18.05 Column graphs, histograms, and pie charts

A column graph always presents a categorical data set. There is one column per category, and the height of each column is the size of that category. There is usually a small gap between each column.

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.

Both chart types always label both axes to provide additional information about the data and tell us what type of values we have.

Consider the histogram below:

We can quickly see that, since the column labelled 1 is the tallest, the mode of the data is 1. We can also see that the column labelled 0 has a value of three, and since column 4 is at the same height, both 0 and 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 and 4 children; three families in each case.

If this histogram feels strangely familiar it is probably because we have already seen this graph in the previous lesson, except in that lesson it was represented as a dot plot.

The reason why these two graphs look so similar, aside from them representing the same data, is because the histogram is essentially a more complex version of the dot plot. Rather than counting dots, the histogram uses a scale to indicate the height of the columns, allowing it to represent larger data sets.

Examples

Example 1

Below is a column graph showing the type of fruit each student in a class brought in for lunch yesterday.

a

What was the most common fruit?

A

Apple

B

Orange

C

Banana

D

Pineapple

E

Mandarin

Worked Solution

Create a strategy

Check the graph which fruit has the tallest column.

Apply the idea

Mandarin has the tallest column, so the correct answer is option E.

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b

How many more mandarins were brought than pineapples?

Worked Solution

Create a strategy

Check the graph to see how many pineapples and mandarins were brought, then find the difference.

Apply the idea

The number of pineapples is 2, and number of mandarin is 8.

\displaystyle \text{Difference}	\displaystyle =	\displaystyle 8-2	Subtract the 2 from 8
	\displaystyle =	\displaystyle 6	Evaluate

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c

Complete the table below using the information from the graph.

Fruit	Number
\text{Banana}	2
\text{Apple}	⬚
\text{Mandarin}	⬚
\text{Pineapple}	⬚
\text{Orange}	⬚

Worked Solution

Create a strategy

Remember, the height of each column represents the number of each fruit that was brought to school.

Apply the idea

By checking the height of each column, we can fill in the numbers of each fruit.

Fruit	Number
\text{Banana}	2
\text{Apple}	3
\text{Mandarin}	8
\text{Pineapple}	2
\text{Orange}	4

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The scale of a column graph 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.

As we can see, the scale of a column graph or histogram is important for reading values. However, there are some cases where the scale may be misleading.

In this column graph, if we look only at the columns and ignore the scale, it certainly seems like the number of albums sold doubled from 2014 to 2015. However, if we look at the scale, we can see that the number of albums sold in 2014 and 2015 were 25 thousand and 30 thousand respectively, so the sales did not double.

The reason why the heights of the columns are so misleading is because the scale doesn't start at zero on this graph. It starts at 20 thousand, so the height of each column indicates how much greater than 20 thousand the value is.

So we should always check the scale to find actual values before making any conclusions about the data.

Examples

Example 2

The table shows the number of people who visited Disneyland between 2011 and 2015.

Year	Number of people in hundreds of thousands
2011	165
2012	164
2013	152
2014	159
2015	168

a

Represent the data in a column graph.

Worked Solution

Create a strategy

We assign the values from the table for each year to make the column graph. The Year should go on the horizontal axis and the Number of people should go on the vertical axis.

Apply the idea

Note that the vertical axis does not start at zero.

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b

A marketing executive examines the histogram and says, "We doubled the number of visitors from 2014 to 2015." Are they correct?

Worked Solution

Create a strategy

Read the values for each column from the graph.

Apply the idea

The column for 2014 goes up to 159 thousand, and the column of 2015 goes up to 168 thousand which is not double 159 thousand.

So the executive is not correct.

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Pie charts are, at first glance, completely different from column graphs and 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.

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\% of the fish are red.

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.

There is a 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\% 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\% so this pie chart is valid.

Examples

Example 3

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

a

Which was the most popular subject?

A

Phys. Ed

B

Maths

C

History

D

Languages

E

Science

F

English

Worked Solution

Create a strategy

Check which subject matches the largest sector.

Apply the idea

Based on the pie chart, English has the largest sector. So the correct answer is option F.

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b

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

A

50\%

B

30\%

C

3\%

D

25\%

Worked Solution

Create a strategy

Together these subjects take up one quarter of the circle.

Apply the idea

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

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c

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

Worked Solution

Create a strategy

Since the Science sector takes up a quarter of the circle, we can multiply the number of Science students by 4.

Apply the idea

\displaystyle \text{Total students}	\displaystyle =	\displaystyle 32 \times 4	Multiply 32 by 4
	\displaystyle =	\displaystyle 128	Evaluate

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