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In any reasonable definition of what a median is then there is either a unique value, or the mean between the two middle elements is a median, so it's perfectly reasonable to take the average (something which you can always do and gives a unique answer).

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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".

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