We propose a novel concept of rationalization, called coarse rationalization, tailored for the analysis of datasets where an agent's choices are imperfectly observed. We characterize those datasets, which are rationalizable in this sense and present an efficient algorithm to verify the characterizing condition. We then demonstrate how our results can be applied through a duality approach to test the rationalizability of datasets with perfectly observed choices but imprecisely observed linear budget sets. For datasets that consist of both perfectly observed feasible sets and choices but are inconsistent with perfect rationality, our results could be used to measure the extent to which choices or prices have to be perturbed to recover rationality.