We characterize the response of a chaotic system by investigating ensembles of, rather than single, trajectories. Time-periodic stimulations are experimentally and numerically investigated. This approach allows detecting and characterizing a broad class of coherent phenomena that go beyond generalized and phase synchronization. In particular, we find that a large average response is not necessarily related to the presence of standard forms of synchronization. Moreover, we study the stability of the response, by introducing an effective method to determine the largest nonzero eigenvalue -γ1 of the corresponding Liouville-type operator, without the need of directly simulating it. The exponent γ1 is a dynamical invariant, which complements the standard characterization provided by the Lyapunov exponents.