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I recently found out that the average gap between a value and the mean of that value for a normal distribution can be expressed as: $\sqrt{\frac{2}\pi}\sigma$ or about $0.7978...\sigma$ where $\sigma$ is the standard deviation.

see: Expected value of normal distribution given that distribution is positive

which I thought was very interesting. This proof was done using the formula for the normal distribution function - $f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ and integrating.

This proof I believe, would imply that:

$\sqrt{\frac{2}\pi}\sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n}} = \frac{\sum_{i=1}^{n} |x_i - \bar{x}|}{n}$

Assuming n goes to infinity and the observations are normally distributed.

Is there a way to prove this fact algebraically without bringing in the normal distribution function? Seems weird when written like that to understand where the pi comes from.

Thanks for the help!

Tom
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  • Your statement needs expectations and limits: for example it is wrong when $n=2$ unless you replace $\bar x$ by $\mu$. Even then, you need something related to the normal distribution, as the $\sqrt{\frac2\pi}$ term is different for other distributions and in some cases the left hand term has infinite expectation while the right hand term has finite expectation – Henry Nov 17 '23 at 09:30

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Assuming that your sample $\{X_i\}_{i=1}^n$ are independent and identically distributed random variables from some normal distribution $N(\mu,\sigma^2)$, then a simple application of the strong law of large numbers would show that $\frac{1}{n}\sum_{i=1}^n|X_i-\overline{X}|\to\mathbb{E}|X_1-\mathbb{E}X_1|=\sqrt{\frac{2\sigma^2}{\pi}}$ almost surely as $n\to\infty$. In the same way we know that $\sqrt{\frac{2}{\pi}\sum_{i=1}^n\frac{(X_i-\overline{X})^2}{n}}\to\sqrt{\frac{2\sigma^2}{\pi}}$ almost surely by the strong law of large numbers and the continuous mapping theorem. This is of course assuming that the first two moments of $X_1$ exist but this is explicitly assumed.

For the rigorous details, you could look at the lecture notes I used during my course in probability theory.

Daan
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