# Quantile absolute deviation of the Normal distribution

In this post, we derive the exact equation for the quantile absolute deviation around the median of the Normal distribution.

## Preparation

We consider the quantile absolute deviation around the median defined as follows:

\[\newcommand{\E}{\mathbb{E}} \newcommand{\PR}{\mathbb{P}} \newcommand{\Q}{\operatorname{Q}} \newcommand{\QAD}{\operatorname{QAD}} \newcommand{\median}{\operatorname{median}} \QAD(X, p) = \Q(|X - \median(X)|, p), \]

where \(\Q\) is a quantile estimator.

We are looking for the asymptotic value of \(\QAD(X, p)\). For simplification, we denote it by \(v_p\):

\[v_p = \lim_{n \to \infty} \E[\Q(|X-M|, p)], \]

where \(M\) is the true median of the distribution.

By the definition of quantiles, this can be rewritten as:

\[\PR(|X_1 - M| < v_p) = p, \]

which is the same as

\[\PR(-v_p < X_1 - M < v_p) = p. \]

Hence,

\[\PR(M - v_p < X_1 < M + v_p) = p. \]

If \(F\) is the CDF of the considered distribution, the above equality can be rewritten as

\[F(M + v_p) - F(M - v_p) = p. \tag{1} \]

## Normal distribution

We consider the standard normal distribution \(\mathcal{N}(0, 1)\) given by the CDF \(F(x)=\Phi(x)\) with the median value \(M=0\). From (1), we have:

\[\Phi(v_p) - \Phi(-v_p) = p. \]

Using \(\Phi(-v_p) = 1 - \Phi(v_p)\), we get:

\[\Phi(v_p) = \frac{p+1}{2}, \]

which is the same as

\[v_p = \Phi^{-1} \Big( \frac{p+1}{2} \Big). \]

Thus, if \(X \sim \mathcal{N}(0, 1)\),

\[\lim_{n \to \infty} \E[\QAD(X, p)] = \Phi^{-1} \Big( \frac{p+1}{2} \Big). \]

Here is the corresponding plot: