Two way tables are tables based on two criteria. The table below that has the two criteria, a student's main writing hand, and whether the student enjoys English or not. The columns show which writing hand is used, and the rows show their enjoyment of English.

	Right	Left
Enjoys English	$4$4	$9$9
Doesn't enjoy English	$2$2	$15$15

To read a two way table look at the column and row that a cell is in. For example there are $9$9 students who are left-handed and enjoy English, and $2$2 students who are right handed that don't enjoy English.

Tables will often include totals of each column, row and the total sum. This is found by adding up every cell in that row or column.

	Right	Left	Total
Enjoys English	$4$4	$9$9	$13$13
Doesn't enjoy English	$2$2	$15$15	$17$17
Total	$6$6	$24$24	$30$30

The categories of the rows and columns should be chosen such that each person or object can only go in one of the cells.

Worked example

A pet store sells three types of animals, which may or may not have spots, shown in the following table. If an animal is randomly selected, what is the probability that a cat will be selected?

	Cat	Dog	Fish
Has spots	$7$7	$4$4	$11$11
No spots	$5$5	$10$10	$8$8

Think: We can use the following formula to find the probability:

$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$Probability=Number of favourable outcomesTotal number of outcomes

We will need to find the total number of animals and the number of fish to substitute into the formula.

Do: The number of fish is given by the last column, $11+8=19$11+8=19. The total number of animals is found by summing every number, $7+4+11+5+10+8=45$7+4+11+5+10+8=45.

The probability will be $\frac{19}{45}$1945.

Two way tables and Venn diagrams

A two way table presents similar information as a Venn diagram. We can convert between a two way table and a Venn diagram, and vice versa, by looking at which categories are represented by the Venn diagram or two way table, and how the different regions or cells match up.

This image shows how a typical two way table and Venn diagram are related.

The column marked "Left" is a category used in the Venn diagram and the row marked "Entered" is the other category used.

The number that is in both categories, $4$4 will go in the overlap of the two circles. The remaining value, $9$9, in the selected row, represents the "Entered and right-handed" students and will go into the "Entered" circle but not in the overlap because they are not "Left-handed". Any cells that are in neither the highlighted row nor the highlighted column will go into the surrounding box.

Worked example

Consider the Venn diagram, assemble a two way table from the Venn diagram given.

Think: Two categories are given so these will form one of my row headings and column headings.

Do: The empty two way table will look like this:

	Cut	Not cut
Dyed
Not dyed

Using the Venn diagram we can fill in the table by looking at the corresponding regions. The top left cell is for the number of people who had their hair cut and dyed, that is $35$35 people. The bottom left cell is for the number of people who had their hair cut but not dyed, which is $12$12.

	Cut	Not cut
Dyed	$35$35	$20$20
Not dyed	$12$12	$3$3

The two way table can be completed in this way for the other two cells.

Practice questions

Question 1

$50$50 students were asked whether or not they were allergic to nuts and dairy. The two way table is provided below.

	Allergic to Nuts	Not Allergic to Nuts
Allergic to Dairy	$6$6	$11$11
Not Allergic to Dairy	$6$6	$27$27

How many students are allergic to nuts?
How many students are allergic to nuts or dairy, or both?
How many students are allergic to at most one of the two things?

Question 2

A student makes a Venn diagram of students who are late to school, and students who catch the bus to school.

Complete the following table:

$Late$ $Not late$

$Caught bus$ $\editable{}$ $\editable{}$

$Didn't catch bus$ $\editable{}$ $\editable{}$

7.07 Tables of data

Two way tables

Worked example

Two way tables and Venn diagrams

Worked example

Practice questions

Question 1

Question 2

What is Mathspace

About Mathspace

	$Late$	$Not late$
$Caught bus$	$\editable{}$	$\editable{}$
$Didn't catch bus$	$\editable{}$	$\editable{}$