Non-Normal Capability and a Two-Reactor Mixture
A specialty chemical plant produces a batch polymer. The key quality characteristic is percent conversion; the customer specification has a lower limit of 92.0% and no upper limit. A Black Belt has 180 batch records from the last six months. The overall mean is 95.8% and the sample standard deviation is 2.1%. A normal probability plot shows a left skew, and the Anderson-Darling test p-value is below 0.01. The normal-based capability report shows Cp = 0.60 and Cpk = 0.60, and the sponsor wants to stop the project because the process is hopeless. However, stratification by reactor shows Reactor 1 has a mean of 97.1% with standard deviation 1.2%, while Reactor 2 has a mean of 94.2% with standard deviation 2.7%. Reactor 2 also has 11 batches below 92%. The two reactors use different impeller designs. Production schedules Reactor 2 for the more viscous grade. The lab assay has a reproducibility standard deviation of 0.35% and no evidence of bias. The Black Belt must present the capability and next steps at a tollgate review.
1. Explain why the normal Cp and Cpk are misleading and how capability should be reported.
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Why normal capability is misleading: The Anderson-Darling p below 0.01 and left skew reject normality. Cp and Cpk assume a normal distribution and a symmetric spread; with a lower-only specification and a left tail, the normal Cpk = 0.60 can misstate the actual fraction below 92.0%. The skew may also come from a mixture of reactors, not a single process. How to report: Fit a suitable distribution such as Weibull, gamma, or lognormal, or transform the data, then report percentile capability and Ppk. Also compute the observed proportion below 92.0% with a confidence interval. Stratify by reactor and present both. Reasoning: Capability is not a single number; it must match the distribution and the process structure. The sponsor needs the predicted defect rate, not an invalid normal index.
2. Interpret the stratification result. What are the next steps and what should not be done?
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Interpretation: The stratification is the key result. Reactor 1 with mean 97.1% and standard deviation 1.2% is much more capable; Reactor 2 with mean 94.2%, standard deviation 2.7%, and 11 batches below 92% is the problem stream. The overall skew is likely a mixture of two reactor distributions, and Reactor 2's impeller and viscosity combination is a candidate special cause. Next steps: Do not average the reactors into one capability claim. Confirm the assay is capable; reproducibility 0.35% is small relative to Reactor 2's 2.7% standard deviation. Stratify Reactor 2 by grade, impeller, batch size, and raw-material lot; run a focused investigation and possibly a DOE or confirmation runs. Protect the customer with screening or adjusted scheduling. Reasoning: Aggregating two processes with different means and variances creates a misleading distribution. The improvement opportunity is Reactor 2, not a plant-wide hopeless conclusion.
3. The sponsor insists on a single overall Ppk number for the customer. How should the Black Belt respond?
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Response to the sponsor: I can provide an overall Ppk, but it must be labeled as a mixture and accompanied by the by-reactor results. A single overall number would hide the 11 nonconforming Reactor 2 batches and could mislead prioritization. Approach: Report the overall observed defect rate and a model-based Ppk with a confidence interval, then show Reactor 1 and Reactor 2 separately. If a customer requires one number, use the worst-case reactor or a weighted capability that reflects production mix, and state the method. Reasoning: The purpose of capability reporting is decision-making. A pooled index that averages a capable and an incapable stream violates the homogeneity assumption and can make the process look better or worse than either stream. The Black Belt should refuse to present a misleading single index without context.