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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.

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.

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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