Given a metric space $X$ and a finite subset $U \subset X$, we define a numerical invariant $\alpha$​, the ε-ultrametricity of $U$, as the proportion of unordered triples in $U$ that form $\epsilon$-ultrametric triangles. This invariant measures how close the set is to being an ultrametric space and, as we prove, is stable under small perturbations of the data. We show that adding a point from outside a cluster decreases this invariant under certain geometric assumptions. We turn this into a classification algorithm and test it on a real medical dataset (≈66% accuracy) and against a Mapper / topological data analysis cross-check. I'll close with an in-progress application to longitudinal cognitive-aging data (CLSA).