The analysis of regression residuals and detection of outliers are discussed, with emphasis on determining how deviant an individual data point must be to be considered an outlier and the impact that multiple suspected outlier data points have on the process of outlier determination and treatment. Only bivariate (one dependent and one independent) models were investigated. A sufficient number of data points (240 pairs) was used to avoid interpretation problems associated with small data sets. The original data set was successively manipulated to include additional data points with increasingly larger degrees of extremeness, and potential outlier data points were added. To meet the demands of the data set modification, a computer program, DrawReg, was written in QuickBASIC. Outlier points could be viewed as belonging to one of five broad categories, and it was apparent that no single residual statistic could adequately account for all types. The use of multiple outlier detection techniques is recommended as part of the least-squares modeling process. The importance of the argument is illustrated through the discussion of the Schrodinger's Cat situation derived from quantum physics. Twelve tables and six figures illustrate the discussion. An appendix describes the DrawReg program. (SLD)