We develop a method using a coarse graining of the energy fluctuations of an equilibrium quantum system which produces simple parametrizations for the behaviour of the system. As an application, we use these methods to gain more understanding on the recently developed Tsallis statistics and we show how energy reparametrizations can be used to produce other similar generalizations of the Boltzmann-Gibbs statistics. We conclude with a brief discussion on the role of entropy and the maximum entropy principle in thermodynamics and in quantum statistics.