The rank aggregation problem has been studied with varying desiderata in varied communities such as Theoretical Computer Science, Statistics, Information Retrieval and Social Welfare Theory. We introduce a variant of this problem we call distributional rank aggregation, where the ranking data is only available via the induced distribution over the set of all permutations. We provide a novel translation of the usual social welfare theory axioms to this setting. As we show this allows for a more quantitative characterization of these axioms: which then are not only less prone to misinterpretation, but also allow simpler proofs for some key impossibility theorems. Most importantly, these quantitative characterizations lead to natural and novel relaxations of these axioms, which as we show, allow us to get around celebrated impossibility results in social choice theory. We are able to completely characterize the class of positional scoring rules with respect to our axioms and show that Borda Count is optimal in a certain sense.