Nucleosome positioning DNA sequences are of increasing interest because of their proposed roles in gene regulation and other chromosome functions in vivo, and because they have revealed new insights into the sequence-dependent structures and mechanics of DNA itself. Here, we describe methods to quantify the relative affinities of histone-DNA interactions in nucleosomes, i.e., the nucleosome positioning power of differing DNA sequences. We review methods developed by others and then discuss in detail our own approach to measurement of histone-DNA interaction free energies. Compared to earlier methods, our dialysis-based approach reduces the possibility that non-equilibrium or irreproducible results could be obtained. It facilitates a direct comparison of free energies for many sequences at the same time and it allows analysis of DNAs having a wide range of relative affinities.