Nine Points Below Center on the 500 mL Fill Line
At a bottling plant a 500 mL fill line has specification limits of 497-503 mL. An Xbar-R chart with n = 5, subgroup every 30 minutes, has run for two years at a mean of 500.0 mL with a within-subgroup sigma of 0.75 mL, giving Cpk = 1.33 and roughly 7 ppm out of specification. Over the past three days the Xbar chart shows nine consecutive points below the centerline, averaging 499.2 mL; the R chart is stable. A rough Cpk from the last 30 subgroups is 0.98. The operator's response to the signal was to leave the line alone, noting that every bottle measured is still inside 497-503 mL. The control plan lists fill volume with a sample size of 5 per hour, 'adjust filler if out of spec', and no chart. The filler has 24 valves; the plant runs three products with different target fills on the same line, changing over twice a day.
1. Was the operator's decision correct? Interpret the chart signal against the applicable rules and state the action required.
Read the worked answer
No - the decision was wrong The operator used the specification limits as the action limit. That is the classic control-versus-specification confusion. Control limits describe the voice of the process; specification limits describe the voice of the customer. A process can sit entirely inside spec and still be out of control, which is exactly what is happening.
The signal Nine consecutive points on one side of the centerline. The conventional run rule triggers at eight (Nelson rule 2; Western Electric rule 3). The probability of nine in a row on the same side by chance is about 0.5^9, roughly 0.2% - about one in 512 subgroups. With a stable R chart, this is a sustained mean shift of about 0.8 mL, not common-cause noise.
Why 'all bottles are in spec' does not protect the line
- At Cpk 0.98 the process is already producing roughly 1,700 ppm out of spec; the shift has consumed most of the low-side margin.
- Only 5 bottles per 30 minutes are measured, so most production is never checked. In-spec samples are not evidence that the population is in spec.
- The purpose of the chart is to catch the shift before it generates nonconforming bottles, not after.
Required action (reaction plan)
- Verify the signal: re-sample immediately, confirm the gauge and the data entry.
- Investigate assignable causes: valve wear or drift on specific valves, product changeover with the wrong target recipe, product temperature/viscosity, supply pressure, pump or filler-head timing. Cross-check the shift start against the changeover log.
- Check valve-level data (a valve-position multi-vari) - a mean drop with stable R commonly means a subset of valves drifted.
- Correct, record the cause and action, and confirm the mean returns; escalate if two corrections fail.
Note the control plan is also defective: 'adjust if out of spec' is not a reaction plan, and there is no chart specified.
2. Quantify the risk. Compute approximate ppm before and after the shift, explain why Cpk changed while the R chart did not, and distinguish centering from variation reduction here.
Read the worked answer
Before the shift Mean 500.0, sigma (within) = 0.75, nearest limit 3.0 mL away. z = 3.0 / 0.75 = 4.0, so tail area about 3.2e-5 per side, roughly 6-7 ppm total. Cpk = 3.0 / (3 x 0.75) = 1.33.
After the shift Mean 499.2, nearest limit is the LSL at 497, only 2.2 mL away. z = 2.2 / 0.75 = 2.93, tail area about 1.7e-3, roughly 1,700 ppm below the LSL; the USL tail is negligible (z = 5.07). Cpk = 2.2 / (3 x 0.75) = 0.98.
Interpretation A pure centering shift of 0.8 mL raised the expected defect rate by roughly 250 times with no change in spread - which is why the R chart stayed flat while Cpk collapsed. At Cpk 1.33 there is almost no margin left on the low side, so the process is fragile to any mean drift.
Centering versus variation reduction
- Centering: move the mean back to target (or re-target the filler). Cheap, immediate, and here it removes most of the risk on its own.
- Variation reduction: reduce sigma from 0.75 to 0.5 and the process would carry Cpk = 2.0 at target, so ordinary drift would no longer be dangerous. This is the durable fix but takes longer and may need valve maintenance, pressure regulation, or nozzle replacement.
Both belong in the improvement plan, centering first. Also: 0.98 is a capability estimate from an out-of-control process - treat it as a risk indicator, not a certified index, and do not report it as capability until the chart is stable again.
3. Redesign the control plan for fill volume, covering chart choice, sampling, reaction plan, ownership, and the complications introduced by 24 valves and three products.
Read the worked answer
Chart choice and limits
- Xbar-R with n = 5 is appropriate if subgroups are formed by time and valve position is recorded; move to Xbar-S if subgroup size exceeds about 9.
- The three products have different targets, so one shared chart would create exactly the mixture problem the I-MR example showed. Use product-specific limits, or a standardized (short-run) chart if changeovers are frequent.
- Re-verify limits after each changeover, after any known process change (new valve, pump, product formulation), and at a defined review interval - never leave stale limits in place for two years without review.
Sampling
- State which valve(s) are sampled and rotate through all 24 valves on a documented plan; sampling valve 1 every time hides valve 7 drifting.
- Define who collects, how the bottle is selected and weighed, and the scale resolution (<=0.1 mL) and calibration schedule.
- Add a daily valve-to-valve check (a multi-vari by valve position) to catch single-valve problems.
Reaction plan (written, with names)
- One point beyond a control limit, or 8 in a row on one side, or 2 of 3 beyond 2 sigma: re-sample and verify; notify the shift lead; check recipe at changeover, valves, pressure, product temperature; adjust; record cause and action.
- Escalate to the process engineer if the second correction fails; stop the line if any bottle is outside 497-503 mL.
Ownership, audit, and mistake-proofing
- Named owner for the characteristic, weekly chart audit, monthly Cpk review per product.
- Recipe interlock tied to product changeover so the wrong fill target cannot be selected.
- Valve maintenance interval written into the control plan.
- Train operators explicitly on control limits versus specification limits - the failure in this case was a training and control-plan failure, not an operator failure.