The analysis of the electrochemical impedance spectroscopy (EIS) response of electrodes and interfaces in terms of a distribution function of relaxation times (DFRT) has become an important topic of basic and applied research. For systems such as oxide films, passive layers, inhibitors, coatings and biological tissues, that distribution may arise as a consequence of normal-to-surface distributions of physical properties such as the electrical resistivity. The aim of this work is to relate the frequency response of those systems to the statistical properties of the distribution function, focusing on frequency dispersion behavior. The theory presented here is successfully applied to well-known models widely employed in the literature to interpret frequency dispersion in films (Young model, CPE behavior and ZARC response), but the analysis is also successful in predicting the response from arbitrary distributions such as the lognormal, Weibull and Cauchy distributions. The results apply mainly to the asymptotic frequency behavior. It is also shown that when the range of values of the distributed property is bounded, then there appear two critical frequencies at which a crossover from frequency dispersion towards ideal response takes place. (C) 2018 Elsevier Ltd. All rights reserved.