We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of nonnegative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We establish fundamental inequalities that relate the symmetric mean of a list of nonnegative real numbers with the barycenter of the measure uniformly supported on these points. As a consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converges to the Halász-Székely barycenter of the corresponding distribution. This is joint work with Jairo Bochi and Mario Ponce.