Assumptions and Effects of Violating Assumptions - Deviation from Normal Distribution

Assumptions. It is assumed that the dependent variable is measured on at least an interval scale level (see Elementary Concepts). Moreover, the dependent variable should be normally distributed within groups. ANOVA/MANOVA contains extensive diagnostic graphics as well as statistics to test the validity of this assumption.

Effects of violations. Overall, the F-test is remarkably robust to deviations from normality (see Lindman, 1974, for a summary). If the kurtosis (see Basic Statistics and Tables) is greater than 0, then the F tends to be too small and we cannot reject the null hypothesis even though it is incorrect. The opposite is the case when the kurtosis is less than 0. The skewness of the distribution usually does not have a sizable effect on the F statistic. If the N per cell is fairly large, then deviations from normality do not matter much at all because of the central limit theorem, according to which the sampling distribution of the mean approximates the normal distribution, regardless of the distribution of the variable in the population. A detailed discussion of the robustness of the F statistic can be found in Box and Anderson (1955), or Lindman (1974).