In a previous post I derived the distribution function for the median of a sample of size \(n = 2k +1\) (with \(k \geq 0\) integer) drawn from a Lorentz (or Cauchy) distribution:

\begin{equation}\label{eq:result}

g(x) = \frac{n!}{(k!)^2 \, \pi ^{n} \, \gamma} \left [ \frac{\pi ^2}{4} - \arctan ^2 \left (\frac{x - x_0}{\gamma} \right )\right ]^k \frac{1}{1+ \left (\frac{x - x_0}{\gamma} \right )^2 }

\end{equation}

I will now consider some of its properties.