middle quantile of a data set or probability distribution
Math formula ( median )
means - What is the general formula for calculating the median? - Mathematics Stack Exchange
Can someone tell me the median formula for even amount of numbers vs odd amount of numbers?
Calculating Median Value - Age Groups
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After some extra research it seems there is no solution for my desire. The best bet probably is to define a function, say $m(x)$, which is defined as
$m\left(x\right) =\begin{cases} x_\frac{n+1}{2} & n\text{ odd}\\ \frac {1}{2}\left(x_{\frac{n}{2}} + x_{\frac{n}{2} + 1}\right) & n \text{ even} \end{cases} $
For a given sample $X_1,\ldots,X_n -$ by which I mean a collection of independent, identically distributed random variables $-$ the median of the sample is either the $\dfrac{n+1}{2}^{th}$ or $\dfrac{n}{2}^{th}$ order-statistic, depending on whether $n$ is odd or even respectively. The sample mean does not give a sufficient amount of information to uniquely determine what these order-statistics are.