Norms about hygiene and violence have both shown a tendency to become increasingly strict, in the sense that the handling of bodily fluids and the use of violence have become increasingly restricted. The generality of this directional change across a large number of societies has not been captured by previous explanations. We propose an explanation of the directional change that is based on the aggregation of everyday interactions. This theory posits that directional norm change can come about if there is an asymmetry in punishment propensity between the people who prefer stricter norms and those who prefer looser norms. Asymmetry in punishment can arise from underlying asymmetry in the threat perceived, where a stricter-than-preferred behaviour is perceived as inherently less threatening than a looser one. We demonstrate the logic of the theory using a formal model and test some of its assumptions through survey experiments.