Prove That The Composite Median Of Two Groups Lies Between Separate Medians of The Groups

Question

Let $M_1$ be the median of $n_1$ observations and $M_2$ be the median of $n_2$ observations. Show that $M$, the median of$(n_1+n_2)$ observations combined, lies within $M_1$ and $M_2$.

Answer 1

Suppose $M_1\le M_2$, without loss of generality. Half of the $n_1$ observations lie below $M_1$, and at most half of the $n_2$ observations lie below $M_1$ (if it were more, then $M_2<M_1$). Hence at most half of the $n_1+n_2$ observations lie below $M_1$ and so their median is at least $M_1$. A similar, but reversed, statement can be made about $M_2$