The median test is a non-parametric test that is used to test whether two (or more) independent groups differ in central tendency - specifically whether the groups have been drawn from a population with the same median. The null hypothesis is that the groups are drawn from populations with the same median. The alternative hypothesis can be either that the two medians are different (two-tailed test) or that one median is greater than the other (one-tailed test).

The principle of the test is that if two samples have the same median, they should have more or less the same proportion of observations above and below that median. This would be true irrespective of their two distributions.
If any scores fall at the value of the combined median they may either be dropped from the analysis, or included with scores less than the median.

The advantage of median test over Mann Whitney is that it only tests for differences in the median irrespective of any differences in the shape of the distribution. However, it has much less power. Siegel & Castellan note there is no alternative to median test when one or more observations are off the scale (e.g. censored observations), it is also more robust to outliers.