Run Rule and Point Beyond Limits on a Bottling Line
A 12-head rotary filler at a beverage plant fills 500 mL PET bottles; the label declares 500 mL and the legal minimum is 497 mL. The analyst plots 30 subgroups of n = 5 taken every 20 minutes over one shift on an X̄-R chart. Control limits were computed from the first 20 subgroups and then frozen. The R chart is in control. On the X̄ chart, the last eight subgroup means fall above the centerline, and one of those means exceeds the upper control limit by 0.6 mL. A new filler operator started three hours ago, and the product tank level was deliberately lowered to reduce changeover waste. The shift produced 9,600 bottles. Production points out that every individual bottle measured is above 497 mL and asks the analyst to sign the release, arguing the chart is 'just noisy.' The next shift starts in 40 minutes.
1. Is the filling process in statistical control? Identify each signal and explain why conforming individual bottles do not settle the question.
Read the worked answer
No - the process is not in statistical control, for two independent reasons.
- Eight consecutive means above the centerline: this violates a run rule and signals a sustained shift in the process mean, not random noise. Runs have a very low probability if the process is stable.
- One mean above the UCL: with limits from 20 in-control subgroups, the false-alarm probability per point is roughly 0.27%, so this is a strong signal of a special cause.
Because the R chart is in control, variability is stable while the location has moved - a classic pattern for a changed input such as lower tank head pressure or a new operator's technique. Every measured bottle being above 497 mL proves nothing about stability: the mean may have shifted toward the legal minimum, and later production could drift below it. Control limits describe the process as characterized; they are not specification limits and cannot be waived by a release request.
2. What immediate actions would you take for the 9,600 bottles, and what data would you collect before any release decision?
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Contain first, decide second.
- Place the 9,600 bottles on hold; do not release on the basis of individual conformance.
- Stratify the shift's measurements by time, filler head, operator, and tank level. The run began after two changes (new operator, lowered tank), so separation should show which factor tracks the shift.
- Using the in-control mean and standard deviation, estimate the proportion of bottles below 497 mL under the shifted mean; compare with the pre-shift period.
- Restore the tank to its qualified level, verify the operator is following the fill-height procedure, and run short-interval subgroups to confirm the mean returns inside the limits.
- Release only when the process is stable and the underfill risk is quantified. If any product shows risk of being below the legal minimum, sort or reject it; legal minimum compliance cannot be traded against yield.
3. Should the analyst recalculate the control limits now? Under what conditions may limits be recalculated?
Read the worked answer
Do not recalculate now. Recomputing limits with the shifted data would absorb the special-cause excursion into the limits, widening them and masking exactly the change the chart is designed to detect. That converts a signal into apparent routine variation.
Limits may be recalculated only when:
- A special cause has been identified and removed (tank level restored, operator retrained, procedure corrected).
- New data - typically 20–25 or more subgroups - have been collected from the stable, corrected process.
- The reason for recalculating is documented (new equipment, deliberate process change, or a verified permanent shift).
- Limits are never widened merely to keep points inside them, and never replaced by specification limits.
If the lowered tank level is a deliberate, validated permanent change, then a new baseline can be established - but only after it is proven stable, with the change documented in the control plan.