A chance encounter between members of a random repertoire and a molecular target is characteristic of different biological systems, including the immune and olfactory pathways as well as combinatorial libraries. In such systems, the affinity between the target and members of the repertoire is distributed with a probability function describing the propensity of obtaining a particular affinity value. We have previously proposed a phenomenological receptor affinity distribution (RAD) formalism, which describes this probability function based on simple statistical considerations. In the present analysis, we use published data from diverse experimental systems, including phage display libraries, immunoglobulins and enzymes, to test the RAD model and to compare it to other affinity distribution formalisms. The RAD model is found to provide the best description for binding data for over eight orders of magnitude on the affinity scale, and to account for a relationship between repertoire size and the maximal obtainable affinity within different repertoires. This approach points to a potential universality of the rules that govern affinity distributions in biology.