Comparison of Some Common Tests for Normality

The normal distribution is the bedrock of many statistical procedures. Inferences and conclusions from parametric statistical analysis may not be valid when the normality assumption is violated. Three common procedures used for evaluating whether a random sample of independent observations come from a population with normal distribution are: graphical methods (histograms, box plots, Q-Q-plots), numerical methods (skewness and kurtosis) and formal normality tests. In this study, the type I error rates and power of four common formal tests of normality: Anderson-Darling (AD) test, Chi-square (CS) test, Kolmogorov-Smirnov (KS) test and Shapiro-Wilk (SW) test were compared. Type I error rate of the four tests were computed via simulation (in R) of sample data generated from the standard normal while power comparisons was conducted using common continuous and discrete type as well as less common mixture normal alternative distributions. Five thousand independent samples of various sample sizes were generated from the different distributions considered. Our findings reveal that Shapiro-Wilk test has the most acceptable type I error rate amongst the four tests, followed by Kolmogorov-Smirnov test, Anderson-Darling test and Chi-square test. The power study revealed that none of the four tests is uniformly most powerful for all types of alternative distributions under consideration. Shapiro-Wilk test is the most powerful amongst the four normality tests for continuous –type alternative distributions while Chi-square test outperforms the other three tests for discrete-type distributions. All four normality tests have significantly low powers under the mixture normal distributions with unequal means and equal variances irrespective of the mixture probabilities while there is an improved performance under mixture normals with unequal means and unequal variances.