If muμ is the mean of the population, the formula for the population standard deviation of the population data x_{1},x_{2},x_{3},\ldots, x_{N} is
sigma=sqrt{\frac{sum_{k=1}^{N}(x_{k}-mu)^{2}}{N}}.
If bar{x} is the mean of a sample, the formula for the sample standard deviation of the sample data x_{1},x_{2},x_{3},\ldots, x_{n} is
s=sqrt{\frac{sum_{k=1}^{n}(x_{k}-bar{x})^{2}}{n-1}}.
The reason this is done is somewhat technical. Doing this makes the sample variance s^{2} a so-called unbiased estimator for the population variance sigma^{2}. In effect, if the population size is really large and you are doing many, many random samples of the same size n from that large population, the mean of the many, many values of s^{2} will have an average very close to the value of sigma^{2} (and, as far as a theoretical perspective goes, the mean of s^{2} as a "random variable" will be exactly sigma^{2}).
The technicalities for why this is true involve lots of algebra with summations, and is usually not worth the time spent for beginning students.
