never2old4school: Sample variance

Wednesday, November 16, 2016

Sample variance

Well, last weekend was great, but I guess I need to get back to studying. I'm now moving from Probability Theory to Statistics. One stat that always seemed weird to me was the sample variance. Sure, I get the whole degrees of freedom thing, but it still seems odd that the sample variance gets the adjustment for the sample mean.

$\bar{X} = \frac{1}{n}\sum X_i\\ \\ S^2 = \frac{1}{n-1}\sum(X_i-\bar{X})^2$

Again, I get it. Because you the sample mean is centered on the sample, not the distribution from which the sample is pulled, dividing by n will bias the variance low. What's always been odd to me is that dividing by n - 1 fixes it. It's one of those results that I understand but still gnaws at me a bit. So, as a public service to anybody else who might be in the same boat, here's the proof that S² is, in fact, an unbiased estimator of the true variance, σ².

$\begin{align*} \textrm{E}(S^2) &= E \left(\frac{1}{1-n}\sum(X_i^2 - n\bar{X}^2) \right )\\ &= \frac{1}{n-1}(n\textrm{E}(X_1^2)-n\textrm{E}(\bar{X}^2))\\ &= \frac{1}{n-1}\left(n(\sigma^2 + \mu^2) - n\left(\frac{\sigma^2}{n}+\mu^2\right)\right)\\ &= \sigma^2 \end{align*}$

Feel better? Neither do I. Just learn it and get on with life.

1 comment:

PadminiprwatechFebruary 24, 2020 at 8:45 PM
Data science is one of the top course in today's career. Your content will going to helpful for all the beginners who are trying to find What is the Sample Variance. Thanks for sharing useful information. keep updating.
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