Introduction

IntroductionInformation about people who are surveyed can be captured in two-way frequency tables. A two-way frequency table is a table of data that separates responses by a characteristic of the respondents.

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4.2.1: Summarizing Data Using Two-Way Frequency Tables

Type of characteristic

Type of responseResponse 1 Response 2

Characteristic 1 a bCharacteristic 1 c d

Introduction, continuedEach cell in the table contains a count of the people with a given characteristic who gave each response. For example, in the table, a, b, c, and d would each be counts for the responses given by people with each characteristic. The sum of all the cells, a + b + c + d, is the total number of respondents. Two-way frequency tables help organize information and provide greater insight into features of a population being surveyed. A trend, or pattern in the data, can be examined using a two-way frequency table.

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Introduction, continuedA joint frequency is the number of responses for a given characteristic. The entries in the cells of a two-way frequency table are joint frequencies. In the sample table, a, b, c, and d are each joint frequencies. A marginal frequency is the total number of times a response was given, or the total number of respondents with a given characteristic. This is the sum of either a row or a column in a two-way frequency table. In the sample table, a + b would be the marginal frequency of people with Characteristic 1.

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Introduction, continuedA conditional relative frequency allows a comparison to be made for multiple responses in a single row, single column, or table. Relative frequencies are expressed as a percentage, usually written as a decimal. They are found by dividing the number of responses by either the total number of people who gave that response, the total number of people with a given characteristic, or the total number of respondents. In the sample table, is the relative frequency of Response 1 for people with Characteristic 1.

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Key Concepts• A two-way frequency table divides survey responses

by characteristics of respondents.

• The number of times a response was given by people with a certain characteristic is called a joint frequency.

• A marginal frequency is the total number of times a response is given, or the total number of people with a certain characteristic.

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Key Concepts, continued• A conditional relative frequency expresses a number

of responses as a percentage of the total number of respondents, the total number of people with a given characteristic, or the total number of times a specific response was given.

• Trends, or patterns of responses, can be identified by looking at the frequency of responses.

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Common Errors/Misconceptions• incorrectly locating frequencies in the table

• incorrectly calculating conditional relative frequencies by being inconsistent in the method used (dividing by the number of times a response was given, the number of people with a given characteristic, or the total number of respondents)

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Guided PracticeExample 2Abigail surveys students in different grades, and asks each student which pet they prefer. The responses are in the table below.

What is the marginal frequency of each type of pet?

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GradePreferred pet

Bird Cat Dog Fish9 3 49 53 2210 7 36 64 10

Guided Practice: Example 2, continued1. Sum the responses of people with each

characteristic for the first pet type, “bird.”3 people in grade 9 preferred birds, and 7 people in grade 10 preferred birds.

3 + 7 = 10 people who preferred birds

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characteristic for the second pet type, “cat.”49 people in grade 9 preferred cats, and 36 people in grade 10 preferred cats.

49 + 36 = 85 people who preferred cats

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characteristic for the third pet type, “dog.”53 people in grade 9 preferred dogs, and 64 people in grade 10 preferred dogs.

53 + 64 = 117 people who preferred dogs

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characteristic for the fourth pet type, “fish.”22 people in grade 9 preferred fish, and 10 people in grade 10 preferred fish.

22 + 10 = 32 people who preferred fish

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Guided Practice: Example 2, continued5. Organize the marginal frequencies in a

two-way frequency table.Create a row and include the marginal frequencies of each response under the name of each response.

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GradePreferred pet

Bird Cat Dog Fish9 3 49 53 2210 7 36 64 10

Total 10 85 117 32 ✔


Guided Practice: Example 2, continued

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Guided PracticeExample 3Ms. Scanlon surveys her students about the time they spend studying. She creates a table showing the amount of time students studied and the score each student earned on a recent test.

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Hours spent studying

Test score0–25 26–50 51–75 76–100

0–2 2 8 12 22–4 0 10 8 244–6 1 0 2 96+ 0 0 1 4

Guided Practice: Example 3, continuedMs. Scanlon wants to understand the distribution of scores among all the students, and to get a sense of how students are performing and how much students are studying. Find the conditional relative frequencies as a percentage of the total number of students.

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Guided Practice: Example 3, continued1. Find the total number of students

represented in the table by summing the joint frequencies.2 + 8 + 12 + 2 + 0 + 10 + 8 + 24 + 1 + 0 + 2 + 9 + 0 + 0 + 1 + 4 = 83

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Guided Practice: Example 3, continued2. Divide each joint frequency by the total

number of students.

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Guided Practice: Example 3, continued3. Represent the conditional joint

frequencies in a new table.Insert each conditional joint frequency in a table set up the same way as the two-way frequency table.

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Hours spent studying

Test score0–25 26–50 51–75 76–100

0–2 0.024 0.096 0.145 0.0242–4 0 0.120 0.096 0.2894–6 0.012 0 0.024 0.1086+ 0 0 0.012 0.048

✔


Guided Practice: Example 3, continued

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Introduction

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