Capability Demonstration With a Small Sample
A quality engineer is qualifying a new grinder. A 30-piece study gives a mean of 12.503 mm and a sample standard deviation of 0.0040 mm on a characteristic with specification limits 12.500 ± 0.015 mm. The calculated Cpk is 1.00. The customer requires Ppk ≥ 1.33. The project manager says, "The sample Cpk is 1.00, close enough; we'll monitor it in production." Another engineer notes that the mean is off-center. The grinder is scheduled for production release in two weeks, and a full process-validation run would take another week.
1. Calculate Cp and Cpk, and explain what each says about the process.
Read the worked answer
USL = 12.515 mm, LSL = 12.485 mm, width = 0.030 mm, σ = 0.0040 mm.
- Cp = 0.030 / (6 × 0.0040) = 0.030 / 0.024 = 1.25.
- Upper side = (12.515 − 12.503) / (3 × 0.0040) = 0.012 / 0.012 = 1.00.
- Lower side = (12.503 − 12.485) / 0.012 = 0.018 / 0.012 = 1.50.
- Cpk = 1.00.
Cp = 1.25 says the spread alone would allow a capable process if centered. Cpk = 1.00 says the process is not centered: the mean is 0.003 mm above target, or 0.75σ. The upper specification is the limiting side. The project manager is reading the point estimate as "close enough," but the process is both off-center and below the requirement.
2. Construct an approximate 95% confidence interval for Cpk and state whether the requirement is demonstrated.
Read the worked answer
Use the approximate standard error:
SE(Cpk) ≈ sqrt[1/(9n) + Cpk²/(2(n−1))] = sqrt[1/(9×30) + 1.00²/(2×29)] = sqrt(0.00370 + 0.01724) = sqrt(0.02094) = 0.145.
95% interval ≈ 1.00 ± 1.96 × 0.145 = 0.72 to 1.28.
The entire interval lies below 1.33. Even the upper bound does not demonstrate the requirement, and the lower bound is far below it. The 30-piece study is too small and too poorly centered to support release. The point estimate of 1.00 is not "close enough"; it is a nonconforming estimate with wide uncertainty.
3. What must happen before production release, and what centering and variation targets are needed?
Read the worked answer
Release should be delayed or conditional pending a valid capability study. Actions:
- Center the process to 12.500 mm. If only centering were improved, Cpk would rise to about Cp = 1.25 - still below 1.33.
- Reduce variation. For a centered process to achieve Cpk = 1.33: σ ≤ 0.015 / (3 × 1.33) = 0.00376 mm. That is about a 6% reduction from 0.0040 mm.
- Collect a proper study: at least several days or lots, covering shifts, operators, and material lots, with a sample large enough to bound Ppk credibly. Ppk should reflect long-term variation, not just 30 consecutive pieces.
- Verify measurement system before trusting either the mean or the sigma.
A short-term Cpk of 1.00 with a confidence interval of 0.72–1.28 is not a release rationale. The correct judgment is to fix centering and variation, then re-demonstrate with adequate data.