Control Chart Limits Calculator
The center lines and control limits for an X̄ (average) and R (range) chart from your subgroup data — the standard Shewhart constants for subgroups of two to ten.
Take subgroups of n consecutive parts, compute each subgroup’s mean and range, and average them for the center lines.
How the control chart limits calculator works
Take subgroups of n consecutive parts, compute each subgroup’s mean and range, and average them for the center lines. The limits come from the average range through tabulated constants: X̄ limits are the grand mean ± A₂ × R̄, and the R chart runs from D₃ × R̄ to D₄ × R̄. Points outside the limits, or runs and trends inside them, signal a special cause worth investigating. Use at least 20–25 subgroups to set the limits.
Formula: UCL/LCL_X̄ = X̿ ± A₂ R̄; UCL_R = D₄ R̄; LCL_R = D₃ R̄
Worked examples
| Inputs | X̄ chart limits | Note |
|---|---|---|
| Ten subgroups of five | 9.8486 to 10.1614 | X̄ 9.85–10.16 |
| Subgroups of two | 49.298 to 50.802 | A₂ = 1.880 |
| A point outside | 4.9523 to 5.2877 | two X̄ signals — 5.6 above and 4.9 below |
FAQFrequently asked questions
Where do A₂, D₃ and D₄ come from?
They are derived from the distribution of the range in samples from a normal population, tabulated for each subgroup size; A₂ = 3 ÷ (d₂ √n). They let you set 3σ limits from ranges without computing standard deviations.
Why do control limits differ from specification limits?
Specification limits say what the customer will accept; control limits say what the process is doing. A process can be in control and still make bad parts — that is what the capability indices measure.
How many subgroups do I need?
At least 20–25 before the limits are trusted; then freeze them and plot new data against them. Recompute only after a deliberate process change.
What if the R chart is out of control?
Fix that first — the X̄ limits depend on R̄, so an unstable range makes the X̄ chart meaningless. Common causes are mixed batches, worn fixtures or an untrained operator.
Where these figures come from
- Nakajima (1988) — Introduction to TPM — the origin of Overall Equipment Effectiveness and its six big losses
- OSHA — the US workplace safety administration
Last checked: September 2026. Definitions follow standard operations-management practice; where plants commonly differ, the page says so.