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Lesson 13

6.SP.2, 6.SP.5.d

Lesson 13 Shape of Data Distributions 57

Main IdeaDescribe a data distribution by its center, spread, and overall shape.Relate the choice of center and spread to the shape of the distribution.

New Vocabularydistributionsymmetric

Shape of Data DistributionsPARASAILING The line plot shows the costs in

dollars for parasailing for different companies

on a certain beach.

1. Find the measures of central

tendency. Round to the nearest

tenth if necessary.

2. Draw a vertical line through

the middle of the data.

What do you notice?

The distribution of a set of data shows the arrangement of data

values. It can be described by its center, spread (variation), and

overall shape. If the left side of a distribution looks like the right

side, then the distribution is symmetric. The distribution below has

a cluster of several data values within the interval 10–12. The gaps 9

and 13 have no data values. The value 10 is a peak because it is the

most frequently occurring value.

98 10 11 12 13 14

×

×

×

×

×

×

×

× ×

Describe the Shape of a Distribution

PARASAILING Refer to the line plot “Parasailing Costs” above.

Use clusters, gaps, peaks, outliers, and symmetry to describe the

shape of the distribution.

The left side of the data looks like the right side, so the shape

of the distribution is symmetric. There is a cluster from $31–$39.

The distribution has a peak in the center at $35. There are no gaps

or outliers.

a. SOLAR ECLIPSES Use clusters,

gaps, peaks, outliers, and

symmetry to describe the

shape of the distribution

at the right.

30 31 32 33 34 35 36 37 38 39 40

Parasailing Costs ($)

× ×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

Solar Eclipses, 2001–201076543210

Num

ber o

f Ecl

ipse

s

0:01–2:30

2:31–5:00

5:01–7:30

7:31–10:00

10:01–12:30

Duration (minutes: seconds)

58 Domain: Statistics and Probability

While you cannot identify gaps, peaks, and clusters in a box-and-

whisker plot, you can still identify symmetry and outliers, as well as

describe the shape of a data distribution.

PARKS The box-and-

whisker plot shows the

number of visitors to a

state park. Describe the

shape of the distribution

using symmetry and outliers.

Each box and whisker has the same length. So, the data is evenly

distributed. The distribution is symmetric since the left side of the

data looks like the right side. There are no outliers.

b. FLIGHTS The box-and-whisker plot shows the distance of several

airplane flights in feet. Describe the shape of the distribution

using symmetry and outliers.

200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200

*

Distance of Flights (ft)

You can also describe the center and spread of a data distribution.

The shape of the distribution tells you which measures are most

appropriate. The mean and mean absolute deviation are affected by

outliers, while the median and interquartile range are resistant to

outliers. If there is an outlier, the distribution is not symmetric.

0 10 20 30 40 50 60 70 80 90 100

Number of Visitors to a State Park

Measures of Center and Spread

Use the following flow chart to decide which measures of center and spread are most appropriate to describe a data distribution.

Is the data distribution symmetric?

Yes No

Use the mean to describe the center. Use the mean absolute deviation to describe the spread.

Use the median to describe the center. Use the interquartile range to describe the spread.

TRAVEL The line plot shows the

number of states students in

Elisa’s social studies class

have visited.

a. Choose the appropriate measures to describe the center and

spread of the distribution. Justify your response based on the

shape of the distribution.

The distribution is not symmetric and there is an outlier, 19. The

median and interquartile range are appropriate measures to use.

b. Write a few sentences describing the center and spread of the

distribution using the appropriate measures.

The median is 12 states. The lower quartile is 11. The upper

quartile is 13. The interquartile range is 13 – 11, or 2 states.

The data are centered around 12 states. The spread of the data

around the center is about 2 states.

c. TENNIS Choose the appropriate

measures to describe the center

and spread of the distribution.

Justify your response based on

the shape of the distribution.

Then describe the center and

spread. Round to the nearest

tenth if necessary.

10 11 12 13 14 15 16 17 18 19 20

Number of States Visited

×

×

×

×

×

×

×

×

×

×

×

×

×

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×

×

×

×

×

24 25 26 27 28 29 30 31 32 33 34

Ages of Tennis Players (yr)

×

×

×

×

×

×

×

×

×

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Real-World LinkIn a recent year, California was the state most visited by U.S. travelers. The top three states visited were California, Florida, and Texas.

RRR

Example 1 1. CONCERTS The histogram shows

the wait times in minutes for

entering a concert. Use clusters,

gaps, peaks, outliers, and

symmetry to describe the

shape of the distribution.

Example 2 2. DOGS The line plot shows the

weights in pounds of several

dogs. Describe the shape of the

distribution using symmetry

and outliers.

Lesson 13 Shape of Data Distributions 59

Concert Entrance Wait Times (min)20181614121086420

Freq

uenc

y

Wait Times (min)

0–910–19

20–2940–49

50–5960–69

70–7980–89

90–9930–39

20 40 60 80 100 120 140 160 180 200

*

Weights of Dogs (lb)

60 Domain: Statistics and Probability

Example 3 3. INTERNET The line plot shows the number

of hours several students spent on the

Internet during the week.

a. Choose the appropriate measures to

describe the center and spread of the

distribution. Justify your response

based on the shape of the distribution.

b. Write a few sentences describing the center and spread of

the distribution using the appropriate measures. Round to

the nearest tenth if necessary.

10 2 3 4 5 6 7 8

Number of HoursSpent on the Internet

×

×

×

×

×

× ×

×

×

×

×

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×

×

×

Example 1 4. ANIMALS The histogram shows 5. DVDS The line plot shows the

the average animal speeds in prices in dollars for several DVDs.

miles per hour of several animals. Use clusters, gaps, peaks, outliers,

Use clusters, gaps, peaks, outliers, and symmetry to describe the

and symmetry to describe the shape of the distribution.

shape of the distribution.

Example 2 6. SCHOOL The box-and-whisker plot

shows the science test scores for

Mrs. Everly’s students. Describe

the shape of the distribution

using symmetry and outliers.

7. DONATIONS The box-and-

whisker plot shows the

donations in dollars to

charity by several people.

Describe the shape

of the distribution

using symmetry and

outliers.

1110 12 13 14 15 16 17 18

DVD Prices ($)

× ×

×

×

× ××

×

×

×

×

×

×

×

Average Animal Speeds20181614121086420

Num

ber o

f Ani

mal

s

Speed (miles per hour)

1–1920–39

40–5980–99

100–119

120–139

140–159

160–179

180–199

200–21960–79

60 65 70 75 80 85 90 95 100

Science Test Scores (%)

20 40 60 80 100 120 140 160 180 200 220

*

Donations to Charity ($)

Example 3 8. TEXT MESSAGING The line plot

shows the number of text messages

sent by different students in one day.

a. Choose the appropriate measures to

describe the center and spread of the

distribution. Justify your response

based on the shape of the distribution.

b. Write a few sentences describing the center and spread of the

distribution using the appropriate measures.

9. RUNNING The line plot shows the

number of miles Elisa ran each week.

a. Choose the appropriate measures to

describe the center and spread of the

distribution. Justify your response

based on the shape of the distribution.

b. Write a few sentences describing the center and spread of the

distribution using the appropriate measures. Round to the nearest

tenth if necessary.

For Exercises 10 and 11, refer to the following information.

A distribution that is not symmetric is called skewed. A distribution

that is skewed left has fewer data values on the left side than the right side.

A distribution that is skewed right has fewer data values on the right side

than the left side.

B 10. TREES The box-and-whisker plot

shows the heights in feet of

several trees.

a. Explain how you know the

distribution is not symmetric.

b. Is the distribution skewed left or skewed right? Explain.

c. Use appropriate measures to describe the center and spread

of the distribution. Justify your choice of measure based on the

shape of the distribution.

11. FAMILY The line plot shows the number of

siblings for 18 students in Jeremiah’s

homeroom.

a. Explain how you know the

distribution is not symmetric.

b. Is the distribution skewed left

or skewed right? Explain.

c. Use appropriate measures to describe the center and spread of the

distribution. Justify your choice of measure based on the shape

of the distribution.

1817 19 20 21 22 23 24 25

Number of TextMessages Sent Today

× ×××

××××

××××××

×××××

××

×

2827 29 30 31 32 33 34 35

Miles Ran Each Week

××

××××

××××

××

××

××××××

××

10 15 20 25 30 35 40 45 50

Height of Trees (ft)

10 2 3 4 5 6 7 8

Number of Siblings

××××

××

×××

×××××××××

Real-World LinkIn a recent year, U.S. teens, ages 13–17, sent and received an average of 58 text messages per day.

RR

Test PracticeTT

C 12. OPEN ENDED Draw a line plot for which the median is the most

appropriate measure to describe the center of the distribution.

13. CHALLENGE Explain why you cannot describe the specific location of the

center and spread of the box-and-whisker plot shown using the most

appropriate measures.

14. REASONING The table gives the average

lengths in millimeters of several insects.

Without creating a graphical display,

describe what the shape of the

distribution would look like.

Justify your response.

15. WRITE MATH Explain how the shape of a data distribution tells you which

measures are most appropriate to describe the center and spread of the

distribution.

40 45 50 55 60 65 70 75 80

Calories in Servings of Fruits

Length of Insects (mm)

22 30 35 28 15 90 27

32 55 36 24 60 20 30

16. Refer to the box-and-whisker plot below.

40 50 60 70 80 90 100 110

*

Roller Coaster Speeds (mph)

Which of the following statements

is false?

A. The distribution is symmetric.

B. The distribution is not symmetric.

C. The distribution has an outlier.

D. The distribution has a gap of data.

17. Refer to the line plot below.

2019 21 22 23 24 25 26 27 28

Gas Mileage (miles per gallon)

×

×

× ×

×

×

×

×

×

×

×

×

×

×

××

Which measure is the most appropriate

to describe the variation (spread) of the

distribution?

F. interquartile range

G. mean

H. mean absolute deviation

I. median