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5.03 Interpreting circle graphs

Lesson

If you've ever seen a poll or a popularity survey, you might be familiar with graphs that look something like this:

In terms of representing data in a visually appealing and digestible manner, a very common tool is circle graph or pie chart.

Circle graphs and the whole

What makes a circle graph so different from other types of displays is that it represents the data as parts of a whole. In a circle graph, all the data is combined to make a single whole with the different sectors representing different categories. The larger the sector, the larger percentage of the data points that category represents.

Exploration

Consider the circle graph below:

We can see from the circle graph (using the legend to check our categories) that the red sector takes up half the circle, while the blue sector takes up a quarter and the yellow and orange sectors both take up one eighth.

The fraction of the circle taken up by each sector indicates what fraction of the total fish are that color. So, in this case, half the fish are red since the red sector takes up half the circle. We can also write this as a percentage: $50%$50% of the fish are red.

If we consider how much of the circle each sector takes up, we can identify what percentage of the total fish are of each color.

Color of fish Fraction of total Percentage
Orange $\frac{1}{8}$18 $12.5%$12.5%
Red $\frac{1}{2}$12 $50%$50%
Blue $\frac{1}{4}$14 $25%$25%
Yellow $\frac{1}{8}$18 $12.5%$12.5%

 

Notice that the sum of our percentages is $100%$100%. This is consistent with the fact that a circle graph represents $100%$100% of the data, one whole, split up into different category sectors.

 

Worked Example

Question 1

Consider the circle graph below.

Circle graph with $4$4 categories

Complete: If there were $3$3 orange fish, how many of each color would there be? Fill in the table below.

Color of fish Number
Orange $3$3
Yellow  
Blue  
Red  

Think: The ratio of Orange:Yellow = $1:1$1:1, the ratio of Orange:Blue=$1:2$1:2, and the ratio of Orange:Red = $1:4$1:4. We can use our proportional reasoning to solve for the number of yellow, blue and red fish.

Do: 

Color of fish Number Reasoning
Orange $3$3 Given
Yellow $3$3 Ratio of $1:1$1:1 means the same
Blue $6$6 Ratio of $1:2$1:2 means the double
Red $12$12 Ratio of $1:4$1:4 means four times


 

Question 2

Consider the circle graph below:

Show that the sector representing basketball takes up $43%$43% of the circle graph.

Think: To show that the basketball sector takes up $43%$43% of the circle graph, we need to show that the number of basketball data points is equal to $43%$43% of the total data points.

Do: We can see from the circle graph that the basketball sector represents $86$86 data points. By adding up the data points from all the different sectors, we find that the total number of data points is:

Total number of data points $=$= $86+27+53+30+4$86+27+53+30+4
  $=$= $200$200

 

So the percentage of the total number of data points represented by basketball is:

Percentage $=$= $\frac{86}{200}\times100%$86200×100%
  $=$= $43%$43%

 

Since basketball represents $43%$43% of the data points, its sector must take up $43%$43% of the circle graph.

Reflect: We can calculate the exact percentage of the circle graph that different sectors take up by finding their number of data points as a percentage of the total.

Pie charts with additional information

A notable drawback of the circle graph is that it doesn't necessarily tell us how many data points belong to each category. This means that, without any additional information, the circle graph can only show us which categories are more or less popular and roughly by how much.

It is for this reason that we will often add some additional information to our circle graphs so that we can show (or at least calculate) the number of data points in each category. There are two main ways to add information to a circle graph:

  • Reveal the total number of data points
  • Reveal the number of data points for each sector

By revealing the total number of data points, we can use the percentages represented by the sector sizes to calculate how many data points each sector represents.

Aside from these two ways to add extra information to a circle graph, there is also the case where the percentage taken up by each sector is shown on the circle graph.

This will often look something like this:

This is very useful as it does a lot of the calculations for us. However, it is important that we always check that the percentages on the graph add up to $100%$100% since a circle graph always represents the whole of the data points, no more and no less.

In this particular case, the percentages do in fact add up to $100%$100% so this circle graph is valid.

Practice question

Question 3

The pie chart below shows the results of a class survey where students were asked to nominate their favorite food:

  1. Which was the most popular food?

    Burgers

    A

    Pizza

    B

    Nuggets

    C

    Noodles

    D

    Burgers

    A

    Pizza

    B

    Nuggets

    C

    Noodles

    D
  2. Which two foods were equally popular? Select both correct options.

    Burgers

    A

    Pizza

    B

    Nuggets

    C

    Noodles

    D

    Burgers

    A

    Pizza

    B

    Nuggets

    C

    Noodles

    D

Question 4

A marketing company conducted a survey to determine the market share of smartphone manufacturers. They surveyed $4000$4000 people, and the results are given in the table below:

Manufacturer Responses
Apple $2051$2051
Samsung $967$967
Huawei $531$531
Other $451$451
  1. Which pie chart represents this data?

    A

    B

    C

    D

    A

    B

    C

    D

Question 5

Every student in year $8$8 was surveyed on their favorite subject, and the results are displayed in this pie chart:

  1. Which was the most popular subject?

    Phys. Ed.

    A

    Math

    B

    History

    C

    Languages

    D

    Science

    E

    English

    F

    Phys. Ed.

    A

    Math

    B

    History

    C

    Languages

    D

    Science

    E

    English

    F
  2. What percentage of the class selected History, Phys. Ed., or Languages?

    $50%$50%

    A

    $30%$30%

    B

    $3%$3%

    C

    $25%$25%

    D

    $50%$50%

    A

    $30%$30%

    B

    $3%$3%

    C

    $25%$25%

    D
  3. You later find out that $32$32 students selected Science. How many students are there in year $8$8?

Outcomes

6.10b

Given a practical situation, make observations and inferences about data represented in a circle graph

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