Biased by Design

When fitting a Gaussian to n samples via maximum likelihood, the MLE for the variance is $\sigma^2_{MLE} = \tfrac{1}{n} \sum_i (x_i - \bar{x})^2$ . Compared to the unbiased sample variance (dividing by $n-1$ ), this estimator is:

Unbiased only when the mean is known in advance

Biased low, systematically underestimating the true variance

Biased high, systematically overestimating the true variance

Unbiased, identical in expectation to the n-1 estimator