Reference

Statistical tables

Critical values and control chart constants, generated rather than copied, with a note on where each one is used in the course.

Every number on this page was computed by verification/tables.py in the course repository, with scipy for the distribution functions and numerical integration for the control chart constants, and then compared entry by entry against published tables: the NIST/SEMATECH handbook's tables 1.3.6.7.1 to 1.3.6.7.4 for the normal, t, chi-square and F values, and the NIST handbook, the AIAG & VDA SPC manual and Montgomery's appendix for the chart constants. Nothing here was typed out of a book. One deliberate difference is noted under Table 5.

Table 1. Standard normal cumulative distribution
Table 2. t distribution critical values
Table 3. Chi-square critical values
Table 4. F distribution critical values, α = 0.05
Table 5. Control chart constants

Standard normal distribution

Φ(z) is the probability that a standard normal variable is less than z, which is the area under the curve to the left. For the tail beyond z, use 1 − Φ(z). The distribution is symmetric, so Φ(−z) = 1 − Φ(z). Capability work uses this table in both directions: forward, to turn a Z into a predicted fraction outside a limit, and backward, to turn an observed defect rate into a sigma level (Module 6).

Table 1. Standard normal cumulative distribution Φ(z), the area to the left of z. Read the units and first decimal of z down the left column and the second decimal across the top: Φ(1.96) is row 1.9, column 0.06.
z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.50000	0.50399	0.50798	0.51197	0.51595	0.51994	0.52392	0.52790	0.53188	0.53586
0.1	0.53983	0.54380	0.54776	0.55172	0.55567	0.55962	0.56356	0.56749	0.57142	0.57535
0.2	0.57926	0.58317	0.58706	0.59095	0.59483	0.59871	0.60257	0.60642	0.61026	0.61409
0.3	0.61791	0.62172	0.62552	0.62930	0.63307	0.63683	0.64058	0.64431	0.64803	0.65173
0.4	0.65542	0.65910	0.66276	0.66640	0.67003	0.67364	0.67724	0.68082	0.68439	0.68793
0.5	0.69146	0.69497	0.69847	0.70194	0.70540	0.70884	0.71226	0.71566	0.71904	0.72240
0.6	0.72575	0.72907	0.73237	0.73565	0.73891	0.74215	0.74537	0.74857	0.75175	0.75490
0.7	0.75804	0.76115	0.76424	0.76730	0.77035	0.77337	0.77637	0.77935	0.78230	0.78524
0.8	0.78814	0.79103	0.79389	0.79673	0.79955	0.80234	0.80511	0.80785	0.81057	0.81327
0.9	0.81594	0.81859	0.82121	0.82381	0.82639	0.82894	0.83147	0.83398	0.83646	0.83891
1.0	0.84134	0.84375	0.84614	0.84849	0.85083	0.85314	0.85543	0.85769	0.85993	0.86214
1.1	0.86433	0.86650	0.86864	0.87076	0.87286	0.87493	0.87698	0.87900	0.88100	0.88298
1.2	0.88493	0.88686	0.88877	0.89065	0.89251	0.89435	0.89617	0.89796	0.89973	0.90147
1.3	0.90320	0.90490	0.90658	0.90824	0.90988	0.91149	0.91309	0.91466	0.91621	0.91774
1.4	0.91924	0.92073	0.92220	0.92364	0.92507	0.92647	0.92785	0.92922	0.93056	0.93189
1.5	0.93319	0.93448	0.93574	0.93699	0.93822	0.93943	0.94062	0.94179	0.94295	0.94408
1.6	0.94520	0.94630	0.94738	0.94845	0.94950	0.95053	0.95154	0.95254	0.95352	0.95449
1.7	0.95543	0.95637	0.95728	0.95818	0.95907	0.95994	0.96080	0.96164	0.96246	0.96327
1.8	0.96407	0.96485	0.96562	0.96638	0.96712	0.96784	0.96856	0.96926	0.96995	0.97062
1.9	0.97128	0.97193	0.97257	0.97320	0.97381	0.97441	0.97500	0.97558	0.97615	0.97670
2.0	0.97725	0.97778	0.97831	0.97882	0.97932	0.97982	0.98030	0.98077	0.98124	0.98169
2.1	0.98214	0.98257	0.98300	0.98341	0.98382	0.98422	0.98461	0.98500	0.98537	0.98574
2.2	0.98610	0.98645	0.98679	0.98713	0.98745	0.98778	0.98809	0.98840	0.98870	0.98899
2.3	0.98928	0.98956	0.98983	0.99010	0.99036	0.99061	0.99086	0.99111	0.99134	0.99158
2.4	0.99180	0.99202	0.99224	0.99245	0.99266	0.99286	0.99305	0.99324	0.99343	0.99361
2.5	0.99379	0.99396	0.99413	0.99430	0.99446	0.99461	0.99477	0.99492	0.99506	0.99520
2.6	0.99534	0.99547	0.99560	0.99573	0.99585	0.99598	0.99609	0.99621	0.99632	0.99643
2.7	0.99653	0.99664	0.99674	0.99683	0.99693	0.99702	0.99711	0.99720	0.99728	0.99736
2.8	0.99744	0.99752	0.99760	0.99767	0.99774	0.99781	0.99788	0.99795	0.99801	0.99807
2.9	0.99813	0.99819	0.99825	0.99831	0.99836	0.99841	0.99846	0.99851	0.99856	0.99861
3.0	0.99865	0.99869	0.99874	0.99878	0.99882	0.99886	0.99889	0.99893	0.99896	0.99900
3.1	0.99903	0.99906	0.99910	0.99913	0.99916	0.99918	0.99921	0.99924	0.99926	0.99929
3.2	0.99931	0.99934	0.99936	0.99938	0.99940	0.99942	0.99944	0.99946	0.99948	0.99950
3.3	0.99952	0.99953	0.99955	0.99957	0.99958	0.99960	0.99961	0.99962	0.99964	0.99965
3.4	0.99966	0.99968	0.99969	0.99970	0.99971	0.99972	0.99973	0.99974	0.99975	0.99976
3.5	0.99977	0.99978	0.99978	0.99979	0.99980	0.99981	0.99981	0.99982	0.99983	0.99983
3.6	0.99984	0.99985	0.99985	0.99986	0.99986	0.99987	0.99987	0.99988	0.99988	0.99989
3.7	0.99989	0.99990	0.99990	0.99990	0.99991	0.99991	0.99992	0.99992	0.99992	0.99992
3.8	0.99993	0.99993	0.99993	0.99994	0.99994	0.99994	0.99994	0.99995	0.99995	0.99995
3.9	0.99995	0.99995	0.99996	0.99996	0.99996	0.99996	0.99996	0.99996	0.99997	0.99997

The course's calculators do not read this table; they call the same function directly in assets/js/stats.js, which agrees with scipy to better than 1 × 10⁻⁹.

t distribution

Used whenever a mean is compared and sigma is estimated from the same sample: one and two sample t-tests, paired t-tests, confidence intervals on a mean, and the slope of a regression (Modules 11 and 12). The values are upper one-sided critical values, so a two-sided test at the 5 % level uses the 0.025 column.

Table 2. Upper one-sided critical values of the t distribution, t_α(df). For a two-sided test at 5 %, use the α = 0.025 column.
df	0.1	0.05	0.025	0.01	0.005	0.001
1	3.078	6.314	12.706	31.821	63.657	318.309
2	1.886	2.92	4.303	6.965	9.925	22.327
3	1.638	2.353	3.182	4.541	5.841	10.215
4	1.533	2.132	2.776	3.747	4.604	7.173
5	1.476	2.015	2.571	3.365	4.032	5.893
6	1.44	1.943	2.447	3.143	3.707	5.208
7	1.415	1.895	2.365	2.998	3.499	4.785
8	1.397	1.86	2.306	2.896	3.355	4.501
9	1.383	1.833	2.262	2.821	3.25	4.297
10	1.372	1.812	2.228	2.764	3.169	4.144
11	1.363	1.796	2.201	2.718	3.106	4.025
12	1.356	1.782	2.179	2.681	3.055	3.93
13	1.35	1.771	2.16	2.65	3.012	3.852
14	1.345	1.761	2.145	2.624	2.977	3.787
15	1.341	1.753	2.131	2.602	2.947	3.733
16	1.337	1.746	2.12	2.583	2.921	3.686
17	1.333	1.74	2.11	2.567	2.898	3.646
18	1.33	1.734	2.101	2.552	2.878	3.61
19	1.328	1.729	2.093	2.539	2.861	3.579
20	1.325	1.725	2.086	2.528	2.845	3.552
21	1.323	1.721	2.08	2.518	2.831	3.527
22	1.321	1.717	2.074	2.508	2.819	3.505
23	1.319	1.714	2.069	2.5	2.807	3.485
24	1.318	1.711	2.064	2.492	2.797	3.467
25	1.316	1.708	2.06	2.485	2.787	3.45
26	1.315	1.706	2.056	2.479	2.779	3.435
27	1.314	1.703	2.052	2.473	2.771	3.421
28	1.313	1.701	2.048	2.467	2.763	3.408
29	1.311	1.699	2.045	2.462	2.756	3.396
30	1.31	1.697	2.042	2.457	2.75	3.385
40	1.303	1.684	2.021	2.423	2.704	3.307
60	1.296	1.671	2	2.39	2.66	3.232
120	1.289	1.658	1.98	2.358	2.617	3.16
∞	1.282	1.645	1.96	2.326	2.576	3.09

Chi-square distribution

Used for tests on counts in a contingency table, for goodness of fit, and for confidence intervals on a variance or a standard deviation (Module 11). α here is the area to the right of the tabulated value.

Table 3. Critical values of the chi-square distribution, χ²_α(df), where α is the area to the *right*. The left-hand columns (0.995 to 0.900) are the ones used for the lower end of a variance confidence interval.
df	0.995	0.99	0.975	0.95	0.9	0.1	0.05	0.025	0.01	0.005
1	0	0	0.001	0.004	0.016	2.706	3.841	5.024	6.635	7.879
2	0.01	0.02	0.051	0.103	0.211	4.605	5.991	7.378	9.21	10.597
3	0.072	0.115	0.216	0.352	0.584	6.251	7.815	9.348	11.345	12.838
4	0.207	0.297	0.484	0.711	1.064	7.779	9.488	11.143	13.277	14.86
5	0.412	0.554	0.831	1.145	1.61	9.236	11.07	12.833	15.086	16.75
6	0.676	0.872	1.237	1.635	2.204	10.645	12.592	14.449	16.812	18.548
7	0.989	1.239	1.69	2.167	2.833	12.017	14.067	16.013	18.475	20.278
8	1.344	1.646	2.18	2.733	3.49	13.362	15.507	17.535	20.09	21.955
9	1.735	2.088	2.7	3.325	4.168	14.684	16.919	19.023	21.666	23.589
10	2.156	2.558	3.247	3.94	4.865	15.987	18.307	20.483	23.209	25.188
11	2.603	3.053	3.816	4.575	5.578	17.275	19.675	21.92	24.725	26.757
12	3.074	3.571	4.404	5.226	6.304	18.549	21.026	23.337	26.217	28.3
13	3.565	4.107	5.009	5.892	7.042	19.812	22.362	24.736	27.688	29.819
14	4.075	4.66	5.629	6.571	7.79	21.064	23.685	26.119	29.141	31.319
15	4.601	5.229	6.262	7.261	8.547	22.307	24.996	27.488	30.578	32.801
16	5.142	5.812	6.908	7.962	9.312	23.542	26.296	28.845	32	34.267
17	5.697	6.408	7.564	8.672	10.085	24.769	27.587	30.191	33.409	35.718
18	6.265	7.015	8.231	9.39	10.865	25.989	28.869	31.526	34.805	37.156
19	6.844	7.633	8.907	10.117	11.651	27.204	30.144	32.852	36.191	38.582
20	7.434	8.26	9.591	10.851	12.443	28.412	31.41	34.17	37.566	39.997
21	8.034	8.897	10.283	11.591	13.24	29.615	32.671	35.479	38.932	41.401
22	8.643	9.542	10.982	12.338	14.041	30.813	33.924	36.781	40.289	42.796
23	9.26	10.196	11.689	13.091	14.848	32.007	35.172	38.076	41.638	44.181
24	9.886	10.856	12.401	13.848	15.659	33.196	36.415	39.364	42.98	45.559
25	10.52	11.524	13.12	14.611	16.473	34.382	37.652	40.646	44.314	46.928
26	11.16	12.198	13.844	15.379	17.292	35.563	38.885	41.923	45.642	48.29
27	11.808	12.879	14.573	16.151	18.114	36.741	40.113	43.195	46.963	49.645
28	12.461	13.565	15.308	16.928	18.939	37.916	41.337	44.461	48.278	50.993
29	13.121	14.256	16.047	17.708	19.768	39.087	42.557	45.722	49.588	52.336
30	13.787	14.953	16.791	18.493	20.599	40.256	43.773	46.979	50.892	53.672
40	20.707	22.164	24.433	26.509	29.051	51.805	55.758	59.342	63.691	66.766
50	27.991	29.707	32.357	34.764	37.689	63.167	67.505	71.42	76.154	79.49
60	35.534	37.485	40.482	43.188	46.459	74.397	79.082	83.298	88.379	91.952
80	51.172	53.54	57.153	60.391	64.278	96.578	101.879	106.629	112.329	116.321
100	67.328	70.065	74.222	77.929	82.358	118.498	124.342	129.561	135.807	140.169

F distribution

Used to compare two variances, and as the overall test in analysis of variance, in gauge R&R by the ANOVA method, and in regression (Modules 4, 11, 12 and 13). Only the 5 % level is tabulated here; the calculators compute any level directly.

Table 4. Upper 5 % critical values of the F distribution, F_0.05(ν₁, ν₂). Numerator degrees of freedom ν₁ across the top, denominator ν₂ down the side. upper critical values at alpha 0.05 (3 decimals) and 0.01 (2 decimals)
ν₂	1	2	3	4	5	6	7	8	9	10	12	15	20	24	30	40	60	120
1	161.448	199.5	215.707	224.583	230.162	233.986	236.768	238.883	240.543	241.882	243.906	245.95	248.013	249.052	250.095	251.143	252.196	253.253
2	18.513	19	19.164	19.247	19.296	19.33	19.353	19.371	19.385	19.396	19.413	19.429	19.446	19.454	19.462	19.471	19.479	19.487
3	10.128	9.552	9.277	9.117	9.013	8.941	8.887	8.845	8.812	8.786	8.745	8.703	8.66	8.639	8.617	8.594	8.572	8.549
4	7.709	6.944	6.591	6.388	6.256	6.163	6.094	6.041	5.999	5.964	5.912	5.858	5.803	5.774	5.746	5.717	5.688	5.658
5	6.608	5.786	5.409	5.192	5.05	4.95	4.876	4.818	4.772	4.735	4.678	4.619	4.558	4.527	4.496	4.464	4.431	4.398
6	5.987	5.143	4.757	4.534	4.387	4.284	4.207	4.147	4.099	4.06	4	3.938	3.874	3.841	3.808	3.774	3.74	3.705
7	5.591	4.737	4.347	4.12	3.972	3.866	3.787	3.726	3.677	3.637	3.575	3.511	3.445	3.41	3.376	3.34	3.304	3.267
8	5.318	4.459	4.066	3.838	3.687	3.581	3.5	3.438	3.388	3.347	3.284	3.218	3.15	3.115	3.079	3.043	3.005	2.967
9	5.117	4.256	3.863	3.633	3.482	3.374	3.293	3.23	3.179	3.137	3.073	3.006	2.936	2.9	2.864	2.826	2.787	2.748
10	4.965	4.103	3.708	3.478	3.326	3.217	3.135	3.072	3.02	2.978	2.913	2.845	2.774	2.737	2.7	2.661	2.621	2.58
11	4.844	3.982	3.587	3.357	3.204	3.095	3.012	2.948	2.896	2.854	2.788	2.719	2.646	2.609	2.57	2.531	2.49	2.448
12	4.747	3.885	3.49	3.259	3.106	2.996	2.913	2.849	2.796	2.753	2.687	2.617	2.544	2.505	2.466	2.426	2.384	2.341
13	4.667	3.806	3.411	3.179	3.025	2.915	2.832	2.767	2.714	2.671	2.604	2.533	2.459	2.42	2.38	2.339	2.297	2.252
14	4.6	3.739	3.344	3.112	2.958	2.848	2.764	2.699	2.646	2.602	2.534	2.463	2.388	2.349	2.308	2.266	2.223	2.178
15	4.543	3.682	3.287	3.056	2.901	2.79	2.707	2.641	2.588	2.544	2.475	2.403	2.328	2.288	2.247	2.204	2.16	2.114
16	4.494	3.634	3.239	3.007	2.852	2.741	2.657	2.591	2.538	2.494	2.425	2.352	2.276	2.235	2.194	2.151	2.106	2.059
17	4.451	3.592	3.197	2.965	2.81	2.699	2.614	2.548	2.494	2.45	2.381	2.308	2.23	2.19	2.148	2.104	2.058	2.011
18	4.414	3.555	3.16	2.928	2.773	2.661	2.577	2.51	2.456	2.412	2.342	2.269	2.191	2.15	2.107	2.063	2.017	1.968
19	4.381	3.522	3.127	2.895	2.74	2.628	2.544	2.477	2.423	2.378	2.308	2.234	2.155	2.114	2.071	2.026	1.98	1.93
20	4.351	3.493	3.098	2.866	2.711	2.599	2.514	2.447	2.393	2.348	2.278	2.203	2.124	2.082	2.039	1.994	1.946	1.896
21	4.325	3.467	3.072	2.84	2.685	2.573	2.488	2.42	2.366	2.321	2.25	2.176	2.096	2.054	2.01	1.965	1.916	1.866
22	4.301	3.443	3.049	2.817	2.661	2.549	2.464	2.397	2.342	2.297	2.226	2.151	2.071	2.028	1.984	1.938	1.889	1.838
23	4.279	3.422	3.028	2.796	2.64	2.528	2.442	2.375	2.32	2.275	2.204	2.128	2.048	2.005	1.961	1.914	1.865	1.813
24	4.26	3.403	3.009	2.776	2.621	2.508	2.423	2.355	2.3	2.255	2.183	2.108	2.027	1.984	1.939	1.892	1.842	1.79
25	4.242	3.385	2.991	2.759	2.603	2.49	2.405	2.337	2.282	2.236	2.165	2.089	2.007	1.964	1.919	1.872	1.822	1.768
26	4.225	3.369	2.975	2.743	2.587	2.474	2.388	2.321	2.265	2.22	2.148	2.072	1.99	1.946	1.901	1.853	1.803	1.749
27	4.21	3.354	2.96	2.728	2.572	2.459	2.373	2.305	2.25	2.204	2.132	2.056	1.974	1.93	1.884	1.836	1.785	1.731
28	4.196	3.34	2.947	2.714	2.558	2.445	2.359	2.291	2.236	2.19	2.118	2.041	1.959	1.915	1.869	1.82	1.769	1.714
29	4.183	3.328	2.934	2.701	2.545	2.432	2.346	2.278	2.223	2.177	2.104	2.027	1.945	1.901	1.854	1.806	1.754	1.698
30	4.171	3.316	2.922	2.69	2.534	2.421	2.334	2.266	2.211	2.165	2.092	2.015	1.932	1.887	1.841	1.792	1.74	1.683
40	4.085	3.232	2.839	2.606	2.449	2.336	2.249	2.18	2.124	2.077	2.003	1.924	1.839	1.793	1.744	1.693	1.637	1.577
60	4.001	3.15	2.758	2.525	2.368	2.254	2.167	2.097	2.04	1.993	1.917	1.836	1.748	1.7	1.649	1.594	1.534	1.467
120	3.92	3.072	2.68	2.447	2.29	2.175	2.087	2.016	1.959	1.91	1.834	1.75	1.659	1.608	1.554	1.495	1.429	1.352
∞	3.841	2.996	2.605	2.372	2.214	2.099	2.01	1.938	1.88	1.831	1.752	1.666	1.571	1.517	1.459	1.394	1.318	1.221

Control chart constants

These constants convert a measure of spread taken from subgroups into an estimate of the process standard deviation, and then into plotting limits. The ones this course uses most: σ̂_within = R̄/d₂ or S̄/c₄; X̄ chart limits X̄̄ ± A₂R̄ or X̄̄ ± A₃S̄; R chart limits D₃R̄ and D₄R̄; S chart limits B₃S̄ and B₄S̄. Every one of them assumes the subgroups were formed rationally (Module 5).

Table 5. Control chart constants for subgroup size n. d₂ and d₃ are the mean and standard deviation of the relative range; c₄ is the bias correction for the sample standard deviation; the rest are the derived plotting constants used in Modules 16 and 17.
n	d₂	d₃	c₄	A₂	A₃	D₃	D₄	B₃	B₄	E₂
2	1.128	0.8525	0.7979	1.88	2.659	0	3.267	0	3.267	2.66
3	1.693	0.8884	0.8862	1.023	1.954	0	2.575	0	2.568	1.772
4	2.059	0.8798	0.9213	0.729	1.628	0	2.282	0	2.266	1.457
5	2.326	0.8641	0.94	0.577	1.427	0	2.114	0	2.089	1.29
6	2.534	0.848	0.9515	0.483	1.287	0	2.004	0.03	1.97	1.184
7	2.704	0.8332	0.9594	0.419	1.182	0.076	1.924	0.118	1.882	1.109
8	2.847	0.8198	0.965	0.373	1.099	0.136	1.864	0.185	1.815	1.054
9	2.97	0.8078	0.9693	0.337	1.032	0.184	1.816	0.239	1.761	1.01
10	3.078	0.7971	0.9727	0.308	0.975	0.223	1.777	0.284	1.716	0.975
11	3.173	0.7873	0.9754	0.285	0.927	0.256	1.744	0.321	1.679	0.945
12	3.258	0.7785	0.9776	0.266	0.886	0.283	1.717	0.354	1.646	0.921
13	3.336	0.7704	0.9794	0.249	0.85	0.307	1.693	0.382	1.618	0.899
14	3.407	0.763	0.981	0.235	0.817	0.328	1.672	0.406	1.594	0.881
15	3.472	0.7562	0.9823	0.223	0.789	0.347	1.653	0.428	1.572	0.864
16	3.532	0.7499	0.9835	0.212	0.763	0.363	1.637	0.448	1.552	0.849
17	3.588	0.7441	0.9845	0.203	0.739	0.378	1.622	0.466	1.534	0.836
18	3.64	0.7386	0.9854	0.194	0.718	0.391	1.609	0.482	1.518	0.824
19	3.689	0.7335	0.9862	0.187	0.698	0.404	1.596	0.497	1.503	0.813
20	3.735	0.7287	0.9869	0.18	0.68	0.415	1.585	0.51	1.49	0.803
21	3.778	0.7242	0.9876	0.173	0.663	0.425	1.575	0.523	1.477	0.794
22	3.819	0.7199	0.9882	0.167	0.647	0.435	1.565	0.534	1.466	0.786
23	3.858	0.7159	0.9887	0.162	0.633	0.443	1.557	0.545	1.455	0.778
24	3.895	0.7121	0.9892	0.157	0.619	0.452	1.548	0.555	1.445	0.77
25	3.931	0.7084	0.9896	0.153	0.606	0.459	1.541	0.565	1.435	0.763

One deliberate difference from the printed tables: D₄ for n = 5 is widely printed as 2.115, and the exact value rounds to 2.114. This course uses 2.114 and says so wherever the constant appears, because the verification scripts compute it rather than copy it. The difference changes an R chart's upper limit by about one part in two thousand and no decision anyone makes.

A note on using tables at all

Nobody needs these tables to compute anything any more: a spreadsheet, this course's calculators, or three lines of Python will do better. They are here for two reasons. The first is reading: supplier reports, standards and textbooks quote critical values, and it helps to be able to check one. The second is understanding: looking up Φ(3) = 0.99865 and seeing where the 0.135 % and the 1350 PPM of Module 7 come from is worth more than trusting a function.