In this article, we study a data-driven control method for the robust output regulation problem. The method exploits the parameterization property of a system from the observed data that do not necessarily satisfy the persistently exciting condition for system identification. It is based on a data informativity theory, which infers a system property for designing a controller for a whole family of systems (including the true system) that can generate the same observed data. The new idea is to explore the necessary and/or sufficient conditions for the data being informative for transmission zeros, stabilizability, and augmented stabilizability, which can be integrated to derive the conditions for data informativity for robust output regulation. Moreover, under these conditions, a dynamic state feedback data-driven controller is explicitly designed for achieving output regulation with robustness against variation of system matrices and noise in observed data.