Statistical Process Control and Process Capability
What the exam tests
Chart selection, control-limit construction, out-of-control rules, and the difference between stability and capability. Numeric Cp/Cpk questions are common.
Core rules
- Variables: Xbar-R for subgroup n = 2 to 9, Xbar-s for n of about 10 or more, I-MR for n = 1. Attributes: p (fraction defective), np (defectives), c (defects per unit with constant area), u (defects per unit with varying area).
- Control limits are plus/minus three standard errors computed from within-subgroup variation, never from specification limits. Rational subgroups must be homogeneous (same shift, same lot, short time window).
- Western Electric/Nelson rules flag runs, trends and stratification, not just points beyond limits. A single point beyond a limit is a signal of a special cause; the process is not in statistical control.
- Capability: Cp = (USL - LSL)/6 sigma; Cpk = min(USL - mean, mean - LSL)/3 sigma. Cp uses within-subgroup sigma and assumes centering; Cpk penalizes off-center means and can never exceed Cp. Pp/Ppk use overall (long-term) sigma.
Worked example
LSL 10, USL 16, mean 12.4, within-subgroup sigma 0.8. Cp = 6/(6x0.8) = 1.25. Upper side: (16-12.4)/(3x0.8) = 1.50. Lower side: (12.4-10)/(3x0.8) = 1.00. Cpk = 1.00. The process is potentially capable but the mean is off-center toward the lower limit; shifting the mean to 13.0 raises Cpk to 1.25, matching Cp.
Traps that cost marks
Plotting specification limits on a control chart. Computing capability before confirming control, which makes the index meaningless. Reporting Cp alone when the mean is off-center. Using Pp/Ppk for a short-term study. Ignoring autocorrelation in I-MR charts. Tampering: adjusting a stable process in response to common-cause variation increases variation.