The Unicorn, The Normal Curve, and Other Improbable Creatures

An investigation of the distributional characteristics of 440 large-sample achievement and psychometric measures found all to be significantly nonnormal at the alpha .01 significance level. Several classes of contamination were found, including tail weights from the uniform to the double exponential, exponential-level asymmetry, severe digit preferences, multimodalities, and modes external to the mean/median interval. Thus, the underlying tenets of normality-assuming statistics appear fallacious for these commonly used types of data. However, findings here also fail to support the types of distributions used in most prior robustness research suggesting the failure of such statistics under nonnormal conditions. A reevaluation of the statistical robustness literature appears appropriate in light of these findings.