= Asymptotic distribution of a sample median
Let $X_1,\ldots,X_n$ be <independent and identically distributed random variables> with unique <median> $m$, and let their <probability distribution> have continuous <probability density function> $f$ with $f(m)>0$. For odd $n$, the <sample median> satisfies
$$
\sqrt n(\widehat m-m)\Longrightarrow N\left(0,\frac1{4f(m)^2}\right).
$$
The <influence function of the sample median> has <mean> zero and <variance> $1/(4f(m)^2)$: each of its two nonzero values occurs with <probability> $1/2$. Its <asymptotic linear representation>, followed by the <central limit theorem>, gives the displayed limit.
For a <standard normal distribution>, $f(0)=1/\sqrt{2\pi}$, so the asymptotic <variance> of the unscaled <sample median> is $\pi/(2n)$. The <sample mean> instead has <variance> $1/n$ in that model. This quantifies an efficiency cost at the reference model without contradicting the <sample median>'s bounded response to an <outlier>.
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