Good parts today, but tomorrow?
A customer approving a new shaft asks for evidence that the bearing seat, 48.00 ± 0.10 mm, will stay within tolerance in series production. Fifty good parts prove that fifty parts were good. The customer wants to know whether the process is capable: whether its normal spread fits comfortably within the tolerance, with room to spare. Capability indices answer that question with a single number each.
Cp: does the spread fit?
Cp compares the width of the tolerance with the width of the process spread, taken as six standard deviations:
Cp = (USL - LSL) / 6σ
Here σ is the short-term standard deviation, estimated from within subgroups as σ̂ = R̄ / d2. Cp = 1 means the ±3σ spread exactly fills the tolerance. Cp = 1.33 means the tolerance is about a third wider, corresponding to ±4σ.
Cp ignores where the process is centred. It describes potential: what the process could do if it were perfectly centred.
Cpk: does it fit where it actually is?
Cpk measures the distance from the mean to the nearest tolerance limit, in units of three standard deviations:
Cpk = min(USL - μ, μ - LSL) / 3σ
where μ is the process mean, in practice X̿. Cpk is never larger than Cp, and is equal to it only when the process is centred. The gap between them shows how much is lost through poor centring.
The figure shows two processes with the same spread and the same Cp. The centred one has the same Cpk; the one shifted towards the upper limit has a Cpk half as large.
Worked example
Illustrative values from 25 subgroups of five shafts: X̿ = 48.004 mm and R̄ = 0.050 mm. With d2 = 2.326 for n = 5, σ̂ = 0.050 / 2.326 ≈ 0.0215 mm.
- Cp = 0.20 / (6 × 0.0215) = 0.20 / 0.129 ≈ 1.55
- Distance to USL: 48.10 - 48.004 = 0.096. Distance to LSL: 48.004 - 47.90 = 0.104. The nearest is 0.096.
- Cpk = 0.096 / (3 × 0.0215) = 0.096 / 0.0645 ≈ 1.49
If the same process drifted to a mean of 48.040, Cp would still be 1.55 but Cpk would be 0.060 / 0.0645 ≈ 0.93.
Pp and Ppk: the overall picture
Pp and Ppk use the same formulas, but with the overall standard deviation s of all individual values instead of the short-term estimate:
Pp = (USL - LSL) / 6s and Ppk = min(USL - X̄, X̄ - LSL) / 3s
The overall standard deviation also includes movement between subgroups: set-ups, material batches, shifts. In the example, if s = 0.026 mm, then Pp = 0.20 / 0.156 ≈ 1.28 and Ppk = 0.096 / 0.078 ≈ 1.23.
Pp and Ppk are called performance indices. They describe what the process has actually delivered over the period, and they are typically used for initial studies of a new process. Comparing Cpk with Ppk is informative: if Ppk is clearly lower, the process moves between subgroups and is not fully stable.
What counts as capable?
There is no universal threshold. In the automotive supply chain, a common requirement is Ppk above 1.67 for an initial study and Cpk of at least 1.33 for ongoing production, as described in the AIAG PPAP manual. If the process is normally distributed, a Cpk of 1.33 means the nearest limit is four standard deviations away, roughly 30 parts per million beyond it; a Cpk of 1.00 means three, roughly 1,350 per million. Always check the requirement in your customer’s documents.
Conditions and common mistakes
- Stability first. Capability predicts the future only if the process is stable. Check the control chart before calculating; see SPC.
- Know the measurement system. Measurement variation is included in σ, so a poor gauge lowers every index. Run a measurement system analysis first.
- Check the distribution. The formulas assume a normal distribution. Skewed characteristics, such as runout or flatness, need a fitted distribution or a percentile method.
- Use enough data. An index from 30 values has a wide uncertainty; 100 or more is common practice.
- Say which index you mean. A Cpk calculated with overall sigma is really a Ppk. Report the method with the number.
In short
Cp tells you whether the spread fits, Cpk whether it fits where the process actually runs, and Pp and Ppk how it has performed overall. Read them together, and only for a stable process measured with a system you trust.
