Two illustrations demonstrate the bias inherent in measuring discrimination by the use of conventional regression techniques. Both cases use a sample of 200 offenders, 32 percent of whom are black. In the first illustration, the entire sample is distributed over four risk classes. Using a single regression equation with a dummy variable for race is shown to overestimate or underestimate the extent of discrimination. In the second case, all blacks within a given risk group are treated the same, and the range of sentences that whites are given is widened. Results reveal that conventional regression techniques introduce a strong possibility of rejecting the hypothesis that discrimination exists when, in fact, the hypothesis is true. In contrast, the residual discrimination methodology represents a legally defensible manner by which to measure discrimination. When this procedure is applied to the U.S. Board of Parole data on released felons in the 1970's, results reveal that blacks served 5 more months in prison compared to identically qualified whites. The technique demonstrates that this disparity cannot be justified empirically on the grounds that blacks are more criminal or are more likely to become recidivists. In short, racial discrimination plays a crucial role in explaining disparities in punishment. Two illustrations, 38 references, and appendixes containing the raw data and equations for the illustrations are provided.