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 $1/2$:

\[\label{eq:median} F_X(\mathrm{median}(X)) = \frac{1}{2} \; .\]

The cumulative distribution function of the normal distribution is

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

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:

\[\label{eq:norm-med-qed} \mathrm{median}(X) = \sqrt{2}\sigma \cdot \mathrm{erf}^{-1}(0) + \mu = \mu \; .\]

Metadata: ID: P16 | shortcut: norm-med | author: JoramSoch | date: 2020-01-09, 15:33.