Control Chart Limits Calculator (X̄ and R)
Shewhart X̄-R control chart limits from subgroup data — centre lines, UCL/LCL for the mean and range using the A2, D3, D4 constants, plus out-of-control points.
How it works
UCL/LCL(X̄) = X̿ ± A₂·R̄; UCL(R) = D₄·R̄, LCL(R) = D₃·R̄; σ̂ = R̄ ÷ d₂
Shewhart charts separate common-cause variation from special causes. Small rational subgroups are measured over time; the R chart checks that within-subgroup spread is stable, and only then are the X̄ limits meaningful. The A2/D3/D4 constants convert the average range into three-sigma limits for subgroup sizes 2–10. Points outside the limits, or runs and trends inside them, signal something changed in the process.
Worked example
Six subgroups of four with X̿ = 10.15 and R̄ = 0.45: X̄ limits 9.82–10.48, R upper limit 1.03, σ̂ = 0.22.
Frequently asked questions
Why check the R chart first?
The X̄ limits are built from R̄; if the spread is unstable the mean limits are unreliable.
What about subgroups larger than 10?
Use X̄-S charts based on the standard deviation (constants A₃, B₃, B₄) — ranges become inefficient.
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This calculator is for educational purposes. Double-check important results before acting on them.