Mean Absolute Deviation

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Objective

SWBAT compare two sets of data through mean absolute deviation.

Big Idea

Students work in pairs and independently to judge consistency of data sets using the mean absolute deviation

Do Now + HW

15 minutes

Students enter silently according to the Daily Entrance Routine.. Students are handed their Do Now assignments at the door. The questions on this assignment are pulled from an online bank of questions created by our data team. These questions are pulled from the same bank used to create unit tests. As stated in previous lessons, my students this year are struggling significantly with the demanding reading expectations in word problem solving, which is the make-up of most of their unit tests. These assessments tend to be more complex in words and terms, making vocabulary review pertinent. As students read through their Do Now they are encourage to:

  • Annotate important terms and information
  • Box any terms or words they do not understand
  • Note any questions they have on the margin of the problem

Some essential questions to focus on when reviewing the answers:

  • What is a measure of center? Measure of variability?
  • What do these values represent?
  • Does this question ask about variation or measure of center?
  • How does this measure answer the question asked? How does it describe variation/center?
  • For question 3: how does this group best represent the population Lakeesha wants to know about?
  • For question 3: describe the population Lakeesha wants more information about?

 

In the last 5 minutes of class we review the answers to the homework from the previous night (SMARTboard resource attached as a pdf file). I make sure to focus the time around the answer to the first question. We discuss and students are responsible for writing the conclusions to the data given as box and whisker plots. These diagrams show the points score in each game of a football season for two teams, the Rams and Patriots. Their interquartile ranges are slightly different (Rams: 11 and Patriots: 14.5) indicating that there was slightly more variability in scoring for the Patriots that the Rams. The Rams had a larger median at 32.5 points versus the Patriots median of 20.5, indicating that the Rams were scoring more points per game. We discuss this information and why it is important using the following guiding questions:

  • What was the median for each team? What do they measure? center
  • Are the “centers” the same? Similar? What does this comparison tell you when comparing the two teams?
  • Which team scores higher? Lower? What does this mean tell you about the ability of the team (how good they are)? Who is the better team?
  • What was the range? The IQR? What do they measure? Variability
  • Are the measures of variability the same? Similar? What does this tell you?
  • What is variability?
  • What does the comparison in their variability tell you about how good the team is at playing football?

*Disclaimer: I am a Giants fan. There might be some bad-mouthing of the Patriots in my notes. I don’t force my students to write that part. :)

Class Notes

15 minutes

After we review answers to the do now and answer any student questions, Class Notes are distributed. The red font is meant to be copied off the board by students. Student begin by filling out their heading and copying the aim. Then, we move on to copying the definition of the first term and answering any questions about term definitions and examples to help students understand. The following may be used to check for understanding of the new terms and review the notes:

  • What is MAD: the mean of the distances of each value from the mean
    • How far is each value from the mean?
    • What is the mean? What does it describe? A measure of center; describes the typical value; gives you a general idea of the number given
      • Examples are very important here!
        • the mean score in a science test is 60%. What does this mean? The average student did well or not so well? What would you expect?
        • the mean score in a science test is 90%. What does this mean? The average student did well or not so well? What would you expect? Compare the two examples given.
        • Graph a sample data list, like the scores on a science test, on a number line. Point out the distance of each value from the mean.

Sample data, avg of 60:

50, 55, 60, 65, 60, 70

  • When you take all of these distances, or differences, and average them out, that is the MAD.
  • This is also a good opportunity to review “distance” as the difference between two units
  • Review the steps for finding mad with one of the examples given in grades on a science test.
  • But why? MAD is a different measure of variability which can inform us about consistency. What does a small MAD number mean? What does a large number mean? Remember, MAD measures the average difference, or distance, of each number from the mean… (students should discuss this before sharing out)

 

Running through the questioning and continuing to push students to analyze what does this value or that difference represents requires access of MP1. In order to understand what he data represents, students may need lots of examples, especially relevant to their own experiences, such as grades, scores on a test, time spent on the phone or on a trip.

Class Work

20 minutes

In the next section of class students receive their class work. The first example shows two teams, the Flamingoes and the Cougars and the number of points they scored at different basketball games in one season. I walk students through finding the mean absolute deviation for one team. And explain what it represents. The mean is 40 point. This means on average they scored 40 points per game. The MAD is about 6.7, which means on average, each value, or score on each game, is 6 points away from the mean.

Students are then asked to follow the same procedure to find the MAD for the cougars, with help from their neighbors. We check together to make sure we all agree on the MAD, about 1.3. I ask students to discuss the different MAD values and what this means about the data. As they are discussing, I ask them to make note of the graph of the data.

  • For students getting closer to the idea that a smaller number means the numbers are closer together, I push them to justify their conclusion with the way the graphs look.
  • I spend more time analyzing data with those students struggling to understand what the MAD means. After calculating through several examples, it can be easy to get lost in the numbers. Step by step questioning about the meaning of each value (the mean, the ranges, the distances from the means, the average of those distances) is usually slow at first, but it is important to make sure students understand through discussion and examples.

The paragraph included with blanks to fill in will hopefully aid in the understanding of these complex ideas as students will not have to create them themselves. They will thus have the time to focus on reading and understanding what the paragraphs say after correctly filling in the blanks.

Closing

5 minutes

Students are asked to work until the end of class, making the last five minutes silent and independent, to complete the class work sheet which I will collect to check over and provide feedback. Students are also advised to ask questions during class and on their worksheet to push further understanding. This is a complex topic that will need lots of spiraling and review in further assignments.