In experimental design, a residual is defined as the difference between

In designed experiments and regression-based analysis, a residual is the difference between the observed value of the response and the value predicted or expected by the model. This is a core statistical concept used heavily in the Analyze Phase. Residuals help the Black Belt assess whether the selected model is appropriate, whether assumptions such as constant variance and normality are reasonable, and whether unusual observations or patterns remain unexplained. The CSSBB treatment of residual analysis emphasizes plotting and studying these differences to detect model inadequacy, outliers, nonlinearity, or heteroscedasticity. The other choices do not describe residuals. The differences in degrees of freedom and sums of squares are ANOVA bookkeeping relationships, not residual definitions. Likewise, the difference between one observation and another observation is not a residual. Residuals always compare actual response values against model-based expected values. That distinction is important because DOE conclusions rely not only on estimated effects, but also on confirming that the model fits the data adequately. Therefore, the correct answer is the observed experimental value minus the expected experimental value.