What is d₂ used for?

Estimating the process standard deviation from ranges: σ̂ = R̄/d₂ for subgrouped data, or σ̂ = MR̄/d₂(2) for individuals. It is the mean range of n standard normal observations, so dividing by it converts an average range back into a standard deviation.

Why is D₃ zero for small subgroups?

D₃ = 1 − 3d₃/d₂, and for n below 7 that expression is negative. A range cannot be negative, so the factor is clamped at zero and the R chart has no lower control limit. Tables print either 0.000 or a dash; they mean the same thing.

Can I use these for subgroup sizes above 25?

Yes — that is most of why this page exists. Nothing changes in the arithmetic above 25; what changes is that no published table covers those rows, so they are marked on the figure as not cross-checkable against a printed source.

Is A₃ the same as A₂?

No. A₂ = 3/(d₂√n) is for limits from R̄; A₃ = 3/(c₄√n) is for limits from s̄. They are close for small subgroups and diverge as n grows, because the range loses efficiency against the standard deviation.

Free · No accountReproduces a published worked example

Control Chart Constants Table — A2, A3, d2, d3, c4, B3, B4, D3, D4 to n = 100

A₂, A₃, d₂, d₃, c₄, B₃, B₄, D₃, D₄ and the rest, for subgroup sizes from 2 to 100 — computed from the normal distribution rather than copied from a table, with the derivation and the source shown for each one.

Why compute them instead of printing a table

Three reasons, and the first is the boring one: the printed tables stop at n = 25. If you are charting subgroups of 30 there is nowhere to look the factor up, and the usual workaround — using the n = 25 row — is simply wrong.

The second is that the tables do not agree with each other. Cross-checking two independent public tables cell by cell turns up a disagreement at D₄ for n = 3: the four-decimal source prints 2.5746 and the three-decimal source prints 2.574, where rounding 2.5746 gives 2.575. One of them derived the factor from its own rounded inputs. That is not a scandal, it is what happens to a table that has been retyped for eighty years, and it is exactly why the /validation page names the exception rather than quietly picking a side.

The third is that a constant with no derivation attached is a number you have to trust. Every symbol on this page carries the expression it comes from, so you can check any cell with a gamma function and a numerical integrator of your own.

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Every constant, n = 2 to n = 100

The figure above is the version that exports. This is the same computation as text — selectable, searchable, and readable on a phone. Every value computed from the standard normal distribution — c₄ in closed form, d₂, d₃ and d₄ by numerical integration of the range distribution. Nothing on this table is copied from a printed table.

All 99 rows and all 16 factors, comma separated.

Shewhart control chart constants for subgroup sizes n = 2 to 100. A dash marks a factor clamped at zero — there is no lower limit, not a missing value. An asterisk marks a subgroup size beyond the largest published table cited below.
n	A	A₂	A₃	c₄	d₂	d₃	d₄	B₃	B₄	B₅	B₆	D₁	D₂	D₃	D₄	E₂
2	2.1213	1.8800	2.6587	0.7979	1.1284	0.8525	0.9539	—	3.2665	—	2.6063	—	3.6859	—	3.2665	2.6587
3	1.7321	1.0233	1.9544	0.8862	1.6926	0.8884	1.5878	—	2.5682	—	2.2760	—	4.3577	—	2.5746	1.7725
4	1.5000	0.7286	1.6281	0.9213	2.0588	0.8798	1.9783	—	2.2660	—	2.0877	—	4.6982	—	2.2821	1.4572
5	1.3416	0.5768	1.4273	0.9400	2.3259	0.8641	2.2569	—	2.0890	—	1.9636	—	4.9182	—	2.1145	1.2898
6	1.2247	0.4832	1.2871	0.9515	2.5344	0.8480	2.4717	0.0304	1.9696	0.0289	1.8742	—	5.0785	—	2.0038	1.1837
7	1.1339	0.4193	1.1819	0.9594	2.7044	0.8332	2.6455	0.1177	1.8823	0.1129	1.8058	0.2047	5.2040	0.0757	1.9243	1.1093
8	1.0607	0.3725	1.0991	0.9650	2.8472	0.8198	2.7908	0.1851	1.8149	0.1786	1.7514	0.3877	5.3067	0.1362	1.8638	1.0537
9	1.0000	0.3367	1.0317	0.9693	2.9700	0.8078	2.9154	0.2391	1.7609	0.2318	1.7068	0.5465	5.3935	0.1840	1.8160	1.0101
10	0.9487	0.3083	0.9754	0.9727	3.0775	0.7971	3.0242	0.2837	1.7163	0.2759	1.6694	0.6864	5.4687	0.2230	1.7770	0.9748
11	0.9045	0.2851	0.9274	0.9754	3.1729	0.7873	3.1205	0.3213	1.6787	0.3134	1.6373	0.8109	5.5348	0.2556	1.7444	0.9455
12	0.8660	0.2658	0.8859	0.9776	3.2585	0.7785	3.2069	0.3535	1.6465	0.3456	1.6095	0.9230	5.5939	0.2833	1.7167	0.9207
13	0.8321	0.2494	0.8495	0.9794	3.3360	0.7704	3.2850	0.3816	1.6184	0.3737	1.5851	1.0247	5.6472	0.3072	1.6928	0.8993
14	0.8018	0.2354	0.8173	0.9810	3.4068	0.7630	3.3562	0.4062	1.5938	0.3985	1.5634	1.1177	5.6958	0.3281	1.6719	0.8806
15	0.7746	0.2231	0.7885	0.9823	3.4718	0.7562	3.4217	0.4282	1.5718	0.4206	1.5440	1.2032	5.7405	0.3466	1.6534	0.8641
16	0.7500	0.2123	0.7626	0.9835	3.5320	0.7499	3.4821	0.4479	1.5521	0.4405	1.5265	1.2823	5.7817	0.3630	1.6370	0.8494
17	0.7276	0.2028	0.7391	0.9845	3.5879	0.7441	3.5383	0.4657	1.5343	0.4585	1.5106	1.3557	5.8200	0.3779	1.6221	0.8361
18	0.7071	0.1943	0.7176	0.9854	3.6401	0.7386	3.5907	0.4818	1.5182	0.4748	1.4960	1.4243	5.8558	0.3913	1.6087	0.8242
19	0.6882	0.1866	0.6979	0.9862	3.6890	0.7335	3.6398	0.4966	1.5034	0.4898	1.4826	1.4885	5.8894	0.4035	1.5965	0.8132
20	0.6708	0.1796	0.6797	0.9869	3.7350	0.7287	3.6859	0.5102	1.4898	0.5036	1.4703	1.5489	5.9210	0.4147	1.5853	0.8032
21	0.6547	0.1733	0.6629	0.9876	3.7783	0.7242	3.7294	0.5228	1.4772	0.5163	1.4589	1.6058	5.9509	0.4250	1.5750	0.7940
22	0.6396	0.1675	0.6473	0.9882	3.8194	0.7199	3.7706	0.5344	1.4656	0.5281	1.4483	1.6596	5.9791	0.4345	1.5655	0.7855
23	0.6255	0.1621	0.6327	0.9887	3.8583	0.7159	3.8097	0.5452	1.4548	0.5391	1.4383	1.7107	6.0060	0.4434	1.5566	0.7775
24	0.6124	0.1572	0.6191	0.9892	3.8953	0.7121	3.8468	0.5553	1.4447	0.5493	1.4291	1.7591	6.0316	0.4516	1.5484	0.7701
25	0.6000	0.1526	0.6063	0.9896	3.9306	0.7084	3.8821	0.5648	1.4352	0.5589	1.4203	1.8053	6.0560	0.4593	1.5407	0.7632
26*	0.5883	0.1484	0.5943	0.9901	3.9643	0.7050	3.9159	0.5737	1.4263	0.5680	1.4121	1.8494	6.0793	0.4665	1.5335	0.7568
27*	0.5774	0.1445	0.5829	0.9904	3.9965	0.7017	3.9482	0.5820	1.4180	0.5765	1.4044	1.8914	6.1016	0.4733	1.5267	0.7506
28*	0.5669	0.1408	0.5722	0.9908	4.0274	0.6986	3.9791	0.5899	1.4101	0.5845	1.3971	1.9318	6.1231	0.4797	1.5203	0.7449
29*	0.5571	0.1373	0.5621	0.9911	4.0570	0.6955	4.0088	0.5974	1.4026	0.5920	1.3902	1.9704	6.1437	0.4857	1.5143	0.7395
30*	0.5477	0.1341	0.5525	0.9914	4.0855	0.6927	4.0373	0.6044	1.3956	0.5992	1.3836	2.0075	6.1635	0.4914	1.5086	0.7343
31*	0.5388	0.1310	0.5433	0.9917	4.1129	0.6899	4.0648	0.6111	1.3889	0.6060	1.3774	2.0432	6.1826	0.4968	1.5032	0.7294
32*	0.5303	0.1281	0.5346	0.9920	4.1393	0.6872	4.0912	0.6175	1.3825	0.6125	1.3714	2.0776	6.2011	0.5019	1.4981	0.7248
33*	0.5222	0.1254	0.5263	0.9922	4.1648	0.6847	4.1168	0.6236	1.3764	0.6187	1.3657	2.1107	6.2189	0.5068	1.4932	0.7203
34*	0.5145	0.1228	0.5184	0.9925	4.1894	0.6822	4.1414	0.6294	1.3706	0.6246	1.3603	2.1427	6.2361	0.5115	1.4885	0.7161
35*	0.5071	0.1204	0.5108	0.9927	4.2132	0.6799	4.1652	0.6349	1.3651	0.6302	1.3551	2.1736	6.2528	0.5159	1.4841	0.7120
36*	0.5000	0.1180	0.5036	0.9929	4.2362	0.6776	4.1883	0.6402	1.3598	0.6356	1.3502	2.2035	6.2690	0.5202	1.4798	0.7082
37*	0.4932	0.1158	0.4966	0.9931	4.2586	0.6754	4.2106	0.6452	1.3548	0.6408	1.3454	2.2324	6.2847	0.5242	1.4758	0.7045
38*	0.4867	0.1137	0.4900	0.9933	4.2802	0.6732	4.2323	0.6501	1.3499	0.6457	1.3408	2.2604	6.2999	0.5281	1.4719	0.7009
39*	0.4804	0.1117	0.4836	0.9934	4.3012	0.6712	4.2533	0.6548	1.3452	0.6505	1.3364	2.2876	6.3147	0.5319	1.4681	0.6975
40*	0.4743	0.1098	0.4774	0.9936	4.3216	0.6692	4.2737	0.6592	1.3408	0.6550	1.3322	2.3140	6.3291	0.5355	1.4645	0.6942
41*	0.4685	0.1079	0.4715	0.9938	4.3414	0.6673	4.2935	0.6636	1.3364	0.6594	1.3281	2.3396	6.3431	0.5389	1.4611	0.6910
42*	0.4629	0.1062	0.4657	0.9939	4.3606	0.6654	4.3128	0.6677	1.3323	0.6637	1.3242	2.3645	6.3568	0.5422	1.4578	0.6880
43*	0.4575	0.1045	0.4602	0.9941	4.3794	0.6636	4.3316	0.6717	1.3283	0.6677	1.3204	2.3887	6.3700	0.5454	1.4546	0.6850
44*	0.4523	0.1028	0.4549	0.9942	4.3976	0.6618	4.3499	0.6756	1.3244	0.6717	1.3167	2.4123	6.3830	0.5485	1.4515	0.6822
45*	0.4472	0.1013	0.4498	0.9943	4.4154	0.6601	4.3677	0.6793	1.3207	0.6755	1.3132	2.4352	6.3956	0.5515	1.4485	0.6794
46*	0.4423	0.0998	0.4448	0.9945	4.4328	0.6584	4.3851	0.6829	1.3171	0.6791	1.3098	2.4576	6.4080	0.5544	1.4456	0.6768
47*	0.4376	0.0983	0.4400	0.9946	4.4497	0.6568	4.4020	0.6864	1.3136	0.6827	1.3065	2.4794	6.4200	0.5572	1.4428	0.6742
48*	0.4330	0.0970	0.4353	0.9947	4.4662	0.6552	4.4185	0.6898	1.3102	0.6861	1.3033	2.5007	6.4318	0.5599	1.4401	0.6717
49*	0.4286	0.0956	0.4308	0.9948	4.4824	0.6536	4.4347	0.6930	1.3070	0.6894	1.3002	2.5214	6.4433	0.5625	1.4375	0.6693
50*	0.4243	0.0943	0.4264	0.9949	4.4981	0.6521	4.4505	0.6962	1.3038	0.6926	1.2972	2.5417	6.4546	0.5651	1.4349	0.6669
51*	0.4201	0.0931	0.4222	0.9950	4.5136	0.6507	4.4659	0.6993	1.3007	0.6958	1.2943	2.5615	6.4656	0.5675	1.4325	0.6647
52*	0.4160	0.0919	0.4181	0.9951	4.5286	0.6492	4.4810	0.7022	1.2978	0.6988	1.2914	2.5809	6.4764	0.5699	1.4301	0.6625
53*	0.4121	0.0907	0.4141	0.9952	4.5434	0.6479	4.4958	0.7051	1.2949	0.7017	1.2887	2.5998	6.4870	0.5722	1.4278	0.6603
54*	0.4082	0.0896	0.4102	0.9953	4.5578	0.6465	4.5102	0.7079	1.2921	0.7046	1.2860	2.6183	6.4973	0.5745	1.4255	0.6582
55*	0.4045	0.0885	0.4064	0.9954	4.5720	0.6452	4.5244	0.7107	1.2893	0.7074	1.2834	2.6365	6.5075	0.5767	1.4233	0.6562
56*	0.4009	0.0874	0.4027	0.9955	4.5858	0.6439	4.5382	0.7133	1.2867	0.7101	1.2808	2.6542	6.5174	0.5788	1.4212	0.6542
57*	0.3974	0.0864	0.3991	0.9955	4.5994	0.6426	4.5518	0.7159	1.2841	0.7127	1.2784	2.6716	6.5272	0.5809	1.4191	0.6523
58*	0.3939	0.0854	0.3957	0.9956	4.6127	0.6413	4.5651	0.7184	1.2816	0.7153	1.2760	2.6887	6.5367	0.5829	1.4171	0.6504
59*	0.3906	0.0844	0.3923	0.9957	4.6258	0.6401	4.5782	0.7209	1.2791	0.7178	1.2736	2.7054	6.5461	0.5848	1.4152	0.6485
60*	0.3873	0.0835	0.3889	0.9958	4.6386	0.6389	4.5910	0.7232	1.2768	0.7202	1.2714	2.7217	6.5554	0.5868	1.4132	0.6468
61*	0.3841	0.0826	0.3857	0.9958	4.6511	0.6378	4.6036	0.7256	1.2744	0.7226	1.2691	2.7378	6.5645	0.5886	1.4114	0.6450
62*	0.3810	0.0817	0.3826	0.9959	4.6635	0.6366	4.6159	0.7278	1.2722	0.7249	1.2670	2.7536	6.5734	0.5905	1.4095	0.6433
63*	0.3780	0.0808	0.3795	0.9960	4.6756	0.6355	4.6281	0.7301	1.2699	0.7271	1.2648	2.7690	6.5821	0.5922	1.4078	0.6416
64*	0.3750	0.0800	0.3765	0.9960	4.6875	0.6344	4.6400	0.7322	1.2678	0.7293	1.2628	2.7842	6.5907	0.5940	1.4060	0.6400
65*	0.3721	0.0792	0.3736	0.9961	4.6992	0.6333	4.6517	0.7343	1.2657	0.7315	1.2607	2.7991	6.5992	0.5957	1.4043	0.6384
66*	0.3693	0.0784	0.3707	0.9962	4.7106	0.6323	4.6632	0.7364	1.2636	0.7336	1.2588	2.8138	6.6075	0.5973	1.4027	0.6369
67*	0.3665	0.0776	0.3679	0.9962	4.7219	0.6313	4.6745	0.7384	1.2616	0.7356	1.2568	2.8282	6.6157	0.5989	1.4011	0.6353
68*	0.3638	0.0769	0.3652	0.9963	4.7330	0.6302	4.6856	0.7404	1.2596	0.7376	1.2549	2.8423	6.6238	0.6005	1.3995	0.6338
69*	0.3612	0.0761	0.3625	0.9963	4.7440	0.6292	4.6965	0.7423	1.2577	0.7396	1.2531	2.8562	6.6317	0.6021	1.3979	0.6324
70*	0.3586	0.0754	0.3599	0.9964	4.7547	0.6283	4.7073	0.7442	1.2558	0.7415	1.2513	2.8699	6.6395	0.6036	1.3964	0.6310
71*	0.3560	0.0747	0.3573	0.9964	4.7653	0.6273	4.7179	0.7460	1.2540	0.7433	1.2495	2.8834	6.6472	0.6051	1.3949	0.6296
72*	0.3536	0.0740	0.3548	0.9965	4.7757	0.6264	4.7283	0.7478	1.2522	0.7452	1.2478	2.8966	6.6548	0.6065	1.3935	0.6282
73*	0.3511	0.0734	0.3523	0.9965	4.7860	0.6254	4.7385	0.7496	1.2504	0.7470	1.2461	2.9096	6.6623	0.6080	1.3920	0.6268
74*	0.3487	0.0727	0.3499	0.9966	4.7960	0.6245	4.7486	0.7513	1.2487	0.7487	1.2444	2.9225	6.6696	0.6093	1.3907	0.6255
75*	0.3464	0.0721	0.3476	0.9966	4.8060	0.6236	4.7586	0.7530	1.2470	0.7504	1.2428	2.9351	6.6769	0.6107	1.3893	0.6242
76*	0.3441	0.0715	0.3453	0.9967	4.8158	0.6227	4.7684	0.7546	1.2454	0.7521	1.2412	2.9475	6.6840	0.6121	1.3879	0.6230
77*	0.3419	0.0709	0.3430	0.9967	4.8254	0.6219	4.7780	0.7563	1.2437	0.7538	1.2396	2.9598	6.6911	0.6134	1.3866	0.6217
78*	0.3397	0.0703	0.3408	0.9968	4.8349	0.6210	4.7875	0.7579	1.2421	0.7554	1.2381	2.9718	6.6980	0.6147	1.3853	0.6205
79*	0.3375	0.0697	0.3386	0.9968	4.8443	0.6202	4.7969	0.7594	1.2406	0.7570	1.2366	2.9837	6.7049	0.6159	1.3841	0.6193
80*	0.3354	0.0691	0.3365	0.9968	4.8535	0.6194	4.8062	0.7610	1.2390	0.7586	1.2351	2.9954	6.7117	0.6172	1.3828	0.6181
81*	0.3333	0.0685	0.3344	0.9969	4.8627	0.6186	4.8153	0.7625	1.2375	0.7601	1.2337	3.0070	6.7183	0.6184	1.3816	0.6169
82*	0.3313	0.0680	0.3323	0.9969	4.8716	0.6178	4.8243	0.7639	1.2361	0.7616	1.2323	3.0184	6.7249	0.6196	1.3804	0.6158
83*	0.3293	0.0675	0.3303	0.9970	4.8805	0.6170	4.8332	0.7654	1.2346	0.7631	1.2309	3.0296	6.7314	0.6208	1.3792	0.6147
84*	0.3273	0.0669	0.3283	0.9970	4.8893	0.6162	4.8419	0.7668	1.2332	0.7645	1.2295	3.0407	6.7378	0.6219	1.3781	0.6136
85*	0.3254	0.0664	0.3264	0.9970	4.8979	0.6154	4.8506	0.7682	1.2318	0.7659	1.2281	3.0516	6.7442	0.6230	1.3770	0.6125
86*	0.3235	0.0659	0.3245	0.9971	4.9064	0.6147	4.8591	0.7696	1.2304	0.7673	1.2268	3.0624	6.7504	0.6242	1.3758	0.6114
87*	0.3216	0.0654	0.3226	0.9971	4.9148	0.6139	4.8675	0.7709	1.2291	0.7687	1.2255	3.0730	6.7566	0.6253	1.3747	0.6104
88*	0.3198	0.0650	0.3207	0.9971	4.9231	0.6132	4.8758	0.7722	1.2278	0.7700	1.2242	3.0835	6.7627	0.6263	1.3737	0.6094
89*	0.3180	0.0645	0.3189	0.9972	4.9313	0.6125	4.8840	0.7735	1.2265	0.7714	1.2230	3.0939	6.7688	0.6274	1.3726	0.6084
90*	0.3162	0.0640	0.3171	0.9972	4.9394	0.6118	4.8921	0.7748	1.2252	0.7727	1.2217	3.1041	6.7747	0.6284	1.3716	0.6074
91*	0.3145	0.0636	0.3154	0.9972	4.9474	0.6111	4.9001	0.7761	1.2239	0.7739	1.2205	3.1142	6.7806	0.6295	1.3705	0.6064
92*	0.3128	0.0631	0.3136	0.9973	4.9553	0.6104	4.9080	0.7773	1.2227	0.7752	1.2193	3.1241	6.7864	0.6305	1.3695	0.6054
93*	0.3111	0.0627	0.3119	0.9973	4.9631	0.6097	4.9158	0.7785	1.2215	0.7764	1.2181	3.1340	6.7922	0.6315	1.3685	0.6045
94*	0.3094	0.0622	0.3103	0.9973	4.9708	0.6090	4.9235	0.7797	1.2203	0.7776	1.2170	3.1437	6.7979	0.6324	1.3676	0.6035
95*	0.3078	0.0618	0.3086	0.9973	4.9784	0.6084	4.9311	0.7809	1.2191	0.7788	1.2158	3.1533	6.8035	0.6334	1.3666	0.6026
96*	0.3062	0.0614	0.3070	0.9974	4.9859	0.6077	4.9387	0.7821	1.2179	0.7800	1.2147	3.1628	6.8091	0.6343	1.3657	0.6017
97*	0.3046	0.0610	0.3054	0.9974	4.9934	0.6071	4.9461	0.7832	1.2168	0.7812	1.2136	3.1722	6.8146	0.6353	1.3647	0.6008
98*	0.3030	0.0606	0.3038	0.9974	5.0007	0.6064	4.9535	0.7843	1.2157	0.7823	1.2125	3.1814	6.8200	0.6362	1.3638	0.5999
99*	0.3015	0.0602	0.3023	0.9975	5.0080	0.6058	4.9608	0.7854	1.2146	0.7834	1.2115	3.1906	6.8254	0.6371	1.3629	0.5990
100*	0.3000	0.0598	0.3008	0.9975	5.0152	0.6052	4.9679	0.7865	1.2135	0.7845	1.2104	3.1996	6.8307	0.6380	1.3620	0.5982

* n > 25: computed on the same footing as every other row, but beyond the range any published table cited here covers, so not cross-checkable against a printed source.

What each symbol is, and how it is derived

A — X̄ limits from a known σ₀: A = 3/√n (closed form)
A₂ — X̄ limits from R̄: A₂ = 3/(d₂√n) (derived from d2, d3)
A₃ — X̄ limits from s̄: A₃ = 3/(c₄√n) (derived from c4)
c₄ — Unbiasing constant for the sample standard deviation: c₄ = √(2/(n−1)) · Γ(n/2) / Γ((n−1)/2) (closed form)
d₂ — Mean of the relative range: d₂ = E[W], W = (X₍n₎ − X₍₁₎)/σ, integrated from the range density n(n−1)∫φ(x)φ(x+w)[Φ(x+w)−Φ(x)]ⁿ⁻² dx (numerical integration)
d₃ — Standard deviation of the relative range: d₃ = √(E[W²] − d₂²), from the same range density (numerical integration)
d₄ — Median of the relative range: the w solving P(W ≤ w) = ½, where P(W ≤ w) = n∫φ(x)[Φ(x+w)−Φ(x)]ⁿ⁻¹ dx (numerical integration)
B₃ — s-chart lower limit from s̄: B₃ = max(0, 1 − 3√(1−c₄²)/c₄) (derived from c4)
B₄ — s-chart upper limit from s̄: B₄ = 1 + 3√(1−c₄²)/c₄ (derived from c4)
B₅ — s-chart lower limit from a known σ₀: B₅ = max(0, c₄ − 3√(1−c₄²)) (derived from c4)
B₆ — s-chart upper limit from a known σ₀: B₆ = c₄ + 3√(1−c₄²) (derived from c4)
D₁ — R-chart lower limit from a known σ₀: D₁ = max(0, d₂ − 3d₃) (derived from d2, d3)
D₂ — R-chart upper limit from a known σ₀: D₂ = d₂ + 3d₃ (derived from d2, d3)
D₃ — R-chart lower limit from R̄: D₃ = max(0, 1 − 3d₃/d₂) (derived from d2, d3)
D₄ — R-chart upper limit from R̄: D₄ = 1 + 3d₃/d₂ (derived from d2, d3)
E₂ — Individuals limits from the average moving range: E₂ = 3/d₂(span) (derived from d2, d3)

What these were checked against

How each family is derived

c₄ is exact: c₄ = √(2/(n−1))·Γ(n/2)/Γ((n−1)/2). It is the expected value of the sample standard deviation of n standard normal observations, and A₃, B₃, B₄, B₅ and B₆ all follow from it algebraically.

d₂ and d₃ have no elementary closed form. They are the mean and the standard deviation of the range of n standard normal observations, and they are obtained here by integrating the range density n(n−1)∫φ(x)φ(x+w)[Φ(x+w)−Φ(x)]ⁿ⁻² dx. A₂, D₁, D₂, D₃, D₄ and E₂ follow from them.

d₄ is the median of that same range distribution — the unbiasing constant for a median moving range — found by solving its CDF for the half-way point. It is not printed by either source cited here, so it is checked against its own closed form at n = 2, where the median range of a pair is √2·Φ⁻¹(0.75) = 0.9539.

The known-sigma family, which most software omits

A, B₅, B₆, D₁ and D₂ are the factors for the case where σ₀ is not estimated from the data but is known in advance — from a long production history, from a specification, or from a previous validated study. Limits are then μ₀ ± A·σ₀ for averages, B₅σ₀ to B₆σ₀ for standard deviations and D₁σ₀ to D₂σ₀ for ranges.

They are rarely offered because most tools assume you are estimating. If you are running a Phase 2 chart against a baseline that was established properly, these are the right factors and the estimated-sigma ones are not.

Naming, and a trap worth knowing about

The symbols collide across sources, and the collisions are not harmless. `d₄` here is the median moving-range constant and has nothing to do with `D₄`, the upper R-chart factor — one is about 1.98 at n = 4 and the other about 2.28. `E₂`, the individuals factor, is routinely mislabelled `A₂` or `A₃`, including inside at least one published international standard. `A₂` for an X̄-R chart and `Ã₂` for a median chart share the value 1.880 at n = 2 and diverge immediately afterwards.

In this site’s source code the constants are named by role at every call site rather than by letter, for exactly this reason.

Common questions

What is d₂ used for?: Estimating the process standard deviation from ranges: σ̂ = R̄/d₂ for subgrouped data, or σ̂ = MR̄/d₂(2) for individuals. It is the mean range of n standard normal observations, so dividing by it converts an average range back into a standard deviation.
Why is D₃ zero for small subgroups?: D₃ = 1 − 3d₃/d₂, and for n below 7 that expression is negative. A range cannot be negative, so the factor is clamped at zero and the R chart has no lower control limit. Tables print either 0.000 or a dash; they mean the same thing.
Can I use these for subgroup sizes above 25?: Yes — that is most of why this page exists. Nothing changes in the arithmetic above 25; what changes is that no published table covers those rows, so they are marked on the figure as not cross-checkable against a printed source.
Is A₃ the same as A₂?: No. A₂ = 3/(d₂√n) is for limits from R̄; A₃ = 3/(c₄√n) is for limits from s̄. They are close for small subgroups and diverge as n grows, because the range loses efficiency against the standard deviation.