We have two estimators for samples dispersion:

$\frac{1}{n}\sum_1^n (X_i - \overline{X_n})^2$ and $\frac{1}{n - 1}\sum_1^n (X_i - \overline{X_n})^2$.

The second estimator unbiased - it has expectation equals to real dispersion.

I know the proof, but I think that there is some logic behind this formula, not only formal derivation. For example, if we have only one sample measure, then unbiased formula doesn't work. And it seems logical because one sample can be thrown from a distribution with any dispersion and we can't estimate it.

Maybe you know some simple explanation of the logic behind these formulas.

  • 1
    $\begingroup$ Your formulas do not appear to be measures of dispersion (though they might be in some special circumstance). Please offer some context for these. Where do they come from? $\endgroup$ – Glen_b May 19 '17 at 11:47
  • $\begingroup$ @Glen_b Sorry, I mixed up formulas. Now it's fine. My question is: why an average of deviations from mean give a biased estimation for dispersion? More precisely, can you give a simple and intuitive explanation of this fact? Because it looks like very simple and right idea, and we know, that for expectation this idea works fine. But here - it doesn't. $\endgroup$ – Nikita Sivukhin May 19 '17 at 13:14