If I have an order statistic, I can rank them from smallest to largest like
$$X_{(1)}, X_{(2)},...,X_{(n)}$$
and there will be some median $X_{m}$.
But I can also have some distribution $f(x)$ and define an integral
$$\int_{0}^m f(x)dx = \frac{1}{2}$$ and, if I solve for $m$, this will be the median for the distribution.
My Question
What is the difference between these two types of medians?