How to compare -50% and +100% This might seem (and be) a really trivial question!
I have this table:
id until_2000 after_2000
1  1          1          1
2  2          1          2
3  3          2          1

Now I want to compare the relative change between the columns after_2000 and until_2000. Just dividing the after_2000 values by the until_2000 (and substract 1?!) values yields the colum rel_change
  id until_2000 after_2000 rel_change
1  1          1          1        0.0
2  2          1          2        1.0
3  3          2          1       -0.5

While the first row stays as is (has a change of 0%), the second row doubles (has a relative change of 100 %). Yet the third row is half the value and thus has a relative value of -50 %.
Now my trivial question is how to correctly compute and average these relative changes. In my line of thinking the values of 0, doubling, and "halfing" should result in an average change of 0.
Yet adding 0 + 1 - 0.5 and dividing it by three yields a value of 0.16666.
What am I conceptually understanding wrong here?

Adding the numbers and then dividing by the count is the arithmetic mean. When dealing with relative change, it often makes more sense to use the geometric mean, which is where you take the product of the numbers and then taking the nth root. So if you take 0.512 and then take the third root, you get 1, which makes more sense as the "average" change. If you had 0.5, 1, 2, and 2, then you'd take the fourth root of 2, which is 1.189
You can also take the logs of the ratio. You can then take the arithmetic mean of the logs. If you want to convert back to relative change, you can then take the exponential of the result. Taking the log, then the arithmetic mean, then taking the exponential of the result gives you the same answer as the geometric mean.
