Summary: Median as a resistant estimator of a distribution symmetry is widely applied in nonparametric problems. Lots of nonparametric tests use this characteristic (for instance: the test of sample randomness, the rank test of homogeneity of many samples, the test of one-dimensional normality).
Here, a median as numerical sample characteristic of position defined by two criteria, an arithmetic one and a geometric one, is presented. Ways of derivation are shown in both cases. Particular attention is paid to arithmetic definition with central and elliptical symmetry of a sample. There are lots of examples. The geometric definition resolves itself into the arithmetic one if a sample fulfils a condition of central symmetry.