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Control chart constants table (n = 2 to 25), with an X̄-R and X̄-S limits calculator

This page has A₂, A₃, d₂, d₃, c₄, D₃, D₄, B₃ and B₄ for every subgroup size from 2 to 25. Enter your grand mean and average range (or average standard deviation) to get the control limits. The row for your n is highlighted in the table. Nothing you type leaves your browser.

Chart

Subgroup size (n)

Grand mean (X̿)

Average range (R̄)

The constants

n is the subgroup size, not the number of subgroups. Scroll sideways on a phone.

n	A₂	A₃	d₂	d₃	c₄	D₃	D₄	B₃	B₄
2	1.880	2.659	1.128	0.853	0.7979	0	3.267	0	3.267
3	1.023	1.954	1.693	0.888	0.8862	0	2.574	0	2.568
4	0.729	1.628	2.059	0.880	0.9213	0	2.282	0	2.266
5	0.577	1.427	2.326	0.864	0.9400	0	2.114	0	2.089
6	0.483	1.287	2.534	0.848	0.9515	0	2.004	0.030	1.970
7	0.419	1.182	2.704	0.833	0.9594	0.076	1.924	0.118	1.882
8	0.373	1.099	2.847	0.820	0.9650	0.136	1.864	0.185	1.815
9	0.337	1.032	2.970	0.808	0.9693	0.184	1.816	0.239	1.761
10	0.308	0.975	3.078	0.797	0.9727	0.223	1.777	0.284	1.716
11	0.285	0.927	3.173	0.787	0.9754	0.256	1.744	0.321	1.679
12	0.266	0.886	3.258	0.778	0.9776	0.283	1.717	0.354	1.646
13	0.249	0.850	3.336	0.770	0.9794	0.307	1.693	0.382	1.618
14	0.235	0.817	3.407	0.763	0.9810	0.328	1.672	0.406	1.594
15	0.223	0.789	3.472	0.756	0.9823	0.347	1.653	0.428	1.572
16	0.212	0.763	3.532	0.750	0.9835	0.363	1.637	0.448	1.552
17	0.203	0.739	3.588	0.744	0.9845	0.378	1.622	0.466	1.534
18	0.194	0.718	3.640	0.739	0.9854	0.391	1.608	0.482	1.518
19	0.187	0.698	3.689	0.734	0.9862	0.403	1.597	0.497	1.503
20	0.180	0.680	3.735	0.729	0.9869	0.415	1.585	0.510	1.490
21	0.173	0.663	3.778	0.724	0.9876	0.425	1.575	0.523	1.477
22	0.167	0.647	3.819	0.720	0.9882	0.434	1.566	0.534	1.466
23	0.162	0.633	3.858	0.716	0.9887	0.443	1.557	0.545	1.455
24	0.157	0.619	3.895	0.712	0.9892	0.451	1.548	0.555	1.445
25	0.153	0.606	3.931	0.708	0.9896	0.459	1.541	0.565	1.435

Values are the standard published constants, as rounded in the usual references. A D₃ or B₃ of 0 means the chart has no lower control limit at that n.

Which constant goes where

Chart	Center line	Control limits	σ estimate
X̄ (with R)	X̿	X̿ ± A₂R̄	σ̂ = R̄ / d₂
R	R̄	UCL = D₄R̄, LCL = D₃R̄	σ̂ = R̄ / d₂
X̄ (with S)	X̿	X̿ ± A₃S̄	σ̂ = S̄ / c₄
S	S̄	UCL = B₄S̄, LCL = B₃S̄	σ̂ = S̄ / c₄
Individuals (I-MR)	x̄, MR̄	x̄ ± 2.66 MR̄, UCL_MR = 3.267 MR̄	σ̂ = MR̄ / 1.128 (d₂ at n = 2)

All of them come from a few definitions. For a normal process, d₂ = E[R]/σ and d₃ = SD[R]/σ for a subgroup of n, and c₄ = E[s]/σ. From those:

A₂ = 3 / (d₂√n) A₃ = 3 / (c₄√n) D₃ = max(0, 1 − 3d₃/d₂) D₄ = 1 + 3d₃/d₂ B₃ = max(0, 1 − 3√(1 − c₄²)/c₄) B₄ = 1 + 3√(1 − c₄²)/c₄ c₄ = √(2/(n − 1)) · Γ(n/2) / Γ((n − 1)/2)

Worked example

Twenty-five subgroups of n = 5 shaft diameters give a grand mean X̿ = 50.0 and an average range R̄ = 4.0. From the n = 5 row, A₂ = 0.577, D₃ = 0, D₄ = 2.114 and d₂ = 2.326.

X̄ chart: UCL = 50.0 + 0.577 × 4.0 = 52.308 LCL = 50.0 − 0.577 × 4.0 = 47.692 R chart: UCL = 2.114 × 4.0 = 8.456 LCL = 0 × 4.0 = 0 (no lower limit at n = 5) σ̂ = R̄ / d₂ = 4.0 / 2.326 = 1.720

Two things to note. The X̄ limits are limits for subgroup averages, not for individual parts, so never compare them with spec limits. And σ̂ is the within-subgroup σ that feeds Cp and Cpk. You can enter it directly in the process capability calculator.

Picking R or S, and common mistakes

R or S. The range is almost as efficient as the standard deviation for small subgroups: NIST puts its relative efficiency at 0.955 for n = 5 and 0.850 for n = 10. Beyond about 10, use X̄-S.
Using the number of subgroups as n. n is the size of each subgroup. With 25 subgroups of 5, look up n = 5.
Mixing A₂ and A₃. A₂ goes with R̄ and A₃ goes with S̄. Using A₂ with S̄ gives limits that are far too tight.
Expecting a lower limit on the R chart for small n. D₃ = 0 up to n = 6, because the formula would go negative and a range can't be negative.
Limits from an unstable baseline. Remove points with assignable causes and recompute before you use the limits going forward. See the run rules and the chart selection guide.

Where the numbers come from

The table shows the values as printed in the standard references, so they match your study materials and most SPC software. Our automated tests recompute every one from first principles:

d₂ comes from numerical integration of the normal range distribution.
d₃ comes from the second moment of the range, using a double integral.
c₄ comes from the gamma function.
The other constants come from the formulas above.

Every printed value is within 0.001 of the exact result.

The cross-check sources are:

The NIST/SEMATECH e-Handbook of Statistical Methods, §6.3.2.1, which gives A₂, D₃ and D₄ for n = 2 to 10.
§6.3.2, which gives the c₄ formula and c₄ = 0.9727 at n = 10.
The widely reproduced n = 2 to 25 table in D. C. Montgomery, Introduction to Statistical Quality Control, Appendix VI.

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More free tools: Process capability calculator · Control chart picker · I-MR chart guide · Printable CQE formula sheet

Constants assume normally distributed data and 3σ limits. Not affiliated with ASQ.