Intent

Students continue exploring methods to measure the spread of a data set. In this activity, they calculate the sum of the absolute deviations from the mean.

Mathematics

As a measure of spread, the sum of absolute deviations from the mean has an advantage over the range in that it uses every data value, or datum, in a set. The sum of the absolute deviations divided by the number of data values is called the mean absolute deviation. This statistic gives the average deviation from the mean of each data value.

Progression

Students will work individually on the activity and then share ideas and discuss the concepts of absolute value and mean absolute deviation as a class.

Approximate Time

25 minutes for activity (at home or in class)

25 minutes for discussion

Classroom Organization

Individuals or groups, followed by whole-class discussion

Doing the Activity

Assign this activity immediately after the discussion of Data Spread.

Discussing and Debriefing the Activity

You might have students convene in their groups to compare their personal schemes for measuring spread (Question 4) as well as the numeric results from the various methods. Then ask each group to report on the method they like best.

If students had trouble finding the appropriate numbers for each data set for Tai’s, Kai’s, and Mai’s methods, review the mechanics of each method.

Then have the class discuss this question: Which data set seems to be the most spread out from the mean? The decision does not have to be made based on a statistical test, and the class need not reach agreement. Focus the discussion on why the class thinks a particular set is the most spread out.

You might ask why students think Mai chose to remove the highest and lowest values. One possible reason is that Mai wants to eliminate the possibility of one or two extreme values skewing the results. If there are any competitive divers in class, you might ask about the scoring for diving competitions (the highest and lowest scores are not considered).

Discussion of spread from the mean provides a nice opportunity to talk about absolute value. For example, ask someone to describe how to apply Kai’s method to data set C: 12, 13, 13, 27, 27, 28. Use values above and below the mean (which is 20) to illustrate that in some cases you subtract the mean from the data item and in other cases you subtract the data item from the mean.

Then ask, How could you find the distance from the mean the same way in all cases? If needed, suggest that students consider the idea of absolute value. Point out that |x – 20| (or |20 – x|) always gives a positive answer and measures how far a number x is from 20, whether x is greater than 20 or less than 20.

Introduce the symbol xˉxˉ size 12{ { bar {x}}} {} (read “x bar”) for the mean, and use notation such as xi for a single piece of data. Then ask how one could write Kai’s method using this notation and the summation symbol.

Students often enjoy seeing all this notation put together, using the summation notation developed in the Patterns unit, as

{} ∑ i = 1 n ∣ x i − x ˉ ∣ ∑ i = 1 n ∣ x i − x ˉ ∣ size 12{ Sum cSub { size 8{i=1} } cSup { size 8{n} } { lline x rSub { size 8{i} } - { bar {x}} rline } } {}

Later you can compare this notation with the symbolic formula for standard deviation.

Point out that, if Kai’s number is divided by n, it gives the average deviation from the mean; it tells, on the average, how far the numbers in a set are from their mean. Introduce the term mean absolute deviation for this average, and have students express it using summation notation as

∑ i = 1 n ∣ x i − x ˉ ∣ n ∑ i = 1 n ∣ x i − x ˉ ∣ n size 12{ { { Sum cSub { size 8{i=1} } cSup { size 8{n} } { lline x rSub { size 8{i} } - { bar {x}} rline } } over {n} } } {} {}

Try to get students to articulate what this expression measures. It is a simplified version of what standard deviation will tell them, which they will learn about next.

Students will need their work on Kai and Mai Spread Data for use during the next activity, The Best Spread.

Key Questions

Which data set seems to be the most spread out from the mean?

How could you find the distance from the mean the same way in all cases?