The standard deviation is defined as a positive square root
of variance. It is calculated if there is available complete set of possible
states of the process (system).
The sample standard deviation is defined as the square root of sample
variance. It is calculated from the selection, i.e. if there is not
available complete set of possible states but just the selection of them.
Therefore, there must be used n-1 (because the one measuring is dependent,
with the calculation of mean value). For a large number of measurements, the
difference between standard deviation and the sample standard deviation
decreases.
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