I received an email inquiry about degrees of freedom. I explain the concept briefly, but this person wanted a more detailed answer to the question, why do we use n-1 in the calculation of the standard deviation and not n?
You can argue that the standard deviation involves the projection onto a subspace of dimension n-1, because the n deviations from the mean have a single constraint--they must sum to zero. That may not be a very satisfying answer.
Another answer is that using n-1 provides a good statistical property, unbiasedness, but that applies to the variance and not to the standard deviation.
Some statisticians (not many) have argued that you should not use n-1, but should use n instead. They make a persuasive argument, but since almost everyone uses n-1, why swim against the tide? There are more important battles to fight, such as the mindless acceptance of an alpha level of .05 for every hypothesis test ever done.
Some web pages that talk about degrees of freedom are:
You can find an earlier version of this page on my original website.