practice lesson 27 measures of center and variability b...practice lesson 27 measures of center and...

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Practice Lesson 27 Measures of Center and Variability

Unit 5

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©Curriculum Associates, LLC Copying is not permitted. 305Lesson 27 Measures of Center and Variability

Name: Measures of Center and Variability

Lesson 27

Vocabularydistribution shows

how spread out or how

clustered data values are.

Prerequisite: Distribution of Data

Study the example showing how to interpret the distribution of data in a line plot. Then solve problems 1–8.

1 The longest insect is how many times as long as the shortest insect? Write an equation to show your solution.

2 Do any insects have a length that is very diff erent from the rest of the insects? If so, what is the length?

3 How would the line plot change if the insect lengths

were clustered between 1 ·· 8 and 1 ·· 4 inch?

Example

There are several rare insects on display at the insect

exhibit at a science museum. The line plot shows the

lengths of the insects to the nearest 1 ·· 8 inch. What is the

distribution of the lengths of the insects?

18

14

38

12

58

34

78

Insect Length (in.)

Insect Exhibit

0 1

X XX

XX

XXXX

XXXX

XXXXX

Most of the data points are between 3 ·· 4 and 1, so the insect

lengths are clustered between 3 ·· 4 and 1 inch.

305305

8 times as long; 1 5 1 ·· 8 3 8

Most of the Xs would be on one or the other of

those two data points.

Yes; 1 ·· 8 inch

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©Curriculum Associates, LLC Copying is not permitted.306 Lesson 27 Measures of Center and Variability

Solve.

Use the line plot for problems 4–8.

The line plot shows the amount of time it took some

runners to finish a charity race to the nearest 1 ·· 8 hour.

4 Tell whether each statement about the data in the line plot is True or False.

a. None of the runners finished

in exactly 2 1 ·· 2 hours. u True u False

b. A time of 2 hours is uncommon. u True u False

c. The most common time

is 2 3 ·· 4 hours. u True u False

d. Most runners finished

in around 2 7 ·· 8 hours. u True u False

5 By how much do the times vary? Explain.

6 Between which two times did most of the runners fi nish?

7 Are any of the times very diff erent from most of the times? If so, which time?

8 Suppose the times for 10 additional runners were recorded, including a new slowest time and a new fastest time. What would be a set of reasonable times based on the data in the line plot? Explain.

Time (hours)

Fund-Raiser Run

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

XX

2 3182

142

382

122

582

342

782

306

3

3

3

3

1 hour; The greatest time is 3 hours and the least time is 2 hours: 3 2 2 5 1.

Possible data: 1 1 ·· 2 hours: 1 runner, 2 5 ·· 8 hours: 2 runners, 2 3 ·· 4 hours: 6 runners, 4 hours:

1 runner; The slowest time had to be more than 3 hours and the fastest time less than

2 hours. The average time was around 2 3 ·· 4 hours, so I gave most of the runners

that time.

2 5 ·· 8 and 3

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Yes; 2 hours

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Name: Lesson 27

Mean

Study the example showing how to describe the center of a data set using mean. Then solve problems 1–8.

1 What is the mean donation? Explain how the graph on the right shows the mean.

2 Complete the equations to show another way to fi nd the mean.

1 1 1 1 1 1 1 1 1 5

4 10 5

3 Are there any outliers in the data? If so, calculate the mean without the outliers and explain how the mean is aff ected.

Example

Ten people were asked how much money they plan to donate to the new library fund. Their answers were $20, $20, $20, $40, $40, $40, $40, $40, $40, and $100. What is the mean, or average, donation?

You can draw a graph that represents how much each person plans to donate. Then, to find the mean, move the symbols to show what each person would donate if all of them donated the same amount.

321 4

Person Person

5 6 7 8 9 10

$$

$$ $ $$

$$$$$

$$

$$

$$

$$

321 4 5 6 7 8 9 10

$$

$$ $$$$

$$

$$

$$

$$

$$

$$

$ = 20 dollars donated

307

$40; Possible explanation: Each dollar symbol stands for $20, and there are two dollar

symbols above each person.

100 is an outlier because all of the other data points are 20 or 40; 20 1 20 1 20 1 40 1

40 1 40 1 40 1 40 1 40 5 300, 300 4 9 33; The mean is lower without the outlier.

20 20 20 40 40 40 40 40 40 100 400

400 40

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Solve.

Use the following situation to solve problems 4–6.

The numbers of hours that 12 students say they typically spend on homework in a week are 0, 2, 8, 8, 8, 8, 8, 8, 10, 10, 10, and 10.

4 What is the mean number of hours?

Show your work.

Solution:

5 What are the outliers in the data set? How do you know that they are outliers?

6 Explain how the outliers aff ect the mean. Then fi nd the mean without the outliers to justify your answer.

7 The mean of six values is 7. There is one outlier that pulls the mean higher than the center. What could the data set be? What is the mean without the outlier?

8 Dee’s scores for a computer game are 20, 18, 22, 12, 25, and 20. Li’s scores are 21, 20, 20, 16, 21, and 16. Who has the higher mean score? By how much?

Show your work.

Solution:

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0 1 2 1 8 1 8 1 8 1 8 1 8 1 8 1 10 1 10 1 10 1 10 5 90 90 4 12 5 7.5

The mean is 7.5 hours.

0 and 2; Possible explanation: All of the other data points are much

greater, either 8 or 10.

Possible explanation: The mean is higher without the outliers.

8 1 8 1 8 1 8 1 8 1 8 1 10 1 10 1 10 1 10 5 88

88 4 10 5 8.8

Possible answer: 6, 6, 6, 6, 6, 12; mean without outlier: 6

Dee: 20 1 18 1 22 1 12 1 25 1 20 5 117; 117 4 6 5 19.5

Li: 21 1 20 1 20 1 16 1 21 1 16 5 114; 114 4 6 5 19

Dee’s mean score is 0.5 higher than Li’s.

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Solve.

Use the following situation to solve problems 4–6.

The heights, in inches, of some small trees for sale at a garden store are 62, 62, 59, 61, 59, 70, 59, and 62.

4 What is the median height?

Show your work.

Solution:

5 What is the mode? Explain.

6 What is the mean height? Explain why the mean height is diff erent from the median.

7 Imena’s math test scores are 83, 93, 78, 89, and 83. She has one more math test this semester. What score must she get on the test to have a median score of 85?

Show your work.

Solution:

8 Does the median of a data set have to be a value of the data set? Explain.

310

59, 59, 59, 61, 62, 62, 62, 70

61 1 62 5 123; 123 4 2 5 61.5

The median height is 61.5 inches.

59 and 62; Possible explanation: Both 59 and 62 appear the same number of times

and more often than the other values in the data set.

No; Possible explanation: When there are an even number of data, the median is the

mean of the 2 middle values. This value may or may not be in the data set.

61.75 inches; Possible explanation: The outlier, 70, makes the mean higher than

the median.

Possible student work: For the median to be 85, Imena’s next score must be greater than 83 and less than 89.78, 83, 83, ?, 89, 93The two middle scores are 83 and Imena’s next score. If the average of these two scores is 85, then their sum equals 170. Subtracting the score of 83: 170 2 83 5 87.

Imena needs a score of 87 on the test for a median of 85.

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Name: Lesson 27

Median and Mode

Study the example showing how to describe the center of a data set using median and mode. Then solve problems 1–8.

1 Jason says that more than half of the values of any data set are above the median. Is he correct? Explain.

2 What would the median score be if 40 were removed from the set of scores?

3 The line plot shows the basketball scores from above. Explain how the line plot shows the mode.

Example

A team’s scores for ten basketball games are 46, 42, 43, 44, 42, 40, 42, 42, 44, and 43. Find the median and the mode of the basketball scores.

List the scores in order from least to greatest to find the median. The median is the middle number. When there are two middle numbers, the median is the mean, or average, of the two numbers.

40, 42, 42, 42, 42, 43, 43, 44, 44, 46

42 1 43 5 85 85 4 2 5 42.5

The median score is 42.5.

The number that appears most often in a data set is the mode. The mode of this data set is 42.

Scores

Basketball Games

X

X

X

X

X

X

X

XX X

40 41 42 43 44 45 46

309

43

The number with the greatest number of Xs is

the mode.

No; Possible explanation: The same number of

values are above and below the median because

the median is the middle number.

B

B

B

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Unit 5



Solve.

Use the table to solve problems 4–7.

The data values in the table represent the ages of children at a park.

4 What is the range of the data values? Write an equation to show your solution.

5 The mean of the data values is 8. Use the mean to complete the table.

6 What is the MAD?

Show your work.

Solution:

7 What does the MAD tell you about the data set?

8 Could the MAD of a data set be a negative value? Explain.

9 The MAD of a set of six data values is 10. The mean is 20. What could the data values be? Show that the mean is 20 and the MAD is 10.

Show your work.

Solution:

Age (years)

Deviation from Mean

Absolute Value of

Deviation

4 24 4

6 22

6 22

8 0

8 0

8 0

10 2

10 2

10 2

10 2

312

6; 10 2 4 5 6

The MAD is 1.6.

Possible answer: On average, the ages are 1.6 years from the mean age of 8.

No; Possible explanation: MAD is the average distance of each data point from the

mean. Distance is always a positive number because it doesn’t depend on direction.

4 1 2 1 2 1 0 1 0 1 0 1 2 1 2 1 2 1 2 5 16; 16 4 10 5 1.6

Possible answer: 10, 10, 10, 30, 30, 30

Mean: 10 1 10 1 10 1 30 1 30 1 30 ························ 6 5 20;

MAD: 10 1 10 1 10 1 10 1 10 1 10 ························ 6 5 10

2

2

0

0

0

2

2

2

2

B

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Possible data values are 10, 10, 10, 30, 30, and 30.


Name:

Variability

Study the example showing how to describe the spread of data sets. Then solve problems 1–9.

1 MAD is another way to measure spread. It is the average distance each value is from the mean. The mean of the donation amounts in the table is 15.

a. Find the absolute value of each deviation. Write the answers in the table.

b. What is the MAD? Show how you found the MAD.

2 What does the MAD tell you about this data set?

3 Compare each deviation to the mean. When is the deviation positive? When is it negative? When is it zero?

Example

Ten people donated the following amounts of money at a school fundraiser: $5, $30, $10, $10, $5, $5, $20, $30, $20, $15. What is the range of the data?

Range is the difference between the greatest and least values. Range gives you an idea of how spread out data values are. Here, the smallest donation is $5 and the largest donation is $30, so the range is $30 2 $5 5 $25.

Lesson 27

Donation Deviation

from Mean

AbsoluteValue of

Deviation

5 210

5 210

5 210

10 25

10 25

15 0

20 5

20 5

30 15

30 15

311

10

10

10

5

5

0

5

5

15

15

The deviation is positive when a value is greater

than the mean, it is negative when a value is less

than the mean, and it is 0 when a value is equal

to the mean.

On average, the data values are $8 away from

the mean donation of $15.

8; 10 1 10 1 10 1 5 1 5 1 0 1 5 1 5 1 15 1

15 5 80; 80 4 10 5 8

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Solve.

4 The line plot shows the ages of eight students on the track team.

Ages of Students

Track Team

XX

XX

XXX X

9 10 11 12 13 14 15

Tell whether each statement is True or False.

a. The modes are 9 and 12. u True u False

b. The median is 12.5. u True u False

c. The mean is 12. u True u False

d. The range is 13. u True u False

5 In a survey, 10 students and 10 teachers were asked how many hours of sleep they get at night. Here are the survey results:

Students: 6, 7, 8, 8, 9, 9, 10, 10, 10, 10

Teachers: 4, 5, 6, 6, 7, 7, 8, 8, 8, 8

What is the mean of each data set? What do the means tell you about each data set? Use the means to compare the data sets.

Show your work.

Solution:

Remember that the mean of a data set is the same as the average value.

What does each X in the line plot represent?

314

33

33

On average, the students get 2 more hours of sleep than

the teachers.

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Students’ mean: 6 1 7 1 8 1 8 1 9 1 9 1 10 1 10 1 10 1 10 ··································· 10 5 8.7

The students surveyed get an average of 8.7 hours of sleep at night.

Teachers’ mean: 4 1 5 1 6 1 6 1 7 1 7 1 8 1 8 1 8 1 8 ······························· 10 5 6.7

The teachers surveyed get an average of 6.7 hours of sleep at night.

8.7 2 6.7 5 2


Name: Lesson 27

Measures of Center and Variability

Solve the problems.

3 A set of six data values have the same mean, median, and mode. The data values are not all the same. What could the data values and the mean, median, and mode be?

2 The line plot shows the number of books that each of 14 students checked out of the library.

Which statements are true? Select all that apply.

A The data points are clustered near 5.

B The range of the data is 6.

C The graph is skewed left.

D There are no outliers in the data set.

1 The lengths, in seconds, of eight commercials are 15, 15, 15, 15, 20, 20, 30, and 30. The mean of the times is 20 seconds. What is the mean absolute deviation (MAD)? What does the MAD tell you?

Show your work.

Solution:

How do you find mean absolute deviation?

What does the distribution of the data in the line plot tell you?

How can you find the median of a set of data with an even number of values?

Number of Books

Library Books

XX

XXX

XXXX

XXXX X

2 3 4 5 6 7 8

313

Deviation: 25, 25, 25 25, 0, 0, 10, 10 Absolute deviation: 5, 5, 5, 5, 0, 0, 10, 10Mean absolute deviation: 5 1 5 1 5 1 5 1 0 1 0 1 10 1 10 5 40, 40 4 8 5 5

The MAD is 5. It tells you that on average the commercial lengths

are 5 seconds from the mean length of 20 seconds.

Possible answer: 4, 6, 8, 8, 10, 12; mean, median,

and mode: 8

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practice lesson 27 measures of center and variability b...practice lesson 27 measures of center and...

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