Capability Indices With an Unstable Range Chart

Within-subgroup (short-term) sigma

sigma_within = Rbar / d2 = 2.4 / 2.326 = 1.032 mL

• Cp = (USL - LSL) / (6 sigma) = 10 / (6 x 1.032) = 1.62
• Cpk = min(USL - Xbar, Xbar - LSL) / (3 sigma) = min(505 - 500.4, 500.4 - 495) / (3 x 1.032) = min(4.6, 5.4) / 3.096 = 1.49

Overall (long-term) sigma = 1.9 mL

• Pp = 10 / (6 x 1.9) = 0.88
• Ppk = 4.6 / (3 x 1.9) = 0.81

Interpretation:

• The process is centred well - Xbar is only 0.4 mL off nominal, so Cp and Cpk are close and centring is not the problem.
• The two pairs tell opposite stories: 1.49 (good) vs 0.81 (poor). That gap is the whole finding. Cp and Ppk differ only in which sigma they use, so a ratio of 1.9/1.03 = 1.84 means the subgroup means are themselves moving around far more than the within-subgroup scatter can explain. Decomposing: sigma_between = sqrt(1.9^2 - 1.03^2) = sqrt(3.61 - 1.06) = 1.60 mL of between-subgroup shift.
• In words: each 30-minute window looks tight, but the process level drifts from window to window - a classic signature of a process that is not in a state of statistical control. Cp assumes control; it is therefore not a valid number here, however attractive it looks.

No. The point estimate of Cpk (1.49) clears 1.33, but certifying on that number would be indefensible on three separate grounds.

1. The process is not stable. The customer's manual requires capability demonstrated from a stable process. The R chart has an out-of-control point, which means the estimate of sigma_within - the denominator of Cp and Cpk - is computed from a process that was not behaving consistently. A capability index computed on an unstable process is not a capability index; it is a snapshot of one afternoon. Capability indices assume that the distribution they describe is the distribution that will exist tomorrow.

2. The Cp/Ppk gap is unexplained process drift. A 1.60 mL between-subgroup sigma on a 10 mL tolerance is a real, physical phenomenon - fill-head-to-fill-head differences, product temperature/viscosity changes, supply pressure swings, or nozzle wear across the shift. The line lead's attribution to 'a new operator' is a hypothesis, not an investigation. Until the drift source is found, the good-looking Cp has no predictive value.

3. The measurement system fails the requirement. GR&R of 18% of tolerance is nearly double the 10% allowed. Rejecting a gauge by specification is not optional here.

What must happen first, in order:

1. Re-run the GR&R on the checkweigher after fixing it (fixturing, zeroing, and repeatability at the actual fill speed). Note that 18% of tolerance means sigma_meas = 0.18 x 10 / 6 = 0.30 mL - small relative to the 1.9 mL observed spread, so measurement error is not the explanation for the Cp/Ppk gap. Fix it anyway, because the customer requires it, but do not expect the capability numbers to change much.
2. Stratify the 25 subgroups. Plot subgroup means against time, fill head, shift, and product lot to locate the drift. The R-chart signal at the end of the sample is a starting point, not the whole story.
3. Eliminate the special cause, then collect a fresh 25 subgroups with no out-of-control signals on either chart.
4. Only then report Cp/Cpk - and report a confidence interval alongside the point estimate. With 125 observations, a Cpk of 1.49 has a lower confidence bound well under 1.33, so even a clean study should not be presented as a bare number.

On deleting the point: there is a legitimate practice of removing a point when an assignable cause has been found and documented and the point is outside the period of interest - but that is a data-handling decision with a paper trail, not an eraser. Here the assignable cause is a guess ('new operator'), the point sits inside the most recent shift, and the Cp/Ppk gap shows drift well beyond that single subgroup. Deleting it would remove the symptom while the disease is untouched, and it would leave a capability study that looks clean precisely because the evidence was discarded. Do not do it.

There is also a mechanical reason it would not help: removing one range value from 25 changes Rbar from 2.4 to about 2.5 (the remaining ranges average slightly higher), so Cp moves down marginally. The exercise would produce almost no numerical benefit at the cost of the entire credibility of the study.

What to tell the customer this week - be specific and forward-looking rather than defensive:

• Report the data honestly: 25 subgroups, X-bar chart in control, one R-chart signal, Cp 1.62 / Cpk 1.49 within-subgroup, Pp 0.88 / Ppk 0.81 overall, gauge GR&R 18% and therefore out of compliance.
• State the finding plainly: the process is centred and locally tight but drifts between 30-minute windows by roughly 1.6 mL, and the gauge must be improved.
• Commit to a dated containment plan: 100% checkweigh verification on outbound pallets in the interim, GR&R re-study within two weeks, drift investigation against fill head and product lot within four weeks, and a fresh 25-subgroup capability study submitted with a confidence interval rather than a single number.
• Do not submit 1.49 as the certified value. Offering a number you cannot defend is the fastest way to lose the account, whereas a documented containment plan with dates is what a competent supplier submission looks like.