There are many statistical systems in which degrees of freedom can be grouped according to their degree of correlation. We develop a cluster-expansion method for these systems based on selecting finite clusters of the most strongly interacting degrees of freedom. The starting point is Kubo's generalized cumulant formalism. We show how this expansion can be renormalized; that is, how the contributions of infinite classes of articulated finite clusters can be resummed. The elimination of articulated clusters from the renormalized expansion generates modified interactions in the finite clusters remaining in the expansion. When truncated at different orders, the final result is a hierarchy of mean-field approximations, which generally reproduce an increasing number of terms in exact series expansions for the system. High-order series-expansion coefficients can be obtained by perturbative renormalization of very large clusters, a convenient alternative to the complicated combinatorial analysis associated with standard expansion techniques for lattice models. We compare our work to previous finite-cluster techniques, such as the cluster-variation method of Kikuchi.