Control Charts

A control chart is a time-ordered plot used in Statistical Process Control (SPC) to determine if a process is in statistical control. It distinguishes common-cause variation (inherent, random) from special-cause variation (assignable, requiring action) by comparing plotted statistics to 3-sigma control limits calculated from the process itself, not specifications. Principle: Form rational subgroups (small samples taken close in time so variation within subgroup is only common cause). Plot subgroup statistic over time with centerline = process average and UCL/LCL = centerline ± 3·σ_statistic. Variables charts for continuous data: X̄-R (mean and range, n=2-10, robust and simple), X̄-S (mean and standard deviation, n>10, more efficient than R), I-MR (individuals and moving range, n=1, for low-rate or destructive testing). X̄ monitors location, R/S/MR monitor dispersion; both must be in control. Attributes charts for discrete data: p-chart monitors proportion defective p = nonconforming/n (binomial model), np-chart monitors count nonconforming n·p (constant n), c-chart monitors count of defects per constant inspection unit (Poisson model), u-chart monitors defects per unit u = c/n when sample size varies. Key parameters: X̄̄ = grand mean, R̄ = average range, S̄ = average std dev, p̄ = average proportion, c̄ = average defects, ū = average defects per unit. Factors A2, D3, D4, B3, B4, E2 depend on n and link R̄ or S̄ to σ. Assumptions: rational subgrouping, measurement system adequate, for variables charts subgroup means ~ normal via CLT, for p/np binomial n·p̄≥5, for c/u Poisson with stable opportunity. Control limits are not spec limits or confidence intervals. Interpretation uses beyond-limits and Western Electric runs rules (e.g., 8 points on one side) to detect small shifts.