Statistical models are mathematical expressions (or sometimes sets of expressions) that either quantify the statistical properties of observed values (e.g., mean, standard deviation, principal components, etc., of the observed values), or quantify empirical relations between two or more observed variables (e.g., regression, analysis of variance, factor analysis, etc.). Statistics is closely associated with probability theory and the uncertainty of events. For example, a key issue for most statistical models is the proportion of variation in the data that is explained by the model. Conversely, the unexplained variation is often used to describe confidence intervals about predicted values. Concepts of verification and validation are important in the context of these kinds of models. Verification provides confirmation that the model faithfully represents the data originally used to derive the model. For example, a plot of residuals does not display a discernable pattern which would suggest some type of systematic error in predictions. In contrast, validation provides confirmation that the model accurately predicts outcomes for new, independent data.
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