How to calculate control limits
Control limits sit 3 standard deviations from the center line. The standard deviation is estimated from the variation within subgroups, and the table constants already do that math.
| Chart | Center line | Limits |
|---|---|---|
| X̄ (with R) | X̿ | X̿ ± A2 × R̄ |
| R | R̄ | D3 × R̄ and D4 × R̄ |
| X̄ (with S) | X̿ | X̿ ± A3 × S̄ |
| S | S̄ | B3 × S̄ and B4 × S̄ |
| I | mean | mean ± 2.66 × MR̄ |
| MR | MR̄ | 0 and 3.267 × MR̄ |
| p | p̄ = total defectives ÷ total inspected | p̄ ± 3 × √(p̄ × (1 − p̄) ÷ n) |
Constants for the most common subgroup sizes (AIAG SPC manual):
| n | A2 | D3 | D4 | d2 |
|---|---|---|---|---|
| 2 | 1.880 | 0 | 3.267 | 1.128 |
| 3 | 1.023 | 0 | 2.574 | 1.693 |
| 4 | 0.729 | 0 | 2.282 | 2.059 |
| 5 | 0.577 | 0 | 2.114 | 2.326 |
Worked example
The data in the calculator is 20 subgroups of 5 shafts, the same as in the Cp and Cpk calculator.
- Mean of the means: X̿ = 10.00985 mm
- Average range: R̄ = 0.0283 mm
- X̄ chart: UCL = 10.00985 + 0.577 × 0.0283 = 10.0262; LCL = 10.00985 − 0.01633 = 9.9935
- R chart: UCL = 2.114 × 0.0283 = 0.0598; LCL = 0
No mean and no range falls outside its limits and there are no suspicious runs: the process is in control. Now Cpk makes sense.
Switch the type to p chart for the second example: 15 days of final inspection, about 200 parts a day. The average proportion is p̄ = 100 ÷ 3,050 = 3.28%. On day 10, 19 of 210 parts were defective (9.05%), above that day’s 6.97% limit. Something different happened there. Once the cause is found and removed, day 10 leaves the calculation and p̄ drops to 81 ÷ 2,840 = 2.85%.
Because the sample size changes from day to day, the p chart limits change too: larger samples get tighter limits.
How to read the chart
The calculator applies the Western Electric rules (also known as Nelson rules) and explains each signal:
- One point outside the control limits. The strongest special-cause signal.
- 2 of 3 points in a row beyond 2σ on the same side. The process level may have shifted.
- 4 of 5 points in a row beyond 1σ on the same side. A smaller but persistent shift.
- 8 points in a row on the same side of the center line. The process mean has moved.
- 6 points in a row going up or down. A trend: tool wear, temperature, drift.
Always read the spread chart first (R, S or MR). If the spread is out of control, the limits of the means chart cannot be trusted, because they depend on R̄, S̄ or MR̄.
Common mistakes
- Using spec limits as control limits. They answer different questions. The control chart tells you whether the process is stable; capability (Cpk) tells you whether it meets the tolerance.
- Computing limits from the overall standard deviation. The chart uses within-subgroup variation. With the overall standard deviation, limits get wide and hide shifts between subgroups.
- Adjusting the process for every point. Tweaking the machine when the point is inside the limits is reacting to common causes, and it increases variation.
- Recalculating limits all the time. Set them from 20 to 25 stable subgroups and keep them. Recalculate only after a real process change.
- Badly formed subgroups. Parts in a subgroup should be made in sequence under the same conditions. Mixing machines or shifts in one subgroup inflates R̄ and widens the limits.