Non-parametric methods are used when the assumptions of normality of the residuals and equal variation in each group, required for parametric methods such as the t-test and ANOVA, are not met and can not be met by a suitable transformation of scale.
The more widely used methods replace each observation by its rank in the total set of observations. This comes with a cost. In general, non-parametric tests are not as powerful as parametric tests because, in transforming the data to ranks, they throw out some useful information. This means that a non-parametric test may fail to detect a true treatment effect which could be detected by a parametric method. Therefore, where possible, parametric methods should be used. Brief notes on three commonly used tests are:
The Mann-Whitney test
This the the non-parametric equivalent of Student's t-test, but it is a test of whether the medians (rather than the means) are the same in each group. Like the t-test it is only appropriate when comparing two groups
The Kruskal-Wallis test
This is the non-parametric equivalent of the one-way ANOVA. It can be used to compare the medians of two or more groups, the null hypothesis being that all are samples from the same population. If the over-all differences are statistically significant, post-hoc comparisons are done by comparing each pair of medians in turn using the same test.
The Friedman's test
This is the non-parametric equivalent of the two-way ANOVA without interaction, appropriate for a randomised block experimental design. Again, it tests the null hypothesis that all treatment groups came from the same population, and is a test of group medians, removing any block effect.
Numerical examples are not given here. The methods are described in most statistical textbooks and are available in most computer packages.