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Academic textbooks, statistical literature, and publication guidelines provide conflicting, ambiguous and often incomplete answers to the question of how researchers should handle the normality assumption for classical general linear model tests when conducting their analyses. Previous studies have shown that normality violations can impact on type I errors, power, parameter estimates and standard error estimates of classical tests. This paper reviews the arguments in favour and against normality testing, the role of the central limit theorem, types of violations that tests within the general linear model are susceptible to, methods for evaluating the normality assumption, and the paradox that normality tests have low power in small sample sizes where the influence of assumption violations are likely to be most profound. A Monte Carlo simulation study was used to evaluate the power of 18 normality tests across 18 alternative distributions, and the effect of normality deviations on estimates of centrality, scatter and regression coefficients. The results demonstrate that the type of normality test and distribution matters, and that a conditional testing procedure utilising normality tests to select between classic, non-parametric and robust tests should not be used. Instead, an alternative procedure for managing the normality assumption is advised, and demonstrated in the supplementary materials using R code and data that are provided.

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