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The formula for standard deviation must not be confused with the formula
The formula for standard deviation must not be confused with the formula
:<math>S_{n} = \sqrt {\sum_n(x - \bar x)^2 \over n - 1}</math>
:<math>S_{n} = \sqrt {\sum_n(X_n - \bar X)^2 \over n - 1}</math>
(where <math>\bar X = {\sum_n X_n \over N}</math> is the [[sample mean]]).
which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.
which is the formula for a [[point estimate]] of the true standard deviation from a sample size of ''n''. As such this estimator itself has a variance which, as the formula indicates, decreases as the sample size increases.