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Control chart constants (A2, A3, B3, B4, D3, D4, d2, c4)

The whole table, n = 2 to 25, with the formula each constant belongs to. The same values the charts on this site are computed from.

Why you are looking this up in the first place

Because Excel does not have them. A₂, A₃, d₂, D₃, D₄, c₄, B₃ and B₄ are not functions — there is no lookup, no add-in in the box, nothing. Every spreadsheet control chart in the world has these numbers typed into it by hand from a table like this one.

Which is the quiet risk. A transposed digit in A₂ does not produce an error, an empty cell or a broken chart. It produces control limits that are wrong by a few per cent, on a chart that looks entirely normal, for as long as the workbook survives.

Nothing you paste leaves this calculation. It is not stored, not logged and not sent anywhere else — the numbers are computed and returned, and that is all.

n A2A3B3B4D3D4d2d3c4
2 1.88 2.659 0 3.267 0 3.267 1.128 0.853 0.7979
3 1.023 1.954 0 2.568 0 2.574 1.693 0.888 0.8862
4 0.729 1.628 0 2.266 0 2.282 2.059 0.88 0.9213
5 0.577 1.427 0 2.089 0 2.114 2.326 0.864 0.94
6 0.483 1.287 0.03 1.97 0 2.004 2.534 0.848 0.9515
7 0.419 1.182 0.118 1.882 0.076 1.924 2.704 0.833 0.9594
8 0.373 1.099 0.185 1.815 0.136 1.864 2.847 0.82 0.965
9 0.337 1.032 0.239 1.761 0.184 1.816 2.97 0.808 0.9693
10 0.308 0.975 0.284 1.716 0.223 1.777 3.078 0.797 0.9727
11 0.285 0.927 0.321 1.679 0.256 1.744 3.173 0.787 0.9754
12 0.266 0.886 0.354 1.646 0.283 1.717 3.258 0.778 0.9776
13 0.249 0.85 0.382 1.618 0.307 1.693 3.336 0.77 0.9794
14 0.235 0.817 0.406 1.594 0.328 1.672 3.407 0.763 0.981
15 0.223 0.789 0.428 1.572 0.347 1.653 3.472 0.756 0.9823
16 0.212 0.763 0.448 1.552 0.363 1.637 3.532 0.75 0.9835
17 0.203 0.739 0.466 1.534 0.378 1.622 3.588 0.744 0.9845
18 0.194 0.718 0.482 1.518 0.391 1.608 3.64 0.739 0.9854
19 0.187 0.698 0.497 1.503 0.403 1.597 3.689 0.734 0.9862
20 0.18 0.68 0.51 1.49 0.415 1.585 3.735 0.729 0.9869
21 0.173 0.663 0.523 1.477 0.425 1.575 3.778 0.724 0.9876
22 0.167 0.647 0.534 1.466 0.434 1.566 3.819 0.72 0.9882
23 0.162 0.633 0.545 1.455 0.443 1.557 3.858 0.716 0.9887
24 0.157 0.619 0.555 1.445 0.451 1.548 3.895 0.712 0.9892
25 0.153 0.606 0.565 1.435 0.459 1.541 3.931 0.708 0.9896

Which constant goes where

X̄ chart (with R)   UCL = x̄̄ + A₂R̄            LCL = x̄̄ − A₂R̄
R chart            UCL = D₄R̄                LCL = D₃R̄
X̄ chart (with s)   UCL = x̄̄ + A₃s̄            LCL = x̄̄ − A₃s̄
s chart            UCL = B₄s̄                LCL = B₃s̄
σ estimate         σ̂ = R̄/d₂   (or s̄/c₄)
I–MR chart         UCL = x̄ + 2.66·MR̄        MR UCL = 3.267·MR̄

Why these numbers exist

A range is a cheap estimate of spread that a shop-floor operator can compute in their head. These constants convert it into the three-sigma distance a normal distribution would give, for each subgroup size. d₂ is the expected range of n standard normal values, so R̄/d₂ estimates sigma; A₂ folds the 3/√n together with 1/d₂ so the X̄ limits fall out in one multiplication.

The part most people get wrong

D₃ is zero below n = 7. That is not a rounding artefact or a missing value: for small subgroups the lower three-sigma bound on the range is negative, and a range cannot be negative, so the chart has no lower limit at all. A range chart with an LCL drawn at zero for n = 5 is decoration.

2.66 is not a special number. It is 3/d₂ at n = 2, because a moving range of two consecutive readings is a subgroup of size 2. Same table, same logic.

Values are the published Shewhart factors (ASTM / ISO 7870 tables), matching Montgomery's Appendix VI. Every chart on this site is computed from exactly these.

What the Excel version will not do

A mistyped constant is invisible
No formula validates it and no chart complains. The limits are simply somewhat wrong, and stay wrong.
The right constant depends on the subgroup size you actually had
A shift that recorded four parts instead of five needs a different A₂. Hard-coded constants silently apply the wrong one.

Questions people ask about this

What is the A2 constant in a control chart?

A₂ converts the mean range into the three-sigma distance for an X̄ chart, so that UCL = x̄̄ + A₂R̄. It folds 3/√n together with 1/d₂, which is why it shrinks as the subgroup size grows: 0.577 at n = 5, 0.308 at n = 10.

What is d2 used for?

d₂ is the expected range of n standard normal values, so R̄/d₂ estimates the process standard deviation from ranges. That estimate is what within-subgroup capability (Cp and Cpk) is built on.

Why is D3 zero for small subgroups?

Because the three-sigma lower bound on a range would be negative for n below seven, and a range cannot be negative. The range chart simply has no lower limit at those subgroup sizes.

Where does the 2.66 in an I-MR chart come from?

It is 3/d₂ at n = 2, because a moving range of two consecutive readings is a subgroup of size two. Same table, same logic as the X̄ charts.

Want this to keep itself up to date?

A calculator answers for the data you pasted. A control chart answers for the data your line produced this morning — limits frozen at a baseline you locked, rules evaluated on every new reading, an alert when one trips.

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Other free tools

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  • Control chart generator — Paste a column of numbers, or rows of subgroups, and get a real control chart: limits from the data, Nelson rules 1–4 evaluated, out-of-control points marked.
  • Nelson & Western Electric rules checker — Every run rule evaluated on your data, each violation named in plain English with what it usually indicates — a shift, a trend, tool wear, two machines mixed.
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  • X̄-R chart calculator — Subgroups in, X̄ and R charts out — limits from A₂, D₃ and D₄, run rules evaluated, and the range chart shown first because it decides whether the averages chart can be trusted.
  • X̄-s chart calculator — For subgroups big enough that the range wastes them. Limits from A₃, B₃ and B₄, sigma recovered with the c₄ correction, and the s chart read first because it decides whether the averages chart can be trusted.
  • I-MR chart calculator — For processes that give you one number at a time — a batch, an oven, a destructive test. Limits from the mean moving range, run rules evaluated, and an honest note about what an individuals chart cannot see.
  • Levey-Jennings chart with Westgard rules — Paste your QC results with the assigned mean and SD from the package insert, and get the chart with every sigma band drawn and all six Westgard rules evaluated — each one named, and labelled random or systematic error.
  • Bland-Altman plot — Paired readings from two methods, plotted as difference against average — with the bias, the limits of agreement, their confidence intervals, and a test for whether the disagreement depends on the magnitude.
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  • Gage R&R calculator (ANOVA) — Paste a crossed study — part, operator, reading — and get the full ANOVA: repeatability and reproducibility separated, the part-by-operator interaction tested rather than assumed away, %GRR, %Tolerance and ndc against AIAG's bands.
  • p, np, u and c chart calculator — Counted data rather than measured: proportion defective, number defective, defects per unit. Limits that step with the sample size instead of pretending it never changed, and an honest warning when the counts are too low for three-sigma limits to mean anything.
  • Grubbs outlier test — The G statistic against its critical value, the p-value, and the point implicated — plus a straight answer about what to do next, which is almost never to delete the reading.
  • Scatter diagram and correlation — Paired readings plotted, with Pearson r, Spearman rho, r², the least-squares line and its p-value — and the two reported together, because when they disagree the disagreement is the finding.
  • Standard deviation calculator — The usual summary statistics, and the thing a spreadsheet will not tell you: the within-subgroup sigma a control chart uses, beside the overall sigma a textbook means. The gap between them is why Cpk and Ppk disagree.