I have a sample $S(x)$ containing $n$ elements:
$S(x)=\{ s_1 x, s_2 x, \ldots, s_n x \},\qquad s_i \in \mathbb{R}, x\in \mathbb{R^{+}}$
Every element in the sample is multiplied by $x$. Now median of this sample is
$\tilde{S}(x)=y,\qquad y\in \mathbb{R^{+}}$
When $y$ is given, how to find $x$?
In other words: If I know median of a sample whose every element is multiplied by a certain factor, how to find this factor? The original sample elements $s_{i}$ are also known.
I think it would be possible to find the value with a search algorithm (to some degree of precision), but maybe there is a simple closed solution.
Note that there may be more that one $x$ satisfying the above equation, since $s_i$ come from $\mathbb{N}$. The solution will more likely be an interval of values.