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Explore Categorical Data and FrequenciesData that can be expressed with numerical measurements are quantitative data. In this lesson you will examine qualitative data, or categorical data, which cannot be expressed using numbers. Data describing animal type, model of car, or favorite song are examples of categorical data.

Circle the categorical data variable. Justify your choice.

temperature weight height color

Identify whether the given data is categorical or quantitative.

large, medium, small

120 ft 2 , 130 ft 2 , 140 ft 2

A frequency table shows how often each item occurs in a set of categorical data. Use the categorical data listed on the left to complete the frequency table.

Ways Students Get to School

bus car walk car car car buswalk walk walk bus bus carbus bus walk bus car bus car

Way Frequency

bus 8

car

Reflect

1. How did you determine the numbers for each category in the frequency column?

2. What must be true about the sum of the frequencies in a frequency table?

Temperature, weight, and height are measured on a numerical scale, so they are

quantitative data. Color cannot be expressed numerically.

You can determine the numbers for each category by counting the number of times each

category is listed in the data.

The sum should equal the total number of items in the data set.

quantitative

walk

7

5

categorical

Module 8 347 Lesson 1

8 . 1 Two-Way Frequency TablesEssential Question: How can categorical data for two categories be summarized?

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Common Core Math StandardsThe student is expected to:

S-ID.5

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Mathematical Practices

MP.7 Using Structure

Language ObjectiveDistinguish between quantitative data and categorical data.

HARDBOUND SEPAGE 279

BEGINS HERE

HARDCOVER PAGES 279290

Turn to these pages to find this lesson in the hardcover student edition.

Two-Way Frequency Tables

ENGAGE Essential Question: How can categorical data for two categories be summarized?You can summarize categorical data for two

categories in a two-way frequency table.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online, and then take a quick, show-of-hands survey to determine the most popular sports among the students in your class. Discuss whether the survey results might have been different had you surveyed the boys and the girls separately. Then preview the Lesson Performance Task.

347

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore Categorical Data and Frequencies

Data that can be expressed with numerical measurements are quantitative data. In this lesson you will examine

qualitative data, or categorical data, which cannot be expressed using numbers. Data describing animal type,

model of car, or favorite song are examples of categorical data.

Circle the categorical data variable. Justify your choice.

temperatureweight

heightcolor

Identify whether the given data is categorical or quantitative.

large, medium, small

120 ft 2 , 130 ft 2 , 140 ft

2

A frequency table shows how often each item occurs in a set of categorical data. Use the

categorical data listed on the left to complete the frequency table.

Ways Students Get to School

bus car walk car car car bus

walk walk walk bus bus car

bus bus walk bus car bus car

WayFrequency

bus8

car

Reflect

1. How did you determine the numbers for each category in the frequency column?

2. What must be true about the sum of the frequencies in a frequency table?

S-ID.5 For the full text of these standards, see the table starting on page CA2.

Temperature, weight, and height are measured on a numerical scale, so they are

quantitative data. Color cannot be expressed numerically.

You can determine the numbers for each category by counting the number of times each

category is listed in the data.

The sum should equal the total number of items in the data set.

quantitative

walk

7

5

categorical

Module 8

347

Lesson 1

8 . 1 Two-Way Frequency Tables

Essential Question: How can categorical data for two categories be summarized?

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347 Lesson 8 . 1

L E S S O N 8 . 1

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Explain 1 Constructing Two-Way Frequency TablesIf a data set has two categorical variables, you can list the frequencies of the paired values in a two-way frequency table.

Example 1 Complete the two-way frequency table.

A high school’s administration asked 100 randomly selected students in the 9th and 10th grades about what fruit they like best. Complete the table.

Preferred Fruit

Grade Apple Orange Banana Total

9th 19 12 23

10th 22 9 15

Total

Row totals:

9th: 19 + 12 + 23 = 54

10th: 22 + 9 + 15 = 46

Preferred Fruit

Grade Apple Orange Banana Total

9th 19 12 23 54

10th 22 9 15 46

Total 41 21 38 100

Jenna asked some randomly selected students whether they preferred dogs, cats, or other pets. She also recorded the gender of each student. The results are shown in the two-way frequency table below. Each entry is the frequency of students who prefer a certain pet and are a certain gender. For instance, 8 girls prefer dogs as pets. Complete the table.

Preferred Pet

Gender Dog Cat Other Total

Girl 8 7 1

Boy 10 5 9

Total

Row totals:

Girl: 8 + 7 + 1 =

Boy: 10 + 5 + 9 =

Column totals:

Apple: 19 + 22 = 41

Orange: 12 + 9 = 21

Banana: 23 + 15 = 38

Column totals:

Dog: 8 + 10 =

Cat: 7 + 5 =

Other: 1 + 9 =

Grand total:

Sum of row totals: 54 + 46 = 100

Sum of column totals: 41 + 21 + 38 = 100

Both sums should equal the grand total.

Grand total:

Sum of row totals: 16 + =

Sum of column totals: 18 + + =

Both sums should equal the grand total.

18 12

16

24

18

12

10

24 40

12 10 40

10 40

24

16





Integrate Mathematical PracticesThis lesson provides an opportunity to use Mathematical Practice MP.7, which asks students to “look for and make use of structure.” In this lesson, students use the structure of two-way frequency tables to analyze data and understand a problem-solving situation.


BEGINS HERE


BEGINS HERE

EXPLORE Categorical Data and Frequencies

INTEGRATE TECHNOLOGYStudents have the option of completing the Explore Activity online or in the book.

CONNECT VOCABULARY Make sure that students understand the distinction between quantitative and categorical data. Quantitative data can be measured on a numbered scale. Categorical data involves either/or choices between two or more descriptive categories. Ask students to provide some examples of quantitative and categorical data.

EXPLAIN 1 Constructing Two-Way Frequency Tables

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 To check students’ understanding of the information presented in a two-way frequency table, have one student ask a question about the data in the table. Have another student answer the question and provide an explanation of how to use the table to arrive at the answer.

QUESTIONING STRATEGIESWhat is the difference between a two-way frequency table and an ordinary frequency

table? In a two-way frequency table, each value

(other than row and column totals) indicates the

number of items that fit two data categories rather

than one, such as the number of boys who prefer

cats. In an ordinary frequency table, data values

correspond to only one data category.

PROFESSIONAL DEVELOPMENT

Two-Way Frequency Tables 348


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Reflect

3. Look at the totals for each row. Was Jenna’s survey evenly distributed among boys and girls? Explain.

4. Look at the totals for each column. Which pet is preferred by the most students? Justify your answer.

Your Turn

Complete the two-way frequency table.

5. Antonio surveyed 60 of his classmates about their participation in school activities and whether they have a part-time job. The results are shown in the two-way frequency table below. Complete the table.

Activities

Job Clubs Only Sports Only Both Neither Total

Yes 12 13 16 4

No 3 5 5 2

Total

6. Jen surveyed 100 students about whether they like baseball or basketball. Complete the table.

Like Basketball

Like Baseball Yes No Total

Yes 61 13

No 16 10

Total

Explain 2 Reading Two-Way Frequency TablesYou can extract information about paired categorical variables by reading a two-way frequency table.

Example 2 Read and complete the two-way frequency table.

Suppose you are given the circled information in the table and instructed to complete the table.

Eat Cereal for Breakfast

Gender Yes No Total

Girl 42 12 54

Boy 36 10 46

Total 78 22 100

Find the total number of boys by subtracting: 100 - 54 = 46 Find the number of boys who do eat cereal by subtracting: 46 - 10 = 36 Add to find the total number of students who eat cereal and the total number of students

who do not eat cereal.

No, there were 24 boys and 16 girls.

Dogs: the total for Dog is greater than the total for Cat and the total for Other.

15 18 21 6

74

26

1002377

60

15

45





BEGINS HERE

COLLABORATIVE LEARNING

Peer-to-Peer ActivityHave students work in pairs. Challenge one in each pair to describe a data set for which the partner must create a two-way frequency table. Students should provide as little direct information as possible, while still making it possible to complete the table. For example, instead of saying, “Twenty boys prefer baseball,” they might say, “The number of boys who prefer baseball is 15 less than the number of girls who prefer softball.” After students complete the frequency tables, have them compare results with their partners. If the students disagree, have them discuss and determine which is correct.

EXPLAIN 2 Reading Two-Way Frequency Tables

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Remind students that there may be more than one process for filling in a particular table. Discuss different ways to fill cells in a table, such as using addition to find the total for a column or row, and working backward to find a missing value that is not a total.

AVOID COMMON ERRORSRemind students to use the row totals and column totals to check that their work is correct. The sum of the row totals should be the same as the sum of the column totals. If the sums are different, students need to review their work to look for errors.

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B One hundred students were surveyed about which beverage they chose at lunch. Some of the results are shown in the two-way frequency table below. Complete the table.

Lunch Beverage

Gender Juice Milk Water Total

Girl 10 17

Boy 15 24 21 60

Total

Find the total number of girls by subtracting: 100 - 60 =

So, the total number of girls is . The number of girls who do not choose milk is + = .

Find the number of girls who chose milk by subtracting: - =

Reflect

7. Which lunch beverage is the least preferred? How do you know?

Your Turn

Read and complete the two-way frequency table.

8. 100 students were asked what fruit they chose at lunch. The two-way frequency table shows some of the results of the survey. Complete the table.

Lunch Fruit

Gender Apple Pear Banana Total

Girl 17 11 49

Boy 10 16

Total

9. 200 high school teachers were asked whether they prefer to use the chalkboard or projector in class. The two-way frequency table shows some of the results of the survey. Complete the table.

Preferred Teaching Aid

Gender Chalkboard Projector Total

Female 56 99

Male 44

Total 87 113 200

13

25 37 38

40

40 17 10 27

40 27

Juice: the total for Juice is less than the total for Milk and the total for Water.

13

100

40

21

25

46 27 27

51

100

43

57 101


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BEGINS HERE

DIFFERENTIATE INSTRUCTION

Cognitive StrategiesWhen students are completing two-way frequency tables that are missing information, discuss how to determine which cells in the table can be completed first. Students should understand that in order for a value to be determined, it must be the only missing value in either its row or its column. By completing a row or column in which only one value is missing, students move one step closer to completing other rows or columns.

QUESTIONING STRATEGIESTo complete a two-way frequency table that is missing information, is it necessary to know

the total of all the values in the table? Explain. No; if

there is enough other information given in the

table, the total can be determined using other

information.



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Elaborate

10. You are making a two-way frequency table of 5 fruit preferences among a survey sample of girls and boys. What are the dimensions of the table you would make? How many entries would you need to fill the table with frequencies and totals?

11. A 3 categories-by-3 categories two-way frequency table has a row with 2 numbers, and no row or column totals. Can you fill the row?

12. Essential Question Check-In How can you summarize categorical data for 2 categories?

• Online Homework• Hints and Help• Extra Practice

1. Identify whether the given data is categorical or quantitative.

gold medal, silver medal, bronze medal

100 m, 200 m, 400 m

2. A theater company asked its members to bring in canned food for a food drive. Use the categorical data to complete the frequency table.

Cans Donated to Food Drivepeas corn peas soup corncorn soup soup corn peaspeas corn soup peas cornpeas corn peas corn soupcorn peas soup corn corn

Cans Frequency

soup

peas

Complete the two-way frequency table.

3. James surveyed some of his classmates about what vegetable they like best.Complete the table.

Preferred Vegetable

Grade Carrots Green Beans Celery Total

9th 30 15 24

10th 32 9 20

Total

Evaluate: Homework and Practice

You would make a 2-by-5 for the frequencies. Adding totals would increase the

dimensions to 3-by-6. You need 3 times 6, or 18, entries to fill the table.

This table would be a 4-by-4 table, including the row of totals. You cannot fill the row

because you need 3 numbers to figure out the last one.

You can use a two-way frequency table.

categorical

quantitative

corn

6

69

61

2462 44 130

8

11


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BEGINS HERE

LANGUAGE SUPPORT

Connect VocabularyThe word quantitative in the phrase quantitative data shares a root with the word quantity, meaning an amount. The word categorical in the phrase categorical data shares a root with the word category, meaning a descriptive grouping. Categorical data can also be referred to as qualitative data because these data describe qualities or characteristics. The word frequency in the phrase frequency table shares a root with the word frequent, meaning often. Each value in a frequency table shows how often that value falls into a given category.

ELABORATE INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Have students discuss why the order of words is important in identifying a cell in a two-way frequency table. For example, students should understand that “people who watch TV but not movies” refers to a different cell than “people who watch movies but not TV. ”

SUMMARIZE THE LESSONWhat information is provided by a two-way frequency table, and how is that information

organized? A two-way frequency table provides

categorical data for two categorical variables, such

as color and size or gender and job. Data for one

variable is organized in rows, and data for the other

variable is organized in columns. The value in each

cell of the table identifies the number of items that

fit the intersection of the two categories.

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4. A high school’s extracurricular committee surveyed a randomly selected group of students about whether they like tennis and soccer. Complete the table.

Like Tennis

Like Soccer Yes No Total

Yes 37 20

No 16 15

Total

5. After a school field trip, Ben surveyed some students about which animals they liked from the zoo. Complete the table.

Preferred Animal at a Zoo

Grade Lion Zebra Monkey Total

11th 9 15 14

12th 4 17 15

Total

6. Jill asked some randomly selected students whether they preferred blue, green, or other colors. She also recorded the gender of each student. The results are shown in the two-way frequency table below. Complete the table.

Preferred Color

Gender Green Blue Other Total

Girl 15 3 10

Boy 3 16 6

Total

7. Kevin surveyed some students about whether they preferred soccer, baseball, or another sport. He also recorded their gender. Complete the table.

Preferred Sport

Gender Soccer Baseball Other Total

Girl 33 7 10

Boy 15 27 7

Total

57

38

31

36

88

74

35

29

53

3213

28

25

53161918

50

49

99173448




IN1_MNLESE389755_U4M08L1.indd 352 01/04/14 8:57 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1–12 1 Recall of Information MP.7 Using Structure

13–14 2 Skills/Concepts MP.3 Logic

15–22 2 Skills/Concepts MP.4 Modeling

23 1 Recall of Information MP.7 Using Structure

24 2 Skills/Concepts MP.3 Logic

25–26 3 Strategic Thinking MP.3 Logic


BEGINS HERE

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreCategorical Data and Frequencies

Exercises 1–2, 23

Example 1Constructing Two-Way Frequency Tables

Exercises 3–12

Example 2Reading Two-Way Frequency Tables

Exercises 13–22, 24–26

INTEGRATE MATHEMATICAL PRACTICESFocus on PatternsMP.8 Have students try to find the smallest number of filled cells in a two-way frequency table that makes it possible to complete the table. Work backward from a completed frequency table by removing values one at a time, making sure it is possible to do so while leaving a row or column with only one missing value. Students may discover that completing a frequency table requires that the table have at least as many filled cells as the number of interior cells.



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8. A school surveyed a group of students about whether they like backgammon and chess. They will use this data to determine whether there is enough interest for the school to compete in these games. Complete the table.

Like Backgammon

Like Chess Yes No Total

Yes 10 61

No 5 3

Total

9. Hugo surveyed some 9th and 10th graders in regard to whether they preferred math, English, or another subject. The results of the survey are in the following table. Complete the table.

Preferred Subject

Grade Math English Other Total

9th 40 35 20

10th 41 32 17

Total

10. Luis surveyed some middle school and high school students about the type of music they prefer. Complete the table.

Preferred Music

School Level Country Pop Other Total

Middle School 18 13 23

High School 7 32 15

Total

11. Natalie surveyed some teenagers and adults on whether they prefer standard cars, vans, or convertibles. Her results are in the following table. Complete the table.

Preferred Car Type

Age Standard Van Convertible Total

Adults 10 25 9

Teenagers 11 7 24

Total

71

8

796415

95

90

185376781

54

54

108384525

44

42

86333221


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BEGINS HERE

MULTIPLE REPRESENTATIONSHave students create a two-question survey whose results can be summarized in a two-way frequency table. Students should understand that the questions they choose for their surveys should not allow for open-ended answers.

AVOID COMMON ERRORSRemind students that in a two-way frequency table, the total of the values in each row and the total of the values in each column must add up to the same number. Students can check their work for errors by comparing the sum of the row totals to the sum of the column totals.

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12. Eli surveyed some teenagers and adults on whether they prefer apples, oranges, or bananas. His results are in the following table. Complete the table.

Preferred Fruit

Age Apple Orange Banana Total

Adults 22 12 10

Teenagers 24 9 9

Total

200 students were asked to name their favorite science class. The results are shown in the two-way frequency table. Use the table for the following questions.

Favorite Science Class

Gender Biology Chemistry Physics Total

Girl 42 39 23 104

Boy 45 32

Total

13. How many boys were surveyed? Explain how you found your answer.

14. Complete the table. How many more girls than boys chose biology as their favorite science class? Explain how you found your answer.

The results of a survey of 150 students about whether they own an electronic tablet or a laptop are shown in the two-way frequency table.

Device

Gender Electronic tablet Laptop Both Neither Total

Girl 15 54 9 88

Boy 35 8 5

Total

15. Complete the table. Do the surveyed students own more laptops or more electronic tablets?

16. Which group had more people answer the survey, boys or students who own an electronic tablet only? Explain.

44

42

86192146

19

558461

96

200

96 boys: 104 of the 200 students were girls, so 200 − 104 = 96 of them were boys.

42 − 19 = 23, so 23 more girls than boys chose biology.

62

1501418

10

8929

14

The number of boys is 150 − 88 = 62.

Girls who own both electronic devices: 88 − 9 − 54 − 15 = 10

Boys who own an electronic tablet: 62 − 5 − 8 − 35 = 14; Students own more laptops.

Boys: 62 boys answered the survey, which is more than the 29 people who own an electronic tablet only.




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BEGINS HERE



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17. The table shows the results of a survey about students’ preferred frozen yogurt flavor. Complete the table, and state the flavors that students preferred the most and the least.

Preferred FlavorGender Vanilla Mint Strawberry Total

Girl 15 18 45

Boy 17 25

Total 100

18. Teresa surveyed 100 students about whether they like pop music or country music. Out of the 100 students surveyed, 42 like only pop, 34 like only country, 15 like both pop and country, and 9 do not like either pop or country. Complete the two-way frequency table.

Like PopLike Country Yes No Total

Yes

No

Total

19. Forty students in a class at an international high school were surveyed about which non-English language they can speak. Complete the table.

Foreign LanguageGender Chinese Spanish French Total

Girl 7 8

Boy 6 7 18

Total

Luis surveyed 100 students about whether they like soccer. The number of girls and the number of boys completing the survey are equal.

20. Complete the table.

Likes SoccerGender Yes No Total

Girl 20

Boy 35

Total 100

21. Twice as many girls like soccer as the number that like tennis. The same number of students like soccer as like tennis. Construct a table containing the tennis data.

12

13 55

29 40 31

Students preferred mint the most and vanilla the least.

15 34

9

43 100

51

49

42

57

5

12 14 14

7 22

40

3015

45 55

50

50

Likes Tennis

Gender Yes No Total

Girl 15 35 50

Boy 30 20 50

Total 45 55 100



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22. A group of 200 high school students were asked about their use of email and text messages. The results are shown in the two-way frequency table. Complete the table.

Text Messages

Email Yes No Total

Yes 72 90

No 45

Total

23. Circle the letter of each data set that is categorical. Select all that apply.

A. 75°, 79°, 77°, 85°

B. apples, oranges, pears

C. male, female

D. blue, green, red

E. 2 feet, 5 feet, 12 feet

F. classical music, country music

G. 1 centimeter, 3 centimeters, 9 centimeters

24. Explain the Error Find the mistake in completing the two-way frequency table for a survey involving 50 students. Then complete the table correctly.

Favorite Foreign Language Class

Gender Russian German Italian Total

Girl 8 8 8 24

Boy 9 7

Total

Correct table:

Favorite Foreign Language Class

Gender Russian German Italian Total

Girl 8 8 8 24

Boy 9 7

Total

42

50

58

65

137 63

110

200

18

10

18 17 15 50

26

Possible answer: 50 was used as the total whose favorite is Russian instead of as the grand total.

Total boys: 50 - 24 = 26 Boys who prefer Russian: 26 - 9 - 7 = 10




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H.O.T. Focus on Higher Order Thinking

25. Justify Reasoning Charles surveyed 100 boys about their favorite color. Of the 100 boys surveyed, 44 preferred blue, 25 preferred green, and 31 preferred red.

a. Explain why it is not possible to make a two-way frequency table from the given data.

b. Suppose Charles also surveyed some girls. Of the girls surveyed, 30 preferred blue and 43 preferred green. Can Charles make a two-way frequency table now? Can he complete it?

26. Persevere in Problem Solving Shown are two different tables about a survey involving students. Each survey had a few questions about musical preferences. All students answered all questions. Complete the tables. What type of music do the students prefer?

Likes Classical Music

Gender Yes No Total

Girl 21

Boy 22

Total 100

Likes Blues Music

Gender Yes No Total

Girl 15 49

Boy 15

Total

Charles only surveyed boys, so there is only one categorical variable: color. You need two categorical variables to make a two-way frequency table.

He can make a table, but he cannot complete it. There are now two categorical variables, color and gender, but there is not enough information about the girls’ preferences. In addition to the information provided, you need to know how many girls were surveyed or how many prefer red to complete the table.

29

50 50

34

36

70 30 100

51

28 49

51

All students answered all the questions, so total girls in both is 49, and the total students in both is 100.

Overall, the students prefer blues music: 70 students like blues, 50 like classical.



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BEGINS HEREJOURNALHave students explain the steps they take to complete a two-way frequency table that has missing values. Students should be sure to include descriptions of how to determine column totals and row totals.

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Lesson Performance Task

Two hundred students were asked about their favorite sport. Of the 200 students surveyed, 98 were female. Some of the results are shown in the following two-way frequency table.

Favorite Sport

Gender Football Baseball Basketball Soccer Total

Female 36 12

Male 38 19

Total 64 36

a. Complete the table.

b. Which sport is the most popular among the students? Which is the least popular? Explain.

c. Which sport is most popular among the females? Which sport is most popular among the males? Explain.

Total number of males: 200 - 98 = 102

Females who chose football: 64 - 38 = 26

Females who chose baseball: 98 - 26 - 36 - 12 = 24

Males who chose soccer: 36 - 12 = 24

Males who chose basketball: 102 - 38 - 19 - 24 = 21

Total baseball: 24 + 19 = 43

Total basketball: 36 + 21 = 57

Most popular is football; 64 is the greatest total.

Least popular is soccer; 36 is the least total.

Most popular among females is basketball; 36 is the greatest number of choices.

Most popular among males is football; 38 is the greatest number of choices.

26 24

21

57

24

98

102

20043



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BEGINS HERE

EXTENSION ACTIVITY

Have students create and conduct a survey of 20 students and then record the results in a two-way frequency table. Students should begin by selecting two categorical variables to record, such as gender and favorite subject, or grade level and music preference. Remind them that quantitative and free-response answers are not appropriate for recording in a frequency table. Encourage students to select a sample of students that is not expected to skew the survey results. Have students share their frequency tables and discuss their findings.

QUESTIONING STRATEGIESIn what order would you fill in the empty spaces in the table? Explain why. Possible

answer: I would first fill in the numbers given in the

text of the problem (200 students in total, and 98

female students). Then I would look for a row or

column with only one empty cell, and find the

missing number (such as the total number of males,

the number of females who chose football, or the

number of males who chose soccer). After I fill in

those numbers, there would still be other rows and

columns with only one empty cell, so I would repeat

the process until the table is completed.

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Discuss with students how the sample of students chosen for the survey could have influenced the survey results. For example, consider whether age, country of origin, participation in sports, or other characteristics of survey participants might influence the results. Have students describe a target population for which soccer is likely to be the most popular sport among those surveyed.

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.



correctionkey=nl-a;ca-a lesson 8 . 1 do not edit--changes ... · s-id.5 summarize categorical data...

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