Mean Absolute Deviation

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Objective

SWBAT calculate the mean absolute deviation (MAD) of a data set and explain the significance of the MAD given the context of the data.

Big Idea

The MAD is a measure that described how much a typical data point in a data set differs from the mean of the data set.

Think About It

7 minutes

Students work on the Think About It problem with their partners today.  On the worksheet, the question of "How did each class do?" is open to interpretation. Students may think about this problem in a variety of ways.  My goal is for my kids to think about the task using all of the tools they've learned to use to analyze data sets so far:  mean, median, mode, range, and inter-quartile range.  Here, you can see a student work sample. 

After pairs have had a few minutes to interact with the data, I ask students to share who they thought did better.  Students might say Class 1 did better because they have more students who got 100%.  Another student might say that Class 2 did better because even though they do not have as many students with 100%, they have fewer students with a score below 60%.

I then plan to ask students for the means of the data sets, which are the same. I'll frame the lesson by telling students that even though these data have the same mean their distributions are very different.  Today we are going to learn how to calculate a statistical measure called the mean absolute deviation that can tell us about variability within a data set.

Intro to New Material

20 minutes

To start the Intro to New Material section, I have students fill in the guided notes with me.  We will focus on the idea that Mean Absolute Deviation (MAD) is a way to examine variation from the mean, or how far away each data point is from the mean.    

The steps to finding the MAD of a data set:

  1. Calculate the mean of the data set.
  2. Subtract the mean from each value in the distribution.  This is the number’s deviation (from the mean).
  3. Take the absolute value of each deviation.  This is called the absolute deviation.
  4. Find the mean of the absolute deviations.  This is the Mean Absolute Deviation.

Before we circle back to find the MAD for the classes in the Think About It, I ask students a series of questions to help them internalize the concept of the MAD:

  • If the MAD is a measure of how DIFFERENT scores are from the mean, which class do you think has a greater MAD? Why? 
    • Class 1 will have a greater MAD because it has more data points that are far away from the mean. 
  • So what does a large MAD tell us?
    • Data in the distribution are more spread out- there is greater variation in the data, therefore the mean is a less reliable summary of the data point. 
  • So what does a SMALL MAD tell us? 
    • Data in the distribution all fall relatively close to the mean. The data are clustered closer to the mean and therefore the mean is a more reliable predictor. 

One key point that I make sure comes out during our conversation is that MAD is a way to examine variation from the mean, whereas range and IQR describe (or summarize) variability from the median. Then we calculate the MAD for the data sets in the Think About It problem. 

Since this concept is challenging for my students, I then guide them through the second example.

Partner Practice

17 minutes

To help students gain some momentum, I next have them work in pairs on the Partner Practice problem set.  As students work, I circulate around the room and check in with each group. I am looking to see that students accurately determine the MAD. They also need to accurately interpret the MAD given the context.  You can see a partner practice sample here.

Teacher's Note: My students have access to calculators as they work.  I am watching to make sure that students are showing their work at each step.  At a minimum, I want to see the numerator and denominator of the fraction, before they divide.  

After 10 minutes of work time, students work on the Check for Understanding problem on their own. As students work on the CFU, I circulate and check in on students' work.  If there are any misconceptions, I'll note the students' names on my clipboard, so that I can check in with those students early on in Independent Practice.

Students who are struggling to interpret the MAD could look at the data on a dot plot. Here they will be able to see more clearly when data is more spread out. Any who struggle with the required computation, I will encourage to cross out each value once they've entered it into their calculators. 

Independent Practice

10 minutes

Next, students work on the Independent Practice problem set. Below are some notes about how students may explore some of the problems on this practice. 

  • Problem 1 is broken down into three clear parts - find the first MAD, the second MAD, and then analyze.
  • Problem 2 is a little more rigorous, in that it simply asks which data set is more consistent.  
  • For Problem 4, students should not make any calculations.  Students should recognize that there is no deviation in the data.  If students are not able to answer this question, it lets me know that they do not fully understand the concepts of consistency or deviation.

Closing and Exit Ticket

8 minutes

After the Independent Work time, students have the opportunity to discuss their responses to Problem_6 with their partners.  Students are able to share their thinking, hear the thinking of a peer, give and receive feedback on their work, and ask clarifying questions.

Students then independently work on the Exit Ticket to close the lesson.

An exemplar response for Problem 2 would resemble: Group 2 was more consistent on their test because the MAD was 3.3%, whereas the MAD for Group 1 was 16.7%.  This means that on average, students in group two were within 3.3% of the average test score, so the scores were clustered more closely. Students in group 1 were on average within 16.7%, which means the scores were more spread out than group 1’s scores.