  We develop a cavity-based method which allows to extract thermodynamic
properties from position information in hard-sphere/disk systems. So far, there
are 'available-volume' and 'free-volume' methods. We add a third one, which we
call 'available-volume-after-takeout', and which is shown to be mathematically
equivalent to the others. In applications, where data sets are finite, all
three methods show limitations, and they do this in different parameter ranges.
We illustrate the principal equivalence and the limitations on data from
molecular dynamics -- In particular, we test robustness against missing data.
We have in mind experimental limitations where there is a small polydispersity,
say 4% in the particle radii, but individual radii cannot be determined. We
observe that, depending on the used method, the errors in such a situation are
easily 100% for the pressure and 10kT for the chemical potentials. Our work is
meant as guideline to the experimentalist for choosing the right one of the
three methods, in order to keep the outcome of experimental data analysis
meaningful.
