Assessing Distributional Assumptions Beyond a Single Normality Test
Do not base normality decisions on one p value. Combine Shapiro–Wilk, Q–Q plots, residual diagnostics, sample size and robust methods.
What needs to be normally distributed?
Not every variable must be normal. In linear regression and ANOVA, conditional errors or residuals are usually the relevant diagnostic; predictors need not be normally distributed. For a paired t test, assess paired differences rather than each measurement separately.
Shapiro–Wilk
A small p value provides evidence of departure from normality. Small samples may miss important departures, while very large samples can detect trivial ones. Therefore p>0.05 does not prove normality.
Use Q–Q plots
Q–Q plots compare observed and theoretical normal quantiles. Systematic curvature can reveal skewness, heavy tails or outliers. Perfect alignment is not expected; judge the pattern in relation to sample size and its influence on the analysis.
Options when assumptions are inadequate
Check coding and measurement errors.
Use a scientifically meaningful transformation.
Consider Welch methods, robust standard errors or bootstrap intervals.
Use generalised models for ordinal, binary or count outcomes.
Use rank methods when the estimand is distributional rather than a mean.
Do not remove outliers merely to make a test non-significant. Use predefined data-quality rules and sensitivity analyses.
References
Accessed 20 June 2026.