This simulation draws repeated random samples from a population. We calculate the variance for each sample using both the biased ($n$ denominator) and unbiased ($n-1$ denominator) formulas. Watch how the running average of the unbiased estimates converges on the true population variance, while the biased estimates systematically fall short.
Restricted to $\ge 30$ to keep the chart scale legible.
How many random samples to generate per click.
The chart shows the running average of the variance estimates as more samples are drawn. Notice how the blue line ($n-1$) converges to the true variance (green dashed line), while the red line ($n$) stays below it. The individual scatter points show the high variability of single samples.
Draw samples to see calculation details here.