It has long been suggested that magnetohydrodynamic (MHD) waves may supply a significant proportion of the energy required to heat the corona and accelerate the solar wind. Depending on the properties of the local plasma, MHD wave modes may exhibit themselves as a variety of incompressible, transverse waves. The local magnetic field and particle density influence the properties of these waves (e.g., amplitude), thus direct measurements of transverse waves provide a mechanism to indirectly probe the local plasma conditions. We present the first statistical approach to magnetoseismology of a localized region of the solar corona, analyzing transverse waves above the south polar coronal hole on 2011 May 23. Automated methods are utilized to examine 4 hr of EUV imaging data to study how the waves evolve as a function of height (i.e., altitude) through the low corona. Between heights of 15 and 35 Mm, we find that the measured wave periods are approximately constant, and that observed displacement and velocity amplitudes increase at rates that are consistent with undamped waves. This enables us to derive a relative density profile for the coronal hole environment in question, without the use of spectroscopic data. Furthermore, our results indicate that between 5 and 15 Mm above the limb, the relative density is larger than that expected from 1D hydrostatic models, and signals a more extended transition region with a gradual change in density. This has implications for self-consistent models of wave propagation from the photosphere to the corona and beyond.