Foundations of Mathematical Reasoning
Preview Assignment 6.A
Preview Assignment 6.A
Preparing for the next class
In the next lesson, you will need to be able to perform basic operations and find the
mean, median, and mode of a set of numbers.
Questions 1–3: Find the sum of each of the following sets of numbers, as well as the
size n of the set of numbers. Try to do the problem without a calculator. Instead, use the
problem to practice techniques for adding numbers quickly.
1)
19, 20, 21
Sum =
n=
2)
101, 73, 49, 27, 24, 36
Sum =
n=
3)
20, 25, 30, 30, 30, 30, 31, 32, 33
Sum =
n=
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Foundations of Mathematical Reasoning
Preview Assignment 6.A
Questions 4–6: Compute the mean, mode, and median for the following set of numbers.
Use the information from Questions 1 through 3. Use a calculator when necessary.
4)
19, 20, 21
Mean =
Mode =
Median =
5)
101, 73, 49, 27, 24, 36,
Mean =
Mode =
Median =
6)
20, 25, 30, 30, 30, 30, 31, 32, 33
Mean =
Mode =
Median =
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Foundations of Mathematical Reasoning
Preview Assignment 6.A
Monitoring your readiness
7)
To effectively plan and use your time wisely, it helps to think about what you know
and do not know. For each of the following, rate how confident you are that you
can successfully do that skill. Use the following descriptions to rate yourself:
5—I am extremely confident I can do this task.
4—I am somewhat confident I can do this task.
3—I am not sure how confident I am.
2—I am not very confident I can do this task.
1—I am definitely not confident I can do this task.
Skills Needed for Lesson 6, Part A
Questions to
check your
understanding
Skill or Concept: I can . . .
Perform basic operations using quantities with
the aid of technology.
1–6
Find the mean, median, and mode of a set of
numbers.
4–6
Rating
from 1 to 5
Now, use the ratings to get ready for your next lesson. If your rating is a 3 or below,
you should get help with the material before class. Remember, your instructor is
going to assume that you are confident with the material and will not take class time to
answer questions about it.
Ways to get help:
•
See your instructor before class for help.
•
Ask your instructor for on-campus resources.
•
Set up a study group with classmates so you can help each other.
•
Work with a tutor.
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Foundations of Mathematical Reasoning
Preview Assignment 6.A
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Foundations of Mathematical Reasoning
Student Pages 6.A, Measures of central tendency
Lesson 6, Part A, Measures of
central tendency
Theme: Personal Finance
Imagine that a sample of individuals
report the balance on their credit card.
The histogram summarizes this report.
1)
How many individuals are included
in the sample?
Objectives for the lesson
You will understand that:
o Mean and median provide different pictures of data.
o Conclusions derived from statistical summaries are subject to misinterpretation.
o Histograms are graphical displays useful for showing the shape of a distribution.
You will be able to:
o Compare mean and median from the shape of a distribution.
o Create a data set that meets certain criteria for measures of central tendency.
2)
Which bin contains the median credit card balance?
3)
Create a data set that matches the histogram. In other words, record a set of dollar
amounts including two values ranging from 0 to $2,499.99, two values ranging
from $2,500 to $4,999.99, and so on.
4)
Compute the mean and median of your data set.
5)
Mark the mean and the median of your data set on the horizontal axis of the
histogram.
6)
Which phrase best describes the median of your data set?
a)
significantly less than the mean
b)
roughly the same as the mean
c)
significantly greater than the mean
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Foundations of Mathematical Reasoning
Student Pages 6.A, Measures of central tendency
Imagine another sample of individual credit card balances summarized by the histogram
below.
7)
The median for this sample is $0. Interpret this statement. Explain what it means
for the median to equal zero dollars.
8)
The mean for the sample is $2,600. Create a data set with a median of $0 and a
mean of $2,600 that matches the histogram. In other words, write down a set of
dollar amounts including fourteen values ranging from 0 to $2,499.99, two values
ranging from $2,500 to $4,999.99, and one value in each of the remaining bins.
Make sure that your 20 values have a median of $0 and a mean of $2,600.
9)
For convenience, name the four individuals in your data set who have credit card
balances in the top four bins as the Spenders and the other individuals as the
Savers. Here is a lame joke:
The Savers are all at a party having a good time. When the Spenders
arrive, the Savers suddenly get depressed. The Spenders say, “Hey, why
is everyone suddenly so glum?” The Savers reply, “Don’t you know what
you four guys do to our average credit card debt?”
Explain the punchline of the joke. Why are the Savers depressed when the
Spenders arrive?
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 6.A, Measures of central tendency
Lesson 6, Part A
Measures of Central Tendency
Overview and student objectives
Overview
Lesson Length: 25
minutes
In this lesson, students understand the impact of skewness on
summary statistics and connect graphical representations of
data with numerical representations.
Prior Lesson: Lesson 5,
Part D, “Shapes of
Distributions”
Objectives
Next Lesson: Lesson 6,
Part B, “Brain Power” (25
minutes)
Students will understand that:
•
Mean and median provide different pictures of data.
•
Conclusions derived from statistical summaries are
subject to misinterpretation.
•
Histograms are graphical displays useful for showing the
shape of a distribution.
Students will be able to:
Constructive
Perseverance Level: 2
Theme: Personal
Finance
Outcomes: N1, N2, N8,
PF1
Goals: Communication,
Reasoning, Evaluation
•
Compare the mean and median from the shape of a distribution.
•
Create a data set that meets certain criteria for measures of central tendency.
Suggested resources and preparation
Materials and technology
•
Computer, projector, document camera
•
Preview Assignment 6.A
•
Student Pages for Lesson 6, Part A
•
Practice Assignment 6.A
•
Resource Mean, Mode, Median
•
Four-function calculator
Prerequisite assumptions
Before beginning this lesson, students should have completed Preview Assignment 6.A.
They should be able to compute mean and median with the aid of technology.
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 6.A, Measures of central tendency
Making connections
This lesson:
•
Connects back to Resource Mean, Mode, Median, which introduces notation for
the arithmetic mean.
•
Connects forward to making decisions based on data and lays the foundation for
descriptive statistics.
Background context
Resource Mean, Mode, Median explains measures of center, defines mean, median,
and mode, and provides examples on how to compute mean, median, and mode. In
Preview Assignment 6.A, students compute mean, median, and mode for small data
sets.
Suggested instructional plan
Frame the lesson
(3 minutes)
Student
Material
•
Display or distribute the Student Pages for this lesson. Show the
histogram for questions 1–6.
•
Have students answer question 1 individually.
•
Quickly check that students count 20 individuals in the sample.
•
Elicit descriptions of the distribution. The histogram is “moundshaped” and approximately symmetric.
•
Transition to the lesson activities by briefly discussing the
Objectives for the lesson.
Lesson activities
(18 minutes)
Questions 2–6
Group Work
Classroom
Culture
•
You may want to remind students that frequency is a count.
•
Illustrate how a line of symmetry helps identify the median bin in
question 2 by asking:
o “How could you draw a vertical line to help you see two
different halves of the data?”
o “Using the line, where do you think the median is?”
•
Share successful student strategies for locating the median bin.
•
Explain that statisticians call the mean, median, and mode
measures of central tendency because these numerical
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 6.A, Measures of central tendency
representations for a data set tend to describe the center of a data
set; that is, they frequently, but not always, describe the center of
a data set.
•
As students begin answering question 3, encourage them to
“stack” credit card balances inside the bars of the histogram to
represent their data set visually.
•
Quickly check that students are computing the mean of their data
set correctly in question 5.
•
In question 6, accept student judgment regarding “significantly
less” versus “roughly the same” and “significantly greater,” but tell
students that measures of central tendency in the same bin are
generally regarded as “roughly the same.” Students may want to
revisit their judgment after completing questions 7–9.
Questions 7–9
•
Students may have difficulty accepting that the median could be
$0, which may open an opportunity to discuss the mode. Small
data sets with highly specific data values (like credit card
balances) are more likely to have a modal bin rather than a
specific mode.
•
Initial attempts by students to record a data set with a mean of
$2,600 are unlikely to succeed. Encourage students to use
information learned from their first attempt to help with subsequent
attempts. If time runs short, accept data sets with means close to
$2,600.
•
The punchline of the joke in question 9 relies both on the mean’s
weakness as a summary statistic when describing a skewed
distribution and on the fact that the mean describes a data set, not
individual data points. Student explanations may concentrate on
these ideas without using statistical jargon.
•
The joke in question 9 also relies on a preconceived notion that
debt is bad, but debt often goes hand-in-hand with wealth. The
Spenders may not need to worry about their credit card balance if
they are wealthy and paying the debt off is only a matter of
transferring funds. This realization may connect to future lessons
involving interest rates.
Wrap-up/transition
(4 minutes)
Wrap-up
•
Have students refer back to the Objectives for the lesson and
check the ones they recognize from the activity. Alternatively,
they may check objectives throughout the lesson.
Transition
•
The next lesson is a student success lesson. After that, students
will continue their study of numerical descriptions of a data set.
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 6.A, Measures of central tendency
•
“We have looked at several ways to give a ‘picture’ of data in a
graphical display. Next, we will make decisions using numerical
descriptions of data.”
•
“But first, we’ll talk about the power of your brain.”
Suggested assessment, assignments, and reflections
•
Give Practice Assignment 6.A.
•
Give the Preview Assignments, if any, for the lesson activities you plan to
complete in the next class meeting.
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 6.A, Measures of central tendency
Lesson 6, Part A, Measures of
central tendency
Theme: Personal Finance
ANSWERS
Imagine that a sample of individuals
report the balance on their credit card.
The histogram summarizes this report.
1)
How many individuals are included
in the sample?
Answer: 20
Objectives for the lesson
You will understand that:
o Mean and median provide different pictures of data.
o Conclusions derived from statistical summaries are subject to misinterpretation.
o Histograms are graphical displays useful for showing the shape of a distribution.
You will be able to:
o Compare mean and median from the shape of a distribution.
o Create a data set that meets certain criteria for measures of central tendency.
2)
Which bin contains the median credit card balance?
Answer: The bin ranging from $7,500 to $9,999.99 contains the median.
3)
Create a data set that matches the histogram. In other words, record a set of dollar
amounts including two values ranging from 0 to $2,499.99, two values ranging
from $2,500 to $4,999.99, and so on.
Answers will vary.
Sample answer: See histogram on the next page.
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 6.A, Measures of central tendency
4)
Compute the mean and median of your data set.
Answers will vary.
Sample answer: The mean for the data set above equals $8,100 and the median
equals $9,000.
5)
Mark the mean and the median of your data set on the horizontal axis of the
histogram.
Answers will vary.
Sample answer: See histogram below.
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 6.A, Measures of central tendency
6)
Which phrase best describes the median of your data set?
a)
significantly less than the mean
b)
roughly the same as the mean
c)
significantly greater than the mean
Answers will vary, but b, “roughly the same as the mean,” is the best description
when the mean and median are in the same bin.
Imagine another sample of individual credit card balances summarized by the histogram
below.
7)
The median for this sample is $0. Interpret this statement. Explain what it means
for the median to equal zero dollars.
Answer: Half of the balances are zero and the other half of the balances are
positive values or zero.
8)
The mean for the sample is $2,600. Create a data set with a median of $0 and a
mean of $2,600 that matches the histogram. In other words, write down a set of
dollar amounts including fourteen values ranging from 0 to $2,499.99, two values
ranging from $2,500 to $4,999.99, and one value in each of the remaining bins.
Make sure that your twenty values have a median of $0 and a mean of $2,600.
Answers will vary.
Sample answer:
{0,0,0,0,0,0,0,0,0,0,0,0,2000,2000,4000,4000,6000,8000,12000,14000}
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Foundations of Mathematical Reasoning
Suggested Instructor Notes 6.A, Measures of central tendency
9)
For convenience, name the four individuals in your data set who have credit card
balances in the top four bins as the Spenders and the other individuals as the
Savers. Here is a lame joke:
The Savers are all at a party having a good time. When the Spenders
arrive, the Savers suddenly get depressed. The Spenders say, “Hey, why
is everyone suddenly so glum?” The Savers reply, “Don’t you know what
you four guys do to our average credit card debt?”
Explain the punchline of the joke. Why are the Savers depressed when the
Spenders arrive?
Answer: The Savers are depressed because the high credit card balances of the
Spenders skew the average credit card balance upward for everyone at the party.
This is funny because the average for everyone at the party actually has no effect
on any single individual’s debt so there is no reason to be depressed.
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