Statistical methods of describing and simulating complex ionized plasmas requires the development of reliable and computationally tractable models. In that spirit, we propose the screened-hydrogenic average atom, augmented with corrections resulting from fluctuations of the occupation probabilities around the mean-field equilibrium, as an approximation to calculate the grand potential and related statistical properties. Our main objective is to check the validity of this approach by comparing its predictions with those given by the superconfiguration accounting method. The latter is well-suited to this purpose. In effect, this method makes it possible to go beyond the mean-field model by using nonperturbative, analytic, and systematic techniques. Besides, it allows us to establish the relationship between the detailed configuration accounting and the average-atom methods. To our knowledge, this is the first time that the superconfiguration description has been used in this context. Finally, this study is also the occasion for presenting a powerful technique from analytic number theory to calculate superconfiguration averaged quantities.