What the study actually establishes
The weld example contains 100 observations in twenty subgroups of five. Cp = 2.576, Cpk = 1.865, Pp = 1.165 and Ppk = 0.843. No observed reading is outside 0.400–0.600 mm, but the time sequence contains a level change. Those are separate findings and should remain separate in the report.
The engineer's job is to connect each finding with evidence and an appropriate action. Passing the collapse window does not settle joint strength or sealing. A high calculated Cpk does not establish stability. A lower Ppk does not, by itself, identify which machine setting changed.
What the four numbers cannot settle
The standard six-sigma-width interpretation assumes a suitable distribution model. A stable but strongly skewed characteristic may need a justified transformation or a nonnormal capability method. A combined sample from two operating levels may resemble a mixture rather than a single normal distribution. Fitting a bell curve over it does not make that model appropriate.
A normal probability plot and process knowledge can help assess the model, but a small-sample normality test is not an automatic permission slip. Check whether the characteristic is bounded, rounded, censored or mixed, and whether consecutive readings are dependent. Autocorrelation can affect both the apparent within spread and the information contained in the sample.
For a characteristic with only an upper specification, use the upper one-sided index, such as 1.865 or 0.843 under the corresponding method. Do not invent a lower specification to obtain Cp or Pp. A target, zero, detection limit or plotting-axis boundary is not automatically an LSL.
A value such as 1.33 is a commonly encountered benchmark, not a universal acceptance rule. The required index, study design, confidence level and minimum criterion come from the applicable customer or quality procedure. A rounded point estimate just above a threshold can conceal substantial sampling uncertainty; confidence bounds may matter for acceptance.
“No rejects,” “Cpk above target” and “stable process” are three different claims. Each needs its own evidence. For our weld example, there are zero observed out-of-spec readings, a calculated within-based Cpk of 1.865, and clear evidence of changing means. Reporting only the second statement would hide information important to the engineering decision.
Reproduce and report the study
In Excel, keep the five sample columns together for each row. Calculate a row average and MAX minus MIN. Average the twenty ranges and divide by 2.326 for the within estimate used here. For overall standard deviation, apply STDEV.S to all 100 sample cells. Exclude subgroup numbers, row averages and ranges from that selection.
Use the grand mean and entered specifications in the four formulas above. In Minitab or another statistical package, select subgroup size five and confirm the within estimator. A default pooled-standard-deviation method can produce a different Cp/Cpk from R̄/d₂. That is a method difference to resolve, not evidence that one program cannot divide correctly.
The Mechatrovich capability trial accepts a small flat set of observations and uses overall sample standard deviation. Its Pp/Ppk results demonstrate the overall calculation; it does not reproduce the within estimate from this 100-reading study. Use the downloadable data and a suitable subgroup-capability workflow to reproduce all four values.
A useful report records the measured characteristic and units, specifications and their authority, machine or stream identity, collection period, sample size, subgroup logic, measurement suitability, estimator, graphs, indices and interpretation limits. Include the observed out-of-spec count separately from any model-based expected rate. Document the investigation and the conditions for a follow-up study.
A report that explains the apparent contradiction
Here is a defensible way to describe this teaching case: “The study contains 100 collapse readings in 20 consecutive-cycle subgroups of five. Specifications are 0.400 and 0.600 mm. The within estimate is based on average range divided by 2.326. Cp is 2.576 and Cpk is 1.865. Overall sample standard deviation gives Pp 1.165 and Ppk 0.843. No observed reading is outside specification. The time sequence and provisional average chart show a level change; therefore these pooled indices are descriptive and do not establish stable future performance.”
That explanation answers why the numbers differ without hiding either pair. It also separates the statistical finding from product disposition. The team still needs the validated acceptance criteria and actual joint checks before deciding whether the affected units can be released. The difference between an average-chart signal and a failed product requirement is especially important when every observed cycle passed the collapse window.
Suppose a follow-up study after a verified correction remains near one consistent operating level. Collect enough data to cover the conditions relevant to the intended claim, rather than choosing only the easiest hour. Recalculate the within and overall estimates from that new study. If their difference narrows, check that the time graph supports the interpretation and that the sampling window includes relevant changes in material, shift or operating condition.
If the gap remains large, examine how the subgroups were formed and whether hidden streams remain. If Cp is high but Cpk is low, examine centering and the closer limit. If both within indices are low, reducing local variation may matter more than moving the mean. These are different engineering problems; one generic instruction to “improve capability” does not tell the team which evidence to collect.
Finally, retain the before-and-after boundaries. Combining pre-correction and post-correction observations can recreate a wider overall distribution and obscure what improved. Conversely, silently deleting the earlier period makes the report look cleaner while losing the explanation. Present the original study and the follow-up separately, with the intervention documented and each conclusion tied to its own data.
References and calculation convention
This article uses the conventional within/overall definitions documented by Minitab, the average-range estimator and X̄–R factors in the NIST/SEMATECH handbook, and the ordinary overall sample standard deviation. The case and graphs are original illustrative calculations.