Questions pupils ask: Why do we divide by n-1?

In these ‘Questions pupils ask’ articles, I often address unusual – even obscure – questions that pupils sometimes ask, because I find them interesting and thoughtprovoking for reflecting on teaching. But here I want to address a question that is not like this – I think this is a question every thoughtful student of statistics is bound to ask. But perhaps this is a feature of how I teach standard deviation, and if I taught it better maybe no one would ask this question?