A two-way table is a frequency table that shows data obtained from one source that belongs to two categories. We will first illustrate how to make a two-way table by using a survey example. Next, the table that we built will help us to explain how we can read a two-way table.

Making a Two-Way Table

Let's consider the results of survey in which $80$ students were asked whether they play a musical instrument and whether they play a sport.

$16$ students do both activities.
$9$ students do not practice any activity.
$41$ students play an instrument.
$46$ students play a sport.

Please note that this is only one survey example of many possibilities. Let's make a two-way table using the above information.

Determine the Categories

We first need to determine the categories of our data. In this case, the categories are Play a Sport and Play an Instrument. Each category is further divided into the possible answers, Yes and No. We can now label our rows and columns.

Create Spaces for Totals

A row total and a column total should be added to the table. These will be filled in with the marginal frequencies and the total number of observations.

Fill the Table With the Given Data

We can now fill the table using the given data. The first entry of the table represents the number of people who practice both activities, which is $16 .$ The second entry is for the people who play a sport but not an instrument, and so on. The total number of people surveyed corresponds to the bottom-right box.

Find Any Missing Frequencies

Using the given frequencies, we can find the missing ones using logical reasoning. For example,

16

out of

46

students play an instrument and a sport. If we find the difference between these values, we can calculate the number of students who play a sport but not an instrument.

46 - 16 = 30

Now, let's add this value to the number of students who do not practice any activity.

30 + 9 = 39

This represents the number of students who do not play an instrument. Continuing this reasoning, we can complete the entire table.

Reading a Two-Way Table

Now, let's see how we can read the two-way table that we built!

two way table with joint and marginal frequencies

Each entry that is not in the Total row or Total column represents the cases with the characteristics described by the intersecting row and column at that entry. These are called joint frequencies.

$16$ students play an instrument and play a sport.
$30$ students do not play an instrument but they play a sport.
$25$ students play an instrument but they do not a sport.
$9$ students neither play an instrument nor play a sport.

The sum of the joint frequencies in a row or column is called the marginal frequency. These frequencies tell us the number of people that fit that criteria in our table.

$41$ students play a sport.
$39$ students do not play a sport.
$46$ students play an instrument.
$34$ students do not play an instrument.

Finally, the bottom-right cell tells us the total number of observations or the number of people surveyed. In this case, $80$ people were surveyed.

Conclusion

For both making and reading a two-way table, we first need to identify the categories. Each entry describes a joint frequency, while the marginal frequencies are placed in the Total row and Total column. The bottom-right cell tells us the total number of observations.

Exercise