A parametric test makes an assumption about the population parameters and the distributions that the data came from. Parametric tests include students’ T-tests and ANOVA, which assume the data comes from a normal distribution. The opposite is a non-parametric test like the Mann-Whitney or Kruskal Wallis, which tend to make fewer assumptions about those population parameters. They are handy when you think you’re going to use a parametric test, which is always preferable, but for one reason or another, usually a small sample size, you can’t. Every parametric test has a non-parametric equivalent shown in this table. For example, if you have parametric data from two independent groups, you can run a two sample T-test to compare means, or for non-parametric data you can run a Mann-Whitney test instead.
Parametric tests are preferable because they’re more powerful, that is they are more likely to detect true differences between samples. they also have a lower probability of type two error, which is where you fail to reject a false null hypothesis. Let’s take a look at some common assumptions for these tests. In order to use a parametric test, your data has to be on the interval or ratio scale. If you have data that’s nominal or ordinal, you need to use the non-parametric version of the test. In general, your data should be normally distributed, if you want to use a parametric test. For a non-parametric test, your data does not have to be normal. If you have outliers in your data, you’ll want to choose a nonparametric test. Parametric tests require equal variances. If your variances are unequal, or if you don’t know those variances, you have to use a non-parametric test. If you have a small sample that’s generally under about 30 items in your sample, then you are going to have to use a non-parametric test.
That’s some general guidelines. Each of these tests is going to have their own individual assumptions. For example, the Mann-Whitney test requires you to have the same distribution between the two samples. For example, both samples might fit a normal distribution or a T distribution. For the Wilcoxon signed rank, both distributions have to be symmetrical. Kruskal Wallis and Friedman’s require the same distribution and equal variances. For Spearman’s R, your data must be interval, ratio or ordinal. In addition, you have to have a monotonic relationship between your data. In general, monotonic means that when one variable goes up, the other also goes up, or one variable goes down, and the other goes down.