### Critical values of R(k,s) for k up to 30 and s up to 20, for alpha values of .1, .05, .025, .01, .005, .0025, .001

The p-product method works with the significance level p computed for each of k mutually independent experiments. The meta-analyst ranks the p's, selects some positive integer s typically much smaller than k, and computes R, the product of the s smallest values of p. R is used to test the composite null hypothesis that all k individual null hypotheses are true. The critical value of R depends on k, s, and alpha. The tables presented here are designed to supplement tables in the first and third documents posted in this section.

Reading the table. For instance, for alpha = .10, the critical value of R(3,2) is .00859. R(k,s) is the product of the smallest s of k independent values of p.

A detailed explanation of the table's derivation appears after the table.

``` alpha = 0.1000
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
k
2   0.0204
3   0.00859   0.00489
4   0.00470   0.00188   0.00123
5   0.00296   0.000913  0.000458  0.000336
6   0.00204   0.000510  0.000207  0.000120  9.17E-05
7   0.00149   0.000314  0.000107  5.15E-05  3.26E-05  2.61E-05
8   0.00113   0.000207  6.05E-05  2.50E-05  1.35E-05  9.20E-06  7.67E-06
9   0.000891  0.000143  3.69E-05  1.33E-05  6.27E-06  3.71E-06  2.66E-06  2.27E-06
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
10   0.000720  0.000104  2.37E-05  7.59E-06  3.17E-06  1.66E-06  1.04E-06  7.77E-07  6.81E-07
11   0.000594  7.73E-05  1.60E-05  4.59E-06  1.72E-06  8.06E-07  4.52E-07  2.99E-07  2.31E-07  2.04E-07
12   0.000498  5.92E-05  1.11E-05  2.91E-06  9.90E-07  4.19E-07  2.12E-07  1.26E-07  8.72E-08  6.90E-08  6.12E-08
13   0.000424  4.64E-05  7.99E-06  1.92E-06  5.96E-07  2.30E-07  1.06E-07  5.76E-08  3.61E-08  2.58E-08  2.08E-08  1.88E-08
14   0.000365  3.70E-05  5.89E-06  1.31E-06  3.74E-07  1.33E-07  5.63E-08  2.80E-08  1.61E-08  1.05E-08  7.74E-09  6.36E-09  5.80E-09
15   0.000318  3.00E-05  4.44E-06  9.15E-07  2.42E-07  7.97E-08  3.13E-08  1.44E-08  7.62E-09  4.59E-09  3.11E-09  2.35E-09  1.97E-09  1.80E-09
16   0.000279  2.46E-05  3.41E-06  6.56E-07  1.62E-07  4.95E-08  1.81E-08  7.75E-09  3.81E-09  2.13E-09  1.34E-09  9.36E-10  7.23E-10  6.10E-10  5.59E-10
17   0.000247  2.05E-05  2.66E-06  4.80E-07  1.11E-07  3.17E-08  1.09E-08  4.35E-09  2.00E-09  1.04E-09  6.11E-10  3.97E-10  2.85E-10  2.23E-10  1.90E-10  1.74E-10
18   0.000220  1.72E-05  2.11E-06  3.58E-07  7.78E-08  2.09E-08  6.73E-09  2.53E-09  1.09E-09  5.34E-10  2.93E-10  1.78E-10  1.19E-10  8.69E-11  6.95E-11  5.98E-11  5.53E-11
19   0.000198  1.46E-05  1.69E-06  2.71E-07  5.56E-08  1.41E-08  4.28E-09  1.52E-09  6.17E-10  2.84E-10  1.46E-10  8.35E-11  5.25E-11  3.59E-11  2.69E-11  2.18E-11  1.89E-11  1.75E-11
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
20   0.000178  1.25E-05  1.37E-06  2.08E-07  4.05E-08  9.71E-09  2.79E-09  9.35E-10  3.60E-10  1.57E-10  7.61E-11  4.09E-11  2.42E-11  1.56E-11  1.10E-11  8.39E-12  6.86E-12  6.00E-12  5.61E-12
21   0.000162  1.08E-05  1.12E-06  1.62E-07  3.00E-08  6.81E-09  1.86E-09  5.91E-10  2.16E-10  8.90E-11  4.10E-11  2.08E-11  1.17E-11  7.12E-12  4.73E-12  3.40E-12  2.62E-12  2.17E-12  1.91E-12
22   0.000147  9.35E-06  9.29E-07  1.28E-07  2.25E-08  4.87E-09  1.27E-09  3.82E-10  1.33E-10  5.20E-11  2.27E-11  1.10E-11  5.82E-12  3.37E-12  2.12E-12  1.44E-12  1.05E-12  8.25E-13  6.88E-13
23   0.000135  8.16E-06  7.75E-07  1.02E-07  1.71E-08  3.53E-09  8.76E-10  2.52E-10  8.35E-11  3.12E-11  1.30E-11  5.96E-12  3.01E-12  1.66E-12  9.89E-13  6.36E-13  4.42E-13  3.28E-13  2.60E-13
24   0.000124  7.17E-06  6.52E-07  8.17E-08  1.32E-08  2.60E-09  6.16E-10  1.70E-10  5.36E-11  1.91E-11  7.58E-12  3.33E-12  1.60E-12  8.41E-13  4.78E-13  2.92E-13  1.93E-13  1.36E-13  1.03E-13
25   0.000114  6.33E-06  5.52E-07  6.62E-08  1.02E-08  1.94E-09  4.40E-10  1.16E-10  3.50E-11  1.19E-11  4.53E-12  1.90E-12  8.73E-13  4.39E-13  2.38E-13  1.39E-13  8.74E-14  5.88E-14  4.22E-14
26   0.000105  5.62E-06  4.70E-07  5.41E-08  8.04E-09  1.46E-09  3.18E-10  8.06E-11  2.33E-11  7.61E-12  2.77E-12  1.11E-12  4.89E-13  2.36E-13  1.22E-13  6.82E-14  4.10E-14  2.63E-14  1.81E-14
27   9.76E-05  5.01E-06  4.03E-07  4.46E-08  6.37E-09  1.12E-09  2.33E-10  5.67E-11  1.58E-11  4.93E-12  1.72E-12  6.64E-13  2.80E-13  1.30E-13  6.45E-14  3.45E-14  1.98E-14  1.22E-14  8.01E-15
28   9.07E-05  4.48E-06  3.48E-07  3.70E-08  5.09E-09  8.60E-10  1.73E-10  4.05E-11  1.08E-11  3.25E-12  1.09E-12  4.04E-13  1.64E-13  7.28E-14  3.49E-14  1.79E-14  9.89E-15  5.84E-15  3.67E-15
29   8.46E-05  4.03E-06  3.01E-07  3.08E-08  4.11E-09  6.69E-10  1.30E-10  2.92E-11  7.50E-12  2.17E-12  7.00E-13  2.50E-13  9.78E-14  4.18E-14  1.93E-14  9.57E-15  5.07E-15  2.88E-15  1.74E-15
30   7.90E-05  3.63E-06  2.63E-07  2.59E-08  3.33E-09  5.26E-10  9.81E-11  2.14E-11  5.27E-12  1.47E-12  4.57E-13  1.58E-13  5.93E-14  2.44E-14  1.09E-14  5.23E-15  2.67E-15  1.46E-15  8.48E-16
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20

alpha = 0.0500
k
2   0.00859
3   0.00350   0.00183
4   0.00188   0.000674  0.000413
5   0.00118   0.000319  0.000149  0.000102
6   0.000804  0.000176  6.57E-05  3.56E-05  2.65E-05
7   0.000585  0.000107  3.33E-05  1.49E-05  9.05E-06  7.00E-06
8   0.000445  7.01E-05  1.86E-05  7.09E-06  3.63E-06  2.36E-06  1.94E-06
9   0.000350  4.84E-05  1.12E-05  3.71E-06  1.64E-06  9.21E-07  6.42E-07  5.43E-07
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
10   0.000282  3.49E-05  7.15E-06  2.09E-06  8.14E-07  4.01E-07  2.43E-07  1.78E-07  1.56E-07
11   0.000233  2.60E-05  4.77E-06  1.25E-06  4.35E-07  1.91E-07  1.02E-07  6.58E-08  5.00E-08  4.38E-08
12   0.000195  1.98E-05  3.31E-06  7.85E-07  2.47E-07  9.73E-08  4.69E-08  2.70E-08  1.81E-08  1.41E-08  1.23E-08
13   0.000166  1.55E-05  2.36E-06  5.13E-07  1.47E-07  5.28E-08  2.31E-08  1.20E-08  7.24E-09  5.05E-09  3.98E-09  3.56E-09
14   0.000143  1.24E-05  1.73E-06  3.46E-07  9.13E-08  3.01E-08  1.20E-08  5.72E-09  3.14E-09  1.99E-09  1.42E-09  1.15E-09  1.04E-09
15   0.000124  1.00E-05  1.30E-06  2.41E-07  5.87E-08  1.79E-08  6.61E-09  2.89E-09  1.46E-09  8.46E-10  5.55E-10  4.10E-10  3.37E-10  3.09E-10
16   0.000109  8.21E-06  9.96E-07  1.72E-07  3.90E-08  1.10E-08  3.78E-09  1.53E-09  7.15E-10  3.84E-10  2.33E-10  1.58E-10  1.19E-10  9.97E-11  8.86E-11
17   9.65E-05  6.82E-06  7.75E-07  1.25E-07  2.66E-08  7.01E-09  2.25E-09  8.49E-10  3.69E-10  1.84E-10  1.04E-10  6.54E-11  4.55E-11  3.51E-11  2.94E-11  2.66E-11
18   8.60E-05  5.72E-06  6.12E-07  9.29E-08  1.85E-08  4.59E-09  1.38E-09  4.88E-10  1.99E-10  9.29E-11  4.89E-11  2.87E-11  1.86E-11  1.33E-11  1.05E-11  8.89E-12  8.40E-12
19   7.71E-05  4.85E-06  4.90E-07  7.02E-08  1.32E-08  3.08E-09  8.71E-10  2.90E-10  1.11E-10  4.88E-11  2.41E-11  1.32E-11  8.03E-12  5.38E-12  3.98E-12  3.17E-12  2.73E-12  2.50E-12
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
20   6.95E-05  4.15E-06  3.96E-07  5.38E-08  9.56E-09  2.11E-09  5.64E-10  1.77E-10  6.43E-11  2.66E-11  1.24E-11  6.40E-12  3.65E-12  2.30E-12  1.60E-12  1.19E-12  9.56E-13  8.28E-13  7.56E-13
21   6.30E-05  3.57E-06  3.24E-07  4.18E-08  7.05E-09  1.47E-09  3.74E-10  1.11E-10  3.83E-11  1.50E-11  6.58E-12  3.22E-12  1.74E-12  1.03E-12  6.73E-13  4.74E-13  3.56E-13  2.91E-13  2.54E-13
22   5.74E-05  3.09E-06  2.68E-07  3.29E-08  5.28E-09  1.05E-09  2.53E-10  7.15E-11  2.34E-11  8.67E-12  3.61E-12  1.67E-12  8.57E-13  4.81E-13  2.97E-13  1.97E-13  1.40E-13  1.08E-13  8.94E-14
23   5.24E-05  2.70E-06  2.23E-07  2.61E-08  4.00E-09  7.59E-10  1.74E-10  4.69E-11  1.46E-11  5.16E-12  2.04E-12  9.00E-13  4.38E-13  2.33E-13  1.36E-13  8.55E-14  5.76E-14  4.19E-14  3.30E-14
24   4.81E-05  2.37E-06  1.88E-07  2.10E-08  3.07E-09  5.57E-10  1.22E-10  3.14E-11  9.33E-12  3.14E-12  1.19E-12  4.98E-13  2.31E-13  1.17E-13  6.46E-14  3.85E-14  2.47E-14  1.71E-14  1.27E-14
25   4.43E-05  2.09E-06  1.59E-07  1.70E-08  2.38E-09  4.15E-10  8.70E-11  2.14E-11  6.07E-12  1.95E-12  7.04E-13  2.82E-13  1.25E-13  6.04E-14  3.17E-14  1.80E-14  1.10E-14  7.23E-15  5.10E-15
26   4.09E-05  1.85E-06  1.35E-07  1.39E-08  1.87E-09  3.12E-10  6.28E-11  1.48E-11  4.02E-12  1.24E-12  4.27E-13  1.64E-13  6.94E-14  3.20E-14  1.60E-14  8.68E-15  5.10E-15  3.18E-15  2.13E-15
27   3.79E-05  1.65E-06  1.16E-07  1.15E-08  1.48E-09  2.38E-10  4.59E-11  1.03E-11  2.70E-12  7.99E-13  2.64E-13  9.71E-14  3.94E-14  1.74E-14  8.33E-15  4.31E-15  2.43E-15  1.45E-15  9.25E-16
28   3.52E-05  1.47E-06  9.99E-08  9.52E-09  1.18E-09  1.83E-10  3.39E-11  7.36E-12  1.85E-12  5.24E-13  1.66E-13  5.87E-14  2.28E-14  9.66E-15  4.44E-15  2.20E-15  1.19E-15  6.80E-16  4.16E-16
29   3.28E-05  1.32E-06  8.66E-08  7.95E-09  9.52E-10  1.42E-10  2.54E-11  5.29E-12  1.28E-12  3.48E-13  1.06E-13  3.61E-14  1.35E-14  5.47E-15  2.42E-15  1.15E-15  5.96E-16  3.29E-16  1.94E-16
30   3.06E-05  1.19E-06  7.54E-08  6.68E-09  7.72E-10  1.11E-10  1.92E-11  3.85E-12  8.95E-13  2.35E-13  6.91E-14  2.26E-14  8.10E-15  3.15E-15  1.35E-15  6.20E-16  3.06E-16  1.64E-16  9.35E-17
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20

alpha = 0.0250
k
2   0.00370
3   0.00149   0.000720
4   0.000797  0.000257  0.000148
5   0.000495  0.000119  5.18E-05  3.44E-05
6   0.000336  6.49E-05  2.24E-05  1.16E-05  8.24E-06
7   0.000244  3.92E-05  1.11E-05  4.70E-06  2.71E-06  2.04E-06
8   0.000185  2.55E-05  6.15E-06  2.19E-06  1.06E-06  6.66E-07  5.43E-07
9   0.000145  1.75E-05  3.66E-06  1.13E-06  4.69E-07  2.53E-07  1.72E-07  1.45E-07
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
10   0.000116  1.25E-05  2.32E-06  6.26E-07  2.29E-07  1.08E-07  6.31E-08  4.52E-08  3.88E-08
11   9.58E-05  9.30E-06  1.54E-06  3.71E-07  1.20E-07  5.05E-08  2.59E-08  1.62E-08  1.21E-08  1.03E-08
12   8.02E-05  7.09E-06  1.06E-06  2.31E-07  6.76E-08  2.54E-08  1.17E-08  6.45E-09  4.25E-09  3.22E-09  2.80E-09
13   6.81E-05  5.53E-06  7.54E-07  1.50E-07  3.99E-08  1.36E-08  5.65E-09  2.81E-09  1.66E-09  1.13E-09  8.79E-10  7.78E-10
14   5.86E-05  4.40E-06  5.51E-07  1.01E-07  2.46E-08  7.70E-09  2.91E-09  1.32E-09  7.05E-10  4.34E-10  3.07E-10  2.44E-10  2.17E-10
15   5.09E-05  3.56E-06  4.12E-07  6.97E-08  1.58E-08  4.54E-09  1.58E-09  6.58E-10  3.22E-10  1.81E-10  1.17E-10  8.44E-11  6.83E-11  6.32E-11
16   4.46E-05  2.92E-06  3.15E-07  4.96E-08  1.04E-08  2.78E-09  8.98E-10  3.46E-10  1.56E-10  8.09E-11  4.79E-11  3.18E-11  2.36E-11  1.96E-11  1.78E-11
17   3.95E-05  2.42E-06  2.45E-07  3.60E-08  7.07E-09  1.76E-09  5.30E-10  1.90E-10  7.95E-11  3.83E-11  2.10E-11  1.28E-11  8.80E-12  6.67E-12  5.57E-12  5.01E-12
18   3.52E-05  2.03E-06  1.93E-07  2.67E-08  4.92E-09  1.15E-09  3.24E-10  1.09E-10  4.24E-11  1.90E-11  9.69E-12  5.52E-12  3.51E-12  2.46E-12  1.90E-12  1.59E-12  1.50E-12
19   3.15E-05  1.72E-06  1.54E-07  2.02E-08  3.50E-09  7.67E-10  2.04E-10  6.41E-11  2.35E-11  9.89E-12  4.70E-12  2.50E-12  1.48E-12  9.68E-13  6.97E-13  5.45E-13  4.67E-13  4.30E-13
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
20   2.84E-05  1.47E-06  1.25E-07  1.54E-08  2.53E-09  5.24E-10  1.31E-10  3.90E-11  1.35E-11  5.33E-12  2.38E-12  1.19E-12  6.61E-13  4.05E-13  2.72E-13  1.99E-13  1.58E-13  1.36E-13  1.25E-13
21   2.57E-05  1.27E-06  1.02E-07  1.20E-08  1.87E-09  3.65E-10  8.67E-11  2.44E-11  7.96E-12  2.98E-12  1.25E-12  5.91E-13  3.08E-13  1.78E-13  1.13E-13  7.74E-14  5.73E-14  4.63E-14  4.02E-14
22   2.34E-05  1.10E-06  8.45E-08  9.44E-09  1.40E-09  2.59E-10  5.85E-11  1.56E-11  4.83E-12  1.71E-12  6.82E-13  3.04E-13  1.50E-13  8.18E-14  4.88E-14  3.16E-14  2.21E-14  1.68E-14  1.38E-14
23   2.14E-05  9.59E-07  7.05E-08  7.51E-09  1.06E-09  1.87E-10  4.02E-11  1.02E-11  3.00E-12  1.01E-12  3.82E-13  1.62E-13  7.56E-14  3.92E-14  2.21E-14  1.35E-14  8.95E-15  6.42E-15  4.98E-15
24   1.97E-05  8.41E-07  5.92E-08  6.04E-09  8.14E-10  1.37E-10  2.81E-11  6.80E-12  1.91E-12  6.12E-13  2.20E-13  8.88E-14  3.94E-14  1.94E-14  1.04E-14  6.04E-15  3.80E-15  2.58E-15  1.89E-15
25   1.81E-05  7.42E-07  5.02E-08  4.91E-09  6.33E-10  1.02E-10  1.99E-11  4.61E-12  1.24E-12  3.78E-13  1.30E-13  5.00E-14  2.12E-14  9.95E-15  5.07E-15  2.80E-15  1.68E-15  1.08E-15  7.52E-16
26   1.67E-05  6.58E-07  4.28E-08  4.02E-09  4.97E-10  7.66E-11  1.43E-11  3.17E-12  8.16E-13  2.39E-13  7.87E-14  2.89E-14  1.17E-14  5.24E-15  2.55E-15  1.34E-15  7.71E-16  4.73E-16  3.11E-16
27   1.55E-05  5.86E-07  3.67E-08  3.31E-09  3.94E-10  5.83E-11  1.04E-11  2.22E-12  5.47E-13  1.53E-13  4.85E-14  1.70E-14  6.63E-15  2.83E-15  1.32E-15  6.64E-16  3.65E-16  2.14E-16  1.34E-16
28   1.44E-05  5.24E-07  3.17E-08  2.76E-09  3.16E-10  4.48E-11  7.70E-12  1.57E-12  3.72E-13  1.00E-13  3.05E-14  1.03E-14  3.84E-15  1.57E-15  6.99E-16  3.38E-16  1.78E-16  9.99E-17  5.96E-17
29   1.34E-05  4.70E-07  2.75E-08  2.31E-09  2.55E-10  3.48E-11  5.75E-12  1.12E-12  2.57E-13  6.65E-14  1.95E-14  6.31E-15  2.27E-15  8.86E-16  3.80E-16  1.76E-16  8.89E-17  4.80E-17  2.74E-17
30   1.25E-05  4.23E-07  2.40E-08  1.94E-09  2.07E-10  2.73E-11  4.33E-12  8.14E-13  1.80E-13  4.47E-14  1.27E-14  3.94E-15  1.37E-15  5.10E-16  2.11E-16  9.43E-17  4.55E-17  2.37E-17  1.30E-17
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20

alpha = 0.0100
k
2   0.00131
3   0.000504  0.000223
4   0.000264  7.73E-05  4.31E-05
5   0.000162  3.52E-05  1.40E-05  8.79E-06
6   0.000110  1.88E-05  5.81E-06  2.83E-06  1.89E-06
7   7.93E-05  1.12E-05  2.83E-06  1.12E-06  6.10E-07  4.46E-07
8   6.00E-05  7.24E-06  1.54E-06  5.09E-07  2.33E-07  1.41E-07  1.10E-07
9   4.70E-05  4.94E-06  9.09E-07  2.58E-07  1.02E-07  5.22E-08  3.39E-08  2.84E-08
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
10   3.78E-05  3.52E-06  5.73E-07  1.42E-07  4.87E-08  2.18E-08  1.21E-08  8.45E-09  7.09E-09
11   3.11E-05  2.60E-06  3.79E-07  8.33E-08  2.53E-08  9.98E-09  4.86E-09  2.92E-09  2.10E-09  1.79E-09
12   2.60E-05  1.97E-06  2.61E-07  5.15E-08  1.40E-08  4.94E-09  2.14E-09  1.13E-09  7.16E-10  5.31E-10  4.60E-10
13   2.21E-05  1.54E-06  1.86E-07  3.33E-08  8.20E-09  2.61E-09  1.02E-09  4.82E-10  2.72E-10  1.79E-10  1.37E-10  1.25E-10
14   1.90E-05  1.22E-06  1.36E-07  2.23E-08  5.01E-09  1.45E-09  5.18E-10  2.22E-10  1.13E-10  6.69E-11  4.62E-11  3.64E-11  3.23E-11
15   1.65E-05  9.83E-07  1.02E-07  1.54E-08  3.18E-09  8.48E-10  2.78E-10  1.09E-10  5.07E-11  2.73E-11  1.71E-11  1.20E-11  9.56E-12  8.93E-12
16   1.44E-05  8.05E-07  7.78E-08  1.09E-08  2.09E-09  5.15E-10  1.56E-10  5.65E-11  2.42E-11  1.20E-11  6.87E-12  4.35E-12  3.16E-12  2.61E-12  2.37E-12
17   1.28E-05  6.67E-07  6.05E-08  7.93E-09  1.41E-09  3.23E-10  9.12E-11  3.07E-11  1.22E-11  5.58E-12  2.96E-12  1.72E-12  1.14E-12  8.55E-13  7.09E-13  6.47E-13
18   1.14E-05  5.59E-07  4.78E-08  5.87E-09  9.76E-10  2.09E-10  5.52E-11  1.74E-11  6.43E-12  2.74E-12  1.35E-12  7.29E-13  4.48E-13  3.06E-13  2.34E-13  1.97E-13  1.81E-13
19   1.02E-05  4.74E-07  3.82E-08  4.43E-09  6.91E-10  1.39E-10  3.45E-11  1.02E-11  3.53E-12  1.41E-12  6.47E-13  3.27E-13  1.87E-13  1.19E-13  8.39E-14  6.51E-14  5.49E-14  4.82E-14
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
20   9.16E-06  4.05E-07  3.09E-08  3.39E-09  4.99E-10  9.47E-11  2.21E-11  6.18E-12  2.01E-12  7.56E-13  3.24E-13  1.55E-13  8.25E-14  4.89E-14  3.22E-14  2.33E-14  1.81E-14  1.53E-14  1.38E-14
21   8.29E-06  3.48E-07  2.53E-08  2.63E-09  3.67E-10  6.58E-11  1.46E-11  3.84E-12  1.18E-12  4.19E-13  1.69E-13  7.63E-14  3.82E-14  2.13E-14  1.31E-14  8.84E-15  6.43E-15  5.17E-15  4.44E-15
22   7.54E-06  3.02E-07  2.09E-08  2.07E-09  2.75E-10  4.67E-11  9.79E-12  2.45E-12  7.14E-13  2.40E-13  9.11E-14  3.91E-14  1.85E-14  9.71E-15  5.61E-15  3.55E-15  2.42E-15  1.84E-15  1.49E-15
23   6.88E-06  2.64E-07  1.74E-08  1.65E-09  2.08E-10  3.36E-11  6.71E-12  1.60E-12  4.42E-13  1.41E-13  5.07E-14  2.07E-14  9.24E-15  4.61E-15  2.51E-15  1.50E-15  9.62E-16  6.86E-16  5.26E-16
24   6.31E-06  2.31E-07  1.46E-08  1.33E-09  1.60E-10  2.46E-11  4.68E-12  1.06E-12  2.80E-13  8.47E-14  2.89E-14  1.13E-14  4.77E-15  2.26E-15  1.17E-15  6.58E-16  4.00E-16  2.68E-16  1.93E-16
25   5.80E-06  2.04E-07  1.24E-08  1.08E-09  1.24E-10  1.83E-11  3.32E-12  7.17E-13  1.81E-13  5.21E-14  1.69E-14  6.28E-15  2.54E-15  1.15E-15  5.60E-16  3.00E-16  1.73E-16  1.09E-16  7.41E-17
26   5.35E-06  1.81E-07  1.05E-08  8.82E-10  9.78E-11  1.38E-11  2.39E-12  4.93E-13  1.19E-13  3.27E-14  1.01E-14  3.59E-15  1.38E-15  5.95E-16  2.77E-16  1.41E-16  7.75E-17  4.61E-17  2.96E-17
27   4.95E-06  1.61E-07  9.03E-09  7.28E-10  7.77E-11  1.05E-11  1.74E-12  3.44E-13  7.96E-14  2.09E-14  6.19E-15  2.09E-15  7.70E-16  3.16E-16  1.41E-16  6.84E-17  3.58E-17  2.02E-17  1.23E-17
28   4.59E-06  1.44E-07  7.78E-09  6.06E-10  6.23E-11  8.09E-12  1.29E-12  2.43E-13  5.40E-14  1.36E-14  3.85E-15  1.24E-15  4.38E-16  1.71E-16  7.29E-17  3.40E-17  1.70E-17  9.14E-18  5.33E-18
29   4.27E-06  1.30E-07  6.74E-09  5.07E-10  5.04E-11  6.29E-12  9.62E-13  1.74E-13  3.72E-14  8.95E-15  2.43E-15  7.47E-16  2.53E-16  9.45E-17  3.86E-17  1.73E-17  8.27E-18  4.27E-18  2.39E-18
30   3.98E-06  1.17E-07  5.86E-09  4.27E-10  4.11E-11  4.95E-12  7.27E-13  1.26E-13  2.59E-14  5.98E-15  1.56E-15  4.57E-16  1.49E-16  5.29E-17  2.08E-17  8.98E-18  4.11E-18  2.05E-18  1.11E-18
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20

alpha = 0.0050
k
2   0.000589
3   0.000227  9.30E-05
4   0.000118  3.14E-05  1.70E-05
5   7.24E-05  1.41E-05  5.41E-06  3.32E-06
6   4.88E-05  7.50E-06  2.21E-06  1.04E-06  6.90E-07
7   3.51E-05  4.46E-06  1.06E-06  4.01E-07  2.14E-07  1.42E-07
8   2.65E-05  2.87E-06  5.73E-07  1.80E-07  7.95E-08  4.51E-08  3.52E-08
9   2.07E-05  1.96E-06  3.36E-07  9.03E-08  3.39E-08  1.66E-08  1.05E-08  8.52E-09
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
10   1.66E-05  1.40E-06  2.10E-07  4.93E-08  1.60E-08  6.86E-09  3.68E-09  2.49E-09  2.06E-09
11   1.37E-05  1.03E-06  1.39E-07  2.88E-08  8.25E-09  3.11E-09  1.45E-09  8.45E-10  6.01E-10  4.89E-10
12   1.14E-05  7.84E-07  9.51E-08  1.77E-08  4.54E-09  1.53E-09  6.31E-10  3.22E-10  2.01E-10  1.45E-10  1.17E-10
13   9.67E-06  6.10E-07  6.75E-08  1.14E-08  2.64E-09  8.00E-10  2.97E-10  1.36E-10  7.55E-11  4.87E-11  3.60E-11  3.24E-11
14   8.30E-06  4.84E-07  4.93E-08  7.61E-09  1.61E-09  4.42E-10  1.49E-10  6.17E-11  3.10E-11  1.80E-11  1.22E-11  9.50E-12  8.10E-12
15   7.20E-06  3.91E-07  3.69E-08  5.25E-09  1.02E-09  2.56E-10  7.93E-11  3.00E-11  1.37E-11  7.26E-12  4.51E-12  3.11E-12  2.46E-12  2.16E-12
16   6.31E-06  3.20E-07  2.81E-08  3.72E-09  6.65E-10  1.55E-10  4.42E-11  1.54E-11  6.49E-12  3.14E-12  1.80E-12  1.12E-12  8.14E-13  6.49E-13  6.09E-13
17   5.57E-06  2.66E-07  2.19E-08  2.69E-09  4.49E-10  9.66E-11  2.57E-11  8.33E-12  3.24E-12  1.44E-12  7.61E-13  4.34E-13  2.91E-13  2.13E-13  1.77E-13  1.55E-13
18   4.96E-06  2.23E-07  1.73E-08  1.99E-09  3.10E-10  6.23E-11  1.55E-11  4.69E-12  1.70E-12  6.99E-13  3.41E-13  1.81E-13  1.12E-13  7.54E-14  5.65E-14  4.66E-14  4.24E-14
19   4.44E-06  1.89E-07  1.38E-08  1.50E-09  2.20E-10  4.13E-11  9.66E-12  2.74E-12  9.26E-13  3.55E-13  1.61E-13  7.95E-14  4.55E-14  2.86E-14  1.96E-14  1.51E-14  1.26E-14  1.13E-14
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
20   3.99E-06  1.61E-07  1.12E-08  1.15E-09  1.59E-10  2.80E-11  6.19E-12  1.65E-12  5.24E-13  1.88E-13  7.94E-14  3.68E-14  1.96E-14  1.15E-14  7.29E-15  5.24E-15  4.03E-15  3.42E-15  3.15E-15
21   3.61E-06  1.39E-07  9.15E-09  8.91E-10  1.17E-10  1.95E-11  4.07E-12  1.03E-12  3.06E-13  1.03E-13  4.08E-14  1.78E-14  8.85E-15  4.86E-15  2.88E-15  1.93E-15  1.38E-15  1.10E-15  9.40E-16
22   3.28E-06  1.20E-07  7.57E-09  7.00E-10  8.71E-11  1.38E-11  2.73E-12  6.53E-13  1.84E-13  5.84E-14  2.17E-14  8.99E-15  4.17E-15  2.16E-15  1.21E-15  7.56E-16  5.05E-16  3.77E-16  3.00E-16
23   3.00E-06  1.05E-07  6.31E-09  5.57E-10  6.60E-11  9.95E-12  1.87E-12  4.25E-13  1.13E-13  3.41E-14  1.19E-14  4.69E-15  2.05E-15  1.00E-15  5.29E-16  3.11E-16  1.95E-16  1.36E-16  1.02E-16
24   2.74E-06  9.19E-08  5.31E-09  4.48E-10  5.07E-11  7.30E-12  1.31E-12  2.83E-13  7.15E-14  2.04E-14  6.76E-15  2.53E-15  1.04E-15  4.82E-16  2.42E-16  1.34E-16  7.93E-17  5.20E-17  3.67E-17
25   2.52E-06  8.10E-08  4.50E-09  3.63E-10  3.94E-11  5.44E-12  9.30E-13  1.92E-13  4.60E-14  1.25E-14  3.94E-15  1.40E-15  5.49E-16  2.41E-16  1.15E-16  6.00E-17  3.38E-17  2.09E-17  1.39E-17
26   2.33E-06  7.18E-08  3.84E-09  2.97E-10  3.10E-11  4.10E-12  6.71E-13  1.32E-13  3.02E-14  7.87E-15  2.35E-15  7.95E-16  2.98E-16  1.24E-16  5.67E-17  2.80E-17  1.50E-17  8.78E-18  5.56E-18
27   2.15E-06  6.39E-08  3.29E-09  2.45E-10  2.46E-11  3.14E-12  4.91E-13  9.24E-14  2.01E-14  5.04E-15  1.44E-15  4.62E-16  1.66E-16  6.61E-17  2.88E-17  1.36E-17  6.95E-18  3.86E-18  2.33E-18
28   2.00E-06  5.71E-08  2.84E-09  2.04E-10  1.97E-11  2.43E-12  3.64E-13  6.56E-14  1.36E-14  3.28E-15  8.97E-16  2.75E-16  9.50E-17  3.61E-17  1.50E-17  6.78E-18  3.33E-18  1.77E-18  1.02E-18
29   1.86E-06  5.12E-08  2.46E-09  1.70E-10  1.59E-11  1.90E-12  2.73E-13  4.71E-14  9.37E-15  2.18E-15  5.71E-16  1.66E-16  5.56E-17  2.02E-17  8.06E-18  3.49E-18  1.65E-18  8.48E-19  4.63E-19
30   1.73E-06  4.61E-08  2.15E-09  1.44E-10  1.30E-11  1.50E-12  2.07E-13  3.43E-14  6.52E-15  1.47E-15  3.70E-16  1.02E-16  3.33E-17  1.16E-17  4.42E-18  1.85E-18  8.43E-19  4.21E-19  2.19E-19
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20

alpha = 0.0025
k
2   0.000277
3   0.000103  4.02E-05
4   5.30E-05  1.33E-05  6.90E-06
5   3.22E-05  5.91E-06  2.17E-06  1.32E-06
6   2.17E-05  3.12E-06  8.72E-07  3.92E-07  2.31E-07
7   1.56E-05  1.85E-06  4.14E-07  1.47E-07  7.17E-08  4.64E-08
8   1.18E-05  1.18E-06  2.21E-07  6.45E-08  2.66E-08  1.48E-08  1.11E-08
9   9.22E-06  8.05E-07  1.28E-07  3.19E-08  1.13E-08  5.47E-09  3.38E-09  2.54E-09
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
10   7.41E-06  5.73E-07  7.96E-08  1.72E-08  5.31E-09  2.25E-09  1.19E-09  7.71E-10  6.68E-10
11   6.09E-06  4.23E-07  5.20E-08  9.98E-09  2.71E-09  1.02E-09  4.67E-10  2.66E-10  1.88E-10  1.49E-10
12   5.10E-06  3.21E-07  3.55E-08  6.11E-09  1.48E-09  4.94E-10  2.02E-10  1.02E-10  6.15E-11  4.35E-11  3.60E-11
13   4.33E-06  2.50E-07  2.50E-08  3.92E-09  8.55E-10  2.56E-10  9.40E-11  4.24E-11  2.26E-11  1.43E-11  1.07E-11  9.08E-12
14   3.72E-06  1.98E-07  1.82E-08  2.61E-09  5.18E-10  1.40E-10  4.67E-11  1.91E-11  9.11E-12  5.22E-12  3.53E-12  2.68E-12  2.30E-12
15   3.23E-06  1.60E-07  1.36E-08  1.80E-09  3.27E-10  8.06E-11  2.46E-11  9.14E-12  3.98E-12  2.07E-12  1.27E-12  8.72E-13  6.84E-13  5.97E-13
16   2.83E-06  1.31E-07  1.03E-08  1.27E-09  2.13E-10  4.82E-11  1.35E-11  4.63E-12  1.86E-12  8.83E-13  4.92E-13  3.08E-13  2.21E-13  1.75E-13  1.62E-13
17   2.50E-06  1.09E-07  8.01E-09  9.20E-10  1.43E-10  2.99E-11  7.79E-12  2.46E-12  9.20E-13  4.01E-13  2.04E-13  1.17E-13  7.67E-14  5.55E-14  4.52E-14  4.10E-14
18   2.23E-06  9.10E-08  6.31E-09  6.81E-10  9.90E-11  1.91E-11  4.65E-12  1.36E-12  4.77E-13  1.92E-13  8.95E-14  4.73E-14  2.86E-14  1.89E-14  1.39E-14  1.15E-14  1.07E-14
19   1.99E-06  7.70E-08  5.04E-09  5.12E-10  7.00E-11  1.26E-11  2.86E-12  7.84E-13  2.58E-13  9.64E-14  4.15E-14  2.03E-14  1.13E-14  6.92E-15  4.64E-15  3.55E-15  2.93E-15  2.59E-15
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
20   1.80E-06  6.58E-08  4.08E-09  3.92E-10  5.05E-11  8.52E-12  1.82E-12  4.66E-13  1.45E-13  5.05E-14  2.01E-14  9.20E-15  4.76E-15  2.69E-15  1.68E-15  1.19E-15  8.92E-16  7.31E-16  6.42E-16
21   1.63E-06  5.66E-08  3.34E-09  3.04E-10  3.71E-11  5.90E-12  1.18E-12  2.86E-13  8.42E-14  2.74E-14  1.02E-14  4.37E-15  2.10E-15  1.11E-15  6.47E-16  4.27E-16  2.98E-16  2.29E-16  1.93E-16
22   1.48E-06  4.91E-08  2.76E-09  2.39E-10  2.78E-11  4.16E-12  7.88E-13  1.80E-13  5.02E-14  1.54E-14  5.38E-15  2.16E-15  9.77E-16  4.82E-16  2.65E-16  1.64E-16  1.08E-16  7.80E-17  6.24E-17
23   1.35E-06  4.28E-08  2.31E-09  1.90E-10  2.11E-11  2.99E-12  5.36E-13  1.16E-13  3.08E-14  8.92E-15  2.94E-15  1.12E-15  4.74E-16  2.20E-16  1.15E-16  6.64E-17  4.15E-17  2.85E-17  2.14E-17
24   1.24E-06  3.76E-08  1.94E-09  1.53E-10  1.62E-11  2.19E-12  3.72E-13  7.68E-14  1.93E-14  5.30E-15  1.66E-15  5.96E-16  2.40E-16  1.05E-16  5.23E-17  2.84E-17  1.70E-17  1.10E-17  7.77E-18
25   1.14E-06  3.31E-08  1.65E-09  1.24E-10  1.27E-11  1.63E-12  2.63E-13  5.18E-14  1.23E-14  3.23E-15  9.62E-16  3.29E-16  1.26E-16  5.25E-17  2.48E-17  1.27E-17  7.27E-18  4.49E-18  2.96E-18
26   1.05E-06  2.94E-08  1.41E-09  1.02E-10  9.98E-12  1.23E-12  1.89E-13  3.56E-14  8.04E-15  2.01E-15  5.74E-16  1.88E-16  6.88E-17  2.72E-17  1.23E-17  5.97E-18  3.25E-18  1.90E-18  1.18E-18
27   9.69E-07  2.61E-08  1.21E-09  8.38E-11  7.96E-12  9.41E-13  1.37E-13  2.49E-14  5.32E-15  1.28E-15  3.51E-16  1.10E-16  3.87E-17  1.47E-17  6.31E-18  2.91E-18  1.51E-18  8.36E-19  4.92E-19
28   8.99E-07  2.34E-08  1.04E-09  6.96E-11  6.41E-12  7.29E-13  1.01E-13  1.77E-14  3.58E-15  8.29E-16  2.20E-16  6.60E-17  2.25E-17  8.20E-18  3.35E-18  1.47E-18  7.22E-19  3.78E-19  2.12E-19
29   8.36E-07  2.10E-08  9.07E-10  5.83E-11  5.20E-12  5.71E-13  7.59E-14  1.27E-14  2.45E-15  5.46E-16  1.41E-16  4.07E-17  1.35E-17  4.73E-18  1.83E-18  7.65E-19  3.55E-19  1.75E-19  9.49E-20
30   7.79E-07  1.89E-08  7.93E-10  4.91E-11  4.26E-12  4.52E-13  5.74E-14  9.32E-15  1.69E-15  3.66E-16  9.21E-17  2.56E-17  8.26E-18  2.81E-18  1.03E-18  4.12E-19  1.79E-19  8.28E-20  4.37E-20
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20

alpha = 0.0010
k
2   0.000104
3   3.76E-05  1.29E-05
4   1.90E-05  4.37E-06  2.16E-06
5   1.14E-05  1.94E-06  6.74E-07  3.87E-07
6   7.64E-06  1.01E-06  2.68E-07  1.14E-07  6.29E-08
7   5.46E-06  5.92E-07  1.25E-07  4.18E-08  1.94E-08  1.29E-08
8   4.10E-06  3.75E-07  6.56E-08  1.80E-08  7.13E-09  3.94E-09  2.94E-09
9   3.19E-06  2.52E-07  3.76E-08  8.75E-09  2.99E-09  1.40E-09  8.51E-10  6.50E-10
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
10   2.56E-06  1.77E-07  2.30E-08  4.65E-09  1.39E-09  5.61E-10  2.88E-10  1.83E-10  1.68E-10
11   2.10E-06  1.30E-07  1.49E-08  2.65E-09  7.01E-10  2.48E-10  1.10E-10  6.00E-11  4.28E-11  2.90E-11
12   1.76E-06  9.76E-08  1.00E-08  1.60E-09  3.79E-10  1.18E-10  4.63E-11  2.21E-11  1.30E-11  8.45E-12  7.42E-12
13   1.49E-06  7.54E-08  7.03E-09  1.02E-09  2.16E-10  6.06E-11  2.12E-11  8.96E-12  4.52E-12  2.76E-12  2.03E-12  1.83E-12
14   1.28E-06  5.95E-08  5.07E-09  6.71E-10  1.30E-10  3.29E-11  1.04E-11  3.95E-12  1.76E-12  9.92E-13  6.32E-13  4.83E-13  4.31E-13
15   1.11E-06  4.78E-08  3.76E-09  4.58E-10  8.09E-11  1.87E-11  5.38E-12  1.86E-12  7.50E-13  3.88E-13  2.19E-13  1.47E-13  1.14E-13  1.06E-13
16   9.76E-07  3.90E-08  2.85E-09  3.22E-10  5.23E-11  1.11E-11  2.94E-12  9.32E-13  3.44E-13  1.63E-13  8.35E-14  5.01E-14  3.48E-14  2.79E-14  2.57E-14
17   8.64E-07  3.23E-08  2.20E-09  2.32E-10  3.49E-11  6.83E-12  1.68E-12  4.91E-13  1.69E-13  7.29E-14  3.43E-14  1.88E-14  1.18E-14  8.41E-15  6.97E-15  6.14E-15
18   7.69E-07  2.70E-08  1.73E-09  1.70E-10  2.39E-11  4.35E-12  9.92E-13  2.70E-13  8.70E-14  3.45E-14  1.50E-14  7.60E-15  4.41E-15  2.83E-15  2.12E-15  1.70E-15  1.73E-15
19   6.90E-07  2.29E-08  1.38E-09  1.28E-10  1.67E-11  2.85E-12  6.08E-13  1.55E-13  4.70E-14  1.71E-14  6.99E-15  3.29E-15  1.77E-15  1.05E-15  7.15E-16  5.26E-16  4.37E-16  4.31E-16
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
20   6.22E-07  1.96E-08  1.12E-09  9.75E-11  1.20E-11  1.92E-12  3.84E-13  9.19E-14  2.64E-14  8.90E-15  3.41E-15  1.51E-15  7.61E-16  4.16E-16  2.61E-16  1.78E-16  1.29E-16  1.07E-16  9.59E-17
21   5.64E-07  1.69E-08  9.15E-10  7.55E-11  8.75E-12  1.32E-12  2.49E-13  5.62E-14  1.53E-14  4.81E-15  1.74E-15  7.27E-16  3.45E-16  1.76E-16  1.02E-16  6.48E-17  4.29E-17  3.19E-17  2.76E-17
22   5.14E-07  1.46E-08  7.58E-10  5.93E-11  6.50E-12  9.29E-13  1.65E-13  3.53E-14  9.19E-15  2.69E-15  9.23E-16  3.65E-16  1.64E-16  7.85E-17  4.25E-17  2.52E-17  1.57E-17  1.09E-17  8.85E-18
23   4.70E-07  1.28E-08  6.33E-10  4.71E-11  4.90E-12  6.66E-13  1.12E-13  2.28E-14  5.65E-15  1.55E-15  5.06E-16  1.90E-16  8.08E-17  3.66E-17  1.86E-17  1.03E-17  6.20E-18  4.07E-18  3.09E-18
24   4.32E-07  1.13E-08  5.34E-10  3.79E-11  3.75E-12  4.85E-13  7.77E-14  1.50E-14  3.55E-15  9.24E-16  2.86E-16  1.02E-16  4.12E-17  1.77E-17  8.46E-18  4.45E-18  2.59E-18  1.64E-18  1.15E-18
25   3.98E-07  9.97E-09  4.54E-10  3.08E-11  2.91E-12  3.60E-13  5.47E-14  1.01E-14  2.28E-15  5.65E-16  1.66E-16  5.66E-17  2.17E-17  8.82E-18  4.00E-18  1.99E-18  1.13E-18  6.88E-19  4.46E-19
26   3.68E-07  8.87E-09  3.89E-10  2.52E-11  2.28E-12  2.71E-13  3.92E-14  6.95E-15  1.49E-15  3.54E-16  9.91E-17  3.21E-17  1.17E-17  4.52E-18  1.95E-18  9.25E-19  5.06E-19  2.98E-19  1.79E-19
27   3.41E-07  7.93E-09  3.36E-10  2.08E-11  1.81E-12  2.06E-13  2.85E-14  4.85E-15  9.91E-16  2.26E-16  6.04E-17  1.86E-17  6.46E-18  2.37E-18  9.81E-19  4.42E-19  2.33E-19  1.30E-19  7.37E-20
28   3.17E-07  7.12E-09  2.92E-10  1.74E-11  1.45E-12  1.59E-13  2.10E-14  3.44E-15  6.69E-16  1.48E-16  3.76E-17  1.10E-17  3.64E-18  1.26E-18  5.05E-19  2.17E-19  1.09E-19  5.73E-20  3.07E-20
29   2.95E-07  6.43E-09  2.55E-10  1.46E-11  1.17E-12  1.24E-13  1.57E-14  2.48E-15  4.58E-16  9.87E-17  2.38E-17  6.62E-18  2.08E-18  6.87E-19  2.65E-19  1.09E-19  5.12E-20  2.49E-20  1.29E-20
30   2.76E-07  5.82E-09  2.24E-10  1.23E-11  9.55E-13  9.81E-14  1.19E-14  1.81E-15  3.17E-16  6.70E-17  1.53E-17  4.05E-18  1.21E-18  3.79E-19  1.42E-19  5.57E-20  2.42E-20  1.07E-20  5.39E-21
s    2         3         4         5         6         7         8         9         10        11        12        13        14        15        16        17        18        19        20
```
These critical values of R(k,s) were generated as follows. For each value of k from 2 to 30, k uniform random numbers from 0 to 1 were generated and sorted from low to high. Then for each value of s from 2 to 20 (or from 2 to k if k < 20), the product R of the first s random numbers was found. That was repeated 201,000 times. Then each set of 201,000 values of R was sorted from low to high, and the 10500th entry (since 201,000 x .05 = 10500) was taken as a single entry in what we shall call the "first draft" of the part of the table for alpha = .05. That was done for each (s, k) combination just described, and for all 7 values of alpha.

The values found in this way for the same value of k but different values of s are not mutually independent, since for instance when k = 10, in each single set of 10 random numbers the 6 numbers multiplied to find R(10,6) for that sample are also used to find R(10,s) for all higher values of s. However, values found for different values of k are mutually independent, since totally new random numbers were used for each new value of k. For each value of alpha, it is reasonable to assume that the values of R(k,s) for a single value of s but different values of k (that is, the entries in a single column of the table) should form a smooth curve. Since the "first draft" values of these entries are mutually independent, it seems reasonable to smooth these values to make each one more accurate.

The smoothing was done as follows. It was observed that when both k and R are expressed in log form, the relation between them is nearly linear except for random fluctuations. In fact, the naked eye cannot see any reliable deviation from linearity. Although it would be presumptuous to assume linearity holds exactly, it does seem reasonable to assume that a cubic polynomial would be well capable of capturing any slight deviations from normality. Therefore a cubic polynomial was fitted to the values of R for different values of k but single values of s and alpha. The values from that cubic polynomial appear in the table.