 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 5, Part E:
Measuring Variation (45 minutes)

In This Part: Mean Absolute Deviation (MAD) | Working with the MAD
Variance and Standard Deviation

 We will now focus on how to measure variation from the mean within a data set. There are several different ways to do this. The first measure we will explore is called the mean absolute deviation, or MAD. Note 7  Problem E1 Three line plots are pictured below; each has 9 values, and the mean of each is 5: Line Plot A Line Plot B Line Plot C Of the three, which line plot's data has the least variation from the mean? Which has the most variation from the mean?  The plot with the most variation will have data values that are, in general, farthest away from the mean. The plot with the least variation will generally have the values closest to the mean.   Close Tip The plot with the most variation will have data values that are, in general, farthest away from the mean. The plot with the least variation will generally have the values closest to the mean. 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:  Number of Coins in Stack (x) Deviation from the Mean (x-5) Absolute Deviation from the Mean |x-5|  2 -3 3 3 -2 2 3 -2 2 4 -1 1 5 0 0 6 +1 1 6 +1 1 8 +3 3 8 +3 3 ______ ______ ______ 45 0 16 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.   Video Segment In this video segment, Professor Kader introduces the MAD as a method for quantifying variation. Watch this segment to review the process of finding the MAD. How can the MAD be used to compare different distributions of data? If you're using a VCR, you can find the first part of this segment on the session video approximately 12 minutes and 8 seconds after the Annenberg Media logo. The second part of this segment begins approximately 16 minutes and 52 seconds after the Annenberg Media logo.    Problem E2 Below is Line Plot B from Problem E1. Create a table like the one above, find the MAD for this allocation, and compare it to the MAD of Line Plot A from the same problem.  Problem E3 Below is Line Plot C from Problem E1. Create another table, find the MAD for this allocation, and compare it to the MADs of Line Plots A and B.    Session 5: Index | Notes | Solutions | Video