Given a set A with median Am = 10 and set B with median Bm = 20 is it true that the median of the combined set C is $10 \le$ Cm$\le 20$ ?
My first thought was that this wasn't true so I tried to find a counter example but I wasnt able to so I am assuming that it likely is true but I havent been able to find any theorem or other proof for this.
Ideally I would like to know if this is true for the general case not just 10,20 I just chose these numbers while trying to find a counter example.
Any help would be greatly appreciated.
Suppose there are $2n$ elements of $A$ and $2m$ elements of $B$. Then there are $n$ elements of $A$ which exceed (or equal) $10$ and $m$ elements of $B$ that exceed (or equal) $20$. of course the latter implies that there are at least $m$ elements of $B$ which exceed (or equal) $10$. Combining those we see that there are at least $n+m$ elements of $C$ which exceed or equal $10$, so the median of $C$ is at least $10$. The odd case is similar: if there are $2n+1$ elements of $A$ then there are at least $n+1$ elements in $A$ which exceed or equal $10$, and so on.