Sample and population Variance and SD Watch

Yes, this is one of those nasty little facts that creeps up behind you and bops you on the head. One of the things that it tells you is that finding unbiased estimators for things can be hard; a second thing that is less often appreciated is that unbiased estimators may not be the nirvana one is searching for. If one is doing predictive estimation, it is not unusual for a biased predictor to give a lower RMS prediction error than an unbiased one.

It's a nice problem in mathematical statistics (which is sometimes used to torture students) to show what the expected value for the sample standard deviation is when dealing with Normal random variables.

I'm not sure what you mean by saying that a sample is "near the mean" - the expected value of the empirical distribution function of a random sample is the population distribution. The usual explanation of the n-1 correction is that you're using the sample mean to estimate the population mean in the calculation of the sample standard deviation - and that needs a correction to take it into account.

The basic point here is that need not be equal to for a general function f. Just write the equations down and you'll see that there's no reason to expect them to be equal.

If f is linear, all is sweetness and light, and if X is specified, it's sometimes possible to calculate how these two will differ.