 3.528 3.358 3.226 3.120 3.032 2.753 2.602 2.506 2.440 2.300 .010 28
28 .025 5.610 4.221 3.626 3.286 3.063 2.903 2.782 2.687 2.611 2.547 2.344 2.232 2.161 2.112 2.007 .025 28
28 .050 4.196 3.340 2.947 2.714 2.558 2.445 2.359 2.291 2.236 2.190 2.041 1.959 1.906 1.869 1.790 .050 28
28 .100 2.894 2.503 2.291 2.157 2.064 1.996 1.943 1.900 1.865 1.836 1.740 1.685 1.650 1.625 1.572 .100 28
29 .005 9.230 6.396 5.276 4.659 4.262 3.983 3.775 3.613 3.483 3.377 3.038 2.855 2.740 2.660 2.492 .005 29
29 .010 7.598 5.420 4.538 4.045 3.725 3.499 3.330 3.198 3.092 3.005 2.726 2.574 2.478 2.412 2.271 .010 29
29 .025 5.588 4.201 3.607 3.267 3.044 2.884 2.763 2.669 2.592 2.529 2.325 2.213 2.142 2.092 1.987 .025 29
29 .050 4.183 3.328 2.934 2.701 2.545 2.432 2.346 2.278 2.223 2.177 2.027 1.945 1.891 1.854 1.775 .050 29
29 .100 2.887 2.495 2.283 2.149 2.057 1.988 1.935 1.892 1.857 1.827 1.731 1.676 1.640 1.616 1.562 .100 29
30 .005 9.180 6.355 5.239 4.623 4.228 3.949 3.742 3.580 3.450 3.344 3.006 2.823 2.708 2.628 2.459 .005 30
30 .010 7.562 5.390 4.510 4.018 3.699 3.473 3.304 3.173 3.067 2.979 2.700 2.549 2.453 2.386 2.245 .010 30
30 .025 5.568 4.182 3.589 3.250 3.026 2.867 2.746 2.651 2.575 2.511 2.307 2.195 2.124 2.074 1.968 .025 30
30 .050 4.171 3.316 2.922 2.690 2.534 2.421 2.334 2.266 2.211 2.165 2.015 1.932 1.878 1.841 1.761 .050 30
30 .100 2.881 2.489 2.276 2.142 2.049 1.980 1.927 1.884 1.849 1.819 1.722 1.667 1.632 1.606 1.552 .100 30
35 .005 8.976 6.188 5.086 4.479 4.088 3.812 3.607 3.447 3.318 3.212 2.876 2.693 2.577 2.497 2.327 .005 35
35 .010 7.419 5.268 4.396 3.908 3.592 3.368 3.200 3.069 2.963 2.876 2.597 2.445 2.348 2.281 2.137 .010 35
35 .025 5.485 4.106 3.517 3.179 2.956 2.796 2.676 2.581 2.504 2.440 2.235 2.122 2.049 1.999 1.890 .025 35
35 .050 4.121 3.267 2.874 2.641 2.485 2.372 2.285 2.217 2.161 2.114 1.963 1.878 1.824 1.786 1.703 .050 35
35 .100 2.855 2.461 2.247 2.113 2.019 1.950 1.896 1.852 1.817 1.787 1.688 1.632 1.595 1.569 1.513 .100 35
Table A7. Upper one-sided percentiles of the F distributions (Continued …)
Numerator degrees of freedom ν1
ν2 p 1 2 3 4 5 6 7 8 9 10 15 20 25 30 50 p ν2
40 .005 8.828 6.066 4.976 4.374 3.986 3.713 3.509 3.350 3.222 3.117 2.781 2.598 2.482 2.401 2.230 .005 40
40 .010 7.314 5.179 4.313 3.828 3.514 3.291 3.124 2.993 2.888 2.801 2.522 2.369 2.271 2.203 2.058 .010 40
40 .025 5.424 4.051 3.463 3.126 2.904 2.744 2.624 2.529 2.452 2.388 2.182 2.068 1.994 1.943 1.832 .025 40
40 .050 4.085 3.232 2.839 2.606 2.449 2.336 2.249 2.180 2.124 2.077 1.924 1.839 1.783 1.744 1.660 .050 40
40 .100 2.835 2.440 2.226 2.091 1.9971.927 1.873 1.829 1.793 1.763 1.662 1.605 1.568 1.541 1.483 .100 40
50 .005 8.626 5.902 4.826 4.232 3.849 3.579 3.376 3.219 3.092 2.988 2.653 2.470 2.353 2.272 2.097 .005 50
50 .010 7.171 5.057 4.199 3.720 3.408 3.186 3.020 2.890 2.785 2.698 2.419 2.265 2.167 2.098 1.949 .010 50
50 .025 5.340 3.975 3.390 3.054 2.833 2.674 2.553 2.458 2.381 2.317 2.109 1.993 1.919 1.866 1.752 .025 50
50 .050 4.034 3.183 2.790 2.557 2.400 2.286 2.199 2.130 2.073 2.026 1.871 1.784 1.727 1.687 1.599 .050 50
50 .100 2.809 2.412 2.197 2.061 1.966 1.895 1.840 1.796 1.760 1.729 1.627 1.568 1.529 1.502 1.441 .100 50
60 .005 8.495 5.795 4.729 4.140 3.760 3.492 3.291 3.134 3.008 2.904 2.570 2.387 2.270 2.187 2.010 .005 60
60 .010 7.077 4.977 4.126 3.649 3.339 3.119 2.953 2.823 2.718 2.632 2.352 2.198 2.098 2.028 1.877 .010 60
60 .025 5.286 3.925 3.343 3.008 2.786 2.627 2.507 2.412 2.334 2.270 2.061 1.944 1.869 1.815 1.699 .025 60
60 .050 4.001 3.150 2.758 2.525 2.368 2.254 2.167 2.097 2.040 1.993 1.836 1.748 1.690 1.649 1.559 .050 60
60 .100 2.791 2.393 2.177 2.041 1.946 1.875 1.819 1.775 1.738 1.707 1.603 1.543 1.504 1.476 1.413 .100 60
70 .005 8.403 5.720 4.661 4.076 3.698 3.431 3.232 3.076 2.950 2.846 2.513 2.329 2.211 2.128 1.949 .005 70
70 .010 7.011 4.922 4.074 3.600 3.291 3.071 2.906 2.777 2.672 2.585 2.306 2.150 2.050 1.980 1.826 .010 70
70 .025 5.247 3.890 3.309 2.975 2.754 2.595 2.474 2.379 2.302 2.237 2.028 1.910 1.833 1.779 1.660 .025 70
70 .050 3.978 3.128 2.736 2.503 2.346 2.231 2.143 2.074 2.017 1.969 1.812 1.722 1.664 1.622 1.530 .050 70
70 .100 2.779 2.380 2.164 2.027 1.931 1.860 1.804 1.760 1.723 1.691 1.587 1.526 1.486 1.457 1.392 .100 70
80 .005 8.335 5.665 4.611 4.029 3.652 3.387 3.188 3.032 2.907 2.803 2.470 2.286 2.168 2.084 1.903 .005 80
80 .010 6.963 4.881 4.036 3.563 3.255 3.036 2.871 2.742 2.637 2.551 2.271 2.115 2.015 1.944 1.788 .010 80
80 .025 5.218 3.864 3.284 2.950 2.730 2.571 2.450 2.355 2.277 2.213 2.003 1.884 1.807 1.752 1.632 .025 80
80 .050 3.960 3.111 2.719 2.486 2.329 2.214 2.126 2.056 1.999 1.951 1.793 1.703 1.644 1.602 1.508 .050 80
80 .100 2.769 2.370 2.154 2.016 1.921 1.849 1.793 1.748 1.711 1.680 1.574 1.513 1.472 1.443 1.377 .100 80
90 .005 8.282 5.623 4.573 3.992 3.617 3.352 3.154 2.999 2.873 2.770 2.437 2.253 2.134 2.051 1.868 .005 90
90 .010 6.925 4.849 4.007 3.535 3.228 3.009 2.845 2.715 2.611 2.524 2.244 2.088 1.987 1.916 1.759 .010 90
90 .025 5.196 3.844 3.265 2.932 2.711 2.552 2.432 2.336 2.259 2.194 1.983 1.864 1.787 1.731 1.610 .025 90
90 .050 3.947 3.098 2.706 2.473 2.316 2.201 2.113 2.043 1.986 1.938 1.779 1.688 1.629 1.586 1.491 .050 90
90 .100 2.762 2.363 2.146 2.008 1.912 1.841 1.785 1.739 1.702 1.670 1.564 1.503 1.461 1.432 1.365 .100 90
100 .005 8.241 5.589 4.542 3.963 3.589 3.325 3.127 2.972 2.847 2.744 2.411 2.227 2.108 2.024 1.840 .005 100
100 .010 6.895 4.824 3.984 3.513 3.206 2.988 2.823 2.694 2.590 2.503 2.223 2.067 1.965 1.893 1.735 .010 100
100 .025 5.179 3.828 3.250 2.917 2.696 2.537 2.417 2.321 2.244 2.179 1.968 1.849 1.770 1.715 1.592 .025 100
100 .050 3.936 3.087 2.696 2.463 2.305 2.191 2.103 2.032 1.975 1.927 1.768 1.676 1.616 1.573 1.477 .050 100
100 .100 2.756 2.356 2.139 2.002 1.906 1.834 1.778 1.732 1.695 1.663 1.557 1.494 1.453 1.423 1.355 .100 100
Table A8. Critical values For Wilcoxon’s signed-rank test
The body of the table contains Aa,n, the critical values for Wilcoxon’s signed-rank test.
Always enter the table with T+, the sum of the ranks of the positive deviations.
If a critical value is missing, the hypothesis can not be rejected for this combination of n and α.
For n >50 use a standard Gaussian approximation.
Tail Probability Tail Probability
n .05 .025 .01 .005 n .05 .025 .01 .005
5 1 28 130 117 102 92
6 2 1 29 141 127 111 100
7 4 2 0 30 152 137 120 109
8 6 4 2 0 31 163 148 130 118
9 8 6 3 2 32 175 159 141 128
10 11 8 5 3 33 188 171 151 138
11 14 11 7 5 34 201 183 162 149
12 17 14 10 7 35 214 195 174 160
13 21 17 13 10 36 228 208 186 171
14 26 21 16 13 37 242 222 198 183
15 30 25 20 16 38 256 235 211 195
16 36 30 24 19 39 271 250 224 208
17 41 35 28 23 40 287 264 238 221
18 47 40 33 28 41 303 279 252 234
19 54 46 38 32 42 319 295 267 248
20 60 52 43 37 43 336 311 281 262
21 68 59 49 43 44 353 327 297 277
22 75 66 56 49 45 371 344 313 292
23 83 73 62 55 46 389 361 329 307
24 92 81 69 61 47 408 379 345 323
25 101 90 77 68 48 427 397 362 339
26 110 98 85 76 49 446 415 380 356
27 120 107 93 84 50 466 434 398 373
Table A9. Critical values for Duncan’s multiple range test
The body of the table represents D
�,(r,v) the critical value for Duncan’s multiple range test for significance level α,
v degrees of freedom and range r.
Example: The D value to compare two means on the 4-range at α = 0.05 with 10 degrees of freedom is 3.37
Range of sample means
v α 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 α v
1 .100 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 8.929 .100 1
1 .050 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 17.97 .050 1
1 .010 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 89.98 .010 1
2 .100 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 4.129 .100 2
2 .050 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 6.085 .050 2
2 .010 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 14.03 .010 2
3 .100 3.328 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 3.330 .100 3
3 .050 4.501 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 4.516 .050 3
3 .010 8.260 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 8.321 .010 3
4 .100 3.015 3.074 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 3.081 .100 4
4 .050 3.927 4.012 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 4.033 .050 4
4 .010 6.511 6.677 6.740 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 6.756 .010 4
5 .100 2.850 2.934 2.964 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 2.969 .100 5
5 .050 3.635 3.749 3.796 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 3.814 .050 5
5 .010 5.702 5.893 5.989 6.040 6.065 6.074 6.074 6.074 6.074 6.074 6.074 6.074 6.074 6.074 6.074 6.074 6.074 6.074 6.074 .010 5
6 .100 2.748 2.846 2.890 2.907 2.911 2.911 2.911 2.911 2.911 2.911 2.911 2.911 2.911 2.911 2.911 2.911 2.911 2.911 2.911 .100 6
6 .050 3.460 3.586 3.649 3.680 3.694 3.697 3.697 3.697 3.697 3.697 3.697 3.697 3.697 3.697 3.697 3.697 3.697 3.697 3.697 .050 6
6 .010 5.243 5.439 5.549 5.615 5.655 5.680 5.694 5.701 5.703 5.703 5.703 5.703 5.703 5.703 5.703 5.703 5.703 5.703 5.703 .010 6
7 .100 2.679 2.785 2.838 2.864 2.876 2.878 2.878 2.878 2.878 2.878 2.878 2.878 2.878 2.878 2.878 2.878 2.878 2.878 2.878 .100 7
7 .050 3.344 3.477 3.548 3.588 3.611 3.622 3.625 3.625 3.625 3.625 3.625 3.625 3.625 3.625 3.625 3.625 3.625 3.625 3.625 .050 7
7 .010 4.948 5.145 5.260 5.333 5.383 5.416 5.439 5.454 5.464 5.470 5.472 5.472 5.472 5.472 5.472 5.472 5.472 5.472 5.472 .010 7
8 .100 2.630 2.741 2.800 2.832 2.849 2.857 2.858 2.858 2.858 2.858 2.858 2.858 2.858 2.858 2.858 2.858 2.858 2.858 2.858 .100 8
8 .050 3.261 3.398 3.475 3.521 3.549 3.566 3.575 3.579 3.579 3.579 3.579 3.579 3.579 3.579 3.579 3.579 3.579 3.579 3.579 .050 8
8 .010 4.745 4.939 5.056 5.134 5.189 5.227 5.256 5.276 5.291 5.302 5.309 5.313 5.316 5.317 5.317 5.317 5.317 5.317 5.317 .010 8
9 .1002.592 2.708 2.771 2.808 2.829 2.840 2.845 2.846 2.846 2.846 2.846 2.846 2.846 2.846 2.846 2.846 2.846 2.846 2.846 .100 9
9 .050 3.199 3.339 3.420 3.470 3.502 3.523 3.536 3.544 3.547 3.547 3.547 3.547 3.547 3.547 3.547 3.547 3.547 3.547 3.547 .050 9
9 .010 4.595 4.787 4.906 4.986 5.043 5.086 5.117 5.142 5.160 5.174 5.185 5.193 5.199 5.202 5.205 5.206 5.206 5.206 5.206 .010 9
10 .100 2.563 2.682 2.748 2.788 2.813 2.827 2.835 2.839 2.839 2.839 2.839 2.839 2.839 2.839 2.839 2.839 2.839 2.839 2.839 .100 10
10 .050 3.151 3.293 3.376 3.430 3.465 3.489 3.505 3.516 3.522 3.525 3.525 3.525 3.525 3.525 3.525 3.525 3.525 3.525 3.525 .050 10
10 .010 4.482 4.671 4.789 4.871 4.931 4.975 5.010 5.036 5.058 5.074 5.087 5.098 5.106 5.112 5.117 5.120 5.122 5.123 5.124 .010 10
11 .100 2.540 2.660 2.729 2.772 2.799 2.817 2.827 2.833 2.835 2.835 2.835 2.835 2.835 2.835 2.835 2.835 2.835 2.835 2.835 .100 11
11 .050 3.113 3.256 3.341 3.397 3.435 3.462 3.480 3.493 3.501 3.506 3.509 3.510 3.510 3.510 3.510 3.510 3.510 3.510 3.510 .050 11
11 .010 4.392 4.579 4.697 4.780 4.841 4.887 4.923 4.952 4.975 4.994 5.009 5.021 5.031 5.039 5.045 5.050 5.054 5.057 5.059 .010 11
Calculated with the ProbMc function of The SAS System, Vs. 6.12.
Table A9. Critical values for Duncan’s multiple range test (Continued …)
Range of sample means
v α 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 α v
12 .100 2.521 2.643 2.714 2.759 2.788 2.808 2.820 2.828 2.832 2.833 2.833 2.833 2.833 2.833 2.833 2.833 2.833 2.833 2.833 .100 12
12 .050 3.081 3.225 3.312 3.370 3.410 3.439 3.459 3.474 3.484 3.491 3.495 3.498 3.498 3.498 3.498 3.498 3.498 3.498 3.498 .050 12
12 .010 4.320 4.504 4.622 4.705 4.765 4.812 4.850 4.882 4.907 4.927 4.944 4.957 4.969 4.978 4.986 4.993 4.998 5.002 5.005 .010 12
13 .100 2.504 2.628 2.701 2.748 2.779 2.800 2.814 2.824 2.829 2.832 2.832 2.832 2.832 2.832 2.832 2.832 2.832 2.832 2.832 .100 13
13 .050 3.055 3.200 3.288 3.348 3.389 3.419 3.441 3.458 3.470 3.478 3.484 3.488 3.490 3.490 3.490 3.490 3.490 3.490 3.490 .050 13
13 .010 4.261 4.442 4.560 4.643 4.706 4.754 4.793 4.824 4.850 4.871 4.889 4.904 4.917 4.927 4.936 4.944 4.950 4.955 4.960 .010 13
14 .100 2.491 2.616 2.689 2.739 2.771 2.794 2.810 2.820 2.827 2.831 2.833 2.833 2.833 2.833 2.833 2.833 2.833 2.833 2.833 .100 14
14 .050 3.033 3.178 3.268 3.328 3.371 3.402 3.426 3.444 3.457 3.467 3.474 3.479 3.482 3.484 3.484 3.484 3.484 3.484 3.484 .050 14
14 .010 4.210 4.390 4.508 4.591 4.654 4.703 4.743 4.775 4.802 4.824 4.843 4.859 4.872 4.884 4.894 4.902 4.909 4.916 4.921 .010 14
15 .100 2.479 2.605 2.681 2.730 2.764 2.788 2.805 2.817 2.825 2.830 2.833 2.834 2.834 2.834 2.834 2.834 2.834 2.834 2.834 .100 15
15 .050 3.014 3.160 3.250 3.312 3.356 3.389 3.413 3.432 3.446 3.457 3.465 3.471 3.476 3.478 3.480 3.480 3.480 3.480 3.480 .050 15
15 .010 4.167 4.346 4.463 4.547 4.610 4.660 4.700 4.733 4.760 4.783 4.803 4.820 4.834 4.846 4.857 4.866 4.874 4.881 4.887 .010 15
16 .100 2.469 2.596 2.672 2.723 2.759 2.784 2.802 2.814 2.824 2.830 2.833 2.835 2.836 2.836 2.836 2.836 2.836 2.836 2.836 .100 16
16 .050 2.998 3.144 3.235 3.297 3.343 3.376 3.402 3.422 3.437 3.449 3.458 3.465 3.470 3.473 3.476 3.477 3.477 3.477 3.477 .050 16
16 .010 4.131 4.308 4.425 4.508 4.572 4.622 4.662 4.696 4.724 4.748 4.768 4.785 4.800 4.813 4.825 4.835 4.843 4.851 4.858 .010 16
17 .100 2.460 2.587 2.665 2.717 2.753 2.779 2.798 2.812 2.822 2.829 2.834 2.837 2.838 2.838 2.838 2.838 2.838 2.838 2.838 .100 17
17 .050 2.984 3.130 3.222 3.285 3.331 3.365 3.392 3.412 3.429 3.441 3.451 3.459 3.465 3.469 3.472 3.474 3.475 3.475 3.475 .050 17
17 .010 4.099 4.275 4.391 4.474 4.538 4.589 4.630 4.664 4.692 4.717 4.737 4.755 4.771 4.785 4.797 4.807 4.816 4.824 4.832 .010 17
18 .100 2.452 2.580 2.659 2.711 2.749 2.776 2.795 2.810 2.821 2.829 2.834 2.838 2.840 2.840 2.840 2.840 2.840 2.840 2.840 .100 18
18 .050 2.971 3.117 3.210 3.274 3.320 3.356 3.383 3.404 3.421 3.435 3.445 3.454 3.460 3.465 3.469 3.472 3.473 3.474 3.474 .050 18
18 .010 4.071 4.246 4.361 4.445 4.509 4.559 4.601 4.635 4.664 4.689 4.710 4.729 4.745 4.759 4.771 4.782 4.792 4.801 4.808 .010 18
19 .100 2.445 2.574 2.653 2.706 2.744 2.772 2.793 2.808 2.820 2.828 2.834 2.839 2.841 2.843 2.843 2.843 2.843 2.843 2.843 .100 19
19 .050 2.960 3.106 3.199 3.264 3.311 3.347 3.375 3.397 3.415 3.429 3.440 3.449 3.456 3.462 3.466 3.469 3.472 3.473 3.474 .050 19
19 .010 4.046 4.220 4.335 4.418 4.483 4.533 4.575 4.610 4.639 4.664 4.686 4.705 4.722 4.736 4.749 4.760 4.771 4.780 4.788 .010 19
20 .100 2.439 2.568 2.648 2.702 2.741 2.769 2.791 2.807 2.819 2.828 2.835 2.839 2.843 2.845 2.845 2.845 2.845 2.845 2.845 .100 20
20 .050 2.950 3.097 3.190 3.255 3.303 3.339 3.368 3.390 3.409 3.423 3.435 3.445 3.452 3.459 3.463 3.467 3.470 3.472 3.473 .050 20
20 .010 4.024 4.197 4.312 4.395 4.459 4.510 4.552 4.587 4.617 4.642 4.664 4.684 4.701 4.716 4.729 4.741 4.751 4.761 4.769 .010 20
25 .100 2.416 2.546 2.628 2.685 2.726 2.758 2.782 2.800 2.815 2.827 2.836 2.843 2.849 2.853 2.856 2.858 2.859 2.860 2.860 .100 25
25 .050 2.913 3.059 3.154 3.221 3.271 3.310 3.341 3.366 3.386 3.403 3.417 3.429 3.439 3.447 3.454 3.459 3.464 3.468 3.471 .050 25
25 .010 3.942 4.112 4.224 4.307 4.371 4.423 4.466 4.502 4.532 4.559 4.582 4.603 4.621 4.638 4.652 4.665 4.677 4.688 4.698 .010 25
30 .100 2.400 2.532 2.615 2.674 2.717 2.750 2.776 2.796 2.813 2.826 2.837 2.846 2.853 2.859 2.863 2.867 2.869 2.871 2.873 .100 30
30 .050 2.888 3.035 3.131 3.199 3.250 3.290 3.322 3.349 3.371 3.389 3.405 3.418 3.429 3.439 3.447 3.454 3.460 3.466 3.470 .050 30
30 .010 3.889 4.056 4.168 4.250 4.314 4.366 4.409 4.445 4.477 4.504 4.528 4.550 4.569 4.586 4.601 4.615 4.628 4.640 4.650 .010 30
50 .100 2.370 2.504 2.590 2.652 2.698 2.735 2.764 2.788 2.808 2.825 2.839 2.851 2.862 2.871 2.879 2.885 2.891 2.896 2.901 .100 50
50 .050 2.841 2.988 3.084 3.154 3.208 3.251 3.286 3.315 3.340 3.362 3.380 3.396 3.410 3.423 3.434 3.444 3.453 3.461 3.468 .050 50
50 .010 3.787 3.948 4.057 4.138 4.202 4.254 4.297 4.335 4.367 4.395 4.421 4.443 4.464 4.482 4.499 4.515 4.529 4.542 4.554 .010 50
60 .100 2.363 2.497 2.584 2.646 2.693 2.731 2.761 2.786 2.807 2.825 2.840 2.853 2.864 2.874 2.883 2.890 2.897 2.903 2.908 .100 60
60 .050 2.829 2.976 3.073 3.143 3.198 3.241 3.277 3.307 3.333 3.355 3.374 3.391 3.406 3.419 3.431 3.441 3.451 3.460 3.468 .050 60
60 .010 3.762 3.922 4.031 4.111 4.174 4.226 4.270 4.307 4.340 4.368 4.394 4.417 4.437 4.456 4.474 4.489 4.504 4.518 4.530 .010 60
100 .100 2.348 2.483 2.571 2.635 2.684 2.723 2.755 2.782 2.805 2.824 2.841 2.856 2.869 2.881 2.891 2.901 2.909 2.917 2.924 .100 100
100 .050 2.806 2.953 3.050 3.122 3.177 3.222 3.259 3.290 3.317 3.341 3.361 3.380 3.396 3.411 3.424 3.436 3.447 3.457 3.467 .050 100
100 .010 3.714 3.871 3.978 4.057 4.120 4.172 4.215 4.253 4.286 4.315 4.341 4.364 4.385 4.405 4.422 4.439 4.454 4.468 4.482 .010 100
smoikawa
Text Box
TableA10_Coeficientes_Polinômios_Ortogonais
Table A11. Upper percentiles of the studentized range distributions
The body of the table represents Qα,k,ϖ, the critical value of the studentized range distribution for right-tail probability α,
v degrees of freedom and range k.
Example: Q0.05,3,9 = 3.95
Range
v α 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 α v
1 .100 8.929 13.44 16.36 18.49 20.15 21.50 22.64 23.62 24.47 25.24 25.92 26.54 27.10 27.62 28.10 28.54 28.96 29.35 29.71 .100 1
1 .050 17.97 26.98 32.82 37.08 40.41 43.12 45.40 47.36 49.07 50.59 51.96 53.19 54.32 55.36 56.32 57.21 58.04 58.82 59.55 .050 1
1 .010 89.98 135.0 164.3 185.6 202.2 215.7 227.1 236.9 245.5 253.1 259.9 266.1 271.8 276.9 281.7 286.2 290.3 294.2 297.9 .010 1
2 .100 4.129 5.733 6.772 7.538 8.139 8.632 9.049 9.409 9.725 10.01 10.26 10.49 10.70 10.89 11.07 11.24 11.39 11.54 11.68 .100 2
2 .050 6.085 8.331 9.798 10.88 11.73 12.43 13.03 13.54 13.99 14.39 14.75 15.08 15.37 15.65 15.91 16.14 16.36 16.57 16.77 .050 2
2 .010 14.03 19.02 22.29 24.72 26.63 28.20 29.53 30.68 31.69 32.59 33.39 34.13 34.8035.42 35.99 36.53 37.03 37.50 37.94 .010 2
3 .100 3.328 4.467 5.199 5.738 6.162 6.511 6.806 7.062 7.287 7.487 7.667 7.831 7.982 8.120 8.248 8.367 8.479 8.584 8.683 .100 3
3 .050 4.501 5.910 6.825 7.502 8.037 8.478 8.852 9.177 9.462 9.717 9.946 10.15 10.35 10.52 10.69 10.84 10.98 11.11 11.24 .050 3
3 .010 8.260 10.62 12.17 13.32 14.24 15.00 15.64 16.20 16.69 17.13 17.52 17.88 18.21 18.52 18.80 19.06 19.31 19.54 19.76 .010 3
4 .100 3.015 3.976 4.586 5.035 5.388 5.679 5.926 6.139 6.327 6.494 6.645 6.783 6.909 7.024 7.132 7.233 7.326 7.414 7.497 .100 4
4 .050 3.927 5.040 5.757 6.287 6.706 7.053 7.347 7.602 7.826 8.027 8.208 8.373 8.524 8.664 8.793 8.914 9.027 9.133 9.233 .050 4
4 .010 6.511 8.118 9.173 9.958 10.58 11.10 11.54 11.93 12.26 12.57 12.84 13.09 13.32 13.53 13.73 13.91 14.08 14.24 14.39 .010 4
5 .100 2.850 3.717 4.264 4.664 4.979 5.238 5.458 5.648 5.816 5.965 6.100 6.223 6.336 6.439 6.536 6.626 6.710 6.788 6.863 .100 5
5 .050 3.635 4.602 5.218 5.673 6.033 6.330 6.582 6.801 6.995 7.167 7.324 7.465 7.596 7.716 7.828 7.932 8.030 8.122 8.208 .050 5
5 .010 5.702 6.976 7.806 8.421 8.913 9.321 9.669 9.971 10.24 10.48 10.70 10.89 11.07 11.24 11.40 11.54 11.68 11.81 11.93 .010 5
6 .100 2.748 3.558 4.065 4.435 4.726 4.966 5.168 5.344 5.499 5.637 5.762 5.875 5.979 6.075 6.164 6.247 6.325 6.398 6.466 .100 6
6 .050 3.460 4.339 4.896 5.305 5.629 5.895 6.122 6.319 6.493 6.649 6.789 6.917 7.034 7.143 7.244 7.338 7.426 7.509 7.587 .050 6
6 .010 5.243 6.331 7.033 7.556 7.974 8.318 8.611 8.869 9.097 9.300 9.485 9.653 9.808 9.951 10.08 10.21 10.32 10.43 10.54 .010 6
7 .100 2.679 3.451 3.931 4.280 4.555 4.780 4.971 5.137 5.283 5.413 5.530 5.637 5.735 5.826 5.910 5.988 6.061 6.130 6.195 .100 7
7 .050 3.344 4.165 4.681 5.060 5.359 5.606 5.815 5.997 6.158 6.302 6.431 6.550 6.658 6.759 6.852 6.939 7.020 7.097 7.169 .050 7
7 .010 4.948 5.919 6.543 7.006 7.373 7.678 7.940 8.167 8.368 8.548 8.711 8.859 8.996 9.124 9.242 9.353 9.456 9.553 9.645 .010 7
8 .100 2.630 3.374 3.834 4.169 4.431 4.646 4.829 4.987 5.126 5.250 5.362 5.464 5.558 5.644 5.724 5.799 5.869 5.935 5.997 .100 8
8 .050 3.261 4.041 4.529 4.886 5.167 5.399 5.596 5.767 5.918 6.053 6.175 6.287 6.389 6.483 6.571 6.653 6.729 6.801 6.870 .050 8
8 .010 4.745 5.635 6.204 6.625 6.960 7.238 7.475 7.681 7.864 8.028 8.177 8.312 8.437 8.552 8.659 8.760 8.854 8.942 9.026 .010 8
9 .100 2.592 3.316 3.761 4.084 4.337 4.545 4.720 4.873 5.007 5.126 5.234 5.333 5.423 5.506 5.583 5.655 5.722 5.786 5.845 .100 9
9 .050 3.199 3.948 4.415 4.755 5.024 5.244 5.432 5.595 5.738 5.867 5.983 6.089 6.186 6.276 6.359 6.437 6.510 6.579 6.644 .050 9
9 .010 4.595 5.428 5.957 6.347 6.658 6.915 7.134 7.326 7.495 7.647 7.785 7.910 8.026 8.133 8.233 8.326 8.413 8.495 8.573 .010 9
10 .100 2.563 3.270 3.704 4.018 4.264 4.465 4.636 4.783 4.913 5.029 5.134 5.229 5.316 5.397 5.472 5.542 5.607 5.668 5.726 .100 10
10 .050 3.151 3.877 4.327 4.654 4.912 5.124 5.304 5.460 5.598 5.722 5.833 5.935 6.028 6.114 6.194 6.269 6.339 6.405 6.467 .050 10
10 .010 4.482 5.270 5.769 6.136 6.428 6.669 6.875 7.055 7.214 7.356 7.485 7.603 7.712 7.813 7.906 7.994 8.076 8.153 8.226 .010 10
11 .100 2.540 3.234 3.658 3.965 4.205 4.401 4.567 4.711 4.838 4.951 5.053 5.145 5.230 5.309 5.382 5.450 5.513 5.573 5.630 .100 11
11 .050 3.113 3.820 4.256 4.574 4.823 5.028 5.202 5.353 5.486 5.605 5.713 5.811 5.901 5.984 6.062 6.134 6.202 6.265 6.325 .050 11
11 .010 4.392 5.146 5.621 5.970 6.247 6.476 6.671 6.842 6.992 7.127 7.250 7.362 7.465 7.560 7.649 7.732 7.810 7.883 7.952 .010 11
12 .100 2.521 3.204 3.621 3.921 4.156 4.349 4.510 4.651 4.776 4.886 4.986 5.076 5.160 5.236 5.308 5.374 5.436 5.495 5.550 .100 12
12 .050 3.081 3.773 4.199 4.508 4.748 4.947 5.116 5.262 5.395 5.510 5.615 5.710 5.797 5.878 5.953 6.023 6.089 6.151 6.209 .050 12
12 .010 4.320 5.046 5.502 5.836 6.101 6.321 6.507 6.670 6.814 6.943 7.060 7.167 7.265 7.356 7.441 7.520 7.594 7.664 7.731 .010 12
13 .100 2.504 3.179 3.589 3.885 4.116 4.304 4.464 4.602 4.724 4.832 4.930 5.019 5.100 5.175 5.245 5.310 5.371 5.429 5.483 .100 13
13 .050 3.055 3.734 4.151 4.453 4.690 4.884 5.049 5.192 5.318 5.431 5.533 5.625 5.711 5.789 5.862 5.930 5.994 6.055 6.112 .050 13
13 .010 4.261 4.964 5.404 5.727 5.981 6.192 6.372 6.528 6.666 6.791 6.903 7.006 7.100 7.188 7.269 7.345 7.417 7.484 7.548 .010 13
Calculated with the ProbMc function of The SAS System, Vs. 6.12.
FCT - Unesp
Realce
FCT - Unesp
Realce
FCT - Unesp
Realce
FCT - Unesp
Realce
smoikawa
Text Box
TableA11_Amplitude_Studentizado
Table A11. Upper percentiles of the studentized range distributions (Continued …)
Range
v α 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 α v
14 .100 2.491 3.158 3.563 3.854 4.081 4.267 4.424 4.560 4.679 4.786 4.882 4.969 5.050 5.124 5.192 5.256 5.316 5.372 5.426 .100 14
14 .050 3.033 3.701 4.111 4.407 4.639 4.829 4.990 5.130 5.253 5.363 5.463 5.554 5.637 5.714 5.785 5.852 5.915 5.973 6.029 .050 14
14 .010 4.210 4.895 5.322 5.634 5.881 6.085 6.258 6.409 6.543 6.664 6.772 6.871 6.962 7.047 7.125 7.199 7.268 7.333 7.394 .010 14
15 .100 2.479 3.140 3.540 3.828 4.052 4.235 4.390 4.524 4.641 4.746 4.841 4.927 5.006 5.079 5.146 5.209 5.268 5.324 5.376 .100 15
15 .050 3.014 3.673 4.076 4.367 4.595 4.782 4.940 5.077 5.198 5.306 5.403 5.492 5.574 5.649 5.719 5.785 5.846 5.904 5.958 .050 15
15 .010 4.167 4.836 5.252 5.556 5.796 5.994 6.162 6.309 6.438 6.555 6.660 6.757 6.845 6.927 7.003 7.074 7.141 7.204 7.264 .010 15
16 .100 2.469 3.124 3.520 3.804 4.026 4.207 4.360 4.492 4.608 4.712 4.805 4.890 4.968 5.040 5.106 5.169 5.227 5.282 5.333 .100 16
16 .050 2.998 3.649 4.046 4.333 4.557 4.741 4.896 5.031 5.150 5.256 5.352 5.439 5.519 5.593 5.662 5.726 5.786 5.843 5.896 .050 16
16 .010 4.131 4.786 5.192 5.488 5.722 5.915 6.079 6.222 6.348 6.461 6.564 6.658 6.743 6.824 6.897 6.967 7.032 7.093 7.151 .010 16
17 .100 2.460 3.110 3.503 3.784 4.003 4.183 4.334 4.464 4.579 4.681 4.774 4.857 4.934 5.005 5.071 5.133 5.190 5.244 5.295 .100 17
17 .050 2.984 3.628 4.020 4.303 4.524 4.705 4.858 4.991 5.108 5.212 5.306 5.392 5.471 5.544 5.612 5.675 5.734 5.790 5.842 .050 17
17 .010 4.099 4.742 5.140 5.430 5.659 5.847 6.007 6.147 6.270 6.380 6.480 6.572 6.656 6.733 6.806 6.873 6.937 6.997 7.053 .010 17
18 .100 2.452 3.098 3.487 3.766 3.984 4.161 4.310 4.440 4.553 4.654 4.746 4.829 4.905 4.975 5.040 5.101 5.158 5.211 5.262 .100 18
18 .050 2.971 3.609 3.997 4.276 4.494 4.673 4.824 4.955 5.071 5.173 5.266 5.351 5.429 5.501 5.567 5.629 5.688 5.743 5.794 .050 18
18 .010 4.071 4.703 5.094 5.379 5.603 5.787 5.944 6.081 6.201 6.309 6.407 6.496 6.579 6.655 6.725 6.791 6.854 6.912 6.967 .010 18
19 .100 2.445 3.087 3.474 3.751 3.966 4.142 4.290 4.418 4.530 4.630 4.721 4.803 4.878 4.948 5.012 5.072 5.129 5.182 5.232 .100 19
19 .050 2.960 3.593 3.977 4.253 4.468 4.645 4.794 4.924 5.037 5.139 5.231 5.314 5.391 5.462 5.528 5.589 5.647 5.701 5.752 .050 19
19 .010 4.046 4.669 5.054 5.333 5.553 5.735 5.888 6.022 6.141 6.246 6.342 6.430 6.510 6.585 6.654 6.719 6.780 6.837 6.891 .010 19
20 .100 2.439 3.077 3.462 3.736 3.950 4.124 4.271 4.398 4.510 4.609 4.699 4.780 4.855 4.923 4.987 5.047 5.103 5.155 5.205 .100 20
20 .050 2.950 3.578 3.958 4.232 4.445 4.620 4.768 4.895 5.008 5.108 5.199 5.282 5.357 5.427 5.492 5.553 5.610 5.663 5.714 .050 20
20 .010 4.024 4.639 5.018 5.293 5.509 5.687 5.839 5.970 6.086 6.190 6.285 6.370 6.449 6.523 6.591 6.654 6.714 6.770 6.823 .010 20
25 .100 2.416 3.041 3.416 3.683 3.890 4.059 4.201 4.324 4.432 4.528 4.614 4.693 4.765 4.831 4.893 4.950 5.004 5.055 5.102 .100 25
25 .050 2.913 3.523 3.890 4.153 4.358 4.526 4.667 4.789 4.897 4.993 5.079 5.158 5.230 5.297 5.359 5.417 5.471 5.522 5.570 .050 25
25 .010 3.942 4.527 4.884 5.143 5.346 5.513 5.654 5.777 5.885 5.982 6.070 6.150 6.223 6.291 6.355 6.414 6.469 6.521 6.571 .010 25
30 .100 2.400 3.017 3.386 3.648 3.851 4.016 4.155 4.275 4.381 4.474 4.559 4.635 4.706 4.770 4.830 4.886 4.939 4.988 5.034 .100 30
30 .050 2.888 3.487 3.845 4.102 4.301 4.464 4.601 4.720 4.824 4.917 5.0015.077 5.147 5.211 5.271 5.327 5.379 5.429 5.475 .050 30
30 .010 3.889 4.454 4.799 5.048 5.242 5.401 5.536 5.653 5.756 5.848 5.932 6.008 6.078 6.142 6.202 6.258 6.311 6.360 6.407 .010 30
40 .100 2.381 2.988 3.348 3.605 3.803 3.963 4.099 4.215 4.317 4.408 4.490 4.564 4.632 4.694 4.752 4.806 4.857 4.904 4.949 .100 40
40 .050 2.858 3.442 3.791 4.039 4.232 4.388 4.521 4.634 4.735 4.824 4.904 4.977 5.044 5.106 5.163 5.216 5.266 5.313 5.358 .050 40
40 .010 3.825 4.367 4.695 4.931 5.114 5.265 5.392 5.502 5.599 5.685 5.764 5.835 5.900 5.961 6.017 6.069 6.118 6.165 6.208 .010 40
50 .100 2.370 2.970 3.326 3.579 3.774 3.932 4.065 4.179 4.279 4.368 4.448 4.521 4.588 4.649 4.706 4.759 4.808 4.855 4.898 .100 50
50 .050 2.841 3.416 3.758 4.002 4.190 4.344 4.473 4.584 4.681 4.768 4.847 4.918 4.983 5.043 5.098 5.150 5.199 5.245 5.288 .050 50
50 .010 3.787 4.316 4.634 4.863 5.040 5.185 5.308 5.414 5.507 5.590 5.665 5.734 5.796 5.854 5.908 5.958 6.005 6.050 6.092 .010 50
60 .100 2.363 2.959 3.312 3.562 3.755 3.911 4.042 4.155 4.254 4.342 4.421 4.493 4.558 4.619 4.675 4.727 4.775 4.821 4.864 .100 60
60 .050 2.829 3.399 3.737 3.977 4.163 4.314 4.441 4.550 4.646 4.732 4.808 4.878 4.942 5.001 5.056 5.107 5.154 5.199 5.241 .050 60
60 .010 3.762 4.282 4.594 4.818 4.991 5.133 5.253 5.356 5.447 5.528 5.601 5.667 5.728 5.784 5.837 5.886 5.931 5.974 6.015 .010 60
80 .100 2.353 2.945 3.294 3.541 3.731 3.885 4.014 4.125 4.223 4.309 4.387 4.457 4.522 4.581 4.636 4.687 4.735 4.780 4.822 .100 80
80 .050 2.814 3.377 3.711 3.947 4.129 4.278 4.402 4.509 4.603 4.686 4.761 4.829 4.892 4.949 5.003 5.052 5.099 5.142 5.183 .050 80
80 .010 3.732 4.241 4.545 4.763 4.931 5.069 5.185 5.284 5.372 5.451 5.521 5.585 5.644 5.698 5.749 5.796 5.840 5.881 5.920 .010 80
100 .100 2.348 2.936 3.283 3.528 3.717 3.870 3.998 4.108 4.204 4.289 4.366 4.436 4.500 4.558 4.612 4.663 4.710 4.755 4.796 .100 100
100 .050 2.806 3.365 3.695 3.929 4.109 4.256 4.379 4.484 4.577 4.659 4.733 4.800 4.862 4.918 4.971 5.020 5.066 5.108 5.149 .050 100
100 .010 3.714 4.216 4.516 4.730 4.896 5.031 5.144 5.242 5.328 5.405 5.474 5.537 5.594 5.648 5.697 5.743 5.786 5.826 5.864 .010 100
120 .100 2.344 2.930 3.276 3.520 3.707 3.859 3.987 4.096 4.191 4.276 4.353 4.422 4.485 4.543 4.597 4.647 4.694 4.738 4.779 .100 120
120 .050 2.800 3.356 3.685 3.917 4.096 4.241 4.363 4.468 4.560 4.641 4.714 4.781 4.842 4.898 4.950 4.998 5.043 5.086 5.126 .050 120
120 .010 3.702 4.200 4.497 4.709 4.872 5.005 5.118 5.214 5.299 5.375 5.443 5.505 5.561 5.614 5.662 5.708 5.750 5.790 5.827 .010 120
 
 
 
TABELA VII 
Valores Críticos da Distribuição da Estatística nD (Kolmogorov-Smirnov) 
 
Os valores tabelados correspondem aos pontos α,nD tais que: P( nD ≥ α,nD )=α. 
 
 αααα αααα 
n 0.20 0.10 0.05 0.02 0.01 n 0.20 0.10 0.05 0.02 0.01 
1 0.900 0.95 0.975 0.990 0.995 21 0.226 0.259 0.287 0.321 0.344 
2 0.684 0.776 0.842 0.900 0.929 22 0.221 0.253 0.281 0.314 0.337 
3 0.565 0.636 0.708 0.785 0.829 23 0.216 0.247 0.275 0.307 0.330 
4 0.493 0.565 0.624 0.689 0.734 24 0.212 0.242 0.269 0.301 0.323 
5 0.447 0.509 0.563 0.627 0.669 25 0.208 0.238 0.264 0.295 0.317 
6 0.410 0.468 0.519 0.577 0.617 26 0.204 0.233 0.259 0.290 0.311 
7 0.381 0.436 0.483 0.538 0.576 27 0.200 0.229 0.254 0.284 0.305 
8 0.358 0.410 0.454 0.407 0.542 28 0.197 0.225 0.250 0.279 0.300 
9 0.339 0.387 0.430 0.480 0.513 29 0.193 0.221 0.246 0.275 0.295 
10 0.323 0.369 0.409 0.457 0.489 30 0.190 0.218 0.242 0.270 0.290 
11 0.308 0.352 0.391 0.437 0.468 31 0.187 0.214 0.238 0.266 0.285 
12 0.296 0.338 0.375 0.419 0.449 32 0.184 0.211 0.234 0.262 0.181 
13 0.285 0.325 0.361 0.404 0.432 33 0.182 0.208 0.231 0.258 0.277 
14 0.275 0.314 0.349 0.390 0.418 34 0.179 0.205 0.227 0.254 0.273 
15 0.266 0.304 0.338 0.377 0.404 35 0.177 0.202 0.224 0.251 0.269 
16 0.258 0.295 0.327 0.366 0.392 36 0.174 0.199 0.221 0.247 0.265 
17 0.250 0.286 0.318 0.355 0.381 37 0.172 0.196 0.218 0.244 0.262 
18 0.244 0.279 0.309 0.346 0.371 38 0.170 0.194 0.215 0.241 0.258 
19 0.237 0.271 0.301 0.337 0.361 39 0.168 0.191 0.213 0.238 0.255 
20 0.232 0.265 0.294 0.329 0.352 40 0.165 0.189 0.210 0.235 0.252 
 
 
 
 
Para n>40 os valores críticos de nD podem ser aproximados pelas seguintes expressões: 
 
 
 αααα 
0.20 0.10 0.05 0.02 0.01 
n
07.1
 
n
22.1
 
n
36.1
 
n
52.1
 
n
63.1
 
 
 
 
 
STATISTICAL TABLES A-17
TABLE A.6
CRITICAL VALUES OF DUNNETTS TWO-TAILED TEST FOR COMPARING
TREATMENTS TO A CONTROL
Error
df
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
24
30
40
a
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
.05
.01
Number of Treatment Means, Including Control (a)
2
2.57
4.03
2.45
3.71
2.36
3.50
2.31
3.36
2.26
3.25
2.23
3.17
2.20
3.11
2.18
3.05
2.16
3.01
2.14
2.98
2.13
2.95
2.12
2.92
2.11
2.90
2.10
2.88
2.09
2.86
2.09
2.85
2.06
2.80
2.04
2.75
2.02
2.70
3
3.03
4.63
2.86
4.21
2.75
3.95
2.67
3.77
2.61
3.63
2.57
3.53
2.53
3.45
2.50
3.39
2.48
3.33
2.46
3.29
2.44
3.25
2.42
3.22
2.41
3.19
2.40
3.17
2.39
3.15
2.38
3.13
2.35
3.07
2.32
3.01
2.29
2.95
4
3.29
4.98
3.10
4.51
2.97
4.21
2.88
4.00
2.81
3.85
2.76
3.74
2.72
3.65
2.68
3.58
2.65
3.52
2.63
3.47
2.61
3.43
2.59
3.39
2.58
3.36
2.56
3.33
2.55
3.31
2.54
3.29
2.51
3.22
2.47
3.15
2.44
3.09
5
3.48
5.22
3.26
4.71
3.12
4.39
3.02
4.17
2.95
4.01
2.89
3.88
2.84
3.79
2.81
3.71
2.78
3.65
2.75
3.59
2.73
3.55
2.71
3.51
2.69
3.47
2.68
3.44
2.66
3.42
2.65
3.40
2.61
3.32
2.58
3.25
2.54
3.19
6
3.62
5.41
3.39
4.87
3.24
4.53
3.13
4.29
3.05
4.12
2.99
3.99
2.94
3.89
2.90
3.81
2.87
3.74
2.84
3.69
2.82
3.64
2.80
3.60
2.78
3.56
2.76
3.53
2.75
3.50
2.73
3.48
2.70
3.40
2.66
3.33
2.62
3.26
7
3.73
5.56
3.49
5.00
3.33
4.64
3.22
4.40
3.14
4.22
3.07
4.08
3.02
3.98
2.98
3.89
2.94
3.82
2.91
3.76
2.89
3.71
2.87
3.67
2.85
3.63
2.83
3.60
2.81
3.57
2.80
3.55
2.76
3.47
2.72
3.39
2.68
3.32
8
3.82
5.69
3.57
5.10
3.41
4.74
3.29
4.48
3.20
4.30
3.14
4.16
3.08
4.05
3.04
3.96
3.00
3.89
2.97
3.83
2.95
3.78
2.92
3.73
2.90
3.69
2.89
3.66
2.87
3.63
2.86
3.60
2.81
3.52
2.77
3.44
2.73
3.37
9
3.90
5.80
3.64
5.20
3.47
4.82
3.35
4.56
3.26
4.37
3.19
4.22
3.14
4.11
3.09
4.02
3.06
3.94
3.02
3.88
3.00
3.83
2.97
3.78
2.95
3.74
2.94
3.71
2.92
3.68
2.90
3.65
2.86
3.57
2.82
3.49
2.77
3.41
10
3.97
5.89
3.71
5.28
3.53
4.89
3.41
4.62
3.32
4.43
3.24
4.28
3.19
4.16
3.14
4.07
3.10
3.99
3.07
3.93
3.04
3.88
3.02
3.83
3.00
3.79
2.98
3.75
2.96
3.72
2.95
3.69
2.90
3.61
2.86
3.52
2.81
3.44
(continued)
TLFeBOOK
Tables 631
Table D.7: Critical values for the two-sided Bonferroni t statistic.
Table entries are tE,ν where Pν(t > tE,ν) = E and E = .05/2/K .
K
ν 2 3 4 5 6 7 8 9 10 15 20 30 50
1 25.5 38.2 50.9 63.7 76.4 89.1 102 115 127 191 255 382 637
2 6.21 7.65 8.86 9.92 10.9 11.8 12.6 13.4 14.1 17.3 20.0 24.5 31.6
3 4.18 4.86 5.39 5.84 6.23 6.58 6.90 7.18 7.45 8.58 9.46 10.9 12.9
4 3.50 3.96 4.31 4.60 4.85 5.07 5.26 5.44 5.60 6.25 6.76 7.53 8.61
5 3.16 3.53 3.81 4.03 4.22 4.38 4.53 4.66 4.77 5.25 5.60 6.14 6.87
6 2.97 3.29 3.52 3.71 3.86 4.00 4.12 4.22 4.32 4.70 4.98 5.40 5.96
7 2.84 3.13 3.34 3.50 3.64 3.75 3.86 3.95 4.03 4.36 4.59 4.94 5.41
8 2.75 3.02 3.21 3.36 3.48 3.58 3.68 3.76 3.83 4.12 4.33 4.64 5.04
9 2.69 2.93 3.11 3.25 3.36 3.46 3.55 3.62 3.69 3.95 4.15 4.42 4.78
10 2.63 2.87 3.04 3.17 3.28 3.37 3.45 3.52 3.583.83 4.00 4.26 4.59
11 2.59 2.82 2.98 3.11 3.21 3.29 3.37 3.44 3.50 3.73 3.89 4.13 4.44
12 2.56 2.78 2.93 3.05 3.15 3.24 3.31 3.37 3.43 3.65 3.81 4.03 4.32
13 2.53 2.75 2.90 3.01 3.11 3.19 3.26 3.32 3.37 3.58 3.73 3.95 4.22
14 2.51 2.72 2.86 2.98 3.07 3.15 3.21 3.27 3.33 3.53 3.67 3.88 4.14
15 2.49 2.69 2.84 2.95 3.04 3.11 3.18 3.23 3.29 3.48 3.62 3.82 4.07
16 2.47 2.67 2.81 2.92 3.01 3.08 3.15 3.20 3.25 3.44 3.58 3.77 4.01
17 2.46 2.65 2.79 2.90 2.98 3.06 3.12 3.17 3.22 3.41 3.54 3.73 3.97
18 2.45 2.64 2.77 2.88 2.96 3.03 3.09 3.15 3.20 3.38 3.51 3.69 3.92
19 2.43 2.63 2.76 2.86 2.94 3.01 3.07 3.13 3.17 3.35 3.48 3.66 3.88
20 2.42 2.61 2.74 2.85 2.93 3.00 3.06 3.11 3.15 3.33 3.46 3.63 3.85
21 2.41 2.60 2.73 2.83 2.91 2.98 3.04 3.09 3.14 3.31 3.43 3.60 3.82
22 2.41 2.59 2.72 2.82 2.90 2.97 3.02 3.07 3.12 3.29 3.41 3.58 3.79
23 2.40 2.58 2.71 2.81 2.89 2.95 3.01 3.06 3.10 3.27 3.39 3.56 3.77
24 2.39 2.57 2.70 2.80 2.88 2.94 3.00 3.05 3.09 3.26 3.38 3.54 3.75
25 2.38 2.57 2.69 2.79 2.86 2.93 2.99 3.03 3.08 3.24 3.36 3.52 3.73
26 2.38 2.56 2.68 2.78 2.86 2.92 2.98 3.02 3.07 3.23 3.35 3.51 3.71
27 2.37 2.55 2.68 2.77 2.85 2.91 2.97 3.01 3.06 3.22 3.33 3.49 3.69
28 2.37 2.55 2.67 2.76 2.84 2.90 2.96 3.00 3.05 3.21 3.32 3.48 3.67
29 2.36 2.54 2.66 2.76 2.83 2.89 2.95 3.00 3.04 3.20 3.31 3.47 3.66
30 2.36 2.54 2.66 2.75 2.82 2.89 2.94 2.99 3.03 3.19 3.30 3.45 3.65
35 2.34 2.51 2.63 2.72 2.80 2.86 2.91 2.96 3.00 3.15 3.26 3.41 3.59
40 2.33 2.50 2.62 2.70 2.78 2.84 2.89 2.93 2.97 3.12 3.23 3.37 3.55
45 2.32 2.49 2.60 2.69 2.76 2.82 2.87 2.91 2.95 3.10 3.20 3.35 3.52
50 2.31 2.48 2.59 2.68 2.75 2.81 2.85 2.90 2.94 3.08 3.18 3.32 3.50
100 2.28 2.43 2.54 2.63 2.69 2.75 2.79 2.83 2.87 3.01 3.10 3.23 3.39
∞ 2.24 2.39 2.50 2.58 2.64 2.69 2.73 2.77 2.81 2.94 3.02 3.14 3.29
 1
Table 17a - Coefficients {an-I+1} for the Shapiro-Wilk W test for normality for n=2(1)50 
-------------------------------------------------------------------------------------------------------------------- 
 n 
 -------------------------------------------------------------------------------------------------------------------- 
 i 2 3 4 5 6 7 8 9 10 
 1 .7070 .6971 .6872 .6646 .6431 .6233 .6052 .5888 .5739 
 2 - .0000 .1677 .2413 .2806 .3031 .3164 .3244 .3291 
 3 - .0000 .0875 .1401 .1743 .1976 .2141 
 4 - .0000 .0561 .0947 .1224 
 5 - .0000 .0399 
 ------------------------------------------------------------------------------------------------------------------- 
 n 
 ------------------------------------------------------------------------------------------------------------------- 
 i 11 12 13 14 15 16 17 18 19 20 
 1 .5601 .5475 .5359 .5251 .5150 .5056 .4968 .4886 .4808 .4734 
 2 .3315 .3325 .3325 .3318 .3306 .3290 .3273 .3253 .3232 .3211 
 3 .2260 .2347 .2412 .2460 .2495 .2521 .2540 .2553 .2561 .2565 
 4 .1429 .1586 .1707 .1802 .1878 .1939 .1988 .2027 .2059 .2085 
 5 .0695 0922 .1099 .1240 .1353 .1447 .1524 .1587 .1641 .1686 
 6 .0000 .0303 .0539 .0727 .0880 .1005 .1109 .1197 .1271 .1334 
 7 - - .0000 .0240 .0433 .0593 .0725 .0837 .0932 .1013 
 8 - - - - .0000 .0196 .0359 .0496 .0612 .0711 
 9 - - - - - .0000 .0163 .0303 .0422 
 10 - - - - .0000 .0140 
 ----------------------------------------------------------------------------------------------------------- 
 n 
 ----------------------------------------------------------------------------------------------------------- 
 i 21 22 23 14 25 26 27 28 29 30 
 1 .4643 .4590 .4542 .4493 .4450 .4407 .4366 .4328 .4291 .4254 
 2 .3185 .3156 .3126 .3098 .3069 .3043 .3018 .2992 .2968 .2944 
 3 .2578 .2571 .2563 .2554 .2543 .2533 .2522 .2510 .2499 .2487 
 4 .2119 .2131 .2139 .2145 .2148 .2151 .2152 .2151 .2150 .2148 
 5 .1736 .1764 .1787 1807 .1822 .1836 .1848 .1857 .1864 .1870 
 6 .1399 .1443 .1480 .1512 .1539 .1563 .1584 .1601 .1616 .1630 
 7 .1092 .1150 .1201 .1245 .1283 .1316 .1346 .1372 .1395 .1415 
 8 .0804 .0878 .0941 .0997 .1046 .1089 .1128 .1162 .1192 .1219 
 9 .0530 .0618 .0696 .0764 .0823 .0876 .0923 .0965 .1002 .1036 
 10 .0263 .0368 .0459 .0539 .0610 .0672 .0728 .0778 .0822 .0862 
 11 .0000 .0122 .0228 .0321 .0403 .0476 .0540 .0598 .0650 .0697 
 12 - - .0000 .0107 .0200 .0284 .0358 .0424 .0483 .0537 
 13 - - - - .0000 .0094 .0178 .0253 .0320 .0381 
 14 - - - - - - .0000 .0084 .0159 .0227 
 15 .0000 .0076 
 ----------------------------------------------------------------------------------------------------------- 
 2
Table 17.b - Percentage points of the W test* for n = 3(1)50 
 ------------------------------------------------------------------------------------------------------- 
 Level 
 ------------------------------------------------------------------------------------------------------- 
n 0.01 0.02 0.05 0.10 0.50 0.90 0.95 0.98 0.99 
 ------------------------------------------------------------------------------------------------------- 
 03 .753 .756 .767 .789 .959 .998 .999 1.00 1.00 
 04 .687 .707 .748 .792 .935 .987 .992 .996 .997 
 05 .686 .715 .762 .806 .927 .979 .986 .991 .993 
 06 .713 .743 .788 .826 .927 .974 .981 .986 .989 
 07 .730 .760 .803 .838 .928 .972 .979 .985 .988 
 08 .749 .778 .818 .851 .932 972 .978 .984 .987 
 09 .764 .791 .829 .859 .935 .972 .978 .984 .986 
 10 .781 .806 .842 .869 .938 972 .978 .983 .986 
 11 .792 .817 .850 .876 .940 .973 .979 .984 .986 
 12 .805 .828 .859 .883 .943 .973 .979 .984 .986 
 13 .814 .837 .866 .889 .945 .974 .979 .984 .986 
 14 .825 .846 .874 .895 .947 .975 .980 .984 .986 
 15 .835 .855 .881 .901 950 .975 .980 .984 .987 
 16 .844 .863 .887 .906 .952 .976 .981 .985 .987 
 17 .851 .869 .892 .910 .954 .977 .981 .985 .987 
 18 .858 .874 .897 .914 .956 .978 .982 .986 .988 
 19 .863 .879 .901 .917 .957 .978 .982 .986 .988 
 20 .868 .884 .905 .920 .959 .979 .983 .986 .988 
 21 .873 .888 .908 .923 .960 .980 .983 .987 .989 
 22 .878 .892 .911 .926 .961 .980 .984 .987 .989 
 23 .881 .895 .914 .928 .962 .981 .984 .987 .989 
 24 .884 .898 .916 .930 .963 .981 .984 .987 .989 
 25 .888 .901 .918 .931 .964 .981 .985 .988 .989 
 26 .891 .904 .920 .933 .965 .982 .985 .988 .989 
 27 .894 .906 .923 .935 .965 .982 .985 .988 .990 
 28 .896 .908 .924 .936 .966 .982 .985 .988 .990 
 29 .898 .910 .926 .937 .966 .982 .985 .988 .990 
 30 .900 .912 .927 .939 .967 .983 .985 .988 .900 
 31 .902 .914 .929 .940 .967 .983 .986 .988 .990 
 32 .904 .915 .930 .941 .968 .983 .986 .988 .990 
 33 .906 .917 .931 .942 .968 .983 .986 .989 .990 
 34 .908 .919 .933 .943 .969 .983 .986 .989 .990 
 35 .910 .920 .934 .944 .969 .984 .986 .989 .990 
 36 .912 .922 .935 .945 .970 .984 .986 .989 .990 
 37 .914 .924 .936 .946 .970 .984 .987 .989 .990 
 38 .916 .925 .938 .947 .971 .984 .987 .989 .990 
 39 .917 .927 .939 .948 .971 .984 .987 .989 .991 
 40 .919 .928 .940 .949 .972 985 .987 .989 .991 
1
 
 
Append ix
A
Durbin-Watson 
Significance Tables 
The Durbin-Watson test statistic tests the null hypothesis that the residuals from an 
ordinary least-squares regression are not autocorrelated against the alternative that the 
residuals follow an AR1 process. The Durbin-Watson statistic ranges in value from 0 
to 4. A value near 2 indicates non-autocorrelation; a value toward 0 indicatespositive 
autocorrelation; a value toward 4 indicates negative autocorrelation. 
Because of the dependence of any computed Durbin-Watson value on the 
associated data matrix, exact critical values of the Durbin-Watson statistic are not 
tabulated for all possible cases. Instead, Durbin and Watson established upper and 
lower bounds for the critical values. Typically, tabulated bounds are used to test the 
hypothesis of zero autocorrelation against the alternative of positive first-order 
autocorrelation, since positive autocorrelation is seen much more frequently in 
practice than negative autocorrelation. To use the table, you must cross-reference the 
sample size against the number of regressors, excluding the constant from the count 
of the number of regressors. 
The conventional Durbin-Watson tables are not applicable when you do not have 
a constant term in the regression. Instead, you must refer to an appropriate set of 
Durbin-Watson tables. The conventional Durbin-Watson tables are also not 
applicable when a lagged dependent variable appears among the regressors. Durbin 
has proposed alternative test procedures for this case. 
Statisticians have compiled Durbin-Watson tables from some special cases, 
including: 
„ Regressions with a full set of quarterly seasonal dummies. 
„ Regressions with an intercept and a linear trend variable (CURVEFIT 
MODEL=LINEAR). 
„ Regressions with a full set of quarterly seasonal dummies and a linear trend 
variable. 
2
Appendix A
In addition to obtaining the Durbin-Watson statistic for residuals from REGRESSION, 
you should also plot the ACF and PACF of the residuals series. The plots might suggest 
either that the residuals are random, or that they follow some ARMA process. If the 
residuals resemble an AR1 process, you can estimate an appropriate regression using 
the AREG procedure. If the residuals follow any ARMA process, you can estimate an 
appropriate regression using the ARIMA procedure. 
In this appendix, we have reproduced two sets of tables. Savin and White (1977) 
present tables for sample sizes ranging from 6 to 200 and for 1 to 20 regressors for 
models in which an intercept is included. Farebrother (1980) presents tables for sample 
sizes ranging from 2 to 200 and for 0 to 21 regressors for models in which an intercept 
is not included. 
Let’s consider an example of how to use the tables. In Chapter 9, we look at the 
classic Durbin and Watson data set concerning consumption of spirits. The sample size 
is 69, there are 2 regressors, and there is an intercept term in the model. The Durbin-
Watson test statistic value is 0.24878. We want to test the null hypothesis of zero 
autocorrelation in the residuals against the alternative that the residuals are positively 
autocorrelated at the 1% level of significance. If you examine the Savin and White 
tables (Table A.2 and Table A.3), you will not find a row for sample size 69, so go to 
the next lowest sample size with a tabulated row, namely N=65. Since there are two 
regressors, find the column labeled k=2. Cross-referencing the indicated row and 
column, you will find that the printed bounds are dL = 1.377 and dU = 1.500. If the 
observed value of the test statistic is less than the tabulated lower bound, then you 
should reject the null hypothesis of non-autocorrelated errors in favor of the hypothesis 
of positive first-order autocorrelation. Since 0.24878 is less than 1.377, we reject the 
null hypothesis. If the test statistic value were greater than dU, we would not reject the 
null hypothesis. 
A third outcome is also possible. If the test statistic value lies between dL and dU, 
the test is inconclusive. In this context, you might err on the side of conservatism and 
not reject the null hypothesis. 
For models with an intercept, if the observed test statistic value is greater than 2, 
then you want to test the null hypothesis against the alternative hypothesis of negative 
first-order autocorrelation. To do this, compute the quantity 4-d and compare this value 
with the tabulated values of dL and dU as if you were testing for positive 
autocorrelation. 
When the regression does not contain an intercept term, refer to Farebrother’s 
tabulated values of the “minimal bound,” denoted dM (Table A.4 and Table A.5), 
instead of Savin and White’s lower bound dL. In this instance, the upper bound is 
3
Durbin-Watson Signif icance Tables
the conventional bound dU found in the Savin and White tables. To test for negative 
first-order autocorrelation, use Table A.6 and Table A.7. 
To continue with our example, had we run a regression with no intercept term, we 
would cross-reference N equals 65 and k equals 2 in Farebrother’s table. The 
tabulated 1% minimal bound is 1.348. 
4
Appendix A
Table A-1
Models with an intercept (from Savin and White)
Durbin-Watson Statistic: 1 Per Cent Significance Points of dL and dU
k’*=1
*k’ is the number of regressors excluding the intercept
k’=2 k’=3 k’=4 k’=5 k’=6 k’=7 k’=8 k’=9 k’=10
n dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU dL dU
6 0.390 1.142 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----
7 0.435 1.036 0.294 1.676 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----
8 0.497 1.003 0.345 1.489 0.229 2.102 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----
9 0.554 0.998 0.408 1.389 0.279 1.875 0.183 2.433 ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----
10 0.604 1.001 0.466 1.333 0.340 1.733 0.230 2.193 0.150 2.690 ----- ----- ----- ----- ----- ----- ----- ----- ----- -----
11 0.653 1.010 0.519 1.297 0.396 1.640 0.286 2.030 0.193 2.453 0.124 2.892 ----- ----- ----- ----- ----- ----- ----- -----
12 0.697 1.023 0.569 1.274 0.449 1.575 0.339 1.913 0.244 2.280 0.164 2.665 0.105 3.053 ----- ----- ----- ----- ----- -----
13 0.738 1.038 0.616 1.261 0.499 1.526 0.391 1.826 0.294 2.150 0.211 2.490 0.140 2.838 0.090 3.182 ----- ----- ----- -----
14 0.776 1.054 0.660 1.254 0.547 1.490 0.441 1.757 0.343 2.049 0.257 2.354 0.183 2.667 0.122 2.981 0.078 3.287 ----- -----
15 0.811 1.070 0.700 1.252 0.591 1.465 0.487 1.705 0.390 1.967 0.303 2.244 0.226 2.530 0.161 2.817 0.107 3.101 0.068 3.374
16 0.844 1.086 0.738 1.253 0.633 1.447 0.532 1.664 0.437 1.901 0.349 2.153 0.269 2.416 0.200 2.681 0.142 2.944 0.094 3.201
17 0.873 1.102 0.773 1.255 0.672 1.432 0.574 1.631 0.481 1.847 0.393 2.078 0.313 2.319 0.241 2.566 0.179 2.811 0.127 3.053
18 0.902 1.118 0.805 1.259 0.708 1.422 0.614 1.604 0.522 1.803 0.435 2.015 0.355 2.238 0.282 2.467 0.216 2.697 0.160 2.925
19 0.928 1.133 0.835 1.264 0.742 1.416 0.650 1.583 0.561 1.767 0.476 1.963 0.396 2.169 0.322 2.381 0.255 2.597 0.196 2.813
20 0.952 1.147 0.862 1.270 0.774 1.410 0.684 1.567 0.598 1.736 0.515 1.918 0.436 2.110 0.362 2.308 0.294 2.510 0.232 2.174
21 0.975 1.161 0.889 1.276 0.803 1.408 0.718 1.554 0.634 1.712 0.552 1.881 0.474 2.059 0.400 2.244 0.331 2.434 0.268 2.625
22 0.997 1.174 0.915 1.284 0.832 1.407 0.748 1.543 0.666 1.691 0.587 1.849 0.510 2.015 0.437 2.188 0.368 2.367 0.304 2.548
23 1.017 1.186 0.938 1.290 0.858 1.407 0.777 1.535 0.699 1.674 0.620 1.821 0.545 1.977 0.473 2.140 0.404 2.308 0.340 2.479
24 1.037 1.199 0.959 1.298 0.881 1.407 0.805 1.527 0.728 1.659 0.652 1.797 0.578 1.944 0.507 2.097 0.439 2.255 0.375 2.417
25 1.055 1.210 0.981 1.305 0.906 1.408 0.832 1.521 0.756 1.645 0.682 1.776 0.610 1.915 0.540 2.059 0.473 2.209 0.409 2.362
26 1.072 1.222 1.000 1.311 0.928 1.410 0.855 1.517 0.782 1.635 0.711 1.759 0.640 1.889 0.572 2.026 0.505 2.168 0.441 2.313
27 1.088 1.232 1.019 1.318 0.948 1.413 0.878 1.514 0.808 1.625 0.738 1.743 0.669 1.867 0.602 1.997 0.536 2.131 0.473 2.269
28 1.104 1.244 1.036 1.325 0.969 1.414 0.901 1.512 0.832 1.618 0.764 1.729 0.696 1.847 0.630 1.970