Cp / CpkProcess Capability Index

Cp and Cpk are short-term process-capability indices — dimensionless numbers that compare the natural variation of a process to the width of its specification. Cp asks 'is the process spread narrow enough to fit inside the spec?' and ignores where the process is centred. Cpk asks the same question but penalises a process whose mean is off-centre toward one specification limit.

Both indices originated in automotive SPC practice (Ford's Q-101 in the 1980s, then AIAG's SPC manual) and were adopted into ISO 22514 for general manufacturing, into FDA's 2011 Process Validation guidance (where Cpk drives PPQ acceptance), and into ICH Q10 as the standard language for continued process verification.

Cp = (USL − LSL) ÷ (6σ_within). Cpk = min[(USL − μ) ÷ 3σ_within, (μ − LSL) ÷ 3σ_within]. The σ_within is the within-subgroup standard deviation — typically estimated from the average range of subgroups divided by d2, or from the pooled within-subgroup variance. This is short-term, instantaneous spread — not the long-term spread you see over weeks of operation, which is captured by the parallel indices Pp and Ppk (using overall standard deviation s).

The 6σ in the denominator represents ±3σ around the mean — the interval that contains 99.73 % of a normal distribution. A Cp of 1.0 means the process spread exactly equals the spec width — 0.27 % of output is theoretically out of spec even if the process is perfectly centred. Cpk = 1.0 means the closer spec limit is exactly 3σ from the mean — same 0.27 % theoretical OOS rate, but only if the process stays centred.

The relationship Cpk ≤ Cp always holds. The gap between them tells the engineer what to fix. If Cp = 1.50 and Cpk = 0.80, the process has plenty of room inside the spec but is badly off-centre — the fix is to recentre the mean, not to reduce variation. If Cp = 0.80 and Cpk = 0.80, the process is centred but the spread is too wide for the spec — the fix is to reduce variation, not to chase the mean. If both are low, you have both problems and need to triage.

The k in Cpk is the centring correction: k = |μ − (USL+LSL)/2| ÷ (USL−LSL)/2, and Cpk = Cp × (1 − k). When k = 0 the process is perfectly centred and Cpk = Cp; when k = 1 the mean sits exactly at one spec limit and Cpk = 0.

• The process is in statistical control — special-cause variation removed. Calculating Cpk on an out-of-control process produces a number, but the number has no predictive value.
• The data are approximately normal. For non-normal data (skewed, bounded, count) use a Box-Cox transformation, a fitted non-normal distribution, or quantile-based capability indices (ISO 22514-4).
• The subgroup structure is rational — items within a subgroup share common cause, items across subgroups capture special cause.
• Sample size is adequate — typically 25 subgroups of 4–5 (≥100 measurements) before Cpk is reported.
• The measurement system is capable — MSA (gauge R&R) consumes less than 10 % of the spec width, ideally less than 30 % of process variation.

FDA's 2011 PV guidance is silent on a specific numeric threshold but expects the protocol to define one, justify it, and demonstrate that PPQ batches meet it. Industry practice settled on Cpk ≥ 1.33 as the minimum acceptance for critical quality attributes during PPQ, with higher targets for life-critical attributes (potency, assay, content uniformity). The protocol must also pre-specify what happens when Cpk falls short — a hold, an extended PPQ, or a Stage 1 (design) revisit.

In Stage 3 CPV the metric shifts: short-term Cpk gives way to long-term Ppk, computed across rolling 12-month windows, and the trend matters more than the absolute value. A Ppk that drifts from 1.50 to 1.10 over six months is a signal to act even if it has not yet crossed the acceptance floor.

1. Cpk reported on out-of-control data — control charts not run before the capability calculation.
2. Subgroup size and frequency arbitrary, not derived from a rational-subgroup analysis.
3. Non-normal data forced through the Cpk formula without transformation or non-normal index.
4. MSA never performed — the measurement system contribution to apparent process variation is unknown.
5. Cpk and Ppk used interchangeably — short-term and long-term spread conflated.
6. Specification limit one-sided (e.g. dissolution NLT 80 %) but two-sided Cpk reported by inventing a meaningless upper limit.
7. PPQ acceptance criterion '≥ 1.33' applied to a CQA where 1.33 leaves an unacceptable patient-safety margin.
8. Cpk improvement claimed after a process change without a recalculation under post-change control conditions.

• Every numeric IPC and release attribute auto-feeds the SPC engine; control limits derived from the validation data, not the spec.
• Cpk and Ppk are computed per attribute per rolling window with normality testing and transformation applied automatically.
• PPQ protocol templates pre-specify the Cpk acceptance and the action plan if missed — built into the validation pack.
• CPV dashboards trend Ppk by attribute; trend toward floor opens a CAPA ticket before an OOS event.
• Specifications carry a one-sided / two-sided flag so Cpk uses the correct numerator without manual intervention.
• Annual Product Quality Review (211.180(e)) pulls the same Cpk / Ppk tables — no separate report.