Mean absolute deviation. What does it mean?

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Objective

SWBAT use mean absolute deviation to make assumptions about the variability in the data.

Big Idea

The students will be working with mean and learning about variablility. They will be making connections to real life applications to help assist them in making a connection to mean absolute deviation.

Do Now

10 minutes

Students will be looking at a set of data in a table and be asked to find the mean.  I will be looking to see if they can add all the data values and divide by the number of values.  I will also be taking note to see who can combine data values to make their calculations easier. (SMP 7).  My goal is to get the students to see that when numbers are repeated they can simplify their calculations by multiplying first and then putting less numbers into the calculator.   After they have found the mean, I want them to put the data into a line plot.  The reason I’m doing this is because the students will need to have a visual of the distance from the mean in order to calculate the mean absolute deviation.  A line plot will be the best visual for this.  You could even ask the students what display would best represent the data.  Since it is numerical they may say stem and leaf, histogram, or line plot.  From there I would ask them which best shows how to find the mean?  They should be able to tell you line plot. (SMP 2)  

Tools: calculator

What does MAD mean?

60 minutes

Vocabulary :  Give students the definition of mean absolute deviation.  Allow them time to write it down and then ask them to translate it into their own words.(SMP 1)  Partner share their version of the definition.

(I’m listening to for them to say:  I know we will have to find the mean and if we are finding the average of the distance from the mean, we will probably have to subtract)  In order to get them to think this way, I may say  

  • How do we find the mean?
  • When looking to find the distance between two points, what action is taking place?
  • If I’m find the average distance, what is the key word here to let us know what is going on?

Direct Instruction:

Use slide #5 as a visual to show the data points in relationship to the mean.  Discuss how far each value is from the mean in both directions.  Ask them if it is possible to have a negative distance? (SMP2)

Slide 6: The steps to finding the mean absolute deviation.  I’ve provided the steps to help the struggling students keep track of where they are in the problem.  The steps will be provided for them in their notes.  Show the students how they already came up with the process on their own when they translated the definition into their own words

Slide 7, 8, 9:  These slides take them through each step to show them how to find the MAD.  Students should be able to complete step 1 on their own (finding the mean).  Be sure to have students tell you what the mean, means.  Understanding what they are answering  and  if their answer is reasonable supports SMP 6 (attending to precision). Next, the students will be finding the distance from the mean.  I would have them use the line plot they created as a visual to “see” the distance.  Finally, step 3 has them finding the average of the distances.  Since students have worked with variability before (quartiles), I would ask them to describe what the MAD represents (SMP 2).  Students should be saying that the mean consistently represents the data because the MAD is close to zero.  Or that there is little variability within the data set. Or there is a small spread of data.

If needed, you may need to remind the students about variability in the box plots.  We looked at box plots and their interquartile range which is another way to describe variability. 

Slide 10:  Now it is time for students to do this on their own.  Before starting, have a whole group discussion on the steps to find MAD.  Allow students time to complete the problem before going over it.  As students complete the work, they can check with a partner.  During this time, the partners should be discussing what they found, how they found it, and whether their answer seems reasonable.  Also, they should provide a description of the data according the MAD.

Activity:

The students can do one of several activities using the MAD activity power point.  The slides can be turned in to an Around the room, Numbered Heads together, or Show down activity.  Each slide has the students calculating MAD.

Question 2:  The students have to find the MAD of 2 data sets and then compare them.  This slide will be good to see if students really understand what the MAD is describing.  Watch to make sure students find the MAD for both data values and then write sentences to compare their variability. 

Question 4:  Students may be confused by this as they are not really finding the MAD.  The question asks them find amount of data points that are 1 standard deviation from them mean.  For students that are having difficulty understanding this, I would have them draw a line plot to “see” the data values.  They will need to find the mean first. 

Question 6:  this question will be a challenge.  Students will need to calculate the MAD.  On top of that, they will need to know what twice the MAD means.  In this case, MAD = 4.5 so twice the MAD = 9.  Then they need to find out if any data values are 9 points from the mean (33.1), yes there are two (48 and 23).  This is a great question to see if students really understand the different numbers.  Working through the language with the students will be helpful.  If using as an ATR, I would have this question by me so I could help them work through the problem.  If using as a team activity, then I would use this question in my final wrap up. 

Tools: Calculator, whiteboards and markers if needed

Closure

15 minutes

Use questions 2, 4, 6 from the MAD activity to go over as whole group instruction. Each of these questions have a little something extra that students had to think about when trying to solve them.   Allow students to come to the board to show how they solved the problems.  Students should be given time to think aloud at the board to discuss their strategies.  Ask the audience (students not at the board) to comment on their classmates work.  Did they solve it the same way?  Did they use a different strategy or did they come up with a different solution?

Today’s lesson objective was to learn how to find the mean absolute deviation.  Wrap up by asking the students:

  • What are the steps to finding the mean absolute deviation.
  • What does the MAD describe? Give an example of a MAD that has little variability?  Give an example of MAD that has a large variability?
  • Use the data set and find the MAD?  (in power point)

Tools: Calculator