Variability and MAD

5 min

Narrative

The purpose of this Warm-up is for students to first reason about the mean of a data set without calculating and then to practice calculating the mean. The context will be used in an upcoming activity in this lesson, so this Warm-up familiarizes students with the context for talking about deviation from the mean. 

In their predictions, students may think that Elena will have the highest mean, because she has a few very high scores (7, 8, and 9 points). They may also think that Lin and Jada will have very close means because they each have 5 higher scores than one another, and their other scores are the same. Even though each player has the same mean, all of these ideas are reasonable things for students to consider when looking at the data. Record and display their predictions without further questions until they have calculated and compared the mean of their individual data sets. 

Launch

Arrange students in groups of 3.

Tell each group member to calculate the mean of the data set for one player in the task, share their work in the small group, and complete the remaining questions. 

Student Task

Elena, Jada, and Lin enjoy playing basketball during recess. Lately, they have been practicing free throws. They record the number of baskets they make out of 10 attempts. Here are their data sets for 12 school days.

Elena

2

2

2

2

4

5

5

6

8

9

9

9

Jada

2

4

5

4

6

6

4

7

3

4

8

7

Lin

3

6

6

4

5

5

3

5

4

6

6

7

  1. Calculate the mean number of baskets each player made, and compare the means. What do you notice?
  2. What do the means tell us in this context?

Sample Response

  1. Elena's mean score is 2+2+2+2+4+5+5+6+8+9+9+912=5.25\frac {2+2+2+2+4+5+5+6+8+9+9+9}{12}=5.25. Jada's mean score is 2+4+5+4+6+6+4+7+3+4+8+712=5\frac {2+4+5+4+6+6+4+7+3+4+8+7}{12}=5. Lin's mean score is 3+6+6+4+5+5+3+5+4+6+6+712=5\frac {3+6+6+4+5+5+3+5+4+6+6+7}{12}=5. I noticed that all three players have close to the same mean score, but Elena's is a little greater.
  2. Sample explanation: The means show that all 3 students make, on average, about half of the 10 attempts to get the basketball in the hoop.

Synthesis

Ask students to share the mean for each player's data set. Record and display their responses for all to see. After each student shares, ask the class if they agree or disagree and what the mean tells us in this context. If the idea that the means show that all three students make, on average, half of the 10 attempts to get the basketball in the hoop does not arise, make that idea explicit. 

If there is time, consider revisiting the predictions and asking how the mean of Elena's data set can be the same as the others when she has more high scores?

Standards
Addressing
  • 6.SP.3·Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
  • 6.SP.5.c·Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
  • 6.SP.A.3·Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
  • 6.SP.B.5.c·Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

15 min

15 min

15 min