Asymptotic distribution of a sample median

Let $X_1,\dots ,X_n$ 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 satisfiesThe 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.