The degree to which consumers treat different options as distinct or differentiated is a key determinant of market competition and pricing. To facilitate the measurement of differentiation, we develop a flexible yet tractable model of random choice in a multi-attribute setting. We show the analyst can separately identify vertical and horizontal differentiation from binary comparison data alone. We characterize the binary choice rules that arise from our model using four easily understood axioms. In multinomial choice, we show the intersection of our model with the classic random utility framework yields random coefficients with an elliptical distribution. We provide applications to consumer demand with differentiated products and to measuring the complexity faced by an agent in individual decision-making problems.