The analytical expressions used in ZAF and ϕ(ρz) calculations give single values for the composition of each element for a single set of intensity measurements from samples and standards. Confidence intervals in composition are established by looking at the variability of repeated measurements. They are usually attributed to x-ray counting statistics or experimental reproducibility factors such as sample repositioning. Uncertainty in the equations themselves or the parameters that go into them are rarely considered. The derivations of ZAF and ϕ(ρz) equations are primarily based on the case where flat single-phase regions, relative to the x-ray excitation volume, are examined using normal electron beam incidence. Use of these equations has been extended to non-normal electron beam incidence as well as the quantitative analysis of layered structures, but usually with less theoretical justification. Finally, special experimental cases including porous structures, rough surfaces, vertical interfaces and small particles are very difficult or impossible to model by the single application of a set of analytical equations to convert measured x-ray intensities to elemental composition.