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Appendix B: Tables

Table I Binomial Probabilities

Tabulated values are $\sum_{x = 0}^{k} p (x)$ $\sum_{x = 0}^{k} p (x)$ . (Computations are rounded at the third decimal place.)

Alternate View

a. $n = 5$ $n = 5$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .951 .774 .590 .328 .168 .078 .031 .010 .002 .000 .000 .000 .000

1 .999 .977 .919 .737 .528 .337 .188 .087 .031 .007 .000 .000 .000

2 1.000 .999 .991 .942 .837 .683 .500 .317 .163 .058 .009 .001 .000

3 1.000 1.000 1.000 .993 .969 .913 .812 .663 .472 .263 .081 .023 .001

4 1.000 1.000 1.000 1.000 .998 .990 .969 .922 .832 .672 .410 .226 .049

b. $n = 6$ $n = 6$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .941 .735 .531 .262 .118 .047 .016 .004 .001 .000 .000 .000 .000

1 .999 .967 .886 .655 .420 .233 .109 .041 .011 .002 .000 .000 .000

2 1.000 .998 .984 .901 .744 .544 .344 .179 .070 .017 .001 .000 .000

3 1.000 1.000 .999 .983 .930 .821 .656 .456 .256 .099 .016 .002 .000

4 1.000 1.000 1.000 .998 .989 .959 .891 .767 .580 .345 .114 .033 .001

5 1.000 1.000 1.000 1.000 .999 .996 .984 .953 .882 .738 .469 .265 .059

c. $n = 7$ $n = 7$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .932 .698 .478 .210 .082 .028 .008 .002 .000 .000 .000 .000 .000

1 .998 .956 .850 .577 .329 .159 .063 .019 .004 .000 .000 .000 .000

2 1.000 .996 .974 .852 .647 .420 .227 .096 .029 .005 .000 .000 .000

3 1.000 1.000 .997 .967 .874 .710 .500 .290 .126 .033 .003 .000 .000

4 1.000 1.000 1.000 .995 .971 .904 .773 .580 .353 .148 .026 .004 .000

5 1.000 1.000 1.000 1.000 .996 .981 .937 .841 .671 .423 .150 .044 .002

6 1.000 1.000 1.000 1.000 1.000 .998 .992 .972 .918 .790 .522 .302 .068

d. $n = 8$ $n = 8$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .923 .663 .430 .168 .058 .017 .004 .001 .000 .000 .000 .000 .000

1 .997 .943 .813 .503 .255 .106 .035 .009 .001 .000 .000 .000 .000

2 1.000 .994 .962 .797 .552 .315 .145 .050 .011 .001 .000 .000 .000

3 1.000 1.000 .995 .944 .806 .594 .363 .174 .058 .010 .000 .000 .000

4 1.000 1.000 1.000 .990 .942 .826 .637 .406 .194 .056 .005 .000 .000

5 1.000 1.000 1.000 .999 .989 .950 .855 .685 .448 .203 .038 .006 .000

6 1.000 1.000 1.000 1.000 .999 .991 .965 .894 .745 .497 .187 .057 .003

7 1.000 1.000 1.000 1.000 1.000 .999 .996 .983 .942 .832 .570 .337 .077

e. $n = 9$ $n = 9$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .914 .630 .387 .134 .040 .010 .002 .000 .000 .000 .000 .000 .000

1 .997 .929 .775 .436 .196 .071 .020 .004 .000 .000 .000 .000 .000

2 1.000 .992 .947 .738 .463 .232 .090 .025 .004 .000 .000 .000 .000

3 1.000 .999 .992 .914 .730 .483 .254 .099 .025 .003 .000 .000 .000

4 1.000 1.000 .999 .980 .901 .733 .500 .267 .099 .020 .001 .000 .000

5 1.000 1.000 1.000 .997 .975 .901 .746 .517 .270 .086 .008 .001 .000

6 1.000 1.000 1.000 1.000 .996 .975 .910 .768 .537 .262 .053 .008 .000

7 1.000 1.000 1.000 1.000 1.000 .996 .980 .929 .804 .564 .225 .071 .003

8 1.000 1.000 1.000 1.000 1.000 1.000 .998 .990 .960 .866 .613 .370 .086

f. $n = 10$ $n = 10$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .904 .599 .349 .107 .028 .006 .001 .000 .000 .000 .000 .000 .000

1 .996 .914 .736 .376 .149 .046 .011 .002 .000 .000 .000 .000 .000

2 1.000 .988 .930 .678 .383 .167 .055 .012 .002 .000 .000 .000 .000

3 1.000 .999 .987 .879 .650 .382 .172 .055 .011 .001 .000 .000 .000

4 1.000 1.000 .998 .967 .850 .633 .377 .166 .047 .006 .000 .000 .000

5 1.000 1.000 1.000 .994 .953 .834 .623 .367 .150 .033 .002 .000 .000

6 1.000 1.000 1.000 .999 .989 .945 .828 .618 .350 .121 .013 .001 .000

7 1.000 1.000 1.000 1.000 .998 .988 .945 .833 .617 .322 .070 .012 .000

8 1.000 1.000 1.000 1.000 1.000 .998 .989 .954 .851 .624 .264 .086 .004

9 1.000 1.000 1.000 1.000 1.000 1.000 .999 .994 .972 .893 .651 .401 .096

g. $n = 15$ $n = 15$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .860 .463 .206 .035 .005 .000 .000 .000 .000 .000 .000 .000 .000

1 .990 .829 .549 .167 .035 .005 .000 .000 .000 .000 .000 .000 .000

2 1.000 .964 .816 .398 .127 .027 .004 .000 .000 .000 .000 .000 .000

3 1.000 .995 .944 .648 .297 .091 .018 .002 .000 .000 .000 .000 .000

4 1.000 .999 .987 .838 .515 .217 .059 .009 .001 .000 .000 .000 .000

5 1.000 1.000 .998 .939 .722 .403 .151 .034 .004 .000 .000 .000 .000

6 1.000 1.000 1.000 .982 .869 .610 .304 .095 .015 .001 .000 .000 .000

7 1.000 1.000 1.000 .996 .950 .787 .500 .213 .050 .004 .000 .000 .000

8 1.000 1.000 1.000 .999 .985 .905 .696 .390 .131 .018 .000 .000 .000

9 1.000 1.000 1.000 1.000 .996 .966 .849 .597 .278 .061 .002 .000 .000

10 1.000 1.000 1.000 1.000 .999 .991 .941 .783 .485 .164 .013 .001 .000

11 1.000 1.000 1.000 1.000 1.000 .998 .982 .909 .703 .352 .056 .005 .000

12 1.000 1.000 1.000 1.000 1.000 1.000 .996 .973 .873 .602 .184 .036 .000

13 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .995 .965 .833 .451 .171 .010

14 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .995 .965 .794 .537 .140

h. $n = 20$ $n = 20$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .818 .358 .122 .012 .001 .000 .000 .000 .000 .000 .000 .000 .000

1 .983 .736 .392 .069 .008 .001 .000 .000 .000 .000 .000 .000 .000

2 .999 .925 .677 .206 .035 .004 .000 .000 .000 .000 .000 .000 .000

3 1.000 .984 .867 .411 .107 .016 .001 .000 .000 .000 .000 .000 .000

4 1.000 .997 .957 .630 .238 .051 .006 .000 .000 .000 .000 .000 .000

5 1.000 1.000 .989 .804 .416 .126 .021 .002 .000 .000 .000 .000 .000

6 1.000 1.000 .998 .913 .608 .250 .058 .006 .000 .000 .000 .000 .000

7 1.000 1.000 1.000 .968 .772 .416 .132 .021 .001 .000 .000 .000 .000

8 1.000 1.000 1.000 .990 .887 .596 .252 .057 .005 .000 .000 .000 .000

9 1.000 1.000 1.000 .997 .952 .755 .412 .128 .017 .001 .000 .000 .000

10 1.000 1.000 1.000 .999 .983 .872 .588 .245 .048 .003 .000 .000 .000

11 1.000 1.000 1.000 1.000 .995 .943 .748 .404 .113 .010 .000 .000 .000

12 1.000 1.000 1.000 1.000 .999 .979 .868 .584 .228 .032 .000 .000 .000

13 1.000 1.000 1.000 1.000 1.000 .994 .942 .750 .392 .087 .002 .000 .000

14 1.000 1.000 1.000 1.000 1.000 .998 .979 .874 .584 .196 .011 .000 .000

15 1.000 1.000 1.000 1.000 1.000 1.000 .994 .949 .762 .370 .043 .003 .000

16 1.000 1.000 1.000 1.000 1.000 1.000 .999 .984 .893 .589 .133 .016 .000

17 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .996 .965 .794 .323 .075 .001

18 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .992 .931 .608 .264 .017

19 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .988 .878 .642 .182

i. $n = 25$ $n = 25$

`p` `k` .01 .05 .10 .20 .30 .40 .50 .60 .70 .80 .90 .95 .99

0 .778 .277 .072 .004 .000 .000 .000 .000 .000 .000 .000 .000 .000

1 .974 .642 .271 .027 .002 .000 .000 .000 .000 .000 .000 .000 .000

2 .998 .873 .537 .098 .009 .000 .000 .000 .000 .000 .000 .000 .000

3 1.000 .966 .764 .234 .033 .002 .000 .000 .000 .000 .000 .000 .000

4 1.000 .993 .902 .421 .090 .009 .000 .000 .000 .000 .000 .000 .000

5 1.000 .999 .967 .617 .193 .029 .002 .000 .000 .000 .000 .000 .000

6 1.000 1.000 .991 .780 .341 .074 .007 .000 .000 .000 .000 .000 .000

7 1.000 1.000 .998 .891 .512 .154 .022 .001 .000 .000 .000 .000 .000

8 1.000 1.000 1.000 .953 .677 .274 .054 .004 .000 .000 .000 .000 .000

9 1.000 1.000 1.000 .983 .811 .425 .115 .013 .000 .000 .000 .000 .000

10 1.000 1.000 1.000 .994 .902 .586 .212 .034 .002 .000 .000 .000 .000

11 1.000 1.000 1.000 .998 .956 .732 .345 .078 .006 .000 .000 .000 .000

12 1.000 1.000 1.000 1.000 .983 .846 .500 .154 .017 .000 .000 .000 .000

13 1.000 1.000 1.000 1.000 .994 .922 .655 .268 .044 .002 .000 .000 .000

14 1.000 1.000 1.000 1.000 .998 .966 .788 .414 .098 .006 .000 .000 .000

15 1.000 1.000 1.000 1.000 1.000 .987 .885 .575 .189 .017 .000 .000 .000

16 1.000 1.000 1.000 1.000 1.000 .996 .946 .726 .323 .047 .000 .000 .000

17 1.000 1.000 1.000 1.000 1.000 .999 .978 .846 .488 .109 .002 .000 .000

18 1.000 1.000 1.000 1.000 1.000 1.000 .993 .926 .659 .220 .009 .000 .000

19 1.000 1.000 1.000 1.000 1.000 1.000 .998 .971 .807 .383 .033 .001 .000

20 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .991 .910 .579 .098 .007 .000

21 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .998 .967 .766 .236 .034 .000

22 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .991 .902 .463 .127 .002

23 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .998 .973 .729 .358 .026

24 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .996 .928 .723 .222

a. $n = 5$ $n = 5$
0	.951	.774	.590	.328	.168	.078	.031	.010	.002	.000	.000	.000	.000
1	.999	.977	.919	.737	.528	.337	.188	.087	.031	.007	.000	.000	.000
2	1.000	.999	.991	.942	.837	.683	.500	.317	.163	.058	.009	.001	.000
3	1.000	1.000	1.000	.993	.969	.913	.812	.663	.472	.263	.081	.023	.001
4	1.000	1.000	1.000	1.000	.998	.990	.969	.922	.832	.672	.410	.226	.049

b. $n = 6$ $n = 6$
0	.941	.735	.531	.262	.118	.047	.016	.004	.001	.000	.000	.000	.000
1	.999	.967	.886	.655	.420	.233	.109	.041	.011	.002	.000	.000	.000
2	1.000	.998	.984	.901	.744	.544	.344	.179	.070	.017	.001	.000	.000
3	1.000	1.000	.999	.983	.930	.821	.656	.456	.256	.099	.016	.002	.000
4	1.000	1.000	1.000	.998	.989	.959	.891	.767	.580	.345	.114	.033	.001
5	1.000	1.000	1.000	1.000	.999	.996	.984	.953	.882	.738	.469	.265	.059

c. $n = 7$ $n = 7$
0	.932	.698	.478	.210	.082	.028	.008	.002	.000	.000	.000	.000	.000
1	.998	.956	.850	.577	.329	.159	.063	.019	.004	.000	.000	.000	.000
2	1.000	.996	.974	.852	.647	.420	.227	.096	.029	.005	.000	.000	.000
3	1.000	1.000	.997	.967	.874	.710	.500	.290	.126	.033	.003	.000	.000
4	1.000	1.000	1.000	.995	.971	.904	.773	.580	.353	.148	.026	.004	.000
5	1.000	1.000	1.000	1.000	.996	.981	.937	.841	.671	.423	.150	.044	.002
6	1.000	1.000	1.000	1.000	1.000	.998	.992	.972	.918	.790	.522	.302	.068

d. $n = 8$ $n = 8$
0	.923	.663	.430	.168	.058	.017	.004	.001	.000	.000	.000	.000	.000
1	.997	.943	.813	.503	.255	.106	.035	.009	.001	.000	.000	.000	.000
2	1.000	.994	.962	.797	.552	.315	.145	.050	.011	.001	.000	.000	.000
3	1.000	1.000	.995	.944	.806	.594	.363	.174	.058	.010	.000	.000	.000
4	1.000	1.000	1.000	.990	.942	.826	.637	.406	.194	.056	.005	.000	.000
5	1.000	1.000	1.000	.999	.989	.950	.855	.685	.448	.203	.038	.006	.000
6	1.000	1.000	1.000	1.000	.999	.991	.965	.894	.745	.497	.187	.057	.003
7	1.000	1.000	1.000	1.000	1.000	.999	.996	.983	.942	.832	.570	.337	.077

e. $n = 9$ $n = 9$
0	.914	.630	.387	.134	.040	.010	.002	.000	.000	.000	.000	.000	.000
1	.997	.929	.775	.436	.196	.071	.020	.004	.000	.000	.000	.000	.000
2	1.000	.992	.947	.738	.463	.232	.090	.025	.004	.000	.000	.000	.000
3	1.000	.999	.992	.914	.730	.483	.254	.099	.025	.003	.000	.000	.000
4	1.000	1.000	.999	.980	.901	.733	.500	.267	.099	.020	.001	.000	.000
5	1.000	1.000	1.000	.997	.975	.901	.746	.517	.270	.086	.008	.001	.000
6	1.000	1.000	1.000	1.000	.996	.975	.910	.768	.537	.262	.053	.008	.000
7	1.000	1.000	1.000	1.000	1.000	.996	.980	.929	.804	.564	.225	.071	.003
8	1.000	1.000	1.000	1.000	1.000	1.000	.998	.990	.960	.866	.613	.370	.086

f. $n = 10$ $n = 10$
0	.904	.599	.349	.107	.028	.006	.001	.000	.000	.000	.000	.000	.000
1	.996	.914	.736	.376	.149	.046	.011	.002	.000	.000	.000	.000	.000
2	1.000	.988	.930	.678	.383	.167	.055	.012	.002	.000	.000	.000	.000
3	1.000	.999	.987	.879	.650	.382	.172	.055	.011	.001	.000	.000	.000
4	1.000	1.000	.998	.967	.850	.633	.377	.166	.047	.006	.000	.000	.000
5	1.000	1.000	1.000	.994	.953	.834	.623	.367	.150	.033	.002	.000	.000
6	1.000	1.000	1.000	.999	.989	.945	.828	.618	.350	.121	.013	.001	.000
7	1.000	1.000	1.000	1.000	.998	.988	.945	.833	.617	.322	.070	.012	.000
8	1.000	1.000	1.000	1.000	1.000	.998	.989	.954	.851	.624	.264	.086	.004
9	1.000	1.000	1.000	1.000	1.000	1.000	.999	.994	.972	.893	.651	.401	.096

g. $n = 15$ $n = 15$
0	.860	.463	.206	.035	.005	.000	.000	.000	.000	.000	.000	.000	.000
1	.990	.829	.549	.167	.035	.005	.000	.000	.000	.000	.000	.000	.000
2	1.000	.964	.816	.398	.127	.027	.004	.000	.000	.000	.000	.000	.000
3	1.000	.995	.944	.648	.297	.091	.018	.002	.000	.000	.000	.000	.000
4	1.000	.999	.987	.838	.515	.217	.059	.009	.001	.000	.000	.000	.000
5	1.000	1.000	.998	.939	.722	.403	.151	.034	.004	.000	.000	.000	.000
6	1.000	1.000	1.000	.982	.869	.610	.304	.095	.015	.001	.000	.000	.000
7	1.000	1.000	1.000	.996	.950	.787	.500	.213	.050	.004	.000	.000	.000
8	1.000	1.000	1.000	.999	.985	.905	.696	.390	.131	.018	.000	.000	.000
9	1.000	1.000	1.000	1.000	.996	.966	.849	.597	.278	.061	.002	.000	.000
10	1.000	1.000	1.000	1.000	.999	.991	.941	.783	.485	.164	.013	.001	.000
11	1.000	1.000	1.000	1.000	1.000	.998	.982	.909	.703	.352	.056	.005	.000
12	1.000	1.000	1.000	1.000	1.000	1.000	.996	.973	.873	.602	.184	.036	.000
13	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.995	.965	.833	.451	.171	.010
14	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.995	.965	.794	.537	.140

h. $n = 20$ $n = 20$
`p` `k`	.01	.05	.10	.20	.30	.40	.50	.60	.70	.80	.90	.95	.99
0	.818	.358	.122	.012	.001	.000	.000	.000	.000	.000	.000	.000	.000
1	.983	.736	.392	.069	.008	.001	.000	.000	.000	.000	.000	.000	.000
2	.999	.925	.677	.206	.035	.004	.000	.000	.000	.000	.000	.000	.000
3	1.000	.984	.867	.411	.107	.016	.001	.000	.000	.000	.000	.000	.000
4	1.000	.997	.957	.630	.238	.051	.006	.000	.000	.000	.000	.000	.000
5	1.000	1.000	.989	.804	.416	.126	.021	.002	.000	.000	.000	.000	.000
6	1.000	1.000	.998	.913	.608	.250	.058	.006	.000	.000	.000	.000	.000
7	1.000	1.000	1.000	.968	.772	.416	.132	.021	.001	.000	.000	.000	.000
8	1.000	1.000	1.000	.990	.887	.596	.252	.057	.005	.000	.000	.000	.000
9	1.000	1.000	1.000	.997	.952	.755	.412	.128	.017	.001	.000	.000	.000
10	1.000	1.000	1.000	.999	.983	.872	.588	.245	.048	.003	.000	.000	.000
11	1.000	1.000	1.000	1.000	.995	.943	.748	.404	.113	.010	.000	.000	.000
12	1.000	1.000	1.000	1.000	.999	.979	.868	.584	.228	.032	.000	.000	.000
13	1.000	1.000	1.000	1.000	1.000	.994	.942	.750	.392	.087	.002	.000	.000
14	1.000	1.000	1.000	1.000	1.000	.998	.979	.874	.584	.196	.011	.000	.000
15	1.000	1.000	1.000	1.000	1.000	1.000	.994	.949	.762	.370	.043	.003	.000
16	1.000	1.000	1.000	1.000	1.000	1.000	.999	.984	.893	.589	.133	.016	.000
17	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.996	.965	.794	.323	.075	.001
18	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.999	.992	.931	.608	.264	.017
19	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.999	.988	.878	.642	.182

i. $n = 25$ $n = 25$
`p` `k`	.01	.05	.10	.20	.30	.40	.50	.60	.70	.80	.90	.95	.99
0	.778	.277	.072	.004	.000	.000	.000	.000	.000	.000	.000	.000	.000
1	.974	.642	.271	.027	.002	.000	.000	.000	.000	.000	.000	.000	.000
2	.998	.873	.537	.098	.009	.000	.000	.000	.000	.000	.000	.000	.000
3	1.000	.966	.764	.234	.033	.002	.000	.000	.000	.000	.000	.000	.000
4	1.000	.993	.902	.421	.090	.009	.000	.000	.000	.000	.000	.000	.000
5	1.000	.999	.967	.617	.193	.029	.002	.000	.000	.000	.000	.000	.000
6	1.000	1.000	.991	.780	.341	.074	.007	.000	.000	.000	.000	.000	.000
7	1.000	1.000	.998	.891	.512	.154	.022	.001	.000	.000	.000	.000	.000
8	1.000	1.000	1.000	.953	.677	.274	.054	.004	.000	.000	.000	.000	.000
9	1.000	1.000	1.000	.983	.811	.425	.115	.013	.000	.000	.000	.000	.000
10	1.000	1.000	1.000	.994	.902	.586	.212	.034	.002	.000	.000	.000	.000
11	1.000	1.000	1.000	.998	.956	.732	.345	.078	.006	.000	.000	.000	.000
12	1.000	1.000	1.000	1.000	.983	.846	.500	.154	.017	.000	.000	.000	.000
13	1.000	1.000	1.000	1.000	.994	.922	.655	.268	.044	.002	.000	.000	.000
14	1.000	1.000	1.000	1.000	.998	.966	.788	.414	.098	.006	.000	.000	.000
15	1.000	1.000	1.000	1.000	1.000	.987	.885	.575	.189	.017	.000	.000	.000
16	1.000	1.000	1.000	1.000	1.000	.996	.946	.726	.323	.047	.000	.000	.000
17	1.000	1.000	1.000	1.000	1.000	.999	.978	.846	.488	.109	.002	.000	.000
18	1.000	1.000	1.000	1.000	1.000	1.000	.993	.926	.659	.220	.009	.000	.000
19	1.000	1.000	1.000	1.000	1.000	1.000	.998	.971	.807	.383	.033	.001	.000
20	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.991	.910	.579	.098	.007	.000
21	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.998	.967	.766	.236	.034	.000
22	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.991	.902	.463	.127	.002
23	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.998	.973	.729	.358	.026
24	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	.996	.928	.723	.222

Table II Normal Curve Areas

Alternate View

`z` .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

**1.0** .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

**1.1** .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

**1.2** .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

**1.3** .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

**1.4** .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

**1.5** .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

**1.6** .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

**1.7** .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

**1.8** .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

**1.9** .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

**2.0** .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817

**2.1** .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857

**2.2** .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890

**2.3** .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916

**2.4** .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936

**2.5** .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952

**2.6** .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964

**2.7** .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974

**2.8** .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981

**2.9** .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986

**3.0** .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990

Source: Abridged from Table I of A. Hald, Statistical Tables and Formulas (New York: Wiley), 1952.

`z`	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
.0	.0000	.0040	.0080	.0120	.0160	.0199	.0239	.0279	.0319	.0359
.1	.0398	.0438	.0478	.0517	.0557	.0596	.0636	.0675	.0714	.0753
.2	.0793	.0832	.0871	.0910	.0948	.0987	.1026	.1064	.1103	.1141
.3	.1179	.1217	.1255	.1293	.1331	.1368	.1406	.1443	.1480	.1517
.4	.1554	.1591	.1628	.1664	.1700	.1736	.1772	.1808	.1844	.1879
.5	.1915	.1950	.1985	.2019	.2054	.2088	.2123	.2157	.2190	.2224
.6	.2257	.2291	.2324	.2357	.2389	.2422	.2454	.2486	.2517	.2549
.7	.2580	.2611	.2642	.2673	.2704	.2734	.2764	.2794	.2823	.2852
.8	.2881	.2910	.2939	.2967	.2995	.3023	.3051	.3078	.3106	.3133
.9	.3159	.3186	.3212	.3238	.3264	.3289	.3315	.3340	.3365	.3389
1.0	.3413	.3438	.3461	.3485	.3508	.3531	.3554	.3577	.3599	.3621
1.1	.3643	.3665	.3686	.3708	.3729	.3749	.3770	.3790	.3810	.3830
1.2	.3849	.3869	.3888	.3907	.3925	.3944	.3962	.3980	.3997	.4015
1.3	.4032	.4049	.4066	.4082	.4099	.4115	.4131	.4147	.4162	.4177
1.4	.4192	.4207	.4222	.4236	.4251	.4265	.4279	.4292	.4306	.4319
1.5	.4332	.4345	.4357	.4370	.4382	.4394	.4406	.4418	.4429	.4441
1.6	.4452	.4463	.4474	.4484	.4495	.4505	.4515	.4525	.4535	.4545
1.7	.4554	.4564	.4573	.4582	.4591	.4599	.4608	.4616	.4625	.4633
1.8	.4641	.4649	.4656	.4664	.4671	.4678	.4686	.4693	.4699	.4706
1.9	.4713	.4719	.4726	.4732	.4738	.4744	.4750	.4756	.4761	.4767
2.0	.4772	.4778	.4783	.4788	.4793	.4798	.4803	.4808	.4812	.4817
2.1	.4821	.4826	.4830	.4834	.4838	.4842	.4846	.4850	.4854	.4857
2.2	.4861	.4864	.4868	.4871	.4875	.4878	.4881	.4884	.4887	.4890
2.3	.4893	.4896	.4898	.4901	.4904	.4906	.4909	.4911	.4913	.4916
2.4	.4918	.4920	.4922	.4925	.4927	.4929	.4931	.4932	.4934	.4936
2.5	.4938	.4940	.4941	.4943	.4945	.4946	.4948	.4949	.4951	.4952
2.6	.4953	.4955	.4956	.4957	.4959	.4960	.4961	.4962	.4963	.4964
2.7	.4965	.4966	.4967	.4968	.4969	.4970	.4971	.4972	.4973	.4974
2.8	.4974	.4975	.4976	.4977	.4977	.4978	.4979	.4979	.4980	.4981
2.9	.4981	.4982	.4982	.4983	.4984	.4984	.4985	.4985	.4986	.4986
3.0	.4987	.4987	.4987	.4988	.4988	.4989	.4989	.4989	.4990	.4990

Table III Critical Values of `t`
Alternate View

Degrees of Freedom $t_{.100}$ $t_{.100}$ $t_{.050}$ $t_{.050}$ $t_{.025}$ $t_{.025}$ $t_{.010}$ $t_{.010}$ $t_{.005}$ $t_{.005}$ $t_{.001}$ $t_{.001}$ $t_{.0005}$ $t_{.0005}$

1 3.078 6.314 12.706 31.821 63.657 318.31 636.62

2 1.886 2.920 4.303 6.965 9.925 22.326 31.598

3 1.638 2.353 3.182 4.541 5.841 10.213 12.924

4 1.533 2.132 2.776 3.747 4.604 7.173 8.610

5 1.476 2.015 2.571 3.365 4.032 5.893 6.869

6 1.440 1.943 2.447 3.143 3.707 5.208 5.959

7 1.415 1.895 2.365 2.998 3.499 4.785 5.408

8 1.397 1.860 2.306 2.896 3.355 4.501 5.041

9 1.383 1.833 2.262 2.821 3.250 4.297 4.781

10 1.372 1.812 2.228 2.764 3.169 4.144 4.587

11 1.363 1.796 2.201 2.718 3.106 4.025 4.437

12 1.356 1.782 2.179 2.681 3.055 3.930 4.318

13 1.350 1.771 2.160 2.650 3.012 3.852 4.221

14 1.345 1.761 2.145 2.624 2.977 3.787 4.140

15 1.341 1.753 2.131 2.602 2.947 3.733 4.073

16 1.337 1.746 2.120 2.583 2.921 3.686 4.015

17 1.333 1.740 2.110 2.567 2.898 3.646 3.965

18 1.330 1.734 2.101 2.552 2.878 3.610 3.922

19 1.328 1.729 2.093 2.539 2.861 3.579 3.883

20 1.325 1.725 2.086 2.528 2.845 3.552 3.850

21 1.323 1.721 2.080 2.518 2.831 3.527 3.819

22 1.321 1.717 2.074 2.508 2.819 3.505 3.792

23 1.319 1.714 2.069 2.500 2.807 3.485 3.767

24 1.318 1.711 2.064 2.492 2.797 3.467 3.745

25 1.316 1.708 2.060 2.485 2.787 3.450 3.725

26 1.315 1.706 2.056 2.479 2.779 3.435 3.707

27 1.314 1.703 2.052 2.473 2.771 3.421 3.690

28 1.313 1.701 2.048 2.467 2.763 3.408 3.674

29 1.311 1.699 2.045 2.462 2.756 3.396 3.659

30 1.310 1.697 2.042 2.457 2.750 3.385 3.646

40 1.303 1.684 2.021 2.423 2.704 3.307 3.551

60 1.296 1.671 2.000 2.390 2.660 3.232 3.460

**120** 1.289 1.658 1.980 2.358 2.617 3.160 3.373

$\infty$ $\infty$ 1.282 1.645 1.960 2.326 2.576 3.090 3.291


1	3.078	6.314	12.706	31.821	63.657	318.31	636.62
2	1.886	2.920	4.303	6.965	9.925	22.326	31.598
3	1.638	2.353	3.182	4.541	5.841	10.213	12.924
4	1.533	2.132	2.776	3.747	4.604	7.173	8.610
5	1.476	2.015	2.571	3.365	4.032	5.893	6.869
6	1.440	1.943	2.447	3.143	3.707	5.208	5.959
7	1.415	1.895	2.365	2.998	3.499	4.785	5.408
8	1.397	1.860	2.306	2.896	3.355	4.501	5.041
9	1.383	1.833	2.262	2.821	3.250	4.297	4.781
10	1.372	1.812	2.228	2.764	3.169	4.144	4.587
11	1.363	1.796	2.201	2.718	3.106	4.025	4.437
12	1.356	1.782	2.179	2.681	3.055	3.930	4.318
13	1.350	1.771	2.160	2.650	3.012	3.852	4.221
14	1.345	1.761	2.145	2.624	2.977	3.787	4.140
15	1.341	1.753	2.131	2.602	2.947	3.733	4.073
16	1.337	1.746	2.120	2.583	2.921	3.686	4.015
17	1.333	1.740	2.110	2.567	2.898	3.646	3.965
18	1.330	1.734	2.101	2.552	2.878	3.610	3.922
19	1.328	1.729	2.093	2.539	2.861	3.579	3.883
20	1.325	1.725	2.086	2.528	2.845	3.552	3.850
21	1.323	1.721	2.080	2.518	2.831	3.527	3.819
22	1.321	1.717	2.074	2.508	2.819	3.505	3.792
23	1.319	1.714	2.069	2.500	2.807	3.485	3.767
24	1.318	1.711	2.064	2.492	2.797	3.467	3.745
25	1.316	1.708	2.060	2.485	2.787	3.450	3.725
26	1.315	1.706	2.056	2.479	2.779	3.435	3.707
27	1.314	1.703	2.052	2.473	2.771	3.421	3.690
28	1.313	1.701	2.048	2.467	2.763	3.408	3.674
29	1.311	1.699	2.045	2.462	2.756	3.396	3.659
30	1.310	1.697	2.042	2.457	2.750	3.385	3.646
40	1.303	1.684	2.021	2.423	2.704	3.307	3.551
60	1.296	1.671	2.000	2.390	2.660	3.232	3.460
120	1.289	1.658	1.980	2.358	2.617	3.160	3.373
$\infty$ $\infty$	1.282	1.645	1.960	2.326	2.576	3.090	3.291

Table IV Critical Values of $χ^{2}$ $χ^{2}$

Alternate View

Degrees of Freedom $χ_{.995}^{2}$ $χ_{.995}^{2}$ $χ_{.990}^{2}$ $χ_{.990}^{2}$ $χ_{.975}^{2}$ $χ_{.975}^{2}$ $χ_{.950}^{2}$ $χ_{.950}^{2}$ $χ_{.900}^{2}$ $χ_{.900}^{2}$

1 .0000393 .0001571 .0009821 .0039321 .0157908

2 .0100251 .0201007 .0506356 .102587 .210720

3 .0717212 .114832 .215795 .351846 .584375

4 .206990 .297110 .484419 .710721 1.063623

5 .411740 .554300 .831211 1.145476 1.61031

6 .675727 .872085 1.237347 1.63539 2.20413

7 .989265 1.239043 1.68987 2.16735 2.83311

8 1.344419 1.646482 2.17973 2.73264 3.48954

9 1.734926 2.087912 2.70039 3.32511 4.16816

10 2.15585 2.55821 3.24697 3.94030 4.86518

11 2.60321 3.05347 3.81575 4.57481 5.57779

12 3.07382 3.57056 4.40379 5.22603 6.30380

13 3.56503 4.10691 5.00874 5.89186 7.04150

14 4.07468 4.66043 5.62872 6.57063 7.78953

15 4.60094 5.22935 6.26214 7.26094 8.54675

16 5.14224 5.81221 6.90766 7.96164 9.31223

17 5.69724 6.40776 7.56418 8.67176 10.0852

18 6.26481 7.01491 8.23075 9.39046 10.8649

19 6.84398 7.63273 8.90655 10.1170 11.6509

20 7.43386 8.26040 9.59083 10.8508 12.4426

21 8.03366 8.89720 10.28293 11.5913 13.2396

22 8.64272 9.54249 10.9823 12.3380 14.0415

23 9.26042 10.19567 11.6885 13.0905 14.8479

24 9.88623 10.8564 12.4011 13.8484 15.6587

25 10.5197 11.5240 13.1197 14.6114 16.4734

26 11.1603 12.1981 13.8439 15.3791 17.2919

27 11.8076 12.8786 14.5733 16.1513 18.1138

28 12.4613 13.5648 15.3079 16.9279 18.9392

29 13.1211 14.2565 16.0471 17.7083 19.7677

30 13.7867 14.9535 16.7908 18.4926 20.5992

40 20.7065 22.1643 24.4331 26.5093 29.0505

50 27.9907 29.7067 32.3574 34.7642 37.6886

60 35.5346 37.4848 40.4817 43.1879 46.4589

70 43.2752 45.4418 48.7576 51.7393 55.3290

80 51.1720 53.5400 57.1532 60.3915 64.2778

90 59.1963 61.7541 65.6466 69.1260 73.2912

**100** 67.3276 70.0648 74.2219 77.9295 82.3581

Degrees of Freedom $χ_{.100}^{2}$ $χ_{.100}^{2}$ $χ_{.050}^{2}$ $χ_{.050}^{2}$ $χ_{.025}^{2}$ $χ_{.025}^{2}$ $χ_{.010}^{2}$ $χ_{.010}^{2}$ $χ_{.005}^{2}$ $χ_{.005}^{2}$

1 2.70554 3.84146 5.02389 6.63490 7.87944

2 4.60517 5.99147 7.37776 9.21034 10.5966

3 6.25139 7.81473 9.34840 11.3449 12.8381

4 7.77944 9.48773 11.1433 13.2767 14.8602

5 9.23635 11.0705 12.8325 15.0863 16.7496

6 10.6446 12.5916 14.4494 16.8119 18.5476

7 12.0170 14.0671 16.0128 18.4753 20.2777

8 13.3616 15.5073 17.5346 20.0902 21.9550

9 14.6837 16.9190 19.0228 21.6660 23.5893

10 15.9871 18.3070 20.4831 23.2093 25.1882

11 17.2750 19.6751 21.9200 24.7250 26.7569

12 18.5494 21.0261 23.3367 26.2170 28.2995

13 19.8119 22.3621 24.7356 27.6883 29.8194

14 21.0642 23.6848 26.1190 29.1413 31.3193

15 22.3072 24.9958 27.4884 30.5779 32.8013

16 23.5418 26.2962 28.8454 31.9999 34.2672

17 24.7690 27.5871 30.1910 33.4087 35.7185

18 25.9894 28.8693 31.5264 34.8053 37.1564

19 27.2036 30.1435 32.8523 36.1908 38.5822

20 28.4120 31.4104 34.1696 37.5662 39.9968

21 29.6151 32.6705 35.4789 38.9321 41.4010

22 30.8133 33.9244 36.7807 40.2894 42.7956

23 32.0069 35.1725 38.0757 41.6384 44.1813

24 33.1963 36.4151 39.3641 42.9798 45.5585

25 34.3816 37.6525 40.6465 44.3141 46.9278

26 35.5631 38.8852 41.9232 45.6417 48.2899

27 36.7412 40.1133 43.1944 46.9630 49.6449

28 37.9159 41.3372 44.4607 48.2782 50.9933

29 39.0875 42.5569 45.7222 49.5879 52.3356

30 40.2560 43.7729 46.9792 50.8922 53.6720

40 51.8050 55.7585 59.3417 63.6907 66.7659

50 63.1671 67.5048 71.4202 76.1539 79.4900

60 74.3970 79.0819 83.2976 88.3794 91.9517

70 85.5271 90.5312 95.0231 100.425 104.215

80 96.5782 101.879 106.629 112.329 116.321

90 107.565 113.145 118.136 124.116 128.299

**100** 118.498 124.342 129.561 135.807 140.169

Degrees of Freedom	$χ_{.995}^{2}$ $χ_{.995}^{2}$	$χ_{.990}^{2}$ $χ_{.990}^{2}$	$χ_{.975}^{2}$ $χ_{.975}^{2}$	$χ_{.950}^{2}$ $χ_{.950}^{2}$	$χ_{.900}^{2}$ $χ_{.900}^{2}$
1	.0000393	.0001571	.0009821	.0039321	.0157908
2	.0100251	.0201007	.0506356	.102587	.210720
3	.0717212	.114832	.215795	.351846	.584375
4	.206990	.297110	.484419	.710721	1.063623
5	.411740	.554300	.831211	1.145476	1.61031
6	.675727	.872085	1.237347	1.63539	2.20413
7	.989265	1.239043	1.68987	2.16735	2.83311
8	1.344419	1.646482	2.17973	2.73264	3.48954
9	1.734926	2.087912	2.70039	3.32511	4.16816
10	2.15585	2.55821	3.24697	3.94030	4.86518
11	2.60321	3.05347	3.81575	4.57481	5.57779
12	3.07382	3.57056	4.40379	5.22603	6.30380
13	3.56503	4.10691	5.00874	5.89186	7.04150
14	4.07468	4.66043	5.62872	6.57063	7.78953
15	4.60094	5.22935	6.26214	7.26094	8.54675
16	5.14224	5.81221	6.90766	7.96164	9.31223
17	5.69724	6.40776	7.56418	8.67176	10.0852
18	6.26481	7.01491	8.23075	9.39046	10.8649
19	6.84398	7.63273	8.90655	10.1170	11.6509
20	7.43386	8.26040	9.59083	10.8508	12.4426
21	8.03366	8.89720	10.28293	11.5913	13.2396
22	8.64272	9.54249	10.9823	12.3380	14.0415
23	9.26042	10.19567	11.6885	13.0905	14.8479
24	9.88623	10.8564	12.4011	13.8484	15.6587
25	10.5197	11.5240	13.1197	14.6114	16.4734
26	11.1603	12.1981	13.8439	15.3791	17.2919
27	11.8076	12.8786	14.5733	16.1513	18.1138
28	12.4613	13.5648	15.3079	16.9279	18.9392
29	13.1211	14.2565	16.0471	17.7083	19.7677
30	13.7867	14.9535	16.7908	18.4926	20.5992
40	20.7065	22.1643	24.4331	26.5093	29.0505
50	27.9907	29.7067	32.3574	34.7642	37.6886
60	35.5346	37.4848	40.4817	43.1879	46.4589
70	43.2752	45.4418	48.7576	51.7393	55.3290
80	51.1720	53.5400	57.1532	60.3915	64.2778
90	59.1963	61.7541	65.6466	69.1260	73.2912
100	67.3276	70.0648	74.2219	77.9295	82.3581

Degrees of Freedom	$χ_{.100}^{2}$ $χ_{.100}^{2}$	$χ_{.050}^{2}$ $χ_{.050}^{2}$	$χ_{.025}^{2}$ $χ_{.025}^{2}$	$χ_{.010}^{2}$ $χ_{.010}^{2}$	$χ_{.005}^{2}$ $χ_{.005}^{2}$
1	2.70554	3.84146	5.02389	6.63490	7.87944
2	4.60517	5.99147	7.37776	9.21034	10.5966
3	6.25139	7.81473	9.34840	11.3449	12.8381
4	7.77944	9.48773	11.1433	13.2767	14.8602
5	9.23635	11.0705	12.8325	15.0863	16.7496
6	10.6446	12.5916	14.4494	16.8119	18.5476
7	12.0170	14.0671	16.0128	18.4753	20.2777
8	13.3616	15.5073	17.5346	20.0902	21.9550
9	14.6837	16.9190	19.0228	21.6660	23.5893
10	15.9871	18.3070	20.4831	23.2093	25.1882
11	17.2750	19.6751	21.9200	24.7250	26.7569
12	18.5494	21.0261	23.3367	26.2170	28.2995
13	19.8119	22.3621	24.7356	27.6883	29.8194
14	21.0642	23.6848	26.1190	29.1413	31.3193
15	22.3072	24.9958	27.4884	30.5779	32.8013
16	23.5418	26.2962	28.8454	31.9999	34.2672
17	24.7690	27.5871	30.1910	33.4087	35.7185
18	25.9894	28.8693	31.5264	34.8053	37.1564
19	27.2036	30.1435	32.8523	36.1908	38.5822
20	28.4120	31.4104	34.1696	37.5662	39.9968
21	29.6151	32.6705	35.4789	38.9321	41.4010
22	30.8133	33.9244	36.7807	40.2894	42.7956
23	32.0069	35.1725	38.0757	41.6384	44.1813
24	33.1963	36.4151	39.3641	42.9798	45.5585
25	34.3816	37.6525	40.6465	44.3141	46.9278
26	35.5631	38.8852	41.9232	45.6417	48.2899
27	36.7412	40.1133	43.1944	46.9630	49.6449
28	37.9159	41.3372	44.4607	48.2782	50.9933
29	39.0875	42.5569	45.7222	49.5879	52.3356
30	40.2560	43.7729	46.9792	50.8922	53.6720
40	51.8050	55.7585	59.3417	63.6907	66.7659
50	63.1671	67.5048	71.4202	76.1539	79.4900
60	74.3970	79.0819	83.2976	88.3794	91.9517
70	85.5271	90.5312	95.0231	100.425	104.215
80	96.5782	101.879	106.629	112.329	116.321
90	107.565	113.145	118.136	124.116	128.299
100	118.498	124.342	129.561	135.807	140.169

Table V Critical Values of $T_{L}$ $T_{L}$ and $T_{U}$ $T_{U}$ for the Wilcoxon Rank Sum Test

Test statistic is the rank sum associated with the smaller sample (if equal sample sizes, either rank sum can be used).

Alternate View

a. $α = .025$ $α = .025$ **one-tailed;** $α = .05$ $α = .05$ **two-tailed**

$n_{1}$ $n_{1}$ $n_{2}$ $n_{2}$ 3 4 5 6 7 8 9 10

$T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$

3 5 16 6 18 6 21 7 23 7 26 8 28 8 31 9 33

4 6 18 11 25 12 28 12 32 13 35 14 38 15 41 16 44

5 6 21 12 28 18 37 19 41 20 45 21 49 22 53 24 56

6 7 23 12 32 19 41 26 52 28 56 29 61 31 65 32 70

7 7 26 13 35 20 45 28 56 37 68 39 73 41 78 43 83

8 8 28 14 38 21 49 29 61 39 73 49 87 51 93 54 98

9 8 31 15 41 22 53 31 65 41 78 51 93 63 108 66 114

10 9 33 16 44 24 56 32 70 43 83 54 98 66 114 79 131

b. $α = .05$ $α = .05$ **one-tailed;** $α = .10$ $α = .10$ **two-tailed**

$n_{1}$ $n_{1}$ $n_{2}$ $n_{2}$ 3 4 5 6 7 8 9 10

$T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$ $T_{L}$ $T_{L}$ $T_{U}$ $T_{U}$

3 6 15 7 17 7 20 8 22 9 24 9 27 10 29 11 31

4 7 17 12 24 13 27 14 30 15 33 16 36 17 39 18 42

5 7 20 13 27 19 36 20 40 22 43 24 46 25 50 26 54

6 8 22 14 30 20 40 28 50 30 54 32 58 33 63 35 67

7 9 24 15 33 22 43 30 54 39 66 41 71 43 76 46 80

8 9 27 16 36 24 46 32 58 41 71 52 84 54 90 57 95

9 10 29 17 39 25 50 33 63 43 76 54 90 66 105 69 111

10 11 31 18 42 26 54 35 67 46 80 57 95 69 111 83 127

Source: From F. Wilcoxon and R. A. Wilcox, “Some Rapid Approximate Statistical Procedures,” 1964, 20–23.

a. $α = .025$ $α = .025$ one-tailed; $α = .05$ $α = .05$ two-tailed
3	5	16	6	18	6	21	7	23	7	26	8	28	8	31	9	33
4	6	18	11	25	12	28	12	32	13	35	14	38	15	41	16	44
5	6	21	12	28	18	37	19	41	20	45	21	49	22	53	24	56
6	7	23	12	32	19	41	26	52	28	56	29	61	31	65	32	70
7	7	26	13	35	20	45	28	56	37	68	39	73	41	78	43	83
8	8	28	14	38	21	49	29	61	39	73	49	87	51	93	54	98
9	8	31	15	41	22	53	31	65	41	78	51	93	63	108	66	114
10	9	33	16	44	24	56	32	70	43	83	54	98	66	114	79	131

b. $α = .05$ $α = .05$ one-tailed; $α = .10$ $α = .10$ two-tailed
$n_{1}$ $n_{1}$ $n_{2}$ $n_{2}$	3	4	5	6	7	8	9	10
	$T_{L}$ $T_{L}$	$T_{U}$ $T_{U}$	$T_{L}$ $T_{L}$	$T_{U}$ $T_{U}$	$T_{L}$ $T_{L}$	$T_{U}$ $T_{U}$	$T_{L}$ $T_{L}$	$T_{U}$ $T_{U}$	$T_{L}$ $T_{L}$	$T_{U}$ $T_{U}$	$T_{L}$ $T_{L}$	$T_{U}$ $T_{U}$	$T_{L}$ $T_{L}$	$T_{U}$ $T_{U}$	$T_{L}$ $T_{L}$	$T_{U}$ $T_{U}$
3	6	15	7	17	7	20	8	22	9	24	9	27	10	29	11	31
4	7	17	12	24	13	27	14	30	15	33	16	36	17	39	18	42
5	7	20	13	27	19	36	20	40	22	43	24	46	25	50	26	54
6	8	22	14	30	20	40	28	50	30	54	32	58	33	63	35	67
7	9	24	15	33	22	43	30	54	39	66	41	71	43	76	46	80
8	9	27	16	36	24	46	32	58	41	71	52	84	54	90	57	95
9	10	29	17	39	25	50	33	63	43	76	54	90	66	105	69	111
10	11	31	18	42	26	54	35	67	46	80	57	95	69	111	83	127

Table VI Critical Values of $T_{0}$ $T_{0}$ in the Wilcoxon Signed Rank Test
Alternate View

One-Tailed Two-Tailed $n = 5$ $n = 5$ $n = 6$ $n = 6$ $n = 7$ $n = 7$ $n = 8$ $n = 8$ $n = 9$ $n = 9$ $n = 10$ $n = 10$

$α = .05$ $α = .05$ $α = .10$ $α = .10$ 1 2 4 6 8 11

$α = .025$ $α = .025$ $α = .05$ $α = .05$ 1 2 4 6 8

$α = .01$ $α = .01$ $α = .02$ $α = .02$ 0 2 3 5

$α = .005$ $α = .005$ $α = .01$ $α = .01$ 0 2 3

$n = 11$ $n = 11$ $n = 12$ $n = 12$ $n = 13$ $n = 13$ $n = 14$ $n = 14$ $n = 15$ $n = 15$ $n = 16$ $n = 16$

$α = .05$ $α = .05$ $α = .10$ $α = .10$ 14 17 21 26 30 36

$α = .025$ $α = .025$ $α = .05$ $α = .05$ 11 14 17 21 25 30

$α = .01$ $α = .01$ $α = .02$ $α = .02$ 7 10 13 16 20 24

$α = .005$ $α = .005$ $α = .01$ $α = .01$ 5 7 10 13 16 19

$n = 17$ $n = 17$ $n = 18$ $n = 18$ $n = 19$ $n = 19$ $n = 20$ $n = 20$ $n = 21$ $n = 21$ $n = 22$ $n = 22$

$α = .05$ $α = .05$ $α = .10$ $α = .10$ 41 47 54 60 68 75

$α = .025$ $α = .025$ $α = .05$ $α = .05$ 35 40 46 52 59 66

$α = .01$ $α = .01$ $α = .02$ $α = .02$ 28 33 38 43 49 56

$α = .005$ $α = .005$ $α = .01$ $α = .01$ 23 28 32 37 43 49

$n = 23$ $n = 23$ $n = 24$ $n = 24$ $n = 25$ $n = 25$ $n = 26$ $n = 26$ $n = 27$ $n = 27$ $n = 28$ $n = 28$

$α = .05$ $α = .05$ $α = .10$ $α = .10$ 83 92 101 110 120 130

$α = .025$ $α = .025$ $α = .05$ $α = .05$ 73 81 90 98 107 117

$α = .01$ $α = .01$ $α = .02$ $α = .02$ 62 69 77 85 93 102

$α = .005$ $α = .005$ $α = .01$ $α = .01$ 55 61 68 76 84 92

$n = 29$ $n = 29$ $n = 30$ $n = 30$ $n = 31$ $n = 31$ $n = 32$ $n = 32$ $n = 33$ $n = 33$ $n = 34$ $n = 34$

$α = .05$ $α = .05$ $α = .10$ $α = .10$ 141 152 163 175 188 201

$α = .025$ $α = .025$ $α = .05$ $α = .05$ 127 137 148 159 171 183

$α = .01$ $α = .01$ $α = .02$ $α = .02$ 111 120 130 141 151 162

$α = .005$ $α = .005$ $α = .01$ $α = .01$ 100 109 118 128 138 149

$n = 35$ $n = 35$ $n = 36$ $n = 36$ $n = 37$ $n = 37$ $n = 38$ $n = 38$ $n = 39$ $n = 39$

$α = .05$ $α = .05$ $α = .10$ $α = .10$ 214 228 242 256 271

$α = .025$ $α = .025$ $α = .05$ $α = .05$ 195 208 222 235 250

$α = .01$ $α = .01$ $α = .02$ $α = .02$ 174 186 198 211 224

$α = .005$ $α = .005$ $α = .01$ $α = .01$ 160 171 183 195 208

$n = 40$ $n = 40$ $n = 41$ $n = 41$ $n = 42$ $n = 42$ $n = 43$ $n = 43$ $n = 44$ $n = 44$ $n = 45$ $n = 45$

$α = .05$ $α = .05$ $α = .10$ $α = .10$ 287 303 319 336 353 371

$α = .025$ $α = .025$ $α = .05$ $α = .05$ 264 279 295 311 327 344

$α = .01$ $α = .01$ $α = .02$ $α = .02$ 238 252 267 281 297 313

$α = .005$ $α = .005$ $α = .01$ $α = .01$ 221 234 248 262 277 292

$n = 46$ $n = 46$ $n = 47$ $n = 47$ $n = 48$ $n = 48$ $n = 49$ $n = 49$ $n = 50$ $n = 50$

$α = .05$ $α = .05$ $α = .10$ $α = .10$ 389 408 427 446 466

$α = .025$ $α = .025$ $α = .05$ $α = .05$ 361 379 397 415 434

$α = .01$ $α = .01$ $α = .02$ $α = .02$ 329 345 362 380 398

$α = .005$ $α = .005$ $α = .01$ $α = .01$ 307 323 339 356 373

Source: From F. Wilcoxon and R. A. Wilcox, “Some Rapid Approximate Statistical Procedures,” 1964, p. 28

One-Tailed	Two-Tailed	$n = 5$ $n = 5$	$n = 6$ $n = 6$	$n = 7$ $n = 7$	$n = 8$ $n = 8$	$n = 9$ $n = 9$	$n = 10$ $n = 10$
$α = .05$ $α = .05$	$α = .10$ $α = .10$	1	2	4	6	8	11
$α = .025$ $α = .025$	$α = .05$ $α = .05$		1	2	4	6	8
$α = .01$ $α = .01$	$α = .02$ $α = .02$			0	2	3	5
$α = .005$ $α = .005$	$α = .01$ $α = .01$				0	2	3
		$n = 11$ $n = 11$	$n = 12$ $n = 12$	$n = 13$ $n = 13$	$n = 14$ $n = 14$	$n = 15$ $n = 15$	$n = 16$ $n = 16$
$α = .05$ $α = .05$	$α = .10$ $α = .10$	14	17	21	26	30	36
$α = .025$ $α = .025$	$α = .05$ $α = .05$	11	14	17	21	25	30
$α = .01$ $α = .01$	$α = .02$ $α = .02$	7	10	13	16	20	24
$α = .005$ $α = .005$	$α = .01$ $α = .01$	5	7	10	13	16	19
		$n = 17$ $n = 17$	$n = 18$ $n = 18$	$n = 19$ $n = 19$	$n = 20$ $n = 20$	$n = 21$ $n = 21$	$n = 22$ $n = 22$
$α = .05$ $α = .05$	$α = .10$ $α = .10$	41	47	54	60	68	75
$α = .025$ $α = .025$	$α = .05$ $α = .05$	35	40	46	52	59	66
$α = .01$ $α = .01$	$α = .02$ $α = .02$	28	33	38	43	49	56
$α = .005$ $α = .005$	$α = .01$ $α = .01$	23	28	32	37	43	49
		$n = 23$ $n = 23$	$n = 24$ $n = 24$	$n = 25$ $n = 25$	$n = 26$ $n = 26$	$n = 27$ $n = 27$	$n = 28$ $n = 28$
$α = .05$ $α = .05$	$α = .10$ $α = .10$	83	92	101	110	120	130
$α = .025$ $α = .025$	$α = .05$ $α = .05$	73	81	90	98	107	117
$α = .01$ $α = .01$	$α = .02$ $α = .02$	62	69	77	85	93	102
$α = .005$ $α = .005$	$α = .01$ $α = .01$	55	61	68	76	84	92
		$n = 29$ $n = 29$	$n = 30$ $n = 30$	$n = 31$ $n = 31$	$n = 32$ $n = 32$	$n = 33$ $n = 33$	$n = 34$ $n = 34$
$α = .05$ $α = .05$	$α = .10$ $α = .10$	141	152	163	175	188	201
$α = .025$ $α = .025$	$α = .05$ $α = .05$	127	137	148	159	171	183
$α = .01$ $α = .01$	$α = .02$ $α = .02$	111	120	130	141	151	162
$α = .005$ $α = .005$	$α = .01$ $α = .01$	100	109	118	128	138	149
		$n = 35$ $n = 35$	$n = 36$ $n = 36$	$n = 37$ $n = 37$	$n = 38$ $n = 38$	$n = 39$ $n = 39$
$α = .05$ $α = .05$	$α = .10$ $α = .10$	214	228	242	256	271
$α = .025$ $α = .025$	$α = .05$ $α = .05$	195	208	222	235	250
$α = .01$ $α = .01$	$α = .02$ $α = .02$	174	186	198	211	224
$α = .005$ $α = .005$	$α = .01$ $α = .01$	160	171	183	195	208
		$n = 40$ $n = 40$	$n = 41$ $n = 41$	$n = 42$ $n = 42$	$n = 43$ $n = 43$	$n = 44$ $n = 44$	$n = 45$ $n = 45$
$α = .05$ $α = .05$	$α = .10$ $α = .10$	287	303	319	336	353	371
$α = .025$ $α = .025$	$α = .05$ $α = .05$	264	279	295	311	327	344
$α = .01$ $α = .01$	$α = .02$ $α = .02$	238	252	267	281	297	313
$α = .005$ $α = .005$	$α = .01$ $α = .01$	221	234	248	262	277	292
		$n = 46$ $n = 46$	$n = 47$ $n = 47$	$n = 48$ $n = 48$	$n = 49$ $n = 49$	$n = 50$ $n = 50$
$α = .05$ $α = .05$	$α = .10$ $α = .10$	389	408	427	446	466
$α = .025$ $α = .025$	$α = .05$ $α = .05$	361	379	397	415	434
$α = .01$ $α = .01$	$α = .02$ $α = .02$	329	345	362	380	398
$α = .005$ $α = .005$	$α = .01$ $α = .01$	307	323	339	356	373

Table VII Percentage Points of the `F`-Distribution, $α = .10$ $α = .10$

Alternate View

`v`₁ Numerator Degrees of Freedom

`v`₂ 1 2 3 4 5 6 7 8 9

1 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86

2 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38

3 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24

4 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94

5 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32

6 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96

7 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72

8 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56

9 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44

10 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35

11 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27

12 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21

13 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16

14 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12

15 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09

16 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06

17 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03

18 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00

19 2.99 2.61 2.40 2.27 2.18 2.11 2.06 2.02 1.98

20 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96

21 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95

22 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93

23 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92

24 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91

25 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89

26 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88

27 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87

28 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87

29 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86

30 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85

40 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79

60 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74

**120** 2.75 2.35 2.13 1.99 1.90 1.82 1.77 1.72 1.68

$\infty$ $\infty$ 2.71 2.30 2.08 1.94 1.85 1.77 1.72 1.67 1.63

`v`₁ Numerator Degrees of Freedom

`v`₂ 10 12 15 20 24 30 40 60 120 $\infty$ $\infty$

1 60.19 60.71 61.22 61.74 62.00 62.26 62.53 62.79 63.06 63.33

2 9.39 9.41 9.42 9.44 9.45 9.46 9.47 9.47 9.48 9.49

3 5.23 5.22 5.20 5.18 5.18 5.17 5.16 5.15 5.14 5.13

4 3.92 3.90 3.87 3.84 3.83 3.82 3.80 3.79 3.78 3.76

5 3.30 3.27 3.24 3.21 3.19 3.17 3.16 3.14 3.12 3.10

6 2.94 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.72

7 2.70 2.67 2.63 2.59 2.58 2.56 2.54 2.51 2.49 2.47

8 2.54 2.50 2.46 2.42 2.40 2.38 2.36 2.34 2.32 2.29

9 2.42 2.38 2.34 2.30 2.28 2.25 2.23 2.21 2.18 2.16

10 2.32 2.28 2.24 2.20 2.18 2.16 2.13 2.11 2.08 2.06

11 2.25 2.21 2.17 2.12 2.10 2.08 2.05 2.03 2.00 1.97

12 2.19 2.15 2.10 2.06 2.04 2.01 1.99 1.96 1.93 1.90

13 2.14 2.10 2.05 2.01 1.98 1.96 1.93 1.90 1.88 1.85

14 2.10 2.05 2.01 1.96 1.94 1.91 1.89 1.86 1.83 1.80

15 2.06 2.02 1.97 1.92 1.90 1.87 1.85 1.82 1.79 1.76

16 2.03 1.99 1.94 1.89 1.87 1.84 1.81 1.78 1.75 1.72

17 2.00 1.96 1.91 1.86 1.84 1.81 1.78 1.75 1.72 1.69

18 1.98 1.93 1.89 1.84 1.81 1.78 1.75 1.72 1.69 1.66

19 1.96 1.91 1.86 1.81 1.79 1.76 1.73 1.70 1.67 1.63

20 1.94 1.89 1.84 1.79 1.77 1.74 1.71 1.68 1.64 1.61

21 1.92 1.87 1.83 1.78 1.75 1.72 1.69 1.66 1.62 1.59

22 1.90 1.86 1.81 1.76 1.73 1.70 1.67 1.64 1.60 1.57

23 1.89 1.84 1.80 1.74 1.72 1.69 1.66 1.62 1.59 1.55

24 1.88 1.83 1.78 1.73 1.70 1.67 1.64 1.61 1.57 1.53

25 1.87 1.82 1.77 1.72 1.69 1.66 1.63 1.59 1.56 1.52

26 1.86 1.81 1.76 1.71 1.68 1.65 1.61 1.58 1.54 1.50

27 1.85 1.80 1.75 1.70 1.67 1.64 1.60 1.57 1.53 1.49

28 1.84 1.79 1.74 1.69 1.66 1.63 1.59 1.56 1.52 1.48

29 1.83 1.78 1.73 1.68 1.65 1.62 1.58 1.55 1.51 1.47

30 1.82 1.77 1.72 1.67 1.64 1.61 1.57 1.54 1.50 1.46

40 1.76 1.71 1.66 1.61 1.57 1.54 1.51 1.47 1.42 1.38

60 1.71 1.66 1.60 1.54 1.51 1.48 1.44 1.40 1.35 1.29

**120** 1.65 1.60 1.55 1.48 1.45 1.41 1.37 1.32 1.26 1.19

$\infty$ $\infty$ 1.60 1.55 1.49 1.42 1.38 1.34 1.30 1.24 1.17 1.00

Denominator Degrees of Freedom

`v`₁	Numerator Degrees of Freedom
`v`₂	1	2	3	4	5	6	7	8	9
1	39.86	49.50	53.59	55.83	57.24	58.20	58.91	59.44	59.86
2	8.53	9.00	9.16	9.24	9.29	9.33	9.35	9.37	9.38
3	5.54	5.46	5.39	5.34	5.31	5.28	5.27	5.25	5.24
4	4.54	4.32	4.19	4.11	4.05	4.01	3.98	3.95	3.94
5	4.06	3.78	3.62	3.52	3.45	3.40	3.37	3.34	3.32
6	3.78	3.46	3.29	3.18	3.11	3.05	3.01	2.98	2.96
7	3.59	3.26	3.07	2.96	2.88	2.83	2.78	2.75	2.72
8	3.46	3.11	2.92	2.81	2.73	2.67	2.62	2.59	2.56
9	3.36	3.01	2.81	2.69	2.61	2.55	2.51	2.47	2.44
10	3.29	2.92	2.73	2.61	2.52	2.46	2.41	2.38	2.35
11	3.23	2.86	2.66	2.54	2.45	2.39	2.34	2.30	2.27
12	3.18	2.81	2.61	2.48	2.39	2.33	2.28	2.24	2.21
13	3.14	2.76	2.56	2.43	2.35	2.28	2.23	2.20	2.16
14	3.10	2.73	2.52	2.39	2.31	2.24	2.19	2.15	2.12
15	3.07	2.70	2.49	2.36	2.27	2.21	2.16	2.12	2.09
16	3.05	2.67	2.46	2.33	2.24	2.18	2.13	2.09	2.06
17	3.03	2.64	2.44	2.31	2.22	2.15	2.10	2.06	2.03
18	3.01	2.62	2.42	2.29	2.20	2.13	2.08	2.04	2.00
19	2.99	2.61	2.40	2.27	2.18	2.11	2.06	2.02	1.98
20	2.97	2.59	2.38	2.25	2.16	2.09	2.04	2.00	1.96
21	2.96	2.57	2.36	2.23	2.14	2.08	2.02	1.98	1.95
22	2.95	2.56	2.35	2.22	2.13	2.06	2.01	1.97	1.93
23	2.94	2.55	2.34	2.21	2.11	2.05	1.99	1.95	1.92
24	2.93	2.54	2.33	2.19	2.10	2.04	1.98	1.94	1.91
25	2.92	2.53	2.32	2.18	2.09	2.02	1.97	1.93	1.89
26	2.91	2.52	2.31	2.17	2.08	2.01	1.96	1.92	1.88
27	2.90	2.51	2.30	2.17	2.07	2.00	1.95	1.91	1.87
28	2.89	2.50	2.29	2.16	2.06	2.00	1.94	1.90	1.87
29	2.89	2.50	2.28	2.15	2.06	1.99	1.93	1.89	1.86
30	2.88	2.49	2.28	2.14	2.05	1.98	1.93	1.88	1.85
40	2.84	2.44	2.23	2.09	2.00	1.93	1.87	1.83	1.79
60	2.79	2.39	2.18	2.04	1.95	1.87	1.82	1.77	1.74
120	2.75	2.35	2.13	1.99	1.90	1.82	1.77	1.72	1.68
$\infty$ $\infty$	2.71	2.30	2.08	1.94	1.85	1.77	1.72	1.67	1.63

`v`₁	Numerator Degrees of Freedom
`v`₂	10	12	15	20	24	30	40	60	120	$\infty$ $\infty$
1	60.19	60.71	61.22	61.74	62.00	62.26	62.53	62.79	63.06	63.33
2	9.39	9.41	9.42	9.44	9.45	9.46	9.47	9.47	9.48	9.49
3	5.23	5.22	5.20	5.18	5.18	5.17	5.16	5.15	5.14	5.13
4	3.92	3.90	3.87	3.84	3.83	3.82	3.80	3.79	3.78	3.76
5	3.30	3.27	3.24	3.21	3.19	3.17	3.16	3.14	3.12	3.10
6	2.94	2.90	2.87	2.84	2.82	2.80	2.78	2.76	2.74	2.72
7	2.70	2.67	2.63	2.59	2.58	2.56	2.54	2.51	2.49	2.47
8	2.54	2.50	2.46	2.42	2.40	2.38	2.36	2.34	2.32	2.29
9	2.42	2.38	2.34	2.30	2.28	2.25	2.23	2.21	2.18	2.16
10	2.32	2.28	2.24	2.20	2.18	2.16	2.13	2.11	2.08	2.06
11	2.25	2.21	2.17	2.12	2.10	2.08	2.05	2.03	2.00	1.97
12	2.19	2.15	2.10	2.06	2.04	2.01	1.99	1.96	1.93	1.90
13	2.14	2.10	2.05	2.01	1.98	1.96	1.93	1.90	1.88	1.85
14	2.10	2.05	2.01	1.96	1.94	1.91	1.89	1.86	1.83	1.80
15	2.06	2.02	1.97	1.92	1.90	1.87	1.85	1.82	1.79	1.76
16	2.03	1.99	1.94	1.89	1.87	1.84	1.81	1.78	1.75	1.72
17	2.00	1.96	1.91	1.86	1.84	1.81	1.78	1.75	1.72	1.69
18	1.98	1.93	1.89	1.84	1.81	1.78	1.75	1.72	1.69	1.66
19	1.96	1.91	1.86	1.81	1.79	1.76	1.73	1.70	1.67	1.63
20	1.94	1.89	1.84	1.79	1.77	1.74	1.71	1.68	1.64	1.61
21	1.92	1.87	1.83	1.78	1.75	1.72	1.69	1.66	1.62	1.59
22	1.90	1.86	1.81	1.76	1.73	1.70	1.67	1.64	1.60	1.57
23	1.89	1.84	1.80	1.74	1.72	1.69	1.66	1.62	1.59	1.55
24	1.88	1.83	1.78	1.73	1.70	1.67	1.64	1.61	1.57	1.53
25	1.87	1.82	1.77	1.72	1.69	1.66	1.63	1.59	1.56	1.52
26	1.86	1.81	1.76	1.71	1.68	1.65	1.61	1.58	1.54	1.50
27	1.85	1.80	1.75	1.70	1.67	1.64	1.60	1.57	1.53	1.49
28	1.84	1.79	1.74	1.69	1.66	1.63	1.59	1.56	1.52	1.48
29	1.83	1.78	1.73	1.68	1.65	1.62	1.58	1.55	1.51	1.47
30	1.82	1.77	1.72	1.67	1.64	1.61	1.57	1.54	1.50	1.46
40	1.76	1.71	1.66	1.61	1.57	1.54	1.51	1.47	1.42	1.38
60	1.71	1.66	1.60	1.54	1.51	1.48	1.44	1.40	1.35	1.29
120	1.65	1.60	1.55	1.48	1.45	1.41	1.37	1.32	1.26	1.19
$\infty$ $\infty$	1.60	1.55	1.49	1.42	1.38	1.34	1.30	1.24	1.17	1.00
Denominator Degrees of Freedom

Table VIII Percentage Points of the `F`-Distribution, $α = .05$ $α = .05$

Alternate View

`v`₁
Numerator Degrees of Freedom

`v`₂ 1 2 3 4 5 6 7 8 9

1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5

2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38

3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81

4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00

5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77

6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10

7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68

8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39

9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18

10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02

11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90

12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80

13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71

14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65

15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59

16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54

17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49

18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46

19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42

20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39

21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37

22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34

23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32

24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30

25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28

26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.77

27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25

28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24

29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22

30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21

40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12

60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04

**120** 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96

$\infty$ $\infty$ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88

`v`₁
Numerator Degrees of Freedom

`v`₂ 10 12 15 20 24 30 40 60 120 $\infty$ $\infty$

1 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3

2 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50

3 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53

4 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63

5 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36

6 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67

7 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23

8 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93

9 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71

10 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54

11 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40

12 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30

13 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21

14 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13

15 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07

16 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01

17 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96

18 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92

19 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88

20 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84

21 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81

22 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78

23 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76

24 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73

25 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71

26 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69

27 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67

28 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65

29 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64

30 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62

40 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51

60 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39

**120** 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25

$\infty$ $\infty$ 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00

Denominator Degrees of Freedom

`v`₁	Numerator Degrees of Freedom
`v`₂	1	2	3	4	5	6	7	8	9
1	161.4	199.5	215.7	224.6	230.2	234.0	236.8	238.9	240.5
2	18.51	19.00	19.16	19.25	19.30	19.33	19.35	19.37	19.38
3	10.13	9.55	9.28	9.12	9.01	8.94	8.89	8.85	8.81
4	7.71	6.94	6.59	6.39	6.26	6.16	6.09	6.04	6.00
5	6.61	5.79	5.41	5.19	5.05	4.95	4.88	4.82	4.77
6	5.99	5.14	4.76	4.53	4.39	4.28	4.21	4.15	4.10
7	5.59	4.74	4.35	4.12	3.97	3.87	3.79	3.73	3.68
8	5.32	4.46	4.07	3.84	3.69	3.58	3.50	3.44	3.39
9	5.12	4.26	3.86	3.63	3.48	3.37	3.29	3.23	3.18
10	4.96	4.10	3.71	3.48	3.33	3.22	3.14	3.07	3.02
11	4.84	3.98	3.59	3.36	3.20	3.09	3.01	2.95	2.90
12	4.75	3.89	3.49	3.26	3.11	3.00	2.91	2.85	2.80
13	4.67	3.81	3.41	3.18	3.03	2.92	2.83	2.77	2.71
14	4.60	3.74	3.34	3.11	2.96	2.85	2.76	2.70	2.65
15	4.54	3.68	3.29	3.06	2.90	2.79	2.71	2.64	2.59
16	4.49	3.63	3.24	3.01	2.85	2.74	2.66	2.59	2.54
17	4.45	3.59	3.20	2.96	2.81	2.70	2.61	2.55	2.49
18	4.41	3.55	3.16	2.93	2.77	2.66	2.58	2.51	2.46
19	4.38	3.52	3.13	2.90	2.74	2.63	2.54	2.48	2.42
20	4.35	3.49	3.10	2.87	2.71	2.60	2.51	2.45	2.39
21	4.32	3.47	3.07	2.84	2.68	2.57	2.49	2.42	2.37
22	4.30	3.44	3.05	2.82	2.66	2.55	2.46	2.40	2.34
23	4.28	3.42	3.03	2.80	2.64	2.53	2.44	2.37	2.32
24	4.26	3.40	3.01	2.78	2.62	2.51	2.42	2.36	2.30
25	4.24	3.39	2.99	2.76	2.60	2.49	2.40	2.34	2.28
26	4.23	3.37	2.98	2.74	2.59	2.47	2.39	2.32	2.77
27	4.21	3.35	2.96	2.73	2.57	2.46	2.37	2.31	2.25
28	4.20	3.34	2.95	2.71	2.56	2.45	2.36	2.29	2.24
29	4.18	3.33	2.93	2.70	2.55	2.43	2.35	2.28	2.22
30	4.17	3.32	2.92	2.69	2.53	2.42	2.33	2.27	2.21
40	4.08	3.23	2.84	2.61	2.45	2.34	2.25	2.18	2.12
60	4.00	3.15	2.76	2.53	2.37	2.25	2.17	2.10	2.04
120	3.92	3.07	2.68	2.45	2.29	2.17	2.09	2.02	1.96
$\infty$ $\infty$	3.84	3.00	2.60	2.37	2.21	2.10	2.01	1.94	1.88

`v`₁	Numerator Degrees of Freedom
`v`₂	10	12	15	20	24	30	40	60	120	$\infty$ $\infty$
1	241.9	243.9	245.9	248.0	249.1	250.1	251.1	252.2	253.3	254.3
2	19.40	19.41	19.43	19.45	19.45	19.46	19.47	19.48	19.49	19.50
3	8.79	8.74	8.70	8.66	8.64	8.62	8.59	8.57	8.55	8.53
4	5.96	5.91	5.86	5.80	5.77	5.75	5.72	5.69	5.66	5.63
5	4.74	4.68	4.62	4.56	4.53	4.50	4.46	4.43	4.40	4.36
6	4.06	4.00	3.94	3.87	3.84	3.81	3.77	3.74	3.70	3.67
7	3.64	3.57	3.51	3.44	3.41	3.38	3.34	3.30	3.27	3.23
8	3.35	3.28	3.22	3.15	3.12	3.08	3.04	3.01	2.97	2.93
9	3.14	3.07	3.01	2.94	2.90	2.86	2.83	2.79	2.75	2.71
10	2.98	2.91	2.85	2.77	2.74	2.70	2.66	2.62	2.58	2.54
11	2.85	2.79	2.72	2.65	2.61	2.57	2.53	2.49	2.45	2.40
12	2.75	2.69	2.62	2.54	2.51	2.47	2.43	2.38	2.34	2.30
13	2.67	2.60	2.53	2.46	2.42	2.38	2.34	2.30	2.25	2.21
14	2.60	2.53	2.46	2.39	2.35	2.31	2.27	2.22	2.18	2.13
15	2.54	2.48	2.40	2.33	2.29	2.25	2.20	2.16	2.11	2.07
16	2.49	2.42	2.35	2.28	2.24	2.19	2.15	2.11	2.06	2.01
17	2.45	2.38	2.31	2.23	2.19	2.15	2.10	2.06	2.01	1.96
18	2.41	2.34	2.27	2.19	2.15	2.11	2.06	2.02	1.97	1.92
19	2.38	2.31	2.23	2.16	2.11	2.07	2.03	1.98	1.93	1.88
20	2.35	2.28	2.20	2.12	2.08	2.04	1.99	1.95	1.90	1.84
21	2.32	2.25	2.18	2.10	2.05	2.01	1.96	1.92	1.87	1.81
22	2.30	2.23	2.15	2.07	2.03	1.98	1.94	1.89	1.84	1.78
23	2.27	2.20	2.13	2.05	2.01	1.96	1.91	1.86	1.81	1.76
24	2.25	2.18	2.11	2.03	1.98	1.94	1.89	1.84	1.79	1.73
25	2.24	2.16	2.09	2.01	1.96	1.92	1.87	1.82	1.77	1.71
26	2.22	2.15	2.07	1.99	1.95	1.90	1.85	1.80	1.75	1.69
27	2.20	2.13	2.06	1.97	1.93	1.88	1.84	1.79	1.73	1.67
28	2.19	2.12	2.04	1.96	1.91	1.87	1.82	1.77	1.71	1.65
29	2.18	2.10	2.03	1.94	1.90	1.85	1.81	1.75	1.70	1.64
30	2.16	2.09	2.01	1.93	1.89	1.84	1.79	1.74	1.68	1.62
40	2.08	2.00	1.92	1.84	1.79	1.74	1.69	1.64	1.58	1.51
60	1.99	1.92	1.84	1.75	1.70	1.65	1.59	1.53	1.47	1.39
120	1.91	1.83	1.75	1.66	1.61	1.55	1.50	1.43	1.35	1.25
$\infty$ $\infty$	1.83	1.75	1.67	1.57	1.52	1.46	1.39	1.32	1.22	1.00
Denominator Degrees of Freedom

Table IX Percentage Points of the `F`-Distribution, $α = .025$ $α = .025$

Alternate View

`v`₁
Numerator Degrees of Freedom

`v`₂ 1 2 3 4 5 6 7 8 9

1 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3

2 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39

3 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47

4 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90

5 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68

6 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52

7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82

8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36

9 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03

10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78

11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59

12 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44

13 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31

14 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21

15 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12

16 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05

17 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98

18 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93

19 5.92 4.51 3.90 3.56 3.33 3.17 3.05 2.96 2.88

20 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84

21 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80

22 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76

23 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73

24 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70

25 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68

26 5.66 4.27 3.67 3.33 3.10 2.94 2.82 2.73 2.65

27 5.63 4.24 3.65 3.31 3.08 2.92 2.80 2.71 2.63

28 5.61 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.61

29 5.59 4.20 3.61 3.27 3.04 2.88 2.76 2.67 2.59

30 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57

40 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45

60 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33

**120** 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22

$\infty$ $\infty$ 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11

`v`₁
Numerator Degrees of Freedom

`v`₂ 10 12 15 20 24 30 40 60 120 $\infty$ $\infty$

1 968.6 976.7 984.9 993.1 997.2 1,001 1,006 1,010 1,014 1,018

2 39.40 39.41 39.43 39.45 39.46 39.46 39.47 39.48 39.49 39.50

3 14.42 14.34 14.25 14.17 14.12 14.08 14.04 13.99 13.95 13.90

4 8.84 8.75 8.66 8.56 8.51 8.46 8.41 8.36 8.31 8.26

5 6.62 6.52 6.43 6.33 6.28 6.23 6.18 6.12 6.07 6.02

6 5.46 5.37 5.27 5.17 5.12 5.07 5.01 4.96 4.90 4.85

7 4.76 4.67 4.57 4.47 4.42 4.36 4.31 4.25 4.20 4.14

8 4.30 4.20 4.10 4.00 3.95 3.89 3.84 3.78 3.73 3.67

9 3.96 3.87 3.77 3.67 3.61 3.56 3.51 3.45 3.39 3.33

10 3.72 3.62 3.52 3.42 3.37 3.31 3.26 3.20 3.14 3.08

11 3.53 3.43 3.33 3.23 3.17 3.12 3.06 3.00 2.94 2.88

12 3.37 3.28 3.18 3.07 3.02 2.96 2.91 2.85 2.79 2.72

13 3.25 3.15 3.05 2.95 2.89 2.84 2.78 2.72 2.66 2.60

14 3.15 3.05 2.95 2.84 2.79 2.73 2.67 2.61 2.55 2.49

15 3.06 2.96 2.86 2.76 2.70 2.64 2.59 2.52 2.46 2.40

16 2.99 2.89 2.79 2.68 2.63 2.57 2.51 2.45 2.38 2.32

17 2.92 2.82 2.72 2.62 2.56 2.50 2.44 2.38 2.32 2.25

18 2.87 2.77 2.67 2.56 2.50 2.44 2.38 2.32 2.26 2.19

19 2.82 2.72 2.62 2.51 2.45 2.39 2.33 2.27 2.20 2.13

20 2.77 2.68 2.57 2.46 2.41 2.35 2.29 2.22 2.16 2.09

21 2.73 2.64 2.53 2.42 2.37 2.31 2.25 2.18 2.11 2.04

22 2.70 2.60 2.50 2.39 2.33 2.27 2.21 2.14 2.08 2.00

23 2.67 2.57 2.47 2.36 2.30 2.24 2.18 2.11 2.04 1.97

24 2.64 2.54 2.44 2.33 2.27 2.21 2.15 2.08 2.01 1.94

25 2.61 2.51 2.41 2.30 2.24 2.18 2.12 2.05 1.98 1.91

26 2.59 2.49 2.39 2.28 2.22 2.16 2.09 2.03 1.95 1.88

27 2.57 2.47 2.36 2.25 2.19 2.13 2.07 2.00 1.93 1.85

28 2.55 2.45 2.34 2.23 2.17 2.11 2.05 1.98 1.91 1.83

29 2.53 2.43 2.32 2.21 2.15 2.09 2.03 1.96 1.89 1.81

30 2.51 2.41 2.31 2.20 2.14 2.07 2.01 1.94 1.87 1.79

40 2.39 2.29 2.18 2.07 2.01 1.94 1.88 1.80 1.72 1.64

60 2.27 2.17 2.06 1.94 1.88 1.82 1.74 1.67 1.58 1.48

**120** 2.16 2.05 1.94 1.82 1.76 1.69 1.61 1.53 1.43 1.31

$\infty$ $\infty$ 2.05 1.94 1.83 1.71 1.64 1.57 1.48 1.39 1.27 1.00

Denominator Degrees of Freedom

`v`₁	Numerator Degrees of Freedom
`v`₂	1	2	3	4	5	6	7	8	9
1	647.8	799.5	864.2	899.6	921.8	937.1	948.2	956.7	963.3
2	38.51	39.00	39.17	39.25	39.30	39.33	39.36	39.37	39.39
3	17.44	16.04	15.44	15.10	14.88	14.73	14.62	14.54	14.47
4	12.22	10.65	9.98	9.60	9.36	9.20	9.07	8.98	8.90
5	10.01	8.43	7.76	7.39	7.15	6.98	6.85	6.76	6.68
6	8.81	7.26	6.60	6.23	5.99	5.82	5.70	5.60	5.52
7	8.07	6.54	5.89	5.52	5.29	5.12	4.99	4.90	4.82
8	7.57	6.06	5.42	5.05	4.82	4.65	4.53	4.43	4.36
9	7.21	5.71	5.08	4.72	4.48	4.32	4.20	4.10	4.03
10	6.94	5.46	4.83	4.47	4.24	4.07	3.95	3.85	3.78
11	6.72	5.26	4.63	4.28	4.04	3.88	3.76	3.66	3.59
12	6.55	5.10	4.47	4.12	3.89	3.73	3.61	3.51	3.44
13	6.41	4.97	4.35	4.00	3.77	3.60	3.48	3.39	3.31
14	6.30	4.86	4.24	3.89	3.66	3.50	3.38	3.29	3.21
15	6.20	4.77	4.15	3.80	3.58	3.41	3.29	3.20	3.12
16	6.12	4.69	4.08	3.73	3.50	3.34	3.22	3.12	3.05
17	6.04	4.62	4.01	3.66	3.44	3.28	3.16	3.06	2.98
18	5.98	4.56	3.95	3.61	3.38	3.22	3.10	3.01	2.93
19	5.92	4.51	3.90	3.56	3.33	3.17	3.05	2.96	2.88
20	5.87	4.46	3.86	3.51	3.29	3.13	3.01	2.91	2.84
21	5.83	4.42	3.82	3.48	3.25	3.09	2.97	2.87	2.80
22	5.79	4.38	3.78	3.44	3.22	3.05	2.93	2.84	2.76
23	5.75	4.35	3.75	3.41	3.18	3.02	2.90	2.81	2.73
24	5.72	4.32	3.72	3.38	3.15	2.99	2.87	2.78	2.70
25	5.69	4.29	3.69	3.35	3.13	2.97	2.85	2.75	2.68
26	5.66	4.27	3.67	3.33	3.10	2.94	2.82	2.73	2.65
27	5.63	4.24	3.65	3.31	3.08	2.92	2.80	2.71	2.63
28	5.61	4.22	3.63	3.29	3.06	2.90	2.78	2.69	2.61
29	5.59	4.20	3.61	3.27	3.04	2.88	2.76	2.67	2.59
30	5.57	4.18	3.59	3.25	3.03	2.87	2.75	2.65	2.57
40	5.42	4.05	3.46	3.13	2.90	2.74	2.62	2.53	2.45
60	5.29	3.93	3.34	3.01	2.79	2.63	2.51	2.41	2.33
120	5.15	3.80	3.23	2.89	2.67	2.52	2.39	2.30	2.22
$\infty$ $\infty$	5.02	3.69	3.12	2.79	2.57	2.41	2.29	2.19	2.11

`v`₁	Numerator Degrees of Freedom
`v`₂	10	12	15	20	24	30	40	60	120	$\infty$ $\infty$
1	968.6	976.7	984.9	993.1	997.2	1,001	1,006	1,010	1,014	1,018
2	39.40	39.41	39.43	39.45	39.46	39.46	39.47	39.48	39.49	39.50
3	14.42	14.34	14.25	14.17	14.12	14.08	14.04	13.99	13.95	13.90
4	8.84	8.75	8.66	8.56	8.51	8.46	8.41	8.36	8.31	8.26
5	6.62	6.52	6.43	6.33	6.28	6.23	6.18	6.12	6.07	6.02
6	5.46	5.37	5.27	5.17	5.12	5.07	5.01	4.96	4.90	4.85
7	4.76	4.67	4.57	4.47	4.42	4.36	4.31	4.25	4.20	4.14
8	4.30	4.20	4.10	4.00	3.95	3.89	3.84	3.78	3.73	3.67
9	3.96	3.87	3.77	3.67	3.61	3.56	3.51	3.45	3.39	3.33
10	3.72	3.62	3.52	3.42	3.37	3.31	3.26	3.20	3.14	3.08
11	3.53	3.43	3.33	3.23	3.17	3.12	3.06	3.00	2.94	2.88
12	3.37	3.28	3.18	3.07	3.02	2.96	2.91	2.85	2.79	2.72
13	3.25	3.15	3.05	2.95	2.89	2.84	2.78	2.72	2.66	2.60
14	3.15	3.05	2.95	2.84	2.79	2.73	2.67	2.61	2.55	2.49
15	3.06	2.96	2.86	2.76	2.70	2.64	2.59	2.52	2.46	2.40
16	2.99	2.89	2.79	2.68	2.63	2.57	2.51	2.45	2.38	2.32
17	2.92	2.82	2.72	2.62	2.56	2.50	2.44	2.38	2.32	2.25
18	2.87	2.77	2.67	2.56	2.50	2.44	2.38	2.32	2.26	2.19
19	2.82	2.72	2.62	2.51	2.45	2.39	2.33	2.27	2.20	2.13
20	2.77	2.68	2.57	2.46	2.41	2.35	2.29	2.22	2.16	2.09
21	2.73	2.64	2.53	2.42	2.37	2.31	2.25	2.18	2.11	2.04
22	2.70	2.60	2.50	2.39	2.33	2.27	2.21	2.14	2.08	2.00
23	2.67	2.57	2.47	2.36	2.30	2.24	2.18	2.11	2.04	1.97
24	2.64	2.54	2.44	2.33	2.27	2.21	2.15	2.08	2.01	1.94
25	2.61	2.51	2.41	2.30	2.24	2.18	2.12	2.05	1.98	1.91
26	2.59	2.49	2.39	2.28	2.22	2.16	2.09	2.03	1.95	1.88
27	2.57	2.47	2.36	2.25	2.19	2.13	2.07	2.00	1.93	1.85
28	2.55	2.45	2.34	2.23	2.17	2.11	2.05	1.98	1.91	1.83
29	2.53	2.43	2.32	2.21	2.15	2.09	2.03	1.96	1.89	1.81
30	2.51	2.41	2.31	2.20	2.14	2.07	2.01	1.94	1.87	1.79
40	2.39	2.29	2.18	2.07	2.01	1.94	1.88	1.80	1.72	1.64
60	2.27	2.17	2.06	1.94	1.88	1.82	1.74	1.67	1.58	1.48
120	2.16	2.05	1.94	1.82	1.76	1.69	1.61	1.53	1.43	1.31
$\infty$ $\infty$	2.05	1.94	1.83	1.71	1.64	1.57	1.48	1.39	1.27	1.00
Denominator Degrees of Freedom

Table X Percentage Points of the `F`-Distribution, $α = .01$ $α = .01$

Alternate View

`v`₁
Numerator Degrees of Freedom

`v`₂ 1 2 3 4 5 6 7 8 9

1 4,052 4,999.5 5,403 5,625 5,764 5,859 5,928 5,982 6,022

2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39

3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35

4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66

5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16

6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98

7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72

8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91

9 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35

10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94

11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63

12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39

13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19

14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03

15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89

16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78

17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68

18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60

19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52

20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46

21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40

22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35

23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30

24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26

25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22

26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18

27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15

28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12

29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09

30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07

40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89

60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72

**120** 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56

$\infty$ $\infty$ 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41

`v`₁
Numerator Degrees of Freedom

`v`₂ 10 12 15 20 24 30 40 60 120 $\infty$ $\infty$

1 6,056 6,106 6,157 6,209 6,235 6,261 6,287 6,313 6,339 6,366

2 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.50

3 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.13

4 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.46

5 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02

6 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88

7 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65

8 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86

9 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31

10 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91

11 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60

12 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36

13 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17

14 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00

15 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87

16 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75

17 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65

18 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57

19 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49

20 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42

21 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36

22 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31

23 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26

24 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21

25 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.17

26 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.13

27 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.10

28 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.06

29 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03

30 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01

40 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80

60 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60

**120** 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38

$\infty$ $\infty$ 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00

Denominator Degrees of Freedom

`v`₁	Numerator Degrees of Freedom
`v`₂	1	2	3	4	5	6	7	8	9
1	4,052	4,999.5	5,403	5,625	5,764	5,859	5,928	5,982	6,022
2	98.50	99.00	99.17	99.25	99.30	99.33	99.36	99.37	99.39
3	34.12	30.82	29.46	28.71	28.24	27.91	27.67	27.49	27.35
4	21.20	18.00	16.69	15.98	15.52	15.21	14.98	14.80	14.66
5	16.26	13.27	12.06	11.39	10.97	10.67	10.46	10.29	10.16
6	13.75	10.92	9.78	9.15	8.75	8.47	8.26	8.10	7.98
7	12.25	9.55	8.45	7.85	7.46	7.19	6.99	6.84	6.72
8	11.26	8.65	7.59	7.01	6.63	6.37	6.18	6.03	5.91
9	10.56	8.02	6.99	6.42	6.06	5.80	5.61	5.47	5.35
10	10.04	7.56	6.55	5.99	5.64	5.39	5.20	5.06	4.94
11	9.65	7.21	6.22	5.67	5.32	5.07	4.89	4.74	4.63
12	9.33	6.93	5.95	5.41	5.06	4.82	4.64	4.50	4.39
13	9.07	6.70	5.74	5.21	4.86	4.62	4.44	4.30	4.19
14	8.86	6.51	5.56	5.04	4.69	4.46	4.28	4.14	4.03
15	8.68	6.36	5.42	4.89	4.56	4.32	4.14	4.00	3.89
16	8.53	6.23	5.29	4.77	4.44	4.20	4.03	3.89	3.78
17	8.40	6.11	5.18	4.67	4.34	4.10	3.93	3.79	3.68
18	8.29	6.01	5.09	4.58	4.25	4.01	3.84	3.71	3.60
19	8.18	5.93	5.01	4.50	4.17	3.94	3.77	3.63	3.52
20	8.10	5.85	4.94	4.43	4.10	3.87	3.70	3.56	3.46
21	8.02	5.78	4.87	4.37	4.04	3.81	3.64	3.51	3.40
22	7.95	5.72	4.82	4.31	3.99	3.76	3.59	3.45	3.35
23	7.88	5.66	4.76	4.26	3.94	3.71	3.54	3.41	3.30
24	7.82	5.61	4.72	4.22	3.90	3.67	3.50	3.36	3.26
25	7.77	5.57	4.68	4.18	3.85	3.63	3.46	3.32	3.22
26	7.72	5.53	4.64	4.14	3.82	3.59	3.42	3.29	3.18
27	7.68	5.49	4.60	4.11	3.78	3.56	3.39	3.26	3.15
28	7.64	5.45	4.57	4.07	3.75	3.53	3.36	3.23	3.12
29	7.60	5.42	4.54	4.04	3.73	3.50	3.33	3.20	3.09
30	7.56	5.39	4.51	4.02	3.70	3.47	3.30	3.17	3.07
40	7.31	5.18	4.31	3.83	3.51	3.29	3.12	2.99	2.89
60	7.08	4.98	4.13	3.65	3.34	3.12	2.95	2.82	2.72
120	6.85	4.79	3.95	3.48	3.17	2.96	2.79	2.66	2.56
$\infty$ $\infty$	6.63	4.61	3.78	3.32	3.02	2.80	2.64	2.51	2.41
`v`₁	Numerator Degrees of Freedom
`v`₂	10	12	15	20	24	30	40	60	120	$\infty$ $\infty$
1	6,056	6,106	6,157	6,209	6,235	6,261	6,287	6,313	6,339	6,366
2	99.40	99.42	99.43	99.45	99.46	99.47	99.47	99.48	99.49	99.50
3	27.23	27.05	26.87	26.69	26.60	26.50	26.41	26.32	26.22	26.13
4	14.55	14.37	14.20	14.02	13.93	13.84	13.75	13.65	13.56	13.46
5	10.05	9.89	9.72	9.55	9.47	9.38	9.29	9.20	9.11	9.02
6	7.87	7.72	7.56	7.40	7.31	7.23	7.14	7.06	6.97	6.88
7	6.62	6.47	6.31	6.16	6.07	5.99	5.91	5.82	5.74	5.65
8	5.81	5.67	5.52	5.36	5.28	5.20	5.12	5.03	4.95	4.86
9	5.26	5.11	4.96	4.81	4.73	4.65	4.57	4.48	4.40	4.31
10	4.85	4.71	4.56	4.41	4.33	4.25	4.17	4.08	4.00	3.91
11	4.54	4.40	4.25	4.10	4.02	3.94	3.86	3.78	3.69	3.60
12	4.30	4.16	4.01	3.86	3.78	3.70	3.62	3.54	3.45	3.36
13	4.10	3.96	3.82	3.66	3.59	3.51	3.43	3.34	3.25	3.17
14	3.94	3.80	3.66	3.51	3.43	3.35	3.27	3.18	3.09	3.00
15	3.80	3.67	3.52	3.37	3.29	3.21	3.13	3.05	2.96	2.87
16	3.69	3.55	3.41	3.26	3.18	3.10	3.02	2.93	2.84	2.75
17	3.59	3.46	3.31	3.16	3.08	3.00	2.92	2.83	2.75	2.65
18	3.51	3.37	3.23	3.08	3.00	2.92	2.84	2.75	2.66	2.57
19	3.43	3.30	3.15	3.00	2.92	2.84	2.76	2.67	2.58	2.49
20	3.37	3.23	3.09	2.94	2.86	2.78	2.69	2.61	2.52	2.42
21	3.31	3.17	3.03	2.88	2.80	2.72	2.64	2.55	2.46	2.36
22	3.26	3.12	2.98	2.83	2.75	2.67	2.58	2.50	2.40	2.31
23	3.21	3.07	2.93	2.78	2.70	2.62	2.54	2.45	2.35	2.26
24	3.17	3.03	2.89	2.74	2.66	2.58	2.49	2.40	2.31	2.21
25	3.13	2.99	2.85	2.70	2.62	2.54	2.45	2.36	2.27	2.17
26	3.09	2.96	2.81	2.66	2.58	2.50	2.42	2.33	2.23	2.13
27	3.06	2.93	2.78	2.63	2.55	2.47	2.38	2.29	2.20	2.10
28	3.03	2.90	2.75	2.60	2.52	2.44	2.35	2.26	2.17	2.06
29	3.00	2.87	2.73	2.57	2.49	2.41	2.33	2.23	2.14	2.03
30	2.98	2.84	2.70	2.55	2.47	2.39	2.30	2.21	2.11	2.01
40	2.80	2.66	2.52	2.37	2.29	2.20	2.11	2.02	1.92	1.80
60	2.63	2.50	2.35	2.20	2.12	2.03	1.94	1.84	1.73	1.60
120	2.47	2.34	2.19	2.03	1.95	1.86	1.76	1.66	1.53	1.38
$\infty$ $\infty$	2.32	2.18	2.04	1.88	1.79	1.70	1.59	1.47	1.32	1.00
Denominator Degrees of Freedom

Table XI Critical Values of Spearman’s Rank Correlation Coefficient

The $α$ $α$ values correspond to a one-tailed test of $H_{0} : ρ = 0.$ $H_{0} : ρ = 0.$ The value should be doubled for two-tailed tests.

Alternate View

`n` $α = .05$ $α = .05$ $α = .025$ $α = .025$ $α = .01$ $α = .01$ $α = .005$ $α = .005$ `n` $α = .05$ $α = .05$ $α = .025$ $α = .025$ $α = .01$ $α = .01$ $α = .005$ $α = .005$

5 .900 − − − 18 .399 .476 .564 .625

6 .829 .886 .943 − 19 .388 .462 .549 .608

7 .714 .786 .893 − 20 .377 .450 .534 .591

8 .643 .738 .833 .881 21 .368 .438 .521 .576

9 .600 .683 .783 .833 22 .359 .428 .508 .562

10 .564 .648 .745 .794 23 .351 .418 .496 .549

11 .523 .623 .736 .818 24 .343 .409 .485 .537

12 .497 .591 .703 .780 25 .336 .400 .475 .526

13 .475 .566 .673 .745 26 .329 .392 .465 .515

14 .457 .545 .646 .716 27 .323 .385 .456 .505

15 .441 .525 .623 .689 28 .317 .377 .448 .496

16 .425 .507 .601 .666 29 .311 .370 .440 .487

17 .412 .490 .582 .645 30 .305 .364 .432 .478

`n`	$α = .05$ $α = .05$	$α = .025$ $α = .025$	$α = .01$ $α = .01$	$α = .005$ $α = .005$	`n`	$α = .05$ $α = .05$	$α = .025$ $α = .025$	$α = .01$ $α = .01$	$α = .005$ $α = .005$
5	.900	−	−	−	18	.399	.476	.564	.625
6	.829	.886	.943	−	19	.388	.462	.549	.608
7	.714	.786	.893	−	20	.377	.450	.534	.591
8	.643	.738	.833	.881	21	.368	.438	.521	.576
9	.600	.683	.783	.833	22	.359	.428	.508	.562
10	.564	.648	.745	.794	23	.351	.418	.496	.549
11	.523	.623	.736	.818	24	.343	.409	.485	.537
12	.497	.591	.703	.780	25	.336	.400	.475	.526
13	.475	.566	.673	.745	26	.329	.392	.465	.515
14	.457	.545	.646	.716	27	.323	.385	.456	.505
15	.441	.525	.623	.689	28	.317	.377	.448	.496
16	.425	.507	.601	.666	29	.311	.370	.440	.487
17	.412	.490	.582	.645	30	.305	.364	.432	.478

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Table of Contents for
Appendix B: Tables

Appendix B: Tables

Table I Binomial Probabilities

Table II Normal Curve Areas

Table III Critical Values of `t`

Table IV Critical Values of $χ^{2}$ $χ^{2}$

Table V Critical Values of $T_{L}$ $T_{L}$ and $T_{U}$ $T_{U}$ for the Wilcoxon Rank Sum Test

Table VI Critical Values of $T_{0}$ $T_{0}$ in the Wilcoxon Signed Rank Test

Table VII Percentage Points of the `F`-Distribution, $α = .10$ $α = .10$

Table VIII Percentage Points of the `F`-Distribution, $α = .05$ $α = .05$

Table IX Percentage Points of the `F`-Distribution, $α = .025$ $α = .025$

Table X Percentage Points of the `F`-Distribution, $α = .01$ $α = .01$

Table XI Critical Values of Spearman’s Rank Correlation Coefficient

Table of Contents for Appendix B: Tables

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Table of Contents for
Appendix B: Tables