If they are real numbers then you can take the arithmetic mean of the two middle elements, but you might prefer to define the median in such a way that the definition is applicable to any set with a total order.
For example, names can be ordered alphabetically, so you can find the median of a set of names! OK, that's a silly example, but I expect someone could come up with a more sensible one.
Another case might be where you're using geometric means everywhere else in your working, but then you want a median in one place, but there are an even number of elements, so why would you suddenly use an arithmetic mean of the two middle elements?
> I expect someone could come up with a more sensible one.
- Imagine you hand out a survey where some answers are on a likert scale. "strongly dislike" is worse than "dislike", which is worse than "neutral" and so on.
- The median ranking over a set of ranking on different criteria.
- When the domain is not the real numbers. "The median employee of this company has 1.5 kids".
For example, names can be ordered alphabetically, so you can find the median of a set of names! OK, that's a silly example, but I expect someone could come up with a more sensible one.
Another case might be where you're using geometric means everywhere else in your working, but then you want a median in one place, but there are an even number of elements, so why would you suddenly use an arithmetic mean of the two middle elements?