For any random variable \(X\) with finite expectation \(EX\) and for any convex function \(f\), the well-known Jensen's inequality \(f(EX) \leq Ef(X)\) holds, playing a significant role in probability and statistics theory. The aim of this note is to present an analogue of Jensen's inequality, where expectation is replaced by median. The author proves that Jensen's inequality for medians is satisfied by a class of functions that encloses the convex functions as a proper subset (the \(C\)-function class is very close to lower semi-continuous functions). The main tool for deriving the new type of inequality is a novel characterization of a median used as a new definition for the median, which entails also the natural extension of the Jensen's inequality to the higher-dimensional form (for multivariate distributions and multivariate \(C\)-functions).