Parametric Statistics

Parametric statistics are a class of statistical methods that make specific assumptions about the distribution of the underlying population from which the data are drawn, most commonly that the data follow a normal (Gaussian) distribution. These methods use population parameters such as means and variances to describe distributions and test hypotheses.

Parametric statistical methods, including the t test, analysis of variance (ANOVA), Pearson correlation, and linear regression, are the workhorses of quantitative research across many disciplines. They are preferred over nonparametric alternatives when their assumptions are met because they have greater statistical power, meaning they are better able to detect true effects and produce narrower confidence intervals with the same sample size. However, when the assumptions of parametric tests are violated, particularly with small samples, the results may be unreliable: significance tests may reject too often or too rarely, and confidence intervals may be too narrow or too wide. Understanding when parametric methods are appropriate and when nonparametric alternatives should be used is a fundamental skill for every quantitative researcher.

Parametric methods derive their name from the fact that they assume a distribution described by a fixed set of population parameters (such as the mean and variance of a normal distribution) and estimate those parameters from sample data. The validity of these estimates and the associated hypothesis tests depend on several assumptions.

The normality assumption requires that the data (or, more precisely, the residuals) follow a normal distribution. This assumption can be assessed visually (using histograms, Q Q plots, or box plots) or statistically (using tests such as the Shapiro Wilk test or Kolmogorov Smirnov test, though these have limitations). Parametric methods are generally considered resistant to moderate departures from normality when sample sizes are large, a property that rests on the central limit theorem, which states that the sampling distribution of the mean approaches normality as the sample size increases for independent observations from any population with a finite variance.

The homogeneity of variance (homoscedasticity) assumption requires that the variability of the data is similar across groups or across the range of a predictor variable. In the context of comparing groups (t tests, ANOVA), this means that the groups have approximately equal variances. In regression, it means that the spread of residuals is roughly constant across predicted values. Levene's test is commonly used to assess this assumption. Violations of this assumption can be addressed through variance stabilizing transformations, the use of Welch's adjustment (for t tests and ANOVA), or heteroscedasticity consistent standard errors (for regression).

The independence assumption requires that observations are not systematically related to one another. Violations occur when data have hierarchical or clustered structures (students within classrooms, measurements within patients) or when repeated measurements are taken from the same individuals. These situations require specialized methods such as mixed effects models or generalized estimating equations rather than standard parametric tests that assume independence.

When parametric assumptions are seriously violated and cannot be remedied through transformations or resistant methods, nonparametric tests (such as the Mann Whitney U test, Kruskal Wallis test, or Spearman correlation) provide alternatives that make fewer distributional assumptions, at the cost of a small loss of power only when the parametric assumptions actually hold.

Use parametric methods when the data reasonably satisfy the required assumptions, when sample sizes are sufficiently large (the central limit theorem provides protection against moderate nonnormality in samples of 30 or more, depending on the degree of skewness), and when you want to estimate population parameters and test hypotheses about means. When assumptions are in doubt, report the results of assumption checks and consider running both parametric and nonparametric analyses as a sensitivity check.

A frequent mistake is applying parametric tests to data that clearly violate assumptions (such as strongly skewed data with small samples) without checking or reporting assumption diagnostics. Another error is reflexively switching to nonparametric methods whenever the Shapiro Wilk test yields a significant result, without considering that significance tests for normality are overly sensitive in large samples and that moderate departures from normality have little practical impact on parametric tests when the sample is large. Failing to check homogeneity of variance when comparing groups is a further oversight. Some researchers also mistakenly believe that the raw data must be normally distributed for parametric tests, when in fact it is the residuals (the deviations from the group means or fitted values) that should be approximately normal, and in large samples the sampling distribution of the test statistic that matters. Finally, applying standard parametric tests to clustered or repeated measures data without accounting for the nonindependence is a serious error that can inflate Type I error rates.

An educational psychologist compares test scores between three teaching methods (lecture, collaborative learning, and flipped classroom) using a one way ANOVA. Before conducting the test, she examines histograms and Q Q plots of the residuals, which show approximately normal distributions. She runs Levene's test, which yields p = 0.34, indicating no significant difference in variances. With these checks raising no concern and a total sample of 120 students (40 per group), the ANOVA is appropriate. The result is F(2, 117) = 7.84, p < 0.001, and she follows up with Tukey's HSD post hoc comparisons to determine which groups differ.