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APPENDIX A
STATISTICAL TABLES

A.1  Area of the Unit Normal Distribution
A.2  Quantiles of the Unit Normal Distribution
A.3  Commonly Used Normal Quantiles
A.4  Quantiles of the t Distribution
A.5  Quantiles of the Chi–Square Distribution
A.6  90–Percentiles of the F(n, m) Distribution
A.7  95–Percentiles of the F(n, m) Distribution
A.8  99–Percentiles of the F(n, m) Distribution
A.9  Quantiles of the K–S Distribution
A.10  Approximation Formulas for Statistical Tables

A.1 AREA OF THE UNIT NORMAL DISTRIBUTION

Table A.1 lists area between 0 and z. For example, the area between z = 0 and z = 1.03 is 0.3485. Due to symmetry of the normal distribution, the area between z = 0 and z = –1.03 is also the same.

TABLE A.1 Area or the Unit Normal Distribution

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2703 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

A.2 QUANTITIES OF THE UNIT NORMAL DISTRIBUTION

Table A.2 lists zp for a given p. For example, for a two–sided confidence interval at 95%, α= 0.05 and p = 1 – α/2 = 0.975. The entry in the row labeled 0.97 and column labeled 0.005 gives zp = 1.960.

TABLE A.2 Quantiles of the Unit Normal Distribution

p 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.5 0.000 0.025 0.050 0.075 0.100 0.126 0.151 0.176 0.202 0.228
0.6 0.253 0.279 0.305 0.332 0.358 0.385 0.412 0.440 0.468 0.496
0.7 0.524 0.553 0.583 0.613 0.643 0.674 0.706 0.739 0.772 0.806
0.8 0.842 0.878 0.915 0.954 0.994 1.036 1.080 1.126 1.175 1.227


p 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

0.90 1.282 1.287 1.293 1.299 1.305 1.311 1.317 1.323 1.329 1.335
0.91 1.341 1.347 1.353 1.359 1.366 1.372 1.379 1.385 1.392 1.398
0.92 1.405 1.412 1.419 1.426 1.433 1.440 1.447 1.454 1.461 1.468
0.93 1.476 1.483 1.491 1.499 1.506 1.514 1.522 1.530 1.538 1.546
0.94 1.555 1.563 1.572 1.580 1.589 1.598 1.607 1.616 1.626 1.635
0.95 1.645 1.655 1.665 1.675 1.685 1.695 1.706 1.717 1.728 1.739
0.96 1.751 1.762 1.774 1.787 1.799 1.812 1.825 1.838 1.852 1.866
0.97 1.881 1.896 1.911 1.927 1.943 1.960 1.977 1.995 2.014 2.034
0.98 2.054 2.075 2.097 2.120 2.144 2.170 2.197 2.226 2.257 2.290


p 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009

0.990 2.326 2.330 2.334 2.338 2.342 2.346 2.349 2.353 2.357 2.362
0.991 2.366 2.370 2.374 2.378 2.382 2.387 2.391 2.395 2.400 2.404
0.992 2.409 2.414 2.418 2.423 2.428 2.432 2.437 2.442 2.447 2.452
0.993 2.457 2.462 2.468 2.473 2.478 2.484 2.489 2.495 2.501 2.506
0.994 2.512 2.518 2.524 2.530 2.536 2.543 2.549 2.556 2.562 2.569
0.995 2.576 2.583 2.590 2.597 2.605 2.612 2.620 2.628 2.636 2.644
0.996 2.652 2.661 2.669 2.678 2.687 2.697 2.706 2.716 2.727 2.737
0.997 2.748 2.759 2.770 2.782 2.794 2.807 2.820 2.834 2.848 2.863
0.998 2.878 2.894 2.911 2.929 2.948 2.968 2.989 3.011 3.036 3.062
0.999 3.090 3.121 3.156 3.195 3.239 3.291 3.353 3.432 3.540 3.719

See Table A.3 for commonly used values.

A.3 COMMONLY USED NORMAL QUANTILES

Table A.3 lists commonly used normal quantiles. The confidence levels listed in the first column are for a two–sided confidence intervals. For example, for a two–sided confidence interval at 99%, α= 0.01, α/2 = 0.005 and z0.995 = 2.576. For a one–sided confidence interval at 99%, α= 0.01, and z1–α = 2.326.

TABLE A.3 Commonly Used Normal Quantiles

Confidence
Level (%)
α α/2 z1–α/2

20 0.8 0.4 0.253
40 0.6 0.3 0.524
60 0.4 0.2 0.842
68.26 0.3174 0.1587 1.000
80 0.2 0.1 1.282
90 0.1 0.05 1.645
95 0.05 0.025 1.960
95.46 0.0454 0.0228 2.000
98 0.02 0.01 2.326
99 0.01 0.005 2.576
99.74 0.0026 0.0013 3.000
99.8 0.002 0.001 3.090
99.9 0.001 0.0005 3.29
99.98 0.0002 0.0001 3.72

A.4 QUANTILES OF THE t DISTRIBUTION

Table A.4 lists t[p;n]. For example, the t[0.95;13] required for a two–sided 90% confidence interval of the mean of a sample of 14 observation is 1.771.

TABLE A.4 Quantiles of the t Distribution

p

n 0.6000 0.7000 0.8000 0.9000 0.9500 0.9750 0.9950 0.9995

1 0.325 0.727 1.377 3.078 6.314 12.706 63.657 636.619
2 0.289 0.617 1.061 1.886 2.920 4.303 9.925 31.599
3 0.277 0.584 0.978 1.638 2.353 3.182 5.841 12.924
4 0.271 0.569 0.941 1.533 2.132 2.776 4.604 8.610
5 0.267 0.559 0.920 1.476 2.015 2.571 4.032 6.869
6 0.265 0.553 0.906 1.440 1.943 2.447 3.707 5.959
7 0.263 0.549 0.896 1.415 1.895 2.365 3.499 5.408
8 0.262 0.546 0.889 1.397 1.860 2.306 3.355 5.041
9 0.261 0.543 0.883 1.383 1.833 2.262 3.250 4.781
10 0.260 0.542 0.879 1.372 1.812 2.228 3.169 4.587
11 0.260 0.540 0.876 1.363 1.796 2.201 3.106 4.437
12 0.259 0.539 0.873 1.356 1.782 2.179 3.055 4.318
13 0.259 0.538 0.870 1.350 1.771 2.160 3.012 4.221
14 0.258 0.537 0.868 1.345 1.761 2.145 2.977 4.140
15 0.258 0.536 0.866 1.341 1.753 2.131 2.947 4.073
16 0.258 0.535 0.865 1.337 1.746 2.120 2.921 4.015
17 0.257 0.534 0.863 1.333 1.740 2.110 2.898 3.965
18 0.257 0.534 0.862 1.330 1.734 2.101 2.878 3.922
19 0.257 0.533 0.861 1.328 1.729 2.093 2.861 3.883
20 0.257 0.533 0.860 1.325 1.725 2.086 2.845 3.850
21 0.257 0.532 0.859 1.323 1.721 2.080 2.831 3.819
22 0.256 0.532 0.858 1.321 1.717 2.074 2.819 3.792
23 0.256 0.532 0.858 1.319 1.714 2.069 2.807 3.768
24 0.256 0.531 0.857 1.318 1.711 2.064 2.797 3.745
25 0.256 0.531 0.856 1.316 1.708 2.060 2.787 3.725
26 0.256 0.531 0.856 1.315 1.706 2.056 2.779 3.707
27 0.256 0.531 0.855 1.314 1.703 2.052 2.771 3.690
28 0.256 0.530 0.855 1.313 1.701 2.048 2.763 3.674
29 0.256 0.530 0.854 1.311 1.699 2.045 2.756 3.659
30 0.256 0.530 0.854 1.310 1.697 2.042 2.750 3.646
60 0.254 0.527 0.848 1.296 1.671 2.000 2.660 3.460
90 0.254 0.526 0.846 1.291 1.662 1.987 2.632 3.402
120 0.254 0.526 0.845 1.289 1.658 1.980 2.617 3.373

A.5 QUANTILES OF THE CHI–SQUARE DISTRIBUTION

Table A.5 lists X2[p;n]. For example, the X2[0.95;13] required for a chi–square test at 95% confidence using 14 cells is 22.362.

TABLE A.5 Quantiles of the Chi–Square Distribution

p

n 0.005 0.010 0.050 0.100 0.200 0.500 0.800 0.900 0.950 0.990 0.995

1 abcd0.064 0.455 1.642 2.706 3.841 6.635 7.879
2 0.010 0.020 0.103 0.211 0.446 1.386 3.219 4.605 5.991 9.210 10.596
3 0.072 0.115 0.352 0.585 1.005 2.366 4.642 6.253 7.817 11.356 12.861
4 0.207 0.297 0.711 1.064 1.649 3.357 5.989 7.779 9.488 13.277 14.861
5 0.412 0.554 1.145 1.610 2.343 4.351 7.289 9.236 11.071 15.086 16.750
6 0.676 0.872 1.635 2.204 3.070 5.348 8.558 10.645 12.592 16.812 18.548
7 0.989 1.239 2.167 2.833 3.822 6.346 9.803 12.017 14.067 18.475 20.278
8 1.344 1.646 2.733 3.490 4.594 7.344 11.030 13.362 15.507 20.090 21.955
9 1.735 2.088 3.325 4.168 5.380 8.343 12.242 14.684 16.919 21.666 23.589
10 2.156 2.558 3.940 4.865 6.179 9.342 13.442 15.987 18.307 23.209 25.188
11 2.603 3.053 4.575 5.578 6.989 10.341 14.631 17.275 19.675 24.725 26.757
12 3.074 3.571 5.226 6.304 7.807 11.340 15.812 18.549 21.026 26.217 28.300
13 3.565 4.107 5.892 7.041 8.634 12.340 16.985 19.812 22.362 27.688 29.820
14 4.075 4.660 6.571 7.790 9.467 13.339 18.151 21.064 23.685 29.141 31.319
15 4.601 5.229 7.261 8.547 10.307 14.339 19.311 22.307 24.996 30.578 32.801
16 5.142 5.812 7.962 9.312 11.152 15.339 20.465 23.542 26.296 32.000 34.267
17 5.697 6.408 8.672 10.085 12.002 16.338 21.615 24.769 27.587 33.409 35.719
18 6.265 7.015 9.390 10.865 12.857 17.338 22.760 25.989 28.869 34.805 37.156
19 6.844 7.633 10.117 11.651 13.716 18.338 23.900 27.204 30.144 36.191 38.582
20 7.434 8.260 10.851 12.443 14.578 19.337 25.038 28.412 31.410 37.566 39.997
21 8.034 8.897 11.591 13.240 15.445 20.337 26.171 29.615 32.671 38.932 41.401
22 8.643 9.542 12.338 14.041 16.314 21.337 27.301 30.813 33.924 40.289 42.796
23 9.260 10.196 13.090 14.848 17.186 22.337 28.429 32.007 35.172 41.638 44.181
24 9.886 10.856 13.848 15.659 18.062 23.337 29.553 33.196 36.415 42.980 45.559
25 10.520 11.524 14.611 16.473 18.940 24.337 30.675 34.382 37.653 44.314 46.928
26 11.160 12.198 15.379 17.292 19.820 25.336 31.795 35.563 38.885 45.642 48.290
27 11.808 12.878 16.151 18.114 20.703 26.336 32.912 36.741 40.113 46.963 49.645
28 12.461 13.565 16.928 18.939 21.588 27.336 34.027 37.916 41.337 48.278 50.993
29 13.121 14.256 17.708 19.768 22.475 28.336 35.139 39.088 42.557 49.588 52.336
30 13.787 14.953 18.493 20.599 23.364 29.336 36.250 40.256 43.773 50.892 53.672
31 14.458 15.655 19.281 21.434 24.255 30.336 37.359 41.422 44.985 52.191 55.003
32 15.134 16.362 20.072 22.271 25.148 31.336 38.466 42.585 46.194 53.486 56.328
33 15.815 17.073 20.866 23.110 26.042 32.336 39.572 43.745 47.400 54.776 57.649
34 16.501 17.789 21.664 23.952 26.938 33.336 40.676 44.903 48.602 56.061 58.964
35 17.192 18.509 22.465 24.797 27.836 34.336 41.778 46.059 49.802 57.342 60.275
36 17.887 19.233 23.269 25.643 28.735 35.336 42.879 47.212 50.998 58.619 61.581
37 18.586 19.960 24.075 26.492 29.635 36.336 43.978 48.363 52.192 59.893 62.883
38 19.289 20.691 24.884 27.343 30.537 37.335 45.076 49.513 53.384 61.162 64.181
39 19.996 21.426 25.695 28.196 31.440 38.335 46.173 50.660 54.572 62.428 65.476
40 20.706 22.164 26.509 29.050 32.345 39.335 47.269 51.805 55.759 63.691 66.766


a3.93 × 10–5
b1.57 × 10–4
c3.93 × 10–3
d1.58 × 10–2

A.6 90–PERCENTILES OF THE F(n, m) DISTRIBUTION

Table A.6 lists F[0.90;n,m]. For example, the F[0.90;9,18] required for an F–test at 90% confidence level is 2.00.

TABLE A.6 90–Pereentiles of the F(n, m) Distribution

Numerator Degrees of Freedom n

m 1 2 3 4 5 6 7 8 9 10

1 39.86 49.50 53.59 55.83 57.24 58.20 58.90 59.44 59.86 60.19
2 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38 9.39
3 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24 5.23
4 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94 3.92
5 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32 3.30
6 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96 2.94
7 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72 2.70
8 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56 2.54
9 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44 2.42
10 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35 2.32
12 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21 2.19
14 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12 2.10
16 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06 2.03
18 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00 1.98
20 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96 1.94
25 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89 1.87
30 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85 1.82
35 2.85 2.46 2.25 2.11 2.02 1.95 1.90 1.85 1.82 1.79
40 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79 1.76
500 2.72 2.31 2.09 1.96 1.86 1.79 1.73 1.68 1.64 1.61

Numerator Degrees of Freedom n

m 12 14 16 18 20 25 30 35 40 500

1 60.70 61.07 61.35 61.56 61.74 62.05 62.26 62.41 62.53 63.26
2 9.41 9.42 9.43 9.44 9.44 9.45 9.46 9.46 9.47 9.49
3 5.22 5.20 5.20 5.19 5.18 5.17 5.17 5.16 5.16 5.13
4 3.90 3.88 3.86 3.85 3.84 3.83 3.82 3.81 3.80 3.76
5 3.27 3.25 3.23 3.22 3.21 3.19 3.17 3.16 3.16 3.10
6 2.90 2.88 2.86 2.85 2.84 2.81 2.80 2.79 2.78 2.72
7 2.67 2.64 2.62 2.61 2.59 2.57 2.56 2.54 2.54 2.47
8 2.50 2.48 2.45 2.44 2.42 2.40 2.38 2.37 2.36 2.29
9 2.38 2.35 2.33 2.31 2.30 2.27 2.25 2.24 2.23 2.16
10 2.28 2.26 2.23 2.22 2.20 2.17 2.16 2.14 2.13 2.06
12 2.15 2.12 2.09 2.08 2.06 2.03 2.01 2.00 1.99 1.90
14 2.05 2.02 2.00 1.98 1.96 1.93 1.91 1.90 1.89 1.80
16 1.99 1.95 1.93 1.91 1.89 1.86 1.84 1.82 1.81 1.72
18 1.93 1.90 1.87 1.85 1.84 1.80 1.78 1.77 1.75 1.66
20 1.89 1.86 1.83 1.81 1.79 1.76 1.74 1.72 1.71 1.61
25 1.82 1.79 1.76 1.74 1.72 1.68 1.66 1.64 1.63 1.52
30 1.77 1.74 1.71 1.69 1.67 1.63 1.61 1.59 1.57 1.46
35 1.74 1.70 1.67 1.65 1.63 1.60 1.57 1.55 1.53 1.41
40 1.71 1.68 1.65 1.62 1.61 1.57 1.54 1.52 1.51 1.38
500 1.56 1.52 1.49 1.46 1.44 1.39 1.36 1.33 1.31 2.16

A.7 95–PERCENTILES OF THE F(n, m) DISTRIBUTION

Table A.7 lists F[0.95;n,m]. For example, the F[0.95;9,14] required for an F–test at 95% confidence level is 2.70.

TABLE A.7 95–Percentiles or the F(n, m) Distribution

Numerator Degrees of Freedom n

m 1 2 3 4 5 6 7 8 9 10

1 161.45 199.50 215.70 224.57 230.15 233.97 236.76 238.87 240.53 241.87
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75
14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16
35 4.12 3.27 2.87 2.64 2.49 2.37 2.29 2.22 2.16 2.11
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08
500 3.86 3.01 2.62 2.39 2.23 2.12 2.03 1.96 1.90 1.85

Numerator Degrees of Freedom n

m 12 14 16 18 20 25 30 35 40 500

1 243.90 245.35 246.45 247.31 248.00 249.25 250.09 250.68 251.13 254.05
2 19.41 19.42 19.43 19.44 19.45 19.46 19.46 19.47 19.47 19.49
3 8.74 8.71 8.69 8.67 8.66 8.63 8.62 8.60 8.59 8.53
4 5.91 5.87 5.84 5.82 5.80 5.77 5.75 5.73 5.72 5.63
5 4.68 4.64 4.60 4.58 4.56 4.52 4.50 4.48 4.46 4.36
6 4.00 3.96 3.92 3.90 3.87 3.83 3.81 3.79 3.77 3.67
7 3.57 3.53 3.49 3.47 3.44 3.40 3.38 3.36 3.34 3.23
8 3.28 3.24 3.20 3.17 3.15 3.11 3.08 3.06 3.04 2.93
9 3.07 3.03 2.99 2.96 2.94 2.89 2.86 2.84 2.83 2.71
10 2.91 2.86 2.83 2.80 2.77 2.73 2.70 2.68 2.66 2.54
12 2.69 2.64 2.60 2.57 2.54 2.50 2.47 2.44 2.43 2.30
14 2.53 2.48 2.44 2.41 2.39 2.34 2.31 2.28 2.27 2.13
16 2.42 2.37 2.33 2.30 2.28 2.23 2.19 2.17 2.15 2.01
18 2.34 2.29 2.25 2.22 2.19 2.14 2.11 2.08 2.06 1.92
20 2.28 2.22 2.18 2.15 2.12 2.07 2.04 2.01 1.99 1.84
25 2.16 2.11 2.07 2.04 2.01 1.96 1.92 1.89 1.87 1.71
30 2.09 2.04 1.99 1.96 1.93 1.88 1.84 1.81 1.79 1.62
35 2.04 1.99 1.94 1.91 1.88 1.82 1.79 1.76 1.74 1.56
40 2.00 1.95 1.90 1.87 1.84 1.78 1.74 1.72 1.69 1.51
500 1.77 1.71 1.66 1.62 1.59 1.53 1.48 1.45 1.42 2.21

A.8 99–PERCENTILES OF THE F(n, m) DISTRIBUTION

Table A.8 lists F[0.99;n,m]. For example, the F[0.99;6,12] required for an F–test at 99% confidence level is 4.82.

TABLE A.8 99–Percentiles or the F(n, m) Distribution

Numerator Degrees of Freedom n

m 1 2 3 4 5 6 7 8 9 10

1 4052.18 4999.50 5403.05 5624.30 5763.37 5858.71 5928.09 5980.80 6022.21 6055.58
2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40
3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.34 27.23
4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55
5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05
6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87
7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62
8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81
9 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26
10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85
12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30
14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03 3.94
16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69
18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51
20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37
25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22 3.13
30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98
35 7.42 5.27 4.40 3.91 3.59 3.37 3.20 3.07 2.96 2.88
40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80
500 6.69 4.65 3.82 3.36 3.05 2.84 2.68 2.55 2.44 2.36

Numerator Degrees of Freedom n

m 12 14 16 18 20 25 30 35 40 500

1 6106.06 6142.42 6169.85 6191.28 6208.48 6239.57 6260.40 6275.32 6286.53 6359.26
2 99.42 99.43 99.44 99.44 99.45 99.46 99.47 99.47 99.47 99.50
3 27.05 26.92 26.83 26.75 26.69 26.58 26.50 26.45 26.41 26.12
4 14.37 14.25 14.15 14.08 14.02 13.91 13.84 13.79 13.75 13.46
5 9.89 9.77 9.68 9.61 9.55 9.45 9.38 9.33 9.29 9.02
6 7.72 7.60 7.52 7.45 7.40 7.30 7.23 7.18 7.14 6.88
7 6.47 6.36 6.28 6.21 6.16 6.06 5.99 5.94 5.91 5.65
8 5.67 5.56 5.48 5.41 5.36 5.26 5.20 5.15 5.12 4.86
9 5.11 5.01 4.92 4.86 4.81 4.71 4.65 4.60 4.57 4.31
10 4.71 4.60 4.52 4.46 4.41 4.31 4.25 4.20 4.17 3.91
12 4.16 4.05 3.97 3.91 3.86 3.76 3.70 3.65 3.62 3.36
14 3.80 3.70 3.62 3.56 3.51 3.41 3.35 3.30 3.27 3.00
16 3.55 3.45 3.37 3.31 3.26 3.16 3.10 3.05 3.02 2.75
18 3.37 3.27 3.19 3.13 3.08 2.98 2.92 2.87 2.84 2.57
20 3.23 3.13 3.05 2.99 2.94 2.84 2.78 2.73 2.69 2.42
25 2.99 2.89 2.81 2.75 2.70 2.60 2.54 2.49 2.45 2.17
30 2.84 2.74 2.66 2.60 2.55 2.45 2.39 2.34 2.30 2.01
35 2.74 2.64 2.56 2.50 2.44 2.35 2.28 2.23 2.19 1.89
40 2.66 2.56 2.48 2.42 2.37 2.27 2.20 2.15 2.11 1.80
500 2.22 2.12 2.04 1.97 1.92 1.81 1.74 1.68 1.63 2.30

A.9 QUANTILES OF THE K–S DISTRIBUTION

Table A.9 lists quantiles K[p;n] of the K–S distribution. For example, the K[0.99;12] required for a K–S test at 99% confidence level is 1.4521.

TABLE A.9 Quantiles of the K–S Distribution

p

n 0.00 0.01 0.05 0.10 0.20 0.50 0.80 0.90 0.95 0.99 0.995

1 0.0050 0.0100 0.0500 0.1000 0.2000 0.5000 0.8000 0.9000 0.9500 0.9900 0.9950
2 0.0067 0.0135 0.0673 0.1296 0.2416 0.5176 0.7818 0.9670 1.0980 1.2728 1.3142
3 0.0081 0.0162 0.0792 0.1471 0.2615 0.5147 0.8187 0.9783 1.1017 1.3589 1.4359
4 0.0093 0.0186 0.0879 0.1590 0.2726 0.5110 0.8248 0.9853 1.1304 1.3777 1.4685
5 0.0103 0.0207 0.0947 0.1675 0.2793 0.5245 0.8277 0.9995 1.1392 1.4024 1.4949
6 0.0113 0.0226 0.1002 0.1739 0.2834 0.5319 0.8343 1.0052 1.1463 1.4144 1.5104
7 0.0121 0.0243 0.1048 0.1787 0.2859 0.5364 0.8398 1.0093 1.1537 1.4246 1.5235
8 0.0130 0.0259 0.1086 0.1826 0.2874 0.5392 0.8431 1.0135 1.1586 1.4327 1.5324
9 0.0137 0.0275 0.1119 0.1856 0.2881 0.5411 0.8455 1.0173 1.1624 1.4388 1.5400
10 0.0145 0.0289 0.1147 0.1880 0.2884 0.5426 0.8477 1.0202 1.1658 1.4440 1.5461
11 0.0151 0.0303 0.1172 0.1900 0.2883 0.5439 0.8498 1.0225 1.1688 1.4484 1.5512
12 0.0158 0.0314 0.1193 0.1916 0.2879 0.5453 0.8519 1.0246 1.1714 1.4521 1.5555
13 0.0164 0.0324 0.1212 0.1929 0.2903 0.5468 0.8537 1.0265 1.1736 1.4553 1.5593
14 0.0171 0.0333 0.1229 0.1940 0.2925 0.5486 0.8551 1.0282 1.1755 1.4581 1.5626
15 0.0176 0.0342 0.1244 0.1948 0.2944 0.5500 0.8564 1.0298 1.1773 1.4606 1.5655
16 0.0182 0.0351 0.1257 0.1955 0.2961 0.5512 0.8576 1.0311 1.1789 1.4629 1.5681
17 0.0188 0.0359 0.1269 0.1961 0.2975 0.5523 0.8587 1.0324 1.1803 1.4649 1.5704
18 0.0193 0.0367 0.1280 0.1965 0.2987 0.5532 0.8597 1.0335 1.1816 1.4667 1.5725
19 0.0198 0.0374 0.1290 0.1968 0.2998 0.5540 0.8607 1.0346 1.1828 1.4683 1.5744
20 0.0203 0.0381 0.1298 0.1971 0.3007 0.5547 0.8616 1.0355 1.1839 1.4699 1.5761
21 0.0208 0.0387 0.1306 0.1973 0.3015 0.5554 0.8624 1.0365 1.1850 1.4712 1.5777
22 0.0213 0.0394 0.1313 0.1974 0.3023 0.5561 0.8631 1.0373 1.1859 1.4725 1.5792
23 0.0217 0.0400 0.1320 0.1974 0.3030 0.5567 0.8639 1.0381 1.1868 1.4737 1.5806
24 0.0221 0.0405 0.1326 0.1974 0.3035 0.5573 0.8645 1.0388 1.1876 1.4748 1.5816
25 0.0225 0.0411 0.1331 0.1974 0.3041 0.5579 0.8651 1.0395 1.1884 1.4758 1.5829
26 0.0228 0.0416 0.1336 0.1977 0.3046 0.5585 0.8657 1.0402 1.1891 1.4768 1.5840
27 0.0231 0.0421 0.1340 0.1985 0.3050 0.5590 0.8663 1.0408 1.1898 1.4777 1.5850
28 0.0235 0.0426 0.1344 0.1992 0.3054 0.5595 0.8668 1.0414 1.1905 1.4786 1.5860
29 0.0238 0.0431 0.1348 0.2000 0.3058 0.5600 0.8673 1.0419 1.1911 1.4794 1.5969
30 0.0241 0.0435 0.1351 0.2006 0.3062 0.5605 0.8678 1.0424 1.1916 1.4801 1.5878

A.10 APPROXIMATION FORMULAS FOR STATISTICAL TABLES

In computer programs, the following approximate formulas may be used in place of statistical tables:

1.  Area Under the Normal Distribution: The area under the unit normal pdf between 0 and z is given approximately by (Hastings, Jr. (1955)):

p = ½ - ½(1+0.196854z + 0.115194z2 + 0.000344z3 + 0.019527z4)-4


Given zκ, this formula can be used to find κ (κ = 0.5 + p).
2.  Unit Normal Quantiles: The unit normal quantile zp for a given p can be calculated approximately by the following formula (Hastings, Jr. (1955)):


where


A simpler but more approximate formula is

Zp = 4.91[p0.14 – (1 – p)0.14]

3.  Chi–Square Quantiles: For large degrees of freedom, the X2(v) quantiles can be calculated approximately from the unit normal quantiles zκ as follows:


Given a X2κ value, this formula can also be used to compute zκ and hence the significance κ.
4.  F–Quantiles: For large degrees of freedoms, the following approximation can be used to compute Fκ from zκ:


Again, given an Fκ value, this formula can also be used to to compute zκ and hence the significance κ.


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