The dependence on the temperature of the population of the "i"th state, "P"[subscript "i"], in the Boltzmann distribution is analyzed by studying its derivative with respect to the temperature, "T." A simple expression is found, involving "P"[subscript "i"], the energy of the state, "E"[subscript "i"], and the average energy, <"E">. This relation is completely general (it has the same form in all the thermodynamic ensembles), and it has a relevant didactic content, given that it predicts the qualitative variation of "P"[subscript "i"] with "T" even in complex systems. The derivation of this relation, the discussion of its properties, and its application to simple problems is appropriate for a statistical thermodynamics course in the chemistry curriculum.