# Two way tables

Here you will learn about two way tables, including what a two way table is, how to construct them, interpret them, and how to calculate the probability of an event using a two way table.

Students will first learn about two way tables as part of statistics and probability in 8 th grade and continue to learn about them in high school.

## What are two way tables?

Two way tables are a type of frequency table used for organizing bivariate data. They are also known as contingency tables.

Two way tables are mostly used for categorical data, though they can be used for numerical data too. Categorical data is data where the items are words rather than numbers. For example, colors, sports, car manufacturers, gender, month, etc.

To construct a two way table, you need two categorical variables. One variable is featured as the top row within the two way table, and the other variable features on the first column of the table.

For example, this two way table shows a data set about what students eat for lunch. One categorical variable is the grade level, the other categorical variable is what type of lunch they have.

There are row and column totals. You can see that there are 35 \, 7 th graders and that 50 students have packed lunch. The total number of students in the table is 90. This is an example that shows the frequency of each group.

You can use the same data in the table above, but show it as a two way relative frequency table using percentages.

You can use tables like this to decide if there is an association between different variables.

For example, the table shows that 8 th graders are associated with packed lunches, while 7 th graders are associated with cooked food. The farther the percentages are apart, the stronger the association.

## Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8 – Statistics and Probability (8.SP.A.4)
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.

Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

## How to construct two way tables

In order to construct two way tables:

1. Fill in the known values into the two way table.
2. Calculate missing values.
3. Calculate the row and column totals.
4. Check the final total.

## Two way tables examples

### Example 1: missing row and column totals

Complete this two way table:

1. Fill in the known values into the two way table.

Here, you know each of the individual values for each category so you can move on to Step 2.

2Calculate missing values.

Here, you do not have any missing values and so you can move on to Step 3.

3Calculate the row and column totals.

Calculate the sum of the rows of the table:

The total number of boys: 18 + 7 = 25

The total number of girls: 17 + 11 = 28

Calculate the sum of the columns of the table:

The total number of students who like soccer: 18 + 17 = 35

The total number of students who do not like soccer: 7 + 11 = 18

Filling in these values into the table, you have:

4Check the final total.

The total number of boys and girls: 25 + 28 = 53

The total number of students who do / not like soccer: 35 + 18 = 53

Filling in the grand total of 53, you have the final solution:

### Example 2: missing values throughout the table

Complete this two way table:

Fill in the known values into the two way table.

Calculate missing values.

Calculate the row and column totals.

Check the final total.

### Example 3: constructing a two way table from a word problem

A school is researching which hand students write with. They survey 90 students in 9 th grade. 17 students in Class A are right-handed.

14 students in Class B are left-handed. 16 students in Class C are left-handed.

There are 44 students who are right-handed, and 30 students in Class B. Construct a two way table to represent this data.

Fill in the known values into the two way table.

Calculate missing values.

Calculate the row and column totals.

Check the final total.

### Example 4: using a two way table to work out a probability

160 middle school students chose to study History or Geography and Spanish or French.

86 of the students chose History.

34 of the students chose Spanish and Geography.

41 of the students chose French and History.

A student is chosen at random. By constructing a two way table, find the probability that the student chose French and History.

Fill in the known values into the two way table.

Calculate missing values.

Calculate the row and column totals.

Check the final total.

## How to identify associations within two way tables

In order to identify associations within two way tables:

1. Fill in the known values into the two way table.
2. Calculate missing values.
3. Calculate the row and column totals.
4. Check the final total.
5. Convert the table to show column or row relative frequency.
6. Compare the percentages to identify any possible associations.

### Example 5: association with three categories with missing totals

Complete this two way table:

Fill in the known values into the two way table.

Calculate missing values.

Calculate the row and column totals.

Check the final total.

Convert the table to show column or row relative frequency.

Compare the percentages to identify any possible associations.

### Example 6: association with a word problem

People can sit in a seat in a box, the balcony or the floor. There are 150 seats altogether. There are 60 children who have seats in the theater.

There are 44 adults in a box and 23 adults in the balcony seats. Altogether there are 65 people in a box and 50 people on the floor.

Is there an association between adults and children and the place they sit in the theater?

Fill in the known values into the two way table.

Calculate missing values.

Calculate the row and column totals.

Check the final total.

Convert the table to show column or row relative frequency.

Compare the percentages to identify any possible associations.

### Teaching tips for two way tables

• Gives students opportunities to collect their own data and create two way tables from scratch.

• Choose worksheets that have a variety of contexts, so that students learn to solve with two way tables in many different ways.

• Provide tutorials, like this page, for struggling students to refer back to examples when they are solving on their own.

### Easy mistakes to make

• Writing non-whole numbers within the frequency cells
The numbers in the two way table are whole numbers, not decimals or fractions. This is because the numbers are frequencies where data has been counted.

• Modeling in math
The situations in math questions have been simplified to make the situation simpler to study. The world is a more complex place to live. This means individuals need to go under one category and one only.

For example, a school asks students to choose football or baseball. In a two way table, you keep these separate. But in real-life, the school may accommodate students who want to choose both or neither. This is where you would use a Venn diagram instead.

• Confusing the frequencies
Each individual is only counted once in the main body of the table. Each of the 14 individuals are only in one of the four cells of the main section of the table.

• Doubling the overall total
Be careful not to double the overall total (or grand total). For example, the ‘grand total’ of this table is 6. Each individual item only counts once towards the ‘grand total’. But it can be calculated out using the row totals or the column totals, but they should not both be added together.

4+2=3+3=6

### Practice two way tables questions

1. Complete the two way table:

The column totals are 12+10=22 for educators and 11+8=19 for students.

The row totals are 12+11=23 for yes and 10+8=18 for no.

The overall total will be 22+19=41 (or 23+18=41 ).

2. Complete the two way table:

Find the missing values in the table by subtracting or adding:

Total number of children: 70-45=25

Adults and that do not drink tea: 45-36=9

Children that drink tea: 47-36=11

Children that do not drink tea: 25-11=14

Total number of people who do not drink tea: 9 + 14 = 23

3. 100 men and women were asked about what sport they play.

Draw a two way table showing the results below:

• 55 people play soccer
• 53 people are women
• 24 people play hockey
• 10 men play hockey
• 18 women play golf

Find the missing values in the table by subtracting or adding:

The information from the question gives us the following incomplete two way table:

The number of females who play hockey: 24-10=14

The number of females who play soccer: 53-(18+14)=21

The number of males who play soccer: 55-21=34

The total number of golf players: 100-(24+55)=21

The number of males who play golf: 21-18=3

The total number of males: 10 + 34 + 3 = 47

4. 40 people were asked if they ate meat or were vegetarian / vegan. 25 women were asked and 10 of them did not eat meat. 13 men ate meat. Calculate the probability of selecting a male at random that does not eat meat.

\cfrac{2}{15}

\cfrac{2}{40}

\cfrac{2}{13}

\cfrac{2}{12}

The original two way table is

Filling in the other values into the two way table, you have:

The number of males who do not eat meat is 2.

The total number of people is 40.

P (Male who is Vegetarian / vegan) = \cfrac{2}{40}

5. 120 students were asked to choose French, German or Spanish. Altogether 45 students chose German and 55 chose Spanish. 13 of the 7 th graders chose French. 17 of the 8 th graders chose German. 64 of the students are 7 th graders.

Complete the table and use it to calculate the probability that a random student selected studies Spanish and is in 8 th grade.

\cfrac{56}{120}

\cfrac{32}{120}

\cfrac{32}{56}

\cfrac{32}{55}

Calculating the values in the two way table, you have:

The number of 8 th graders who study Spanish: 32

The total number of students: 120

P ( 8 th grader who studies Spanish) = \cfrac{32}{120}

6. Below is a two way table showing data about the number of students in 3 classes that wear glasses (or not).

Which statement about association is true?

Class   A   is associated with glasses.

Class B is associated with glasses.

Class A and B are associated with glasses.

Class C is associated with glasses.

First, calculate the row relative frequencies.

Now compare the column frequencies. Class A and Class B have similar percentages.

However, Class C is different. Since 68 \% is larger than the other classes, you can say “Class C is associated with glasses.”

This also means that you can say “Class A and B are associated with no glasses.”

## Two way tables FAQs

What are joint frequencies and marginal frequencies?

Joint frequencies are the frequencies that combine the two variables. For example, the total of glasses in Class A (from Practice Question 6 ).

Marginal frequency is the total of one variable. For example, the total students in Class A.

What is conditional relative frequency?

This is the ratio of one row or column to the total in that column or row. For example, the ratio of glasses to no glasses in Class A (from Practice \# 6 ).

What is the marginal relative frequency?

This is the ratio of the total of row or column to overall total. For example, the ratio of Class A to the total students (from Practice Question 6 ).

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