Interpreting Graphs
Every question with a fully worked solution.
- Interpreting Graphs - Video - Reading bar graph examples | Measurement and data Watch
Theory
A graph is a picture of data. To interpret it, read the title, the labels and the scale, then read off the values you need. This Year 7 maths page shows column, picture, line, sector and divided bar graphs, trends, and how a graph can mislead.
Types of data. Numerical data are numbers that are counted or measured (goals scored, heights). Categorical data are groups or names (pets, colours, ways to travel).
Parts of a graph. The title says what the graph shows. Each axis has a label, with units. The scale shows the values. Some graphs also have a key or a source.
Reading a value. Work out the scale first: each grid line \(=\) (gap between two labels) \(\div\) (number of spaces). Then go to the top of the column or to the line, and read across to the scale.
Parts of a whole. In a sector graph, amount \(= \dfrac{\text{sector angle}}{360^\circ} \times \text{total}\). In a divided bar graph, amount \(=\) percentage \(\times\) total. To draw a sector graph, angle \(= \dfrac{\text{amount}}{\text{total}} \times 360^\circ\).
Picture graphs. Read the key first. Half a symbol is half the key value. To show an amount, divide it by the key value.
Trends. A trend is the overall pattern. If a steady trend continues, you can use it to make a prediction.
Which graph to use:
| Graph | Best for showing |
|---|---|
| column graph | comparing groups |
| picture graph | counts, using a key |
| line graph | change over time |
| sector graph | parts of a whole (angles) |
| divided bar graph | parts of a whole (%) |
| dot plot, stem-and-leaf plot | every value of numerical data |
How to read a graph
- Read the title and the axis labels (with units).
- Work out the scale: what each grid line is worth.
- Read each value: up to the column or line, then across.
- Answer the question, then check it makes sense.
How to spot a misleading graph
- Check that the vertical axis starts at \(0\).
- Check that the scale goes up in equal steps.
- Compare the actual numbers, not the sizes of columns or sectors.
(i) Which region has the most koalas?
(ii) Which region has the fewest?
(iii) Estimate the total number of koalas.
| \(32 + 36 + 78 + 180\) | \(=\) | \(326\) |
(i) Region D (about \(180\,000\))
(ii) Region A (about \(32\,000\))
(iii) About \(326\,000\) koalas
(i) on Wednesday
(ii) altogether?
The labels go up by \(10\) with one grid line between them, so each grid line is \(5\).
| \(25 + 15 + 35 + 20 + 30\) | \(=\) | \(125\) |
(i) \(35\) books
(ii) \(125\) books
(i) How tall was it after \(2\) hours?
(ii) How much shorter does it get each hour?
(iii) If the trend continues, when will it be all gone?
| \(24 - 21\) | \(=\) | \(3\) |
| \(24 \div 3\) | \(=\) | \(8\) |
(i) \(18\) cm
(ii) \(3\) cm each hour
(iii) After \(8\) hours (red line)
(i) the beach
(ii) the zoo?
| \(\dfrac{90}{360} \times 120\) | \(=\) | \(30\) |
| \(\dfrac{150}{360} \times 120\) | \(=\) | \(50\) |
(i) \(30\) students
(ii) \(50\) students. Check: \(150 + 90 + 60 + 60 = 360^\circ\).
| \(\dfrac{9}{36} \times 360\) | \(=\) | \(90\) |
\(90^\circ\) (one quarter of the circle)
(i) How much more did 7B raise?
(ii) 7B’s column looks how many times as tall as 7A’s?
(iii) Why is the graph misleading?
| \(92 - 84\) | \(=\) | \(8\) |
| \((92 - 80) \div (84 - 80)\) | \(=\) | \(3\) |
(i) \(\$8\)
(ii) \(3\) times
(iii) The vertical axis starts at \(80\), not \(0\), so a small difference looks huge.
Common pitfalls
Frequently asked questions
What is the difference between numerical and categorical data?
Numerical data are numbers that are counted or measured; categorical data are names or groups.
How do you read a sector graph?
Amount \(=\) sector angle \(\div 360^\circ \times\) total. A \(90^\circ\) sector of \(120\) students is \(30\) students.
How do you work out the scale on a graph?
Divide the gap between two labels by the number of spaces between them.
What is a trend?
The overall pattern in a graph, such as a steady rise or fall over time.
How can a graph be misleading?
The vertical axis may not start at 0, or the scale may be uneven, so differences look bigger than they are.