Introduction Information about people who are surveyed can
Introduction Information 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. Type of response Type of characteristic Response 1 Response 2 Characteristic 1 a b Characteristic 1 c d 1 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Introduction, continued Each 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. 2 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Introduction, continued A 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. 3 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Introduction, continued A 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. 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables 4
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. 5 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
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. 6 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
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) 7 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice Example 2 Abigail surveys students in different grades, and asks each student which pet they prefer. The responses are in the table below. Grade Preferred pet Bird Cat Dog Fish 9 3 49 53 22 10 7 36 64 10 What is the marginal frequency of each type of pet? 8 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 2, continued 1. 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 9 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 2, continued 2. Sum the responses of people with each 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 10 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 2, continued 3. Sum the responses of people with each 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 11 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 2, continued 4. Sum the responses of people with each 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 12 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 2, continued 5. 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. Grade Preferred pet Bird Cat Dog Fish 9 3 49 53 22 10 7 36 64 10 Total 10 85 117 32 ✔ 13 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 2, continued 14 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice Example 3 Ms. 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. Hours spent studying Test score 0– 25 26– 50 51– 75 76– 100 0– 2 2 8 12 2 2– 4 0 10 8 24 4– 6 1 0 2 9 6+ 0 0 1 4 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables 15
Guided Practice: Example 3, continued Ms. 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. 16 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 3, continued 1. Find the total number of students represented in the table by summing the joint frequencies. 2 + 8 + 12 + 0 + 10 + 8 + 24 + 1 + 0 + 2 + 9 + 0 + 1 + 4 = 83 17 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 3, continued 2. Divide each joint frequency by the total number of students. 18 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 3, continued 3. 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. Hours spent studying Test score 0– 25 26– 50 51– 75 76– 100 0– 2 0. 024 0. 096 0. 145 0. 024 2– 4 0 0. 120 0. 096 0. 289 4– 6 0. 012 0 0. 024 0. 108 6+ 0 0 0. 012 0. 048 ✔ 19 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
Guided Practice: Example 3, continued 20 4. 2. 1: Summarizing Data Using Two-Way Frequency Tables
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