Download Data and Statistics for Middle School

Data and Statistics for Middle School
Introduction to Statistics
Your PLC has been asked to determine the effects of two different math programs. The programs were
assigned to Teacher A and Teacher B. The following table shows the final test scores of the last test in
two classes. Which class did better on the test? Which program had better results? Use as many pieces
of evidence that you know about to provide evidence for your claim.
Teacher A
Teacher B
100
100
95
100
67
98
85
97
45
45
80
46
77
50
77
73
75
65
50
73
100
85
80
94
60
58
70
52
84
57
82
85
90
75
94
80
47
70
60
66
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Data and Statistics for Middle School
Statistics in 6-8th Grade
Common Core Standards
Statistics and Probability 6.SP
Develop understanding of statistical variability.
1. Recognize a statistical question as one that anticipates variability in the data related to the question
and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but
“How old are the students in my school?” is a statistical question because one anticipates variability
in students’ ages.
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.
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.
Summarize and describe distributions.
4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
5. Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured and
its units of measurement.
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.
d. Relating the choice of measures of center and variability to the shape of the data distribution
and the context in which the data were gathered.
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Data and Statistics for Middle School
Statistics and Probability 7.SP
Use random sampling to draw inferences about a population.
1. Understand that statistics can be used to gain information about a population by examining a
sample of the population; generalizations about a population from a sample are valid only if the
sample is representative of that population. Understand that random sampling tends to produce
representative samples and support valid inferences.
2. Use data from a random sample to draw inferences about a population with an unknown
characteristic of interest. Generate multiple samples (or simulated samples) of the same size to
gauge the variation in estimates or predictions. For example, estimate the mean word length in a
book by randomly sampling words from the book; predict the winner of a school election based on
randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Draw informal comparative inferences about two populations.
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.
4. Use measures of center and measures of variability for numerical data from random samples to
draw informal comparative inferences about two populations. For example, decide whether the
words in a chapter of a seventh-grade science book are generally longer than the words in a chapter
of a fourth-grade science book.
Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the
likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0
indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor
likely, and a probability near 1 indicates a likely event.
6. Approximate the probability of a chance event by collecting data on the chance process that
produces it and observing its long-run relative frequency, and predict the approximate relative
frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3
or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a
model to observed frequencies; if the agreement is not good, explain possible sources of the
discrepancy.
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Data and Statistics for Middle School
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the
model to determine probabilities of events. For example, if a student is selected at random from
a class, find the probability that Jane will be selected and the probability that a girl will be
selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data
generated from a chance process. For example, find the approximate probability that a spinning
penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes
for the spinning penny appear to be equally likely based on the observed frequencies?
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction
of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables
and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”),
identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use
random digits as a simulation tool to approximate the answer to the question: If 40% of donors
have type A blood, what is the probability that it will take at least 4 donors to find one with type
A blood?
Statistics and Probability 8.SP
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of
association between two quantities. Describe patterns such as clustering, outliers, positive or
negative association, linear association, and nonlinear association.
2. Know that straight lines are widely used to model relationships between two quantitative variables.
For scatter plots that suggest a linear association, informally fit a straight line, and informally assess
the model fit by judging the closeness of the data points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate measurement data,
interpreting the slope and intercept. For example, in a linear model for a biology experiment,
interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated
with an additional 1.5 cm in mature plant height.
4. Understand that patterns of association can also be seen in bivariate categorical data by displaying
frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table
summarizing data on two categorical variables collected from the same subjects. Use relative
frequencies calculated for rows or columns to describe possible association between the two
variables. For example, collect data from students in your class on whether or not they have a curfew
on school nights and whether or not they have assigned chores at home. Is there evidence that those
who have a curfew also tend to have chores?
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Data and Statistics for Middle School
Answer the following questions in your group:
1. Which standards are completely new to you?
2. Which standards do you think will be particularly difficult for students?
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Data and Statistics for Middle School
Progressions
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Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
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Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
8
Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
9
Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
10
Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
11
Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
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Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
13
Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
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Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
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Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
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Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
17
Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
18
Retrieved from http://ime.math.arizona.edu/progressions/
https://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf
Data and Statistics for Middle School
Mathematical Practices
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Data and Statistics for Middle School
Framework of Statistical Literacy
The GAISE report published by the American Statistical Association was referenced in Developing
Essential Understanding of Statistics: Grades 6-8. Excerpts from the report will be used in this
professional development. The full document can be accessed via http://tinyurl.com/k9mqkev or
20
American Statistical Association. (2007). Guidelines for assessment and instruction in statistics education
(GAISE) report. Alexandria, VA: Author. Retrieved from http://www.amstat.org/education/gaise/GAISEPreK12_Full.pdf
Data and Statistics for Middle School
Framework of Statistical Literacy
21
American Statistical Association. (2007). Guidelines for assessment and instruction in statistics education
(GAISE) report. Alexandria, VA: Author. Retrieved from http://www.amstat.org/education/gaise/GAISEPreK12_Full.pdf
Data and Statistics for Middle School
22
American Statistical Association. (2007). Guidelines for assessment and instruction in statistics education
(GAISE) report. Alexandria, VA: Author. Retrieved from http://www.amstat.org/education/gaise/GAISEPreK12_Full.pdf
Data and Statistics for Middle School
23
American Statistical Association. (2007). Guidelines for assessment and instruction in statistics education
(GAISE) report. Alexandria, VA: Author. Retrieved from http://www.amstat.org/education/gaise/GAISEPreK12_Full.pdf
Data and Statistics for Middle School
What is a statistical question?
Which of the following are statistical questions? (A statistical question is one that can be
answered by collecting data and where there will be variability in that data.)
a. How many days are in March?
b. How old is your dog?
c. How old are the dogs on this street?
d. What percent of people like watermelons?
e. Do you like watermelons?
f. How many bricks are in this wall?
g. What was the highest temperature today in town?
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Identifying Statistical Questions; Retrieved from Illustrative Mathematics:
https://www.illustrativemathematics.org/content-standards/6/SP/A/1/tasks/703
Data and Statistics for Middle School
What is a statistical question? – Revisited
For the questions, you deemed to be statistical questions, explain ways you would collect data
to answer the question.
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Identifying Statistical Questions; Retrieved from Illustrative Mathematics:
https://www.illustrativemathematics.org/content-standards/6/SP/A/1/tasks/703
Data and Statistics for Middle School
Spinner Activity
This page is left blank for calculations, examples or notes.
http://illuminations.nctm.org/adjustablespinner/ or http://tinyurl.com/pyjpkcs or
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Data and Statistics for Middle School
Probability Games Activity
Formative Assessment Lesson
Record any information here that may be helpful for you to remember when using this FAL with
students:
Create a statistical question using the data you collected in the Probability Games Activity.
Is this the appropriate method of creating a statistical question? Why or why not?
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Probability Games, Retrieved from:
http://map.mathshell.org/materials/lessons.php?taskid=596&subpage=concept
http://tinyurl.com/lvzz7sj
Data and Statistics for Middle School
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Probability Games, Retrieved from:
http://map.mathshell.org/materials/lessons.php?taskid=596&subpage=concept
http://tinyurl.com/lvzz7sj
Data and Statistics for Middle School
Race Results Sheet
Explain your results. Are they different from what you expect? Why is this?
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Probability Games, Retrieved from:
http://map.mathshell.org/materials/lessons.php?taskid=596&subpage=concept
http://tinyurl.com/lvzz7sj
Data and Statistics for Middle School
Finishing Places of Horses
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Probability Games, Retrieved from:
http://map.mathshell.org/materials/lessons.php?taskid=596&subpage=concept
http://tinyurl.com/lvzz7sj
Data and Statistics for Middle School
Grand Canyon Temperature Graphing Activity
Temperatures were taken over a 15-day period at the Grand Canyon and are shown in the table
below. Using the data, create the following graphs.
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87
70
99
87
79
98
89
78
98
84
83
117
83
95
1. Find the mean, median, mode, and range.
Mean = _________ Mode = _________ Median = __________ Range = _________
2. Create a frequency table.
3. Create a dot plot.
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Data and Statistics for Middle School
4. Create a histogram.
5. Create a box plot.
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Data and Statistics for Middle School
This page is left blank for calculations, examples or notes.
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Data and Statistics for Middle School
Day 2
Let’s review information from the first day.
1. What are the steps to statistical problem solving?
2. How can we use different measures of center to find out more information about a
dataset?
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Data and Statistics for Middle School
Purposeful Pedagogy
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Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf
Data and Statistics for Middle School
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Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf
Data and Statistics for Middle School
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Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf
Data and Statistics for Middle School
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Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf
Data and Statistics for Middle School
39
Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf
Data and Statistics for Middle School
40
Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf
Data and Statistics for Middle School
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Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf
Data and Statistics for Middle School
Froot Loops Activity
Name________________________________________________________________
Materials: Fruit Loops, paper towels, and Froot Loops Activity handout
Query: How many Fruit Loops are there in a “handful”?
Instructions:
1. Each student grabs a handful of Froot Loops and counts them.
2. Record the class data using the table below.
Student # Cereal #
Student # Cereal #
1
16
2
17
3
18
4
19
5
20
6
21
7
22
8
23
9
24
10
25
11
26
12
27
13
28
14
29
15
30
Using your class data:
1. Find the measures of central tendency.
Mean = _________
Mode = _________
Median = __________
Range = __________
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This activity was retrieved from
http://www.regentsprep.org/regents/math/algebra/AD3/FruitLoopsActivity.pdf
(Copyright © Regents Exam Prep Center)
Data and Statistics for Middle School
2. Which measure of central tendency do you feel best represents the number of pieces of
cereal in a handful for this class? Explain.
3. The Principal, a former basketball player, comes into class and takes a handful. With his
entry, the mean increases. What can be said about the number of cereal pieces in the
Principal’s handful?
4. Peggy Sue comes into class and grabs a handful. With her entry, the median does not change.
What can be said about the number of cereal pieces in Peggy Sue’s handful?
5. Mrs. Smith, the librarian, and her pre-school daughter, Ashley, come in and grab handfuls.
When Mrs. Smith’s entry is added, the median decreases. What can be said about the
number of cereal pieces in Mrs. Smith’s handful?
6a. Little Ashley’s entry is added. Ashley is a very small little girl. With her entry, what would
you predict would happen to the mean?
6b. Again, with Ashley’s entry, what would you predict would happen to the median?
43
This activity was retrieved from
http://www.regentsprep.org/regents/math/algebra/AD3/FruitLoopsActivity.pdf
(Copyright © Regents Exam Prep Center)
Data and Statistics for Middle School
7. Specify the five statistical summary for your class data:
minimum = ______________
maximum = ______________
1st quartile = ______________
2nd quartile = ______________
3rd quartile = ______________
8. Construct a box plot for the class data.
44
This activity was retrieved from
http://www.regentsprep.org/regents/math/algebra/AD3/FruitLoopsActivity.pdf
(Copyright © Regents Exam Prep Center)
Data and Statistics for Middle School
Froot Loop Activity Extension
(for Pre-AP, Honors or other Advanced classes)
1. Predict what color you think will occur most often. Explain your reasoning.
2. Use your handfuls of Froot Loops to answer the following statistical question: What are the
ratios of colors in a typical box of Froot Loops?
3. Create a display for your data.
4. Determine the ratio of each color using percentages.
5. Compare your results to the results of other groups. Are your results similar? Explain:
6. What could you do to make your predictions more valid?
45
This activity was retrieved from
http://www.regentsprep.org/regents/math/algebra/AD3/FruitLoopsActivity.pdf
(Copyright © Regents Exam Prep Center)
Data and Statistics for Middle School
A New Park
Does Little Rock need a new city park? To seek support in determining if Little Rock needs a
new park, 20 people were asked if they would vote for the park. Their gender and political
party association were also collected. Using the data, create a table to organize the data so
that it could be used to make a decision.
Gender
Political
Party
Vote
M
D
Y
M
D
N
F
R
Y
M
R
Y
F
R
Y
F
R
Y
F
D
N
M
R
Y
F
D
N
M
R
N
M
R
N
M
R
Y
F
R
N
M
R
Y
F
D
Y
M
D
Y
F
D
Y
M
R
Y
F
D
Y
M
R
N
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Data and Statistics for Middle School
This page is for calculations, notes and explanations.
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Data and Statistics for Middle School
Value of Used Subaru Forester I
Jane wants to sell her Subaru Forester, but doesn’t know what the listing price should be. She checks on
craigslist.com and finds 22 Subarus listed. The table below shows age (in years), mileage (in miles), and
listed price (in dollars) for these 22 Subarus. (Collected on June 6th, 2012 for the San Francisco Bay
Area.)
Age
Mileage
Price
8
109,428
12,995
5
84,804
14,588
3
55,321
20,994
3
57,474
18,991
1
11,696
19,981
13
125,260
6,888
10
67,740
9,888
11
97,500
6,950
6
36,967
19,700
12
148,000
3,995
2
29,836
18,990
3
32,349
21,995
10
161,460
5,995
4
68,075
12,999
3
30,007
22,900
8
66,000
13,995
10
93,450
8,488
3
35,518
22,995
3
30,047
20,850
8
107,506
11,988
11
89,207
8,995
13
141,235
5,977
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This activity was modified from S-ID Used Subaru Foresters I; retrieved from Illustrative Mathematics:
https://www.illustrativemathematics.org/content-standards/HSS/ID/B/6/tasks/941
Data and Statistics for Middle School
1. Make appropriate plots with well-labeled axes that would allow you to see if there is a relationship
between price and age and between price and mileage. Describe the direction, strength and form of
the relationships that you observe. Does either mileage or age seem to be a good predictor of price?
49
This activity was modified from S-ID Used Subaru Foresters I; retrieved from Illustrative Mathematics:
https://www.illustrativemathematics.org/content-standards/HSS/ID/B/6/tasks/941
Data and Statistics for Middle School
2. If appropriate, describe the strength of each relationship using the correlation coefficient. Do the
values of the correlation coefficients agree with what you see in the plots?
3. Find the equation that describes each of the relationships.
4. If Jane’s car is 9 years old with 95000 miles on it, what listing price would you suggest? Explain how
you arrived at this price.
50
This activity was modified from S-ID Used Subaru Foresters I; retrieved from Illustrative Mathematics:
https://www.illustrativemathematics.org/content-standards/HSS/ID/B/6/tasks/941
Data and Statistics for Middle School
Central Tendency and Shapes of Data
For each set of data, find the mean, median, and mode.
A.
3, 6, 8, 8, 5, 2, 3, 6, 4
Mean_____
Median_____
Mode______
B. 5, 6, 5, 6, 4, 5, 4, 5, 5
Mean_____
Median_____
Mode______
C. 10, 10, 1, 1, 1, 1, 1, 10, 10
Mean_____
Median_____
Mode______
Create a dot plot for each set:
Dataset A:
Dataset B:
Dataset C:
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Data and Statistics for Middle School
1. What do each of the data sets have in common?
2. Compare the “shapes” of the data on the dot plots. What do you see?
3. What does this tell you about using Mean, Median and Mode as descriptors for data?
4. Why would the median be a better descriptor of a data set than the mean in some cases?
5. Create a data set where the median is a better descriptor than the mean:
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Data and Statistics for Middle School
Mean Absolute Deviation (MAD)
CCSS will require students to not only calculate mean absolute deviation, but to also understand,
interpret and describe how it helps to identify variability in data.
To calculate mean absolute deviation:
1. Find the mean of the data set.
2. Calculate the difference between the mean and each data point.
3. Take the absolute value of each difference.
4. Add the differences together and find the mean.
Example
Data set: 5, 7, 4, 13, 4, 10, 10, 7, 3
Mean: 7
Data point
3
4
4
5
7
7
10
10
13
Difference from mean
Absolute value of difference
-4
4
-3
3
-3
3
-2
2
0
0
0
0
3
3
3
3
6
6
Sum of the absolute value of the differences= 24
24/9 = 2.67, which is MAD
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Data and Statistics for Middle School
Now, go back and find the mean absolute deviation for the three sets of data in the Central Tendency
and Shapes of Data activity.
How does the MAD help you describe the data set more accurately than just mean or median?
Dataset A: 3, 6, 8, 8, 5, 2, 3, 6, 4
Dataset B: 5, 6, 5, 6, 4, 5, 4, 5, 5
Dataset C: 10, 10, 1, 1, 1, 1, 1, 10, 10
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Data and Statistics for Middle School
The Best Basketball Shooter Award
Coach Jameson has a problem. He has to pick one of his basketball players to receive the Best Shooter
Award at the athletic celebration next week. He has it narrowed down to 2 players, Jamal and Steven.
Here is the list of games and scores each player made:
Game 1
Game 2
Game 3
Game 4
Game 5
Game 6
Game 7
Game 8
Game 9
Game 10
Game 11
Game 12
Game 13
Game 14
Game 15
Jamal’s
Points Scored
11
10
14
12
8
4
15
14
13
8
9
14
11
15
10
Steven’s Points
Scored
10
13
28
0
4
8
10
16
16
4
10
0
20
14
15
1. What is Coach Jameson’s problem? What measure of center do you think he is using?
2. Create a statistical question for Coach Jameson’s problem:
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Data and Statistics for Middle School
The coach decided to go talk to Mrs. Smith, the math teacher, to find out if there was another way to
determine which player should get the award. Mrs. Smith told Coach Jameson that he should try Mean
Absolute Deviation.
3. Why did Mrs. Smith suggest using Mean Absolute Deviation?
4. Determine the Mean Absolute Deviation for each set of scores.
5. Which player would you recommend get the award? Explain your reasoning including the
mathematical reasoning you used to find your answer:
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Data and Statistics for Middle School
Returning to the First Dataset
Now that you know more about statistical analysis, let’s return to the first dataset. Which teacher has
better scores? Why? Be sure to explain using evidence.
Teacher A
Teacher B
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Data and Statistics for Middle School
This page is left blank for calculations, notes and explanations.
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Data and Statistics for Middle School
Variance and Standard Deviation
Variance and Standard Deviation give us another piece of information about datasets.
Here are the steps:
1) Find the mean of your dataset.
2) Subtract the mean from each datapoint (this is the deviation from the mean).
3) Square each deviation.
4) Take the sum of all the squared deviation scores.
5) A) For a population, you will divide the sum of the squared deviation scores by the number of
data points in your dataset. This is the variation.
B) If your dataset is a sample from a population, you have to use a correction. You will subtract
one from the number of data points in your dataset. This is the variation.
6) Take the square root of the variance. This is the standard deviation.
Here is an example:
Score
7
6
4
2
1
Deviation
from
Squared
mean
Deviation
7-4=3
9
6-4=2
4
4-4=0
0
2-4=-2
4
1-4=-3
3
Mean=4
Sum: 26
Since this is a sample, we need to subtract one from the total number of data points before we divide.
Variance: 26/4 = 6.5
Standard deviation: √6.5 = 2.55
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Data and Statistics for Middle School
Now, find the variance and standard deviation for all three datasets from the Central Tendency and
Shapes of Data activity.
Dataset A: 3, 6, 8, 8, 5, 2, 3, 6, 4
Dataset B: 5, 6, 5, 6, 4, 5, 4, 5, 5
Dataset C: 10, 10, 1, 1, 1, 1, 1, 10, 10
How does standard deviation help you compare data? How is it different from the mean absolute
deviation?
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Data and Statistics for Middle School
Comparisons of data with the same mean, but different standard deviations:
Describe the different datasets.
The normal curve:
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Data and Statistics for Middle School
The normal curve shows that approximately 68% of data points fall between +1 and -1 standard
deviation. We use this to figure out which scores are less normal than the rest.
Teacher A
Teacher B
100
100
95
67
100
98
85
97
45
45
80
77
46
50
77
73
75
65
50
100
73
85
80
94
60
58
70
84
52
57
82
85
90
75
94
47
80
70
60
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The following tables are descriptive statistics from the datasets above (these are the same datasets we
used yesterday). These were found using Excel. How does using variance and standard deviation help
you compare data sets?
Teacher A
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Range
Minimum
Maximum
Sum
Count
Teacher B
75.9
3.754225689
78.5
100
16.78940769
281.8842105
55
45
100
1518
20
62
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Range
Minimum
Maximum
Sum
Count
73.45
4.143145893
73
100
18.52871172
343.3131579
55
45
100
1469
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Data and Statistics for Middle School
With a partner, one of you develop a normal curve for the class of Teacher A and the other develop a
normal curve for the class of Teacher B.
Which scores fall into the top 5%?
Which are in the bottom 5%?
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Data and Statistics for Middle School
Fizz, Fizz
This activity was adapted from the Antacids in Developing Essential Understanding of Statistics:
Grades 6-8, pgs. 42-50.
Name________________________________________________________________
Materials: Two brands of effervescent tablets, stop watch, like cups to dissolve the tablets,
enough water at the same temperature to fill the cups equally
Scenario: Cold medicine comes in different forms: tablets, capsules, liquids, and effervescent
tablets. Effervescent tablets are tablets, that when dropped in water, dissolve. You then drink
the solution. If you are sick, you want to take the medicine as soon as possible to start relieving
your cold.
Instructions: In this activity, you want the students to engage in all four components of
statistical problem solving: 1) write a statistical question, 2) collect data, 3) analyze data, and 4)
interpret data. Depending on the level of your students more guidance may need to be
provided for the components.
Divide students into groups of 2-3 students. There will need to be someone recording the data
and taking measurements. These jobs can be shared between students, but this could cause
variability in the data collected. Not all students may interpret “dissolve” time the same. Use
this as a discussion topic at the end when groups are sharing their results.
Distribute materials to groups. When providing students with the effervescent tables, do not
let them know which brand they are receiving. Since you have selected the cups and water, it is
important to talk about how this reduces variability. The discussion could be done while you
are walking around listening to the groups as they determine how they are going to collect data
or it could be done when groups are sharing their results.
At the end of the activity, be sure to discuss the following with the students if not previously
addressed.
 Students were not told which brand they were observing. Why?
 Would it have been better to have one student to measure the time? Explain.
 Explain why using the same “cup/glass” was important.
 Would different amounts of water used to dissolve the tablet affected the outcome?
 Would having water of different temperatures make a difference in dissolve times?
Explain.
 What are other factors that could affect the results of the fizz time? Why?
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Data and Statistics for Middle School
This page is left blank for calculations, notes and explanations.
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Data and Statistics for Middle School
Day 1 Exit Slip
1. What are 3 things you have learned?
2. What are 2 things you still don’t understand?
3. What was your favorite activity today?
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Data and Statistics for Middle School
Day 2 Exit Slip
1. What are 3 things you have learned?
2. What are 2 things you still don’t understand?
3. What was your favorite activity today?
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Data and Statistics for Middle School
Materials List
Activity
Teacher
Two Sets of Test Scores – pg. 1
Data & Statistics Grades 6-8 Standards – pgs. 2-5
Progressions for Probability & Statistics – pgs. 6-18
Purposeful Pedagogy – pgs. 19-25
Statistical Literacy Framework – pgs. 26-27
Collecting Data – pgs. 28-29
What is a Statistical Question – pg. 31
What is a Statistical Question (Revisited) – pg. 32
Spinner Activity – pg. 33
Probability Games – pgs. 34-37
Probability Games Recording Sheet– pg. 34
Race Card – pg. 35
Race Results Sheet – pg. 36
Finish Places of Horses – pg. 37
Counters
A pair of die
Grand Canyon Temperatures – pgs. 38-40
Opening Activity Day 2 – pg. 41
Froot Loops – pgs. 42-45
Paper towels
Plastic Gloves or Hand-sanitizer
Boxes of Froot Loops
Recording Handout – pgs. 42-44
Extension Activity – pg. 45
A New Park – pgs. 46-47
Used Subaru Forester I – pgs. 48-50
Central Tendency & Shapes of Data – pgs. 51-52
Mean Absolute Deviation – pgs. 53-54
Best Basketball Shooter Award – pgs. 55-56
Return to Two Sets of Test Scores – pgs. 57-58
Variance and Standard Deviation – pgs. 59-63
Fizz, Fizz – pgs.64-67
Pitcher to pour water
Glass with water
Two brands of effervescent tablets
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1 per student
1 per student
1 per student
1 per student
1 per student
1 per student
1 per student
11 per group
1 pair per group
1 per student
Enough for class
1 box per 20 students
1 per student
1 per student - optional
1 per student
1 per student
1 per student
1 per student
1 per student
1 per student
1 per student
1 per student
1 per group ( all glasses should the same and
filled equally with water of same temperature)
1 of Brand A per group for half of the groups
1 of Brand B per group for other half of groups
Data and Statistics for Middle School
Resources
American Statistical Association. (2007). Guidelines for assessment and instruction in statistics education
(GAISE) report. Alexandria, VA: Author. Retrieved from
http://www.amstat.org/education/gaise/GAISEPreK-12_Full.pdf
Common Core State Standards retrieved from http://www.corestandards.org/wpcontent/uploads/Math_Standards.pdf
Fizz, Fizz activity was adapted from the Antacids in Developing Essential Understanding of
Statistics: Grades 6-8, pgs. 42-50.
Froot Loops activity was retrieved from
http://www.regentsprep.org/regents/math/algebra/AD3/FruitLoopsActivity.pdf (Copyright © Regents
Exam Prep Center)
Identifying Statistical Questions; Retrieved from Illustrative Mathematics:
https://www.illustrativemathematics.org/content-standards/6/SP/A/1/tasks/703
Mathematical Practices retrieved from http://www.corestandards.org/Math/Practice/
Probability Games, retrieved from:
http://map.mathshell.org/materials/lessons.php?taskid=596&subpage=concept
Progression Documents retrieved from
http://achievethecore.org/content/upload/Draft%206%E2%80%938%20Progression%20on%20Statistics
%20and%20Probability.pdf
Purposeful Pedagogy retrieved from http://commoncore.aetn.org/mathematics/ccss-mathematics4/CCSS%20Math%204-Purposeful%20Pedagogy%20and%20Discourse%20Model.pdf
Used Subaru Foresters I activity was modified from S-ID Used Subaru Foresters I; retrieved from
Illustrative Mathematics:
https://www.illustrativemathematics.org/content-standards/HSS/ID/B/6/tasks/941
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