The sample variance is an unbiased estimator of the parametric variance.
Only if you define sample variance as the estimator that irchans has given above which divides by N-1. Many books define the sample variance as the one that divides by N. That is a biased estimator, and this is a constant source of confusion.
I found an online "math encyclopedia" reference that erroneously states that the square root of the unbiased estimator of the variance is an unbiased estimator of the the standard deviation.
Idiots!