Past exam of the experimental design and methods course of the year 2 of the psychology (experimental) course of the University of Oxford / 2022 / 4 / b

The normality assumption concerns the model's error distribution, not necessarily the pooled raw observations. In an independent-groups analysis of variance, errors around each group mean are modeled as following a normal distribution; combining groups with genuinely different means can produce a nonnormal pooled distribution even when this assumption holds. In a paired t-test, the relevant distribution is that of within-pair differences. Linear regression conventionally assumes normal errors conditional on the predictors for exact small-sample Student t-tests and F-tests.^[ref]

Inspect suitable within-group or fitted-model regression residuals using a quantile-quantile plot and examine outliers and asymmetry. A formal normality test is not a decisive automatic gatekeeper: with small samples it may miss important departures, while large samples can detect minor departures. Approximate robustness depends on the sample size, balance, tail behavior and intended inference. The central limit theorem concerns sampling distributions and does not make every raw observation normal or repair dependence. Depending on the question, a justified transformation, a better-fitting probability model, robust inference or an appropriate bootstrap may be preferable. The normality assumption, independence and equal variance are distinct assumptions.