# Find factor of sample elements given the median

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.

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@MichaelHardy I don't think this edit is correct. If I read it correctly, s$_i$x were the observed samples and you know that the median of the sample of s$_i$x =y. Then find x. – Michael Chernick Sep 23 '12 at 16:32
Who is right? me or Michael Hardy? – Michael Chernick Sep 23 '12 at 16:35
@MichaelChernick You first comment is alright. – Libor Sep 23 '12 at 16:40
My only edit to this posting was to change "..." to "\ldots". Is that was is being called incorrect? – Michael Hardy Sep 23 '12 at 16:45
I haven't found any problems with the edit. Using '\ldots' instead of '...' is OK. – Libor Sep 23 '12 at 16:54

Sort your sequence $s_1,\ldots,s_n$ to obtain a new sequence $t_1,\ldots,t_n$.

If $n$ is odd, take $t=t_{(n+1)/2}$.
If $n$ is even, take $t=(t_{n/2}+t_{(n/2)+1})/2$

Solve for $x$ the following equation

$$tx=y$$

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What does "Solve for $tx=y$" mean? One could say "Solve $tx=y$ for $x$" or "Solve $tx-y$ for $t$", and I would understand it. – Michael Hardy Sep 23 '12 at 16:28
But his sequence is not the t$_i$s it is the t$_i$s multiplied by x, You need to take account of the fact that the t$_i$s are integers. – Michael Chernick Sep 23 '12 at 16:29
@MichaelHardy: i'm not English, excuse me. – enzotib Sep 23 '12 at 16:36
@MichaelChernick: the median shouldn't be integer: Medians for samples. – enzotib Sep 23 '12 at 16:37
I totally forgot that position of median value does not change by multiplying the elements, hence can be computed in this simple manner... – Libor Sep 23 '12 at 16:38

The fact that y is the median only tells that approximately n/2 elements are below y and n/2 are above. So it seems that the most information comes from knowing y/x is an integer. So the only thing I think you can do is search through values less than y give y/x as an integer. There will be at least one but there could be more than 1 in which case you will get a discrete list of possible values.

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