Given a sample from any of a variety of theoretical distributions, the problem of choosing one for the analysis of the given data has long been a major concern to both the theoretical and the applied statistician. The major topic of this paper addresses this topic as applied to classes of distributions. More specifically, given the data, we wish to classify the parent distribution as either normal-tailed, long-tailed, or skewed in a particular direction. The basis for the classification is a characterization of the normal distribution. Essentially, the independence of the sample means and variance is under consideration here. Graphical methods indicate that the relationship between these variables can distinguish the above three classes. A two-dimensional statistics, (R₁, R₂), is developed from a second order, linear regression model. Empirical power investigations, using the normal, chi-square with two degrees of freedom, and Cauchy distributions as representatives of their respective classes, exhibit the strength of this statistic.