SPC: control charts and the Nelson rules
Statistical process control (SPC) aims to separate the natural, random variation of a process from changes caused by special causes, so you can react before scrap is made. Its main tool is the control chart, which plots values calculated from samples together with a centre line and control limits at ±3σ.
Control limits are calculated from process data, not from the drawing tolerance. A process can be stable but not capable (tight tolerance), or capable but not stable.
Which chart when?
| Data | Chart | When |
|---|---|---|
| Variable, subgroup size 2–9 | x̄–R | the most common, for example 5 parts measured every hour |
| Variable, subgroup size ≥ 10 | x̄–s | large subgroups, automatic gauging |
| Variable, individual values | I–MR (X–mR) | slow processes, destructive tests, batch production |
| Attribute: proportion defective | p (variable subgroup) / np (constant) | go/no-go judgements |
| Attribute: count of defects | c (constant) / u (variable units) | defects per unit, for example paint defects |
The 8 Nelson rules
- Rule 1: 1 point beyond the ±3σ limits.
- Rule 2: 9 points in a row on the same side of the centre line.
- Rule 3: 6 points in a row steadily increasing or decreasing.
- Rule 4: 14 points in a row alternating up and down.
- Rule 5: 2 out of 3 points in a row beyond 2σ on the same side.
- Rule 6: 4 out of 5 points in a row beyond 1σ on the same side.
- Rule 7: 15 points in a row within 1σ on either side of the centre line (suspiciously low variation).
- Rule 8: 8 points in a row beyond 1σ on either side of the centre line (mixed processes).
Constants for x̄–R charts
| n | A₂ | D₃ | D₄ | d₂ |
|---|---|---|---|---|
| 2 | 1.880 | 0.000 | 3.267 | 1.128 |
| 3 | 1.023 | 0.000 | 2.574 | 1.693 |
| 4 | 0.729 | 0.000 | 2.282 | 2.059 |
| 5 | 0.577 | 0.000 | 2.114 | 2.326 |
| 6 | 0.483 | 0.000 | 2.004 | 2.534 |
| 7 | 0.419 | 0.076 | 1.924 | 2.704 |
| 8 | 0.373 | 0.136 | 1.864 | 2.847 |
| 9 | 0.337 | 0.184 | 1.816 | 2.970 |
| 10 | 0.308 | 0.223 | 1.777 | 3.078 |
Limits: x̄ chart: x̿ ± A₂·R̄; R chart: lower D₃·R̄, upper D₄·R̄. Estimated process standard deviation: σ ≈ R̄ / d₂.