We consider an inverse elasticity problem in which forces and displacements are known on the boundary and the material property distribution inside the body is to be found. In other words, we need to estimate the distribution of constitutive properties using the finite boundary data sets. Uniqueness of the solution to this problem is proved in the literature only under certain assumptions for a given complete Dirichlet-to-Neumann map. Another complication in the numerical solution of this problem is that the number of boundary data sets needed to establish uniqueness is not known even under the restricted cases where uniqueness is proved theoretically. In this paper, we present a numerical technique that can assess the sufficiency of given boundary data sets by computing the rank of a sensitivity matrix that arises in the Gauss-Newton method used to solve the problem. Numerical experiments are presented to illustrate the method.