Why "n-1" in empirical variance? A simulation.

It is well-known that the empirical variance underestimates the population variance. Specifically, the empirical variance is defined as: (var_{emp} = \frac{\sum_i (x_i - \bar{x})^2}{n-1}). But why (n-1), why not just (n), as intuition (of some) dictates? Put shortly, as the variance of a sample tends to underestimate the population variance we have to inflate it artificially, to enlarge it, that’s why we do put a smaller number (the “n-1”) in the denominator, resulting in a larger value of the whole …