Process capability indices

The formula

Potential capability, assuming the process sits dead center:

Cp = (USL − LSL) / (6·σ)

Spec width divided by process width. A Cp of 1.0 means the ±3σ spread of the process exactly fills the tolerance band — zero margin.

Actual capability, accounting for where the mean really sits:

Cpk = min( (USL − μ) / (3·σ) ,  (μ − LSL) / (3·σ) )

Distance from the process mean to the nearest spec limit, measured in 3σ units. The min() is the whole point — the near wall is the one you hit.

The two are related through the centering fraction k (how far off-center the mean is, as a fraction of the half-tolerance):

Cpk = Cp · (1 − k)      where  k = |m − μ| / ((USL − LSL)/2),  m = (USL + LSL)/2

Pp and Ppk are the same formulas with a different σ. Cp/Cpk use the within-subgroup estimate — typically σ̂ = R̄/d₂ from a control chart (d₂ = 2.326 for subgroups of 5, 1.128 for a moving range of 2). Pp/Ppk use the plain overall sample standard deviation s = √( Σ(xᵢ − x̄)² / (n−1) ) of everything you made ^[7]^[8]^[9].

The index-to-fallout conversion, for a normal, centered, in-control process:

Cp = 1.00  →  2700 ppm outside spec
Cp = 1.33  →  ~63 ppm
Cp = 1.67  →  ~0.6 ppm
Cp = 2.00  →  ~0.002 ppm

For an off-center process, use the near-side tail: Cpk = 1.33 puts the mean 4σ from the nearest limit, about 32 ppm out that side ^[2].

Where you meet it

PPAP submission for an automotive or aerospace customer. The Production Part Approval Process paperwork wants an initial capability study — commonly Ppk ≥ 1.67 on the short pre-production run, Cpk ≥ 1.33 ongoing — before your part number ships in quantity ^[1]^[7]. No index, no source approval.
On the CMM bench after a first article. Thirty parts off the new fixture, a critical bore diameter, and the quality engineer runs the histogram. Whether the machine keeps the job or the process gets re-tooled turns on whether that Cpk clears 1.33.
At the ATE station. Electrical test limits are spec limits like any other. A capability study on measured gain or offset across a lot tells you whether you're screening a capable process or shipping the lucky half of an incapable one.
At the production readiness review. Someone from the customer side asks "what's the Cpk on the safety-critical characteristics?" — and expects 1.67, not a story ^[1].

How it works

The index is a margin statement. Cpk = 1.0 means the mean sits exactly 3σ from the nearest limit; every increment of 0.33 buys roughly another sigma of guard band. That is all it is — which is why customers like it (one number, one threshold) and why it gets abused (one number hides a lot).

Cp vs Cpk is a free diagnosis. High Cp with low Cpk means the spread is fine and the process is just aimed wrong. Re-centering the mean — a tool offset, a setpoint change — is usually cheap. Closing the spread almost never is. Check both numbers before spending money.

Cpk vs Ppk is a stability check. Cpk is built on within-subgroup sigma, so it captures only short-term, common-cause variation — the best the process can do. Ppk is built on the overall sigma of everything produced, drift and shift included — what the customer actually receives ^[7]^[8]^[9]. When Ppk lags Cpk badly, the machine is fine and the process is wandering: tool wear, lot-to-lot material changes, warm-up drift. Fix the wandering before touching the machine.

The number is meaningless without statistical control. Capability math assumes a stable process; run the control chart first. An index computed over a process that's shifting is a fiction with two decimal places ^[1]^[2].

Normality is load-bearing. The ppm conversions above come straight off the normal tail. Skewed or bounded data — runout, flatness, leak rate, anything one-sided at zero — will lie to you. The honest fixes are a Box-Cox transform or percentile-based indices that replace 6σ with the observed 99.865th-minus-0.135th percentile spread ^[2]. One-sided specs get the one-sided indices, Cpu = (USL − μ)/(3σ) or Cpl = (μ − LSL)/(3σ), not a fake two-sided calculation.

Small samples make confident-looking nonsense. A Cpk of 1.33 computed from 30 parts carries a 95% confidence interval of roughly 0.97 to 1.69 — spanning "reject" to "excellent." At n = 100 it tightens to about 1.13 to 1.53. NIST's guidance is a minimum of 50 values and preferably 100 or more for a capability study ^[2]. Anyone quoting Cpk to three decimals from a 30-piece run is reporting noise.

And the classic gaming move: measure 30 consecutive parts from one warmed-up setup, use within-subgroup sigma, and the index flatters everyone. That's why initial studies demand Ppk over a run long enough to see real variation.

Worked check, so the arithmetic is concrete: spec 10.00 ± 0.05 mm, process mean 10.01, σ = 0.01. Cp = 0.10/0.06 = 1.67, but the mean sits 0.04 from the upper limit, so Cpk = 0.04/0.03 = 1.33 — about 32 ppm over the top limit, essentially nothing under the bottom.

History

The vocabulary is American; the indices are Japanese. Western Electric's 1956 Statistical Quality Control Handbook — the Bell System book that codified control charting — used the phrase "process capability" but never boiled it down to an index ^[3]. That same year, at a Japanese Society for Quality Control conference, M. Kato and T. Otsu proposed converting Bell's percentage-based capability measure into a single ratio: the Cp index, built to rank the machines on their own factory floor ^[3]^[4].

The fix for off-center processes came in 1967, when T. Ishiyama proposed an index of process bias at another JSQC conference — the ancestor of Cpk, whose trailing "k" is not a variable but an abbreviation of katayori, Japanese for offset or bias ^[3]^[4].

The indices crossed the Pacific with the auto industry. Victor E. Kane's 1986 paper "Process Capability Indices" in the Journal of Quality Technology laid out Cp, Cpu, Cpl, k, and Cpk for a Western audience, noting that Japanese industry was already using all five and that U.S. automakers had begun adopting them ^[5]^[6]. Ford wrote capability requirements into its Q-101 supplier standard, and when Ford, GM, and Chrysler merged their quality systems into QS-9000 in the 1990s, the AIAG statistical process control manual fixed the convention still in force under IATF 16949: Cp/Cpk from within-subgroup sigma, Pp/Ppk from overall sigma ^[7]^[8].

Motorola's Six Sigma program then turned the index into a slogan: design margin of Cp = 2.0, an assumed long-term mean shift of 1.5σ, hence a delivered Cpk of 1.5 — which is where the famous 3.4 defects per million comes from, the one-sided normal tail beyond 4.5σ ^[1]^[10]. Taguchi's target-value thinking later produced the Cpm variant, which penalizes distance from the target rather than distance from the limits ^[1]^[3].

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →

The formula

Where you meet it

How it works

History

Sources

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