Six Sigma relies on a broad toolkit of statistical methods to analyze data and verify improvements. The empirical rule—68% within ±1σ, 95% within ±2σ, and 99.7% within ±3σ—underpins control chart limits and capability calculations when data are approximately normal. When data are non-normal, a Box-Cox transformation can mathematically reshape them toward normality using a power parameter lambda, enabling valid use of standard tools. Normal probability plots and histograms reveal whether this assumption holds, displaying shape, center, spread, and outliers.
Hypothesis testing quantifies whether observed differences are statistically significant. The null hypothesis is rejected when the p-value falls below alpha, typically 0.05. A Type I error occurs when a true null is rejected, while a Type II error, denoted beta, occurs when a false null is missed; statistical power equals 1 minus beta. Confidence intervals express parameter uncertainty as a range, and tests may be one-tailed (detecting an effect in one direction) or two-tailed (detecting any difference). For comparing means, t-tests handle one or two groups, while ANOVA compares three or more groups by partitioning between-group and within-group variation into an F-statistic. Chi-square tests analyze categorical data for independence or goodness of fit.
Design of Experiments (DOE) provides a structured approach to uncovering cause-and-effect relationships by intentionally varying multiple factors simultaneously. Full factorial designs test every combination, while fractional factorial designs confound some effects to reduce runs—useful for screening. Replication, randomization, and blocking improve precision and validity. Signal factors are controllable inputs whose effects are desired, while noise factors are uncontrollable variables the design should make robust against. Response Surface Methodology (RSM) extends DOE into the optimization realm using second-order designs to model curvature and locate optimal settings. Regression analysis complements DOE by quantifying how independent variables predict a continuous response, supporting both prediction and optimization across the Analyze and Improve phases.