Two-Way Frequency Tables

Key Concepts

• Understand what are joint frequencies
• Understand what are marginal frequencies
• Understand what are conditional relative frequencies.

Explore and Reason 

Baseball teams at a high school and a college play at the same stadium. Results for every game 

last seasons are given for both teams. There were no ties. 

1. How could you organize the data in table form? 
2. Look for relationships 

How would you analyze the data to determine whether the data? Support the claim that the team 

that plays at home is more likely to win? 

Answer: 

1. One way to organize data would be in a two-way table. Each entry is in the form of the tuple, where the first element refers to the number of wins and the second element refers total games played in a category.  

2. The table for winning rate (wins/no. of games) is as follows 

We can see that for both high schools, the winning rate at home is greater than the winning 

rate away from home. The data suggests that the team that plays at home is more likely 

to win 

Example 1: Interpret a Two-way frequency table Owners of a major food chain are planning to add a vegetarian item to its menu. Customers were asked to choose one of two vegetarian items. The results are shown in the table. What trends do the results suggest? 

Joint frequencies indicate the frequency of a single option for one category: for example 

The frequency of males choosing a veggie burger. 

Marginal frequencies indicate the total frequency for each option or category, such as the total frequency of female respondents. 

The joint frequencies suggest that male customers prefer veggie burgers over veggie pizzas and female customers prefer veggie pizzas over veggie burgers. 

The marginal frequencies suggest that all of the respondents showed only a slight preference for veggie pizza over a veggie burger. They also indicate that more females than males were surveyed. 

Try it 

1. What do the marginal frequencies tell you about the number of male and female respondents? 

Solution: 

• Marginal frequency corresponding t male and female respondents are equal to the number of male and female respondents, respectively. 
• The marginal frequency of male respondents is the same as the number of male respondents 

Example 2: Interpret a two-way relative frequency table 

What do the survey results reveal about male and female customer preferences for veggie burgers? 

Solution: 

Joint relative frequency is the ratio, or percent, of the joint frequency to the total. 

Marginal relative frequency is the ratio, or percent, of the marginal frequency to the total of the customers surveyed, about 22% were males who selected veggie burgers and about 27% were females who selected veggie burgers. So, a greater percentage of females than males selected veggie burgers. 

Try it  

2. How can you tell whether a greater percentage of customers surveyed selected a veggie pizza or a veggie burger? 

Solution: 

Step 1: 

To answer whether the greater percentage of customers surveyed selected veggie burger or 

veggie pizza, we should compare marginals corresponding to the veggie burger and veggie pizza  

in a two-way relative frequency table. 

As marginal for veggie pizza (51%) is higher than marginal for a veggie burger (49%), the greater  

percentage of customers surveyed selected veggie pizza 

Example 3: Calculate the conditional relative frequency  

Using data from examples 1 and 2, a marketing team concludes that females prefer veggie burgers 

More than men do. Does the survey support this conclusion? 

Solution: 

Conditional relative frequency is the ratio of the joint frequency and the related marginal frequency. 

Calculating the conditional relative frequency for each row will adjust for differences in the number 

of male and female customers surveyed.  

• The results do not support this conclusion. The conditional relative frequencies show that  

while about 56% of the male surveyed prefer veggie burgers, only about 44% of the female  

prefer veggie burgers 

Try it:  

3. What conclusion could the marketing team make about male and female preferences for veggie Pizza? Justify your answer. 

Solution: 

• By looking at conditional relative frequency, we observe that males prefer veggie burgers (56%) over veggie pizza (44%), whereas females prefer veggie pizza (56%) over veggie burgers (44%) 

• Males prefer veggie burgers, whereas females prefer veggie pizza. 

Example 4: Interpret conditional relative frequency 

The marketing team also concludes that there is a greater variation between the percent of 

Men and women who like veggie pizza more than there is for those who prefer veggie burgers. 

Do the survey results support this conclusion? 

Solution: 

Calculating the conditional relative frequency for each column allows you to analyze male and female preferences within each food choice category. 

The conclusion is supported by the survey results.  

Conditional relative frequencies show that of customers who prefer pizza, 65% are female and 

only 35% are male. Of those who prefer veggie burgers, 55% are female and 45% are male 

Try it 

4. What conclusion could you draw if the percentages for male and female customers were the same across the rows in this table? 

Solution: 

• The percentages in the row represent the preferences of male and female customers regarding the veggie burger and veggie pizza 
• If the percentages for male and female customers were the same across the rows, that would  

mean that each gender prefers both veggie burgers and veggie pizzas equally 

Example 5: Interpret Data frequencies 

A random sample of spectators entering a stadium was asked whether they were cheering  

for the bears or the tigers in a championship game. The sample was categorized according to 

gender and team. 

1. What does the joint relative frequency 64/230  represent in this context? 

Joint relative frequency is the ratio of the joint frequency to the total. 

Find the joint frequency 65 in the table. 

You can see that 65 males cheered for the Tigers, so

64/230 represents the ratio of male  

Tiger fans to the total number of people surveyed 

2. What does the conditional relative frequency 49/93 represent in this context? 

Conditional relative frequency is the ratio of the joint frequency to the related marginal  

Frequency. 

The number  49/93 represents the ratio of female bears fans to the number of females surveyed 

Try it 

3. What does the conditional relative frequency 72/137 represent in this context? 

Solution: 

Conditional relative frequency

72/137

Represents the ratio of male bear fans to the total number of fans. In other words,

72/137 is the share of males surveyed who were bear fans. 

Concept Summary

Words 

Two-way frequency tables show relationships between two sets of categorical data. 

Entries can be frequency counts or relative frequencies. Entries in the body of the table are 

joint frequencies (counts) or joint relative frequencies(ratios). Entries in the totals column or row are marginal frequencies or marginal relative frequencies 

Conditional relative frequencies show the frequency of responses for a given condition or the ratio of the joint frequencies to the corresponding marginal frequency. 

Tables 

Check your knowledge 

1. In a survey, music club members select their preference between Song A or Song B. Song A is selected by 30 teens and 10 adults. Of 20 who select song B, five are teens. 

Make a two – way frequency table to organize the data. 

1. Is it reasonable to say that more people surveyed prefer song A? Explain. 
2. Is it reasonable to say that more adults than teens participated in the survey? Explain. 

Calculate conditional relative frequencies. 

3. Is it reasonable to say that teens prefer song A more than adults do? Explain. 
4. Is a member who prefers song B significantly more likely to be an adult than a teen? Explain 

2. In the two-way frequency table, frequencies are shown on the top of each cell in blue, and relative 

Frequencies are shown at the bottom in red. Most of the frequencies are missing. 

1. Complete the table 
2. Calculate conditional relative frequencies for yes and no. 
3. Calculate conditional relative frequencies for choices A or B? Explain. 
4. Is a high school graduate more likely to prefer choice A or B? Explain.  
5. Is someone preferring choice A more likely to be a high school graduate than not? Explain.  
6. What does the joint relative frequency 64/200 represent in this context? 
7. What does the conditional relative frequency 96/120 represent in this context? 

Answers: 

1. 

Solution: 

The two-way frequency table looks like the following when we just fill in the information explicitly given in the question: 

A. 

Using the fact that totals (Marginal frequencies) are the sum of rows or columns entries (joint  

frequencies), we can find the missing values to complete the table.  

40 people like song A whereas 20 people like song B. Thus, it is responsible to say that more people like Song A 

B. 

Solution: 

The two – way frequency table is as follows: 

It is not reasonable to say that more adults than teens participated in the survey, as 35 teens  

participated in comparison to 25 adults. 

C. 

Step 1: 

The conditional relative frequency table is obtained by dividing joint frequency with the  

Relevant marginal frequency. 

The following is a conditional relative frequency table for teens and adults 

It is reasonable to say that more teens prefer song A as a higher number of teens (85.7%) prefer song A as compared to adults (40%) 

D. 

The conditional relative frequency table is obtained by dividing the joint frequency by the relevant marginal frequency. 

The following is a conditional relative frequency table for song A and song B.  

Among those who like song B, 75% are adults. This, it significantly more likely that a member  

who likes song B is an adult than a teen?  

Yes, since among those who like song B, 75% are adults. 

2. 

Solution: 

A. 

Using the fact that totals (marginals frequencies or marginal relative frequencies) are the sum of rows or columns entries(joint frequencies or joint relative frequencies) and we can find the missing values to complete the table. 

B. 

Solution: 

The conditional relative frequency table is obtained by dividing the joint frequency by the relevant marginal frequency. 

The following is a conditional relative frequency table for yes or no.  

C. 

Solution: 

The conditional frequency table is obtained by the divided joint frequency with the  

relevant marginal frequency, in this case, marginals of choice A and Choice B. 

The following is a conditional relative frequency table for Choice A and Choice B. 

D. 

Solution: 

The following is a conditional relative frequency table for yes and no. 

Among the high graduates, 85.7% (>50%) prefer choice B. thus, a high school graduate  

is more likely to prefer choice B. 

E. 

The following is a conditional relative frequency table for choice A and Choice B. 

Among those who prefer choice A, 80% (>50%) are high school graduates. Thus, someone who prefers choice A is more likely to be a high school graduate.

Exercise

1. Fill in the blanks
2. …………… indicate the frequency of a single option for one category.
3. ………………. indicate the total frequency for each option or category.
4. …………….. is the ratio, or percent, of joint frequency to the total.
5. …………………… is the ratio, or percent, of the marginal frequency to the total.
6. In a two-way frequency table, the joint frequency in a cell is 8 and the marginal frequency in the same row is 32. What is the conditional relative frequency for the cell?
7. An equal number of juniors and seniors were surveyed about whether they prefer lunch item a or B. Is it reasonable to infer from the table that more juniors prefer lunch item B while more seniors prefer lunch item A? Explain.

Concept Map