The sensitivity of surface plasmon band location in noble metal nanoparticles to the refractive index, n, of the medium is investigated using closed form approximations to the particle polarizability. Within its range of validity, quasi-static analysis indicates that single component nanoparticles, including hollow nanoshells, have peak wavelength sensitivities that are determined exclusively by band location and dielectric 'constants' of the metal, ε, and medium, n2. Among particle plasmons that peak in the frequency range where the real part of the metal dielectric function varies linearly with wavelength and the imaginary part is small and slowly varying, the sensitivity of the peak wavelength, λ*, to refractive index, n, is found to be a linearly increasing function of λ*, regardless of the structural features of the particle that determine λ*. The dependence of the sensitivity on band position is determined by the wavelength dependence of the real part of the particle dielectric function. The results are applicable to all particle shapes, including rods, disks, hexagons, chopped tetrahedra, and hollow nanoshells and are not limited to dipolar resonances. Modification of the quasi-static analysis to account for electrodynamic effects to second order in the size parameter indicates that the structural independence of the refractive index sensitivity extends to larger nanoparticles than those accurately represented by quasi-static theory. The bulk refractive index sensitivity yielded by the theory serves as an upper bound to sensitivities of nanoparticles on dielectric substrates and sensitivities of nanoparticles to local refractive index changes, such as those associated with biomolecule sensing.