It is well known that Bessel's correction creates an unbiased estimator of variance. What it basically does is divide by $n-1$ instead of $n$.
Now what I did is that I chose a few number, like $1,2,3,4,5,60$ and calculated it's population variance which is $452.92$. I then took all possible 4-combinations (15 altogether) and calculated their sample variance (dividing by $n-1$) respectively. The average sample variance is $543.5$ which is off by $90.58$.
When I take the population variance (dividing by $n$) of the samples instead I get an average variance of $407.63$ which is off by only $45.29$!
I did several other experiments of the same kind with different numbers, population and sample sizes, all with this strange result that the population variance of the samples is less biased than the so called unbiased sample variance.
How can that be? What am I missing?
Because of the illuminating discussion in the comments I posted this follow-up question:
Unbiased estimator of variance for samples *without* replacement