How to test normality distribution for a variable: a real example and a simulation study

Many commonly used statistical methods require that the population distribution be nearly normal. Unfortunately, in some papers the one-sample Kolmogorov-Smirnov test has been used for testing normality while the assumptions of applying this test are not satisfied. To conduct this test, it is assumed that the population distribution is fully specified. In practical situation where the mean and SD of population distribution is not specified in advance, one can use a modification of the K-S test for checking the normality assumption which is called, Lilliefors test. In this paper, we explain the method of computing this test with some common statistical softwares such as SPSS, S-PLUS, R and StatXact and utilize a dermatology dataset from Skin Research Center of Shohada-e-Tajrish hospital to illustrate how the use of the one-sample K-S (with the mean and SD estimated from the sample) instead of its modification can be misleading in practice. We also use Monte Carlo simulation to compare the approximate power of the one-sample K-S test (with the estimated population mean and SD) with Lilliefors test in some common specified continuous distributions. The result indicates that one should not use the one-sample K-S test for assessing the normality assumption in practical situation.