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.

What are two way tables?

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:

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

You have quite a few missing values in the two way table, so you need to find these values. You need to look for a row or column where you know all but one of the missing values.

Here, you know the number of females who can swim, and the total number of people who can swim. This means, you can calculate the number of males by subtracting the number of females from the row total:

49 - 23 = 26.

Now you can calculate the number of males that cannot swim as you know the number of males that can swim, and the total number of males:

30 - 26 = 4

Now you can calculate the number of females who cannot swim:

31 - 4 = 27

The total number of females: 23 + 27 = 50

Filling in this value, you have:

The total number of males and females: 30 + 50 = 80

The total number of people can / not swim: 49 + 31 = 80

Filling in the grand total of 80, you have the final solution:

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.

Here, you know that you have 3 classes (A, B, C) and you are comparing which hand they write with. Drawing the two way table, you have:

Completing the information given from the question, fill in the known values into the correct section in the table:

You have quite a few missing values in the two way table, so you need to find these values. You need to look for a row or column where you know all but one of the missing values.

Here, you know that out of the 30 students in Class B, \, 14 are left-handed, and so you can find the number of right-handed students in Class B\text{:}

30 - 14 = 16

You now know all but one value in the right-handed row.

44 - (17 + 16) = 11

For the number of left-handed students in Class A, you need to know the row total:

90 - 44 = 46

The number of left-handed students in Class A\text{:} \, 46 - (14 + 16) = 16

The total number of students in Class A\text{:} \, 16 + 17 = 33

The total number of students in Class C\text{:} 16 + 11 = 27

Filling in these values, you have:

The total number of students in the three classes: 27 + 30 + 23 = 90

The total number of left / right handed people: 46 + 44 = 90

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.

As 160 students chose to study History or Geography, this is our overall total.

86 students chose History:

34 of the students chose Spanish and Geography:

41 of the students studied French and History:

This completes the table with all the information you know so far.

Inspecting the table, you need to find the row or column where you only have one unknown value. This is the column for History, where you can calculate the number of students who also chose Spanish.

86 - 41 = 45

To calculate the number of students who chose French and Geography, you need to know the total number of students who chose Geography.

160 - 86 = 74

The number of student who chose French and Geography is therefore:

74 - 34 = 40

The total number of students who chose Spanish: 45 + 34 = 79

The total number of students who chose French: 41 + 40 = 81

The total number of students who chose Spanish or French: 79 + 81 = 160

The total number of students who chose History or Geography: 86 + 74 = 160

Now that you have the two way table, you need to use this to calculate the probability of selecting a student choosing both French and History.

Looking at the two way table, there are 41 students who chose French and History out of a total of 160 students, so

P (student who chose French and History) = \cfrac{41}{160}.

Note: unless the question states, leave the answer to a probability as a fraction without simplifying it – it may be useful for the next question.

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:

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

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

The total number of morning customers: 5 + 12 + 4 = 21

The total number of afternoon customers: 14 + 3 + 6 = 23

The total number of customers who drink tea: 5 + 14 = 19

The total number of customers who drink coffee: 12 + 3 = 15

The total number of customers who drink juice: 4 + 6 = 10

Filling in these values into the table, you have:

The total number of morning and afternoon customers: 21 + 23 = 44

The total number of customers who drink tea, coffee, and juice: 19 + 15 + 10 = 44

Filling in the grand total of 44, you have the final solution:

Now you can calculate column relative frequency. Place the number of each cell over the total of the column. Then divide to convert it to a percent.

To decide if there is a possible association, compare the row percentages. This compares the morning to the afternoon. The greater the difference, the stronger the association.

• Tea: 74 \% \, – \, 26 \% = 48 \%
  This is a significant difference. You can say “Tea drinkers are associated with the afternoon.”
  
  
• Coffee: 80 \% \, – \, 20 \% = 60 \%
  This is a significant difference. You can say “Coffee drinkers are associated with the morning.”
  
  
• Juice: 60 \% \, – \, 40 \% = 20 \%
  This is less than the others, so there is some evidence of an association, but it is weaker than the previous drinks. You can say “Juice drinkers are associated with the morning, but not as strongly as the other drinks.”

Note: This page does not address how to identify the level of significance of 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?

As there are 150 seats altogether, the overall total is 150.

The total number of children is 60.

As there are 44 adults in the box, and 23 adults in the balcony, you can fill in these two values in the adult row:

As there are 65 people in the box and 50 people in the floor, you can also fill in these values:

The row or column that has one missing value is the column for the box where you do not know the number of children.

65 - 44 = 21

To calculate the number of children in the balcony, you need to know the total number of people in the balcony.

150 - (65 + 50) = 35

The number of children in the balcony is:

35 - 23 = 12

The total number of children in the floor is:

60 - (21 + 12) = 27

The number of adults in the floor is:

50 - 27 = 23

The only total you need to calculate to complete the two way table is the number of adults:

44+23+23 = 90

The total number of people: 90 + 60 = 150

The total number of people in the box, balcony and floor: 65 + 35 + 50 = 150

Now you can calculate row relative frequency. Place the number of each cell over the total of the row. Then divide to convert it to a percent.

To decide if there is a possible association, compare the column percentages. This compares the adults to the children. The greater the difference, the stronger the association.

The greatest difference is in the floor seat column. You could say “Children are associated with floor seats” since the percentage of children is almost 20 \% more than adults.

Notice there is not much difference in the balcony column, so there is no evidence of association there. You could also say “Adults tend to be in the box more than children” since there is almost a 15 \% difference in this column.

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.
  
  See also: Venn diagram

• 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
  
  

Related representing data lessons

• Stem and leaf plot
• Box plot
• Quartile
• Interquartile range
• Scatterplot
• Line graph
• Time series graph
• Pie chart

The next lessons are

• Frequency table
• Frequency graph
• Types of sampling

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