Data Science for Business and Decision Making

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Table J

Critical Values of U_c for the Mann-Whitney U Test Considering P(U_cal < U_c) = α
P(U_cal < U_c) = 0.05
N₂ N₁	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
3	0	0	1	2	2	3	4	4	5	5	6	7	7	8	9	9	10	11
4	0	1	2	3	4	5	6	7	8	9	10	11	12	14	15	16	17	18
5	1	2	4	5	6	8	9	11	12	13	15	16	18	19	20	22	23	25
6	2	3	5	7	8	10	12	14	16	17	19	21	23	25	26	28	30	32
7	2	4	6	8	11	13	15	17	19	21	24	26	28	30	33	35	37	39
8	3	5	8	10	13	15	18	20	23	26	28	31	33	36	39	41	44	47
9	4	6	9	12	15	18	21	24	27	30	33	36	39	42	45	48	51	54
10	4	7	11	14	17	20	24	27	31	34	37	41	44	48	51	55	58	62
11	5	8	12	16	19	23	27	31	34	38	42	46	50	54	57	61	65	69
12	5	9	13	17	21	26	30	34	38	42	47	51	55	60	64	68	72	77
13	6	10	15	19	24	28	33	37	42	47	51	56	61	65	70	75	80	84
14	7	11	16	21	26	31	36	41	46	51	56	61	66	71	77	82	87	92
15	7	12	18	23	28	33	39	44	50	55	61	66	72	77	83	88	94	100
16	8	14	19	25	30	36	42	48	54	60	65	71	77	83	89	95	101	107
17	9	15	20	26	33	39	45	51	57	64	70	77	83	89	96	102	109	115
18	9	16	22	28	35	41	48	55	61	68	75	82	88	95	102	109	116	123
19	10	17	23	30	37	44	51	58	65	72	80	87	94	101	109	116	123	130
20	11	18	25	32	39	47	54	62	69	77	84	92	100	107	115	123	130	138

Table J

P(U_cal < U_c) = 0.025
N₂ N₁	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
3	–	0	0	1	1	2	2	3	3	4	4	5	5	6	6	7	7	8
4	–	0	1	2	3	4	4	5	6	7	8	9	10	11	11	12	13	14
5	0	1	2	3	5	6	7	8	9	11	12	13	14	15	17	18	19	20
6	1	2	3	5	6	8	10	11	13	14	16	17	19	21	22	24	25	27
7	1	3	5	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34
8	2	4	6	8	10	13	15	17	19	22	24	26	29	31	34	36	38	41
9	2	4	7	10	12	15	17	20	23	26	28	31	34	37	39	42	45	48
10	3	5	8	11	14	17	20	23	26	29	33	36	39	42	45	48	52	55
11	3	6	9	13	16	19	23	26	30	33	37	40	44	47	51	55	58	62
12	4	7	11	14	18	22	26	29	33	37	41	45	49	53	57	61	65	69
13	4	8	12	16	20	24	28	33	37	41	45	50	54	59	63	67	72	76
14	5	9	13	17	22	26	31	36	40	45	50	55	59	64	67	74	78	83
15	5	10	14	19	24	29	34	39	44	49	54	59	64	70	75	80	85	90
16	6	11	15	21	26	31	37	42	47	53	59	64	70	75	81	86	92	98
17	6	11	17	22	28	34	39	45	51	57	63	67	75	81	87	93	99	105
18	7	12	18	24	30	36	42	48	55	61	67	74	80	86	93	99	103	112
19	7	13	19	25	32	38	45	52	58	65	72	78	85	92	99	106	113	119
20	8	14	20	27	34	41	48	55	62	69	76	83	90	98	105	112	119	127

Unlabelled Table

P(U_cal < U_c ) = 0.01
N₂ N₁	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
3	–	0	0	0	0	0	1	1	1	2	2	2	3	3	4	4	4	5
4	–	–	0	1	1	2	3	3	4	5	5	6	7	7	8	9	9	10
5	–	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
6	–	1	2	3	4	6	7	8	9	11	12	13	15	16	18	19	20	22
7	0	1	3	4	6	7	9	11	12	14	16	17	19	21	23	24	26	28
8	0	2	4	6	7	9	11	13	15	17	20	22	24	26	28	30	32	34
9	1	3	5	7	9	11	14	16	18	21	23	26	28	31	33	36	38	40
10	1	3	6	8	11	13	16	19	22	24	27	30	33	36	38	41	44	47
11	1	4	7	9	12	15	18	22	25	29	31	34	37	41	44	47	50	53
12	2	5	8	11	14	17	21	24	28	31	35	38	42	46	49	53	56	60
13	2	5	9	12	16	20	23	27	31	35	39	43	47	51	55	59	63	67
14	2	6	10	13	17	22	26	30	34	38	43	47	51	56	60	65	69	73
15	3	7	11	15	19	24	28	33	37	42	47	51	56	61	66	70	75	80
16	3	7	12	16	21	26	31	36	41	46	51	56	61	66	71	76	82	87
17	4	8	13	18	23	28	33	38	44	49	55	60	66	71	77	82	88	93
18	4	9	14	19	24	30	36	41	47	53	59	65	70	76	82	88	94	100
19	4	9	15	20	26	32	38	44	50	56	63	69	75	82	88	94	101	107
20	5	10	16	22	28	34	40	47	53	60	67	73	80	87	93	100	107	114

Unlabelled Table

P(U_cal < U_c ) = 0.005
N₂ N₁	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
3	–	0	0	0	0	0	0	0	0	1	1	1	2	2	2	2	3	3
4	–	–	0	0	0	1	1	2	2	3	3	4	5	5	6	6	7	8
5	–	–	0	1	1	2	3	4	5	6	7	7	8	9	10	11	12	13
6	–	0	1	2	3	4	5	6	7	9	10	11	12	13	15	16	17	18
7	–	0	1	3	4	6	7	9	10	12	13	15	16	18	19	21	22	24
8	–	1	2	4	6	7	9	11	13	15	17	18	20	22	24	26	28	30
9	0	1	3	5	7	9	11	13	16	18	20	22	24	27	29	31	33	36
10	0	2	4	6	9	11	13	16	18	21	24	26	29	31	34	37	39	42
11	0	2	5	7	10	13	16	18	21	24	27	30	33	36	39	42	45	48
12	1	3	6	9	12	15	18	21	24	27	31	34	37	41	44	47	51	54
13	1	3	7	10	13	17	20	24	27	31	34	38	42	45	49	53	56	60
14	1	4	7	11	15	18	22	26	30	34	38	42	46	50	54	58	63	67
15	2	5	8	12	16	20	24	29	33	37	42	46	51	55	60	64	69	73
16	2	5	9	13	18	22	27	31	36	41	45	50	55	60	65	70	74	79
17	2	6	10	15	19	24	29	34	39	44	49	54	60	65	70	75	81	86
18	2	6	11	16	21	26	31	37	42	47	53	58	64	70	75	81	87	92
19	3	7	12	17	22	28	33	39	45	51	56	63	69	74	81	87	93	99
20	3	8	13	18	24	30	36	42	48	54	60	67	73	79	86	92	99	105

Unlabelled Table

Table K

Critical Values for the Friedman’s Test Considering P(F_cal > F_c) = α
k	N	α ≤ 0.10	α ≤ 0.05	α ≤ 0.01
3	3	6.00	6.00	–
	4	6.00	6.50	8.00
	5	5.20	6.40	8.40
	6	5.33	7.00	9.00
	7	5.43	7.14	8.86
	8	5.25	6.25	9.00
	9	5.56	6.22	8.67
	10	5.00	6.20	9.60
	11	4.91	6.54	8.91
	12	5.17	6.17	8.67
	13	4.77	6.00	9.39
	∞	4.61	5.99	9.21
4	2	6.00	6.00	–
	3	6.60	7.40	8.60
	4	6.30	7.80	9.60
	5	6.36	7.80	9.96
	6	6.40	7.60	10.00
	7	6.26	7.80	10.37
	8	6.30	7.50	10.35
	∞	6.25	7.82	11.34
5	3	7.47	8.53	10.13
	4	7.60	8.80	11.00
	5	7.68	8.96	11.52
	∞	7.78	9.49	13.28

Table K

Table L

Critical Values for the Kruskal-Wallis Test Considering P(H_cal > H_c) = α
Sample Sizes			α
n₁	n₂	n₃	0.10	0.05	0.01	0.005	0.001
2	2	2	4.25
3	2	1	4.29
3	2	2	4.71	4.71
3	3	1	4.57	5.14
3	3	2	4.56	5.36
3	3	3	4.62	5.60	7.20	7.20
4	2	1	4.50
4	2	2	4.46	5.33
4	3	1	4.06	5.21
4	3	2	4.51	5.44	6.44	7.00
4	3	3	4.71	5.73	6.75	7.32	8.02
4	4	1	4.17	4.97	6.67
4	4	2	4.55	5.45	7.04	7.28
4	4	3	4.55	5.60	7.14	7.59	8.32
4	4	4	4.65	5.69	7.66	8.00	8.65
5	2	1	4.20	5.00
5	2	2	4.36	5.16	6.53
5	3	1	4.02	4.96
5	3	2	4.65	5.25	6.82	7.18
5	3	3	4.53	5.65	7.08	7.51	8.24
5	4	1	3.99	4.99	6.95	7.36
5	4	2	4.54	5.27	7.12	7.57	8.11
5	4	3	4.55	5.63	7.44	7.91	8.50
5	4	4	4.62	5.62	7.76	8.14	9.00
5	5	1	4.11	5.13	7.31	7.75
5	5	2	4.62	5.34	7.27	8.13	8.68
5	5	3	4.54	5.71	7.54	8.24	9.06
5	5	4	4.53	5.64	7.77	8.37	9.32
5	5	5	4.56	5.78	7.98	8.72	9.68
Large samples			4.61	5.99	9.21	10.60	13.82

Table L

Table M

Critical Values of the Cochran’s C Statistic Considering P(C_cal > C_c) = α
α = 5%
ν/k	2	3	4	5	6	7	8	9	10	12	15	20	24	30	40	60	120
1	0.9985	0.9669	0.9065	0.8412	0.7808	0.7271	0.6798	0.6385	0.6020	0.5410	0.4709	0.3894	0.3434	0.2929	0.2370	0.1737	0.0998
2	0.9750	0.8709	0.7679	0.6838	0.6161	0.5612	0.5157	0.4775	0.4450	0.3924	0.3346	0.2705	0.2354	0.1980	0.1567	0.1131	0.0632
3	0.9392	0.7977	0.6841	0.5981	0.5321	0.4800	0.4377	0.4027	0.3733	0.3264	0.2758	0.2205	0.1907	0.1593	0.1259	0.0895	0.0495
4	0.9057	0.7457	0.6287	0.5441	0.4803	0.4307	0.3910	0.3584	0.3311	0.2880	0.2419	0.1921	0.1656	0.1377	0.1082	0.0765	0.0419
5	0.8772	0.7071	0.5895	0.5065	0.4447	0.3974	0.3595	0.3286	0.3029	0.2624	0.2195	0.1735	0.1493	0.1237	0.0968	0.0682	0.0371
6	0.8534	0.6771	0.5598	0.4783	0.4184	0.3726	0.3362	0.3067	0.2823	0.2439	0.2034	0.1602	0.1374	0.1137	0.0887	0.0623	0.0337
7	0.8332	0.6530	0.5365	0.4564	0.3980	0.3535	0.3185	0.2901	0.2666	0.2299	0.1911	0.1501	0.1286	0.1061	0.0827	0.0583	0.0312
8	0.8159	0.6333	0.5175	0.4387	0.3817	0.3384	0.3043	0.2768	0.2541	0.2187	0.1815	0.1422	0.1216	0.1002	0.0780	0.0552	0.0292
9	0.8010	0.6167	0.5017	0.4241	0.3682	0.3259	0.2926	0.2659	0.2439	0.2098	0.1736	0.1357	0.1160	0.0958	0.0745	0.0520	0.0279
10	0.7880	0.6025	0.4884	0.4118	0.3568	0.3154	0.2829	0.2568	0.2353	0.2020	0.1671	0.1303	0.1113	0.0921	0.0713	0.0497	0.0266
16	0.7341	0.5466	0.4366	0.3645	0.3135	0.2756	0.2462	0.2226	0.2032	0.1737	0.1429	0.1108	0.0942	0.0771	0.0595	0.0411	0.0218
36	0.6602	0.4748	0.3720	0.3066	0.2612	0.2278	0.2022	0.1820	0.1655	0.1403	0.1144	0.0879	0.0743	0.0604	0.0462	0.0316	0.0165
144	0.5813	0.4031	0.3093	0.2513	0.2119	0.1833	0.1616	0.1446	0.1308	0.1100	0.0889	0.0675	0.0567	0.0457	0.0347	0.0234	0.0120
∞	0.5000	0.3333	0.2500	0.2000	0.1667	0.1429	0.1250	0.1111	0.1000	0.0833	0.0667	0.0500	0.0417	0.0333	0.0250	0.0167	0.0083

Table M

α = 1%
ν/k	2	3	4	5	6	7	8	9	10	12	15	20	24	30	40	60	120
1	0.9999	0.9933	0.9676	0.9279	0.8828	0.8376	0.7945	0.7544	0.7175	0.6528	0.5747	0.4799	0.4247	0.3632	0.2940	0.2151	0.1225
2	0.9950	0.9423	0.8643	0.7885	0.7218	0.6644	0.6152	0.5727	0.5358	0.4751	0.4069	0.3297	0.2821	0.2412	0.1915	0.1371	0.0759
3	0.9794	0.8831	0.7814	0.6957	0.6258	0.5685	0.5209	0.4810	0.4469	0.3919	0.3317	0.2654	0.2295	0.1913	0.1508	0.1069	0.0585
4	0.9586	0.8335	0.7212	0.6329	0.5635	0.5080	0.4627	0.4251	0.3934	0.3428	0.2882	0.2288	0.1970	0.1635	0.1281	0.0902	0.0489
5	0.9373	0.7933	0.6761	0.5875	0.5195	0.4659	0.4226	0.3870	0.3572	0.3099	0.2593	0.2048	0.1759	0.1454	0.1135	0.0796	0.0429
6	0.9172	0.7606	0.6410	0.5531	0.4866	0.4347	0.3932	0.3592	0.3308	0.2861	0.2386	0.1877	0.1608	0.1327	0.1033	0.0722	0.0387
7	0.8988	0.7335	0.6129	0.5259	0.4608	0.4105	0.3704	0.3378	0.3106	0.2680	0.2228	0.1748	0.1495	0.1232	0.0957	0.0668	0.0357
8	0.8823	0.7107	0.5897	0.5037	0.4401	0.3911	0.3522	0.3207	0.2945	0.2535	0.2104	0.1646	0.1406	0.1157	0.0898	0.0625	0.0334
9	0.8674	0.6912	0.5702	0.4854	0.4229	0.3751	0.3373	0.3067	0.2813	0.2419	0.2002	0.1567	0.1388	0.1100	0.0853	0.0594	0.0316
10	0.8539	0.6743	0.5536	0.4697	0.4084	0.3616	0.3248	0.2950	0.2704	0.2320	0.1918	0.1501	0.1283	0.1054	0.0816	0.0567	0.0302
16	0.7949	0.6059	0.4884	0.4094	0.3529	0.3105	0.2779	0.2514	0.2297	0.1961	0.1612	0.1248	0.1060	0.0867	0.0668	0.0461	0.0242
36	0.7067	0.5153	0.4057	0.3351	0.2858	0.2494	0.2214	0.1992	0.1811	0.1535	0.1251	0.0960	0.0810	0.0658	0.0503	0.0344	0.0178
144	0.6062	0.4230	0.3251	0.2644	0.2229	0.1929	0.1700	0.1521	0.1376	0.1157	0.0934	0.0709	0.0595	0.0480	0.0363	0.0245	0.0125
∞	0.5000	0.3333	0.2500	0.2000	0.1667	0.1429	0.1250	0.1111	0.1000	0.0833	0.0667	0.0500	0.0417	0.0333	0.0250	0.0167	0.0083

Unlabelled Table

Table N

Critical Values of Hartley F_max Statistic Considering P(F_max.cal > F_max.c) = α
α = 5%
ν/k	2	3	4	5	6	7	8	9	10	11	12
2	39	87.5	142	202	266	333	403	475	550	626	704
3	15.4	27.8	39.2	50.7	62	72.9	83.5	93.9	104	114	124
4	9.6	15.5	20.6	25.2	29.5	33.6	37.5	41.1	44.6	48	51.4
5	7.15	10.8	13.7	16.3	18.7	20.8	22.9	24.7	26.5	28.2	29.9
6	5.82	8.38	10.4	12.1	13.7	15	16.3	17.5	18.6	19.7	20.7
7	4.99	6.94	8.44	9.7	10.8	11.8	12.7	13.5	14.3	15.1	15.8
8	4.43	6	7.18	8.12	9.03	9.78	10.5	11.1	11.7	12.2	12.7
9	4.03	5.34	6.31	7.11	7.8	8.41	8.95	9.45	9.91	10.3	10.7
10	3.72	4.85	5.67	6.34	6.92	7.42	7.87	8.28	8.66	9.01	9.34
12	3.28	4.16	4.79	5.3	5.72	6.09	6.42	6.72	7	7.25	7.48
15	2.86	3.54	4.01	4.37	4.68	4.95	5.19	5.4	5.59	5.77	5.93
20	2.46	2.95	3.29	3.54	3.76	3.94	4.1	4.24	4.37	4.49	4.59
30	2.07	2.4	2.61	2.78	2.91	3.02	3.12	3.21	3.29	3.36	3.39
60	1.67	1.85	1.96	2.04	2.11	2.17	2.22	2.26	2.3	2.33	2.36
∞	1	1	1	1	1	1	1	1	1	1	1

Table N

α = 1%
ν/k	2	3	4	5	6	7	8	9	10	11	12
2	199	448	729	1036	1362	1705	2069	2432	2813	3204	3605
3	47.5	85	120	151	184	216	249	281	310	337	361
4	23.2	37	49	59	69	79	89	97	106	113	120
5	14.9	22	28	33	38	42	46	50	54	57	60
6	11.1	15.5	19.1	22	25	27	30	32	34	36	37
7	8.89	12.1	14.5	16.5	18.4	20	22	23	24	26	27
8	7.5	9.9	11.7	13.2	14.5	15.8	16.9	17.9	18.9	19.8	21
9	6.54	8.5	9.9	11.1	12.1	13.1	13.9	14.7	15.3	16	16.6
10	5.85	7.4	8.6	9.6	10.4	11.1	11.8	12.4	12.9	13.4	13.9
12	4.91	6.1	6.9	7.6	8.2	8.7	9.1	9.5	9.9	10.2	10.6
15	4.07	4.9	5.5	6	6.4	6.7	7.1	7.3	7.5	7.8	8
20	3.32	3.8	4.3	4.6	4.9	5.1	5.3	5.5	5.6	5.8	5.9
30	2.63	3	3.3	3.4	3.6	3.7	3.8	3.9	4	4.1	4.2
60	1.96	2.2	2.3	2.4	2.4	2.5	2.5	2.6	2.6	2.7	2.7
∞	1	1	1	1	1	1	1	1	1	1	1

Unlabelled Table

Table O

Control Chart Constants $\bar{X}$ $\bar{X}$
				$\bar{X} and R Charts$ $\bar{X} and R Charts$			$\bar{X} and S Charts$ $\bar{X} and S Charts$
n	d₂	d₃	C₄	A₂	D₃	D₄	A₃	B₃	B₄
2	1.128	0.853	0.798	1.880	–	3.267	2.659	–	3.267
3	1.693	0.888	0.886	1.023	–	2.574	1.954	–	2.568
4	2.059	0.880	0.921	0.729	–	2.282	1.628	–	2.266
5	2.326	0.880	0.940	0.577	–	2.114	1.427	–	2.089
6	2.534	0.848	0.952	0.483	–	2.004	1.287	0.030	1.970
7	2.704	0.833	0.959	0.419	0.076	1.924	1.182	0.118	1.882
8	2.847	0.820	0.965	0.373	0.136	1.864	1.099	0.185	1.815
9	2.970	0.808	0.969	0.337	0.184	1.816	1.032	0.239	1.761
10	3.078	0.797	0.973	0.308	0.223	1.777	0.975	0.284	1.716
11	3.173	0.787	0.975	0.285	0.256	1.744	0.927	0.321	1.679
12	3.258	0.779	0.978	0.266	0.283	1.717	0.886	0.354	1.646
13	3.336	0.770	0.979	0.249	0.307	1.693	0.850	0.382	1.618
14	3.407	0.763	0.981	0.235	0.328	1.672	0.817	0.406	1.594
15	3.472	0.756	0.982	0.223	0.347	1.653	0.789	0.428	1.572
16	3.532	0.750	0.984	0.212	0.363	1.637	0.763	0.448	1.552
17	3.588	0.744	0.985	0.203	0.378	1.662	0.739	0.466	1.534
18	3.640	0.739	0.985	0.194	0.391	1.607	0.718	0.482	1.518
19	3.689	0.734	0.986	0.187	0.403	1.597	0.698	0.497	1.503
20	3.735	0.729	0.987	0.180	0.415	1.585	0.680	0.510	1.490
21	3.778	0.727	0.988	0.173	0.425	1.575	0.663	0.523	1.477
22	3.819	0.720	0.988	0.167	0.434	1.566	0.647	0.534	1.466
23	3.858	0.716	0.989	0.162	0.443	1.557	0.633	0.545	1.455
24	3.895	0.712	0.989	0.157	0.451	1.548	0.619	0.555	1.445
25	3.931	0.708	0.990	0.153	0.459	1.541	0.606	0.565	1.435

Table O

For n > 25:

$A = \frac{3}{\sqrt{n}} A_{3} = \frac{3}{c_{4} \sqrt{n}} c_{4} ≅ \frac{4 (n - 1)}{4 n - 3}$ $A = \frac{3}{\sqrt{n}} A_{3} = \frac{3}{c_{4} \sqrt{n}} c_{4} ≅ \frac{4 (n - 1)}{4 n - 3}$

si11_e

$B_{3} = 1 - \frac{3}{c_{4} \sqrt{2 (n - 1)}} B_{4} = 1 + \frac{3}{c_{4} \sqrt{2 (n - 1)}}$ $B_{3} = 1 - \frac{3}{c_{4} \sqrt{2 (n - 1)}} B_{4} = 1 + \frac{3}{c_{4} \sqrt{2 (n - 1)}}$

si12_e

$B_{5} = c_{4} - \frac{3}{\sqrt{2 (n - 1)}} B_{6} = c_{4} + \frac{3}{\sqrt{2 (n - 1)}}$ $B_{5} = c_{4} - \frac{3}{\sqrt{2 (n - 1)}} B_{6} = c_{4} + \frac{3}{\sqrt{2 (n - 1)}}$

si13_e

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Data Science for Business and Decision Making