18.07 Pie charts
If you've ever seen a poll or a popularity survey, you might be familiar with graphs that look something like these:
In terms of representing data in a visually appealing and digestible manner, three of the most common tools are bar charts, histograms and pie charts.
Unlike the dot plot and stem-and-leaf plot, these graphs focus more on representing the relation between different results visually while worrying less about displaying the exact values of the survey. It is for this reason that these charts are often used to represent large data sets.
Pie charts and the whole
What makes a pie chart so different is that it represents the data as parts of a whole. In a pie chart, 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 pie chart below:
We can see from the pie chart (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 colour. 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 colour.
| Colour 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 pie chart represents $100%$100% of the data, one whole, split up into different category sectors.
Pie charts with additional information
A notable drawback of the pie chart is that it doesn't necessarily tell us how many data points belong to each category. This means that, without any additional information, the pie chart 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 pie charts 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 pie chart:
- 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.
Worked example
example 1
Consider the pie chart below:
If there are $48$48 fish in total, how many of them are either blue or yellow?
Think: We found in the exploration above that $25%$25% of the fish are blue and $12.5%$12.5% are yellow. Together this represents $37.5%$37.5% of the $48$48 fish.
Do: We can find the number of blue or yellow fish by multiplying the total number of fish by the percentage taken up by these two colours.
| Blue or yellow fish | $=$= | $48\times37.5%$48×37.5% |
| $=$= | $48\times\frac{3}{8}$48×38 | |
| $=$= | $18$18 |
As shown, $18$18 fish are either blue or yellow.
Reflect: By relating the sizes of sectors to fractions or percentages, we can calculate the number of data points belonging to a category by multiplying that fraction (or percentage) by the total number of data points.
Revealing the total number of data points is useful for calculating the value represented by each sector, but this is only if we can interpret the exact size of each sector from the pie chart.
In the case where it is not so obvious what percentage of the pie chart each sector represents, we can instead add information by explicitly stating how many data points each sector represents. This can be written either on the sectors or the legend, as shown below.
Worked example
example 2
Consider the pie chart below:
Show that the sector representing basketball takes up $43%$43% of the pie chart.
Think: To show that the basketball sector takes up $43%$43% of the pie chart, 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 pie chart 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 pie chart.
Reflect: We can calculate the exact percentage of the pie chart that different sectors take up by finding their number of data points as a percentage of the total.
Aside from these two ways to add extra information to a pie chart, there is also the case where the percentage taken up by each sector is shown on the pie chart.
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 pie chart 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 pie chart is valid.
Practice question
Question 1
Every student in year $8$8 was surveyed on their favourite subject, and the results are displayed in this pie chart:
A pie chart entitled "Favourite subject" and is divided into six sectors, each representing a different school subject. The colors of the sectors are red, green, yellow, dark blue, orange and light blue, arranged in a clockwise manner. Next to the chart, there is a legend that matches the colors to the food items: red for English, green for Science, yellow for Maths, dark blue for History, orange for Phys. Ed. and light blue for Languages.
Which was the most popular subject?
Phys. Ed.
AMaths
BHistory
CLanguages
DScience
EEnglish
FWhat percentage of the class selected History, Phys. Ed., or Languages?
$50%$50%
A$30%$30%
B$3%$3%
C$25%$25%
DYou later find out that $32$32 students selected Science. How many students are there in year $8$8?