Eight Above the Line and a Convenient Re-measurement
An X̄–R chart with subgroups of n = 5 monitors a bore diameter of 25.000 ± 0.100 mm. The chart was established three months ago and has been stable. Center line X̄ = 25.00 mm, average range R̄ = 0.42 mm, with A2 = 0.577, D3 = 0 and D4 = 2.114. During the night shift the last eight consecutive subgroup means fell above the center line, and the final subgroup mean was 25.31 mm. The operator re-measured the retained part, obtained 25.05 mm, and asked the inspector to overwrite the 25.31 with the re-measurement so the chart would not stop production. The inspector has the original part and the same gauge available and knows that the night shift changed to a new bar-stock supplier last week.
1. Compute the control limits and confirm whether 25.31 mm is out of control.
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Limits
- UCL = X̄ + A2·R̄ = 25.00 + 0.577 × 0.42 = 25.00 + 0.242 = 25.242 mm
- LCL = X̄ − A2·R̄ = 24.758 mm
- R chart: UCL_R = D4·R̄ = 2.114 × 0.42 = 0.888; LCL_R = D3·R̄ = 0
Sigma of the subgroup means
sigma_xbar = R̄ / (d2·√n) = 0.42 / (2.326 × 2.236) = 0.42 / 5.201 = 0.0808 mm
(Consistent check: 3 × 0.0808 = 0.242, matching A2·R̄.)
Assessment of 25.31 mm
Deviation from center line = 0.31 mm = 0.31 / 0.0808 = 3.84 sigma_xbar. The point is beyond the UCL and is a genuine signal, not a rounding artifact; the chance of a single point beyond 3 sigma from common cause alone is about 0.0027.
Check the R chart first. If the ranges are in control, the shift is a change in location. If a range point is also out, suspect the gauge, the fixture or the measurement method before blaming the process - and note this is the same gauge on which a re-measurement gave a different answer.
2. Interpret the run of eight consecutive points above the center line. What is its probability and what should be investigated?
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Probability. For a process centered on the center line, the chance that a given point falls above it is 0.5. A run of 8 on one side:
(0.5)^8 = 1/256 ≈ 0.0039 for a specified side, or about 0.0078 for a run of 8 on either side. This is a designed sensitivity rule; the point is that a pattern this unlikely from common cause deserves an investigation.
Interpretation. The run and the single out-of-limit point are two signals of the same event: a sustained location shift, not a blip. The last eight subgroup means are running above the center line and the latest exceeds the 3-sigma limit, so the process moved and did not come back. A subgroup mean shift of about 0.1–0.2 mm with unchanged within-subgroup ranges is the classic signature.
What to investigate, in order:
- The new bar-stock supplier - material grade, hardness, machinability, incoming diameter, or a different heat lot can move bore size directly. This is the most recent change and the timing matches.
- Setup and tooling changes on the night shift: tool change point, fixture clamp force, coolant, spindle warm-up.
- The measurement system: gauge master check, probe tip, temperature (night shift is cooler), and whether the gauge itself has been dropped or replaced.
- Operator and method changes, if any.
Action: stop and quarantine per the reaction plan, notify the supervisor, and do not restart until the assignable cause is identified and the chart is back in control. The part at 25.31 mm is within the 25.000 ± 0.100 mm tolerance only up to 25.100, so it is out of specification as well as out of control - an actual nonconforming part, needing disposition.
3. The operator wants to overwrite 25.31 with 25.05 from a re-measurement. How should the inspector respond, and what is the legitimate path?
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Refuse the overwrite. Replacing a recorded result because of its value is data falsification. It also corrupts the chart - the apparent variation narrows, the limits become artificially tight, and the very signal that prevents escapes is deleted. Records must retain the original reading; any correction must be documented, dated, justified and leave the original legible.
Why the request does not even solve the problem. With 25.05 the part would still sit 0.05 mm above nominal, and the run of eight points above the center line is untouched. If the re-measurement is correct, the plant has two problems - a measurement-system error and a real process shift - not zero.
The legitimate path, in this order:
- Verify the gauge against its master or check standard before doing anything else. If the gauge fails, the readings since its last verification are suspect and the issue is a measurement-system nonconformity; the affected subgroups must be flagged, not deleted.
- Check the part and the method: burrs, dirt, temperature, measuring force, correct measuring point, probe zero.
- If the gauge and method are good, record a legitimate re-measurement under the documented measurement procedure: both values retained, the reason stated, the gauge identified, and the inspector notified. Treat the discrepancy as evidence about the gauge's GRR.
- Escalate the out-of-control signal: the reaction plan is a stop-and-quarantine decision, not a data-editing decision. Do not recalculate the limits and do not drop the subgroup.
- Follow through on the 3.84-sigma point and the run of eight as a single investigation, starting with the bar-stock change.
The judgment call is this: a re-measurement is a legitimate measurement-system check performed before the result is used, and it is never a way to change a result after seeing it.