upper control limit

**UCL** (Upper Control Limit) is the **upper boundary on an SPC control chart, set at the process mean plus three standard deviations** — $UCL = ar{x} + 3sigma$ (for an X-bar chart) or calculated using appropriate factors for other chart types (R-chart, S-chart, p-chart). **UCL for Different Chart Types** - **X-bar Chart**: $UCL = ar{ar{x}} + A_2 ar{R}$ — using range-based sigma estimation. - **R Chart**: $UCL = D_4 ar{R}$ — upper limit for the range chart. - **Individuals Chart**: $UCL = ar{x} + 2.66 ar{MR}$ — using moving range. - **p-Chart**: $UCL = ar{p} + 3sqrt{ar{p}(1-ar{p})/n}$ — for proportion defective. **Why It Matters** - **Alarm**: Any point above UCL triggers an out-of-control alarm — immediate investigation required. - **Action**: UCL violations indicate a special cause — something changed in the process (tool, material, recipe). - **Recalculation**: UCL should be recalculated when the process changes — after improvement, limits tighten. **UCL** is **the ceiling of normal** — the upper boundary of expected process variation above which a special cause is indicated.