22.07 Histograms and 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 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.
What are histograms?
A histogram always presents a numerical data set. There is one column per number (or class of numbers), and the height of each column is the number of times that number appears in the set (or in the class). There is a gap equal to half a column's width between the vertical axis and the first column, and there are no gaps between columns.
Histograms always label both axes to provide additional information about the data and tell us what type of values we have.
Exploration
Consider the histogram below:
We can quickly see that, since the column labelled $1$1 is the tallest, the mode of the data is $1$1. We can also see that the column labelled $0$0 has a value of three, and since column $4$4 is at the same height, both $0$0 and $4$4 have a value of three.
The vertical axis label tells us that the values represent the "number of families" while the horizontal axis label tells us that each column represents a specific "number of children in the family".
Putting this information together, we can see that in the survey there were an equal number of families that had $0$0 and $4$4 children; three families in each case.
Practice question
Question 1
The length of time that consumers spent in a supermarket on a particular day is shown in the frequency table below.
| Time spent (mins) | Frequency |
|---|---|
| $0\le t<10$0≤t<10 | $30$30 |
| $10\le t<20$10≤t<20 | $75$75 |
| $20\le t<30$20≤t<30 | $100$100 |
| $30\le t<40$30≤t<40 | $65$65 |
| $40\le t<50$40≤t<50 | $40$40 |
| $50\le t<60$50≤t<60 | $15$15 |
| $60\le t<70$60≤t<70 | $5$5 |
Construct a histogram to represent this data.
How many shoppers visited the supermarket that day?
What percentage of the shoppers spent between $30$30 and $40$40 minutes at the supermarket?
Round your answer to two decimal places.
What percentage of the shoppers spent an hour or longer at the supermarket?
Round your answer to two decimal places.
Reading the scale on a bar chart or histogram
The scale of a bar chart or histogram (indicated by the numbers and ticks on the vertical axis of the graph) is a very useful feature for this data presentation, so we should learn how to read it.
The scale can always be read as if it were a number line, with the marked numbers indicating the value at certain heights, and the ticks between them can be used to determine the values at other heights.
Practice question
Question 2
The frequency table below shows the average time spent travelling to work for fifty two people.
| Commute time | Frequency |
|---|---|
| $0-19$0−19 minutes | $15$15 |
| $20-39$20−39 minutes | $17$17 |
| $40-59$40−59 minutes | $10$10 |
| $60-79$60−79 minutes | $6$6 |
| $80-99$80−99 minutes | $4$4 |
| Total | $52$52 |
Construct a histogram to display the data shown in the frequency table:
Which of the following statements about the data is most accurate?
The data suggests that people don't care too much about how far away from work they live. Roughly equal portions of people live less than $40$40 minutes away and more than $40$40 minutes away.
AThe data suggests that people prefer a shorter commute to work. A majority live within $40$40 minutes travel, and in general the longer the commute the less people there are in that category.
BThe data shows that everyone lives within an hours travel from their work, with the peak amount of people living between $20$20 and $40$40 minutes away.
CThe data shows that most people travel to work by car or by walking, since most travel times are fairly short, and only a few people travel by bus or train.
D
Pie charts and the whole
Pie charts are, at first glance, completely different from histograms. The main similarity is that the mode of a pie chart is clearly visible, just as it is on a histogram.
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 examples
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.
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 3
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?