9.3
Measures of Center
In what other ways can you describe an
average of a data set?
1
ACTIVITY: Finding a Median
Work with a partner.
a. Write the total number of letters in the first and
last names of 19 celebrities, historical figures, or
people you know. Organize your data in a table.
One person is already listed for you.
b. Order the values in your data set from least to
greatest. Then write the data on a strip of grid
paper with 19 boxes.
c. Place a finger on the square at each end of the strip. Move your fingers
toward the center of the ordered data set until your fingers touch.
On what value do your fingers touch?
Statistics
In this lesson, you will
● understand the concept
of measures of center.
● find the median and mode
of data sets.
d. Now take your strip of grid paper and fold it in half. On what number
is the crease? What do you notice? This value is called the median.
How would you describe to another student what the median of a
data set represents?
e. How many values are greater than the median? How many are less
than the median?
f.
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Why do you think the median is considered an average of a data set?
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2
ACTIVITY: Adding a Value to a Data Set
Work with a partner.
a. How many total letters are in your firstt
name and last name? Add this value to
o
the ordered data set in Activity 1. How
w
many values are now in your data set?
b. Write the ordered data, including
your new value from part (a), on a
strip of grid paper.
c. Repeat parts (c) and (d) from Activity 1. Explain your findings.
How do you think you can find the median of this data set?
d. Compare the medians in Activities 1 and 2. Then answer the
following questions. Explain your reasoning.
3
●
Do you think the median always has to be a value in the data set?
●
Do you think the median always has to be a whole number?
ACTIVITY: Finding a Mode
Work with a partner.
Math
Practice
Use a Graph
How can you use
the dot plot to find
the mode?
a. Make a dot plot for the data set in Activity 2. Describe the distribution of
the data.
b. Which value occurs most often in the data set? This value is called the mode.
c. Do you think a data set can have no mode or more than one mode? Explain.
d. Do you think the mode always has to be a value in the data set? Explain.
e. Why do you think the mode is considered an average of a data set?
4. IN YOUR OWN WORDS In what other ways can you describe an average of a
data set?
5. Find the mean of your data set in Activity 2. Then compare the mean,
median, and mode. Is there one measure that you think best represents
your data set? Explain your reasoning.
Use what you learned about the median of a data set to complete
Exercises 5 and 6 on page 407.
Section 9.3
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Measures of Center
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Lesson
9.3
Lesson Tutorials
A measure of center is a measure that describes the typical value
of a data set. The mean is one type of measure of center. Here are
two others.
Key Vocabulary
measure of center,
p. 404
median, p. 404
mode, p. 404
Median
Words
Order the data. For a set with an odd number of values, the
median is the middle value. For a set with an even number of
values, the median is the mean of the two middle values.
Numbers
Data: 5, 8, 9, 12, 14
The median is 9.
Data: 2, 3, 5, 7, 10, 11
5+7
2
The median is —, or 6.
Study Tip
The mode is the only
measure of center that
you can use to describe
a set of data that is not
made up of numbers.
Mode
Words
The mode of a data set is the value or values that occur most
often. Data can have one mode, more than one mode, or no
mode. When all values occur only once, there is no mode.
Numbers
Data: 11, 13, 15, 15, 18, 21, 24, 24
The modes are 15 and 24.
EXAMPLE
1
Find the median and mode of the bowling scores.
Bowling Scores
120 135 160 125
Finding the Median and Mode
90
90, 105, 120, 125, 135, 145, 160, 160, 175, 205
Order the data.
205 160 175 105 145
135 + 145
2
280
2
Median:
M
d
— = —, or 140
Add the two middle values and divide by 2.
Mode: 90, 105, 120, 125, 135, 145, 160, 160, 175, 205
The value 160 occurs most often.
The median is 140. The mode is 160.
Find the median and mode of the data.
Exercises 7–12
404
Chapter 9
ms_green pe_0903.indd 404
1. 20, 4, 17, 8, 12, 9, 5, 20, 13
2. 100, 75, 90, 80, 110, 102
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EXAMPLE
2
Favorite Types of Movies
Finding the Mode
The list shows the favorite types of movies for students in a class.
Organize the data in a frequency table. Then find the mode.
Comedy
Drama
Horror
Horror
Drama
Horror
Comedy
Comedy
Action
Action
Action
Comedy
Action
Comedy
Horror
Drama
Comedy
Drama
Comedy
Comedy
Horror
Horror
Horror
Comedy
Action
Horror
Action
Drama
Type
Tally
̇̇̇̇ ̇
∣∣∣∣
∣̇
∣∣̇∣∣∣ ∣̇∣∣
∣∣∣∣
∣∣̇∣∣∣ ∣̇∣
Frequency
The number of tally
marks is the frequency.
5
̇̇̇̇
̇̇̇̇ ̇̇̇
̇̇̇̇ ̇̇̇
̇̇̇̇
̇̇̇̇
̇̇̇̇ ̇̇
̇̇̇̇ ̇̇
8
4
7
Make a tally for each vote.
Comedy received the most votes.
So, the mode is comedy.
3. One member of the class was absent and ends up voting
for horror. Does this change the mode? Explain.
Exercises 14–15
EXAMPLE
3
Choosing the Best Measure of Center
Find the mean, median, and mode of the sneaker prices. Which
measure best represents the data?
$20
$31
20 + 31 + 122 + 48 + 37 + 20 + 45 + 65
8
388
8
Mean: ———— = —, or 48.5
$122
$48
Median: 20, 20, 31, 37, 45, 48, 65, 122
37 + 45
2
Order from least to greatest.
82
2
— = —, or 41
$37
$20
Mode: 20, 20, 31, 37, 45, 48, 65, 122
The value 20 occurs most often.
$45
$65
20
30
40
Mode: 20 Median: 41
50
60
80
70
90
100
110
120
130
Price
(dollars)
Mean: 48.5
The median best represents the data. The mode is less than most of
the data, and the mean is greater than most of the data.
Exercises 17–20
Find the mean, median, and mode of the data. Choose the measure
that best represents the data. Explain your reasoning.
4. 1, 93, 46, 48, 34, 194, 67, 55
5. 96, 150, 102, 87, 150, 75
Section 9.3
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EXAMPLE
4
Removing an Outlier
Identify the outlier in Example 3. Find the mean, median, and mode
without the outlier. Which measure does the outlier affect the most?
The price of $122 is much greater than any other price. So, it is
the outlier.
With Outlier (Example 3)
Mean
Median
Mode
48.5
41
20
38
37
20
Without Outlier
The mean is affected the most by the outlier.
6. The times (in minutes) it takes six students to travel to school
are 8, 10, 10, 15, 20, and 45. Identify the outlier. Find the mean,
median, and mode with and without the outlier. Which measure
does the outlier affect the most?
Exercises 21–22
EXAMPLE
5
Changing the Values of a Data Set
The prices of six video games at an online store
are shown in the table. The price of each game
increases by $4.98 when a shipping charge is
included. How does this increase affect the mean,
median, and mode?
Video Game
Prices with
Shipping Charge
$58.40
$40.67
$23.97
$30.11
$32.95
$58.40
Video Game
Prices
$53.42
$35.69
$18.99
$25.13
$27.97
Make a new table by adding $4.98 to each price.
Then find the mean, median, and mode of both data sets.
$53.42
Mean
Median
Mode
Original Price
35.77
31.83
53.42
Price with
Shipping Charge
40.75
36.81
58.4
Compare:
Mean: 40.75 − 35.77 = 4.98
Median: 36.81 − 31.83 = 4.98
Mode: 58.4 − 53.42 = 4.98
By increasing each video game price by $4.98 for shipping, the
mean, median, and mode all increase by $4.98.
7. WHAT IF? The store decreases the price of each video game by
$3. How does this decrease affect the mean, median, and mode?
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Exercises
9.3
Help with Homework
1. NUMBER SENSE Give an example of a data set that has no mode.
2. WRITING Which is affected most by an outlier: the mean, median, or mode? Explain.
3. WHICH ONE DOESN’T BELONG Which word does not belong with the other three?
Explain.
median
outlier
mode
mean
4. NUMBER SENSE A data set has a mean of 7, a median of 5, and a mode of 8.
Which of the numbers 7, 5, and 8 must be in the data set? Explain.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Use grid paper to find the median of the data.
5. 9, 7, 2, 4, 3, 5, 9, 6, 8, 0, 3, 8
6. 16, 24, 13, 36, 22, 26, 22, 28, 25
Find the median and mode(s) of the data.
1
7. 3, 5, 7, 9, 11, 3, 8
8. 14, 19, 16, 13, 16, 14
9. 93, 81, 94, 71, 89, 92, 94, 99
11. 12, 33, 18, 28, 29, 12, 17, 4, 2
10. 44, 13, 36, 52, 19, 27, 33
12. 55, 44, 40, 55, 48, 44, 58, 67
13. ERROR ANALYSIS Describe and correct the error
in finding the median of the data.
✗
The median is 58.
63, 55, 49, 58, 50, 59, 51
Find the mode(s) of the data.
2 14.
15.
Shirt Color
Talent Show Acts
Black
Blue
Red
Singing
Dancing
Comedy
Pink
Black
Black
Singing
Singing
Dancing
Gray
Green
Blue
Juggling
Dancing
Singing
Blue
Blue
Red
Singing
Poetry
Dancing
Yellow
Blue
Blue
Comedy
Magic
Dancing
Black
Orange
Black
Poetry
Singing
Singing
Black
16. REASONING In Exercises 14 and 15, can you find the mean and median
of the data? Explain.
Section 9.3
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Measures of Center
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Find the mean, median, and mode(s) of the data. Choose the measure that best
represents the data. Explain your reasoning.
3 17. 48, 12, 11, 45, 48, 48, 43, 32
18. 12, 13, 40, 95, 88, 7, 95
19. 2, 8, 10, 12, 56, 9, 5, 2, 4
20. 126, 62, 144, 81, 144, 103
Find the mean, median, and mode(s) of the data with and without the outlier.
Describe the effect of the outlier on the measures of center.
4 21. 45, 52, 17, 63, 57, 42, 54, 58
22. 85, 77, 211, 88, 91, 84, 85
Find the mean, median, and mode(s) of the data.
1
2
5
8
1
8
3
4
5
8
1
4
5
8
1
2
24. 8 —, 6 —, 3 —, 5 —, 6 —, 5 —, 10 —, 4 —
23. 4.7, 8.51, 6.5, 7.42, 9.64, 7.2, 9.3
25. WEATHER The weather forecast for a week is shown.
Sun
Mon
Tue
High
90º F
91º F
89º F
Low
74º F
78º F
77º F
Thu
Fri
Sat
97º F
101º F
99º F
91º F
77º F
83º F
78º F
72º F
Wed
a. Find the mean, median, and mode(s) of the high temperatures. Which
measure best represents the data? Explain your reasoning.
b. Repeat part (a) for the low temperatures.
26. RESEARCH Find the unit costs of 10 different kinds of cereal. Choose one
cereal whose unit cost will be an outlier.
a. Find the mean, median, and mode(s) of the data. Which measure best
represents the data? Explain your reasoning.
b. Identify the outlier in the data set. Find the mean, median, and mode(s)
of the data set without the outlier. Which measure does the outlier affect
the most?
Volunteering at an Animal Shelter
18
16
15
14
Hours
12
11
10
10
8
6
9
7
6
4
2
0
1
2
4
3
5
Week
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6
27. PROBLEM SOLVING The bar graph shows
the numbers of hours you volunteered at
an animal shelter. What is the minimum
number of hours you need to work in the
seventh week to justify that you worked an
average of 10 hours for the 7 weeks? Explain
your answer using measures of center.
28. REASONING Why do you think the mode
is the least frequently used measure to
describe a data set? Explain.
Statistical Measures
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29. MOTOCROSS The ages of the racers in a bicycle motocross race
are 14, 22, 20, 25, 26, 17, 21, 30, 27, 25, 14, and 29. The 30-year-old
drops out of the race and is replaced with a 15-year-old. How are
the mean, median, and mode of the ages affected?
30. CAMERAS The data are
the prices of several
digital cameras at a store.
$130 $170 $230 $130
$250 $275 $130 $185
a. Does the price shown in the advertisement represent the prices well?
Explain.
b. Why might the store use this advertisement?
c. In this situation, why might a person want to know the mean?
the median? the mode? Explain.
31. SALARIES The table shows the
monthly salaries for employees at a
company.
Monthly Salaries (dollars)
1940
1660
1860
2100
1720
1540
1760
1940
1820
1600
a. Find the mean, median, and mode
of the data.
b. Each employee receives a 5% raise. Find the mean, median, and mode of
the data with the raise. How does this increase affect the mean, median,
and mode of the data?
c. Use the original monthly salaries to calculate the annual salaries. Find the
mean, median, and mode of the annual salaries. How are these values related
to the mean, median, and mode of the monthly salaries?
32.
Consider the algebraic expressions 3x, 9x, 4x, 23x, 6x, and 3x.
Assume x > 0.
a. Find the mean, median, and mode.
b. Is there an outlier? If so, what is it?
Find the value of the expression. (Section 1.1)
33. 48 − 35
34. 188 − 123
35. 416 − 297
36. 6249 − 3374
37. MULTIPLE CHOICE A shelf in your room can hold at most 30 pounds.
There are 12 pounds of books already on it. Which inequality represents
the number of pounds you can add to the shelf? (Section 7.6)
A x < 18
○
B x ≥ 18
○
C x ≤ 42
○
Section 9.3
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D x ≤ 18
○
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