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Stage 5.1-3

9.02 Venn diagrams and two-way tables

Lesson

Venn diagrams

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

A Venn diagram with 2 sets: hockey and gymnastics that overlap. Ask your teacher for more information.

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.

A Venn diagram for Hockey and gymnastics. The circle for Hockey is shaded. Ask your teacher for more information.

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.

A Venn diagram for Hockey and gymnastics. The area outside the circles is shaded. Ask your teacher for more information.

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.

A Venn diagram for 2 sets Hockey and gymnastics. The intersection is shaded. Ask your teacher for more information.

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

A Venn diagram for Hockey and gymnastics. Hockey is shaded but not the intersection. Ask your teacher for more information.

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 venn diagram showing students that have a sister or a brother or neither. Ask your teacher for more information.
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+9Add the numbers
\displaystyle =\displaystyle 20Evaluate
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+9Add the numbers
\displaystyle =\displaystyle 14Evaluate
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.

A Venn diagram with two overlapping circles. Everything outside the left circle is shaded.
Apply the idea
\displaystyle \text{Number of students}\displaystyle =\displaystyle 9+8Add the numbers
\displaystyle =\displaystyle 17Evaluate
Idea summary

Venn diagrams are a type of diagram that uses circles to group and organise things.

Three circle Venn diagram

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.

A Venn Diagram with 3 overlapping circles labeled Comedy, Action, and Horror.

Here is a diagram representing three movie genres.

A Venn diagram with 3 overlapping circles. The upper left circle is shaded.

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

A Venn diagram with 3 sets overlapping.  The shaded part is the intersection of the 3 sets.

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

A Venn diagram with 3 sets overlapping. The intersection of 2 sets is shaded. Ask your teacher for more information.

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 Venn diagram with 3 sets overlaps: comedy, action, and horror. Ask your teacher for more information.
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 + 6Add the numbers inside horror
\displaystyle =\displaystyle 27Evaluate
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:

A Venn diagram with 3 sets overlapping. All sets are shaded excluded the intersection. Ask your teacher for more information.
Apply the idea
\displaystyle \text{Number of movies}\displaystyle =\displaystyle 10+20+13Add the numbers
\displaystyle =\displaystyle 43Evaluate
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:

A Venn diagram with 3 sets overlapping. Ask your teacher for more information.
Apply the idea
\displaystyle \text{Number of movies}\displaystyle =\displaystyle 4 + 5 + 6Add the numbers
\displaystyle =\displaystyle 15Evaluate
Idea summary

Three circle Venn diagram is a diagram shows how components of three sets are related using three overlapping circles.

Two-way tables

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

RightLeft
Enjoys English 4 9
Doesn't enjoy English 2 15

The table 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.

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.

RightLeftTotal
Enjoys English4913
Doesn't enjoy English21517
Total62430

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

Example 3

This table describes the departures of trains out of a train station for the months of May and June.

Departed on timeDeparted late
May11331
June108 33
a

How many trains departed during May and June?

Worked Solution
Create a strategy

Add all the numbers in the table

Apply the idea
\displaystyle \text{Number of trains}\displaystyle =\displaystyle 113+108+31+33Add the numbers
\displaystyle =\displaystyle 285Evaluate
b

What percentage of the trains in June were delayed? Write your answer as a percentage to one decimal place.

Worked Solution
Create a strategy

Find the number of delayed trains in June as a percentage of the total trains departing in June.

Apply the idea
\displaystyle \text{June trains}\displaystyle =\displaystyle 108+33Add all the June trains
\displaystyle =\displaystyle 141Evaluate

Now we can find the 33 delayed June trains as a percentage of the total June trains:

\displaystyle \text{Percentage delayed}\displaystyle =\displaystyle \dfrac{33}{141}\times 100\%Find 33 as a percentage of 141
\displaystyle \approx\displaystyle 23.4\%Evaluate and round
Reflect and check

This percentage also tells us that if a train in June was randomly selected, the probability that it would have been delayed is 23.4\%.

c

What fraction of the total number of trains during the 2 months were ones that departed on time in May?

Worked Solution
Create a strategy

Find the number of on time trains in May as a fraction of the total trains.

Apply the idea

From part (a) we found that there are 285 total trains. We can see from the table that 113 departed on time in May.

\displaystyle \text{Fraction}\displaystyle =\displaystyle \dfrac{113}{285}Write as a fraction
Reflect and check

This fraction also tells us that the probability that a train selected at random departed on time in May is \dfrac{113}{285}.

Idea summary

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

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.

A two-way table with columns left and right and rows entered and didn't enter. Ask your teacher for more information.

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

A Venn diagram with 2 overlapping circles called left-handed and entered. Ask your teacher for more information.

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.

A Venn diagram with 2  overlapping sets late to school and caught bus. Ask your teacher for more information.

Using the Venn diagram, complete the following table:

LateNot 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.

LateNot late
Caught bus13
Didn't catch bus9

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.

LateNot late
Caught bus133
Didn't catch bus79

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.

Idea summary

We can convert between a two-way table and a Venn diagram by matching up their different parts.

Outcomes

MA5.1-13SP

calculates relative frequencies to estimate probabilities of simple and compound events

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