Statistical Assumptions

Core Concept: Why Assumptions Are Necessary

Statistical tests are derived to work correctly under specific mathematical conditions, called assumptions. The four primary assumptions are: (1) normality — residuals (or measurements) follow a normal distribution; (2) homoscedasticity — variance is constant across groups or predictor levels; (3) independence — one observation does not influence another; and (4) linearity — in regression, the relationship between predictor and outcome is linear. When these conditions fail, the theoretical basis of the test is undermined.

How Are Assumptions Checked?

Normality is typically assessed with Q-Q plots and the Shapiro-Wilk test, which is especially informative in small samples. Homoscedasticity is examined visually through residual-versus-fitted plots, or statistically with Levene's and Bartlett's tests. Independence is usually evaluated from the research design itself; in time-series data the Durbin-Watson statistic measures autocorrelation (expected value ≈ 2; values near 0 indicate positive autocorrelation). Linearity is inspected through scatter plots of predictors against the outcome.

Common Misconceptions

The most common misconception is failing to recognize that the normality assumption applies to residuals, not to raw data; in regression, the dependent variable itself need not be normally distributed. A second misconception is that in large samples, formal normality tests (e.g., Kolmogorov-Smirnov) flag trivially small deviations as statistically significant: statistical significance does not equal practical importance. Additionally, by the central limit theorem, t-tests and ANOVA are quite robust to normality violations in large samples — this does not make assumption checking unnecessary, but it does affect how findings should be prioritized.

Importance in Research Practice and Remedies

If violations are ignored, results can be misleading: Type I and Type II error rates may deviate from their nominal levels, confidence intervals may be distorted, and coefficient estimates may be biased. Key remedies include logarithmic or square-root transformations (for skewed distributions), robust methods such as Welch's t-test or HC standard errors (for variance heterogeneity), and non-parametric tests such as Mann-Whitney U or Kruskal-Wallis. The appropriate remedy depends on the nature of the data, sample size, and research question.

Sources