Process capability analysis compares the process spread to the specification limits — Cp/Cpk from the within-subgroup σ, Pp/Ppk from the overall σ, with ppm defective and a capability histogram.
Capability indices summarize how comfortably a process fits inside its spec limits. Cp = (USL − LSL)/6σ is the potential (assuming perfect centering); Cpk = min(Cpu, Cpl) penalizes off-center processes by using the distance to the nearer limit. The 'within' σ (from subgroup ranges or the moving range) reflects short-term, common-cause variation; the overall σ includes drift between subgroups, giving the long-term Pp/Ppk.
Cpk ≥ 1.33 (4σ to the nearer limit) is the conventional 'capable' threshold; six-sigma programs target 2.0. A Ppk well below Cpk is a diagnostic: the process is capable in the short term but drifts.
The indices are normal-theory. For skewed data, the Box-Cox option transforms the measurements and the spec limits with the same λ (maximum-likelihood estimate, or a fixed value like 0 = log) and runs the analysis on the transformed scale, where the normal assumption holds — the standard workflow for non-normal capability.
When no Box-Cox power works (e.g. heavy tails or bounded data), the Johnson option fits the Johnson system: for a grid of z values, four symmetric normal quantiles select the family — SB (bounded), SL (lognormal-type), or SU (unbounded) — and the Slifker-Shapiro percentile formulas give its parameters; the candidate whose transformed data maximizes the Shapiro-Wilk p wins (Chou-Polansky-Mason). A spec limit outside the fitted support cannot produce defects and is dropped from the indices.
One measurement column + subgroup size (1 = individuals, within-σ from the moving range) + LSL and/or USL + optional target + optional Box-Cox (λ fixed or ML-estimated) or Johnson (SB/SL/SU) transformation.
Cp/Cpk with 95% CIs, Pp/Ppk, within and overall σ, observed and expected ppm out of spec, a Shapiro-Wilk normality check (before/after when transformed), and a histogram with spec limits and both normal overlays.
Cp ≫ Cpk means the process is off-center: re-centering recovers capability without reducing variation. Expected ppm far from observed ppm suggests non-normality.
With Box-Cox or Johnson, all indices and the histogram are on the transformed scale; the observed ppm still counts the raw data against the raw limits.
Prefer Box-Cox when it normalizes (simpler, more stable in the tails); reach for Johnson when the transformed Shapiro-Wilk still rejects.