A general methodology is presented for the consideration of both data and model uncertainty in the determination of the response of geometrically nonlinear structural dynamic systems. The approach is rooted in the availability of reduced order models of these nonlinear systems with a deterministic basis extracted from a reference model (the mean model). Uncertainty, both from data and model, is introduced by randomizing the coefficients of the reduced order model in a manner that guarantees the physical appropriateness of every realization of the reduced order model, i.e. while maintaining the fundamental properties of symmetry and positive definiteness of every such reduced order model. This randomization is achieved not by postulating a specific joint statistical distribution of the reduced order model coefficients but rather by deriving this distribution through the principle of maximization of the entropy constrained to satisfy the necessary symmetry and positive definiteness properties. Several desirable features of this approach are that the uncertainty can be characterized by a single measure of dispersion, affects all coefficients of the reduced order model, and is computationally easily achieved. The reduced order modeling strategy and this stochastic modeling of its coefficients are presented in details and several applications to a beam undergoing large displacement are presented. These applications demonstrate the appropriateness and computational efficiency of the method to the broad class of uncertain geometrically nonlinear dynamic systems.