I am looking for a derivation of the fact that $\frac{1}{\Phi^{-1}(3/4)}$ is the multiplier needed for the Median Absolute Deviation (MAD) to be an unbiased estimator of $\sigma$ when $x_i\sim N(0, \sigma^2)$. Recall the MAD is defined as:

$$ \lambda = b\times Median(| X-Median(X)|). $$

the claim is stated in many places (wikipedia and journal articles) but I cannot find a proof.

I am looking for a derivation of the fact that $\frac{1}{\Phi^{-1}(3/4)}$ is the multiplier needed for the Median Absolute Deviation (MAD) to be an unbiased estimator of $\sigma$ when $x_i\sim N(0, \sigma^2)$. Recall the MAD is defined as:

$$ \lambda = b\times Median(| X-Median(X)|). $$

for some $b$ chosen to meet a given criteria, (e.g. often unbiased-ness) the claim is stated in many places (wikipedia and journal articles) but I cannot find a proof.