Appendixes

  1. Appendix A-1 Cumulative binomial distribution
  2. Appendix A-2 Cumulative Poisson distribution
  3. Appendix A-3 Cumulative standard normal distribution
  4. Appendix A-4 Values of t for a specified right-tail area
  5. Appendix A-5 Chi-squared values for a specified right-tail area
  6. Appendix A-6 Values of F for a specified right-tail area
  7. Appendix A-7 Factors for computing centerline and three-sigma control limits
  8. Appendix A-8 Uniform random numbers

Appendix A-1 Cumulative binomial distribution

p = Probability of Occurrence
n X 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
2 0 0.903 0.810 0.723 0.640 0.563 0.490 0.423 0.360 0.303 0.250
1 0.998 0.990 0.978 0.960 0.938 0.910 0.878 0.840 0.798 0.750
3 0 0.857 0.729 0.614 0.512 0.422 0.343 0.275 0.216 0.166 0.125
1 0.993 0.972 0.939 0.896 0.844 0.784 0.718 0.648 0.575 0.500
2 1.000 0.999 0.997 0.992 0.984 0.973 0.957 0.936 0.909 0.875
4 0 0.815 0.656 0.522 0.410 0.316 0.240 0.179 0.130 0.092 0.063
1 0.986 0.948 0.890 0.819 0.738 0.652 0.563 0.475 0.391 0.313
2 1.000 0.996 0.988 0.973 0.949 0.916 0.874 0.821 0.759 0.688
3 1.000 0.999 0.998 0.996 0.992 0.985 0.974 0.959 0.938
5 0 0.774 0.590 0.444 0.328 0.237 0.168 0.116 0.078 0.050 0.031
1 0.977 0.919 0.835 0.737 0.633 0.528 0.428 0.337 0.256 0.188
2 0.999 0.991 0.973 0.942 0.896 0.837 0.765 0.683 0.593 0.500
3 1.000 1.000 0.998 0.993 0.984 0.969 0.946 0.913 0.869 0.813
4 1.000 1.000 0.999 0.998 0.995 0.990 0.982 0.969
6 0 0.735 0.531 0.377 0.262 0.178 0.118 0.075 0.047 0.028 0.016
1 0.967 0.886 0.776 0.655 0.534 0.420 0.319 0.233 0.164 0.109
2 0.998 0.984 0.953 0.901 0.831 0.744 0.647 0.544 0.442 0.344
3 1.000 0.999 0.994 0.983 0.962 0.930 0.883 0.821 0.745 0.656
4 1.000 1.000 0.998 0.995 0.989 0.978 0.959 0.931 0.891
5 1.000 1.000 0.999 0.998 0.996 0.992 0.984
7 0 0.698 0.478 0.321 0.210 0.133 0.082 0.049 0.028 0.015 0.008
1 0.956 0.850 0.717 0.577 0.445 0.329 0.234 0.159 0.102 0.063
2 0.996 0.974 0.926 0.852 0.756 0.647 0.532 0.420 0.316 0.227
3 1.000 0.997 0.988 0.967 0.929 0.874 0.800 0.710 0.608 0.500
4 1.000 0.999 0.995 0.987 0.971 0.944 0.904 0.847 0.773
5 1.000 1.000 0.999 0.996 0.991 0.981 0.964 0.938
6 1.000 1.000 0.999 0.998 0.996 0.992
8 0 0.663 0.430 0.272 0.168 0.100 0.058 0.032 0.017 0.008 0.004
1 0.943 0.813 0.657 0.503 0.367 0.255 0.169 0.106 0.063 0.035
2 0.994 0.962 0.895 0.797 0.679 0.552 0.428 0.315 0.220 0.145
3 1.000 0.995 0.979 0.944 0.886 0.806 0.706 0.594 0.477 0.363
4 1.000 0.997 0.990 0.973 0.942 0.894 0.826 0.740 0.637
5 1.000 0.999 0.996 0.989 0.975 0.950 0.912 0.855
6 1.000 1.000 0.999 0.996 0.991 0.982 0.965
7 1.000 1.000 0.999 0.998 0.996
9 0 0.630 0.387 0.232 0.134 0.075 0.040 0.021 0.010 0.005 0.002
1 0.929 0.775 0.599 0.436 0.300 0.196 0.121 0.071 0.039 0.020
2 0.992 0.947 0.859 0.738 0.601 0.463 0.337 0.232 0.150 0.090
3 0.999 0.992 0.966 0.914 0.834 0.730 0.609 0.483 0.361 0.254
4 1.000 0.999 0.994 0.980 0.951 0.901 0.828 0.733 0.621 0.500
5 1.000 0.999 0.997 0.990 0.975 0.946 0.901 0.834 0.746
6 1.000 1.000 0.999 0.996 0.989 0.975 0.950 0.910
7 1.000 1.000 0.999 0.996 0.991 0.980
8 1.000 1.000 0.999 0.998
10 0 0.599 0.349 0.197 0.107 0.056 0.028 0.013 0.006 0.003 0.001
1 0.914 0.736 0.544 0.376 0.244 0.149 0.086 0.046 0.023 0.011
2 0.988 0.930 0.820 0.678 0.526 0.383 0.262 0.167 0.100 0.055
3 0.999 0.987 0.950 0.879 0.776 0.650 0.514 0.382 0.266 0.172
4 1.000 0.998 0.990 0.967 0.922 0.850 0.751 0.633 0.504 0.377
5 1.000 0.999 0.994 0.980 0.953 0.905 0.834 0.738 0.623
6 1.000 0.999 0.996 0.989 0.974 0.945 0.898 0.828
7 1.000 1.000 0.998 0.995 0.988 0.973 0.945
8 1.000 0.999 0.998 0.995 0.989
9 1.000 1.000 1.000 0.999
11 0 0.569 0.314 0.167 0.086 0.042 0.020 0.009 0.004 0.001 0.000
1 0.898 0.697 0.492 0.322 0.197 0.113 0.061 0.030 0.014 0.006
2 0.985 0.910 0.779 0.617 0.455 0.313 0.200 0.119 0.065 0.033
3 0.998 0.981 0.931 0.839 0.713 0.570 0.426 0.296 0.191 0.113
4 1.000 0.997 0.984 0.950 0.885 0.790 0.668 0.533 0.397 0.274
5 1.000 0.997 0.988 0.966 0.922 0.851 0.753 0.633 0.500
6 1.000 0.998 0.992 0.978 0.950 0.901 0.826 0.726
7 1.000 0.999 0.996 0.988 0.971 0.939 0.887
8 1.000 0.999 0.998 0.994 0.985 0.967
9 1.000 1.000 0.999 0.998 0.994
10 1.000 1.000 1.000
12 0 0.540 0.282 0.142 0.069 0.032 0.014 0.006 0.002 0.001 0.000
1 0.882 0.659 0.443 0.275 0.158 0.085 0.042 0.020 0.008 0.003
2 0.980 0.889 0.736 0.558 0.391 0.253 0.151 0.083 0.042 0.019
3 0.998 0.974 0.908 0.795 0.649 0.493 0.347 0.225 0.134 0.073
4 1.000 0.996 0.976 0.927 0.842 0.724 0.583 0.438 0.304 0.194
5 0.999 0.995 0.981 0.946 0.882 0.787 0.665 0.527 0.387
6 1.000 0.999 0.996 0.986 0.961 0.915 0.842 0.739 0.613
7 1.000 0.999 0.997 0.991 0.974 0.943 0.888 0.806
8 1.000 1.000 0.998 0.994 0.985 0.964 0.927
9 1.000 0.999 0.997 0.992 0.981
10 1.000 1.000 0.999 0.997
11 1.000 1.000
13 0 0.513 0.254 0.121 0.055 0.024 0.010 0.004 0.001 0.000 0.000
1 0.865 0.621 0.398 0.234 0.127 0.064 0.030 0.013 0.005 0.002
2 0.975 0.866 0.692 0.502 0.333 0.202 0.113 0.058 0.027 0.011
3 0.997 0.966 0.882 0.747 0.584 0.421 0.278 0.169 0.093 0.046
4 1.000 0.994 0.966 0.901 0.794 0.654 0.501 0.353 0.228 0.133
5 0.999 0.992 0.970 0.920 0.835 0.716 0.574 0.427 0.291
6 1.000 0.999 0.993 0.976 0.938 0.871 0.771 0.644 0.500
7 1.000 0.999 0.994 0.982 0.954 0.902 0.821 0.709
8 1.000 0.999 0.996 0.987 0.968 0.930 0.867
9 1.000 0.999 0.997 0.992 0.980 0.954
10 1.000 1.000 0.999 0.996 0.989
11 1.000 0.999 0.998
12 1.000 1.000
14 0 0.488 0.229 0.103 0.044 0.018 0.007 0.002 0.001 0.000 0.000
1 0.847 0.585 0.357 0.198 0.101 0.047 0.021 0.008 0.003 0.001
2 0.970 0.842 0.648 0.448 0.281 0.161 0.084 0.040 0.017 0.006
3 0.996 0.956 0.853 0.698 0.521 0.355 0.220 0.124 0.063 0.029
4 1.000 0.991 0.953 0.870 0.742 0.584 0.423 0.279 0.167 0.090
5 0.999 0.988 0.956 0.888 0.781 0.641 0.486 0.337 0.212
6 1.000 0.998 0.988 0.962 0.907 0.816 0.692 0.546 0.395
7 1.000 0.998 0.990 0.969 0.925 0.850 0.741 0.605
8 1.000 0.998 0.992 0.976 0.942 0.881 0.788
9 1.000 0.998 0.994 0.982 0.957 0.910
10 1.000 0.999 0.996 0.989 0.971
11 1.000 0.999 0.998 0.994
12 1.000 1.000 0.999
13 1.000
15 0 0.463 0.206 0.087 0.035 0.013 0.005 0.002 0.000 0.000 0.000
1 0.829 0.549 0.319 0.167 0.080 0.035 0.014 0.005 0.002 0.000
2 0.964 0.816 0.604 0.398 0.236 0.127 0.062 0.027 0.011 0.004
3 0.995 0.944 0.823 0.648 0.461 0.297 0.173 0.091 0.042 0.018
4 0.999 0.987 0.938 0.836 0.686 0.515 0.352 0.217 0.120 0.059
5 1.000 0.998 0.983 0.939 0.852 0.722 0.564 0.403 0.261 0.151
6 1.000 0.996 0.982 0.943 0.869 0.755 0.610 0.452 0.304
7 0.999 0.996 0.983 0.950 0.887 0.787 0.654 0.500
8 1.000 0.999 0.996 0.985 0.958 0.905 0.818 0.696
9 1.000 0.999 0.996 0.988 0.966 0.923 0.849
10 1.000 0.999 0.997 0.991 0.975 0.941
11 1.000 1.000 0.998 0.994 0.982
12 1.000 0.999 0.996
13 1.000 1.000
16 0 0.440 0.185 0.074 0.028 0.010 0.003 0.001 0.000 0.000 0.000
1 0.811 0.515 0.284 0.141 0.063 0.026 0.010 0.003 0.001 0.000
2 0.957 0.789 0.561 0.352 0.197 0.099 0.045 0.018 0.007 0.002
3 0.993 0.932 0.790 0.598 0.405 0.246 0.134 0.065 0.028 0.011
4 0.999 0.983 0.921 0.798 0.630 0.450 0.289 0.167 0.085 0.038
5 1.000 0.997 0.976 0.918 0.810 0.660 0.490 0.329 0.198 0.105
6 0.999 0.994 0.973 0.920 0.825 0.688 0.527 0.366 0.227
7 1.000 0.999 0.993 0.973 0.926 0.841 0.716 0.563 0.402
8 1.000 0.999 0.993 0.974 0.933 0.858 0.744 0.598
9 1.000 0.998 0.993 0.977 0.942 0.876 0.773
10 1.000 0.998 0.994 0.981 0.951 0.895
11 1.000 0.999 0.995 0.985 0.962
12 1.000 0.999 0.997 0.989
13 1.000 0.999 0.998
14 1.000 1.000
17 0 0.418 0.167 0.063 0.023 0.008 0.002 0.001 0.000 0.000 0.000
1 0.792 0.482 0.252 0.118 0.050 0.019 0.007 0.002 0.001 0.000
2 0.950 0.762 0.520 0.310 0.164 0.077 0.033 0.012 0.004 0.001
3 0.991 0.917 0.756 0.549 0.353 0.202 0.103 0.046 0.018 0.006
4 0.999 0.978 0.901 0.758 0.574 0.389 0.235 0.126 0.060 0.025
5 1.000 0.995 0.968 0.894 0.765 0.597 0.420 0.264 0.147 0.072
6 0.999 0.992 0.962 0.893 0.775 0.619 0.448 0.290 0.166
7 1.000 0.998 0.989 0.960 0.895 0.787 0.641 0.474 0.315
8 1.000 0.997 0.988 0.960 0.901 0.801 0.663 0.500
9 1.000 0.997 0.987 0.962 0.908 0.817 0.685
10 0.999 0.997 0.988 0.965 0.917 0.834
11 1.000 0.999 0.997 0.989 0.970 0.928
12 1.000 0.999 0.997 0.991 0.975
13 1.000 1.000 0.998 0.994
14 1.000 0.999
15 1.000
18 0 0.397 0.150 0.054 0.018 0.006 0.002 0.000 0.000 0.000 0.000
1 0.774 0.450 0.224 0.099 0.039 0.014 0.005 0.001 0.000 0.000
2 0.942 0.734 0.480 0.271 0.135 0.060 0.024 0.008 0.003 0.001
3 0.989 0.902 0.720 0.501 0.306 0.165 0.078 0.033 0.012 0.004
4 0.998 0.972 0.879 0.716 0.519 0.333 0.189 0.094 0.041 0.015
5 1.000 0.994 0.958 0.867 0.717 0.534 0.355 0.209 0.108 0.048
6 0.999 0.988 0.949 0.861 0.722 0.549 0.374 0.226 0.119
7 1.000 0.997 0.984 0.943 0.859 0.728 0.563 0.391 0.240
8 0.999 0.996 0.981 0.940 0.861 0.737 0.578 0.407
9 1.000 0.999 0.995 0.979 0.940 0.865 0.747 0.593
10 1.000 0.999 0.994 0.979 0.942 0.872 0.760
11 1.000 0.999 0.994 0.980 0.946 0.881
12 1.000 0.999 0.994 0.982 0.952
13 1.000 0.999 0.995 0.985
14 1.000 0.999 0.996
15 1.000 0.999
16 1.000
19 0 0.377 0.135 0.046 0.014 0.004 0.001 0.000 0.000 0.000 0.000
1 0.755 0.420 0.198 0.083 0.031 0.010 0.003 0.001 0.000 0.000
2 0.933 0.705 0.441 0.237 0.111 0.046 0.017 0.005 0.002 0.000
3 0.987 0.885 0.684 0.455 0.263 0.133 0.059 0.023 0.008 0.002
4 0.998 0.965 0.856 0.673 0.465 0.282 0.150 0.070 0.028 0.010
5 1.000 0.991 0.946 0.837 0.668 0.474 0.297 0.163 0.078 0.032
6 0.998 0.984 0.932 0.825 0.666 0.481 0.308 0.173 0.084
7 1.000 0.996 0.977 0.923 0.818 0.666 0.488 0.317 0.180
8 0.999 0.993 0.971 0.916 0.815 0.667 0.494 0.324
9 1.000 0.998 0.991 0.967 0.913 0.814 0.671 0.500
10 1.000 0.998 0.989 0.965 0.912 0.816 0.676
11 1.000 0.997 0.989 0.965 0.913 0.820
12 0.999 0.997 0.988 0.966 0.916
13 1.000 0.999 0.997 0.989 0.968
14 1.000 0.999 0.997 0.990
15 1.000 0.999 0.998
16 1.000 1.000
20 0 0.358 0.122 0.039 0.012 0.003 0.001 0.000 0.000 0.000 0.000
1 0.736 0.392 0.176 0.069 0.024 0.008 0.002 0.001 0.000 0.000
2 0.925 0.677 0.405 0.206 0.091 0.035 0.012 0.004 0.001 0.000
3 0.984 0.867 0.648 0.411 0.225 0.107 0.044 0.016 0.005 0.001
4 0.997 0.957 0.830 0.630 0.415 0.238 0.118 0.051 0.019 0.006
5 1.000 0.989 0.933 0.804 0.617 0.416 0.245 0.126 0.055 0.021
6 0.998 0.978 0.913 0.786 0.608 0.417 0.250 0.130 0.058
7 1.000 0.994 0.968 0.898 0.772 0.601 0.416 0.252 0.132
8 0.999 0.990 0.959 0.887 0.762 0.596 0.414 0.252
9 1.000 0.997 0.986 0.952 0.878 0.755 0.591 0.412
10 0.999 0.996 0.983 0.947 0.872 0.751 0.588
11 1.000 0.999 0.995 0.980 0.943 0.869 0.748
12 1.000 0.999 0.994 0.979 0.942 0.868
13 1.000 0.998 0.994 0.979 0.942
14 1.000 0.998 0.994 0.979
15 1.000 0.998 0.994
16 1.000 0.999
17 1.000

Appendix A-2 Cumulative Poisson Distribution

λ = Mean
X 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0 0.990 0.951 0.905 0.819 0.741 0.670 0.607 0.549 0.497 0.449 0.407
1 1.000 0.999 0.995 0.982 0.963 0.938 0.910 0.878 0.844 0.809 0.772
2 1.000 1.000 0.999 0.996 0.992 0.986 0.977 0.966 0.953 0.937
3 1.000 1.000 0.999 0.998 0.997 0.994 0.991 0.987
4 1.000 1.000 1.000 0.999 0.999 0.998
5 1.000 1.000 1.000
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
0 0.368 0.333 0.301 0.273 0.247 0.223 0.202 0.183 0.165 0.150 0.135
1 0.736 0.699 0.663 0.627 0.592 0.558 0.525 0.493 0.463 0.434 0.406
2 0.920 0.900 0.879 0.857 0.833 0.809 0.783 0.757 0.731 0.704 0.677
3 0.981 0.974 0.966 0.957 0.946 0.934 0.921 0.907 0.891 0.875 0.857
4 0.996 0.995 0.992 0.989 0.986 0.981 0.976 0.970 0.964 0.956 0.947
5 0.999 0.999 0.998 0.998 0.997 0.996 0.994 0.992 0.990 0.987 0.983
6 1.000 1.000 1.000 1.000 0.999 0.999 0.999 0.998 0.997 0.997 0.995
7 1.000 1.000 1.000 1.000 0.999 0.999 0.999
8 1.000 1.000 1.000
2.2 2.4 2.6 2.8 3.0 3.5 4.0 4.5 5.0 5.5 6.0
0 0.111 0.091 0.074 0.061 0.050 0.030 0.018 0.011 0.007 0.004 0.002
1 0.355 0.308 0.267 0.231 0.199 0.136 0.092 0.061 0.040 0.027 0.017
2 0.623 0.570 0.518 0.469 0.423 0.321 0.238 0.174 0.125 0.088 0.062
3 0.819 0.779 0.736 0.692 0.647 0.537 0.433 0.342 0.265 0.202 0.151
4 0.928 0.904 0.877 0.848 0.815 0.725 0.629 0.532 0.440 0.358 0.285
5 0.975 0.964 0.951 0.935 0.916 0.858 0.785 0.703 0.616 0.529 0.446
6 0.993 0.988 0.983 0.976 0.966 0.935 0.889 0.831 0.762 0.686 0.606
7 0.998 0.997 0.995 0.992 0.988 0.973 0.949 0.913 0.867 0.809 0.744
8 1.000 0.999 0.999 0.998 0.996 0.990 0.979 0.960 0.932 0.894 0.847
9 1.000 1.000 0.999 0.999 0.997 0.992 0.983 0.968 0.946 0.916
10 1.000 1.000 0.999 0.997 0.993 0.986 0.975 0.957
11 1.000 0.999 0.998 0.995 0.989 0.980
12 1.000 0.999 0.998 0.996 0.991
13 1.000 0.999 0.998 0.996
14 1.000 0.999 0.999
15 1.000 0.999
16 1.000
6.5 7.0 7.5 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
0 0.002 0.001 0.001 0.000 0.000
1 0.011 0.007 0.005 0.003 0.001 0.000 0.000
2 0.043 0.030 0.020 0.014 0.006 0.003 0.001
3 0.112 0.082 0.059 0.042 0.021 0.010 0.002 0.000
4 0.224 0.173 0.132 0.100 0.055 0.029 0.008 0.002 0.000
5 0.369 0.301 0.241 0.191 0.116 0.067 0.020 0.006 0.001 0.000
6 0.527 0.450 0.378 0.313 0.207 0.130 0.046 0.014 0.004 0.001 0.000
7 0.673 0.599 0.525 0.453 0.324 0.220 0.090 0.032 0.010 0.003 0.001
8 0.792 0.729 0.662 0.593 0.456 0.333 0.155 0.062 0.022 0.007 0.002
9 0.877 0.830 0.776 0.717 0.587 0.458 0.242 0.109 0.043 0.015 0.005
10 0.933 0.901 0.862 0.816 0.706 0.583 0.347 0.176 0.077 0.030 0.011
11 0.966 0.947 0.921 0.888 0.803 0.697 0.462 0.260 0.127 0.055 0.021
6.5 7.0 7.5 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
12 0.984 0.973 0.957 0.936 0.876 0.792 0.576 0.358 0.193 0.092 0.039
13 0.993 0.987 0.978 0.966 0.926 0.864 0.682 0.464 0.275 0.143 0.066
14 0.997 0.994 0.990 0.983 0.959 0.917 0.772 0.570 0.368 0.208 0.105
15 0.999 0.998 0.995 0.992 0.978 0.951 0.844 0.669 0.467 0.287 0.157
16 1.000 0.999 0.998 0.996 0.989 0.973 0.899 0.756 0.566 0.375 0.221
17 1.000 0.999 0.998 0.995 0.986 0.937 0.827 0.659 0.469 0.297
18 1.000 0.999 0.998 0.993 0.963 0.883 0.742 0.562 0.381
19 1.000 0.999 0.997 0.979 0.923 0.812 0.651 0.470
20 1.000 0.998 0.988 0.952 0.868 0.731 0.559
21 0.999 0.994 0.971 0.911 0.799 0.644
22 1.000 0.997 0.983 0.942 0.855 0.721
23 0.999 0.991 0.963 0.899 0.787
24 0.999 0.995 0.978 0.932 0.843
25 1.000 0.997 0.987 0.955 0.888
26 0.999 0.993 0.972 0.922
27 0.999 0.996 0.983 0.948
28 1.000 0.998 0.990 0.966
29 0.999 0.994 0.978
30 0.999 0.997 0.987
31 1.000 0.998 0.992
32 0.999 0.995
33 1.000 0.997
34 0.999
35 0.999
36 1.000

Appendix A-3 Cumulative Standard Normal Distribution

Figure depicting a normal distribution curve where point Z is present near the right tail. Area of curve on the left of Z is shaded.
Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
−3.40 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
−3.30 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
−3.20 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
−3.10 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
−3.00 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
−2.90 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
−2.80 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
−2.70 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
−2.60 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
−2.50 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
−2.40 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
−2.30 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
−2.20 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
−2.10 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
−2.00 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
−1.90 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
−1.80 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
−1.70 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
−1.60 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
−1.50 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
−1.40 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
−1.30 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
−1.20 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
−1.10 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
−1.00 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
−0.90 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
−0.80 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
−0.70 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
−0.60 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
−0.50 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
−0.40 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
−0.30 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
−0.20 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
−0.10 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
−0.00 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
−0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
−0.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
−0.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
−0.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
−0.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
−0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
−0.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
−0.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
−0.80 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
−0.90 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
−1.00 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
−1.10 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
−1.20 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
−1.30 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
−1.40 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
−1.50 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
−1.60 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
−1.70 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
−1.80 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
−1.90 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
−2.00 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
−2.10 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
−2.20 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
−2.30 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
−2.40 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
−2.50 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
−2.60 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
−2.70 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
−2.80 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
−2.90 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
−3.00 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
−3.10 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
−3.20 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
−3.30 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
−3.40 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
Z F(z) Z F(z) Z F(z)
3.50 0.99976 73709 4.35 0.99999 31931 5.20 0.99999 99003
3.55 0.99980 73844 4.40 0.99999 45875 5.25 0.99999 99239
3.60 0.99984 08914 4.45 0.99999 57065 5.30 0.99999 99420
3.65 0.99986 88798 4.50 0.99999 66023 5.35 0.99999 99560
3.70 0.99989 22003 4.55 0.99999 73177 5.40 0.99999 99666
3.75 0.99991 15827 4.60 0.99999 78875 5.45 0.99999 99748
3.80 0.99992 76520 4.65 0.99999 83403 5.50 0.99999 99810
3.85 0.99994 09411 4.70 0.99999 86992 5.55 0.99999 99857
3.90 0.99995 19037 4.75 0.99999 89829 5.60 0.99999 99892
3.95 0.99996 09244 4.80 0.99999 92067 5.65 0.99999 99919
4.00 0.99996 83288 4.85 0.99999 93827 5.70 0.99999 99940
4.05 0.99997 43912 4.90 0.99999 95208 5.75 0.99999 99955
4.10 0.99997 93425 4.95 0.99999 96289 5.80 0.99999 99966
4.15 0.99998 33762 5.00 0.99999 97133 5.85 0.99999 99975
4.20 0.99998 66543 5.05 0.99999 97790 5.90 0.99999 99981
4.25 0.99998 93115 5.10 0.99999 98301 5.95 0.99999 99986
4.30 0.99999 14601 5.15 0.99999 98697 6.00 0.99999 99990

Appendix A-4 Values of t for a Specified Right-Tail Area

Figure depicting a normal distribution curve where point tα,v is present near the right tail. Area of curve on the right of tα,v is shaded.
df α = Right-Tail Area
ν 0.250 0.200 0.100 0.050 0.025 0.010 0.005 0.001
1 1.000 1.376 3.078 6.314 12.706 31.821 63.657 318.309
2 0.816 1.061 1.886 2.920 4.303 6.965 9.925 22.327
3 0.765 0.978 1.638 2.353 3.182 4.541 5.841 10.215
4 0.741 0.941 1.533 2.132 2.776 3.747 4.604 7.173
5 0.727 0.920 1.476 2.015 2.571 3.365 4.032 5.893
6 0.718 0.906 1.440 1.943 2.447 3.143 3.707 5.208
7 0.711 0.896 1.415 1.895 2.365 2.998 3.499 4.785
8 0.706 0.889 1.397 1.860 2.306 2.896 3.355 4.501
9 0.703 0.883 1.383 1.833 2.262 2.821 3.250 4.297
10 0.700 0.879 1.372 1.812 2.228 2.764 3.169 4.144
11 0.697 0.876 1.363 1.796 2.201 2.718 3.106 4.025
12 0.695 0.873 1.356 1.782 2.179 2.681 3.055 3.930
13 0.694 0.870 1.350 1.771 2.160 2.650 3.012 3.852
14 0.692 0.868 1.345 1.761 2.145 2.624 2.977 3.787
15 0.691 0.866 1.341 1.753 2.131 2.602 2.947 3.733
16 0.690 0.865 1.337 1.746 2.120 2.583 2.921 3.686
17 0.689 0.863 1.333 1.740 2.110 2.567 2.898 3.646
18 0.688 0.862 1.330 1.734 2.101 2.552 2.878 3.610
19 0.688 0.861 1.328 1.729 2.093 2.539 2.861 3.579
20 0.687 0.860 1.325 1.725 2.086 2.528 2.845 3.552
21 0.686 0.859 1.323 1.721 2.080 2.518 2.831 3.527
22 0.686 0.858 1.321 1.717 2.074 2.508 2.819 3.505
23 0.685 0.858 1.319 1.714 2.069 2.500 2.807 3.485
24 0.685 0.857 1.318 1.711 2.064 2.492 2.797 3.467
25 0.684 0.856 1.316 1.708 2.060 2.485 2.787 3.450
26 0.684 0.856 1.315 1.706 2.056 2.479 2.779 3.435
27 0.684 0.855 1.314 1.703 2.052 2.473 2.771 3.421
28 0.683 0.855 1.313 1.701 2.048 2.467 2.763 3.408
29 0.683 0.854 1.311 1.699 2.045 2.462 2.756 3.396
30 0.683 0.854 1.310 1.697 2.042 2.457 2.750 3.385
35 0.682 0.852 1.306 1.690 2.030 2.438 2.724 3.340
40 0.681 0.851 1.303 1.684 2.021 2.423 2.704 3.307
45 0.680 0.850 1.301 1.679 2.014 2.412 2.690 3.281
50 0.679 0.849 1.299 1.676 2.009 2.403 2.678 3.261
55 0.679 0.848 1.297 1.673 2.004 2.396 2.668 3.245
60 0.679 0.848 1.296 1.671 2.000 2.390 2.660 3.232
65 0.678 0.847 1.295 1.669 1.997 2.385 2.654 3.220
70 0.678 0.847 1.294 1.667 1.994 2.381 2.648 3.211
80 0.678 0.846 1.292 1.664 1.990 2.374 2.639 3.195
90 0.677 0.846 1.291 1.662 1.987 2.368 2.632 3.183
100 0.677 0.845 1.290 1.660 1.984 2.364 2.626 3.174
110 0.677 0.845 1.289 1.659 1.982 2.361 2.621 3.166
120 0.677 0.845 1.289 1.658 1.980 2.358 2.617 3.160
∞ 0.674 0.842 1.282 1.645 1.960 2.326 2.576 3.090

Appendix A-5 Chi-Squared Values for a Specified Right-Tail Area

Figure depicting a Chi-squared curve where a small area on the right tail is shaded and denotes α.
df α = Right-Tail Area
ν 0.999 0.995 0.990 0.975 0.950 0.900 0.100 0.050 0.025 0.010 0.005 0.001
1 0.00 0.00 0.00 0.00 0.00 0.02 2.71 3.84 5.02 6.63 7.88 10.83
2 0.00 0.01 0.02 0.05 0.10 0.21 4.61 5.99 7.38 9.21 10.60 13.82
3 0.02 0.07 0.11 0.22 0.35 0.58 6.25 7.81 9.35 11.34 12.84 16.27
4 0.09 0.21 0.30 0.48 0.71 1.06 7.78 9.49 11.14 13.28 14.86 18.47
5 0.21 0.41 0.55 0.83 1.15 1.61 9.24 11.07 12.83 15.09 16.75 20.52
6 0.38 0.68 0.87 1.24 1.64 2.20 10.64 12.59 14.45 16.81 18.55 22.46
7 0.60 0.99 1.24 1.69 2.17 2.83 12.02 14.07 16.01 18.48 20.28 24.32
8 0.86 1.34 1.65 2.18 2.73 3.49 13.36 15.51 17.53 20.09 21.95 26.12
9 1.15 1.73 2.09 2.70 3.33 4.17 14.68 16.92 19.02 21.67 23.59 27.88
10 1.48 2.16 2.56 3.25 3.94 4.87 15.99 18.31 20.48 23.21 25.19 29.59
11 1.83 2.60 3.05 3.82 4.57 5.58 17.28 19.68 21.92 24.72 26.76 31.26
12 2.21 3.07 3.57 4.40 5.23 6.30 18.55 21.03 23.34 26.22 28.30 32.91
13 2.62 3.57 4.11 5.01 5.89 7.04 19.81 22.36 24.74 27.69 29.82 34.53
14 3.04 4.07 4.66 5.63 6.57 7.79 21.06 23.68 26.12 29.14 31.32 36.12
15 3.48 4.60 5.23 6.26 7.26 8.55 22.31 25.00 27.49 30.58 32.80 37.70
16 3.94 5.14 5.81 6.91 7.96 9.31 23.54 26.30 28.85 32.00 34.27 39.25
17 4.42 5.70 6.41 7.56 8.67 10.09 24.77 27.59 30.19 33.41 35.72 40.79
18 4.90 6.26 7.01 8.23 9.39 10.86 25.99 28.87 31.53 34.81 37.16 42.31
19 5.41 6.84 7.63 8.91 10.12 11.65 27.20 30.14 32.85 36.19 38.58 43.82
20 5.92 7.43 8.26 9.59 10.85 12.44 28.41 31.41 34.17 37.57 40.00 45.31
21 6.45 8.03 8.90 10.28 11.59 13.24 29.62 32.67 35.48 38.93 41.40 46.80
22 6.98 8.64 9.54 10.98 12.34 14.04 30.81 33.92 36.78 40.29 42.80 48.27
23 7.53 9.26 10.20 11.69 13.09 14.85 32.01 35.17 38.08 41.64 44.18 49.73
24 8.08 9.89 10.86 12.40 13.85 15.66 33.20 36.42 39.36 42.98 45.56 51.18
25 8.65 10.52 11.52 13.12 14.61 16.47 34.38 37.65 40.65 44.31 46.93 52.62
26 9.22 11.16 12.20 13.84 15.38 17.29 35.56 38.89 41.92 45.64 48.29 54.05
27 9.80 11.81 12.88 14.57 16.15 18.11 36.74 40.11 43.19 46.96 49.64 55.48
28 10.39 12.46 13.56 15.31 16.93 18.94 37.92 41.34 44.46 48.28 50.99 56.89
29 10.99 13.12 14.26 16.05 17.71 19.77 39.09 42.56 45.72 49.59 52.34 58.30
30 11.59 13.79 14.95 16.79 18.49 20.60 40.26 43.77 46.98 50.89 53.67 59.70
32 12.81 15.13 16.36 18.29 20.07 22.27 42.58 46.19 49.48 53.49 56.33 62.49
34 14.06 16.50 17.79 19.81 21.66 23.95 44.90 48.60 51.97 56.06 58.96 65.25
36 15.32 17.89 19.23 21.34 23.27 25.64 47.21 51.00 54.44 58.62 61.58 67.99
38 16.61 19.29 20.69 22.88 24.88 27.34 49.51 53.38 56.90 61.16 64.18 70.70
40 17.92 20.71 22.16 24.43 26.51 29.05 51.81 55.76 59.34 63.69 66.77 73.40
42 19.24 22.14 23.65 26.00 28.14 30.77 54.09 58.12 61.78 66.21 69.34 76.08
44 20.58 23.58 25.15 27.57 29.79 32.49 56.37 60.48 64.20 68.71 71.89 78.75
46 21.93 25.04 26.66 29.16 31.44 34.22 58.64 62.83 66.62 71.20 74.44 81.40
48 23.29 26.51 28.18 30.75 33.10 35.95 60.91 65.17 69.02 73.68 76.97 84.04
50 24.67 27.99 29.71 32.36 34.76 37.69 63.17 67.50 71.42 76.15 79.49 86.66
55 28.17 31.73 33.57 36.40 38.96 42.06 68.80 73.31 77.38 82.29 85.75 93.17
60 31.74 35.53 37.48 40.48 43.19 46.46 74.40 79.08 83.30 88.38 91.95 99.61
65 35.36 39.38 41.44 44.60 47.45 50.88 79.97 84.82 89.18 94.42 98.11 105.99
70 39.04 43.28 45.44 48.76 51.74 55.33 85.53 90.53 95.02 100.43 104.21 112.32
75 42.76 47.21 49.48 52.94 56.05 59.79 91.06 96.22 100.84 106.39 110.29 118.60
80 46.52 51.17 53.54 57.15 60.39 64.28 96.58 101.88 106.63 112.33 116.32 124.84
85 50.32 55.17 57.63 61.39 64.75 68.78 102.08 107.52 112.39 118.24 122.32 131.04
90 54.16 59.20 61.75 65.65 69.13 73.29 107.57 113.15 118.14 124.12 128.30 137.21
95 58.02 63.25 65.90 69.92 73.52 77.82 113.04 118.75 123.86 129.97 134.25 143.34
100 61.92 67.33 70.06 74.22 77.93 82.36 118.50 124.34 129.56 135.81 140.17 149.45

Appendix A-6 Values of F for a Specified Right-Tail Area

Figure depicting a Chi-squared curve where a small area on the right tail is shaded and denotes α. The point is represented as Fα,v1,v2.
ν1 = Degrees of Freedom for Numerator
ν2 α 1 2 3 4 5 6 7 8 9 10 11
0.100 39.9 49.5 53.6 55.8 57.2 58.2 58.9 59.4 59.9 60.2 60.5
0.050 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.0
1 0.025 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3 968.6 973.0
0.010 4052.2 4999.5 5403.4 5624.6 5763.6 5859.0 5928.4 5981.1 6022.5 6055.8 6083.3
0.005 16210.7 19999.5 21614.7 22499.6 23055.8 23437.1 23714.6 23925.4 24091.0 24224.5 24334.4
0.001 405284.1 499999.5 540379.2 562499.6 576404.6 585937.1 592873.3 598144.2 602284.0 605621.0 608367.7
0.100 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38 9.39 9.40
0.050 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.40
2 0.025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.41
0.010 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.41
0.005 198.50 199.00 199.17 199.25 199.30 199.33 199.36 199.37 199.39 199.40 199.41
0.001 998.50 999.00 999.17 999.25 999.30 999.33 999.36 999.37 999.39 999.40 999.41
0.100 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24 5.23 5.22
0.050 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.76
3 0.025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.37
0.010 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.13
0.005 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.88 43.69 43.52
0.001 167.03 148.50 141.11 137.10 134.58 132.85 131.58 130.62 129.86 129.25 128.74
0.100 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94 3.92 3.91
0.050 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.94
4 0.025 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.79
0.010 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.45
0.005 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 21.14 20.97 20.82
0.001 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 48.47 48.05 47.70
0.100 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32 3.30 3.28
0.050 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.70
5 0.025 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.57
0.010 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.96
0.005 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.77 13.62 13.49
0.001 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 27.24 26.92 26.65
0.100 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96 2.94 2.92
0.050 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.03
6 0.025 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.41
0.010 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.79
0.005 18.63 14.54 12.92 12.03 11.46 11.07 10.79 10.57 10.39 10.25 10.13
0.001 35.51 27.00 23.70 21.92 20.80 20.03 19.46 19.03 18.69 18.41 18.18
0.100 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72 2.70 2.68
0.050 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.60
7 0.025 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.71
0.010 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.54
0.005 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.51 8.38 8.27
0.001 29.25 21.69 18.77 17.20 16.21 15.52 15.02 14.63 14.33 14.08 13.88
0.100 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56 2.54 2.52
0.050 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.31
8 0.025 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.24
0.010 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.73
0.005 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.34 7.21 7.10
0.001 25.41 18.49 15.83 14.39 13.48 12.86 12.40 12.05 11.77 11.54 11.35
0.100 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44 2.42 2.40
0.050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.10
9 0.025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.91
0.010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.18
0.005 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.54 6.42 6.31
0.001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 10.11 9.89 9.72
0.100 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35 2.32 2.30
0.050 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.94
10 0.025 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.66
0.010 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.77
0.005 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.97 5.85 5.75
0.001 21.04 14.91 12.55 11.28 10.48 9.93 9.52 9.20 8.96 8.75 8.59
0.100 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27 2.25 2.23
0.050 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.82
11 0.025 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53 3.47
0.010 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.46
0.005 12.23 8.91 7.60 6.88 6.42 6.10 5.86 5.68 5.54 5.42 5.32
0.001 19.69 13.81 11.56 10.35 9.58 9.05 8.66 8.35 8.12 7.92 7.76
0.100 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21 2.19 2.17
0.050 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.72
12 0.025 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.32
0.010 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.22
0.005 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 5.20 5.09 4.99
0.001 18.64 12.97 10.80 9.63 8.89 8.38 8.00 7.71 7.48 7.29 7.14
0.100 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09 2.06 2.04
0.050 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.51
15 0.025 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06 3.01
0.010 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.73
0.005 10.80 7.70 6.48 5.80 5.37 5.07 4.85 4.67 4.54 4.42 4.33
0.001 16.59 11.34 9.34 8.25 7.57 7.09 6.74 6.47 6.26 6.08 5.94
0.100 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00 1.98 1.95
0.050 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.37
18 0.025 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93 2.87 2.81
0.010 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.43
0.005 10.22 7.21 6.03 5.37 4.96 4.66 4.44 4.28 4.14 4.03 3.94
0.001 15.38 10.39 8.49 7.46 6.81 6.35 6.02 5.76 5.56 5.39 5.25
0.100 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96 1.94 1.91
0.050 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.31
20 0.025 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.72
0.010 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.29
0.005 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.96 3.85 3.76
0.001 14.82 9.95 8.10 7.10 6.46 6.02 5.69 5.44 5.24 5.08 4.94
0.100 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89 1.87 1.84
0.050 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.20
25 0.025 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68 2.61 2.56
0.010 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22 3.13 3.06
0.005 9.48 6.60 5.46 4.84 4.43 4.15 3.94 3.78 3.64 3.54 3.45
0.001 13.88 9.22 7.45 6.49 5.89 5.46 5.15 4.91 4.71 4.56 4.42
0.100 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85 1.82 1.79
0.050 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.13
30 0.025 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57 2.51 2.46
0.010 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.91
0.005 9.18 6.35 5.24 4.62 4.23 3.95 3.74 3.58 3.45 3.34 3.25
0.001 13.29 8.77 7.05 6.12 5.53 5.12 4.82 4.58 4.39 4.24 4.11
0.100 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79 1.76 1.74
0.050 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.04
40 0.025 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45 2.39 2.33
0.010 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.73
0.005 8.83 6.07 4.98 4.37 3.99 3.71 3.51 3.35 3.22 3.12 3.03
0.001 12.61 8.25 6.59 5.70 5.13 4.73 4.44 4.21 4.02 3.87 3.75
0.100 2.81 2.41 2.20 2.06 1.97 1.90 1.84 1.80 1.76 1.73 1.70
0.050 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 1.99
50 0.025 5.34 3.97 3.39 3.05 2.83 2.67 2.55 2.46 2.38 2.32 2.26
0.010 7.17 5.06 4.20 3.72 3.41 3.19 3.02 2.89 2.78 2.70 2.63
0.005 8.63 5.90 4.83 4.23 3.85 3.58 3.38 3.22 3.09 2.99 2.90
0.001 12.22 7.96 6.34 5.46 4.90 4.51 4.22 4.00 3.82 3.67 3.55
0.100 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74 1.71 1.68
0.050 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.95
60 0.025 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33 2.27 2.22
0.010 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.56
0.005 8.49 5.79 4.73 4.14 3.76 3.49 3.29 3.13 3.01 2.90 2.82
0.001 11.97 7.77 6.17 5.31 4.76 4.37 4.09 3.86 3.69 3.54 3.42
0.100 2.76 2.36 2.14 2.00 1.91 1.83 1.78 1.73 1.69 1.66 1.64
0.050 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97 1.93 1.89
100 0.025 5.18 3.83 3.25 2.92 2.70 2.54 2.42 2.32 2.24 2.18 2.12
0.010 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.59 2.50 2.43
0.005 8.24 5.59 4.54 3.96 3.59 3.33 3.13 2.97 2.85 2.74 2.66
0.001 11.50 7.41 5.86 5.02 4.48 4.11 3.83 3.61 3.44 3.30 3.18
0.100 2.71 2.30 2.08 1.95 1.85 1.77 1.72 1.67 1.63 1.60 1.57
0.050 3.84 3.00 2.61 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.79
∞ 0.025 5.03 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05 1.99
0.010 6.64 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.25
0.005 7.88 5.30 4.28 3.72 3.35 3.09 2.90 2.75 2.62 2.52 2.43
0.001 10.83 6.91 5.43 4.62 4.11 3.75 3.48 3.27 3.10 2.96 2.85
ν1 = Degrees of Freedom for Numerator
ν2 α 12 15 20 40 50 60 75 100 120 ∞
0.100 60.7 61.2 61.7 62.5 62.7 62.8 62.9 63.0 63.1 63.32
0.050 243.9 245.9 248.0 251.1 251.8 252.2 252.6 253.0 253.3 254.30
1 0.025 976.7 984.9 993.1 1005.6 1008.1 1009.8 1011.5 1013.2 1014.0 1018.21
0.010 6106.3 6157.3 6208.7 6286.8 6302.5 6313.0 6323.6 6334.1 6339.4 6365.55
0.005 24426.4 24630.2 24836.0 25148.2 25211.1 25253.1 25295.3 25337.5 25358.6 25463.18
0.001 610667.8 615763.7 620907.7 628712.0 630285.4 631336.6 632389.5 633444.3 633972.4 636587.61
0.100 9.41 9.42 9.44 9.47 9.47 9.47 9.48 9.48 9.48 9.49
0.050 19.41 19.43 19.45 19.47 19.48 19.48 19.48 19.49 19.49 19.50
2 0.025 39.41 39.43 39.45 39.47 39.48 39.48 39.48 39.49 39.49 39.50
0.010 99.42 99.43 99.45 99.47 99.48 99.48 99.49 99.49 99.49 99.50
0.005 199.42 199.43 199.45 199.47 199.48 199.48 199.49 199.49 199.49 199.50
0.001 999.42 999.43 999.45 999.47 999.48 999.48 999.49 999.49 999.49 999.50
0.100 5.22 5.20 5.18 5.16 5.15 5.15 5.15 5.14 5.14 5.13
0.050 8.74 8.70 8.66 8.59 8.58 8.57 8.56 8.55 8.55 8.53
3 0.025 14.34 14.25 14.17 14.04 14.01 13.99 13.97 13.96 13.95 13.90
0.010 27.05 26.87 26.69 26.41 26.35 26.32 26.28 26.24 26.22 26.13
0.005 43.39 43.08 42.78 42.31 42.21 42.15 42.09 42.02 41.99 41.83
0.001 128.32 127.37 126.42 124.96 124.66 124.47 124.27 124.07 123.97 123.48
0.100 3.90 3.87 3.84 3.80 3.80 3.79 3.78 3.78 3.78 3.76
0.050 5.91 5.86 5.80 5.72 5.70 5.69 5.68 5.66 5.66 5.63
4 0.025 8.75 8.66 8.56 8.41 8.38 8.36 8.34 8.32 8.31 8.26
0.010 14.37 14.20 14.02 13.75 13.69 13.65 13.61 13.58 13.56 13.46
0.005 20.70 20.44 20.17 19.75 19.67 19.61 19.55 19.50 19.47 19.33
0.001 47.41 46.76 46.10 45.09 44.88 44.75 44.61 44.47 44.40 44.06
0.100 3.27 3.24 3.21 3.16 3.15 3.14 3.13 3.13 3.12 3.11
0.050 4.68 4.62 4.56 4.46 4.44 4.43 4.42 4.41 4.40 4.37
5 0.025 6.52 6.43 6.33 6.18 6.14 6.12 6.10 6.08 6.07 6.02
0.010 9.89 9.72 9.55 9.29 9.24 9.20 9.17 9.13 9.11 9.02
0.005 13.38 13.15 12.90 12.53 12.45 12.40 12.35 12.30 12.27 12.15
0.001 26.42 25.91 25.39 24.60 24.44 24.33 24.22 24.12 24.06 23.79
0.100 2.90 2.87 2.84 2.78 2.77 2.76 2.75 2.75 2.74 2.72
0.050 4.00 3.94 3.87 3.77 3.75 3.74 3.73 3.71 3.70 3.67
6 0.025 5.37 5.27 5.17 5.01 4.98 4.96 4.94 4.92 4.90 4.85
0.010 7.72 7.56 7.40 7.14 7.09 7.06 7.02 6.99 6.97 6.88
0.005 10.03 9.81 9.59 9.24 9.17 9.12 9.07 9.03 9.00 8.88
0.001 17.99 17.56 17.12 16.44 16.31 16.21 16.12 16.03 15.98 15.75
0.100 2.67 2.63 2.59 2.54 2.52 2.51 2.51 2.50 2.49 2.47
0.050 3.57 3.51 3.44 3.34 3.32 3.30 3.29 3.27 3.27 3.23
7 0.025 4.67 4.57 4.47 4.31 4.28 4.25 4.23 4.21 4.20 4.14
0.010 6.47 6.31 6.16 5.91 5.86 5.82 5.79 5.75 5.74 5.65
0.005 8.18 7.97 7.75 7.42 7.35 7.31 7.26 7.22 7.19 7.08
0.001 13.71 13.32 12.93 12.33 12.20 12.12 12.04 11.95 11.91 11.70
0.100 2.50 2.46 2.42 2.36 2.35 2.34 2.33 2.32 2.32 2.29
0.050 3.28 3.22 3.15 3.04 3.02 3.01 2.99 2.97 2.97 2.93
8 0.025 4.20 4.10 4.00 3.84 3.81 3.78 3.76 3.74 3.73 3.67
0.010 5.67 5.52 5.36 5.12 5.07 5.03 5.00 4.96 4.95 4.86
0.005 7.01 6.81 6.61 6.29 6.22 6.18 6.13 6.09 6.06 5.95
0.001 11.19 10.84 10.48 9.92 9.80 9.73 9.65 9.57 9.53 9.34
0.100 2.38 2.34 2.30 2.23 2.22 2.21 2.20 2.19 2.18 2.16
0.050 3.07 3.01 2.94 2.83 2.80 2.79 2.77 2.76 2.75 2.71
9 0.025 3.87 3.77 3.67 3.51 3.47 3.45 3.43 3.40 3.39 3.33
0.010 5.11 4.96 4.81 4.57 4.52 4.48 4.45 4.41 4.40 4.31
0.005 6.23 6.03 5.83 5.52 5.45 5.41 5.37 5.32 5.30 5.19
0.001 9.57 9.24 8.90 8.37 8.26 8.19 8.11 8.04 8.00 7.82
0.100 2.28 2.24 2.20 2.13 2.12 2.11 2.10 2.09 2.08 2.06
0.050 2.91 2.85 2.77 2.66 2.64 2.62 2.60 2.59 2.58 2.54
10 0.025 3.62 3.52 3.42 3.26 3.22 3.20 3.18 3.15 3.14 3.08
0.010 4.71 4.56 4.41 4.17 4.12 4.08 4.05 4.01 4.00 3.91
0.005 5.66 5.47 5.27 4.97 4.90 4.86 4.82 4.77 4.75 4.64
0.001 8.45 8.13 7.80 7.30 7.19 7.12 7.05 6.98 6.94 6.76
0.100 2.21 2.17 2.12 2.05 2.04 2.03 2.02 2.01 2.00 1.97
0.050 2.79 2.72 2.65 2.53 2.51 2.49 2.47 2.46 2.45 2.41
11 0.025 3.43 3.33 3.23 3.06 3.03 3.00 2.98 2.96 2.94 2.88
0.010 4.40 4.25 4.10 3.86 3.81 3.78 3.74 3.71 3.69 3.60
0.005 5.24 5.05 4.86 4.55 4.49 4.45 4.40 4.36 4.34 4.23
0.001 7.63 7.32 7.01 6.52 6.42 6.35 6.28 6.21 6.18 6.00
0.100 2.15 2.10 2.06 1.99 1.97 1.96 1.95 1.94 1.93 1.90
0.050 2.69 2.62 2.54 2.43 2.40 2.38 2.37 2.35 2.34 2.30
12 0.025 3.28 3.18 3.07 2.91 2.87 2.85 2.82 2.80 2.79 2.73
0.010 4.16 4.01 3.86 3.62 3.57 3.54 3.50 3.47 3.45 3.36
0.005 4.91 4.72 4.53 4.23 4.17 4.12 4.08 4.04 4.01 3.91
0.001 7.00 6.71 6.40 5.93 5.83 5.76 5.70 5.63 5.59 5.42
0.100 2.02 1.97 1.92 1.85 1.83 1.82 1.80 1.79 1.79 1.76
0.050 2.48 2.40 2.33 2.20 2.18 2.16 2.14 2.12 2.11 2.07
15 0.025 2.96 2.86 2.76 2.59 2.55 2.52 2.50 2.47 2.46 2.40
0.010 3.67 3.52 3.37 3.13 3.08 3.05 3.01 2.98 2.96 2.87
0.005 4.25 4.07 3.88 3.58 3.52 3.48 3.44 3.39 3.37 3.26
0.001 5.81 5.54 5.25 4.80 4.70 4.64 4.57 4.51 4.47 4.31
0.100 1.93 1.89 1.84 1.75 1.74 1.72 1.71 1.70 1.69 1.66
0.050 2.34 2.27 2.19 2.06 2.04 2.02 2.00 1.98 1.97 1.92
18 0.025 2.77 2.67 2.56 2.38 2.35 2.32 2.30 2.27 2.26 2.19
0.010 3.37 3.23 3.08 2.84 2.78 2.75 2.71 2.68 2.66 2.57
0.005 3.86 3.68 3.50 3.20 3.14 3.10 3.05 3.01 2.99 2.87
0.001 5.13 4.87 4.59 4.15 4.06 4.00 3.93 3.87 3.84 3.67
0.100 1.89 1.84 1.79 1.71 1.69 1.68 1.66 1.65 1.64 1.61
0.050 2.28 2.20 2.12 1.99 1.97 1.95 1.93 1.91 1.90 1.84
20 0.025 2.68 2.57 2.46 2.29 2.25 2.22 2.20 2.17 2.16 2.09
0.010 3.23 3.09 2.94 2.69 2.64 2.61 2.57 2.54 2.52 2.42
0.005 3.68 3.50 3.32 3.02 2.96 2.92 2.87 2.83 2.81 2.69
0.001 4.82 4.56 4.29 3.86 3.77 3.70 3.64 3.58 3.54 3.38
0.100 1.82 1.77 1.72 1.63 1.61 1.59 1.58 1.56 1.56 1.52
0.050 2.16 2.09 2.01 1.87 1.84 1.82 1.80 1.78 1.77 1.71
25 0.025 2.51 2.41 2.30 2.12 2.08 2.05 2.02 2.00 1.98 1.91
0.010 2.99 2.85 2.70 2.45 2.40 2.36 2.33 2.29 2.27 2.17
0.005 3.37 3.20 3.01 2.72 2.65 2.61 2.56 2.52 2.50 2.38
0.001 4.31 4.06 3.79 3.37 3.28 3.22 3.15 3.09 3.06 2.89
0.100 1.77 1.72 1.67 1.57 1.55 1.54 1.52 1.51 1.50 1.46
0.050 2.09 2.01 1.93 1.79 1.76 1.74 1.72 1.70 1.68 1.62
30 0.025 2.41 2.31 2.20 2.01 1.97 1.94 1.91 1.88 1.87 1.79
0.010 2.84 2.70 2.55 2.30 2.25 2.21 2.17 2.13 2.11 2.01
0.005 3.18 3.01 2.82 2.52 2.46 2.42 2.37 2.32 2.30 2.18
0.001 4.00 3.75 3.49 3.07 2.98 2.92 2.86 2.79 2.76 2.59
0.100 1.71 1.66 1.61 1.51 1.48 1.47 1.45 1.43 1.42 1.38
0.050 2.00 1.92 1.84 1.69 1.66 1.64 1.61 1.59 1.58 1.51
40 0.025 2.29 2.18 2.07 1.88 1.83 1.80 1.77 1.74 1.72 1.64
0.010 2.66 2.52 2.37 2.11 2.06 2.02 1.98 1.94 1.92 1.81
0.005 2.95 2.78 2.60 2.30 2.23 2.18 2.14 2.09 2.06 1.93
0.001 3.64 3.40 3.14 2.73 2.64 2.57 2.51 2.44 2.41 2.23
0.100 1.68 1.63 1.57 1.46 1.44 1.42 1.41 1.39 1.38 1.33
0.050 1.95 1.87 1.78 1.63 1.60 1.58 1.55 1.52 1.51 1.44
50 0.025 2.22 2.11 1.99 1.80 1.75 1.72 1.69 1.66 1.64 1.55
0.010 2.56 2.42 2.27 2.01 1.95 1.91 1.87 1.82 1.80 1.68
0.005 2.82 2.65 2.47 2.16 2.10 2.05 2.00 1.95 1.93 1.79
0.001 3.44 3.20 2.95 2.53 2.44 2.38 2.31 2.25 2.21 2.03
0.100 1.66 1.60 1.54 1.44 1.41 1.40 1.38 1.36 1.35 1.29
0.050 1.92 1.84 1.75 1.59 1.56 1.53 1.51 1.48 1.47 1.39
60 0.025 2.17 2.06 1.94 1.74 1.70 1.67 1.63 1.60 1.58 1.48
0.010 2.50 2.35 2.20 1.94 1.88 1.84 1.79 1.75 1.73 1.60
0.005 2.74 2.57 2.39 2.08 2.01 1.96 1.91 1.86 1.83 1.69
0.001 3.32 3.08 2.83 2.41 2.32 2.25 2.19 2.12 2.08 1.89
0.100 1.61 1.56 1.49 1.38 1.35 1.34 1.32 1.29 1.28 1.22
0.050 1.85 1.77 1.68 1.52 1.48 1.45 1.42 1.39 1.38 1.28
100 0.025 2.08 1.97 1.85 1.64 1.59 1.56 1.52 1.48 1.46 1.35
0.010 2.37 2.22 2.07 1.80 1.74 1.69 1.65 1.60 1.57 1.43
0.005 2.58 2.41 2.23 1.91 1.84 1.79 1.74 1.68 1.65 1.49
0.001 3.07 2.84 2.59 2.17 2.08 2.01 1.94 1.87 1.83 1.62
0.100 1.55 1.49 1.42 1.30 1.26 1.24 1.22 1.19 1.17 1.00
0.050 1.75 1.67 1.57 1.40 1.35 1.32 1.28 1.25 1.22 1.00
∞ 0.025 1.95 1.83 1.71 1.49 1.43 1.39 1.35 1.30 1.27 1.00
0.010 2.19 2.04 1.88 1.59 1.53 1.48 1.42 1.36 1.33 1.00
0.005 2.36 2.19 2.00 1.67 1.59 1.54 1.47 1.40 1.37 1.00
0.001 2.75 2.52 2.27 1.84 1.74 1.66 1.58 1.50 1.45 1.00

Appendix A-7 Factors for Computing Centerline and Three-Sigma Control Limits

Observations in Sample, n X-Charts s-Charts R-Charts
Factors for Control Limits Factors for Centerline Factors for Control Limits Factors for Centerline Factors for Control Limits
A A2 A3 c4 1/c4 B3 B4 B5 B6 d2 1/d2 d3 D1 D2 D3 D4
2 2.121 1.880 2.659 0.7979 1.2533 0 3.267 0 2.606 1.128 0.8865 0.853 0 3.686 0 3.267
3 1.732 1.023 1.954 0.8862 1.1284 0 2.568 0 2.276 1.693 0.5907 0.888 0 4.358 0 2.574
4 1.500 0.729 1.628 0.9213 1.0854 0 2.266 0 2.088 2.059 0.4857 0.880 0 4.698 0 2.282
5 1.342 0.577 1.427 0.9400 1.0638 0 2.089 0 1.964 2.326 0.4299 0.864 0 4.918 0 2.114
6 1.225 0.483 1.287 0.9515 1.0510 0.030 1.970 0.029 1.874 2.534 0.3946 0.848 0 5.078 0 2.004
7 1.134 0.419 1.182 0.9594 1.0423 0.118 1.882 0.113 1.806 2.704 0.3698 0.833 0.204 5.204 0.076 1.924
8 1.061 0.373 1.099 0.9650 1.0363 0.185 1.815 0.179 1.751 2.847 0.3512 0.820 0.388 5.306 0.136 1.864
9 1.000 0.337 1.032 0.9693 1.0317 0.239 1.761 0.232 1.707 2.970 0.3367 0.808 0.547 5.393 0.184 1.816
10 0.949 0.308 0.975 0.9727 1.0281 0.284 1.716 0.276 1.669 3.078 0.3249 0.797 0.687 5.469 0.223 1.777
11 0.905 0.285 0.927 0.9754 1.0252 0.321 1.679 0.313 1.637 3.173 0.3152 0.787 0.811 5.535 0.256 1.744
12 0.866 0.266 0.886 0.9776 1.0229 0.354 1.646 0.346 1.610 3.258 0.3069 0.778 0.922 5.594 0.283 1.717
13 0.832 0.249 0.850 0.9794 1.0210 0.382 1.618 0.374 1.585 3.336 0.2998 0.770 1.025 5.647 0.307 1.693
14 0.802 0.235 0.817 0.9810 1.0194 0.406 1.594 0.399 1.563 3.407 0.2935 0.763 1.118 5.696 0.328 1.672
15 0.775 0.223 0.789 0.9823 1.0180 0.428 1.572 0.421 1.544 3.472 0.2880 0.756 1.203 5.741 0.347 1.653
16 0.750 0.212 0.763 0.9835 1.0168 0.448 1.552 0.440 1.526 3.532 0.2831 0.750 1.282 5.782 0.363 1.637
17 0.728 0.203 0.739 0.9845 1.0157 0.466 1.534 0.458 1.511 3.588 0.2787 0.744 1.356 5.820 0.378 1.622
18 0.707 0.194 0.718 0.9854 1.0148 0.482 1.518 0.475 1.496 3.640 0.2747 0.739 1.424 5.856 0.391 1.608
19 0.688 0.187 0.698 0.9862 1.0140 0.497 1.503 0.490 1.483 3.689 0.2711 0.734 1.487 5.891 0.403 1.597
20 0.671 0.180 0.680 0.9869 1.0133 0.510 1.490 0.504 1.470 3.735 0.2677 0.729 1.549 5.921 0.415 1.585
21 0.655 0.173 0.663 0.9876 1.0126 0.523 1.477 0.516 1.459 3.778 0.2647 0.724 1.605 5.951 0.425 1.575
22 0.640 0.167 0.647 0.9882 1.0119 0.534 1.466 0.528 1.448 3.819 0.2618 0.720 1.659 5.979 0.434 1.566
23 0.626 0.162 0.633 0.9887 1.0114 0.545 1.455 0.539 1.438 3.858 0.2592 0.716 1.710 6.006 0.443 1.557
24 0.612 0.157 0.619 0.9892 1.0109 0.555 1.445 0.549 1.429 3.895 0.2567 0.712 1.759 6.031 0.451 1.548
25 0.600 0.153 0.606 0.9896 1.0105 0.565 1.435 0.559 1.420 3.931 0.2544 0.708 1.806 6.056 0.459 1.541
Source: © American Society for Testing and Materials (ASTM). Reprinted with permission.

Appendix A-8 Uniform Random Numbers

97878 44645 60468 86596 29743 98439 64428 50357
64600 47935 55776 38732 70498 61832 11372 30484
21734 98488 94734 99531 25282 24016 50366 11021
71402 60964 28456 72686 45225 00076 81351 82365
00358 30750 23487 57473 60720 34874 64186 80531
40782 28908 67197 86933 87919 65522 30539 71547
76722 69512 95964 74114 44095 12630 81913 13102
11137 92332 34009 04099 92740 95264 04667 40145
23984 99243 90979 95199 25357 84703 07202 51970
30262 22720 64621 91122 00940 07670 24647 58469
41609 32500 37060 75444 99182 59388 38806 14520
17469 07308 37637 75591 67256 74415 97339 84763
85130 96392 70015 42639 36885 35159 20170 98134
04204 59281 44421 93374 42647 75392 25164 45359
02387 34404 31376 52748 41546 70173 03409 96409
98770 39674 03575 97601 91398 39995 10671 61442
68709 98636 78218 25617 46255 22234 88613 31217
56293 30564 01867 87371 50834 67311 96809 72744
35017 39984 13007 16757 37348 09247 22734 01217
03193 76349 97895 26047 80563 29319 70426 64120
67759 40380 74450 91825 03074 66039 28096 22809
18755 25573 14639 38260 47489 58234 50219 32596
49269 80057 24228 09605 34931 01224 76877 87911
97434 82611 58899 30042 16356 01293 31830 69230
68837 56094 82048 93441 72467 62565 38400 99459
03797 28132 17109 57402 62259 98531 14472 35450
57828 30374 98465 13151 43132 32193 78184 50939
32393 43266 85401 41622 45396 80588 53661 83531
54176 40988 97983 10019 97341 14550 47511 18987
82120 01302 88694 94271 22290 39296 63110 02916
25874 68357 81583 69948 24461 25326 60400 54006
76479 94047 52701 61100 21783 11980 48124 30173
29314 93695 41337 67648 70136 83831 54796 56998
08217 12006 91741 94542 68309 73687 66076 51625
83677 66495 97931 36805 92243 35255 38746 20177
59042 24587 47017 23384 29948 40591 17259 78456
27293 16782 69543 20522 15442 37452 62532 86340
44154 11467 26355 64544 00741 63428 49008 79631
56887 08762 65555 94922 56064 72304 13685 13303
02523 15422 14344 90718 98416 79818 15141 90939
74141 88274 34566 57489 34419 85325 64590 71890
28175 40295 48870 65330 13591 31724 96311 09956
99599 73883 62419 89694 87842 72571 09171 47657
84925 48510 18369 35883 88947 33572 51302 43004
46361 94992 31390 72792 18506 42730 72923 20226
91086 96394 97533 42656 20758 94888 83053 80529
60435 33374 18112 25029 62553 32646 84162 21814
30737 01996 59146 76739 63951 31707 04183 04168
33045 97426 48039 92940 47647 19067 75199 72413
06223 02998 98687 40526 32807 89534 02039 76278
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