Standardized median estimation Is there a 'standard' way to calculate an estimated median value for a given data-set that either never references a data point in the set, or is unlikely to?  I'm familiar with the median of medians approach, but I'm not sure what it's limitations are.
Example:
Assume each element in the data-set is associated with a unique individual (e.g it can be mapped back 1-1)
Data = {1, 2, 3, 4, 5, 6}

Note than when the number of elements in the dataset is even, the median is defined as (3+4)/2=3.5, which does not correlate with a value in the data-set, and hence does not map back to an individual.
The same cannot be said when the number of elements is odd.
Data = {1, 2, 3, 4, 5}

Here, the median is 3, which can be mapped back to an individual.  So perhaps there's an interpolation technique that can be applied when the number of elements is odd?
Hopefully this clears things up.
 A: The simple answer to your question is that your suggested calculation of the median in a sample having even $n$ is a good one, but certainly not the only one nor is it universally agreed upon as the best one. 
If you think of the sample median as the empirical quantile function evaluated at 0.5, you will always select the $ceiling(n/2)$ order statistic as the median. This has the property that the median is always a member of the sample. We choose the lower value in even $n$ samples because the empirical distribution function and empirical quantile function are left continuous.
Suppose in R we take a sample of normal variates:
set.seed(123)
x <- rnorm(6)
m <- median(x)
plot(ecdf(x))
segments(min(x), 0.5, m, 0.5, col='red')
segments(m, 0.5, m, 0, col='red')
text(m, .25, 'Sample median', cex=0.5, srt=90, pos=2)

Gives us:

Changing 6 to 7

A radical idea is that instead of the median being a single number, it could be whole range of values which interpolates the $n/2$ and $n/2+1$ domain when n is even, making a kind of functional inverse.
