Obviously writing out 10 numbers like so:

1 2 3 4 5 6 7 8 9 10

Clearly shows that the median is between 5 and 6 since that would mean you have the same number of items above and below.

But if we think of 10 as a length of 10 on a number line for instance and try to find the "spot" where half the data is above and below, I would think it would be 5. i.e. 10/2 (halving the length).

Does this have something to do with how we define each item from a discrete sense (writing out 10 integers like 1-10) vs the continuous representation of drawing a line?

Since with the number line, each unit (or discrete object if we need to think of that) is actually a span that stretches between two integers, do we need to find a middle span and not just a middle point?

I guess I'm trying to reconcile what the differences are between these two models and when they lead to different or non-intuitive results.

Can anybody please comment / let me know what they think as I think this kind of has to do with how I represent / think of what the nature of a number really is.

Related to this question is, can somebody help me understand the formula (n+1)/2 as the position of the median from a geometric (length based) representation? It's not clicking with me.