A two-way frequency table is a table that displays the frequencies for two categorical variables.
For example, the following two-way table shows the results of a survey that asked 100 people which sport they liked best: baseball, basketball, or football.
The rows display the gender of the respondent and the columns show which sport they chose:
The marginal frequencies are the frequencies shown in the margins of the table:
These values tell us the total values for each variable. For example:
- 36 total respondents chose baseball as their favorite sport
- 31 total respondents chose basketball as their favorite sport
- 33 total respondents chose football as their favorite sport
We can also see:
- 48 total respondents were male
- 52 total respondents were female
The joint frequencies are the frequencies shown in the cells of the table:
These values are known as “joint” frequencies because they tell us the frequency of two values that occur jointly.
For example, we can see:
- There were a total of 13 respondents who were male and preferred baseball.
- There were a total of 15 respondents who were male and preferred basketball.
- There were a total of 20 respondents who were male and preferred football.
- There were a total of 23 respondents who were female and preferred baseball.
- There were a total of 16 respondents who were female and preferred basketball.
- There were a total of 13 respondents who were female and preferred football.
Notice that the sum of all the joint frequencies adds up to the total number of survey respondents:
Total survey respondents = 13 + 15 + 20 + 23 + 16 + 13 = 100.
What Are Joint Relative Frequencies?
A joint relative frequency tells us the frequency of one variable relative to another variable.
For example, consider our two-way table from earlier:
Question 1: What is the joint relative frequency that a survey respondent prefers baseball, given that the respondent is a female?
To answer this, we will only consider the row that contains female responses. We’ll then take the number of females who prefer baseball and divide by the total number of females.
This turns out to be 23/52 = 0.4423 = 44.23%
In other words, 44.23% of all female survey respondents prefer baseball as their favorite sport.
Question 2: What is the joint relative frequency that a survey respondent is male, given that they prefer football as their favorite sport?
To answer this, we will only consider the column that contains football as the favorite sport. We’ll then take the number of males who prefer football and divide by the total number of respondents who preferred football.
This turns out to be 20/33 = 0.606 = 60.6%
In other words, 60.6% of all survey respondents who prefer football are male.
Additional Resources
What is a Marginal Distribution?
What is a Conditional Distribution
What is a Joint Probability Distribution?
A joint relative frequency tells us the frequency of one variable relative to another variable.
it is calculated taking into consideration sum of total subject not row or columns
Hi,
I really liked that you kept the explanations simple for students to understand. I looked at :
Question 1: What is the joint relative frequency that a survey respondent prefers baseball, given that the respondent is a female?
To answer this, we will only consider the row that contains female responses. We’ll then take the number of females who prefer baseball and divide by the total number of females.
This turns out to be 23/52 = 0.4423 = 44.23%
This looks to be the same as conditional probability – where given females, we are looking for an event A given B has occurred. (Baseball given a female.) Can you please elucidate between the two.
Hi LP…You are thinking about it in exactly the right way. What you calculated is essentially the same idea as conditional probability, but the terminology comes from two slightly different ways of looking at the data.
Joint relative frequency usually comes from a two way table. It simply describes how often two characteristics occur together in the data. For example, the combination of female and prefers baseball. When we focus only on the female row and compute 23 out of 52, we are looking at the relative frequency of baseball preference within the female group.
Conditional probability is the formal probability language for that same idea. It asks for the probability of event A given that event B has already occurred. In your example, A is preferring baseball and B is being female. So the conditional probability P baseball given female is calculated by taking the number of females who prefer baseball and dividing by the total number of females.
So in practice, what you did using the table is the empirical estimate of the conditional probability based on the sample data. The phrase joint relative frequency tends to emphasize the data in the table, while conditional probability emphasizes the probability relationship between two events. The calculation itself ends up being the same in this case.