Total Quality 3D

The X-bar chart

A process makes a dimension with target 0 and standard deviation σ (all values here are in units of σ). Subgroups of n consecutive parts are averaged; the means have standard deviation σ/√n, so the three-sigma control limits are ±3/√n. In the 3D line each bead on the wall chart is the mean of the n shafts that just passed the inspection arch; a bead outside the limits is a signal of a special cause (a false alarm has probability 0.0027 per sample).

Tool drift and detection power

After sample 30 the lathe tool has drifted (the carriage moves in the picture) and the process mean moves by δ. The chance that a subgroup mean falls beyond a limit is 1 - Φ(3 - δ√n) + Φ(-3 - δ√n) and the mean wait for a signal is its reciprocal (the third graph). Bigger subgroups see a small drift sooner, at the cost of measuring more parts.

Capability and the podium

The specification window [LSL, USL] = ±wσ is compared with the spread of individual parts: Cp = 2w/6 and Cpk = (w - |μ|)/3. The fraction outside specification is Φ(-w-μ) + 1 - Φ(w-μ). On the podium the green solid is the density of individual parts, the red ends are the out-of-spec tails (those shafts are kicked into the red bin), the glass walls are LSL and USL, and the translucent blue ridge behind is the much narrower distribution of subgroup means (peak scaled for visibility). A Cpk of 1.33 is a common supplier requirement.

Independent numerical checks

Check 1 draws 20 000 individual values from the shifted process (same seed and draw order as the 2D page) and compares the out-of-spec fraction with the exact normal tails within four binomial standard errors. Check 2 is a separate experiment with another seed: 6 000 fresh subgroups from the shifted process, counting how many means exceed the control limits, against the closed-form detection probability.

Honest simplifications

Normal, independent, known σ and a perfectly centred target; the limits use the known σ instead of an estimate from the data; only the single three-sigma rule is used (no run rules); the shaft diameters in the picture are exaggerated by the factor on the slider so that a one-sigma difference is visible; the lathe, conveyor and bins are illustrations, not a machining model.

Things to try

• Set the shift to 1 sigma and compare n = 1, 4 and 9: how many samples pass before the beacon lights?
• Set the spec half-width to 3 sigma with zero shift: Cpk = 1.00 and about 2 700 ppm are rejected, yet the chart stays in control.
• Start off-centre: the chart is silent (its limits are around the target) but the capability index is already poor.

Educational model; not a substitute for a real quality-engineering study.