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Lesson 28
Part 1: Introduction
CCLS
7.SP.3
Using Mean and Mean Absolute Deviation
to Compare Data
In this lesson, you will compare two distributions by looking at their centers and
variabilities. Take a look at this problem.
Marcus is the basketball reporter at Chesapeake State University. He wants to write a
story comparing the heights of players on the men’s team to the heights of players on
the women’s team. He made two dot plots to help him compare heights.
Players’ Heights
Players’ Heights
72
74
76
78
80
Men (in inches)
82
84
66
68
70
72
74
Women (in inches)
76
78
Explore It
Use the math you already know to solve this problem.
What are some heights that you see only on the men’s team? What are some heights you
see only on the women’s team?
What are some heights that you see on both the men’s team and the women’s team?
Estimate the typical height for the men’s team. Mark it on the graph. Do the same for the
women’s team. How does the typical height for the men’s team compare to the typical height for the
women’s team? How does the spread for the distribution of heights on the men’s team compare to that
of the women’s team? 264
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Part 1: Introduction
Lesson 28
Find Out More
For data sets like the ones on the previous page, it can be helpful to compute the measures
of center and compare them against your estimates.
One measure of center you can use for each data set is the mean. For the men’s basketball
team, the mean height is 77.2 inches (rounded to the nearest tenth). For the women’s
basketball team, the mean height is 71.5 inches (rounded to the nearest tenth).
You can also compare estimates of spread against a formal measure. One measure of spread
is the mean absolute deviation (MAD). The MAD describes the average distance between
the data values in a distribution and the mean of the distribution. To compute the MAD for
the men’s team, start by subtracting the value of each data point from the mean.
Height
Difference
from Mean
78
76
73
76
80
77
75
20.8
1.2
4.2
1.2
22.8
0.2
2.2
83
79
79
25.8 21.8 21.8
75
73
80
2.2
4.2
22.8
Then, you can average the absolute values of all of those results to get the MAD:
1.2 1 2.8 1 0.2 1 2.2  1 5.8 1 1.8
1 1.8 1 2.2 
1 4.2 1 2.8​ 
​ 0.8 1 1.2 1 4.2 1
      
 
5 2.4
13
·························································
You can use the same procedure to compute the women’s team’s MAD, which in this case
would equal 2.3 (rounded to the nearest tenth).
The difference between the mean heights of the teams is about 6 (77.2 2 71.5).
The MAD for the two distributions are about the same (between 2.3 and 2.4).
6 is a little more than two times 2.4, so the difference between the means is a little more
than twice the value of the MAD for each distribution. If the difference between the means
was three or more times as great as the MAD for each distribution, there would be an even
greater difference between the mean heights. A dot plot would show fewer heights in
common between the men’s and women’s teams.
Reflect
1 How would the distributions change if the difference between the means was the same
as the MAD for each distribution?
L28: Using Mean and Mean Absolute Deviation to Compare Data
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265
Part 2: Modeled Instruction
Lesson 28
Read the problem below. Then explore ways to compare data sets that have similar
variabilities but different centers.
A lot of the basketball players at Chesapeake State had graduated from Central Middle
School. How can Marcus compare the heights of players on the boys’ team at Central
Middle School to those on the men’s team at Chesapeake State?
Model It
You can compare the dot plots for each distribution.
The mean height of the men’s team is 77.2. Using the data for the boy’s team, you can find
the mean:
62 1 62 1 63 1 63 1 64  1 65 1 66 
1 66 1 67 1  
67 1 67 1 69​ 
​ 60 1 60 1 61 1       
5 64.1
15
····························································
You can use vertical segments to mark the mean for each data set.
Players’ Heights
68
70
72
74
76
78
Men (in inches)
80
82
84
72
74
76
Players’ Heights
60
266
62
64
66
68
70
Boys (in inches)
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Part 2: Guided Instruction
Lesson 28
Connect It
Now you will use these representations to compare the data sets.
2 What is the difference between the mean height for men on the Chesapeake State team
and the mean height for boys on the middle school team? 3 Compute the mean absolute deviation for boys’ height, to the nearest tenth. How does it
compare to 2.4, the mean absolute deviation for men’s height?
4 By what number would you have to multiply the MAD to get the difference between the
mean heights you found in problem 1? Round your answer to the nearest tenth.
5 What would the dot plot look like if you combined the data from boys at Central Middle
School with the data from men at Chesapeake State to make one big dot plot?
6 You would have to multiply the MAD by more than 3 to get the difference between the
means in these distributions. When the means of distributions are more than 3 MADs
apart, do you expect them to have a lot of values in common? Why or or why not?
Try It
Use what you just learned about comparing distributions to solve this problem.
7 The Central Middle School girls’ basketball team has a mean height of 62.3 inches, and a
MAD of 2.3 (the same MAD as the Chesapeake University women’s team). How many
MADs greater is the mean height for the Chesapeake University women’s team?
(Remember: the mean height for the women was 71.5 inches).
L28: Using Mean and Mean Absolute Deviation to Compare Data
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267
Part 3: Guided Practice
Lesson 28
Read the situation described below. Then solve problems 8–10.
Student Model
The student divided the
difference in the means by
the mean absolute
deviation to solve the
problem.
Sara was curious about how many text messages students with cell
phones send each day. She surveyed a random sample of students
who own cell phones at the middle school and another at the high
school. For the middle school data set, the mean number of texts per
day was 60. For the high school data set, the mean number was 76.
The MADs for the data sets were the same. Both MADs were 5.
How does the difference of the means compare to the mean absolute
deviations?
Look at how you can use the information in the problem to answer
the question.
Pair/Share
Why is it good to
compare data sets using
MADs along with looking
at the differences
between the means of
the data sets?
What does it mean when
the MADs of two sets of
data are about the same?
The difference in the means is 16.
16 4 5 5 3.2
The difference of the means is a little more than 3 times
Solution:
the mean absolute deviations.
8 How would the dot plots of the data described in the student model
differ? How might they be the same? Explain.
Show your work.
Pair/Share
What are some kinds of
data sets that you might
want to compare that
would NOT have similar
MADs?
268
Solution: L28: Using Mean and Mean Absolute Deviation to Compare Data
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Part 3: Guided Practice
9 Do you think that the two data sets described on the previous page
have a lot of values in common? Why or why not?
Show your work.
Lesson 28
What information tells me
whether the data sets
have a lot of values in
common?
Solution: 10 Which pair of data sets is most likely to have the greatest number of
values in common?
A Data set 1: mean 5 7
Data set 2: mean 5 15;
MAD for both data sets is 8
Pair/Share
Why do data sets usually
have a lot of shared
values when their means
are one MAD or less
apart?
What values should I be
comparing?
B Data set 1: mean 5 7
Data set 2: mean 5 15
MAD for both data sets is 4
C Data set 1: mean 5 10
Data set 2: mean 5 18
MAD for both data sets is 2
D Data set 1: mean 5 10
Data set 2: mean 5 15
MAD for both data sets is 1
Bryce chose D as the correct answer. How did he get that answer?
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Pair/Share
How would you help
Bryce understand his
error?
269
Part 4: Common Core Practice
Lesson 28
Answer Form
1  B C D
2  B C D
3  B C D
Solve the problems. Mark your answers to problems 1–3 on
the Answer Form to the right. Be sure to show your work.
1 2 Number
Correct
3
Two data distributions with similar MADs and many values in common often have which of
the following?
A
small MADs
B
large MADs
C
means that are many MADs apart from one another
D
means that are not many MADs apart from one another
Choose the data set with the greatest MAD.
A
mean = 81.9389
72
74
76
78
80
Height (in inches)
82
84
mean = 76.8109
B
72
74
76
78
80
Height (in inches)
82
84
mean = 76.9897
C
72
74
76
78
80
Height (in inches)
82
84
mean = 77.3007
D
72
270
74
76
78
80
Height (in inches)
82
84
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Part 4: Common Core Practice
3 4 Lesson 28
The mean absolute deviation is best described as:
A
The value obtained when you add all the values in a data set and divide by the number
of values.
B
The value obtained when averaging the distances between each data point in a
distribution and the mean.
C
The difference between the means of two similar distributions.
D
An approximation of the center of a statistical distribution.
Make up two data sets. List all the values in each data set and write a story to describe where
they may have originated. The data sets should meet the following conditions:
• The means should be different.
• The MADs should be similar.
• The means should be more than one MAD apart.
Be sure to show work to demonstrate that your data sets meet the above conditions.
Self Check Go back and see what you can check off on the Self Check on page 247.
L28: Using Mean and Mean Absolute Deviation to Compare Data
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271
Lesson 28
(Student Book pages 264–271)
Using Mean and Mean Absolute Deviation
to Compare Data
Lesson Objectives
The Learning Progression
•Use visual representations, such as dot plots, to
compare two real-world numerical data sets with
similar and differing variabilities.
In Grade 7, students continue their work with data
display and analysis. In previous grades, they have
described the shape of a data set and calculated mean,
median, mode, and range. They have worked with dot
plots and box plots. In Grade 7, students learn
measures of variability and use them to compare data
sets. They assess the degree of visual overlap of two
numerical data distributions with similar variabilities,
measuring the difference between the centers by
expressing it as a multiple of a measure of variability.
•Compare data sets and measure the difference
between the centers.
•Represent the difference between centers of data sets
by using the mean.
•Step through the calculations necessary to find the
mean absolute deviation for each of two data sets.
•Describe the variation in data sets.
Prerequisite Skills
•Find the mean, median, mode, and range of a data
set.
•Read and create data displays, including dot plots.
In this lesson, students will use dot plots to compare
two real-world numerical data sets with similar and
differing variabilities, measure the difference between
centers, and represent the difference in the means.
Students will step through the calculations necessary
to find the mean absolute deviation for and describe
the variation in each data set.
•Find the mean absolute deviation of a data set.
Toolbox
Vocabulary
Teacher-Toolbox.com
Prerequisite
Skills
mean: the average of the numbers; the sum of the
values divided by the number of values
Ready Lessons
mean absolute deviation (MAD): the average distance
of each data point from the mean
Tools for Instruction
Interactive Tutorials
✓
✓
7.SP.3
✓
✓
✓
CCLS Focus
7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the
difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players
on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute
deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1–7 (see page A9 for full text)
L28: Using Mean and Mean Absolute Deviation to Compare Data
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271
Part 1: Introduction
Lesson 28
At a Glance
Students explore a problem and are guided to compare
two distributions by looking at centers and variabilities.
Lesson 28
Step By Step
72
•Ask the pairs to measure each other’s heights to
the nearest inch.
•Record the measurements for each student on the
board, sorting the measures of boys from girls.
•Have the pairs create one dot plot for girls and one
for boys.
•Ask pairs to compare the shapes of the graphs, the
averages of the typical heights, and the spreads.
•Discuss how the students’ data compares with the
data of the players’ heights.
272
76
78
80
Men (in inches)
82
84
66
68
70
72
74
Women (in inches)
76
78
use the math you already know to solve this problem.
What are some heights that you see only on the men’s team? What are some heights you
see only on the women’s team?
men’s team: 78, 79, 80, and 83; women’s team: 67, 69, 70, 71, and 72
What are some heights that you see on both the men’s team and the women’s team?
Possible answers: some heights that you see on both teams are 73, 75, and 77.
Estimate the typical height for the men’s team. Mark it on the graph. Do the same for the
women’s team. Possible answer: men’s team about 77; women’s team about 71
How does the typical height for the men’s team compare to the typical height for the
women’s team? the typical height for men is about 6 inches taller than the
ELL Support
Have students work in pairs and save their work for
the Hands-On Activity on page 275.
74
explore it
•Ask student pairs or groups to answer the last
question and share their explanation with the class.
Materials: yard sticks, rulers
Players’ Heights
Players’ Heights
•Give students time to make observations based on
the dot plots, prompting them as necessary to
describe clumps, gaps, and spreads. Encourage
students to debate and justify their observations.
Make dot plots to compare height data.
7.sP.3
Marcus is the basketball reporter at Chesapeake State University. He wants to write a
story comparing the heights of players on the men’s team to the heights of players on
the women’s team. He made two dot plots to help him compare heights.
•Work through Explore It as a class.
Hands-On Activity
CCLs
in this lesson, you will compare two distributions by looking at their centers and
variabilities. take a look at this problem.
•Tell students that this page guides them in
understanding how to compare two distributions by
looking at their centers and variabilities. Have
students read the problem at the top of the page and
examine the dot plots.
On this page, some everyday words are used in a
math context: typical, center, spread, distribution.
Make a chart with the words, their everyday
meanings, and their meanings as math terms.
Part 1: introduction
using Mean and Mean absolute Deviation
to Compare Data
typical height for women.
How does the spread for the distribution of heights on the men’s team compare to that
of the women’s team? Possible answer the spreads are pretty similar because the
tallest height for the men is 10 inches greater than the shortest height. the
tallest height for the women is also 10 inches greater than the shortest height.
264
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Mathematical Discourse
•How did you estimate the typical heights of the men
and the women? On what did you base your
estimates?
Students may have different ways to estimate
the “typical” height. Follow up by asking
whether others used a different method. Listen
for ideas about “most” heights.
•How would the shape of the dot plots change if you
added data from this class?
Student observations may include a change in
the mean, a greater range, a greater spread
distribution, and changes in the shape of the
dot plot.
•Describe a scenario in which information from the
problem may be of value.
Knowing the typical heights of chosen players.
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Part 1: Introduction
Lesson 28
At a Glance
Students examine how to determine the mean absolute
deviations of two data sets to assess the degree of
overlap.
Step By Step
•Read Find Out More as a class.
•Review briefly what the mean is and how it is
calculated. Explain that for this data set, we’ll use the
mean as the “typical” height.
•Read aloud the third paragraph. Ask three or four
students to explain in their own words what the
mean absolute deviation (MAD) is and what it tells
about a data set. Ask others to agree or add on to the
explanations.
•Direct students’ attention to the final steps in
determining the MAD. Here, all the differences are
combined and then divided by the total number of
absolute values.
•Discuss how to find the difference between the mean
heights and how to compare it to the MADs for the
distributions.
Part 1: introduction
Lesson 28
Find out More
For data sets like the ones on the previous page, it can be helpful to compute the measures
of center and compare them against your estimates.
One measure of center you can use for each data set is the mean. For the men’s basketball
team, the mean height is 77.2 inches (rounded to the nearest tenth). For the women’s
basketball team, the mean height is 71.5 inches (rounded to the nearest tenth).
You can also compare estimates of spread against a formal measure. One measure of spread
is the mean absolute deviation (MaD). The MAD describes the average distance between
the data values in a distribution and the mean of the distribution. To compute the MAD for
the men’s team, start by subtracting the value of each data point from the mean.
height
Difference
from Mean
78
76
73
76
80
77
75
20.8
1.2
4.2
1.2
22.8
0.2
2.2
83
79
SMP Tip: Students reason abstractly when they
consider how the distributions or dot plots would
change if there were different MAD values (SMP 2).
•Have pairs or groups do Reflect and share their
answers with the class.
ELL Support
To help students understand the abstract meanings
of such words as mean absolute deviation, compare
the words to their everyday uses:
mean (not to be confused with cruel): average, in
the middle
absolute: unlimited; a dictator has absolute
power; an absolute number ignores its positive or
negative signs
deviate, deviation: go away from
L28: Using Mean and Mean Absolute Deviation to Compare Data
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75
73
80
2.2
4.2
22.8
Then, you can average the absolute values of all of those results to get the MAD:
0.8 1 1.2 1 4.2 1 1.2 1 2.8 1 0.2 1 2.2 1 5.8 1 1.8 1 1.8 1 2.2 1 4.2 1 2.8
 
 
 
5 2.4
13
·························································
You can use the same procedure to compute the women’s team’s MAD, which in this case
would equal 2.3 (rounded to the nearest tenth).
The difference between the mean heights of the teams is about 6 (77.2 2 71.5).
The MAD for the two distributions are about the same (between 2.3 and 2.4).
6 is a little more than two times 2.4, so the difference between the means is a little more
than twice the value of the MAD for each distribution. If the difference between the means
was three or more times as great as the MAD for each distribution, there would be an even
greater difference between the mean heights. A dot plot would show fewer heights in
common between the men’s and women’s teams.
reflect
1 How would the distributions change if the difference between the means was the same
as the MAD for each distribution?
Possible answer: there would be a lot more overlap between the distributions.
you would see more heights in common between the men’s team and the
women’s team.
L28: Using Mean and Mean Absolute Deviation to Compare Data
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•Emphasize that a data set with a smaller MAD has
data values that are closer to the mean than a data
set with a greater MAD. Create a dot plot on the
board with combined data to support this reasoning.
79
25.8 21.8 21.8
265
Real-World Connection
Ask students to think of situations in which it would
be helpful to know the mean absolute deviations of
two data sets. If they have trouble with this, suggest
this situation:
Suppose a teacher writes two versions of a test covering
the same subject. To be a good test, students should
score about the same on both tests. How would knowing
the mean absolute deviations of the scores help? [The
MADs of the scores on the two tests would show
whether students as a group were scoring roughly
the same on both tests.]
273
Part 2: Modeled Instruction
Lesson 28
At a Glance
Part 2: Modeled instruction
Students will use dot plots to compare the mean
average deviation of two data sets.
read the problem below. then explore ways to compare data sets that have similar
variabilities but different centers.
Step By Step
A lot of the basketball players at Chesapeake State had graduated from Central Middle
School. How can Marcus compare the heights of players on the boys’ team at Central
Middle School to those on the men’s team at Chesapeake State?
•Read the problem at the top of the page as a class.
Model it
•Direct students to Model It. You will discuss with
students what they can determine from casual
observation of the dot plots.
you can compare the dot plots for each distribution.
The mean height of the men’s team is 77.2. Using the data for the boy’s team, you can find
the mean:
60 1 60 1 61 1 62 1 62 1 63 1 63 1 64 1 65 1 66 1 66 1 67 1 67 1 67 1 69
 
 
 
5 64.1
15
····························································
•Ask, Are the height data of the boys tightly clustered or
generally spread out? [The boys’ height data are
spread out, ranging from 60 to 69 inches.]
You can use vertical segments to mark the mean for each data set.
Players’ Heights
68
• Ask, How do the height data for the boys compare to the
height data for the men? [The boys’ height data range
from 60 to 69 inches and appear to have a mean of
about 64 inches. The men’s height data range from 73
to 83 inches and appear to have a mean of about
77 inches.]
• Ask, Do the boys’ height data overlap with the men’s
height data? How do you know? [No, the data sets do
not overlap. The greatest height for the boys is
69 inches; the least height for the men is 73 inches.
Caution students not to be fooled by the apparent
overlap caused by the dot plots being so close to each
other.]
SMP Tip: When students describe the centers and
spread of data without using a formula, they are
reasoning inductively and making reasonable
conjectures (SMP 2). Periodically ask students to
describe and make conjectures before doing formal
calculations.
274
Lesson 28
70
72
74
76
78
Men (in inches)
80
82
84
72
74
76
Players’ Heights
60
266
62
64
66
68
70
Boys (in inches)
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Mathematical Discourse
•What predictions can you make about the dot plots?
Explain and justify this prediction.
Listen for explanations which include no
overlapping of values when all of the data are
combined on a single dot plot.
•If you made a dot plot of players’ heights for a men’s
team from another university, how would it compare
to the dot plot for the players’ heights from
Chesapeake State?
Students will likely say that the data values for
both universities will be similar. Some say that
the data would overlap if graphed on a single
dot plot.
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Part 2: Guided Instruction
Lesson 28
At a Glance
Students revisit the problem on page 266 to compare
data sets by assessing the degree of overlap by using the
mean absolute deviation.
Part 2: guided instruction
Lesson 28
Connect it
now you will use these representations to compare the data sets.
2 What is the difference between the mean height for men on the Chesapeake State team
Step By Step
•Read Connect It as a class. Point out that the
questions refer to the problem on page 266 and also
the team data from the beginning of the lesson. Point
out that the data points for the men’s team have not
changed and still have a mean of 77.2.
•As you work through problems 2–4 with the class,
have students describe how to solve the problems
and share their answers.
•For problems 5 and 6, use the dot plot on page 266 to
help students measure the distance between the
mean heights. Ask, How can the number of MAD
segments between the two means be determined? [find
the difference between the two means and then
divide by the MAD]
•Read Try It as a class. Have students explain how
they got their answers to the class.
and the mean height for boys on the middle school team?
compare to 2.4, the mean absolute deviation for men’s height?
4.1 1 4.1 1 3.1 1 2.1 1 2.1 1 1.1 1 1.1 1 0.1 1 0.9 1 1.9 1 1.9 1 2.9 1 2.9 1 2.9 1 4.9
 
 
 
 
5 2.4;
15
·····································································
it is about the same as the MaD for the men.
4 By what number would you have to multiply the MAD to get the difference between the
mean heights you found in problem 1? Round your answer to the nearest tenth.
13.1 4 2.4 < 5.458; about 5.5 inches
5 What would the dot plot look like if you combined the data from boys at Central Middle
School with the data from men at Chesapeake State to make one big dot plot?
Possible answer: there wouldn’t be any overlap.
6 You would have to multiply the MAD by more than 3 to get the difference between the
means in these distributions. When the means of distributions are more than 3 MADs
apart, do you expect them to have a lot of values in common? Why or or why not?
Possible answer: no, i would not expect many common values. as data sets are
more MaDs apart, there is less overlap.
try it
use what you just learned about comparing distributions to solve this problem.
7 The Central Middle School girls’ basketball team has a mean height of 62.3 inches, and a
MAD of 2.3 (the same MAD as the Chesapeake University women’s team). How many
MADs greater is the mean height for the Chesapeake University women’s team?
(Remember: the mean height for the women was 71.5 inches).
Possible answer: 71.5 2 62.3 5 9.2 inches difference between the means. if you
divide 9.2 by 2.3, you get 4. so, the mean heights are 4 MaDs apart.
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Hands-On Activity
Calculate means and MADs to compare data.
Materials: dot plots from Hands-On Activity page
272
•Have students work in the same pairs they did
earlier.
•Ask the pairs to determine the mean of each data
set, the MAD of each data set, and the difference
of the means using MAD.
77.2 2 64.1 5 13.1
3 Compute the mean absolute deviation for boys’ height, to the nearest tenth. How does it
267
Try It Solution
7 Solution: 4 MADs apart; Students may determine
the difference of the mean heights by calculating;
71.5 2 62.3 5 9.2 inches. Divide 9.2 by 2.3 5 4. So,
the mean heights are 4 MADS apart.
ERROR ALERT: Students who wrote 27.1 MADs did
not determine the difference between the mean
heights.
•Have the pairs describe in writing the meaning of
the differences in the means using MAD.
•Have students add data points to the data sets to
change the outcome of the differences in the
means using the MAD.
•Discuss everyone’s results and analyses as a class.
L28: Using Mean and Mean Absolute Deviation to Compare Data
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275
Part 3: Guided Practice
Part 3: guided Practice
Lesson 28
Lesson 28
read the situation described below. then solve problems 8–10.
StudentModel
The student divided the
difference in the means by
the mean absolute
deviation to solve the
problem.
Sara was curious about how many text messages students with cell
phones send each day. She surveyed a random sample of students
who own cell phones at the middle school and another at the high
school. For the middle school data set, the mean number of texts per
day was 60. For the high school data set, the mean number was 76.
The MADs for the data sets were the same. Both MADs were 5.
How does the difference of the means compare to the mean absolute
deviations?
Look at how you can use the information in the problem to answer
the question.
Pair/share
Why is it good to
compare data sets using
MADs along with looking
at the differences
between the means of
the data sets?
What does it mean when
the MADs of two sets of
data are about the same?
the difference in the means is 16.
Part 3: guided Practice
9 Do you think that the two data sets described on the previous page
have a lot of values in common? Why or why not?
Show your work.
values in common. When data sets are more than 3 MaDs apart,
they don’t usually overlap much.
10 Which pair of data sets is most likely to have the greatest number of
values in common?
a
Data set 1: mean 5 7
Data set 2: mean 5 15;
MAD for both data sets is 8
b
Data set 1: mean 5 7
Data set 2: mean 5 15
MAD for both data sets is 4
C
Data set 1: mean 5 10
Data set 2: mean 5 18
MAD for both data sets is 2
D
Data set 1: mean 5 10
Data set 2: mean 5 15
MAD for both data sets is 1
the mean absolute deviations.
8 How would the dot plots of the data described in the student model
differ? How might they be the same? Explain.
Show your work.
What information tells me
whether the data sets
have a lot of values in
common?
Solution: Possible answer: no, i don’t think there are a lot of
16 4 5 5 3.2
Solution: the difference of the means is a little more than 3 times
Lesson 28
Pair/share
Why do data sets usually
have a lot of shared
values when their means
are one MAD or less
apart?
What values should I be
comparing?
Bryce chose D as the correct answer. How did he get that answer?
bryce probably chose D because the difference between the
means is the lowest. but bryce probably didn’t see that the
means for choice D are 5 MaDs apart. the means for choice a are
Pair/share
What are some kinds of
data sets that you might
want to compare that
would NOT have similar
MADs?
268
Solution:
one dot plot would center around 60, the other
would center around 76. since the MaDs are the same, the
only 1 MaD apart, so there is a better chance the two data sets
will have more overlap.
spread of the data would be similar.
L28: Using Mean and Mean Absolute Deviation to Compare Data
Pair/share
How would you help
Bryce understand his
error?
L28: Using Mean and Mean Absolute Deviation to Compare Data
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269
At a Glance
Solutions
Students will compare data sets by assessing the
overlap of two numerical data distributions. Students
will also measure the difference between the centers of
data sets by expressing it as a multiple of a measure of
variability.
Ex Solution: Find the difference of the means and
divide that difference by the MAD.
Step By Step
9 Solution: No; When data sets are more than 3 MADs
apart, the overlap is minimal.
•Ask students to solve the problems individually by
using mean absolute deviation to compare data sets.
•When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner or in a group.
276
8 Solution: One dot plot would center around 60, the
other around 76; The MADs are the same, so the
spread of the data would be similar.
10 Solution A; The difference between the means is the
same as the MAD for each distribution, so there are
most likely many data values in common.
Explain to students why the other two answer
choices are not correct:
B is not correct because MADs of length 4 would fit
into the difference of the means twice, which is not
as good as the overlap in A.
C is not correct because MADs of length 2 would fit
into this difference of the means 4 times. Data
points would not.
L28: Using Mean and Mean Absolute Deviation to Compare Data
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Part 4: Common Core Practice
Part 4: Common Core Practice
Lesson 28
Lesson 28
Part 4: Common Core Practice
Lesson 28
answer Form
1 A B C D
2 A B C D
3 A B C D
Solve the problems. Mark your answers to problems 1–3 on
the Answer Form to the right. Be sure to show your work.
1
2
number
Correct
3
3
The mean absolute deviation is best described as:
A
The value obtained when you add all the values in a data set and divide by the number
of values.
Two data distributions with similar MADs and many values in common often have which of
the following?
B
The value obtained when averaging the distances between each data point in a
distribution and the mean.
A
small MADs
C
The difference between the means of two similar distributions.
B
large MADs
D
An approximation of the center of a statistical distribution.
C
means that are many MADs apart from one another
D
means that are not many MADs apart from one another
4
Choose the data set with the greatest MAD.
A
mean = 81.9389
72
74
76
78
80
Height (in inches)
82
Make up two data sets. List all the values in each data set and write a story to describe where
they may have originated. The data sets should meet the following conditions:
•
The means should be different.
•
The MADs should be similar.
•
The means should be more than one MAD apart.
Be sure to show work to demonstrate that your data sets meet the above conditions.
84
Check to see that students’ work meets each of the conditions described above.
mean = 76.8109
B
72
74
76
78
80
Height (in inches)
82
84
mean = 76.9897
C
72
74
76
78
80
Height (in inches)
82
84
mean = 77.3007
D
72
74
76
78
80
Height (in inches)
82
84
self Check Go back and see what you can check off on the Self Check on page 247.
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L28: Using Mean and Mean Absolute Deviation to Compare Data
L28: Using Mean and Mean Absolute Deviation to Compare Data
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271
At a Glance
Solutions
Students solve problems that might appear on a
mathematics test. They compare data sets by assessing
the degree of overlap of two numerical data
distributions with similar variabilities, and measure the
difference between the centers by expressing it as a
multiple of a measure of variability.
1 Solution: D; A lot of values in common means the
overlapping of data values and mean absolute
deviations will have little difference.
Step By Step
•First tell students that they will solve problems about
data sets and mean absolute deviation. Then have
students read the directions and answer the
questions independently. Remind students to fill in
the correct answer choices on the Answer Form.
2 Solution: D; Students may recognize the spread of
the data horizontally.
3 Solution: B; Students identify the definition of mean
absolute deviation.
4 Solution: Check to see that students’ work meets
each of the conditions described.
•After students have completed the Common Core
Practice problems, review and discuss correct
answers. Have students record the number of correct
answers in the box provided.
L28: Using Mean and Mean Absolute Deviation to Compare Data
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277
Differentiated Instruction
Lesson 28
Assessment and Remediation
•Ask students to compare these data sets to determine whether the distributions overlap. Have them show
their work and write a few sentences to explain. Round answers to the nearest whole number.
Set A: 58, 38, 54, 48, 26, 36
Set B: 58, 42, 64, 62, 70, 40
[Means: 43, 56; MADs: 10, 10. The MADs are similar, which means the spread of data is similar. The difference
of the means is 13, and the MADs indicate a good chance that the two data sets will have some overlap.]
•For students who are struggling, use the chart below to guide remediation.
•After providing remediation, check students’ understanding. Ask students to compare data sets by assessing
the degree of overlap of these data sets:
Set A: 12, 11, 9, 9, 7
Set B: 10, 11, 8, 8, 6
•If a student is still having difficulty, use Ready Instruction, Level 6, Lesson 27.
If the error is . . .
Students may . . .
To remediate . . .
MADs of 43 and 56
have confused MADs and the
means of the data sets.
Review the process of determining MADs. Student will also
benefit from using a dot plot to see the locations of MADs and
the means. Ask the student to write out the steps for comparing
data sets using the mean absolute deviations.
MADs of –2 and 0
not understand that they need
to take the absolute value of
the differences and then find
the average.
Work with students to build two tables showing the MADs as the
average distance from the mean.
overlapping of data
have confused the MAD of the
data sets with the distance
between the means.
Create dot plots of the data. Focus on the number of MADs
between the means as the determining factor in overlapping
data.
Hands-On Activity
Challenge Activity
Describe variations and overlap in two data
sets.
Create and compare data sets.
Give students the following data sets:
Set A: 88, 116, 94, 114, 112, 124
Set B: 100, 127, 105, 120, 130, 135
Have students create a dot plot to represent each set of
data. Tell them to calculate the mean of each set and
the MADs to compare the sets. Tell students to round
their answers to the nearest whole number. Then have
students write a few sentences to describe and
compare the data sets using mean absolute deviation.
Tell students to create two sets of data that satisfy the
following conditions:
•The mean absolute deviation of the first set is less
than the mean absolute deviation of the second set.
•The mean of the first set is greater than the mean
of the second set.
Have students write a few sentences to describe and
compare the data sets. Have students exchange their
data sets with another student who will determine if
the data sets meet the criteria.
Means: 108, 120; MADs: 11, 11. The MADs are similar,
which means the spread of the data is similar. The
difference of the means is 12. The MADs indicate a
good chance that the two data sets will have some
overlap.
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L28: Using Mean and Mean Absolute Deviation to Compare Data
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