| n | d2 | d3 | c4 | A2 | D3 | D4 | A3 | B3 | B4 |
|---|---|---|---|---|---|---|---|---|---|
| 2 | 1.128 | 0.853 | 0.7979 | 1.881 | 0.000 | 3.269 | 2.659 | 0.000 | 3.267 |
| 3 | 1.693 | 0.888 | 0.8862 | 1.023 | 0.000 | 2.574 | 1.954 | 0.000 | 2.568 |
| 4 | 2.059 | 0.880 | 0.9213 | 0.729 | 0.000 | 2.282 | 1.628 | 0.000 | 2.266 |
| 5 | 2.326 | 0.864 | 0.9400 | 0.577 | 0.000 | 2.114 | 1.427 | 0.000 | 2.089 |
| 6 | 2.534 | 0.848 | 0.9515 | 0.483 | 0.000 | 2.004 | 1.287 | 0.030 | 1.970 |
| 7 | 2.704 | 0.833 | 0.9594 | 0.419 | 0.076 | 1.924 | 1.182 | 0.118 | 1.882 |
| 8 | 2.847 | 0.820 | 0.9650 | 0.373 | 0.136 | 1.864 | 1.099 | 0.185 | 1.815 |
| 9 | 2.970 | 0.808 | 0.9693 | 0.337 | 0.184 | 1.816 | 1.032 | 0.239 | 1.761 |
| 10 | 3.078 | 0.797 | 0.9727 | 0.308 | 0.223 | 1.777 | 0.975 | 0.284 | 1.716 |
| 11 | 3.173 | 0.787 | 0.9754 | 0.285 | 0.256 | 1.744 | 0.927 | 0.321 | 1.679 |
| 12 | 3.258 | 0.778 | 0.9776 | 0.266 | 0.284 | 1.716 | 0.886 | 0.354 | 1.646 |
| 13 | 3.336 | 0.770 | 0.9794 | 0.249 | 0.308 | 1.692 | 0.850 | 0.382 | 1.618 |
| 14 | 3.407 | 0.763 | 0.9810 | 0.235 | 0.328 | 1.672 | 0.817 | 0.406 | 1.594 |
| 15 | 3.472 | 0.756 | 0.9823 | 0.223 | 0.347 | 1.653 | 0.789 | 0.428 | 1.572 |
| 16 | 3.532 | 0.750 | 0.9835 | 0.212 | 0.363 | 1.637 | 0.763 | 0.448 | 1.552 |
| 17 | 3.588 | 0.744 | 0.9845 | 0.203 | 0.378 | 1.622 | 0.739 | 0.466 | 1.534 |
| 18 | 3.640 | 0.739 | 0.9854 | 0.194 | 0.391 | 1.609 | 0.718 | 0.482 | 1.518 |
| 19 | 3.689 | 0.734 | 0.9862 | 0.187 | 0.403 | 1.597 | 0.698 | 0.497 | 1.503 |
| 20 | 3.735 | 0.729 | 0.9869 | 0.180 | 0.414 | 1.586 | 0.680 | 0.510 | 1.490 |
| 21 | 3.778 | 0.724 | 0.9876 | 0.173 | 0.425 | 1.575 | 0.663 | 0.523 | 1.477 |
| 22 | 3.819 | 0.720 | 0.9882 | 0.167 | 0.434 | 1.566 | 0.647 | 0.534 | 1.466 |
| 23 | 3.858 | 0.716 | 0.9887 | 0.162 | 0.443 | 1.557 | 0.633 | 0.545 | 1.455 |
| 24 | 3.895 | 0.712 | 0.9892 | 0.157 | 0.452 | 1.548 | 0.619 | 0.555 | 1.445 |
| 25 | 3.931 | 0.708 | 0.9896 | 0.153 | 0.460 | 1.540 | 0.606 | 0.565 | 1.435 |
Formulas
- Sigma within: σ = R̄ / d2 (X-bar / R), S̄ / c4 (X-bar / S), MR̄ / 1.128 (I-MR).
- X-bar chart: UCL / LCL = X̿ ± A2 · R̄, or X̿ ± A3 · S̄.
- R chart: UCL = D4 · R̄, LCL = D3 · R̄. S chart: UCL = B4 · S̄, LCL = B3 · S̄.
- Individuals chart: UCL / LCL = X̄ ± 2.66 · MR̄; MR chart UCL = 3.267 · MR̄.
The capability tool applies these constants automatically; read the capability guide for the theory.
Frequently asked questions
What is d2 for n = 5?
d2 = 2.326 for subgroups of 5, so sigma within = R-bar / 2.326.
When do I use c4 instead of d2?
Use c4 with the average standard deviation S-bar (X-bar / S chart), typically for subgroups larger than 9. Use d2 with the average range R-bar (X-bar / R chart) for small subgroups, and d2 = 1.128 with the average moving range for individuals.