Many physical properties of concentrated dispersions of immiscible fluids are captured by the concept of an osmotic pressure, which measures how much energy is required to deform the bubbles or drops upon compaction. This pressure has a strong impact on the flow and drainage behavior of dispersions. Nevertheless, theoretical models describing its variation with the volume fraction phi of the continuous phase are so far available only in the limits of low or high phi and experimental data are scarce. We report an experimental study of osmotic pressure in foams and emulsions, showing how the effects of phi, disorder, grain size, polydispersity and interfacial tension can all be captured by a single law which satisfies previously established theoretical constraints. Building on this result, we propose the first equation which accurately describes the variation of the volume fraction with the height of a fluid dispersion under gravity.