In what sense is the mean of any distribution the "best guess" of the score of any single individual selected at random from the group?

I. In a series of such guesses, the sum of the errors in one direction will balance the sum of the errors in the other direction.
II. The mean score will occur more often than any other single score.
III. The chances are 50-50 that any individual will be above or below the mean.

Explanation: II and III are not necessarily true with distributions other than normal distributions.