In literature it is common to think about segregation as the outcome of a stochastic allocation process, characterized by the preferences and the constraints of both the individuals and the allocation units. In this perspective, several authors have already documented, using simulations, that the observed value of the Duncan & Duncan dissimilarity index is an upward biased estimator of the true population counterpart. In this paper, following a multinomial framework, we provide a mathematical proof of the nonnegativity of this bias.