How the control limits are calculated
A Shewhart chart puts limits three standard errors either side of the center line. The standard deviation is estimated from variation within subgroups, so that shifts between subgroups show up as signals instead of widening the limits. With \(k\) subgroups of size \(n\), grand mean \(\bar{\bar x}\), average range \(\bar R\), and average standard deviation \(\bar s\):
The constants come from the distribution of the range and standard deviation of \(n\) normal values: \(\sigma = \bar R/d_2 = \bar s/c_4\), so \(A_2 = 3/(d_2\sqrt n)\) and \(A_3 = 3/(c_4\sqrt n)\), and \(D_3, D_4 = 1 \mp 3d_3/d_2\), with negative values set to zero. With a standard given (Phase II), the limits are \(\mu_0 \pm 3\sigma_0/\sqrt n\). The range chart then uses \(D_1\sigma_0\), \(d_2\sigma_0\), and \(D_2\sigma_0\), and the S chart uses \(B_5\sigma_0\), \(c_4\sigma_0\), and \(B_6\sigma_0\).
| n | A₂ | D₃ | D₄ | d₂ | A₃ | B₃ | B₄ | c₄ |
|---|---|---|---|---|---|---|---|---|
| 2 | 1.880 | 0 | 3.267 | 1.128 | 2.659 | 0 | 3.267 | 0.7979 |
| 3 | 1.023 | 0 | 2.574 | 1.693 | 1.954 | 0 | 2.568 | 0.8862 |
| 4 | 0.729 | 0 | 2.282 | 2.059 | 1.628 | 0 | 2.266 | 0.9213 |
| 5 | 0.577 | 0 | 2.114 | 2.326 | 1.427 | 0 | 2.089 | 0.9400 |
| 6 | 0.483 | 0 | 2.004 | 2.534 | 1.287 | 0.030 | 1.970 | 0.9515 |
| 8 | 0.373 | 0.136 | 1.864 | 2.847 | 1.099 | 0.185 | 1.815 | 0.9650 |
| 10 | 0.308 | 0.223 | 1.777 | 3.078 | 0.975 | 0.284 | 1.716 | 0.9727 |
The calculator holds the full table for n = 2–25. Each value was checked against the defining integrals of \(d_2\) and \(d_3\) and against \(c_4 = \sqrt{2/(n-1)}\,\Gamma(n/2)/\Gamma\bigl((n-1)/2\bigr)\).
Run rules
Points beyond the control limits are not the only sign of a special cause. The zone rules from the Western Electric handbook catch smaller sustained shifts sooner. They divide each side of the center line into zones at 1σ and 2σ of the plotted statistic:
- Rule 1: one point beyond a 3σ control limit.
- Rule 2: two of three consecutive points beyond 2σ, on the same side.
- Rule 3: four of five consecutive points beyond 1σ, on the same side.
- Rule 4: eight consecutive points on the same side of the center line.
- Trend: six consecutive points steadily increasing or decreasing (Nelson's rule 3).
The tool applies all of them to the X̄ or individuals chart. It applies only rule 1 to the range, S, and moving-range charts, because their distributions are skewed and the symmetric zone rules do not apply. Each extra rule raises the false-alarm rate. For a stable, normally distributed process, rule 1 alone gives a false alarm about once every 370 points. Rules 1–4 together bring that down to about once every 92 points (Champ and Woodall), and adding the trend rule to about once every 78. Investigate signals; do not adjust the process for every one.
Worked example
The default data are 25 subgroups of 5 ablation-depth measurements. The grand mean is 12.138 µm and \(\bar R = 0.703\) µm, so the X̄ limits are \(12.138 \pm 0.577 \times 0.703 = 11.732\) to \(12.544\) µm. The R-chart UCL is \(2.114 \times 0.703 = 1.487\) µm, and the within-subgroup σ is \(0.703/2.326 = 0.302\) µm.
No point crosses a control limit, but the zone rules catch a run of four low means (subgroups 8–12, rule 3). Late in the run they catch the upward shift: rule 2 at subgroup 24, rule 3 at 24, and rule 4 at 25. The realized shift was about 0.3 µm, roughly 1 process σ. Because the Phase I grand mean includes the shifted subgroups, those subgroups sit only about 1.5σ of X̄ above the center line, and rule 1 alone would have missed the shift. In Phase I, find the cause, remove the affected subgroups, and recalculate the limits before using them for monitoring.
Assumptions and limits
- Rational subgroups. Measurements within a subgroup should be made close together under the same conditions, so within-subgroup variation reflects only common causes.
- Independence. Strongly autocorrelated data, such as slow thermal drift or high-rate sensor streams, produce too many false alarms. Subsample the data, or chart the residuals of a time-series model.
- Approximate normality. X̄ charts are robust to non-normal data because subgroup means tend toward normal. Individuals charts are much less robust, so check the distribution.
- Enough data. Limits from fewer than about 20–25 subgroups (or 50–100 individuals) are uncertain. Treat them as trial limits and revise them as data accumulate.
- Control limits are not specification limits. Control limits describe the process; specification limits describe the customer requirement. Use the capability calculator to compare the two.
References
- Western Electric Co., Statistical Quality Control Handbook (1956).
- AIAG, Statistical Process Control (SPC) Reference Manual, 2nd ed. (2005).
- D. C. Montgomery, Introduction to Statistical Quality Control, 8th ed., Wiley (Appendix VI, factors for constructing variables control charts).
- L. S. Nelson, “The Shewhart control chart—tests for special causes,” Journal of Quality Technology 16(4), 237–239 (1984).
- C. W. Champ and W. H. Woodall, “Exact results for Shewhart control charts with supplementary runs rules,” Technometrics 29(4), 393–399 (1987).