The power of the classical Linear Discriminant Function (LDF) is compared, using Monte Carlo techniques with five other procedures for classifying observations from certain non-normal distributions. The alternative procedures considered are the Quadratic Discriminant Function, a Nearest Neighbor Procedure with Probability Blocks, and three density estimators. Comparisons of misclassifications are examined for varying sample sizes for two and three dimensional models. Three types of distributions are considered: finite range (Logit Normal), semi-infinite range (Log Normal), and infinite range (Inverse Hyperbolic Sine Normal). Results indicate that certain alternatives to the LDF classify observations correctly in a greater proportion than does the LDF for non-normal data, and that different procedures are best for different types of non-normality. (Author)