Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Median Test

The median test (Mood's median test) is a nonparametric test that tests the null hypothesis that the medians of the populations from which two or more samples are drawn are identical. The test statistic is calculated by counting the number of data points which are below or above the common median of the groups. The resulting contingency table can be used to decide whether the observed frequencies deviate from the expected frequencies.

The test statistic is chi-square distributed. However, the chi-square distribution is a continuous distribution while the contigency table for a small number of observation results in discrete probability values which can differ considerably from the probabilities obtained from the chi-square distribution. Yates has therefore proposed a "continuity correction" [Yates 1934] which results in a better approximation to the true values obtained by Fisher's exact test.

In general it is recommended to apply Fisher's exact test for small samples (number of observations below 20) or heavily unbalanced samples.

The assumptions of the median test are that the observations are at least at an ordinal scale. Please note that the power of the median test is rather low compared to parametric tests.