Parametric vs non-parametric tests, and correlation hypotheses

In short

• A parametric test concerns a population parameter (such as $\rho$) and/or relies on a distributional assumption, usually normality.
• A non-parametric test is not about a parameter, or needs only minimal assumptions about the population.
• Go non-parametric when the data break the distributional assumptions, contain outliers, come as ranks, or the hypothesis is not about a parameter.
• The Pearson correlation $r = s_{XY}/(s_X s_Y)$ measures linear association; its sign comes from the covariance.
• Correlation hypotheses compare $\rho$ with zero: two-sided ($\rho \neq 0$), right-sided ($\rho > 0$) or left-sided ($\rho < 0$).
• This module has three tests: the t-test on Pearson r (parametric), the Spearman rank test and the chi-square test of independence (both non-parametric).