Distributions of binding energies of a water molecule in the water liquid-vapor interface are obtained on the basis of molecular simulation with the SPC/E model of water. These binding energies together with the observed interfacial density profile are used to test a minimally conditioned Gaussian quasi-chemical statistical thermodynamic theory. Binding energy distributions for water molecules in that interfacial region clearly exhibit a composite structure. A minimally conditioned Gaussian quasi-chemical model that is accurate for the free energy of bulk liquid water breaks down for water molecules in the liquid-vapor interfacial region. This breakdown is associated with the fact that this minimally conditioned Gaussian model would be inaccurate for the statistical thermodynamics of a dilute gas. Aggressive conditioning greatly improves the performance of that Gaussian quasi-chemical model. The analogy between the Gaussian quasi-chemical model and dielectric models of hydration free energies suggests that naive dielectric models without the conditioning features of quasi-chemical theory will be unreliable for these interfacial problems. Multi-Gaussian models that address the composite nature of the binding energy distributions observed in the interfacial region might provide a mechanism for correcting dielectric models for practical applications.