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.
Student Lesson in Spanish
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.
Activity
None
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
Calculate the mean number of baskets each player made, and compare the means. What do you notice?
What do the means tell us in this context?
Student Response
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Building on Student Thinking
Activity 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?
11.2
Activity
Standards Alignment
Building On
Addressing
6.SP.A.2
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
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.
In this activity, students turn their focus to variability while continuing to develop their understanding of what could be considered typical for a group. Students compare distributions with the same mean but different spreads and interpret them in the context of a situation. The context given here (basketball scores) prompts them to connect the mean to the notion of how well a player plays in general, and deviations from the mean to how “consistently” that player plays.
They encounter the idea of calculating the average absolute deviation from the mean as a way to describe variability in data.
Launch
Arrange students in groups of 3–4. Give groups 6–7 minutes to answer the questions, and follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for calculating the mean from a dot plot before they begin. Students can speak quietly to themselves, or share with a partner. Supports accessibility for: Organization, Conceptual Processing, Language
MLR2 Collect and Display. Collect the language that students use to describe how well the players play. Display words and phrases such as “consistent,” “typical,” and “most.” During the synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.
Advances: Conversing, Reading
Activity
None
Here are the dot plots showing the number of baskets that Elena, Jada, and Lin each made over 12 school days.
On each dot plot, mark the location of the mean with a triangle. Then, contrast the dot plot distributions. Write 2–3 sentences to describe the shape and spread of each distribution.
A dot plot, number of baskets Elena made, 0 to 10 by ones. Beginning at 1 number of dots above each increment is 4, 0, 1, 2, 1, 0, 3.
A dot plot, number of baskets Jada made, 0 to 10 by ones. Beginning at 2 the number of dots above each increment is 1, 1, 4, 1, 2, 2, 1.
A dot plot, number of baskets Lin made, 0 to 10 by ones. Beginning at 3 the number of dots above each increment is 2, 2, 3, 4, 1.
Discuss these questions with your group. Explain your reasoning.
Would you say that all three students play equally well?
Would you say that all three students play equally consistently?
If you could choose one player to be on your basketball team based on their records, whom would you choose?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of the discussion is to highlight that the center of the distribution is not always the only consideration when discussing data. The variability or spread can also influence how we understand the data.
There are many ways to answer the second set of questions. Invite students or groups who have different interpretations of playing well and playing consistently to share their thinking. Allow as many interpretations to be shared as time permits. Discuss:
“How might we use the given data to quantify playing well and playing consistently?” (The mean can represent how many baskets a player typically makes. The greater the mean, the better a player typically plays. The spread or variability indicates how consistent a player is. The greater the spread, the less consistent a player is.)
“There are many considerations when selecting a teammate. What is not captured by the data here? How do data help with the decision?” (Position on the team, how they get along with the other players and coach, and other skills like assists or rebounds are important for basketball. If the other aspects are the same or similar, then these data can help make a choice that is not based only on something like who I like.)
11.3
Activity
Standards Alignment
Building On
Addressing
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.
In this activity, students learn the term mean absolute deviation (MAD) as a way to quantify variability and calculate it by finding distances between the mean and each data value. Students compare data sets with the same mean but different MADs and interpret the variability in context.
While this process of calculating MAD involves taking the absolute value of the difference between each data point and the mean, this formal language is downplayed here. Instead, the idea of “finding the distance,” which is always positive, is used. This is done for a couple of reasons. One reason is to focus students' attention on the statistical work rather than on terminology or symbolic work. Another reason is that finding these differences may involve operations with signed numbers, which are not expected in this course.
Launch
Remind students that earlier they found the distance between each data point and the mean, and found that the sum of those distances on the left and the sum on the right are equal, which allows us to think of the mean as the balancing point, or the center, of the data. Explain that the distance between each point and the mean can be used to tell us something else about a distribution.
Arrange students in groups of 2. Give students 4–5 minutes to complete the first set of questions with their partner, and then 4–5 minutes of quiet time to complete the remaining questions. Follow with a whole-class discussion.
Activity
None
The tables show the number of baskets made by Jada and Lin in several games. Recall that the mean of Jada and Lin’s data is 5.
Record the distance between the number of baskets Jada made in each game and the mean.
Jada
2
4
5
4
6
6
4
7
3
4
8
7
distance from 5
Now find the average of the distances in the table. Show your reasoning, and round your answer to the nearest tenth.
This value is the mean absolute deviation (MAD) of Jada’s data. Jada’s MAD: _________
Find the mean absolute deviation of Lin’s data. Round it to the nearest tenth.
Lin
3
6
6
4
5
5
3
5
4
6
6
7
distance from 5
Lin’s MAD: _________
Elena’s distribution has a MAD of about 2.5. Compare the MADs and dot plots of the three students’ data. Do you see a relationship between each student’s MAD and the distribution on her dot plot? Explain your reasoning.
A dot plots, number of baskets Elena made, 0 to 10 by ones. Beginning at 2 number of dots above each increment is 4, 0, 1, 2, 1, 0, 1, 3, triangle at 5.25.
A dot plot, number of baskets Jada made, 0 to 10 by ones. Beginning at 2 the number of dots above each increment is 1, 1, 4, 1, 2, 2, 1, triangle at 5.
A dot plot, number of baskets Lin made, 0 to 10 by ones. Beginning at 3 the number of dots above each increment is 2, 2, 3, 4, 1, triangle at 5.
Student Response
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Building on Student Thinking
Students may recall an earlier lesson about thinking of the mean as a balance point and think that the MAD should always be zero because the left and right distances should be equal. Remind them that distances are always positive, so the average of these distances to the mean can be zero only if all the data points are exactly at the mean.
Activity Synthesis
During discussion, highlight that finding how far away, on average, the data points are from the mean is a way to describe the variability of a distribution. Discuss:
“What can we say about a data set whose data points have very small distances from the mean?” (That means most of the points are near the mean, so there is not much spread.)
“What about a data set with points that show large distances from the mean?” (That means most of the points are not close to the mean, so there is a large spread.)
“Does a data set with smaller distances (and therefore smaller average distances) show less or more variability?” (There is less variability when the points have smaller distances from the mean.)
Representation: Develop Languageand Symbols. Maintain a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of variability. Terms may include “mean absolute deviation” (MAD).
Supports accessibility for: Conceptual Processing, Language
MLR8 Discussion Supports. Display a sentence frame to support a whole-class discussion: “The largest (or smallest) mean absolute deviation value matches with the dot plot with the _____ spread because . . . .”Advances: Conversing, Speaking
11.4
Activity
Optional
Standards Alignment
Building On
Addressing
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.
This optional activity uses a game to help students develop the idea of variability. Use this activity if students could benefit from more concrete experiences with the idea of distance from the mean. Students will draw from a standard deck of playing cards and find the sum of the values of the drawn cards. They will determine how far their sum is from “22.” After playing 5 rounds, the player with the least mean distance from 22 wins the game.
Launch
Have students work in groups of 2–3. Provide a deck of standard playing cards to each group. Play 5 rounds.
Representation: Access for Perception. Begin by demonstrating the game for the whole class and doing the steps in the activity to support understanding of the context. Supports accessibility for: Conceptual Processing, Language
Activity
None
Your teacher will give your group a deck of cards. Shuffle the cards, and put the deck face down on the playing surface.
To play: Draw 3 cards and add up the values. An ace is a 1. A jack, queen, and king are each worth 10. Cards 2–10 are each worth their face value. If your sum is anything other than 22 (either above or below 22), say: “My sum deviated from 22 by ____ ,” or “My sum was off from 22 by ____ .”
To keep score: Record each sum and each distance from 22 in the table. After five rounds, calculate the average of the distances. The player with the lowest average distance from 22 wins the game.
player A
round 1
round 2
round 3
round 4
round 5
sum of cards
distance from 22
Average distance from 22: ____________
player B
round 1
round 2
round 3
round 4
round 5
sum of cards
distance from 22
Average distance from 22: ____________
player C
round 1
round 2
round 3
round 4
round 5
sum of cards
distance from 22
Average distance from 22: ____________
Whose average distance from 22 is the smallest? Who won the game?
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to think about how average distance from a number can be used to summarize variability, and invite a couple of students to share their thinking.
In the game, we can think of the player with the least average distance from 22 as having cards that are, on the whole, closest to 22 or the “least different” from 22. By the same token, a player with the greatest average distance from 22 can be seen as having cards that are, on the whole, farthest away from 22 or the “most different” from 22. Connect this to the idea that a data set with a large MAD means it has many values that vary from what we could consider a typical member of the group.
Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine with 2–3 successive pair shares to give students a structured opportunity to revise and refine their response to “How can the average distance from a number to the mean be used to summarize variability?” Provide students with prompts for feedback (for example, “Could you give a specific example from a round of ‘Game of 22’?” or, “Could you use the term ‘mean absolute deviation’ to explain your example further?”). Students can borrow ideas and language from each other to strengthen their final product. This will help students strengthen their ideas and clarify their language. Design Principle(s): Optimize output (for explanation); Cultivate conversation
Lesson Synthesis
The purpose of the discussion is to understand how a measure of spread can be used to quantify variability for a distribution. Ask students:
“How did we measure spread?” (By finding the average of how far each value is from the balance point.)
“We learned about a measure called the ‘mean absolute deviation’ (or MAD). What is the meaning of this term?” (The typical distance from each data value to the mean.)
“What does the MAD tell us?” (The average distance between data points and the mean. It tells us how spread out the data values are.)
“How do we find the MAD?” (Find the distance from each point to the mean. Then find the mean of those distances.)
“How is MAD related to the variability of a data set?” (The greater the MAD, the more variability in the data.)
Student Lesson Summary
We use the mean of a data set as a "measure of center" of its distribution, but two data sets with the same mean could have very different distributions.
This dot plot shows the number of stickers on each page of a 22-page sticker book.
A dot plot for “number of stickers on a page.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 stickers. The data are as follows: 18 stickers, 1 dot; 19 stickers, 3 dots; 20 stickers, 4 dots; 21 stickers, 5 dots; 22 stickers, 6 dots; 23 stickers, 2 dots, 24 stickers, 1 dot.
The mean number of stickers is 21. All the pages have within 3 stickers of the mean, and most of them are even closer. These pages are all fairly close in the number of stickers on them.
This dot plot shows the number of stickers on each page of another sticker book that has 30 pages.
A dot plot for “number of stickers on a page.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 stickers. The data are as follows: 9 stickers, 1 dot; 10 stickers, 1 dot; 11 stickers, 2 dots; 12 stickers, 1 dot; 14 stickers, 1 dot; 16 stickers, 2 dots; 17 stickers, 1 dot; 18 stickers, 2 dots; 19 stickers, 1 dot; 20 stickers, 3 dots; 21 stickers, 1 dot; 22 stickers, 3 dots; 23 stickers, 1 dot; 24 stickers, 2 dots; 26 stickers, 2 dots; 28 stickers, 1 dot; 30 stickers, 1 dot; 32 stickers, 2 dots; 33 stickers, 1 dot; 34 stickers, 1 dot.
In this sticker book, the mean number of stickers on each page is also 21, but some pages have less than half that number of stickers and others have more than one-and-a-half times as many. There is a lot more variability in the number of stickers.
There is a number that we can use to describe how far away, or how spread out, data points generally are from the mean. This measure of spread is called the mean absolute deviation (MAD).
To find the MAD, we find the distance between each data value and the mean, and then calculate the mean of those distances. For instance, the point that represents 18 stickers is 3 units away from the mean of 21 stickers.
A dot plot for “number of stickers on a page.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 stickers. There are two perpendicular lines drawn, one at 18 stickers and the other at 21 stickers. A horizontal line between the two lines is labeled 3. The data are as follows: 18 stickers, 1 dot; 19 stickers, 3 dots; 20 stickers, 4 dots; 21 stickers, 5 dots; 22 stickers, 6 dots; 23 stickers, 2 dots, 24 stickers, 1 dot.
We can find the distance between each point and the mean of 21 stickers and then organize the distances into a table, as shown.
The values in the first row of the table are the number of stickers on each page in the first book. Their mean, 21, is the mean number of stickers on a page.
The values in the second row of the table are the distances, or absolute deviation, between the values in the first row and 21. The mean of these distances is the MAD of the number of stickers on a page, about 1.2 stickers.
What can we learn from the averages of these distances once they are calculated?
In the first book, the distances are all between 0 and 3. The MAD is 1.2 stickers, which tells us that the number of stickers are typically within 1.2 of the mean number, 21. We could say that a typical page has between 19.8 and 22.2 stickers.
In the second book, the distances are all between 0 and 13. The MAD is 5.6 stickers, which tells us that the number of stickers are typically within 5.6 of the mean number, 21. We could say that a typical page has between 15.4 and 26.6 stickers.
The MAD is also called a measure of the variability of the distribution. In these examples, it is easy to see that a higher MAD suggests a distribution that is more spread out, showing more variability.
Standards Alignment
Building On
Addressing
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.
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.
