Out-of-Control Range Chart on a Machining Cell
A CNC turning cell runs a 25.40 ± 0.05 mm bore. Every hour a subgroup of five consecutive parts is measured. After 25 subgroups, the Xbar chart shows all points inside the control limits, but the last nine points are on one side of the centerline. The R chart has one point above its upper control limit at subgroup 22. The operator says the bore is "in control" because no Xbar point is outside the limits and wants to keep running. The last 200 parts have a sample mean of 25.415 mm and a sample standard deviation of 0.021 mm. The customer requires Cpk ≥ 1.33.
1. Is the process in statistical control? Explain what each chart signal means and why the operator's reasoning is incomplete.
Read the worked answer
No - the process is not in statistical control.
- The R-chart point above the upper control limit at subgroup 22 signals that within-subgroup variation increased. The Xbar control limits were calculated from an earlier baseline of common-cause variation; once the R chart is out of control, those Xbar limits are no longer a valid yardstick.
- The nine points on one side of the Xbar centerline form a run, a standard signal of a sustained shift in the mean. Even if every point is inside the limits, a run is an assignable-cause signal.
All points inside the limits is not the same as in control. The operator is applying only the "point beyond limit" rule and ignoring both the range signal and the run rule. The correct sequence is to investigate the variation change first, because a change in within-subgroup variation undermines the mean chart.
2. Calculate Cp and Cpk from the 200-part data and interpret the result against the customer requirement.
Read the worked answer
Specification width = 25.45 − 25.35 = 0.10 mm. Process sigma estimate = 0.021 mm.
- Cp = (USL − LSL) / (6σ) = 0.10 / (6 × 0.021) = 0.10 / 0.126 = 0.79.
- Upper-side capability = (25.45 − 25.415) / (3 × 0.021) = 0.035 / 0.063 = 0.56.
- Lower-side capability = (25.415 − 25.35) / 0.063 = 0.065 / 0.063 = 1.03.
- Cpk = min(0.56, 1.03) = 0.56.
Interpretation: the process is both too variable (Cp = 0.79) and off-center high (mean 25.415 vs. target 25.40). Cpk = 0.56 is far below 1.33. Because the process is not in control, this capability estimate is also provisional - but it is more than sufficient to show the process cannot meet the requirement as it stands. Continue running is not defensible.
3. What immediate and corrective actions should be taken?
Read the worked answer
Immediate: stop releasing product from the cell, quarantine parts since the last known-good subgroup, and inspect or sort the suspect window. Do not adjust the mean based only on a single Xbar point.
Corrective sequence:
- Investigate the R-chart signal: tool wear, fixture clamping, material lot change, coolant, or measurement repeatability. A range increase can create apparent mean shifts.
- Investigate the nine-point run: setup change, tool offset, or a new operator.
- Recalculate control limits only after the process is stable and the assignable causes are removed.
- Center the process near 25.400 mm and reduce variation. At the current σ = 0.021 mm, even perfect centering gives Cpk ≈ 0.79, so variation reduction is mandatory.
- Re-study capability with adequate data across shifts and lots, then compare to Cpk ≥ 1.33 before release.
The key judgment is that capability cannot be "inspected in" by sorting alone; the variation and centering causes must be corrected.