There are, in fact, two different formulas for standard deviation here: The population standard deviation and the sample standard deviation .

If denote all values from a population, then the (population) standard deviation is where is the mean of the population.

If denote values from a sample, however, then the (sample) standard deviation is where is the mean of the sample.

The reason for the change in formula with the sample is this: When you're calculating you are normally using (the sample variance) to estimate (the population variance). The problem, though, is that if you don't know you generally don't know the population mean , either, and so you have to use in the place in the formula where you normally would use . Doing so introduces a slight bias into the calculation: Since is calculated from the sample, the values of are on average closer to than they would be to , and so the sum of squares turns out to be smaller on average than . It just so happens that that bias can be corrected by dividing by instead of . (Proving this is a standard exercise in an advanced undergraduate or beginning graduate course in statistical theory.) The technical term here is that (because of the division by ) is an unbiased estimator of .

Another way to think about it is that with a sample you have independent pieces of information. However, since is the average of those pieces, if you know , you can figure out what is. So when you're squaring and adding up the residuals , there are really only independent pieces of information there. So in that sense perhaps dividing by rather than makes sense. The technical term here is that there are degrees of freedom in the residuals .

For more information, see Wikipedia's article on the sample standard deviation.

There are, in fact, two different formulas for standard deviation here: The population standard deviation and the sample standard deviation .

If denote all values from a population, then the (population) standard deviation is where is the mean of the population.

If denote values from a sample, however, then the (sample) standard deviation is where is the mean of the sample.

The reason for the change in formula with the sample is this: When you're calculating you are normally using (the sample variance) to estimate (the population variance). The problem, though, is that if you don't know you generally don't know the population mean , either, and so you have to use in the place in the formula where you normally would use . Doing so introduces a slight bias into the calculation: Since is calculated from the sample, the values of are on average closer to than they would be to , and so the sum of squares turns out to be smaller on average than . It just so happens that that bias can be corrected by dividing by instead of . (Proving this is a standard exercise in an advanced undergraduate or beginning graduate course in statistical theory.) The technical term here is that (because of the division by ) is an unbiased estimator of .

Another way to think about it is that with a sample you have independent pieces of information. However, since is the average of those pieces, if you know , you can figure out what is. So when you're squaring and adding up the residuals , there are really only independent pieces of information there. So in that sense perhaps dividing by rather than makes sense. The technical term here is that there are degrees of freedom in the residuals .

For more information, see Wikipedia's article on the sample standard deviation.