I know the expected absolute deviation from the mean of a normal distribution $E[|X-\mu_x|]$. From this I want to derive the variance $\sigma^2$ of said distribution. This is done to tune a filter of $X$ that expects an estimate of the current variance at each time step. I don't know the sign of the deviation, just the magnitude.
I identified two approaches to derive the variance, but I am unsure which one is the better estimate of the variance:
Approach 1 - Re-arranging the mean absolute deviation (MAD):
Here I interpret $E[|X-\mu_x|]$ as the known expected deviation from the mean. For a normal distribution: $$E[|X-\mu_x|] = \sigma_x \sqrt{\frac{2}{\pi}}$$
The MAD can be re-arranged to yield the variance:
$${\sigma_x}^2 = \frac{\pi}{2} \cdot {E[|X-\mu_x|]}^2$$
Approach 2 - Calculating ${\sigma_x}^2$ directly from one sample
$${\sigma_x}^2 = \frac{1}{n} \sum_{i=1}^n (X_i-\mu_x)^2$$
with n=1:
$${\sigma_x}^2 = (X_i -\mu_x)^2$$
Which approach is valid? The argument for approach 1 seems to be straight forward. In approach $2$ I can't even apply Bessel's correction without dividing by zero (And I don't know the mean, just the deviation from it!).
Edit:
To clarify my use case: I know the deviation, which is derived from a separate measurement. I trust this measurement to be the expected deviation (mean absolute deviation). I do not know the sign of the deviation and as such can not directly correct X. Instead I am filtering it with a variable estimate of the variance.