1. 4.2.1: Summarizing Data Using Two-Way Frequency Tables
Unit 4 Part B
Concept: Frequencies
EQ: How do we record and use various types of frequencies?
Vocabulary:
Joint Frequency
Marginal Frequency
Trend
Conditional Relative Frequency
4.2.1: Summarizing Data Using Two-Way Frequency Tables

2. 4.2.1: Summarizing Data Using Two-Way Frequency Tables
Standard….
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.
4.2.1: Summarizing Data Using Two-Way Frequency Tables

3. 4.2.1: Summarizing Data Using Two-Way Frequency Tables

4. Type of characteristic
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 characteristic
Type of response
Response 1
Response 2
Characteristic 1 a b c d
4.2.1: Summarizing Data Using Two-Way Frequency Tables

5. 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.
4.2.1: Summarizing Data Using Two-Way Frequency Tables

6. 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.
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

7. 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.
4.2.1: Summarizing Data Using Two-Way Frequency Tables

8. 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.
4.2.1: Summarizing Data Using Two-Way Frequency Tables

9. Guided Practice Example 1
Abigail 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?
Grade
Preferred pet
Bird
Cat
Dog
Fish 9 3 49 53 22 10 7 36 64
4.2.1: Summarizing Data Using Two-Way Frequency Tables

10. Guided Practice: Example 1, continued
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
4.2.1: Summarizing Data Using Two-Way Frequency Tables

11. Guided Practice: Example 1, continued
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.
= 85 people who preferred cats
4.2.1: Summarizing Data Using Two-Way Frequency Tables

12. Guided Practice: Example 1, continued
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.
= 117 people who preferred dogs
4.2.1: Summarizing Data Using Two-Way Frequency Tables

13. Guided Practice: Example 1, continued
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.
= 32 people who preferred fish
4.2.1: Summarizing Data Using Two-Way Frequency Tables

14. ✔ Guided Practice: Example 1, continued
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
Total 85
117 32 ✔
4.2.1: Summarizing Data Using Two-Way Frequency Tables

15. Guided Practice: Example 1, continued
4.2.1: Summarizing Data Using Two-Way Frequency Tables

16. Guided Practice Example 2
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 students earned on a recent test.
Time spent studying
in hours
Test score
0–25
26–50
51–75
76–100
0–2 2 8 12
2–4 10 24
4–6 1 9 6+ 4
4.2.1: Summarizing Data Using Two-Way Frequency Tables

17. Guided Practice: Example 2, 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.
4.2.1: Summarizing Data Using Two-Way Frequency Tables

18. Guided Practice: Example 2, continued
Find the total number of students represented in the table by summing the joint frequencies.
= 83
4.2.1: Summarizing Data Using Two-Way Frequency Tables

19. Guided Practice: Example 2, continued
Divide each joint frequency by the total number of students.
4.2.1: Summarizing Data Using Two-Way Frequency Tables

20. Guided Practice: Example 2, continued
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.
Time spent studying
in hours
Test score
0–25
26–50
51–75
76–100
0–2
0.024
0.096
0.145
2–4
0.120
0.289
4–6
0.012
0.108 6+
0.048
4.2.1: Summarizing Data Using Two-Way Frequency Tables

21. Guided Practice: Example 2, continued
4.2.1: Summarizing Data Using Two-Way Frequency Tables