Process Capability and Control Charts

What is modelled

A machining cell makes parts whose size varies by chance, close to a normal distribution. The tolerance limits say what the customer accepts; the control limits say what the process normally does. Capability indices compare spread and position with the tolerance, and control charts of subgroup means and ranges warn when the process moves or widens, before many parts fall outside tolerance.

Equations

• Cp = (USL − LSL) / 6σ, Cpk = min(USL − μ, μ − LSL) / 3σ; fraction out of tolerance Q((USL − μ)/σ) + Q((μ − LSL)/σ).
• Mean chart limits ± 3σ0 / √n (equal to A2 R̄), range chart D3 d2 σ0 to D4 d2 σ0, with d2 the expected range of n standard normal values.
• Average run length ARL = 1 / (1 − β), β = Φ[(UCL − μ)/(σ/√n)] − Φ[(LCL − μ)/(σ/√n)].

Checks run in the page

• The chart constants d2 and A2 by numerical integration.
• The out-of-tolerance fraction by Simpson integration against the closed form.
• The average run length by seeded Monte Carlo against the formula.
• The mean range of seeded subgroups against d2.

Honest simplifications

Measurements are independent and exactly normal, the chart limits are computed from a known design sigma of 6 micrometres rather than estimated from a first run, only the two simplest rules (a point outside the limits) are used, there is no measurement error or autocorrelation, and the drift is perfectly linear. It teaches the logic of SPC and does not replace a measurement system study.

• Press Off-centre by 2 sigma and watch Cpk fall far below Cp while the charts stay quiet.
• Press Slow tool wear and see the dots creep upwards until the mean chart signals.
• Press Large subgroup, small shift to see a larger subgroup catch a small shift sooner (shorter run length).