Obviously, in most circumstances, you would just take the mean of the two middle values.
However, imagine a dataset of 100 countries. You want to know which country has the median household income. You could easily find the median household income by taking the mean of the two middle values, but how do you choose the median country within the sample?
Is there a rule for choosing one? Or is it appropriate to report both?
This is one of those never-ending debates. First, I think that it depends heavily on the field you are in, if there is some established practice you would want to follow that.
In Economics (to my knowledge) there is no such standard. In that case, I think it depends on: “Does it really matter”? Consider a sample of (say) 100 people, and you would like to find the median income, assume $n_{50} = 50.000$ and $n_ {51} = 70.000$ then a median of $60.000$ might seem “fair”, but really it is just interpolation. Then perhaps it makes sense to report both values (but I have not seen any paper doing that - at least in my field).
Perhaps $60.000$ is not even a realistic number - due to the specifics of your data. Suppose instead that the difference between the two is just $1$ - then I think most would agree that it does not matter (and you should be free to report what you would like).
What I am trying to say is that you are making some assumption about the data, and that should be based on the analysis of that specific data. Of course there is always the pragmatically solution of just reporting whatever standard is coded into the software, or only sampling an odd number of observations.
