The idea of variation from the mean is related to the idea of fairness in an allocation of coins, which you discussed earlier in this session. Think back to the method of determining the unfairness of an allocation -- counting the number of moves required to transform an ordered allocation to an equal-shares allocation. Here's the connection: The sum of the absolute deviations is equal to twice this required number of moves.
The absolute deviations occur in pairs, since the mean is the "balance point" for the set. Half the absolute deviations are above the mean, and half are below. When a move is made, one coin is moved from a value above the mean to a value below the mean. This removes two of the deviations; the deviation above the mean is reduced, and the deviation below the mean is reduced. Since each move reduces the absolute deviation by two, the sum of the absolute deviations must be twice the required number of moves.
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