Normality Test

Normality testing checks whether a distribution is close enough to normal for parametric procedures such as t-tests, ANOVA and regression. Note that these tests assume approximately normal residuals rather than normally distributed raw data, a distinction that trips up many first-time analysts.

The Shapiro-Wilk test is preferred for samples up to roughly 2,000, with Kolmogorov-Smirnov plus the Lilliefors correction as the older alternative. A significant result indicates a departure from normality. Both are strongly influenced by sample size: in large samples trivial deviations turn significant, and in small samples genuine ones go undetected.

For that reason, judge normality on several sources of evidence together — skewness and kurtosis (commonly acceptable within plus or minus 2), histograms and Q-Q plots, alongside the formal test. If normality genuinely fails, the options are transformation, bootstrapping, or a non-parametric test such as Mann-Whitney or Kruskal-Wallis.