The composition of ultra-high energy cosmic rays is an important issue in astroparticle physics research, and additional experimental results are required for further progress. Here we investigate what can be learned from the statistical correlation factor r between the depth of shower maximum and the muon shower size, when these observables are measured simultaneously for a set of air showers. The correlation factor r contains the lowest-order moment of a two-dimensional distribution taking both observables into account, and it is independent of systematic uncertainties of the absolute scales of the two observables. We find that, assuming realistic measurement uncertainties, the value of r can provide a measure of the spread of masses in the primary beam. Particularly, one can differentiate between a well-mixed composition (i.e., a beam that contains large fractions of both light and heavy primaries) and a relatively pure composition (i.e., a beam that contains species all of a similar mass). The number of events required for a statistically significant differentiation is ~ 200. This differentiation, though diluted, is maintained to a significant extent in the presence of uncertainties in the phenomenology of high energy hadronic interactions. Testing whether the beam is pure or well-mixed is well motivated by recent measurements of the depth of shower maximum.
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