Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ Normal distribution ▷ Median

Theorem: Let $X$ be a random variable following a normal distribution:

$\label{eq:norm} X \sim \mathcal{N}(\mu, \sigma^2) \; .$

Then, the median of $X$ is

$\label{eq:norm-median} \mathrm{median}(X) = \mu \; .$

Proof: The median is the value at which the cumulative distribution function is $1/2$:

$\label{eq:median} F_X(\mathrm{median}(X)) = \frac{1}{2} \; .$ $\label{eq:norm-cdf} F_X(x) = \frac{1}{2} \left[ 1 + \mathrm{erf} \left( \frac{x-\mu}{\sqrt{2}\sigma} \right) \right]$

where $\mathrm{erf}(x)$ is the error function. Thus, the inverse CDF is

$\label{eq:norm-cdf-inv} x = \sqrt{2}\sigma \cdot \mathrm{erf}^{-1}(2p-1) + \mu$

where $\mathrm{erf}^{-1}(x)$ is the inverse error function. Setting $p = 1/2$, we obtain:

$\label{eq:norm-med-qed} \mathrm{median}(X) = \sqrt{2}\sigma \cdot \mathrm{erf}^{-1}(0) + \mu = \mu \; .$
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Metadata: ID: P16 | shortcut: norm-med | author: JoramSoch | date: 2020-01-09, 15:33.