Facts About Statistics

One use of statistics is winning bets. I'm going to pick one of the elements of L (uniformly at random). You get to pick one number.

• Lose if your number is smaller than mine: maximum.
• Lose if your number is bigger than mine: minimum.
• Lose if you get the wrong number: mode.
• Lose proportional to the square of the difference between the numbers: mean.
• Lose proportional to the absolute difference between the numbers: median.

Extra credit for proving either or both of the last two statements!