Control Limit (UCL / LCL)

Action limits on a control chart, usually at ±3σ from the centre line — NOT the same as specification limits.

Also known as: UCL, LCL, control limit, action limit

Control limits (Upper/Lower Control Limit) mark the natural spread a stable process "normally" produces. They are derived from the observed process data itself — not set by the customer.

UCL = \overset{x}{ˉ} + 3 \overset{σ}{^}, LCL = \overset{x}{ˉ} - 3 \overset{σ}{^}

±3σ standard — per-point false-alarm probability approx. 0.27 % (under normality)
Chart-specific — X̄-R: $A_{2} \overset{ˉ}{R}$ ; I-MR: $E_{2} \overset{ˉ}{M R}$ ; p chart: $\overset{p}{ˉ} \pm 3 \overset{p}{ˉ} (1 - \overset{p}{ˉ}) / n$
Compute from at least 20–25 subgroups or 100+ individuals

Used in

In the Algorithm Lab

control-charts/control-chart

Sources

Shewhart, W. A. — Economic Control of Quality of Manufactured Product, 1931
Montgomery, D. C. — Introduction to Statistical Quality Control, 8th Edition, Chapter 5
AIAG — SPC Reference Manual, 2nd Edition