Normality tests

Parametric tests such as t test and ANOVA assume that you have sampled data from populations that follow a normal (Gaussian bell-shaped) distribution. If the sample sizes are large (more than 50 or 100), 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.

An alternative approach does not assume that data follow a Gaussian distribution. In this approach, values are ranked from low to high and the analyses are based on the distribution of ranks. These tests, called non-parametric tests, are appealing because they make fewer assumptions about the distribution of the data. Nonparametric methods do not rely on the estimation of parameters (such as the mean) describing the distribution of the variable in the population. Therefore, these methods are also called distribution-free methods.

Nonparametric tests are less powerful than the parametric tests. This means that it is harder to detect real differences as being statistically significant.

Shapiro-Wilk W test is the preferred test of normality because of its good power properties as compared to a wide range of alternative tests. If the W statistic is significant, then the hypothesis that the respective distribution is normal should be rejected.

Each of normality tests provides evidence for certain types of "non-normality" it does not guarantee "normality".