Comments on Statistical Issues in January 2016

increases. Taking this characteristic into consideration, we could confront the following circumstances: when the sample size is very small, we obtain a P-value of the normality test greater than 0.05 in most cases, so we conclude that the data are from a normally distributed population even though it apparently looks like a non-normal distribution. And when the sample size is very large, we frequently obtain a P-value less than 0.05 and conclude that the data are from a non-normally distributed population even though it looks very close to a normal distribution. To resolve this problem, we should not rely entirely on the results of the normality test but should to recall the central limit theorem which appears in most introductory statistics textbooks as follows: for any population (even a non-normal distribution) with mean μ and standard deviation σ, distribution of the sample mean is approximately normal with mean μ and standard deviation σ/sqrt(n) under the condition that the sample size n is sufficiently large. Therefore, it is proper to use the normality test only when the sample size is less than 30 and use the non-parametric tests if the P-value is smaller than 0.05. When the sample size is less than 30 and the P-value of the normality test is greater than 0.05, we should not hastily conclude that the data are from a normal population, and carefully decide their distributions after referring to the previous research which used the same response variable. Finally, we do not need to perform the normality test and can use the parametric tests when the sample size is sufficiently large.