Statistical Process Control (SPC) uses control charts to distinguish common cause variation, which is inherent and predictable, from special cause variation, which comes from external sources and creates non-random patterns. Control limits are typically set at three standard deviations from the center line, providing a 99.73% confidence band under common cause assumptions. For X-bar charts, the upper control limit equals the grand mean plus A2 times the average range. The R-chart plots subgroup ranges to monitor variability and is paired with X-bar charts when subgroup size n is at most 10; for larger subgroups, an S-chart using standard deviations is more accurate. When subgroups cannot be formed, an Individuals and Moving Range (I-MR) chart monitors each measurement and its consecutive variation.
Attribute charts handle discrete pass/fail and count data. A p-chart tracks the proportion of defective items in subgroups of varying size, an np-chart tracks the number of defective items in constant-size samples, a c-chart monitors the count of defects per unit when sample size is constant, and a u-chart monitors defects per unit when sample sizes vary. Western Electric rules add sensitivity by flagging non-random patterns such as eight consecutive points on one side of the centerline. Run tests further detect trends, shifts, or cycles that suggest special causes even when points stay within control limits. Stratification and Multi-Vari analysis help reveal patterns hidden in aggregated data by separating observations into subgroups by shift, machine, operator, or material.
Process capability studies connect SPC with quality outcomes. Short-term capability indices such as Cp and Cpk reflect inherent variation under controlled conditions, while long-term indices Pp and Ppk use sample standard deviation across routine operation, typically showing wider spread. Process yield equals good units divided by total units started, and Rolled Throughput Yield (RTY) multiplies the first-pass yield of each step to show the probability that a single unit passes an entire multi-step process defect-free. The Taguchi loss function goes further by quantifying quality loss as a quadratic function of deviation from target, arguing that any deviation—even within specification—carries a cost to society and customer. Together, SPC and capability analysis sustain and quantify the gains achieved through DMAIC projects.