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Suppose we have $n$ samples, with mean $\mu$.

Calculate the average absolute distance from $\mu$, i.e., $$ y = \frac{1}{n} \sum_{i=1}^n |X_i - \mu| \>. $$

Then, take as an estimate of the standard deviation $$ \tilde \sigma = \frac{1}{\sqrt{2\pi (1-1/n)}} y \>. $$

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So, presumably $\mu$ is known a priori? – cardinal Jul 30 '12 at 13:55
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Hint: Can you find the expectation $\mathbb E |X - \mu|$? It is a scaled version of $\sigma$. Find the scale factor. – cardinal Jul 30 '12 at 13:57
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I think I proved it was unbiased estimator. E|X-u| = standard deviation * sqrt(2/pi) – steviekm3 Jul 30 '12 at 14:04
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I was wondering though why we don't see this formula that often for estimating standard deviation? – steviekm3 Jul 30 '12 at 14:04
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Good start, stevie. You have the scale factor correct, but not the conclusion regarding the unbiasedness. Your follow-up question is a good one, with a fairly deep answer. You might consider adding it to the question to make it more prominent. :) – cardinal Jul 30 '12 at 14:32
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