From working on Problem E1, you probably got an intuitive sense of the variation in the three data sets. But is there a way to measure exactly how much the values in a line plot differ from the mean?
Recall that in Part B, we described the unfairness of allocations of coins by counting the number of moves required to transform an ordered allocation into an equal-shares allocation. The idea of variation from the mean is related to the idea of fairness in the coin allocation.
For example, consider this ordered allocation:
Eight moves are required to make this allocation fair. This is true because there is an excess of 8 coins above the mean from 4 stacks (+1, +1, +3, +3), and a deficit of 8 coins below the mean from 4 other stacks (-3, -2, -2, -1).
The number of moves required to make an allocation fair tells us how much the original allocation differs from the fair allocation and thus gives us a measure of the variation in our data. (The fair allocation has no variation -- no moves = no variation.)
Here is the line plot that corresponds to this allocation:
Here are the deviations from the mean for each value in the set (i.e., how much each value differs from the mean):
Now consider only the magnitude of these deviations -- that is, forget for the moment whether they are positive or negative. These are called the absolute deviations. The absolute deviations for this set are plotted below:
We are now going to find the mean of these absolute deviations, which is an indicator, on average, of how far (what distance) the values in our data are from the mean. As usual, find the mean by adding all the absolute deviations and then dividing by how many there are. Here is a table for this calculation:
The mean of these absolute deviations -- the MAD (Mean Absolute Deviation) -- is 16 / 9 = 1 7/9, or approximately 1.78. This measure tells us how much, on average, the values in a line plot differ from the mean. If the MAD is small, it tells us that the values in the set are clustered closely around the mean. If it is large, we know that at least some values are quite far away from the mean.