What could go wrong ignoring the even size case in computing the median value?

Question

Given a list of $N$ numbers I need to compute the median.

The book Numerical Recipes(second edition, section 8.5) says that:

When $N$ is odd, the median is the $k$th element, with $k=\frac{N+1}{2}$. When $N$ is even, statistics books define the median as the arithmetic mean of the elements $k=\frac{N}{2}$ and $k=1+\frac{N}{2}$ (that is, $\frac{N}{2}$ from the bottom and $\frac{N}{2}$ from the top). If you accept such pedantry, you must perform two separate selections to find these elements. For $N > 100$ we usually define $k=\frac{N}{2}$ to be the median element, pedants be damned.

I can define myself as pedant and so when I need to code an (exact) median algorithm I usually consider the two cases for $N$ but what could possibly go wrong defining $k=\frac{N}{2}$ to be the median element when $N$ is not odd?

My use of the median value is usually just for summary purposes, like in the following R snippet:

v=rnorm(10)
summary(v)

Answer 1

For practical purposes, things will rarely go wrong, for most reasonable values of "wrong", unless you have very strange distributions.

Note that this recipe is just one possibility. Hyndman & Fan (The American Statistician, 1996) give no less than nine different definitions of sample quantiles and recommend number 8. R's quantile() function follows their tabulation and implements all of them, with the default type=7. I would very much recommend that article, which also discusses possibilities for things to go "wrong".