SPC Process Stability: Control Limits, Signals and Capability

What the showcase models

The lab uses an individuals-style control chart to separate estimated common-cause behavior from an injected step change. The first 16 observations form the baseline. Their moving ranges estimate short-term variation; those baseline limits are then frozen while later observations are evaluated.

That distinction matters. Recomputing limits after injecting a shift lets the abnormal state influence the definition of normality and can partially hide the signal.

Equations

For consecutive baseline observations, MRᵢ = |xᵢ − xᵢ₋₁|. The lab estimates σ = MR̄ / 1.128 and uses UCL = x̄baseline + 3σ and LCL = x̄baseline − 3σ. Specification limits are independent requirements, not control limits.

Displayed Cpk uses the estimated baseline σ and the overall displayed process mean: min[(USL−x̄)/(3σ),(x̄−LSL)/(3σ)]. That capability number should only be interpreted after stability has been assessed.

Controls and interpretation

Shift changes the post-baseline process level; tolerance changes specification width. A point outside the frozen control limits is evidence inconsistent with the estimated baseline process. A point inside specification is not automatically statistically stable, and a stable process is not automatically capable.

Assumptions and limits

The miniature example assumes a reasonably stationary baseline and uses the moving-range estimator appropriate to individual observations. It does not implement the full Western Electric/Nelson run-rule family, rational subgrouping, autocorrelation correction or measurement-system analysis.

Validation checks

With zero injected shift, later observations should remain centered on the same baseline mechanism. Increasing the shift must move post-baseline observations without moving the frozen control limits. Widening specification tolerance must change capability without changing control limits. Those invariants are more important than whether the chart merely looks plausible.