Process capability: Cp and Cpk

Why it matters

A control chart says whether a process is stable; it does not say whether the stable process is good enough for the drawing. Process capability compares the natural spread of the process with the tolerance the designer allowed, in a single number that buyers, suppliers and auditors all understand. Automotive and aerospace customers routinely demand Cpk ≥ 1.33 (or 1.67 for critical features) before they approve a supplier's process.

Key ideas

Natural tolerance. For a stable, normally distributed process, almost all output (99.73 %) lies within μ ± 3σ. The width 6σ is called the natural tolerance or process spread. It belongs to the process; the specification width USL − LSL belongs to the design.

Cp: potential capability. Cp compares the two widths. It ignores where the process is centred, so it tells you what the process could achieve if it were perfectly centred.

• Cp < 1: the spread is wider than the tolerance; some out-of-spec parts are unavoidable even when centred.
• Cp = 1: spread equals tolerance; about 0.27 % out of spec when centred.
• Cp ≥ 1.33: generally accepted as capable; Cp = 2 corresponds to six-sigma quality (±6σ fits inside the tolerance).

Cpk: actual capability. Cpk measures the distance from the mean to the nearer specification limit in units of 3σ. It accounts for both spread and centring.

• Cpk = Cp only when the process is centred at the mid-point of the tolerance; otherwise Cpk < Cp.
• Cpk can be zero (mean on a limit) or negative (mean outside the limits).
• A large gap between Cp and Cpk means the cheap fix is to re-centre; Cp itself below the target means the variation has to be reduced (better machine, fixture, material or method).

One-sided specifications. If only an upper limit exists (e.g. surface roughness, runout), only Cpu is meaningful; for only a lower limit (e.g. strength), only Cpl.

Conditions for a valid study.

1. The process must be in statistical control (X̄–R charts) first; capability of an unstable process is meaningless because it cannot be predicted.
2. The data should be approximately normal; otherwise the link between index and ppm defective breaks down.
3. σ is the within-subgroup (short-term) estimate, usually σ̂ = R̄/d₂ from the R chart. When the overall standard deviation of all data is used instead, the indices are called performance indices Pp and Ppk; Ppk ≤ Cpk in practice because it includes between-subgroup drift.
4. Measurement error must be small compared with the tolerance (check the gauge first).

Cpm (Taguchi index). Uses the deviation from the target T as well as σ, so it penalises off-target processes even when they are inside the limits.

Formulas

Cp = (USL − LSL) / (6σ)

• USL, LSL = upper and lower specification limits; σ = process standard deviation (all in the same unit, e.g. mm). Cp is dimensionless.

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

• μ = process mean.

Cpk = Cp (1 − k) with k = |M − μ| / [(USL − LSL)/2]

• M = (USL + LSL)/2 = mid-point of the tolerance; k = fractional off-centring.

σ̂ = R̄ / d₂

• Short-term σ from an R chart (d₂ = 2.326 for n = 5, 2.059 for n = 4).

Fraction above USL = 1 − Φ[(USL − μ)/σ], fraction below LSL = Φ[(LSL − μ)/σ]

• Φ = standard normal cumulative distribution (from the normal table).

Cpm = (USL − LSL) / (6·√[σ² + (μ − T)²])

• T = target value.

Worked examples

Example 1 (standard): Cp, Cpk and ppm out of specification Given: shaft diameter specification 25.00 ± 0.05 mm; the stable process has μ = 25.01 mm and σ = 0.012 mm.

1. Cp = (25.05 − 24.95) / (6 × 0.012) = 0.10 / 0.072 = 1.389.
2. Cpu = (25.05 − 25.01) / (3 × 0.012) = 0.04 / 0.036 = 1.111.
3. Cpl = (25.01 − 24.95) / 0.036 = 0.06 / 0.036 = 1.667.
4. Cpk = min(1.111, 1.667) = 1.111. Check: k = 0.01/0.05 = 0.2, Cp(1 − k) = 1.389 × 0.8 = 1.111.
5. Above USL: z = 0.04/0.012 = 3.33, fraction = 1 − Φ(3.33) ≈ 0.00043 (about 430 ppm). Below LSL: z = −5.0, negligible (about 0.3 ppm).

Cp = 1.39, Cpk = 1.11; about 430 ppm oversize. Re-centring to 25.00 mm would raise Cpk to 1.39 without touching the machine.

Example 2 (GATE level): capability from control-chart data Given: an X̄–R chart (n = 5) is in control with X̿ = 50.02 mm and R̄ = 0.093 mm. Specification 50.00 ± 0.15 mm; d₂ = 2.326. Find Cp, Cpk and the largest σ that would give Cp = 1.33.

1. σ̂ = R̄ / d₂ = 0.093 / 2.326 = 0.0400 mm.
2. Cp = (50.15 − 49.85) / (6 × 0.0400) = 0.30 / 0.24 = 1.25.
3. Cpu = (50.15 − 50.02) / (3 × 0.0400) = 0.13 / 0.12 = 1.083; Cpl = (50.02 − 49.85) / 0.12 = 0.17 / 0.12 = 1.417.
4. Cpk = 1.083.
5. For Cp = 1.33: σ_max = (USL − LSL) / (6 × 1.33) = 0.30 / 7.98 = 0.0376 mm.

Cp = 1.25, Cpk = 1.08; σ must be cut from 0.040 mm to about 0.0376 mm (and the process centred) to reach 1.33. Here re-centring alone gives Cpk = Cp = 1.25, so the spread also has to be reduced.

Common mistakes

• Using control limits (or the X̄ chart's σ/√n) in place of the individual-part σ; capability is about individual parts.
• Calculating capability from an unstable process.
• Taking Cp as the actual capability of an off-centre process; always check Cpk.
• Using the half-tolerance in the Cp numerator, or 3σ in its denominator.
• Forgetting that a one-sided specification has only Cpu or Cpl.
• Mixing units (μm for σ and mm for limits).

For GATE PI

Expect numericals: Cp and Cpk from given limits, mean and σ; finding σ from R̄/d₂ first; finding the mean or σ needed for a target Cpk; fraction defective using the normal table; and conceptual MCQs on the relation Cpk ≤ Cp, centring, and the meaning of Cp = 1. Practise working quickly with z-values of 3, 4 and 4.5.

Quick check

1. USL = 10.6, LSL = 9.4, σ = 0.15, μ = 10.1 (mm). Find Cp and Cpk.
2. When does Cpk equal Cp?
3. What does a negative Cpk mean?
4. A process has Cp = 2.0. What sigma level does that correspond to when centred?

Answers: 1. Cp = 1.33, Cpk = 1.11. 2. When the mean is at the mid-point of the specification. 3. The mean lies outside a specification limit. 4. Six sigma (±6σ inside the limits).