A circle graph is different from a bar chart or line plot, because it does not show the count or frequency of each category. Instead it shows the proportion of the data that is in a category as parts of a whole.

A circle graph is sometimes called a pie chart because each sector could be a piece of pie.

A pie chart showing the data of fish colors in a tank. Ask your teacher for more information.

A circle graph:

Is a circle broken into pieces called sectors
Has a key or labels to show what each sector represents
Has a title that tells you what the graph is about
Will have sectors that are proportional to the percentage of the data that is in each category. For example, half of the circle is red, so 50\% of the fish in the tank would be red.

Fish colors in tank
	Fraction of total	Percentage
Orange	\dfrac{1}{8}	12.5\%
Red	\dfrac{1}{2}	50\%
Blue	\dfrac{1}{4}	25\%
Yellow	\dfrac{1}{8}	12.5\%

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

The sum of the percentages should always be 100\%, because they represent parts of a whole.

We may not always be able determine the percentages just by looking at it.

We often label the percentages on each sector, so that we can compare more easily and do calculations. For example:

A pie chart showing the data on fruits purchased from the grocery. Ask your teacher for more information.

Sometimes, we will show the number labels instead of percentages, for example, if 300 people were surveyed, this circle graph shows the same information as the one before:

A circle graph about fruits purchased from the grocery. The following are the sectors and the corresponding labels: Apples 81, oranges 39, grapes 120, limes 24 and blueberries 36.

It is important that we always check that the percentages on the graph add up to 100\% since a circle graph always represents the whole of the data points.

Circle graphs are not helpful for representing data with large numbers of categories because they get hard to read with too many sectors.

We can use circle graphs in the "Organize and Represent" stage of the data cycle. They can be helpful for questions that ask about a relationship of the parts of a whole.

Circle graphs can show us the probability of the events they represent. Recall that \text{Probability of an event}=\dfrac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

In a circle graph the favorable outcomes are represented by sectors of the graph and the total outcomes are the entire circle. So the percentage of the circle that the sector(s) makes up is the probability of that event occurring. Probabilities of an event can be described as:

Impossible - if no sectors represent that event
Unlikely - if the sector(s) for that event make up much less than half of circle
Equally likely - if the sector(s) for the event make up half of the circle
Likely - if the sector(s) for the event make up much more than half of the circle
Certain - if the sector(s) for the event make up the entire circle

A circle graph showing the reasons for being at the arena. The following are the sectors and percentages: Working 5%, playing ringette 10%, playing hockey 15%, skating lessons 20% and watching as a spectator 50%.

For example, this circle graph represents the reasons why people are at the arena. If we randomly select one person from the arena, the probability is:

Impossible that they are there for soccer
Unlikely that they are working
Equally likely that they are a spectator
Likely that they are not playing ringette
Certain that they are in the arena

Examples

Example 1

For each of the following questions for the data cycle, determine if the data can be well represented using a circle graph. If yes, explain why. If not, suggest a different type of display and explain why you chose it.

How much time do students spend on homework, in hours?

Worked Solution

Create a strategy

If we are comparing parts of a whole then a circle graph would be a good choice. We don't to use a circle graph if there are a large number of categories or possible answers. Decide if this question would lead to seeing how big one part is compared to the others.

Apply the idea

No, a circle graph isn't the best choice. There are too many possible answers and no need to compare parts of a whole.

A bar graph would be a good choice as it can have more categories for the different possible answers.

Reflect and check

A line graph shows changes over time, so total amount of time in a day would not be displayed well in a line graph.

What was the most and least popular kind of candy that was sold at the school fair?

Worked Solution

Apply the idea

Yes, we can use a circle graph. This is about the kinds of candy sold. We can use pieces of the circle graph to show how much of each candy was sold.

Reflect and check

If we are just looking at totals, a bar graph could work too. We can compare how many of each kind are sold to the total using a circle graph.

Which Neapolitan ice cream flavors (chocolate, vanilla, or strawberry) do most students like?

Worked Solution

Apply the idea

Yes, a circle graph would appropriately represent the data. We're seeing which ice cream flavors are most liked. The circle graph can show which flavor most students pick.

Reflect and check

A stem-and-leaf plot would not work well because this is a question based on categories.

Example 2

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

A circle graph about favorite foods: pizza, noodles, burger and nuggets. Ask you teacher for more information.

Which was the most popular food?

Burgers

Pizza

Nuggets

Noodles

Worked Solution

Create a strategy

Choose the largest section in the graph.

Apply the idea

The most popular food was pizza, because it largest section in the graph. So the correct answer is option B.

Which two foods were equally popular?

Burgers

Pizza

Nuggets

Noodles

Worked Solution

Create a strategy

Choose the food with the same size section.

Apply the idea

Noodles and nuggets have the same size section, so nuggets are as popular as noodles. So the correct answers is options C and D.

Example 3

Every student in 6\text{th} grade was surveyed on their favorite subject, and the results are displayed in this circle graph:

A circle graph about favorite subject. There are 6 subjects:Languages, physical education, history, math, science and english. Ask your teacher for more information

Which was the most popular subject?

Phys. Ed

Math

History

Languages

Science

English

Worked Solution

Create a strategy

Check which subject matches the largest sector.

Apply the idea

Based on the circle graph, English has the largest sector. So the correct answer is option F.

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

50\%

75\%

3\%

25\%

Worked Solution

Create a strategy

Together these subjects take up about one quarter of the circle.

Apply the idea

We know that one quarter is 25\% in percentage. So the correct answer is option D.

You later find out that there are 200 students in 6th grade. Approximately how many students selected Science as their favorite subject.

Worked Solution

Create a strategy

The Science sector takes up about a quarter of the circle, so will be about \dfrac{1}{4} of the students in 6th grade.

Apply the idea

\displaystyle \text{Number of students who chose Science}	\displaystyle =	\displaystyle \dfrac{1}{4} \text{ of the total number of students}
	\displaystyle =	\displaystyle \dfrac{1}{4}\cdot 200
	\displaystyle =	\displaystyle \dfrac{200}{4}
	\displaystyle =	\displaystyle 50

There are 50 students who chose Science.

Reflect and check

We can check this using the half of a half is a quarter. Half of 200 is 100, and half of that is 50.

Idea summary

Circle graphs represent the data as parts of a whole. Each sector of a circle graph represents a different category. The larger the sector, the larger the percentage of data in that category.

A circle graph should include:

A title to explain what the graph is about
A key to explain how to read the graph
Percents or number labels for each category

Circle graphs are good for representing categorical or countable numerical data with only a few categories.

Create circle graphs

Circle graphs can be created by hand or using technology. Some programs like Excel or Google Sheets refer to circle graphs as pie charts.

Using technology, we can:

Enter the data as a list or a frequency table
Highlight the data
Insert a chart and select pie chart or circle graph

Season	Number of students
\text{Winter}	5
\text{Spring}	15
\text{Summer}	30
\text{Fall}	10

Suppose we formulate the question "What proportion of students at my school prefer each season?" and collect this data from a sample of 60 students.

Season	Number of students	Fraction	Percent
\text{Winter}	5	\dfrac{5}{60}=\dfrac{1}{12}	8.\overline{3}\%
\text{Spring}	15	\dfrac{15}{60}=\dfrac{1}{4}	25\%
\text{Summer}	30	\dfrac{30}{60}=\dfrac{1}{2}	50\%
\text{Fall}	10	\dfrac{10}{60}=\dfrac{1}{6}	16.\overline{6}\%

We can convert each category to a fraction or a percentage.

Then we can divide up the circle by first cutting it in half, then splitting one half into two quarters, then splitting one quarter into twelfth.

A circle graph divided into two equal parts, half is colored yellow and labeled 'Summer'.

One half of the circle represents summer

A circle graph. Half is labeled 'summer' and a quarter is labeled 'spring'.

One quarter of the circle represents spring

A circle graph. Half is labeled 'summer', a quarter is labeled 'spring' and the remaining quarter is divided into 3 - of which 2/3 is labeled 'fall'.

We can split the remain quarter into three pieces that will each represent \dfrac{1}{12}

A circle graph. Half is labeled 'summer', a quarter is labeled 'spring' and the remaining quarter is divided into 3 - of which 2/3 is labeled 'fall' and 1/3 is labeled 'winter'.

\,\,\,\,\,\,\,\,\,\dfrac{1}{12} represents winter and \dfrac{2}{12}=\dfrac{1}{6} represents fall

Examples

Example 4

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

Manufacturer	Responses
\text{Brand A}	2051
\text{Brand B}	967
\text{Brand C }	531
\text{Other}	451

Which circle graph most accurately represents this data?

The circle graph shows the results of a survey on smartphone market share. Ask your teacher for more information.

Worked Solution

Create a strategy

Determine the percentage of market share for each manufacturer by dividing the number of responses for each manufacturer by the total number of responses and multiply the decimal by 100.

Apply the idea

Let's start by looking at the market share of Brand A. \dfrac{2051}{4000} \cdot 100 \approx 51\%

51\% is more than half of 4000, so the sector corresponding to Brand A should take up more than half the circle.

For Brand B, we have \dfrac{967}{4000} \cdot 100 \approx 24\% which is a little under a quarter of a circle.

For Brand C, \dfrac{531}{4000} \cdot 100 \approx 13\%For others we have \dfrac{451}{4000} \cdot 100 \approx 11\%So the the sector for Brand C and the others should be about the same size.

Here is the circle graph that most accurately represents the data would look like.

So the correct answer is option C.

Reflect and check

We could create the circle graph using technology to check.

Write a conclusion that the marketing company could make based on the data and representation.

Worked Solution

Create a strategy

Circle graphs show the proportion of each category, so the conclusion will be about which brands are the most or least popular.

Apply the idea

Brand A has the majority of the market share with more than 50\% of those surveyed using Brand A.

Brand B has about a quarter of the market share, so a similar number of people use Brand B as all other brands, not including Brand A.

Reflect and check

We are assuming that this sample is representative of the population. If the sample was not well selected, then this conclusion would only be valid for the sample and not the population.

Idea summary

To create a circle graph, we can use technology, or create it by hand. To make it by hand, we can:

Create a table with the categories, their count, fraction, and percentage
Draw a circle
Divide the circle into segments that match the proportions for each category
Check that the proportions add up to 1 or 100\%

Outcomes

6.PS.1

The student will apply the data cycle (formulate questions; collect or acquire data; organize and represent data; and analyze data and communicate results) with a focus on circle graphs.

6.PS.1a

Formulate questions that require the collection or acquisition of data with a focus on circle graphs.

6.PS.1b

Determine the data needed to answer a formulated question and collect the data (or acquire existing data) using various methods (e.g., observations, measurement, surveys, experiments).

6.PS.1d

Organize and represent data using circle graphs, with and without the use of technology tools. The number of data values should be limited to allow for comparisons that have denominators of 12 or less or those that are factors of 100 (e.g., in a class of 20 students, 7 choose apples as a favorite fruit, so the comparison is 7 out of 20, 7/20, or 35%).

6.PS.1e

Analyze data represented in a circle graph by making observations and drawing conclusions.

9.02 Create and interpret circle graphs

Ideas

Circle graphs

Exploration

Examples

Example 1

Example 2

Example 3

Create circle graphs

Examples

Example 4

Outcomes

6.PS.1

6.PS.1a

6.PS.1b

6.PS.1d

6.PS.1e

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