normaltheory

Normal theory is a framework in statistics that describes classical parametric methods based on the assumption that the underlying population or the model errors are normally distributed. This approach enables exact or approximate inferential results for a range of procedures, with many results derived from the properties of the normal distribution or closely related distributions.

Core assumptions of normal theory typically include linear relationships between variables, independence of observations, homoscedasticity (constant error variance), and normality of residuals. Under these assumptions, estimators such as least squares or maximum likelihood have well-defined sampling distributions, often normal in finite samples or asymptotically, which supports hypothesis testing and interval estimation.

Common methods within normal theory include t-tests for comparing means, F-tests used in analysis of variance (ANOVA) and regression, and confidence intervals for model parameters. These methods underpin much of traditional experimental design, regression analysis, and other parametric analyses. Diagnostics in this framework often focus on assessing residual normality, homoscedasticity, and model specification to ensure the validity of the inferences.

Limitations of normal theory arise when assumptions are violated. Sensitivity to outliers and deviations from normality or constant variance can compromise test validity. In such cases, nonparametric or robust alternatives, as well as transformations or model adjustments, may be preferred. Normal theory remains a foundational reference point in classical statistics, contrasting with nonparametric, robust, and Bayesian approaches, and it is widely taught in statistical education and used in many applied settings.