Process Capability: Cp, Cpk, and What They Tell You
In process improvement work, two questions look similar but are fundamentally different: "Is this process stable?" and "Can this process meet its specifications?" Process control — typically assessed with control charts — answers the first question. Process capability — measured by Cp and Cpk — answers the second. Confusing the two is one of the most common mistakes in quality management.
Process Control vs. Process Capability
A stable process is one whose variation is predictable. Control charts track this by plotting output measurements over time and flagging when variation exceeds the bounds expected from random causes alone. A stable process behaves consistently — but consistently could mean consistently bad.
A capable process is one whose natural variation fits within the specification limits set by the customer or the engineering design. A process can be stable and incapable (consistently producing output outside the spec), capable but unstable (meeting spec today but unpredictably so), or — the goal — both stable and capable.
Capability indices should only be calculated on a stable process. If the process is out of control, the indices are meaningless because the variation they measure is not predictable.
What Cp Measures
Cp (potential capability) measures the ratio of the specification width to the natural spread of the process. The formula is:
Cp = (USL - LSL) / (6σ)
Where USL is the upper specification limit, LSL is the lower specification limit, and σ (sigma) is the process standard deviation estimated from the control chart (typically as R-bar/d2 for subgroup data). The denominator, 6σ, represents the spread of 99.73% of the process output under a normal distribution.
A Cp of 1.0 means the process spread exactly equals the specification width. A Cp of 2.0 means the process spread is half the specification width — there is plenty of room. Cp tells you how much potential the process has, but it ignores whether the process is centred between the specification limits.
What Cpk Measures — and Why It Matters More
Cpk (actual capability) accounts for both the spread and the centring of the process. A process that is highly capable in terms of spread (high Cp) but running off-centre will still produce defects — Cpk captures this. The formula is:
Cpk = min[(USL - μ) / (3σ), (μ - LSL) / (3σ)]
Where μ is the process mean. Cpk takes the smaller of the two values — the distance to the nearest specification limit, divided by 3σ. A process perfectly centred between the limits will have Cpk equal to Cp. Any off-centring reduces Cpk below Cp.
A Worked Example
Suppose a machined shaft must have a diameter between 24.90 mm and 25.10 mm (LSL = 24.90, USL = 25.10, specification width = 0.20 mm). Process data from a stable process gives a mean (μ) of 25.02 mm and a standard deviation (σ) of 0.020 mm.
Cp = (25.10 - 24.90) / (6 × 0.020) = 0.20 / 0.12 = 1.67 — the process spread looks capable
Cpk = min[(25.10 - 25.02) / (3 × 0.020), (25.02 - 24.90) / (3 × 0.020)]
= min[0.08 / 0.06, 0.12 / 0.06] = min[1.33, 2.00] = 1.33
Cpk is 1.33 — capable but not excellent — because the mean is slightly above centre
The gap between Cp (1.67) and Cpk (1.33) reveals off-centring. Shifting the mean toward 25.00 mm would raise Cpk toward Cp without changing the process spread at all.
Interpreting Cpk Values
Cpk < 1.00 — incapable: the process is producing defects. Immediate action required.
Cpk 1.00 to 1.33 — marginal: the process barely meets specifications. Sensitive to any shift or drift.
Cpk > 1.33 — capable: generally accepted as the minimum target for existing processes.
Cpk > 1.67 — excellent: the standard for new processes or safety-critical applications.
Cpk > 2.00 — Six Sigma level: fewer than 3.4 defects per million opportunities (short-term)
Relationship Between Cpk and Sigma Level
A Cpk of 1.00 corresponds roughly to a 3-sigma process — about 2,700 defects per million opportunities. A Cpk of 1.33 corresponds to 4 sigma (~63 defects per million). A Cpk of 1.67 corresponds to 5 sigma (~233 defects per billion). A Cpk of 2.00 corresponds to 6 sigma (~2 defects per billion), though the famous "3.4 DPMO" figure used in Six Sigma assumes a 1.5-sigma long-term shift in the mean.
Limitations to Keep in Mind
Capability indices assume the process is stable — out-of-control variation makes the indices unreliable. They also assume the output follows a normal (Gaussian) distribution. Non-normal data requires either transformation or alternative indices (such as Pp and Ppk for overall performance, or non-parametric methods). Finally, a high Cpk does not guarantee zero defects — it estimates the probability of defects under current conditions, which can change.