Chapter 13 Appendix

13.1 Statistical Tables

13.1.1 Critical Values

13.1.1.1 \(r\)

	One-tailed Test			Two-tailed Test
df	0.1	0.05	0.01	0.1	0.05	0.01
2	0.90	0.95	0.99	0.95	0.98	0.99
3	0.81	0.88	0.96	0.88	0.92	0.96
4	0.73	0.81	0.92	0.81	0.87	0.92
5	0.67	0.75	0.87	0.75	0.82	0.87
6	0.62	0.71	0.83	0.71	0.77	0.83
7	0.58	0.67	0.80	0.67	0.73	0.80
8	0.55	0.63	0.76	0.63	0.70	0.76
9	0.52	0.60	0.73	0.60	0.67	0.73
10	0.50	0.58	0.71	0.58	0.64	0.71
11	0.48	0.55	0.68	0.55	0.62	0.68
12	0.46	0.53	0.66	0.53	0.59	0.66
13	0.44	0.51	0.64	0.51	0.57	0.64
14	0.43	0.50	0.62	0.50	0.56	0.62
15	0.41	0.48	0.61	0.48	0.54	0.61
16	0.40	0.47	0.59	0.47	0.53	0.59
17	0.39	0.46	0.58	0.46	0.51	0.58
18	0.38	0.44	0.56	0.44	0.50	0.56
19	0.37	0.43	0.55	0.43	0.49	0.55
20	0.36	0.42	0.54	0.42	0.48	0.54
21	0.35	0.41	0.53	0.41	0.47	0.53
22	0.34	0.40	0.52	0.40	0.46	0.52
23	0.34	0.40	0.51	0.40	0.45	0.51
24	0.33	0.39	0.50	0.39	0.44	0.50
25	0.32	0.38	0.49	0.38	0.43	0.49
26	0.32	0.37	0.48	0.37	0.42	0.48
27	0.31	0.37	0.47	0.37	0.42	0.47
28	0.31	0.36	0.46	0.36	0.41	0.46
29	0.30	0.36	0.46	0.36	0.40	0.46
30	0.30	0.35	0.45	0.35	0.40	0.45
31	0.29	0.34	0.44	0.34	0.39	0.44
32	0.29	0.34	0.44	0.34	0.38	0.44
33	0.28	0.33	0.43	0.33	0.38	0.43
34	0.28	0.33	0.42	0.33	0.37	0.42
35	0.27	0.32	0.42	0.32	0.37	0.42
36	0.27	0.32	0.41	0.32	0.36	0.41
37	0.27	0.32	0.41	0.32	0.36	0.41
38	0.26	0.31	0.40	0.31	0.35	0.40
39	0.26	0.31	0.40	0.31	0.35	0.40
40	0.26	0.30	0.39	0.30	0.35	0.39
41	0.25	0.30	0.39	0.30	0.34	0.39
42	0.25	0.30	0.38	0.30	0.34	0.38
43	0.25	0.29	0.38	0.29	0.33	0.38
44	0.25	0.29	0.38	0.29	0.33	0.38
45	0.24	0.29	0.37	0.29	0.33	0.37
46	0.24	0.28	0.37	0.28	0.32	0.37
47	0.24	0.28	0.36	0.28	0.32	0.36
48	0.24	0.28	0.36	0.28	0.32	0.36
49	0.23	0.28	0.36	0.28	0.31	0.36
50	0.23	0.27	0.35	0.27	0.31	0.35
51	0.23	0.27	0.35	0.27	0.31	0.35
52	0.23	0.27	0.35	0.27	0.30	0.35
53	0.22	0.27	0.34	0.27	0.30	0.34
54	0.22	0.26	0.34	0.26	0.30	0.34
55	0.22	0.26	0.34	0.26	0.30	0.34
56	0.22	0.26	0.34	0.26	0.29	0.34
57	0.22	0.26	0.33	0.26	0.29	0.33
58	0.21	0.25	0.33	0.25	0.29	0.33
59	0.21	0.25	0.33	0.25	0.29	0.33
60	0.21	0.25	0.32	0.25	0.28	0.32
61	0.21	0.25	0.32	0.25	0.28	0.32
62	0.21	0.25	0.32	0.25	0.28	0.32
63	0.21	0.24	0.32	0.24	0.28	0.32
64	0.20	0.24	0.31	0.24	0.28	0.31
65	0.20	0.24	0.31	0.24	0.27	0.31
66	0.20	0.24	0.31	0.24	0.27	0.31
67	0.20	0.24	0.31	0.24	0.27	0.31
68	0.20	0.24	0.31	0.24	0.27	0.31
69	0.20	0.23	0.30	0.23	0.27	0.30
70	0.20	0.23	0.30	0.23	0.26	0.30
71	0.19	0.23	0.30	0.23	0.26	0.30
72	0.19	0.23	0.30	0.23	0.26	0.30
73	0.19	0.23	0.30	0.23	0.26	0.30
74	0.19	0.23	0.29	0.23	0.26	0.29
75	0.19	0.22	0.29	0.22	0.26	0.29
76	0.19	0.22	0.29	0.22	0.25	0.29
77	0.19	0.22	0.29	0.22	0.25	0.29
78	0.19	0.22	0.29	0.22	0.25	0.29
79	0.18	0.22	0.28	0.22	0.25	0.28
80	0.18	0.22	0.28	0.22	0.25	0.28
81	0.18	0.22	0.28	0.22	0.25	0.28
82	0.18	0.21	0.28	0.21	0.24	0.28
83	0.18	0.21	0.28	0.21	0.24	0.28
84	0.18	0.21	0.28	0.21	0.24	0.28
85	0.18	0.21	0.27	0.21	0.24	0.27
86	0.18	0.21	0.27	0.21	0.24	0.27
87	0.18	0.21	0.27	0.21	0.24	0.27
88	0.17	0.21	0.27	0.21	0.24	0.27
89	0.17	0.21	0.27	0.21	0.23	0.27
90	0.17	0.20	0.27	0.20	0.23	0.27
91	0.17	0.20	0.27	0.20	0.23	0.27
92	0.17	0.20	0.26	0.20	0.23	0.26
93	0.17	0.20	0.26	0.20	0.23	0.26
94	0.17	0.20	0.26	0.20	0.23	0.26
95	0.17	0.20	0.26	0.20	0.23	0.26
96	0.17	0.20	0.26	0.20	0.23	0.26
97	0.17	0.20	0.26	0.20	0.23	0.26
98	0.17	0.20	0.26	0.20	0.22	0.26
99	0.16	0.20	0.26	0.20	0.22	0.26
100	0.16	0.19	0.25	0.19	0.22	0.25

13.1.1.2 \(t\)

One-tailed Test						Two-tailed Test
df	0.05	0.025	0.01	0.005	0.0025	df	0.1	0.05	0.02	0.01	0.005
1	6.314	12.706	31.821	63.657	127.321	1	6.314	12.706	31.821	63.657	127.321
2	2.920	4.303	6.965	9.925	14.089	2	2.920	4.303	6.965	9.925	14.089
3	2.353	3.182	4.541	5.841	7.453	3	2.353	3.182	4.541	5.841	7.453
4	2.132	2.776	3.747	4.604	5.598	4	2.132	2.776	3.747	4.604	5.598
5	2.015	2.571	3.365	4.032	4.773	5	2.015	2.571	3.365	4.032	4.773
6	1.943	2.447	3.143	3.707	4.317	6	1.943	2.447	3.143	3.707	4.317
7	1.895	2.365	2.998	3.499	4.029	7	1.895	2.365	2.998	3.499	4.029
8	1.860	2.306	2.896	3.355	3.833	8	1.860	2.306	2.896	3.355	3.833
9	1.833	2.262	2.821	3.250	3.690	9	1.833	2.262	2.821	3.250	3.690
10	1.812	2.228	2.764	3.169	3.581	10	1.812	2.228	2.764	3.169	3.581
11	1.796	2.201	2.718	3.106	3.497	11	1.796	2.201	2.718	3.106	3.497
12	1.782	2.179	2.681	3.055	3.428	12	1.782	2.179	2.681	3.055	3.428
13	1.771	2.160	2.650	3.012	3.372	13	1.771	2.160	2.650	3.012	3.372
14	1.761	2.145	2.624	2.977	3.326	14	1.761	2.145	2.624	2.977	3.326
15	1.753	2.131	2.602	2.947	3.286	15	1.753	2.131	2.602	2.947	3.286
16	1.746	2.120	2.583	2.921	3.252	16	1.746	2.120	2.583	2.921	3.252
17	1.740	2.110	2.567	2.898	3.222	17	1.740	2.110	2.567	2.898	3.222
18	1.734	2.101	2.552	2.878	3.197	18	1.734	2.101	2.552	2.878	3.197
19	1.729	2.093	2.539	2.861	3.174	19	1.729	2.093	2.539	2.861	3.174
20	1.725	2.086	2.528	2.845	3.153	20	1.725	2.086	2.528	2.845	3.153
21	1.721	2.080	2.518	2.831	3.135	21	1.721	2.080	2.518	2.831	3.135
22	1.717	2.074	2.508	2.819	3.119	22	1.717	2.074	2.508	2.819	3.119
23	1.714	2.069	2.500	2.807	3.104	23	1.714	2.069	2.500	2.807	3.104
24	1.711	2.064	2.492	2.797	3.091	24	1.711	2.064	2.492	2.797	3.091
25	1.708	2.060	2.485	2.787	3.078	25	1.708	2.060	2.485	2.787	3.078
26	1.706	2.056	2.479	2.779	3.067	26	1.706	2.056	2.479	2.779	3.067
27	1.703	2.052	2.473	2.771	3.057	27	1.703	2.052	2.473	2.771	3.057
28	1.701	2.048	2.467	2.763	3.047	28	1.701	2.048	2.467	2.763	3.047
29	1.699	2.045	2.462	2.756	3.038	29	1.699	2.045	2.462	2.756	3.038
30	1.697	2.042	2.457	2.750	3.030	30	1.697	2.042	2.457	2.750	3.030
31	1.696	2.040	2.453	2.744	3.022	31	1.696	2.040	2.453	2.744	3.022
32	1.694	2.037	2.449	2.738	3.015	32	1.694	2.037	2.449	2.738	3.015
33	1.692	2.035	2.445	2.733	3.008	33	1.692	2.035	2.445	2.733	3.008
34	1.691	2.032	2.441	2.728	3.002	34	1.691	2.032	2.441	2.728	3.002
35	1.690	2.030	2.438	2.724	2.996	35	1.690	2.030	2.438	2.724	2.996
36	1.688	2.028	2.434	2.719	2.990	36	1.688	2.028	2.434	2.719	2.990
37	1.687	2.026	2.431	2.715	2.985	37	1.687	2.026	2.431	2.715	2.985
38	1.686	2.024	2.429	2.712	2.980	38	1.686	2.024	2.429	2.712	2.980
39	1.685	2.023	2.426	2.708	2.976	39	1.685	2.023	2.426	2.708	2.976
40	1.684	2.021	2.423	2.704	2.971	40	1.684	2.021	2.423	2.704	2.971
41	1.683	2.020	2.421	2.701	2.967	41	1.683	2.020	2.421	2.701	2.967
42	1.682	2.018	2.418	2.698	2.963	42	1.682	2.018	2.418	2.698	2.963
43	1.681	2.017	2.416	2.695	2.959	43	1.681	2.017	2.416	2.695	2.959
44	1.680	2.015	2.414	2.692	2.956	44	1.680	2.015	2.414	2.692	2.956
45	1.679	2.014	2.412	2.690	2.952	45	1.679	2.014	2.412	2.690	2.952
46	1.679	2.013	2.410	2.687	2.949	46	1.679	2.013	2.410	2.687	2.949
47	1.678	2.012	2.408	2.685	2.946	47	1.678	2.012	2.408	2.685	2.946
48	1.677	2.011	2.407	2.682	2.943	48	1.677	2.011	2.407	2.682	2.943
49	1.677	2.010	2.405	2.680	2.940	49	1.677	2.010	2.405	2.680	2.940
50	1.676	2.009	2.403	2.678	2.937	50	1.676	2.009	2.403	2.678	2.937
51	1.675	2.008	2.402	2.676	2.934	51	1.675	2.008	2.402	2.676	2.934
52	1.675	2.007	2.400	2.674	2.932	52	1.675	2.007	2.400	2.674	2.932
53	1.674	2.006	2.399	2.672	2.929	53	1.674	2.006	2.399	2.672	2.929
54	1.674	2.005	2.397	2.670	2.927	54	1.674	2.005	2.397	2.670	2.927
55	1.673	2.004	2.396	2.668	2.925	55	1.673	2.004	2.396	2.668	2.925
56	1.673	2.003	2.395	2.667	2.923	56	1.673	2.003	2.395	2.667	2.923
57	1.672	2.002	2.394	2.665	2.920	57	1.672	2.002	2.394	2.665	2.920
58	1.672	2.002	2.392	2.663	2.918	58	1.672	2.002	2.392	2.663	2.918
59	1.671	2.001	2.391	2.662	2.916	59	1.671	2.001	2.391	2.662	2.916
60	1.671	2.000	2.390	2.660	2.915	60	1.671	2.000	2.390	2.660	2.915
61	1.670	2.000	2.389	2.659	2.913	61	1.670	2.000	2.389	2.659	2.913
62	1.670	1.999	2.388	2.657	2.911	62	1.670	1.999	2.388	2.657	2.911
63	1.669	1.998	2.387	2.656	2.909	63	1.669	1.998	2.387	2.656	2.909
64	1.669	1.998	2.386	2.655	2.908	64	1.669	1.998	2.386	2.655	2.908
65	1.669	1.997	2.385	2.654	2.906	65	1.669	1.997	2.385	2.654	2.906
66	1.668	1.997	2.384	2.652	2.904	66	1.668	1.997	2.384	2.652	2.904
67	1.668	1.996	2.383	2.651	2.903	67	1.668	1.996	2.383	2.651	2.903
68	1.668	1.995	2.382	2.650	2.902	68	1.668	1.995	2.382	2.650	2.902
69	1.667	1.995	2.382	2.649	2.900	69	1.667	1.995	2.382	2.649	2.900
70	1.667	1.994	2.381	2.648	2.899	70	1.667	1.994	2.381	2.648	2.899
71	1.667	1.994	2.380	2.647	2.897	71	1.667	1.994	2.380	2.647	2.897
72	1.666	1.993	2.379	2.646	2.896	72	1.666	1.993	2.379	2.646	2.896
73	1.666	1.993	2.379	2.645	2.895	73	1.666	1.993	2.379	2.645	2.895
74	1.666	1.993	2.378	2.644	2.894	74	1.666	1.993	2.378	2.644	2.894
75	1.665	1.992	2.377	2.643	2.892	75	1.665	1.992	2.377	2.643	2.892
76	1.665	1.992	2.376	2.642	2.891	76	1.665	1.992	2.376	2.642	2.891
77	1.665	1.991	2.376	2.641	2.890	77	1.665	1.991	2.376	2.641	2.890
78	1.665	1.991	2.375	2.640	2.889	78	1.665	1.991	2.375	2.640	2.889
79	1.664	1.990	2.374	2.640	2.888	79	1.664	1.990	2.374	2.640	2.888
80	1.664	1.990	2.374	2.639	2.887	80	1.664	1.990	2.374	2.639	2.887
81	1.664	1.990	2.373	2.638	2.886	81	1.664	1.990	2.373	2.638	2.886
82	1.664	1.989	2.373	2.637	2.885	82	1.664	1.989	2.373	2.637	2.885
83	1.663	1.989	2.372	2.636	2.884	83	1.663	1.989	2.372	2.636	2.884
84	1.663	1.989	2.372	2.636	2.883	84	1.663	1.989	2.372	2.636	2.883
85	1.663	1.988	2.371	2.635	2.882	85	1.663	1.988	2.371	2.635	2.882
86	1.663	1.988	2.370	2.634	2.881	86	1.663	1.988	2.370	2.634	2.881
87	1.663	1.988	2.370	2.634	2.880	87	1.663	1.988	2.370	2.634	2.880
88	1.662	1.987	2.369	2.633	2.880	88	1.662	1.987	2.369	2.633	2.880
89	1.662	1.987	2.369	2.632	2.879	89	1.662	1.987	2.369	2.632	2.879
90	1.662	1.987	2.368	2.632	2.878	90	1.662	1.987	2.368	2.632	2.878
91	1.662	1.986	2.368	2.631	2.877	91	1.662	1.986	2.368	2.631	2.877
92	1.662	1.986	2.368	2.630	2.876	92	1.662	1.986	2.368	2.630	2.876
93	1.661	1.986	2.367	2.630	2.876	93	1.661	1.986	2.367	2.630	2.876
94	1.661	1.986	2.367	2.629	2.875	94	1.661	1.986	2.367	2.629	2.875
95	1.661	1.985	2.366	2.629	2.874	95	1.661	1.985	2.366	2.629	2.874
96	1.661	1.985	2.366	2.628	2.873	96	1.661	1.985	2.366	2.628	2.873
97	1.661	1.985	2.365	2.627	2.873	97	1.661	1.985	2.365	2.627	2.873
98	1.661	1.984	2.365	2.627	2.872	98	1.661	1.984	2.365	2.627	2.872
99	1.660	1.984	2.365	2.626	2.871	99	1.660	1.984	2.365	2.626	2.871
100	1.660	1.984	2.364	2.626	2.871	100	1.660	1.984	2.364	2.626	2.871
101	1.660	1.984	2.364	2.625	2.870	101	1.660	1.984	2.364	2.625	2.870
102	1.660	1.983	2.363	2.625	2.869	102	1.660	1.983	2.363	2.625	2.869
103	1.660	1.983	2.363	2.624	2.869	103	1.660	1.983	2.363	2.624	2.869
104	1.660	1.983	2.363	2.624	2.868	104	1.660	1.983	2.363	2.624	2.868
105	1.659	1.983	2.362	2.623	2.868	105	1.659	1.983	2.362	2.623	2.868
106	1.659	1.983	2.362	2.623	2.867	106	1.659	1.983	2.362	2.623	2.867
107	1.659	1.982	2.362	2.623	2.866	107	1.659	1.982	2.362	2.623	2.866
108	1.659	1.982	2.361	2.622	2.866	108	1.659	1.982	2.361	2.622	2.866
109	1.659	1.982	2.361	2.622	2.865	109	1.659	1.982	2.361	2.622	2.865
110	1.659	1.982	2.361	2.621	2.865	110	1.659	1.982	2.361	2.621	2.865
111	1.659	1.982	2.360	2.621	2.864	111	1.659	1.982	2.360	2.621	2.864
112	1.659	1.981	2.360	2.620	2.864	112	1.659	1.981	2.360	2.620	2.864
113	1.658	1.981	2.360	2.620	2.863	113	1.658	1.981	2.360	2.620	2.863
114	1.658	1.981	2.360	2.620	2.863	114	1.658	1.981	2.360	2.620	2.863
115	1.658	1.981	2.359	2.619	2.862	115	1.658	1.981	2.359	2.619	2.862
116	1.658	1.981	2.359	2.619	2.862	116	1.658	1.981	2.359	2.619	2.862
117	1.658	1.980	2.359	2.619	2.861	117	1.658	1.980	2.359	2.619	2.861
118	1.658	1.980	2.358	2.618	2.861	118	1.658	1.980	2.358	2.618	2.861
119	1.658	1.980	2.358	2.618	2.860	119	1.658	1.980	2.358	2.618	2.860
120	1.658	1.980	2.358	2.617	2.860	120	1.658	1.980	2.358	2.617	2.860

13.1.1.3 \(F\)

13.1.1.3.1 \(\alpha = 0.05\)

	\(df_{numerator}\)
\(df_{denominator}\)	1	2	3	4	5	6	7	8	9	10
1	161.45	199.50	215.71	224.58	230.16	233.99	236.77	238.88	240.54	241.88
2	18.51	19.00	19.16	19.25	19.30	19.33	19.35	19.37	19.38	19.40
3	10.13	9.55	9.28	9.12	9.01	8.94	8.89	8.85	8.81	8.79
4	7.71	6.94	6.59	6.39	6.26	6.16	6.09	6.04	6.00	5.96
5	6.61	5.79	5.41	5.19	5.05	4.95	4.88	4.82	4.77	4.74
6	5.99	5.14	4.76	4.53	4.39	4.28	4.21	4.15	4.10	4.06
7	5.59	4.74	4.35	4.12	3.97	3.87	3.79	3.73	3.68	3.64
8	5.32	4.46	4.07	3.84	3.69	3.58	3.50	3.44	3.39	3.35
9	5.12	4.26	3.86	3.63	3.48	3.37	3.29	3.23	3.18	3.14
10	4.96	4.10	3.71	3.48	3.33	3.22	3.14	3.07	3.02	2.98
11	4.84	3.98	3.59	3.36	3.20	3.09	3.01	2.95	2.90	2.85
12	4.75	3.89	3.49	3.26	3.11	3.00	2.91	2.85	2.80	2.75
13	4.67	3.81	3.41	3.18	3.03	2.92	2.83	2.77	2.71	2.67
14	4.60	3.74	3.34	3.11	2.96	2.85	2.76	2.70	2.65	2.60
15	4.54	3.68	3.29	3.06	2.90	2.79	2.71	2.64	2.59	2.54
16	4.49	3.63	3.24	3.01	2.85	2.74	2.66	2.59	2.54	2.49
17	4.45	3.59	3.20	2.96	2.81	2.70	2.61	2.55	2.49	2.45
18	4.41	3.55	3.16	2.93	2.77	2.66	2.58	2.51	2.46	2.41
19	4.38	3.52	3.13	2.90	2.74	2.63	2.54	2.48	2.42	2.38
20	4.35	3.49	3.10	2.87	2.71	2.60	2.51	2.45	2.39	2.35
21	4.32	3.47	3.07	2.84	2.68	2.57	2.49	2.42	2.37	2.32
22	4.30	3.44	3.05	2.82	2.66	2.55	2.46	2.40	2.34	2.30
23	4.28	3.42	3.03	2.80	2.64	2.53	2.44	2.37	2.32	2.27
24	4.26	3.40	3.01	2.78	2.62	2.51	2.42	2.36	2.30	2.25
25	4.24	3.39	2.99	2.76	2.60	2.49	2.40	2.34	2.28	2.24
26	4.23	3.37	2.98	2.74	2.59	2.47	2.39	2.32	2.27	2.22
27	4.21	3.35	2.96	2.73	2.57	2.46	2.37	2.31	2.25	2.20
28	4.20	3.34	2.95	2.71	2.56	2.45	2.36	2.29	2.24	2.19
29	4.18	3.33	2.93	2.70	2.55	2.43	2.35	2.28	2.22	2.18
30	4.17	3.32	2.92	2.69	2.53	2.42	2.33	2.27	2.21	2.16
31	4.16	3.30	2.91	2.68	2.52	2.41	2.32	2.25	2.20	2.15
32	4.15	3.29	2.90	2.67	2.51	2.40	2.31	2.24	2.19	2.14
33	4.14	3.28	2.89	2.66	2.50	2.39	2.30	2.23	2.18	2.13
34	4.13	3.28	2.88	2.65	2.49	2.38	2.29	2.23	2.17	2.12
35	4.12	3.27	2.87	2.64	2.49	2.37	2.29	2.22	2.16	2.11
36	4.11	3.26	2.87	2.63	2.48	2.36	2.28	2.21	2.15	2.11
37	4.11	3.25	2.86	2.63	2.47	2.36	2.27	2.20	2.14	2.10
38	4.10	3.24	2.85	2.62	2.46	2.35	2.26	2.19	2.14	2.09
39	4.09	3.24	2.85	2.61	2.46	2.34	2.26	2.19	2.13	2.08
40	4.08	3.23	2.84	2.61	2.45	2.34	2.25	2.18	2.12	2.08
41	4.08	3.23	2.83	2.60	2.44	2.33	2.24	2.17	2.12	2.07
42	4.07	3.22	2.83	2.59	2.44	2.32	2.24	2.17	2.11	2.06
43	4.07	3.21	2.82	2.59	2.43	2.32	2.23	2.16	2.11	2.06
44	4.06	3.21	2.82	2.58	2.43	2.31	2.23	2.16	2.10	2.05
45	4.06	3.20	2.81	2.58	2.42	2.31	2.22	2.15	2.10	2.05
46	4.05	3.20	2.81	2.57	2.42	2.30	2.22	2.15	2.09	2.04
47	4.05	3.20	2.80	2.57	2.41	2.30	2.21	2.14	2.09	2.04
48	4.04	3.19	2.80	2.57	2.41	2.29	2.21	2.14	2.08	2.03
49	4.04	3.19	2.79	2.56	2.40	2.29	2.20	2.13	2.08	2.03
50	4.03	3.18	2.79	2.56	2.40	2.29	2.20	2.13	2.07	2.03
51	4.03	3.18	2.79	2.55	2.40	2.28	2.20	2.13	2.07	2.02
52	4.03	3.18	2.78	2.55	2.39	2.28	2.19	2.12	2.07	2.02
53	4.02	3.17	2.78	2.55	2.39	2.28	2.19	2.12	2.06	2.01
54	4.02	3.17	2.78	2.54	2.39	2.27	2.18	2.12	2.06	2.01
55	4.02	3.16	2.77	2.54	2.38	2.27	2.18	2.11	2.06	2.01
56	4.01	3.16	2.77	2.54	2.38	2.27	2.18	2.11	2.05	2.00
57	4.01	3.16	2.77	2.53	2.38	2.26	2.18	2.11	2.05	2.00
58	4.01	3.16	2.76	2.53	2.37	2.26	2.17	2.10	2.05	2.00
59	4.00	3.15	2.76	2.53	2.37	2.26	2.17	2.10	2.04	2.00
60	4.00	3.15	2.76	2.53	2.37	2.25	2.17	2.10	2.04	1.99
61	4.00	3.15	2.76	2.52	2.37	2.25	2.16	2.09	2.04	1.99
62	4.00	3.15	2.75	2.52	2.36	2.25	2.16	2.09	2.03	1.99
63	3.99	3.14	2.75	2.52	2.36	2.25	2.16	2.09	2.03	1.98
64	3.99	3.14	2.75	2.52	2.36	2.24	2.16	2.09	2.03	1.98
65	3.99	3.14	2.75	2.51	2.36	2.24	2.15	2.08	2.03	1.98
66	3.99	3.14	2.74	2.51	2.35	2.24	2.15	2.08	2.03	1.98
67	3.98	3.13	2.74	2.51	2.35	2.24	2.15	2.08	2.02	1.98
68	3.98	3.13	2.74	2.51	2.35	2.24	2.15	2.08	2.02	1.97
69	3.98	3.13	2.74	2.50	2.35	2.23	2.15	2.08	2.02	1.97
70	3.98	3.13	2.74	2.50	2.35	2.23	2.14	2.07	2.02	1.97
71	3.98	3.13	2.73	2.50	2.34	2.23	2.14	2.07	2.01	1.97
72	3.97	3.12	2.73	2.50	2.34	2.23	2.14	2.07	2.01	1.96
73	3.97	3.12	2.73	2.50	2.34	2.23	2.14	2.07	2.01	1.96
74	3.97	3.12	2.73	2.50	2.34	2.22	2.14	2.07	2.01	1.96
75	3.97	3.12	2.73	2.49	2.34	2.22	2.13	2.06	2.01	1.96
76	3.97	3.12	2.72	2.49	2.33	2.22	2.13	2.06	2.01	1.96
77	3.97	3.12	2.72	2.49	2.33	2.22	2.13	2.06	2.00	1.96
78	3.96	3.11	2.72	2.49	2.33	2.22	2.13	2.06	2.00	1.95
79	3.96	3.11	2.72	2.49	2.33	2.22	2.13	2.06	2.00	1.95
80	3.96	3.11	2.72	2.49	2.33	2.21	2.13	2.06	2.00	1.95
81	3.96	3.11	2.72	2.48	2.33	2.21	2.12	2.05	2.00	1.95
82	3.96	3.11	2.72	2.48	2.33	2.21	2.12	2.05	2.00	1.95
83	3.96	3.11	2.71	2.48	2.32	2.21	2.12	2.05	1.99	1.95
84	3.95	3.11	2.71	2.48	2.32	2.21	2.12	2.05	1.99	1.95
85	3.95	3.10	2.71	2.48	2.32	2.21	2.12	2.05	1.99	1.94
86	3.95	3.10	2.71	2.48	2.32	2.21	2.12	2.05	1.99	1.94
87	3.95	3.10	2.71	2.48	2.32	2.20	2.12	2.05	1.99	1.94
88	3.95	3.10	2.71	2.48	2.32	2.20	2.12	2.05	1.99	1.94
89	3.95	3.10	2.71	2.47	2.32	2.20	2.11	2.04	1.99	1.94
90	3.95	3.10	2.71	2.47	2.32	2.20	2.11	2.04	1.99	1.94
91	3.95	3.10	2.70	2.47	2.31	2.20	2.11	2.04	1.98	1.94
92	3.94	3.10	2.70	2.47	2.31	2.20	2.11	2.04	1.98	1.94
93	3.94	3.09	2.70	2.47	2.31	2.20	2.11	2.04	1.98	1.93
94	3.94	3.09	2.70	2.47	2.31	2.20	2.11	2.04	1.98	1.93
95	3.94	3.09	2.70	2.47	2.31	2.20	2.11	2.04	1.98	1.93
96	3.94	3.09	2.70	2.47	2.31	2.19	2.11	2.04	1.98	1.93
97	3.94	3.09	2.70	2.47	2.31	2.19	2.11	2.04	1.98	1.93
98	3.94	3.09	2.70	2.46	2.31	2.19	2.10	2.03	1.98	1.93
99	3.94	3.09	2.70	2.46	2.31	2.19	2.10	2.03	1.98	1.93
100	3.94	3.09	2.70	2.46	2.31	2.19	2.10	2.03	1.97	1.93

13.1.1.3.2 \(\alpha = 0.01\)

	\(df_{numerator}\)
\(df_{denominator}\)	1	2	3	4	5	6	7	8	9	10
1	4052.18	4999.50	5403.35	5624.58	5763.65	5858.99	5928.36	5981.07	6022.47	6055.85
2	98.50	99.00	99.17	99.25	99.30	99.33	99.36	99.37	99.39	99.40
3	34.12	30.82	29.46	28.71	28.24	27.91	27.67	27.49	27.35	27.23
4	21.20	18.00	16.69	15.98	15.52	15.21	14.98	14.80	14.66	14.55
5	16.26	13.27	12.06	11.39	10.97	10.67	10.46	10.29	10.16	10.05
6	13.75	10.92	9.78	9.15	8.75	8.47	8.26	8.10	7.98	7.87
7	12.25	9.55	8.45	7.85	7.46	7.19	6.99	6.84	6.72	6.62
8	11.26	8.65	7.59	7.01	6.63	6.37	6.18	6.03	5.91	5.81
9	10.56	8.02	6.99	6.42	6.06	5.80	5.61	5.47	5.35	5.26
10	10.04	7.56	6.55	5.99	5.64	5.39	5.20	5.06	4.94	4.85
11	9.65	7.21	6.22	5.67	5.32	5.07	4.89	4.74	4.63	4.54
12	9.33	6.93	5.95	5.41	5.06	4.82	4.64	4.50	4.39	4.30
13	9.07	6.70	5.74	5.21	4.86	4.62	4.44	4.30	4.19	4.10
14	8.86	6.51	5.56	5.04	4.69	4.46	4.28	4.14	4.03	3.94
15	8.68	6.36	5.42	4.89	4.56	4.32	4.14	4.00	3.89	3.80
16	8.53	6.23	5.29	4.77	4.44	4.20	4.03	3.89	3.78	3.69
17	8.40	6.11	5.18	4.67	4.34	4.10	3.93	3.79	3.68	3.59
18	8.29	6.01	5.09	4.58	4.25	4.01	3.84	3.71	3.60	3.51
19	8.18	5.93	5.01	4.50	4.17	3.94	3.77	3.63	3.52	3.43
20	8.10	5.85	4.94	4.43	4.10	3.87	3.70	3.56	3.46	3.37
21	8.02	5.78	4.87	4.37	4.04	3.81	3.64	3.51	3.40	3.31
22	7.95	5.72	4.82	4.31	3.99	3.76	3.59	3.45	3.35	3.26
23	7.88	5.66	4.76	4.26	3.94	3.71	3.54	3.41	3.30	3.21
24	7.82	5.61	4.72	4.22	3.90	3.67	3.50	3.36	3.26	3.17
25	7.77	5.57	4.68	4.18	3.85	3.63	3.46	3.32	3.22	3.13
26	7.72	5.53	4.64	4.14	3.82	3.59	3.42	3.29	3.18	3.09
27	7.68	5.49	4.60	4.11	3.78	3.56	3.39	3.26	3.15	3.06
28	7.64	5.45	4.57	4.07	3.75	3.53	3.36	3.23	3.12	3.03
29	7.60	5.42	4.54	4.04	3.73	3.50	3.33	3.20	3.09	3.00
30	7.56	5.39	4.51	4.02	3.70	3.47	3.30	3.17	3.07	2.98
31	7.53	5.36	4.48	3.99	3.67	3.45	3.28	3.15	3.04	2.96
32	7.50	5.34	4.46	3.97	3.65	3.43	3.26	3.13	3.02	2.93
33	7.47	5.31	4.44	3.95	3.63	3.41	3.24	3.11	3.00	2.91
34	7.44	5.29	4.42	3.93	3.61	3.39	3.22	3.09	2.98	2.89
35	7.42	5.27	4.40	3.91	3.59	3.37	3.20	3.07	2.96	2.88
36	7.40	5.25	4.38	3.89	3.57	3.35	3.18	3.05	2.95	2.86
37	7.37	5.23	4.36	3.87	3.56	3.33	3.17	3.04	2.93	2.84
38	7.35	5.21	4.34	3.86	3.54	3.32	3.15	3.02	2.92	2.83
39	7.33	5.19	4.33	3.84	3.53	3.30	3.14	3.01	2.90	2.81
40	7.31	5.18	4.31	3.83	3.51	3.29	3.12	2.99	2.89	2.80
41	7.30	5.16	4.30	3.81	3.50	3.28	3.11	2.98	2.87	2.79
42	7.28	5.15	4.29	3.80	3.49	3.27	3.10	2.97	2.86	2.78
43	7.26	5.14	4.27	3.79	3.48	3.25	3.09	2.96	2.85	2.76
44	7.25	5.12	4.26	3.78	3.47	3.24	3.08	2.95	2.84	2.75
45	7.23	5.11	4.25	3.77	3.45	3.23	3.07	2.94	2.83	2.74
46	7.22	5.10	4.24	3.76	3.44	3.22	3.06	2.93	2.82	2.73
47	7.21	5.09	4.23	3.75	3.43	3.21	3.05	2.92	2.81	2.72
48	7.19	5.08	4.22	3.74	3.43	3.20	3.04	2.91	2.80	2.71
49	7.18	5.07	4.21	3.73	3.42	3.19	3.03	2.90	2.79	2.71
50	7.17	5.06	4.20	3.72	3.41	3.19	3.02	2.89	2.78	2.70
51	7.16	5.05	4.19	3.71	3.40	3.18	3.01	2.88	2.78	2.69
52	7.15	5.04	4.18	3.70	3.39	3.17	3.00	2.87	2.77	2.68
53	7.14	5.03	4.17	3.70	3.38	3.16	3.00	2.87	2.76	2.68
54	7.13	5.02	4.17	3.69	3.38	3.16	2.99	2.86	2.76	2.67
55	7.12	5.01	4.16	3.68	3.37	3.15	2.98	2.85	2.75	2.66
56	7.11	5.01	4.15	3.67	3.36	3.14	2.98	2.85	2.74	2.66
57	7.10	5.00	4.15	3.67	3.36	3.14	2.97	2.84	2.74	2.65
58	7.09	4.99	4.14	3.66	3.35	3.13	2.96	2.83	2.73	2.64
59	7.08	4.98	4.13	3.65	3.34	3.12	2.96	2.83	2.72	2.64
60	7.08	4.98	4.13	3.65	3.34	3.12	2.95	2.82	2.72	2.63
61	7.07	4.97	4.12	3.64	3.33	3.11	2.95	2.82	2.71	2.63
62	7.06	4.96	4.11	3.64	3.33	3.11	2.94	2.81	2.71	2.62
63	7.06	4.96	4.11	3.63	3.32	3.10	2.94	2.81	2.70	2.62
64	7.05	4.95	4.10	3.63	3.32	3.10	2.93	2.80	2.70	2.61
65	7.04	4.95	4.10	3.62	3.31	3.09	2.93	2.80	2.69	2.61
66	7.04	4.94	4.09	3.62	3.31	3.09	2.92	2.79	2.69	2.60
67	7.03	4.94	4.09	3.61	3.30	3.08	2.92	2.79	2.68	2.60
68	7.02	4.93	4.08	3.61	3.30	3.08	2.91	2.78	2.68	2.59
69	7.02	4.93	4.08	3.60	3.29	3.08	2.91	2.78	2.68	2.59
70	7.01	4.92	4.07	3.60	3.29	3.07	2.91	2.78	2.67	2.59
71	7.01	4.92	4.07	3.60	3.29	3.07	2.90	2.77	2.67	2.58
72	7.00	4.91	4.07	3.59	3.28	3.06	2.90	2.77	2.66	2.58
73	7.00	4.91	4.06	3.59	3.28	3.06	2.89	2.77	2.66	2.57
74	6.99	4.90	4.06	3.58	3.28	3.06	2.89	2.76	2.66	2.57
75	6.99	4.90	4.05	3.58	3.27	3.05	2.89	2.76	2.65	2.57
76	6.98	4.90	4.05	3.58	3.27	3.05	2.88	2.75	2.65	2.56
77	6.98	4.89	4.05	3.57	3.26	3.05	2.88	2.75	2.65	2.56
78	6.97	4.89	4.04	3.57	3.26	3.04	2.88	2.75	2.64	2.56
79	6.97	4.88	4.04	3.57	3.26	3.04	2.87	2.75	2.64	2.55
80	6.96	4.88	4.04	3.56	3.26	3.04	2.87	2.74	2.64	2.55
81	6.96	4.88	4.03	3.56	3.25	3.03	2.87	2.74	2.63	2.55
82	6.95	4.87	4.03	3.56	3.25	3.03	2.87	2.74	2.63	2.54
83	6.95	4.87	4.03	3.55	3.25	3.03	2.86	2.73	2.63	2.54
84	6.95	4.87	4.02	3.55	3.24	3.02	2.86	2.73	2.63	2.54
85	6.94	4.86	4.02	3.55	3.24	3.02	2.86	2.73	2.62	2.54
86	6.94	4.86	4.02	3.55	3.24	3.02	2.85	2.73	2.62	2.53
87	6.94	4.86	4.02	3.54	3.24	3.02	2.85	2.72	2.62	2.53
88	6.93	4.85	4.01	3.54	3.23	3.01	2.85	2.72	2.62	2.53
89	6.93	4.85	4.01	3.54	3.23	3.01	2.85	2.72	2.61	2.53
90	6.93	4.85	4.01	3.53	3.23	3.01	2.84	2.72	2.61	2.52
91	6.92	4.85	4.00	3.53	3.23	3.01	2.84	2.71	2.61	2.52
92	6.92	4.84	4.00	3.53	3.22	3.00	2.84	2.71	2.61	2.52
93	6.92	4.84	4.00	3.53	3.22	3.00	2.84	2.71	2.60	2.52
94	6.91	4.84	4.00	3.53	3.22	3.00	2.84	2.71	2.60	2.52
95	6.91	4.84	3.99	3.52	3.22	3.00	2.83	2.70	2.60	2.51
96	6.91	4.83	3.99	3.52	3.21	3.00	2.83	2.70	2.60	2.51
97	6.90	4.83	3.99	3.52	3.21	2.99	2.83	2.70	2.60	2.51
98	6.90	4.83	3.99	3.52	3.21	2.99	2.83	2.70	2.59	2.51
99	6.90	4.83	3.99	3.51	3.21	2.99	2.83	2.70	2.59	2.51
100	6.90	4.82	3.98	3.51	3.21	2.99	2.82	2.69	2.59	2.50

13.1.1.4 \(q\)

13.1.1.4.1 \(\alpha = 0.05\)

	\(k\) (number of groups)
\(df_\epsilon\)	2	3	4	5	6	7	8	9	10
5	3.64	4.60	5.22	5.67	6.03	6.33	6.58	6.80	6.99
6	3.46	4.34	4.90	5.30	5.63	5.90	6.12	6.32	6.49
7	3.34	4.16	4.68	5.06	5.36	5.61	5.82	6.00	6.16
8	3.26	4.04	4.53	4.89	5.17	5.40	5.60	5.77	5.92
9	3.20	3.95	4.41	4.76	5.02	5.24	5.43	5.59	5.74
10	3.15	3.88	4.33	4.65	4.91	5.12	5.30	5.46	5.60
11	3.11	3.82	4.26	4.57	4.82	5.03	5.20	5.35	5.49
12	3.08	3.77	4.20	4.51	4.75	4.95	5.12	5.27	5.39
13	3.06	3.73	4.15	4.45	4.69	4.88	5.05	5.19	5.32
14	3.03	3.70	4.11	4.41	4.64	4.83	4.99	5.13	5.25
15	3.01	3.67	4.08	4.37	4.59	4.78	4.94	5.08	5.20
16	3.00	3.65	4.05	4.33	4.56	4.74	4.90	5.03	5.15
17	2.98	3.63	4.02	4.30	4.52	4.70	4.86	4.99	5.11
18	2.97	3.61	4.00	4.28	4.49	4.67	4.82	4.96	5.07
19	2.96	3.59	3.98	4.25	4.47	4.65	4.79	4.92	5.04
20	2.95	3.58	3.96	4.23	4.45	4.62	4.77	4.90	5.01
21	2.94	3.56	3.94	4.21	4.42	4.60	4.74	4.87	4.98
22	2.93	3.55	3.93	4.20	4.41	4.58	4.72	4.85	4.96
23	2.93	3.54	3.91	4.18	4.39	4.56	4.70	4.83	4.94
24	2.92	3.53	3.90	4.17	4.37	4.54	4.68	4.81	4.92
25	2.91	3.52	3.89	4.15	4.36	4.53	4.67	4.79	4.90
26	2.91	3.51	3.88	4.14	4.35	4.51	4.65	4.77	4.88
27	2.90	3.51	3.87	4.13	4.33	4.50	4.64	4.76	4.86
28	2.90	3.50	3.86	4.12	4.32	4.49	4.62	4.74	4.85
29	2.89	3.49	3.85	4.11	4.31	4.47	4.61	4.73	4.84
30	2.89	3.49	3.85	4.10	4.30	4.46	4.60	4.72	4.82
31	2.88	3.48	3.84	4.09	4.29	4.45	4.59	4.71	4.81
32	2.88	3.48	3.83	4.09	4.28	4.45	4.58	4.70	4.80
33	2.88	3.47	3.83	4.08	4.28	4.44	4.57	4.69	4.79
34	2.87	3.47	3.82	4.07	4.27	4.43	4.56	4.68	4.78
35	2.87	3.46	3.81	4.07	4.26	4.42	4.56	4.67	4.77
36	2.87	3.46	3.81	4.06	4.25	4.41	4.55	4.66	4.76
37	2.87	3.45	3.80	4.05	4.25	4.41	4.54	4.66	4.76
38	2.86	3.45	3.80	4.05	4.24	4.40	4.53	4.65	4.75
39	2.86	3.45	3.79	4.04	4.24	4.39	4.53	4.64	4.74
40	2.86	3.44	3.79	4.04	4.23	4.39	4.52	4.63	4.73
41	2.86	3.44	3.79	4.03	4.23	4.38	4.51	4.63	4.73
42	2.85	3.44	3.78	4.03	4.22	4.38	4.51	4.62	4.72
43	2.85	3.43	3.78	4.03	4.22	4.37	4.50	4.62	4.72
44	2.85	3.43	3.78	4.02	4.21	4.37	4.50	4.61	4.71
45	2.85	3.43	3.77	4.02	4.21	4.36	4.49	4.61	4.70
46	2.85	3.42	3.77	4.01	4.20	4.36	4.49	4.60	4.70
47	2.85	3.42	3.77	4.01	4.20	4.36	4.48	4.60	4.69
48	2.84	3.42	3.76	4.01	4.20	4.35	4.48	4.59	4.69
49	2.84	3.42	3.76	4.00	4.19	4.35	4.48	4.59	4.69
50	2.84	3.42	3.76	4.00	4.19	4.34	4.47	4.58	4.68
51	2.84	3.41	3.76	4.00	4.19	4.34	4.47	4.58	4.68
52	2.84	3.41	3.75	4.00	4.18	4.34	4.47	4.58	4.67
53	2.84	3.41	3.75	3.99	4.18	4.33	4.46	4.57	4.67
54	2.84	3.41	3.75	3.99	4.18	4.33	4.46	4.57	4.67
55	2.83	3.41	3.75	3.99	4.18	4.33	4.46	4.57	4.66
56	2.83	3.40	3.74	3.99	4.17	4.32	4.45	4.56	4.66
57	2.83	3.40	3.74	3.98	4.17	4.32	4.45	4.56	4.66
58	2.83	3.40	3.74	3.98	4.17	4.32	4.45	4.56	4.65
59	2.83	3.40	3.74	3.98	4.17	4.32	4.44	4.55	4.65
60	2.83	3.40	3.74	3.98	4.16	4.31	4.44	4.55	4.65
61	2.83	3.40	3.74	3.98	4.16	4.31	4.44	4.55	4.64
62	2.83	3.40	3.73	3.97	4.16	4.31	4.44	4.55	4.64
63	2.83	3.39	3.73	3.97	4.16	4.31	4.43	4.54	4.64
64	2.83	3.39	3.73	3.97	4.15	4.30	4.43	4.54	4.64
65	2.82	3.39	3.73	3.97	4.15	4.30	4.43	4.54	4.63
66	2.82	3.39	3.73	3.97	4.15	4.30	4.43	4.54	4.63
67	2.82	3.39	3.73	3.96	4.15	4.30	4.42	4.53	4.63
68	2.82	3.39	3.72	3.96	4.15	4.30	4.42	4.53	4.63
69	2.82	3.39	3.72	3.96	4.15	4.29	4.42	4.53	4.62
70	2.82	3.39	3.72	3.96	4.14	4.29	4.42	4.53	4.62
71	2.82	3.39	3.72	3.96	4.14	4.29	4.42	4.52	4.62
72	2.82	3.38	3.72	3.96	4.14	4.29	4.41	4.52	4.62
73	2.82	3.38	3.72	3.96	4.14	4.29	4.41	4.52	4.62
74	2.82	3.38	3.72	3.95	4.14	4.29	4.41	4.52	4.61
75	2.82	3.38	3.72	3.95	4.14	4.28	4.41	4.52	4.61
76	2.82	3.38	3.71	3.95	4.13	4.28	4.41	4.52	4.61
77	2.82	3.38	3.71	3.95	4.13	4.28	4.41	4.51	4.61
78	2.82	3.38	3.71	3.95	4.13	4.28	4.40	4.51	4.61
79	2.81	3.38	3.71	3.95	4.13	4.28	4.40	4.51	4.60
80	2.81	3.38	3.71	3.95	4.13	4.28	4.40	4.51	4.60
81	2.81	3.38	3.71	3.95	4.13	4.28	4.40	4.51	4.60
82	2.81	3.38	3.71	3.94	4.13	4.27	4.40	4.51	4.60
83	2.81	3.37	3.71	3.94	4.13	4.27	4.40	4.50	4.60
84	2.81	3.37	3.71	3.94	4.12	4.27	4.40	4.50	4.60
85	2.81	3.37	3.71	3.94	4.12	4.27	4.39	4.50	4.60
86	2.81	3.37	3.71	3.94	4.12	4.27	4.39	4.50	4.59
87	2.81	3.37	3.70	3.94	4.12	4.27	4.39	4.50	4.59
88	2.81	3.37	3.70	3.94	4.12	4.27	4.39	4.50	4.59
89	2.81	3.37	3.70	3.94	4.12	4.27	4.39	4.50	4.59
90	2.81	3.37	3.70	3.94	4.12	4.27	4.39	4.50	4.59
91	2.81	3.37	3.70	3.94	4.12	4.26	4.39	4.49	4.59
92	2.81	3.37	3.70	3.94	4.12	4.26	4.39	4.49	4.59
93	2.81	3.37	3.70	3.93	4.12	4.26	4.39	4.49	4.58
94	2.81	3.37	3.70	3.93	4.11	4.26	4.38	4.49	4.58
95	2.81	3.37	3.70	3.93	4.11	4.26	4.38	4.49	4.58
96	2.81	3.37	3.70	3.93	4.11	4.26	4.38	4.49	4.58
97	2.81	3.37	3.70	3.93	4.11	4.26	4.38	4.49	4.58
98	2.81	3.37	3.70	3.93	4.11	4.26	4.38	4.49	4.58
99	2.81	3.37	3.70	3.93	4.11	4.26	4.38	4.49	4.58
100	2.81	3.36	3.70	3.93	4.11	4.26	4.38	4.48	4.58
101	2.81	3.36	3.69	3.93	4.11	4.25	4.38	4.48	4.58
102	2.81	3.36	3.69	3.93	4.11	4.25	4.38	4.48	4.57
103	2.80	3.36	3.69	3.93	4.11	4.25	4.38	4.48	4.57
104	2.80	3.36	3.69	3.93	4.11	4.25	4.37	4.48	4.57
105	2.80	3.36	3.69	3.93	4.11	4.25	4.37	4.48	4.57
106	2.80	3.36	3.69	3.92	4.10	4.25	4.37	4.48	4.57
107	2.80	3.36	3.69	3.92	4.10	4.25	4.37	4.48	4.57
108	2.80	3.36	3.69	3.92	4.10	4.25	4.37	4.48	4.57
109	2.80	3.36	3.69	3.92	4.10	4.25	4.37	4.48	4.57
110	2.80	3.36	3.69	3.92	4.10	4.25	4.37	4.48	4.57
111	2.80	3.36	3.69	3.92	4.10	4.25	4.37	4.47	4.57
112	2.80	3.36	3.69	3.92	4.10	4.25	4.37	4.47	4.57
113	2.80	3.36	3.69	3.92	4.10	4.25	4.37	4.47	4.56
114	2.80	3.36	3.69	3.92	4.10	4.24	4.37	4.47	4.56
115	2.80	3.36	3.69	3.92	4.10	4.24	4.37	4.47	4.56
116	2.80	3.36	3.69	3.92	4.10	4.24	4.37	4.47	4.56
117	2.80	3.36	3.69	3.92	4.10	4.24	4.36	4.47	4.56
118	2.80	3.36	3.69	3.92	4.10	4.24	4.36	4.47	4.56
119	2.80	3.36	3.69	3.92	4.10	4.24	4.36	4.47	4.56
120	2.80	3.36	3.68	3.92	4.10	4.24	4.36	4.47	4.56

13.1.1.4.2 \(\alpha = 0.01\)

	k (number of groups)
\(df_\epsilon\)	2	3	4	5	6	7	8	9	10
5	5.70	6.98	7.80	8.42	8.91	9.32	9.67	9.97	10.24
6	5.24	6.33	7.03	7.56	7.97	8.32	8.61	8.87	9.10
7	4.95	5.92	6.54	7.00	7.37	7.68	7.94	8.17	8.37
8	4.75	5.64	6.20	6.62	6.96	7.24	7.47	7.68	7.86
9	4.60	5.43	5.96	6.35	6.66	6.91	7.13	7.33	7.49
10	4.48	5.27	5.77	6.14	6.43	6.67	6.87	7.05	7.21
11	4.39	5.15	5.62	5.97	6.25	6.48	6.67	6.84	6.99
12	4.32	5.05	5.50	5.84	6.10	6.32	6.51	6.67	6.81
13	4.26	4.96	5.40	5.73	5.98	6.19	6.37	6.53	6.67
14	4.21	4.89	5.32	5.63	5.88	6.08	6.26	6.41	6.54
15	4.17	4.84	5.25	5.56	5.80	5.99	6.16	6.31	6.44
16	4.13	4.79	5.19	5.49	5.72	5.92	6.08	6.22	6.35
17	4.10	4.74	5.14	5.43	5.66	5.85	6.01	6.15	6.27
18	4.07	4.70	5.09	5.38	5.60	5.79	5.94	6.08	6.20
19	4.05	4.67	5.05	5.33	5.55	5.73	5.89	6.02	6.14
20	4.02	4.64	5.02	5.29	5.51	5.69	5.84	5.97	6.09
21	4.00	4.61	4.99	5.26	5.47	5.65	5.79	5.92	6.04
22	3.99	4.59	4.96	5.22	5.43	5.61	5.75	5.88	5.99
23	3.97	4.57	4.93	5.20	5.40	5.57	5.72	5.84	5.95
24	3.96	4.55	4.91	5.17	5.37	5.54	5.69	5.81	5.92
25	3.94	4.53	4.89	5.14	5.35	5.51	5.65	5.78	5.89
26	3.93	4.51	4.87	5.12	5.32	5.49	5.63	5.75	5.86
27	3.92	4.49	4.85	5.10	5.30	5.46	5.60	5.72	5.83
28	3.91	4.48	4.83	5.08	5.28	5.44	5.58	5.70	5.80
29	3.90	4.47	4.81	5.06	5.26	5.42	5.56	5.67	5.78
30	3.89	4.45	4.80	5.05	5.24	5.40	5.54	5.65	5.76
31	3.88	4.44	4.79	5.03	5.23	5.38	5.52	5.63	5.74
32	3.87	4.43	4.77	5.02	5.21	5.37	5.50	5.61	5.72
33	3.87	4.42	4.76	5.00	5.20	5.35	5.48	5.60	5.70
34	3.86	4.41	4.75	4.99	5.18	5.34	5.47	5.58	5.68
35	3.85	4.40	4.74	4.98	5.17	5.32	5.45	5.57	5.67
36	3.85	4.40	4.73	4.97	5.16	5.31	5.44	5.55	5.65
37	3.84	4.39	4.72	4.96	5.15	5.30	5.43	5.54	5.64
38	3.83	4.38	4.71	4.95	5.13	5.29	5.41	5.53	5.62
39	3.83	4.37	4.70	4.94	5.12	5.28	5.40	5.51	5.61
40	3.82	4.37	4.70	4.93	5.11	5.26	5.39	5.50	5.60
41	3.82	4.36	4.69	4.92	5.11	5.26	5.38	5.49	5.59
42	3.82	4.35	4.68	4.91	5.10	5.25	5.37	5.48	5.58
43	3.81	4.35	4.67	4.91	5.09	5.24	5.36	5.47	5.57
44	3.81	4.34	4.67	4.90	5.08	5.23	5.35	5.46	5.56
45	3.80	4.34	4.66	4.89	5.07	5.22	5.34	5.45	5.55
46	3.80	4.33	4.66	4.89	5.07	5.21	5.34	5.44	5.54
47	3.80	4.33	4.65	4.88	5.06	5.21	5.33	5.44	5.53
48	3.79	4.32	4.64	4.87	5.05	5.20	5.32	5.43	5.52
49	3.79	4.32	4.64	4.87	5.05	5.19	5.31	5.42	5.51
50	3.79	4.32	4.63	4.86	5.04	5.19	5.31	5.41	5.51
51	3.78	4.31	4.63	4.86	5.03	5.18	5.30	5.41	5.50
52	3.78	4.31	4.63	4.85	5.03	5.17	5.29	5.40	5.49
53	3.78	4.30	4.62	4.85	5.02	5.17	5.29	5.39	5.49
54	3.78	4.30	4.62	4.84	5.02	5.16	5.28	5.39	5.48
55	3.77	4.30	4.61	4.84	5.01	5.16	5.28	5.38	5.47
56	3.77	4.29	4.61	4.83	5.01	5.15	5.27	5.38	5.47
57	3.77	4.29	4.60	4.83	5.00	5.15	5.27	5.37	5.46
58	3.77	4.29	4.60	4.83	5.00	5.14	5.26	5.37	5.46
59	3.76	4.29	4.60	4.82	5.00	5.14	5.26	5.36	5.45
60	3.76	4.28	4.59	4.82	4.99	5.13	5.25	5.36	5.45
61	3.76	4.28	4.59	4.81	4.99	5.13	5.25	5.35	5.44
62	3.76	4.28	4.59	4.81	4.98	5.12	5.24	5.35	5.44
63	3.76	4.27	4.59	4.81	4.98	5.12	5.24	5.34	5.43
64	3.75	4.27	4.58	4.80	4.98	5.12	5.24	5.34	5.43
65	3.75	4.27	4.58	4.80	4.97	5.11	5.23	5.33	5.42
66	3.75	4.27	4.58	4.80	4.97	5.11	5.23	5.33	5.42
67	3.75	4.26	4.57	4.79	4.97	5.11	5.22	5.33	5.42
68	3.75	4.26	4.57	4.79	4.96	5.10	5.22	5.32	5.41
69	3.75	4.26	4.57	4.79	4.96	5.10	5.22	5.32	5.41
70	3.74	4.26	4.57	4.79	4.96	5.10	5.21	5.31	5.40
71	3.74	4.26	4.56	4.78	4.95	5.09	5.21	5.31	5.40
72	3.74	4.25	4.56	4.78	4.95	5.09	5.21	5.31	5.40
73	3.74	4.25	4.56	4.78	4.95	5.09	5.20	5.30	5.39
74	3.74	4.25	4.56	4.78	4.95	5.08	5.20	5.30	5.39
75	3.74	4.25	4.56	4.77	4.94	5.08	5.20	5.30	5.39
76	3.74	4.25	4.55	4.77	4.94	5.08	5.20	5.30	5.38
77	3.74	4.25	4.55	4.77	4.94	5.08	5.19	5.29	5.38
78	3.73	4.24	4.55	4.77	4.94	5.07	5.19	5.29	5.38
79	3.73	4.24	4.55	4.76	4.93	5.07	5.19	5.29	5.38
80	3.73	4.24	4.55	4.76	4.93	5.07	5.18	5.28	5.37
81	3.73	4.24	4.54	4.76	4.93	5.07	5.18	5.28	5.37
82	3.73	4.24	4.54	4.76	4.93	5.06	5.18	5.28	5.37
83	3.73	4.24	4.54	4.76	4.92	5.06	5.18	5.28	5.36
84	3.73	4.23	4.54	4.75	4.92	5.06	5.17	5.27	5.36
85	3.73	4.23	4.54	4.75	4.92	5.06	5.17	5.27	5.36
86	3.73	4.23	4.54	4.75	4.92	5.06	5.17	5.27	5.36
87	3.72	4.23	4.53	4.75	4.92	5.05	5.17	5.27	5.35
88	3.72	4.23	4.53	4.75	4.92	5.05	5.17	5.27	5.35
89	3.72	4.23	4.53	4.75	4.91	5.05	5.16	5.26	5.35
90	3.72	4.23	4.53	4.74	4.91	5.05	5.16	5.26	5.35
91	3.72	4.23	4.53	4.74	4.91	5.05	5.16	5.26	5.35
92	3.72	4.22	4.53	4.74	4.91	5.04	5.16	5.26	5.34
93	3.72	4.22	4.52	4.74	4.91	5.04	5.16	5.25	5.34
94	3.72	4.22	4.52	4.74	4.90	5.04	5.15	5.25	5.34
95	3.72	4.22	4.52	4.74	4.90	5.04	5.15	5.25	5.34
96	3.72	4.22	4.52	4.74	4.90	5.04	5.15	5.25	5.34
97	3.72	4.22	4.52	4.73	4.90	5.04	5.15	5.25	5.33
98	3.72	4.22	4.52	4.73	4.90	5.03	5.15	5.25	5.33
99	3.71	4.22	4.52	4.73	4.90	5.03	5.15	5.24	5.33
100	3.71	4.22	4.52	4.73	4.90	5.03	5.14	5.24	5.33
101	3.71	4.22	4.52	4.73	4.89	5.03	5.14	5.24	5.33
102	3.71	4.21	4.51	4.73	4.89	5.03	5.14	5.24	5.32
103	3.71	4.21	4.51	4.73	4.89	5.03	5.14	5.24	5.32
104	3.71	4.21	4.51	4.73	4.89	5.02	5.14	5.24	5.32
105	3.71	4.21	4.51	4.72	4.89	5.02	5.14	5.23	5.32
106	3.71	4.21	4.51	4.72	4.89	5.02	5.14	5.23	5.32
107	3.71	4.21	4.51	4.72	4.89	5.02	5.13	5.23	5.32
108	3.71	4.21	4.51	4.72	4.89	5.02	5.13	5.23	5.32
109	3.71	4.21	4.51	4.72	4.88	5.02	5.13	5.23	5.31
110	3.71	4.21	4.51	4.72	4.88	5.02	5.13	5.23	5.31
111	3.71	4.21	4.50	4.72	4.88	5.02	5.13	5.23	5.31
112	3.71	4.21	4.50	4.72	4.88	5.01	5.13	5.22	5.31
113	3.71	4.20	4.50	4.72	4.88	5.01	5.13	5.22	5.31
114	3.70	4.20	4.50	4.71	4.88	5.01	5.12	5.22	5.31
115	3.70	4.20	4.50	4.71	4.88	5.01	5.12	5.22	5.31
116	3.70	4.20	4.50	4.71	4.88	5.01	5.12	5.22	5.30
117	3.70	4.20	4.50	4.71	4.88	5.01	5.12	5.22	5.30
118	3.70	4.20	4.50	4.71	4.87	5.01	5.12	5.22	5.30
119	3.70	4.20	4.50	4.71	4.87	5.01	5.12	5.22	5.30
120	3.70	4.20	4.50	4.71	4.87	5.01	5.12	5.21	5.30

13.1.2 Upper-tail Quantiles of the \(\chi^2\) Distribution

	Upper-tail Probabilities
df	0.99	0.975	0.95	0.90	0.80	0.50	0.20	0.10	0.05	0.025	0.01
1	0.00	0.00	0.00	0.02	0.06	0.45	1.64	2.71	3.84	5.02	6.63
2	0.02	0.05	0.10	0.21	0.45	1.39	3.22	4.61	5.99	7.38	9.21
3	0.11	0.22	0.35	0.58	1.01	2.37	4.64	6.25	7.81	9.35	11.34
4	0.30	0.48	0.71	1.06	1.65	3.36	5.99	7.78	9.49	11.14	13.28
5	0.55	0.83	1.15	1.61	2.34	4.35	7.29	9.24	11.07	12.83	15.09
6	0.87	1.24	1.64	2.20	3.07	5.35	8.56	10.64	12.59	14.45	16.81
7	1.24	1.69	2.17	2.83	3.82	6.35	9.80	12.02	14.07	16.01	18.48
8	1.65	2.18	2.73	3.49	4.59	7.34	11.03	13.36	15.51	17.53	20.09
9	2.09	2.70	3.33	4.17	5.38	8.34	12.24	14.68	16.92	19.02	21.67
10	2.56	3.25	3.94	4.87	6.18	9.34	13.44	15.99	18.31	20.48	23.21
11	3.05	3.82	4.57	5.58	6.99	10.34	14.63	17.28	19.68	21.92	24.72
12	3.57	4.40	5.23	6.30	7.81	11.34	15.81	18.55	21.03	23.34	26.22
13	4.11	5.01	5.89	7.04	8.63	12.34	16.98	19.81	22.36	24.74	27.69
14	4.66	5.63	6.57	7.79	9.47	13.34	18.15	21.06	23.68	26.12	29.14
15	5.23	6.26	7.26	8.55	10.31	14.34	19.31	22.31	25.00	27.49	30.58
16	5.81	6.91	7.96	9.31	11.15	15.34	20.47	23.54	26.30	28.85	32.00
17	6.41	7.56	8.67	10.09	12.00	16.34	21.61	24.77	27.59	30.19	33.41
18	7.01	8.23	9.39	10.86	12.86	17.34	22.76	25.99	28.87	31.53	34.81
19	7.63	8.91	10.12	11.65	13.72	18.34	23.90	27.20	30.14	32.85	36.19
20	8.26	9.59	10.85	12.44	14.58	19.34	25.04	28.41	31.41	34.17	37.57
21	8.90	10.28	11.59	13.24	15.44	20.34	26.17	29.62	32.67	35.48	38.93
22	9.54	10.98	12.34	14.04	16.31	21.34	27.30	30.81	33.92	36.78	40.29
23	10.20	11.69	13.09	14.85	17.19	22.34	28.43	32.01	35.17	38.08	41.64
24	10.86	12.40	13.85	15.66	18.06	23.34	29.55	33.20	36.42	39.36	42.98
25	11.52	13.12	14.61	16.47	18.94	24.34	30.68	34.38	37.65	40.65	44.31
26	12.20	13.84	15.38	17.29	19.82	25.34	31.79	35.56	38.89	41.92	45.64
27	12.88	14.57	16.15	18.11	20.70	26.34	32.91	36.74	40.11	43.19	46.96
28	13.56	15.31	16.93	18.94	21.59	27.34	34.03	37.92	41.34	44.46	48.28
29	14.26	16.05	17.71	19.77	22.48	28.34	35.14	39.09	42.56	45.72	49.59
30	14.95	16.79	18.49	20.60	23.36	29.34	36.25	40.26	43.77	46.98	50.89

13.2 Greek Letters

13.2.1 Symbols and names

Name	Lower-case Greek	Upper-case Greek
alpha	\(\alpha\)	\(\text{A}\)
beta	\(\beta\)	\(\text{B}\)
gamma	\(\gamma\)	\(\Gamma\)
delta	\(\delta\)	\(\Delta\)
epsilon	\(\epsilon\)	\(\text{E}\)
zeta	\(\zeta\)	\(\text{Z}\)
eta	\(\eta\)	\(\text{H}\)
theta	\(\theta\)	\(\Theta\)
iota	\(\iota\)	\(\text{I}\)
kappa	\(\kappa\)	\(\text{K}\)
lamdba	\(\lambda\)	\(\Lambda\)
mu	\(\mu\)	\(\text{M}\)
nu	\(\nu\)	\(\text{N}\)
xi	\(\xi\)	\(\Xi\)
omicron	\(\omicron\)	\(\text{O}\)
pi	\(\pi\)	\(\Pi\)
rho	\(\rho\)	\(\text{P}\)
sigma	\(\sigma\)	\(\Sigma\)
tau	\(\tau\)	\(\text{T}\)
upsilon	\(\upsilon\)	\(\Upsilon\)
phi	\(\phi\)	\(\Phi\)
chi	\(\chi\)	\(\text{X}\)
psi	\(\psi\)	\(\Psi\)
omega	\(\omega\)	\(\Omega\)

13.2.2 Usages

Lower-case Greek Letter	Name	Usage(s)
\(\alpha\)	alpha	False alarm rate
\(\beta\)	beta	Miss rate
\(\gamma\)	gamma
\(\delta\)	delta	difference
\(\epsilon\)	epsilon	Error
\(\zeta\)	zeta
\(\eta\)	eta	\(\eta^2\) is an effect-size measure for ANOVA
\(\theta\)	theta	Generic unknown parameter symbol
\(\iota\)	iota
\(\kappa\)	kappa	Cohen’s \(\kappa\) is a test of inter-rater agreement
\(\lambda\)	lamdba	Indicates a rate parameter in Poisson regression; also used to denote a hazard function
\(\mu\)	mu	mean
\(\nu\)	nu	degrees of freedom
\(\xi\)	xi
\(\omicron\)	omicron
\(\pi\)	pi	population proportion parameter
\(\rho\)	rho	the name of a nonparametric correlation coefficient; population correlation parameter
\(\sigma\)	sigma	\(\sigma=\) population standard deviation; \(\sigma^2 =\) population variance
\(\tau\)	tau	the name of another nonparametric correlation coefficient
\(\upsilon\)	upsilon
\(\phi\)	phi
\(\chi\)	chi	The \(\chi^2\) test is a nonparametric test of association
\(\psi\)	psi
\(\omega\)	omega	\(\omega^2\) is an effect-size measure for ANOVA

Upper-case Greek Letter	Name	Usage(s)
\(\text{A}\)	alpha
\(\text{B}\)	beta
\(\Gamma\)	gamma
\(\Delta\)	delta	difference
\(\text{E}\)	epsilon
\(\text{Z}\)	zeta
\(\text{H}\)	eta
\(\Theta\)	theta
\(\text{I}\)	iota
\(\text{K}\)	kappa
\(\Lambda\)	lamdba
\(\text{M}\)	mu
\(\text{N}\)	nu
\(\Xi\)	xi
\(\text{O}\)	omicron
\(\Pi\)	pi	product
\(\text{P}\)	rho
\(\Sigma\)	sigma	summation
\(\text{T}\)	tau
\(\Upsilon\)	upsilon
\(\Phi\)	phi
\(\text{X}\)	chi
\(\Psi\)	psi
\(\Omega\)	omega	sample space