8.02 Venn diagrams and two-way tables

Venn diagrams are a type of diagram that uses circle to group and organise things, as seen below:

A Venn diagram is a helpful tool in displaying information about two categories, especially if things can be in both categories. Some Venn diagrams will use the diagram to sort the different objects. The one shown above uses numbers in each section to show how many objects fit into that section.

We can look at various parts of the a Venn diagram to understand it more. The circle on the left represents everything that fits into that category, so it will be all the students that play hockey. It will be the sum of the two smaller regions of the circle, 5+6=11, 11 students play hockey. Similarly the full right circle will be all the gymnastics students.

Everything in the outside box, does not fit into either category.

There are 15 students who do not participate in either hockey or gymnastics.

Some Venn diagrams wont have the outside box. Depending on the context this means that they are only considering the two circles or everything they are interested could be placed in at least one of the two circles.

The very middle section highlighted here represents students that are in both categories. It represents the 6 students who play hockey and gymnastics.

The part of the circle highlighted here is all the students who play hockey but do not do gymnastics. We could refer to these as "the students who only play hockey".

Examples

Example 1

A group of students were asked about their siblings. The two categories show if they have at least one brother, and if they have at least one sister.

a

How many of the students have at least one sibling?

Worked Solution

Create a strategy

Add all the numbers inside the circles.

Apply the idea

\displaystyle \text{Number of students}	\displaystyle =	\displaystyle 6+5+9	Add the numbers
	\displaystyle =	\displaystyle 20	Evaluate

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b

How many of the students have at least one brother?

Worked Solution

Create a strategy

Add the numbers inside the "Have a brother" circle.

Apply the idea

\displaystyle \text{Number of students}	\displaystyle =	\displaystyle 5+9	Add the numbers
	\displaystyle =	\displaystyle 14	Evaluate

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c

How many of the students don't have a sister?

Worked Solution

Create a strategy

Add the numbers outside of the "Have a sister" circle.

Apply the idea

\displaystyle \text{Number of students}	\displaystyle =	\displaystyle 9+8	Add the numbers
	\displaystyle =	\displaystyle 17	Evaluate

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We can also use Venn diagrams with three circles. This uses the same concepts as a two circle Venn diagram but adds another circle which represents a third category. We can work things out in the same way, however there are more places the categories can overlap.

Here is a diagram representing three movie genres.

As before each circle represents a category, the highlighted circle below will be for all the people that like comedy movies.

There are also some new possibilities such as, people that liked all three movie genres.

And people that only like two of the three genres.

Examples

Example 2

Joanne is struggling to decide what to watch online. She decides to pick one movie at random from the streaming website. A Venn diagram of her options sorts movies into three categories based on their genre: Comedy, Action and Horror.

a

How many of the movies are horror films?

Worked Solution

Create a strategy

Add all the numbers inside the "Horror" circle.

Apply the idea

\displaystyle \text{Horror films}	\displaystyle =	\displaystyle 13 + 3 + 5 + 6	Add the numbers inside horror
	\displaystyle =	\displaystyle 27	Evaluate

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b

How many of the movies fit into only one genre?

Worked Solution

Create a strategy

Add the numbers in these parts where the movies fit only one genre:

Apply the idea

\displaystyle \text{Number of movies}	\displaystyle =	\displaystyle 10+20+13	Add the numbers
	\displaystyle =	\displaystyle 43	Evaluate

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c

How many of the movies are an action film combined with at least one other genre?

Worked Solution

Create a strategy

Add the numbers in these parts where the movies are action combined with at least one other genre:

Apply the idea

\displaystyle \text{Number of movies}	\displaystyle =	\displaystyle 4 + 5 + 6	Add the numbers
	\displaystyle =	\displaystyle 15	Evaluate

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Two-way tables represent data that is classified by two criteria.

To read a two-way table look at the column and row that a cell is in. For example there are 9 students who are left-handed and enjoy English, and 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.

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

Examples

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 will go in the overlap of the two circles. The remaining value, 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.

Examples

Example 4

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

Using the Venn diagram, complete the following table:

	Late	Not late
Caught bus
Didn't catch bus

Worked Solution

Create a strategy

We can use the Venn diagram to construct the table by looking at the corresponding regions.

Apply the idea

The categories are Late/Not late and Caught bus/Didn't catch bus.

	Late	Not late
Caught bus	13
Didn't catch bus		9

The top left cell is for the number of people who are late to school and caught the bus, that is 13 people. The 9 outside the circles are the students that did not caught the bus or not late to school, so it goes in the bottom right cell.

	Late	Not late
Caught bus	13	3
Didn't catch bus	7	9

The is the number of students that caught the bus but not late, so it goes in the top right cell. The 7 is the number of students that did not catch the bus but late, so it goes in the bottom left cell.

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