MAIN IDEADescribe a set of data using mean, median, mode, and range.

New Vocabularymeasures of central

tendencymeanmedianmode

Math Online

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Key ConceptMean

Words The mean of a set of data is the sum of the data divided by the number of items in the data set. The mean is also referred to as average.

Examples data set: 1 cm, 1 cm, 5 cm, 2 cm, 2 cm, 4 cm, 2 cm, 5 cm

mean: 1 + 1 + 5 + 2 + 2 + 4 + 2 + 5 ___

8 or 2.75 cm

8-2

The number of pennies in each cup represents Jack’s scores for

five math quizzes.

8 7 9 6 10

Move the pennies among the cups so that each cup has the same

number of pennies.

1. What was Jack’s average score for the five quizzes? 8 points

2. If Jack scores 14 points on the next quiz, how many pennies would

be in each cup? 9

A number used to describe the center of a set of data is a measure of

central tendency. The most common of these measures is the mean.

Find the Mean

1 QUIZ SCORES The table shows the quiz scores Quiz Scores

47 40 43 45

49 41 49 44

43 41 44 49

44 50 41 44

for 16 students. Find the mean.

mean = 47 + 40 + ... + 44 __

16

sum of data number of data items

= 714 _

16 or 44.625

a. MONEY Adam earned $14, $10, $12, $15, and $13 by doing chores

around the house. What is the mean amount Adam earned doing

these chores? $12.80

Measures of Central Tendency and Range

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Key ConceptMedian

Words In a data set that has been ordered from least to greatest, themedian is the middle number if there is an odd number of data items. If there is an even number of data items, the median is the mean of the two numbers closest to the middle.

Example data set: 7 yd, 11 yd, 15 yd, 17 yd, 20 yd, 20 yd

median: 15 + 17 _

2 or 16 yd The median divides

the data in half.

ModeWords The mode of a set of data is the number that occurs most

often. If there are two or more numbers that occur most often, all of them are modes.

Example data set: 50 mi, 45 mi, 45 mi, 52 mi, 49 mi, 56 mi, 56 mi

modes: 45 mi and 56 mi

Vocabulary LinkMedian

Everyday Use the middle paved or planted section of a highway, as in median strip.

Math Use the middle number of the ordered data.

Two other common measures of central tendency are median and mode.

Find the Mean, Median, and Mode

2 MOVIE RENTALS The number

of DVDs rented during one

week at Star Struck Movie

Rental is shown in the table.

What are the mean, median,

and mode of the data?

mean: 55 + 34 + 35 + 34 + 57 + 78 + 106

___ 7 = 399

_ 7 or 57

median: 34, 34, 35, 55, 57, 78, 106 First, write the data in order.

median

mode: 34 It is the only value that occurs more than once.

The mean is 57 DVDs, the median is 55 DVDs, and the mode is

34 DVDs.

b. BICYCLES The sizes of the bicycles

owned by the students in Ms. Garcia’s

class are listed in the table. What are the

mean, median, and mode of the data?

c. FOOTBALL The points scored in each game by Darby Middle

School’s football team for 9 games are 21, 35, 14, 17, 28, 14, 7, 21,

and 14. Find the mean, median, and mode. 19, 17, 14

24.25 in., 24 in., 24 in.24.25 in., 24 in., 24 in.

Star Struck Movie RentalDaily DVD Rentals

S M T W TH F S

55 34 35 34 57 78 106

Students’ Bicycle Sizes (in.)

20 24 20 26

24 24 24 26

24 29 26 24

Lesson 8-2 Measures of Central Tendency and Range 403

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Comparing Measures Another way to solve Example 3 is to find the measures before 98 is added to the data set. Then find the measures after 98 is added to the data set. Then compare.

3 The maximum length in feet of several whales is listed below.

46, 53, 33, 53, 79

If the maximum length of the Blue Whale, 98 feet, is added to this

list, which of the following statements would be true?

A The mode would decrease. C The mean would increase.

B The median would decrease. D The mean would decrease.

Read the ItemYou are asked to identify which statement would be true if the data

value 98 was added to the data set.

Solve the ItemUse number sense to eliminate possibilities.

The mode, 53, will remain unchanged since the new data value

occurs only once. So, eliminate answer choice A.

Since the new data value is greater than each value in the data set, the

median will not decrease. So, eliminate answer choice B.

The remaining two answer choices refer to the mean. Since 98 is

greater than each value in the data set, the mean will increase, not

decrease. So, the answer is C.

d. If the maximum length of the Orca Whale, 30 feet, is added to the

list in Example 3, which of the following statements would be true?

F The mode would decrease. H The mean would increase.

G The median would increase. J The mean would decrease.

In addition to the mean, median, and mode, you can also use the range to

describe a set of data. Below are some guidelines for using these measures.

Concept SummaryMean, Median, Mode, and Range

Measure Most Useful When...

Mean · data set has no outliers

Median · data set has outliers· there are no big gaps in the middle of the data

Mode · data set has many identical numbers

Range · describing the spread of the data

JJ

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Choose Mean, Median, Mode, or Range

4 PLANTS The line plot

shows the height of

desert cacti. Would the

mean, median, mode,

or range best represent

the heights?

mean: 1 + 2 + 2 + ... + 30

__ 14

or 8.8

median: 7th term + 8th term

__ 2 =

5 + 5 _

2 or 5

mode: 5

range: 30 - 1 or 29

The mean of 8.8 misrepresents the score. The median or mode

represents the height of the cacti well.

e. GAMES The table shows the cost of

various board games. Would the

mean, median, mode, or range best

represent the costs? Explain.

Examples 1, 2(pp. 402–403)

Find the mean, median, and mode for each set of data. Round to the nearest

tenth if necessary.

1. Miles traveled on the weekend: 29, 14, 80, 59, 78, 30, 59, 69, 55, 50

2. Team Number of Wins

Eagles 10

Hawks 8

Zipps 9

Falcons 11

3.

Example 3(p. 404)

4. MULTIPLE CHOICE During the week, the daily low temperatures were 52°F,

45°F, 51°F, 45°F, and 48°F. If Saturday’s low temperature of 51°F is added,

which statement about the data set would be true? C

A The mean would decrease. C The mode would increase.

B The median would decrease. D The mode would decrease.

Example 4(p. 405)

5. SHOES The line plot shows

80655045 10060 95908570 7555

× ×

×

×

×

×

×

×

×

×

×

×

×

×

× ××

×

×

×

×

×

×

×

×the price of athletic shoes.

Which measure best describes

the data: mean, median, mode,

or range? Explain. See margin.

e. The mean is $21.50,

the median is $18.50,

the mode is $40, and

the range is $28. The

mean or median could

be used to represent

the data.

e. The mean is $21.50,

the median is $18.50,

the mode is $40, and

the range is $28. The

mean or median could

be used to represent

the data.

★ indicates multi-step problem★ indicates multi-step problem

1. 52.3; 57; 591. 52.3; 57; 59

2. 9.5; 9.5; none

3. 26.5, 25.5, 23

and 25

2. 9.5; 9.5; none

3. 26.5, 25.5, 23

and 25

★★

Median Median When there is an odd number of data, the median is the middle number of the ordered set. When there is an even number of data, the median is the mean of the two middle numbers. 14820 18 22 26 306 281610 124 20 24

× ×

×

×

×

××

×

×

×

×

×

× ×

29262322 3225 313027 2824

×

×

× ×

×

×

×

×

×

×

× ×

×

×

×

×

Board Game Costs ($)

12 15 40 22

14 40 15 17

20 18 40 19

16 21 19 16

Lesson 8-2 Measures of Central Tendency and Range 405

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Real-World LinkThe International Space Station measures 356 feet by 290 feet, and contains almost an acre of solar panels.Source: The World Almanac

Find the mean, median, and mode for each set of data. Round to the nearest

tenth if necessary.

6. Number of dogs groomed each week: 65, 56, 57, 75, 76, 66, 64 65.6; 65; none

7. Daily number of boats in a harbor: 93, 84, 80, 91, 94, 90, 78, 93, 80

8. Scores earned on a math test: 95, 90, 92, 94, 91, 90, 98, 88, 89, 100

For Exercises

6–1130–3212–13

See Examples

1, 234

HOMEWORK HELPHELP

9. Prices of books: $10, $18, $11, $6, $6, $5, $10, $11, $46, $7, $6, $8 $12; $9; $6

10. Cost Number of Coats

$75 8

$80 3

$85 6

11.

12. MUSIC The line plot shows the number of weeks that songs have been on

the Top 20 Country Songs list. Would the mean, median, mode, or range

best represent the data? Explain. See margin.

6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

Country Songs Number of Weeks in Top 20

× ×× × ××× × × × ××

× × ×

× × × × ×

13. SPACE Twenty-seven countries have sent people into space. The table

shows the number of individuals from each country. Which measure best

describes the data: mean, median, mode, or range? Explain. See margin.

People in Space

267 1 9 8 1 1 1 1 1

97 1 1 1 3 1 1 2 1

11 2 1 1 5 1 1 1 1

Source: The World Almanac

Find the mean, median, and mode for each set of data. Round to the nearest

tenth if necessary.

14. Weight in ounces of various insects: 6.1, 5.2, 7.2, 7.2, 3.6, 9.0, 6.5, 7.4, 5.4

15. Prices of magazines: $3.50, $3.75, $3.50, $4.00, $3.00, $3.50, $3.25

16. Daily low temperatures: -2°F, -8°F, -2°F, 0°F, -1°F, 1°F, -2°F, -1°F

REASONING Determine whether each statement is always, sometimes, or never

true about the data set {8, 12, 15, 23}. Explain your reasoning.

17. If a value greater than 23 is added, the mean will increase.

18. If a value less than or equal to 8 is added, the mean will decrease.

19. If a value between 8 and 23 is added, the mean will remain unchanged.

Exercise Levels

A: 6–13

B: 14–24

C: 25–29

Exercise Levels

A: 6–13

B: 14–24

C: 25–29

11. 46, 45, 4411. 46, 45, 44

★★

★★

14. 6.4, 6.5, 7.2 15. $3.50, $3.50, $3.5014. 6.4, 6.5, 7.2 15. $3.50, $3.50, $3.50

-1.9°F, -1.5°F, -2°F-1.9°F, -1.5°F, -2°F

17–19.

See margin.

17–19.

See margin.

50443836 5642 545246 4840

× × ×

××

×

× × × × × ×× ××

×

7. 87; 90; 80 and 93 8. 92.7; 91.5; 90

$79.40; $80; $75

406 Chapter 8 Statistics: Analyzing Data

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See pages 687, 711.EXTRA PRACTICEPRACTICE

DINOSAURS For Exercises 20–22, use the lengths of the dinosaurs

shown below.

Tyrannosaurus RexLength: 480 inches

CorythosaurusLength: 396 inches

ParasaurolophusLength: 480 inches

20. What is the mean length of the dinosaurs? 452 inches

21. One of the largest dinosaurs ever is the Brachiosaurus. Its length was

960 inches. If this data value is added to the lengths of the dinosaurs

above, how will it affect the mean? Explain your reasoning.

22. Which measure best describes the data if the length of the Brachiosaurus is

included: mean, median, mode, or range? If the length of the Brachiosaurus

is not included? Explain any similarities or differences.

23. SPORTS The table shows the points scored Hawks Lacrosse Team Points Scored

11 15 12 10 10 10 13

14 13 13 10 15 12

by a lacrosse team so far this season. The

team will play 14 games this season. How

many points need to be scored during the

last game so that the average number of

points scored this season is 12? Explain. 10 points; See margin for explanation.

24. FIND THE DATA Refer to the Data File on pages 16–19. Choose some data and

then describe it using the mean, median, mode, and range. See students’ work.

H.O.T. Problems 25. OPEN ENDED Give an example of a set of data in which the mean is not the

best representation of the data set. Explain why not.

26. Which One Doesn’t Belong? Identify the term that does not have the same

characteristic as the other three. Explain your reasoning.

mean median range mode

27. REASONING Determine whether the median is sometimes, always, or never

part of the data set. Explain your reasoning. See margin.

28. CHALLENGE Without calculating, would the mean, median, or mode be most

affected by eliminating 1,000 from the data shown? Which would be the

least affected? Explain your reasoning. See margin.

50, 100, 75, 60, 75, 1,000, 90, 100

29. MATHWRITING IN According to the U.S. Census Bureau, the typical

number of family members per household is 2.59. State whether this

measure is a mean or mode. Explain how you know.

21. Sample answer:

A length of 960

inches is much

greater than the

other pieces of

data. So, if this

piece of data is

added, the mean

will increase.

21. Sample answer:

A length of 960

inches is much

greater than the

other pieces of

data. So, if this

piece of data is

added, the mean

will increase.

22. Median or

mode; Mean,

median, or mode;

Sample answer:

The length of the

Brachiosaurus

is an outlier, so

the mean

misrepresents the

data if this value

is included.

22. Median or

mode; Mean,

median, or mode;

Sample answer:

The length of the

Brachiosaurus

is an outlier, so

the mean

misrepresents the

data if this value

is included.

25. Sample answer:

4, 5, 2, 2, 3, 3, 1, 0,

1, 2, 4, 68, 5; the

mean, 7.7, is

not the best

representation

since it is greater

than all the data

items except one.

25. Sample answer:

4, 5, 2, 2, 3, 3, 1, 0,

1, 2, 4, 68, 5; the

mean, 7.7, is

not the best

representation

since it is greater

than all the data

items except one.

26. range; mean,

median, and mode

are used to

describe the center

of data. Range

describes the

spread of the data.

26. range; mean,

median, and mode

are used to

describe the center

of data. Range

describes the

spread of the data.

See margin. See margin.

Lesson 8-2 Measures of Central Tendency and Range 407

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30. The table below shows the number of

soup labels collected in one week by

each homeroom in grade 7.

Classroom Number of Soup Labels

Mr. Martin 138

Ms. Davis 125

Mr. Cardona 89

Mrs. Turner 110

Mr. Wilhelm 130

Mrs. LaBash ?

Which number could be added to the

set of data in order for the mode and

median of the set to be equal? C

A 89 C 125

B 110 D 130

31. An antique dealer purchased

5 antiques for a total of $850.00.

He later bought another antique

for $758.00. What is the mean cost

of all the antiques? H

F $151.60 H $268.00

G $170.00 J $321.60

32. Gina found the mean and median of

the following list of numbers.

5, 7, 7

If the number 11 was added to this list,

which of the following statements

would be true? A

A The mean would increase.

B The mean would decrease.

C The median would increase.

D The median would decrease.

33. TEMPERATURE The table shows record high temperatures for July Temperatures

99 98 96 98 98

97 100 103 103 103

100 95 100 103 105

Kentucky in July. Make a line plot of the data. (Lesson 8-1)

Find the simple interest earned to the nearest cent for each

principal, interest rate, and time. (Lesson 7-8)

34. $1,250, 3.5%, 2 years $87.50

35. $569, 5.5%, 4 months $10.43

36. FOOD The United States produced almost 11 Uses of Apples in the United States

FreshFruit59%

Juice andCider23%

CannedFruit11%

Other7%

Source: U.S. Apple Association

billion pounds of apples in a recent year. Use

the information in the graph to find how many

pounds of apples were used to make juice and

cider. (Lesson 7-1) about 2.53 billion

37. Name the property shown by the statement

4 × 6 = 6 × 4. (Lesson 1-8) Commutative (×)

PREREQUISITE SKILL Name the place value of the underlined digit. (p. 669)

38. 581 hundreds 39. 6,295 ones 40. 4,369 thousands 41. 2.84 tenths

See margin.See margin.

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