Two-way tables organize data based on two categorical variables.

Two way frequency tables

Two-way frequency tables show how many data points fit in each category.
Here's an example:
PreferenceMaleFemale
Prefers dogs3622
Prefers cats826
No preference26
The columns of the table tell us whether the student is a male or a female. The rows of the table tell us whether the student prefers dogs, cats, or doesn't have a preference.
Each cell tells us the number (or frequency) of students. For example, the 36 is in the male column and the prefers dogs row. This tells us that there are 36 males who preferred dogs in this dataset.
Notice that there are two variables—gender and preference—this is where the two in two-way frequency table comes from.
Want a review of making two-way frequency tables? Check out this video.
Want to practice making frequency tables? Check out this exercise.
Want to practice reading frequency tables? Check out this exercise

Two way relative frequency tables

Two-way relative frequency tables show what percent of data points fit in each category. We can use row relative frequencies or column relative frequencies, it just depends on the context of the problem.
For example, here's how we would make column relative frequencies:
Step 1: Find the totals for each column.
PreferenceMaleFemale
Prefers dogs3622
Prefers cats826
No preference26
Total4654
Step 2: Divide each cell count by its column total and convert to a percentage.
PreferenceMaleFemale
Prefers dogsstart fraction, 36, divided by, 46, end fraction, approximately equals, 78, percentstart fraction, 22, divided by, 54, end fraction, approximately equals, 41, percent
Prefers catsstart fraction, 8, divided by, 46, end fraction, approximately equals, 17, percentstart fraction, 26, divided by, 54, end fraction, approximately equals, 48, percent
No preferencestart fraction, 2, divided by, 46, end fraction, approximately equals, 4, percentstart fraction, 6, divided by, 54, end fraction, approximately equals, 11, percent
Totalstart fraction, 46, divided by, 46, end fraction, equals, 100, percentstart fraction, 54, divided by, 54, end fraction, equals, 100, percent
Notice that sometimes your percentages won't add up to 100, percent even though we rounded properly. This is called round-off error, and we don't worry about it too much.
Two-way relative frequency tables are useful when there are different sample sizes in a dataset. In this example, more females were surveyed than males, so using percentages makes it easier to compare the preferences of males and females. From the relative frequencies, we can see that a large majority of males preferred dogs left parenthesis, 78, percent, right parenthesis compared to a minority of females left parenthesis, 41, percent, right parenthesis.
Want a review of making two-way relative frequency tables? Check out this video.
Want to practice making relative frequency tables? Check out this exercise.
Want to practice reading relative frequency tables? Check out this exercise