Detecting Departures from Normality: A Monte Carlo Simulation of a New Omnibus Test Based on Moments.

The assumption of normality underlies much of the standard statistical methodology. Knowing how to determine whether a sample of measurements is from a normally distributed population is crucial both in the development of statistical theory and in practice. W. Ware and J. Ferron have developed a new test statistic, modeled after the K-squared test of R. D'Agostino and E. Pearson (1973), the g-squared test statistic. This statistic has been used to estimate critical values for sample sizes up to 100, but a more extensive derivation and validation of the critical values are required, and the power of g-squared against a wide range of alternative distributions requires study. Monte Carlo simulations were performed to investigate these areas. The main advantage of g-squared is its conceptual and computational simplicity. The power study shows that g-squared is sensitive to a wide range of alternative distributions, especially peaked distributions, having absolute power for many distributions with a large "n." G-squared could be valuable for testing univariate normality in statistical routines, but it does have some weaknesses. One of its main disadvantages is its low power with small sample sizes except for peaked distributions. While gsquared can tell you about a departure from normality, it can not tell if the departure is due to a single outlier. It is recommended that when testing for departures from normality, g-squared should be used as a supplemental quantitative measure of normality to the information obtained from histograms, box plots, stem and leaf diagrams, and normality plots. (Contains 7 figures, 12 tables, and 28 references.) (SLD) ******************************************************************************** Reproductions supplied by EDRS are the best that can be made from the original document. ******************************************************************************** Detecting Departures from Normality: A Monte Carlo Simulation of a New Omnibus Test Based on Moments Linda Akel Althouse Columbia Assessment Services, Inc. William B. Ware University of North Carolina at Chapel Hill John M. Ferron University of South Florida Paper presented at the Annual Meeting of the American Educational Research Association San Diego, California April 13-17, 1998 PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIAL HAS BEEN GRANTED BY ame,, 4-fAttk_s_ TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) 1 U.S. DEPARTMENT OF EDUCATION dice of Educational Research and Improvement EDU ATIONAL RESOURCES INFORMATION CENTER (ERIC) This document has been reproduced as received from the person or organization originating it. 1:1 Minor changes have been made to improve reproduction quality. Points of view or opinions stated in this document do not necessarily represent official OERI position or policy. Detecting Departures from Normality: A Monte Carlo Simulation of a New Omnibus Test Based on Moments Linda Akel Althouse Columbia Assessment Services, Inc. William B. Ware University of North Carolina at Chapel Hill John M. Ferron University of South Florida Introduction The assumption of normality underlies much of the standard statistical methodology employed for several reasons. First, many test statistics are assumed to be asymptotically normally distributed due to the applicability of large sample theorems such as the Central Limit Theorem. Second, the normal distribution is often assumed to be the appropriate mathematical model for underlying phenomena that the researcher may be investigating. That is, scores on many measures in the behavioral and social sciences are normally distributed so that the bell curve shape of the normal distribution provides a reasonable good fit to the frequency distributions of the scores. When using inferential statistics, this knowledge becomes useful as one can think of the distribution of the true magnitudes of a trait as being normally distributed in a population. Third, the normal curve provides a good approximation of other theoretical distributions that are more difficult to work with when determining probabilities. (D'Agostino, 1986; Glass & Hopkins, 1984; Shavelson, 1988). With the assumption of normality yielding a rich set of mathematical consequences, it is no surprise that the normal distribution is the most widely used distribution in statistics. Therefore, knowing how to determine whether a sample of measurements is from a normally distributed population is crucial both in the development of statistical theory and in practice. As a result, much effort has been exerted in developing techniques solely for the purpose of detecting departures from normality. This effort began as early as the late 19th century with Pearson's (1895) work on moments, particularly the third and fourth moments which are commonly referred to as the skewness and kurtosis coefficients, respectively. However, while many tests currently exist, there is no gold standard among them as there is no one test which is both sensitive to a wide range of alternative distributions and