Capability, Gauge Error, and a 10 Micron Escape on a 25 Micron Bore
A 6061-T6 hydraulic manifold bore is specified at 40.000/40.025 mm. The supplier gauges 100% of production with an air-plug gauge mastered with a ring at 20 C, in a shop held at 28 C; parts are gauged within a few minutes of machining. The supplier's SPC chart reports Cpk = 1.33 and ships everything. Within a month the customer's CMM, in a 20 C room, rejects about 4% of parts at 40.028-40.032 mm; every one of those parts passed the supplier's gauge, which had read them at 40.019-40.022 mm. A gauge R&R study at the supplier gives 8 microns (6 sigma) GRR. The customer is demanding 100% sorting at the supplier's expense. The supplier's quality engineer argues that Cpk = 1.33 and %GRR of 32% both clear the internal acceptance criteria.
1. Evaluate the thermal expansion argument as an explanation for the escape. Use alpha = 23.6e-6 per C for 6061-T6 and a 40 mm bore.
Read the worked answer
Thermal error = alpha x L x dT = 23.6e-6 x 40 mm x (28 - 20) C = 7.6 microns
The magnitude is right - 7.6 microns is comparable to the 8-10 micron discrepancy, which is why the thermal story sounds convincing. But the sign is wrong. The part at 28 C is larger than its 20 C nominal, so a gauge mastered at 20 C reads the hot bore too large. Thermal expansion would make the supplier's gauge over-reject, not pass undersized parts. Here the supplier read smaller than the CMM by about 8-10 microns, which is the opposite direction.
So thermal error is real and should be eliminated (soak parts, or apply a size correction, or gauge at 20 C), but it does not explain this escape. Something is producing a systematic bias in the gauge reading in the passing direction - a worn or mis-stated master ring, a drifted air gauge, a taper or ovality that the two-jet plug averages differently than the CMM, or a dirty jet. Chase bias, not scatter.
2. Interpret the Cpk = 1.33 and %GRR = 32%. Does either actually support the claim that this gauge and process are acceptable?
Read the worked answer
%GRR = 8 microns / 25 microns tolerance = 32%. By AIAG convention, under 10% is acceptable, 10-30% is conditionally acceptable depending on the application, and over 30% is unacceptable - a 32% gauge is not fit for sorting a 25 micron tolerance. Worse, 32% clears nothing; it is on the wrong side of the line and the QE is quoting it as a pass.
The more serious error is conceptual: GRR measures precision, not accuracy. Gauge R&R cannot detect a constant bias. A gauge that reads 8 microns low every single time will produce an excellent GRR and still ship 100% defective parts. The entire escape here is a bias phenomenon, and the study performed was structurally incapable of finding it. A calibration/bias and linearity check against a certified master ring at multiple sizes, not another GRR, is the required test.
Cpk = 1.33 uses the observed data, which includes gauge error. It is not process capability; it is process-plus-gauge capability. With a 32% gauge on a 25 micron tolerance, a meaningful fraction of the reported 1.33 is measurement noise, and the number tells you almost nothing about the parts. Neither argument survives scrutiny.
3. The customer's data show parts at 40.026-40.032 mm, suggesting the population is not centered where the Cpk assumed. Estimate the true process sigma and the true Cpk, and state the immediate containment and corrective action.
Read the worked answer
Separate measurement variation from process variation. Observed (centered) Cpk = 1.33 with a 0.025 mm tolerance gives 3*sigma_obs = 0.0125/1.33 = 0.00940 mm, so sigma_obs = 3.13 microns.
Gauge sigma = 8/6 = 1.33 microns. Then: sigma_true = sqrt(3.13^2 - 1.33^2) = sqrt(9.80 - 1.77) = sqrt(8.03) = 2.83 microns
If the process were centered at 40.0125, true Cpk = 0.0125/(3 x 0.00283) = 1.47 - comfortably capable. So the spread is fine; the problem is entirely the mean.
The customer data put the population around 40.026-40.032, i.e., a mean near 40.026, which is already above the 40.025 USL. Cpk at that mean = min(40.025 - 40.026, 40.026 - 40.000)/(3 x 0.00283) = -0.001/0.0085, i.e., negative. That is consistent with a 4% reject rate in the upper tail and with a gauge reading 8 microns low. The real defect is a process-centering (tool wear offset / thermal growth at the machine) plus a biased gauge.
Immediate containment: sort 100% with a calibrated gauge or CMM in a 20 C room, not the air-plug, and quarantine the suspect window.
Corrective action, in order: (1) recalibrate/replace the air-plug master and verify bias and linearity at 40.000/40.015/40.025; (2) re-center the machining process and put the mean on 40.0125 with a tool-offset check; (3) apply a guardband at inspection - accept 40.003-40.022 rather than 40.000-40.025 - sized from the 8 micron gauge uncertainty; (4) report Cpk on data from a gauge whose %GRR is under 10% of tolerance. Note that the mean shift, not the gauge, is the primary defect; the biased gauge is why it escaped.