The average for a set of data corresponds to the equal-shares allocation or fair allocation of the data. For example, suppose that each of 9 people has several dollars and altogether they have $45. The mean of $5 represents the number of dollars each of the 9 people would get if they combined all their money and then redistributed it fairly (i.e., equally).
As seen in Problem A3, the fair allocation of 45 coins into 9 stacks is to place 5 coins in each stack, as follows:
Here is a second allocation of the 45 coins, which is almost fair:
The above allocation is almost fair because most stacks have 5 coins, and the others have close to 5. But, the following allocation of 45 coins doesn't seem fair at all: