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 — $UCL = \bar{x} + 3\sigma$ (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: $UCL = \bar{\bar{x}} + A_2 \bar{R}$ — using range-based sigma estimation.
• R Chart: $UCL = D_4 \bar{R}$ — upper limit for the range chart.
• Individuals Chart: $UCL = \bar{x} + 2.66 \bar{MR}$ — using moving range.
• p-Chart: $UCL = \bar{p} + 3\sqrt{\bar{p}(1-\bar{p})/n}$ — 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.