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.
Answer from Mike Spivey on Stack ExchangeVideos
What Is Population Standard Deviation?
The population standard deviation measures the variability of data in a population. It is usually an unknown constant. σ (Greek letter sigma) is the symbol for the population standard deviation.
What Is The Formula of Population Standard Deviation?
The following is the population standard deviation formula:
Where:
σ = population standard deviation
x1, ..., xN = the population data set
μ = mean of the population data set
N = size of the population data set
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.
Answer from Mike Spivey on Stack Exchange