1. Two-Way Frequency Tables
Joint, Marginal, and Conditional Frequencies

2. Standards:
Common Core State Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Attend to precision. MCC9-12.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.

3. By the end of this lesson, you should be able to:
Interpret data from a two-way frequency table
Identify and interpret joint, marginal, and conditional relative frequencies
Create a conditional distribution table
Create a relative frequency chart

4. Terms
Two-way Frequency table: a table listing two categorical variables who values have been paired
Joint relative frequency: is found by dividing a frequency that is not in the Total row or the Total column by the grand total.
Marginal relative frequency: is found by dividing a row total or a column total by the grand total.
Two-Way relative frequency table: displays both the joint relative and marginal relative frequencies.
Conditional relative frequency: is found by dividing a frequency that is not in the Total row or Total column by the frequency’s row total or column total.

5. Two-Way Frequency Table
Watch this:

6. Conditional Distributions
Watch:

7. Creating a Relative Frequency Table:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Creating a Relative Frequency Table:
Jenna asked 40 randomly selected students whether they preferred dogs, cats or other pets.
Complete the table for both decimals and percents
Preferred Pet
Dog
Cat
Other
Total
Frequency 18 12 10 40
Relative Frequency Decimals
Relative Frequency Percents
= .45
= .30
= .25
= 1.00
45%
30%
25%
100%

8. Creating a Two-Way Frequency Table:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Creating a Two-Way Frequency Table:
Jenna also recorded the gender of each student.
The results are shown in the two-way frequency table. Find the totals and answer the questions.
Preferred Pet
Dog
Cat
Other
Total
Boys 10 5 9 24
Girls 8 7 1 16
Totals 18 12 10 40
1) How many total students took the survey? 40
2) How many total boys took the survey? 24
3) How many students said they like dogs as pets? 18
4) How many boys said they like dogs as pets? 10

9. Creating a Two-Way Relative Frequency Table:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Creating a Two-Way Relative Frequency Table:
Jenna also recorded the gender of each student surveyed, complete the two-way frequency table below.
Preferred Pet
Dog
Cat
Other
Total
Boys 10 5 9 24
Girls 8 7 1 16
Totals 18 12 10 40
Preferred Pet
Dog
Cat
Other
Total
Boys
Girls
Totals
= .25
= .125
= .225
= .60
= .20
= .175
= .025
= .40
= .45
= .30
= .25
= 1.00

10. Creating a Two-Way Relative Frequency Table:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Creating a Two-Way Relative Frequency Table:
Preferred Pet
Dog
Cat
Other
Total
Boys
Girls
Totals
= .25
= .125
= .225
= .60
= .20
= .175
= .025
= .40
= .45
= .30
= .25
= 1.00
State the percents for the table above.
Preferred Pet
Dog
Cat
Other
Total
Boys
Girls
Totals
25%
12.5%
22.5%
60%
20%
17.5%
2.5%
40%
45%
30%
25%
100%

11. Creating a Two-Way Relative Frequency Table:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Creating a Two-Way Relative Frequency Table:
Preferred Pet
Dog
Cat
Other
Total
Boys
25%
12.5%
22.5%
60%
Girls
20%
17.5%
2.5%
40%
Totals
45%
30%
100%
1) Find the joint relative frequency of students surveyed who are girls and prefer dogs as pets.
20%
2) Find the joint relative frequency of students surveyed who are boys and prefer cats as pets.
12.5%
3) Find the marginal RF of students surveyed who prefer dogs as pets and then prefer cats as pets.
dogs - 45%
cats - 30%

12. Calculating a Conditional Relative Frequency:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Calculating a Conditional Relative Frequency:
Preferred Pet
Dog
Cat
Other
Total
Boys 10 5 9 24
Girls 8 7 1 16
Totals 18 12 10 40
Find the conditional relative frequency that a student surveyed prefers cats as pets, given the student is a girl.
Number of girls who prefer cats: 7
= .4375
= 43.75%
Number of girls: 16

13. Calculating a Conditional Relative Frequency:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Calculating a Conditional Relative Frequency:
Preferred Pet
Dog
Cat
Other
Total
Boys 10 5 9 24
Girls 8 7 1 16
Totals 18 10 40 12
Find the conditional relative frequency that a student surveyed is a girl, given that the student prefers cats as pets.
Number of girls who prefer cats: 7
= .583
= 58.3%
Number of students who prefer cats: 12

14. Calculating a Conditional Relative Frequency:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Calculating a Conditional Relative Frequency:
Preferred Pet
Dog
Cat
Other
Total
Boys 10 5 9 24
Girls 8 7 1 16
Totals 18 12 10 40
Find the conditional relative frequency that a student surveyed prefers cats as pets, given the student is a boy.
Number of boys who prefer cats: 5
= .208
= 20.8%
Number of boys: 24

15. Calculating a Conditional Relative Frequency:
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Calculating a Conditional Relative Frequency:
Preferred Pet
Dog
Cat
Other
Total
Boys 10 5 9 24
Girls 8 7 1 16
Totals 18 10 40 12
Find the conditional relative frequency that a student surveyed is a boy, given that the student prefers dogs as pets.
Number of boys who prefer dogs: 10
= .556
= 55.6%
Number of students who prefer dogs: 18

16. Finding Possible Associations and Trends
Old Holt 4-5/4-6; New Holt 10-1/10-2/ On-Core: 9-5 Two Way Frequency Tables
Finding Possible Associations and Trends
Jenna survey results (numbers vs percents).
Dog
Cat
Other
Total
Boys
10/25%
5/12.5%
9/22.5%
24/60%
Girls
8/20%
7/17.5%
1/2.5%
16/40%
Totals
18/45%
12/30%
40/100%
1) Does the table reflect gender bias towards pets? NO
But, it does imply, boys like “other” pets more than girls: % to 2.5%,
2) If so, which gender, likes which pets the best?
N/A
3) What are some examples of “other” pets?
Hamsters, gerbils, mice, rats, snakes, rabbits, pigs, etc.,
4) Girls tend not to like which types of pets?
“other” types

17. Make sure you try the 10 practice problems.
Relative Frequency
Go here:
Make sure you try the 10 practice problems.

18. Review: Key concepts covered in this presentation:
How to read a two-way frequency table
Marginal and joint distributions
Conditional distribution tables
Relative frequency