Choosing Data Displays
Theory
Different data need different displays. This Year 9 maths page shows how to choose and justify a display for a given type of data, and how to draw and read sector graphs, divided bar graphs, column graphs, dot plots, stem-and-leaf plots, histograms and line graphs, including sector angles, lengths of bar sections and trends over time. Curriculum: VC2M9ST04 / AC9M9ST04.
Data are categorical (labels, such as transport or music genre) or numerical (numbers). Categorical data are nominal (no order) or ordinal (an order, such as T-shirt sizes S, M, L, XL). Numerical data are discrete (counted, such as kittens in a litter) or continuous (measured, such as height or time).
A frequency table organises raw data: tally each category or value, then count. Every display starts from one.
Categorical data: a column graph compares the counts, with gaps between the columns. A sector graph (also called a pie chart or circle graph) and a divided bar graph show how a whole is split into parts.
Numerical data: a dot plot shows each value as a dot and suits a small set of discrete data. A stem-and-leaf plot keeps every value and suits up to about \(50\) scores. A histogram suits a large set or continuous data grouped into classes; its columns touch.
Data over time: a line graph shows how a value changes, with time on the horizontal axis. Its overall direction is the trend.
To justify a choice, name the type of data and say what the display shows well: parts of a whole, every value, the shape, or change over time.
From a percentage, \(1\% = 3.6^\circ\): a \(40\%\) sector has an angle of \(40 \times 3.6^\circ = 144^\circ\). If all \(n\) categories are equal, each angle is \(360^\circ \div n\).
How to choose and read a display
- Classify the data: categorical (nominal or ordinal) or numerical (discrete or continuous). Is it recorded over time?
- Choose: column, sector or divided bar graph for categories; dot plot or stem-and-leaf plot for a small numerical set; histogram for a large or continuous set; line graph for change over time.
- Justify the choice in one sentence: the type of data and what the display shows well.
- Draw from a frequency table: sector angles out of \(360^\circ\), bar sections out of the bar length, even scales on every axis, and a title and labels.
- Read the key, labels and scale first, then answer from the display.
Each student is \(360^\circ \div 40 = 9^\circ\), so multiply each frequency by \(9^\circ\).
| Fruit | Frequency | Angle |
|---|---|---|
| apple | \(12\) | \(108^\circ\) |
| banana | \(10\) | \(90^\circ\) |
| mango | \(9\) | \(81^\circ\) |
| orange | \(6\) | \(54^\circ\) |
| grapes | \(3\) | \(27^\circ\) |
| Total | \(40\) | \(360^\circ\) |
Check that the angles add to \(360^\circ\).
(i) How many students travel by bus?
(ii) The same data are drawn as a divided bar graph \(12\) cm long. How long is the Bus section?
| \(\text{(i)} \ \dfrac{150}{360} \times 72\) | \(=\) | \(\mathbf{30} \text{ students}\) |
| \(\text{(ii)} \ \dfrac{30}{72} \times 12\) | \(=\) | \(\mathbf{5} \text{ cm}\) |
Both graphs show the same fraction of the whole: \(\dfrac{150}{360} = \dfrac{5}{12}\) of the circle and \(\dfrac{5}{12}\) of the bar.
(i) The daily top temperature in Hobart for one month.
(ii) The type of pet owned by each of \(30\) students.
(iii) The birth masses of \(120\) babies.
(iv) \(18\) quiz marks, when the teacher wants to see every mark.
(i) Numerical data recorded over time: a line graph, which shows the rises, falls and trend.
(ii) Categorical (nominal): a column graph to compare the counts, or a sector graph to show each pet’s share of the whole.
(iii) Numerical (continuous) with many values: a histogram with equal classes, which shows the shape.
(iv) Numerical (discrete), a small set: a stem-and-leaf plot or dot plot, which keeps every mark.
(i) Between which two years was the increase largest?
(ii) Describe the trend.
(iii) Find the percentage increase from 2019 to 2024.
(i) The yearly changes are \(+6\), \(+9\), \(-4\), \(+7\), \(+6\). The largest increase, \(9\) thousand, was from 2020 to 2021.
(ii) The overall trend is upward: visitor numbers rose from \(30\) to \(54\) thousand, with one dip, from 2021 to 2022.
| \(\text{(iii)} \ \dfrac{54 - 30}{30} \times 100\%\) | \(=\) | \(\mathbf{80\%}\) |
Common pitfalls
Frequently asked questions
What is the best graph for categorical data?
A column graph to compare the counts, or a sector graph or divided bar graph to show how the whole is shared between the categories.
What is the best graph for numerical data?
A dot plot or stem-and-leaf plot for a small set where every value matters, and a histogram for a large set or continuous data grouped into classes.
How do you work out the angles for a sector graph?
Divide each frequency by the total and multiply by \(360^\circ\). For \(9\) out of \(36\), the angle is \(\dfrac{9}{36} \times 360^\circ = 90^\circ\).
What is another name for a sector graph?
A pie chart or a circle graph. A divided bar graph shows the same kind of information as a rectangle instead of a circle.
When should you use a line graph?
When the data show how one quantity changes over time, such as rainfall each month or prices each year. Time goes on the horizontal axis.