What are Two-Way Tables?
Two-way tables are used to display the frequency information of two categories of variables collected from a common source. The table has labels for the categories at the top and on the left, and the frequency data is shown in four or more of the cells in the table. The ‘total’ of each row can be seen at the right and ‘total’ of each column at the bottom. This table is also known as a contingency table. These tables are often used to analyze survey results.
Here is an example of a two-way table that shows the number of males and females who like various games:
Each entry in the table is called joint frequency and the sum or total of rows and columns is called the marginal frequency. Here, 13, 15, 23, ……. are joint frequencies, and the sums, that is, 48, 52, 36, 31,….. are the marginal frequencies.
How do we Create a Two-Way Table?
Here are the steps to construct a two-way table.
Step 1: Identify the two relevant variables.
Step 2: Determine the possible values of each variable.
Step 3: Choose a variable for the rows and another variable for the columns.
Step 4: Add frequencies in the inner cells of the table for each set of variables.
Rapid Recall
Solved Examples
Example 1: Mindy surveyed students who participated in a monthly fundraiser. The two-way table shows the results.
a.) What is the count of female students who participated in the fundraiser?
b.) How many male students did not participate in the fundraiser?
Solution:
a.) Observe the table and look for the entry in the cell corresponding to ‘Yes’ and ‘Female’. The entry is 51.
So, from the given table, we can say that the number of female students who participated in the fundraiser is 51.
So, 51 female students participated in the fundraiser.
b.) Similarly, look for the entry in the cell corresponding to ‘No’ and ‘Male’. The entry is 30.
So, 30 male students did not take part in the fundraiser.
Hence, 30 male students did not participate in the fundraiser.
Example 2: Look at the information in the table and interpret the marginal frequencies.
So, the marginal frequencies are: 172, 310, 253, 229 and 482.
Hence, we can interpret that:
- 78 + 94 = 172 users who avail the limited data plan.
- 175 + 135 = 310 users who avail the unlimited data plan.
- 78 + 175 = 253 users who use the data services of company A.
- 94 + 135 = 229 users use the data services of company B.
- 229 + 253 = 172 + 310 = 482 users who avail data services from both companies.
Example 3: The following table shows the number of junior and senior students who attended the school play and those who did not.
a.) How many students attended the school play?
b.) How many students did not attend the school play?
Solution:
From the table:
Total number of juniors who attended the play = 41
Total number of seniors who attended the play = 52
Total number of juniors who did not attend the play = 30
Total number of seniors who did not attend the play = 23
a.)
Total number of students who attended the play = Number of juniors who attended the play + Number of seniors who attended the play,
= 41 + 52
= 93
Hence, 93 students attended the school play.
b.)
Total number of students who did not attend the play = Number of juniors who did not attend the play + Number of seniors who did not attend the play,
= 30 + 23
= 53
Hence, 53 students did not attend the school play.
[Note: 93 and 53 are the marginal frequencies corresponding to the columns in the two-way table.]
A two-way table is helpful when the dataset has variable sample sizes.
Joint frequencies are the data in each cell of the table.
Marginal frequencies refer to the cells that contain the total or sum of the rows and columns.
For one-way data, we have a single independent variable and one or more dependent variables. Whereas in a two-way table, we have two independent categories on which the variables are dependent.