Cp and Cpk vs Pp and Ppk, What Capability Indices Hide

Pull-quote: “A capability index is a point estimate wearing the costume of a verdict. A Cpk of 1.41 from thirty parts is honestly stated as: somewhere between roughly 1.0 and 1.8, we think.”

Why this matters

Capability indices are the currency of supplier quality. PPAP submissions carry them, customer scorecards rank on them, and sourcing decisions cite them to two decimal places. The trouble is that the decimals are fiction unless two questions are answered first: which variation did you measure, and how much data stands behind the number. The Cp and Cpk versus Pp and Ppk distinction answers the first question. Confidence intervals answer the second. Most capability reports answer neither, and the numbers travel anyway.

Four indices, two kinds of variation

All four indices compare the voice of the process to the voice of the customer: six standard deviations of process spread against the width of the tolerance. They differ in which standard deviation they use. Cp and Cpk estimate sigma from within-subgroup variation, typically through the average range, so they describe the short-term spread the process is capable of under one setup, one lot, one stretch of stable running. Pp and Ppk use the overall standard deviation of every measurement in the study, which absorbs every setup change, lot change, and drift along the way. The k versions penalize off-center running; the plain versions describe a centering nobody actually has.

The gap between Cpk and Ppk is itself a measurement. A process holding Cpk 1.8 and Ppk 1.1 is a capable process being run unstably: within-subgroup spread is fine, and something between subgroups, setups, lots, shifts, tools, is moving the center. That diagnosis is more useful than either number alone, and it costs nothing to read.

value
  │  subgrp 1   subgrp 2   subgrp 3   subgrp 4
  │   ▪▪▪
  │    ▪▪        ▪▪▪                              within-subgroup
  │               ▪▪         ▪▪▪                  spread → Cp / Cpk
  │                           ▪▪        ▪▪▪
  │                                      ▪▪
  │  └───────── overall spread → Pp / Ppk ────┘
  └──────────────────────────────────────────► time
       tight within subgroups; the center wanders

What all four hide

Three things no index reports. Stability: capability arithmetic assumes a stable process, and an index computed over an out-of-control run describes the past, not the future. Normality: mapping six sigma to a defect rate leans on a distributional assumption that machining, molding, and plating routinely violate, so tail estimates deserve suspicion before belief. And sampling error, the quietest one: an index is a statistic, and a statistic from thirty parts has error bars wide enough to change the verdict.

The interval is the honest answer

The approximate confidence interval for Cpk widens quickly as sample size falls. For a measured Cpk of 1.33:

Read the first row again. A supplier reporting 1.35 against a 1.33 requirement from a thirty-piece study has reported a coin flip, not a capability. Below roughly a hundred measurements, the interval spans pass and fail for most acceptance thresholds in common automotive use, which means the study is not finished, however clean the point estimate looks. The operational fix is procedural: compute Cp, Cpk, Pp, and Ppk with confidence intervals attached by default, and gate capability studies on demonstrated chart stability first, because an index over an unstable run is an anecdote with decimals.

Closing

Read four numbers, not one, and read the interval around each. Cpk states the short-term entitlement. Ppk states delivered reality. The gap between them tells you where to work: on the process spread, or on whatever moves between subgroups. And the confidence interval tells you whether you know any of this yet. If it spans your acceptance threshold, collect more data; no arrangement of decimal places will finish the study for you.