What's the relationship between half of a value x, and the midpoint of that value?

Question

Say you have 4 apples:

A A A A

Where is the middle of this, I mean, on paper it's obvious, but what is the numerical value of this? Halving this value (or rather any value) is easy, 4/2 = 2, but is taking half of something the same as finding its centre?

If you were to find the medium of this, then it's said that the medium is 2.5, which seems correct, as it lies halfway between 2 and 3, with 1 and 4 to the left/right:

A A A A
   ^
   |
 Here!

But if you have 4 of something, then 2.5 of that something represents a larger sum than half it's value. At the same time, if you point to the second A and say that this is the midpoint, this is equally absurd! And to me this is kinda mind blowing, I had always thought that if you took half of a value, this is the same as its midpoint.

How is this actually possible though, it seems quite paradoxical, where does the extra 0.5 come from, and to which side, (or half?) does it belong to?

In all honesty I feel silly asking as i've long moved on from basic stats, but I can't wrap my head around this ridiculously simple/complicated concept.

Answer 1

A A A A
   ^
   |

At that midpoint, how many complete A's are on the left? 2. On the right 2. You do not have 0.5 A's at the midpoint. ie

| A | A | A | A |
+---+---+---+---+
0   1   2   3   4

The line is now measuring (the width of) the A's. The midpoint is at 2.

| A | A | A | A |
--+---+---+---+
  1   2   3   4

This is NOT how you should draw the line - if you are thinking of partial A's (like 0.5 A). Why would 1 be aligned with the middle of the first A - if you were walking along from left to right, once you are there, you are really only half way across the A. So

| A | A | A | A |
--+---+---+---+
 0.5 1.5 2.5 3.5

which is

| A | A | A | A |
+---+---+---+---+
0   1   2   3   4

When counting boxcars on trains as they past, don't count the boxcar until it is completely past you.