Capability Index Discrepancy and an Out-of-Control Grinding Process
A CNC grinding cell produces a bearing bore specified as 25.000 ± 0.050 mm. Control is an X̄–R chart with n = 5 and k = 25 subgroups taken over three days. Charts show subgroup 18 above the upper control limit and a run of seven points above the centerline. From all 25 subgroups the engineer computes X̄̄ = 25.012 mm, R̄ = 0.021 mm, and overall sample standard deviation s = 0.0120 mm. Using R̄/d2 (d2 = 2.326), within-subgroup sigma is 0.0090 mm; this yields Cpk = 1.40 and Ppk = 1.05. The customer's PPAP requirement is Cpk ≥ 1.33 and Ppk ≥ 1.33 from a stable process with at least 100 data points. The production manager argues the part is fine because Cpk passes and wants to submit the study. Two weeks earlier a new coolant concentrate was introduced at the same time as a night-shift staffing change. Records show subgroup 18 was taken 20 minutes after a wheel dress.
1. Interpret the difference between Cpk 1.40 and Ppk 1.05. What does it indicate about the process, and can the study be submitted?
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Cpk uses the within-subgroup sigma estimated from R̄/d2, which reflects short-term, common-cause variation inside a subgroup. Ppk uses the overall standard deviation, which includes drift between subgroups, shifts, and any special causes retained in the data. The ratio here is about 1.33 (0.0120/0.0090), meaning between-subgroup variation is a substantial part of total variation. The process is not behaving as one homogeneous stream.
Check the arithmetic: the mean is 0.012 mm above nominal, so the distance to the upper limit is 0.038 mm and to the lower limit 0.062 mm. Cpk = 0.038/(3 × 0.0090) = 1.40; Ppk = 0.038/(3 × 0.0120) = 1.05. The gap is not a rounding artifact; it is a signal.
The study cannot be submitted as conforming. The requirement is explicitly from a stable process, and the chart is not stable: one point beyond the upper control limit plus a run of seven above the centerline is a sustained shift, not a single outlier. Capability indices computed on an unstable process are not meaningful because the distribution is not predictable. Submitting Cpk 1.40 while withholding the control-chart signal would also misrepresent the process to the customer. The correct response is to investigate the shift first, then re-baseline.
2. Subgroup 18 is linked to a documented wheel dress. May it be removed and the indices recomputed? Describe the analysis and the decision rule.
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A point may be excluded only when an assignable cause is documented, the cause is understood, and there is a defensible reason it will not recur in the reported period. A wheel dress is a routine, planned event, so 'the dress caused it' does not by itself justify exclusion; if a normal dress can push a point outside the limit, the process is not in control under normal operation.
More important, removing subgroup 18 does not address the run of seven points above the centerline. That run indicates a level shift, plausibly from the coolant change or the night-shift staffing change. The data may therefore represent two different processes.
The defensible analysis is: (1) plot the data in time order and mark the coolant change and staffing change; (2) split the data at the coolant change, or analyze pre- and post-change subsets separately; (3) recompute X̄–R limits for each subset, not with the original limits; (4) check that each subset is stable using run rules and limits; (5) compute Cpk and Ppk only within a stable subset.
Decision rule: if both subsets are stable and neither shows a point beyond its own limits, report capability per condition and choose the current condition for PPAP. If instability persists, capability is not reportable; the process must be corrected and a new 100-plus point study collected after the fix. Document any excluded point with cause and evidence in the PPAP package.
3. After segregation, the best stable subset gives Ppk = 1.18 with a mean 0.010 mm above nominal. Outline the actions and what you would communicate to the customer.
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At Ppk 1.18 the process misses the 1.33 requirement, so the study cannot be submitted as a pass. Decompose the gap before acting. With overall sigma near 0.012 mm, centering the mean from 25.010 to 25.000 would move the nearest limit distance from 0.040 to 0.050 mm and raise the index to about 0.050/(3 × 0.012) = 1.39. That suggests two levers: an offset correction and a reduction in between-subgroup variation.
Actions: adjust the machine offset and verify with a short run; control wheel-dress frequency and coolant concentration; and reduce shift-to-shift differences through standardized setup checks. Then collect a fresh 125-point study with the control chart attached. Do not calculate capability on the old mixed-condition data.
Interim containment while the study is rebuilt: 100% inspection or an expanded sampling plan on the bore, with clear traceability and a documented containment level. If shipped product is at risk, notify the customer under the quality agreement rather than waiting.
Communication to the customer should state the facts: an out-of-control period associated with a coolant and staffing change, the segregated subsets, the current Ppk of 1.18, the centering and variation-reduction plan with dates, and the containment in force. Request a temporary deviation with a defined expiry and a resubmission date. Do not present the original Cpk 1.40 as representative; a customer who later discovers the control-chart signal will treat it as a data-integrity issue, which is far more costly than an honest interim plan.