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? Red Blue Black Shirts Ordered on One Day White Green EXPLORE ACTIVITY ESSENTIAL QUESTION Describing Categorical Data Some data are quantitative, such as height or number of siblings of all students in a class. Other data are qualitative, such as eye color or favorite type of music. Categorical data are data that are sorted into categories on the basis of qualitative characteristics. You can use the mode to summarize or describe categorical data. The mode of a categorical data set is the category that occurs most often. If all categories have the same frequency, there is no mode. Karl sells red, blue, black, white, and green shirts online. One day Karl received orders for 4 red, 5 blue, 6 black, 6 white, and 3 green shirts. Complete the dot plot of Karl’s shirt orders for the day. Which shirt color or colors were the most and least popular that day? Identify the mode(s) of the data. Reflect 1. Justify Reasoning Is it possible to find the mean or median of Karl’s data set? Explain. A B C LESSON 17.5 Categorical Data How can you summarize and describe categorical data? Math Talk Mathematical Processes 6.12.D Measurement and data—6.12.D Summarize categorical data with numerical and graphical summaries, including the mode, … relative frequency table, and the percent bar graph, and use these … to describe the data distribution. Could the colors in the dot plot be displayed in a different order? Would this change the overall results? Explain. 511 Lesson 17.5 © Houghton Mifflin Harcourt Publishing Company

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Page 1: TX-DUSA-6 MTXESE051676 U6M17L5 Combined · Math On the Spot my.hrw.com Math Trainer Online Assessment and Intervention Personal my.hrw.com Summarizing Categorical Data with a Relative

?

Red Blue Black

Shirts Ordered on One Day

White Green

EXPLORE ACTIVITY

ESSENTIAL QUESTION

Describing Categorical DataSome data are quantitative, such as height or number of siblings of all students

in a class. Other data are qualitative, such as eye color or favorite type of music.

Categorical data are data that are sorted into categories on the basis of

qualitative characteristics.

You can use the mode to summarize or describe categorical data. The mode of

a categorical data set is the category that occurs most often. If all categories have

the same frequency, there is no mode.

Karl sells red, blue, black, white, and green shirts online. One day Karl

received orders for 4 red, 5 blue, 6 black, 6 white, and 3 green shirts.

Complete the dot plot of Karl’s shirt orders for the day.

Which shirt color or colors were the most and least popular that day?

Identify the mode(s) of the data.

Reflect 1. Justify Reasoning Is it possible to find the mean or median of Karl’s

data set? Explain.

A

B

C

L E S S O N

17.5 Categorical Data

How can you summarize and describe categorical data?

Math TalkMathematical Processes

6.12.D

Measurement and data—6.12.D Summarize categorical data with numerical and graphical summaries, including the mode, … relative frequency table, and the percent bar graph, and use these … to describe the data distribution.

Could the colors in the dot plot be displayed in a

different order? Would this change the overall

results? Explain.

511Lesson 17.5

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Math On the Spotmy.hrw.com

Summarizing Categorical Data with a Relative Frequency TableIn the Explore Activity, you plotted how many times a particular color shirt was ordered. The number of times a color was ordered is called its frequency.

The relative frequency of a category is the ratio of its frequency to the sum

of the frequencies for all categories. Relative frequency is often written as a

fraction or a percent. For example, because 6 out of 24 orders were for black

shirts, the relative frequency of black shirts is or 25%.

The table shows the results of a survey of students about their favorite

summer sport. Make a relative frequency table of the data.

Favorite Summer Sport

Sport basketball baseball swimming soccer track softball

Frequency 5 7 4 3 2 4

Find the sum of the frequencies for all categories.

5 + 7 + 4 + 3 + 2 + 4 = 25

Make a frequency table. Write the relative frequency of each

category as a fraction of the total, 25, and as a percent.

Favorite Summer Sport

Sport basketball baseball swimming soccer track softball

Relative Frequency

6 __ 24

EXAMPLE 1

STEP 1

STEP 2

4 __ 25

= 16% 4 __ 25

= 16% 3 __ 25

= 12% 2 __ 25

= 8% 7 __ 25

= 28% 5 __ 25

= 20%

2. Chuy has 40 dimes, 20 pennies, 10 nickels, and 10 quarters in his coin jar.

Make a relative frequency table of the coins in the jar.

Coins in a Jar

Type of coin

Relative frequency

YOUR TURN

Math TalkMathematical Processes

6.12.D

What is the sum of the relative frequencies in

fraction form? in percent form? Why?

Multiply numerator and denominator by 4: 5 × 4 ______ 25 × 4 = 20 ____ 100 = 20%.

512 Unit 6

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Math On the Spot

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35

30

25

20

15

10

5

0

Rela

tive

freq

uen

cy (

%)

Instrument

drums guitar bass saxo-phone

trumpet clarinet

Music Class Survey Results

Math On the Spot

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Using a Percent Bar Graph to Describe a Data DistributionYou can graph relative frequencies in a percent bar graph to help you visualize

how the data are distributed.

The table shows the results of a survey taken in Mr. Jenk’s music class

about the students’ favorite musical instruments. Make a percent bar

graph. Then find the mode and describe how the data are distributed.

Favorite Musical Instrument

Instrument drums guitar bass saxophone trumpet clarinet

Frequency 6 5 3 3 2 1

Relative frequency

= 30% = 25% = 15% = 15% = 10% = 5%

Make a percent bar graph of the relative frequencies. The height of each

bar shows the percent of students surveyed who favor that instrument.

The instrument mentioned most often is drums, so the mode is

drums, although the guitar is nearly as popular. More than half the

students (55%) favor drums or the guitar. About a third of students

(30%) favor the bass or saxophone. All other students (15%) favor

the trumpet or clarinet.

Reflect 3. Analyze Relationships Why is it helpful to arrange the categories in

the order of their relative frequencies?

EXAMPLEXAMPLE 2

6 __ 20

5 __ 20

3 __ 20

3 __ 20

2 __ 20

1 __ 20

STEP 1

STEP 2

6.12.D

The saxophone is the favorite musical instrument of 15% of the students surveyed.

513Lesson 17.5

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Math On the Spotmy.hrw.com

Reading Movies Sports

Favorite Summer Activities

Travel

Math Trainer

Online Assessment and Intervention

Personal

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0

10

Toma

toes

Kale

Strawbe

rries

Zucchini

Melons

30

40

20

Summer Garden Plots

Plant

Rela

tive

freq

uenc

y (%

)

Devon’s Tulips50

40

30

20

10

0Rela

tive

freq

uenc

y (%

)

Tulip ColorRed Yellow Purple

4. Devon is growing tulips in his garden. He has

24 red tulips, 11 yellow tulips, and 15 purple

tulips. Make a percent bar graph and describe

the distribution.

YOUR TURN

Guided Practice

Mrs. Valentine surveyed her class about their favorite

summer activity. Four students chose reading, 7 chose

movies, 7 chose sports, and 5 chose travel. (Explore Activity)

1. Make a dot plot of the data.

2. Identify the mode(s) of the data set.

The garden club is planning their spring and summer garden. They have

20 plots. Tomatoes will be in 3 plots, kale will be in 5 plots, strawberries

will be in 6 plots, zucchini will be in 2 plots, and melons will be in 4 plots.

(Examples 1 and 2)

3. Make a relative frequency table of the data that shows both fractions

and percents.

Summer Garden Plots

Plant tomatoes kale strawberries zucchini melons

Relative frequency

4. Make a percent bar graph of the relative frequencies of the

garden plots.

5. How can you calculate relative frequencies as percents?

ESSENTIAL QUESTION CHECK-IN??

514 Unit 6

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Gerbil Fish Mouse Lizard

Name Class Date

Independent Practice17.5

Mr. Anderson’s fifth grade class is getting a class pet. Seven students vote

to get a gerbil, 3 vote for a fish, 6 vote for a mouse, and 4 vote for a lizard.

6. What is the mode of the data set? What does it mean for this situation?

7. If each pet had received 9 votes, what would the mode have been?

8. Analyze Relationships Make a dot plot of the data. Use the dot plot to

describe the data.

9. The service club sells snacks at school basketball games. In the first

quarter they sell 6 servings of nachos, 4 bags of popcorn, 7 pieces of fruit,

and 3 bags of nuts.

a. Make a relative frequency table. Include both fractions and percents.

Snacks Sold in the First Quarter

Snack nachos popcorn fruit nuts

Relative frequency

b. Analyze Relationships Deloria says she can find the relative

frequency of nuts based on another relative frequency. What might

she be doing?

c. Draw Conclusions If 9 bags of popcorn were sold rather than 4,

which relative frequencies would be affected? Explain.

6.12.D

515Lesson 17.5

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Page 6: TX-DUSA-6 MTXESE051676 U6M17L5 Combined · Math On the Spot my.hrw.com Math Trainer Online Assessment and Intervention Personal my.hrw.com Summarizing Categorical Data with a Relative

Work Area

Eye Colors

Blue5%

Brown60%

Green16%

Hazel19%

Rela

tive

freq

uenc

y (%

)

Colorgreen blue brown hazel

0

10

20

30

40

50

60

70

Eye ColorThe percent bar graph shows the relative frequencies that

resulted from of a survey about eye color.

10. Draw Conclusions Can you tell from the bar graph how many

people were surveyed? Why or why not?

11. Communicate Mathematical Ideas Describe how the data

are distributed.

12. What If? Suppose 250 people were surveyed to create the eye

color data shown in the graph for Exercises 10 and 11. How

many people would have brown eyes? Justify your answer.

13. Multiple Representations Describe how this circle graph is

similar to and different from the percent bar graph shown for

Exercises 10–11.

14. Justify Reasoning Jayshree says the number of dots in a dot plot for

Exercise 8 shows that the data is numeric and not categorical. Is she right?

Explain. If not, what is her mistake?

FOCUS ON HIGHER ORDER THINKING

Unit 6516

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