The screened-hydrogenic average-atom model is well suited to describe multicharged ion plasmas in local thermodynamic equilibrium (LTE) for in-line plasma physics calculations. Using general principles of statistical mechanics, this model is shown to be properly defined and thermodynamically consistent. The grand canonical partition function ${Z}_{G}$ of the bound electrons is written as a multidimensional integral. Its saddle-point evaluation gives the intuitive average-atom equations. Using this formalism, a method for accounting the various ionization stages of a LTE plasma is proposed. It can be used to estimate the integer charge stage distribution in this type of medium from any average-atom model. Once the model is well established, simpler formulas, more suitable for fast computations, are derived in the framework of the classical theory of fluctuations. Numerical results are presented and discussed.