An SPC Chart That Signals Every Afternoon
After a six-month DMAIC project, fill-volume variation on a bottling line fell from σ = 2.8 ml to σ = 1.10 ml against a 500 ± 6 ml specification, and the team reports Cpk = 1.82. The control plan specifies an Xbar-R chart with subgroups of five consecutive bottles pulled every two hours, with limits frozen from a 25-subgroup baseline of the improved process. The reaction plan reads: 'Operator: if any point is outside the control limits, adjust the filler head and restart.' The nozzle wears, and mean fill drifts upward by about 2.4 ml across each eight-hour shift before maintenance resets it at shift change. In practice the operators see the chart go out of control every afternoon and now adjust at the first two-of-three trend; the last three weekly Cpk values have fallen to 1.3–1.5. Production wants the chart removed because 'it creates the variation it is supposed to detect'.
1. Explain why the Xbar chart signals almost every shift, using the subgroup data given. Is the process out of control?
Read the worked answer
The limits are built from the wrong variation. Five consecutive bottles pulled in a couple of minutes measure short-term repeatability only. If within-subgroup σ ≈ 0.40 ml, the Xbar limits are ±3 × 0.40/√5 = ±0.537 ml around the centre line.
The process mean moves far more than that between subgroups. Drift is 2.4 ml over 8 hours, and with subgroups every 2 hours the subgroup means climb by about 0.6 ml per interval - larger than the entire upper limit. From the second or third subgroup of a shift the chart exceeds its limits as a matter of arithmetic, not of process misbehaviour. That is exactly the 'every afternoon' pattern reported.
Is it out of control? Yes, but the signal is a genuine, predictable special cause - tool wear - not randomness, and not something an operator should chase point by point. The chart conflates two variation components: within-subgroup (repeatability, 0.40 ml) and between-subgroup drift (about 0.69 ml if the 2.4 ml drift is roughly linear, since 2.4/√12 ≈ 0.69). Total observed σ of 1.10 ml is dominated by the drift and other between-shift effects, not by bottle-to-bottle noise.
The design flaw is rational subgrouping: consecutive parts cannot see the drift they are supposed to detect, so the limits are far too tight for the Xbar statistic.
2. The team's Cpk of 1.82 was computed from total σ. What would happen if the within-subgroup σ were used, and what capability figures should be reported?
Read the worked answer
Using σ_within = 0.40 ml: Cpk = 6 / (3 × 0.40) = 5.0. That is the classic inflated figure produced by short-term-only sigma; it would be reported to a customer as a spectacular process and would be pure fiction. The σ_within estimate deliberately excludes drift, so it can never be used for capability against a specification.
Using σ_total = 1.10 ml: Cpk = 6 / (3 × 1.10) = 1.82 - the number the team reported, and the correct one for the specification decision. Report it alongside Ppk, which uses the overall standard deviation including between-shift and between-batch shifts; if Cpk and Ppk diverge materially, the process is not stable between subgroups, which is exactly this case.
Value of removing the drift: if nozzle wear were compensated and drift eliminated, σ would fall to about √(1.10² − 0.69²) = 0.85 ml, giving Cpk = 6 / (3 × 0.85) ≈ 2.3. That is a large capability gain available from a maintenance and compensation change rather than from another DOE - quantify it before the team spends more project budget on variation reduction.
Report structure: state which sigma was used, on what data, over what period, and include a control chart of the data behind the capability number. A Cpk without its chart and its sigma definition is not evidence.
3. Rewrite the reaction plan and the control strategy. How do you stop the over-adjustment, and how do you hand this over so the gains hold?
Read the worked answer
Separate the two signals the chart is mixing.
- Drift (wear): handle preventively, not reactively. Define a nozzle replacement trigger on fill count or run hours, not on a chart point, and compensate the mean offset on a verified wear curve at fixed intervals. If a monitoring chart is wanted for drift, use an EWMA (λ ≈ 0.2) on individual bottles or subgroup means, which detects small sustained shifts early and does not fire on single-point noise.
- Spikes (nozzle clog, valve fault, product change): keep an individuals or Xbar chart sized to catch them, with subgroups taken across the shift (e.g. one bottle every 30 minutes, combined into a rational subgroup) rather than five consecutive bottles, so between-subgroup drift is inside the limits instead of outside them.
Stop the over-adjustment explicitly. Operators adjusting on two-of-three trends are adding variance - classic tampering on a drifting process. Rewrite the reaction plan into three named triggers with owners: investigate (single point beyond limits after a spike check), compensate (EWMA drift signal - apply the documented offset, no trial-and-error), escalate (two consecutive out-of-limit points or any out-of-spec bottle - stop, quarantine, call maintenance). Ban adjustment on trend rules unless the EWMA rule has fired.
Handover to hold the gain:
- Update the control plan, FMEA detection controls and the operator SOP with the new triggers and a photograph of the guardbanded limits.
- Train each shift, then audit the control plan at 30/60/90 days - check that adjustments are being logged against the documented triggers, which is the real test of whether over-adjustment stopped.
- Track two metrics for six months: rolling Cpk (with sigma definition stated) and the number of adjustments per 1,000 bottles, which should fall as capability holds.
- Hand ownership to the line lead with a named process owner and a documented response to the 'remove the chart' complaint: the chart was mis-specified, not useless.