You are waiting for an elevator. There are four elevators, equally spaced along a wall. The third elevator is out of order. Where would you stand, and why?
Many people would choose to stand at the mean position, and think that doing so minimizes the average walking distance. If working elevators are at positions 1, 2, and 4, then the mean position would be (1+2+4)/3=2⅓, or slightly to the right of the second elevator.
But standing at the mean minimizes the average squared distance instead. To minimize average distance, one would need to stand at the median.
In the above example, average walking distance from position 2⅓ is (1⅓+⅓+1⅔)/3=(3⅓)/3, and average walking distance from position 2 (the median position) is (1+0+2)/3=3/3.
Source: Hanley, J. A., and Lippman, A. (1999), “Where Do You Stand? Notions of the Statistical ‘Centre’”, Teaching Statistics, 21, 49–51. PDF