A View on the Validity of Central Limit Theorem: An Empirical Study Using Random Samples from Uniform Distribution

We derive the exact distribution of summation for random samples from uniform distribution and then compare the exact distribution with the approximated normal distribution obtained by the central limit theorem. To check the similarity between two distributions, we consider five existing normality tests based on the difference between the target normal distribution and empirical distribution: Anderson-Darling test, Kolmogorov-Smirnov test, Cramer-von Mises test, Shapiro-Wilk test and Shaprio-Francia test. For the purpose of comparison, those normality tests are applied to the simulated data. It can sometimes be difficult to derive an exact distribution. Thus, we try two different transformations to find out which transform is easier to get the exact distribution in terms of calculation complexity. We compare two transformations and comment on the advantages and disadvantages for each transformation.