Two-way tables

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Even if you didn't know their name, you've probably seen one-way

frequency tables

which tell the number of times something occurs. For example, the frequency table below shows the frequency of eye colors in Marcy's science class. In this example, the variable is eye color. The frequency tells how many students have each eye color.

Eye color	Frequency
Blue	4
Brown	12
Green	3
Hazel	5

Two-way tables show information for two categorical, or qualitative, variables. For example, the two-way table below shows the eye color and dominant hand of the students in Marcy's science class. In this example, the variables are eye color and dominant hand.

	Right-handed	Left-handed	Total
Blue eyes	3	1	4
Brown eyes	10	2	12
Green eyes	3	0	3
Hazel eyes	4	1	5
Total	20	4	24

Two-way tables can be frequency tables or relative frequency tables. Read on to learn more about each type of two-way table.

Two-way frequency tables

Two-way frequency

tables,

also called contingency

tables,

show the number of times something occurs for two categorical variables. A two-way frequency table can help you see possible relationships between two sets of categorical data.

For example, the two-way frequency table below shows the results of a survey of 108 students. The students were asked whether they play a sport and whether they play an instrument.

	Plays an instrument	Does not play an instrument	Total
Plays a sport	42	27	69
Does not play a sport	24	15	39
Total	66	42	108

Notice that one category, playing a sport, is represented by rows. The other category, playing an instrument, is represented by columns. The totals for each row are at the end of each row, and the totals for each column are at the bottom of each column.

Values in the body of a two-way frequency table are called joint

frequencies.

Values in the total row and column, with the exception of the total for the entire table, are called marginal

frequencies.

	Plays an instrument	Does not play an instrument	Total
Plays a sport	42	27	69
Does not play a sport	24	15	39
Total	66	42	108

Each joint frequency tells about a combination of the two variables. For example, 42 of the students surveyed play an instrument and play a sport.

Each marginal frequency tells the total for a category. For example, 66 of the students surveyed play an instrument. The last number in the bottom row is the total number of data values. In this example, 108 students were surveyed in total.

Interpreting two-way frequency tables

You can find relationships between groups of categorical data using a two-way frequency table.

Let's try it! Customers at Percy's Pizza were asked if they prefer thin-crust pizza or thick-crust pizza. Responses were separated by whether the customer was a child or an adult. The results were recorded in the two-way frequency table below.

	Thin-crust	Thick-crust	Total
Children	6	24	30
Adults	54	36	90
Total	60	60	120

Compare the joint frequencies in each row of the table.

	Thin-crust	Thick-crust	Total
Children	6	24	30
Adults	54	36	90
Total	60	60	120

In the top row of joint frequencies, you can see that 6 children prefer thin-crust pizza and 24 children prefer thick-crust pizza. So, according to this survey, children are more likely to prefer thick-crust pizza.

In the bottom row of joint frequencies, you can see that 54 adults prefer thin-crust pizza and 36 adults prefer thick-crust pizza. So, according to this survey, adults are more likely to prefer thin-crust pizza.

But be careful!

Not all comparisons will result in valid conclusions. For example, you might be tempted to compare the joint frequencies in each column.

	Thin-crust	Thick-crust	Total
Children	6	24	30
Adults	54	36	90
Total	60	60	120

If you did that, you might notice that in the column for people who prefer thick-crust pizza, 24 were children and 36 were adults. This might lead you to conclude that people who prefer thick-crust pizza are more likely to be adults.

However, those numbers on their own do not tell the whole story. Notice that 30 children and 90 adults were surveyed. Since the number of adults surveyed is 3 times the number of children surveyed, you would need to do more calculations to see if people who prefer thick-crust pizza are more likely to be adults.

Skills

Find probabilities using two-way frequency tables

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