In this paper, we will study the statistical theory of shape for ordered finite point configurations, or otherwise stated, the uncertainty of geometric invariants. A general approach for defining shape and finding its density, expressed in the densities for the individual points, is developed. Some examples, that can be computed analytically, are given, including both affine and positive similarity shape. Projective shape and projective invariants are important topics in computer vision and are discussed at the end of the paper.