At the intersection of topology and computer science lies the field of topological data analysis(TDA). TDA uses the concept of persistence to identify low dimensional topological features of data embedded in high dimensions. We look at a specific tool in TDA called persistent homology for point cloud data, which modifies the classical theory of simplicial homology to study the topology of point clouds collected on some surface in R^N. For a point cloud in R^N, persistent homology allows us to compute a barcode for the data, a collection of intervals representing the changes in homology of an increasing simplicial complex based on the point cloud. We show, with a reformulation of a stability theorem, that we can significantly improve computational speed of an algorithm used to compute the persistent homology, with relatively little difference in the resulting barcode, by excluding points in the point cloud.