Why do you compute the standard deviation s of a sample set by dividing a summation by N-1, instead of dividing it by N, as you would do in computing the mean of this very same sample set?

Here is why:

Because the computation of s involves an inherent comparison of this sample set of N elements

A = { x_{1}, x_{2}, …, x_{N} }

with a nonexisting but presupposed sample set of a single element:

B = { x̄ }

i.e. with a set that only contains the mean point of these N samples.

Just like we have to subtract x̄from each of the x_{i}s in the numerator, we need to subtract the 1 that is the count of the x̄from the N that is the count of these x_{i}s in the denominator.

In other words, we begin counting not from an empty set of 0 samples where standard deviation is undefined, but from a set of 1 sample^{*} where standard deviation is zero by definition.

Therefore, a more general equation for standard deviation would be^{†}:

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