When you want to test for differences between two groups, the independent-samples t test comes naturally to mind. However, despite its simplicity, power, and robustness, the independent-samples t test is invalid when certain critical assumptions are not met. These assumptions center around the parameters of the test variable (in this case, the mean and variance) and the distribution of the variable itself.

Most important, the t test assumes that the sample mean is a valid measure of center. While the mean is valid when the distance between all scale values is equal, it's a problem when your test variable is ordinal because in ordinal scales the distances between the values are arbitrary. Furthermore, because the variance is calculated using squared distances from the mean, it too is invalid if those distances are arbitrary. Finally, even if the mean is a valid measure of center, the distribution of the test variable may be so non-normal that it makes you suspicious of any test that assumes normality.

If any of these circumstances is true for your analysis, you should consider using the nonparametric procedures designed to test for the significance of the difference between two groups. They are called nonparametric because they make no assumptions about the parameters of a distribution, nor do they assume that any particular distribution is being used. Two popular nonparametric tests of location--the Mann-Whitney and Wilcoxon tests--and a test of location and shape--the two-sample Kolmogorov-Smirnov test--are illustrated.

Mann-Whitney and Wilcoxon tests. You can use the Mann-Whitney and Wilcoxon statistics to test the null hypothesis that two independent samples come from the same population. Their advantage over the independent-samples t test is that Mann-Whitney and Wilcoxon do not assume normality and can be used to test ordinal variables.

Two-sample Kolmogorov-Smirnov test. Alternatively, you can use the two-sample Kolmogorov-Smirnov test to test the null hypothesis that two samples have the same distribution. The test variable is assumed to be continuous; however, its cumulative distribution function (CDF) can assume any shape at all.

In addition to their standard output, the Mann-Whitney and Wilcoxon and the two-sample Kolmogorov-Smirnov tests display descriptive statistics and/or quartiles of the test variable.