Organising and Displaying Data
Organising and Displaying Data in NSW Year 8 maths (NESA outcome MA4‑DAT‑C‑01) starts with a frequency table, using tallies and class intervals of equal width that do not overlap, then turns the table into a graph.
You will draw and read sector graphs, with each angle found from \(\frac{\text{frequency}}{\text{total}} \times 360^\circ\), divided bar graphs, dot plots and stem-and-leaf plots, find a modal class and choose the best graph for a set of data. You will also spot a misleading picture graph and give every display a title, labels, a scale and a source. Rainfall, travel to school and household water use are typical examples. It is part of Year 8 Investigating Data.
Theory
Organise raw data in a frequency table first, then choose a display that suits it. In a sector graph, each angle is \(\dfrac{\text{frequency}}{\text{total}} \times 360^\circ\): \(10\) white jelly beans out of \(40\) take \(\dfrac{10}{40} \times 360^\circ = 90^\circ\). This Year 8 maths page shows how.
A frequency table lists each value or category once, with a tally and a frequency. The frequencies add to the number of pieces of data, which is a useful check.
Class intervals. When there are many different numbers, group them into classes of equal width, such as \(0\)–\(9\), \(10\)–\(19\), \(20\)–\(29\). Classes must not overlap, so every value lands in exactly one class. For measured data, \(2\)–\({<}3\) means from \(2\) up to but not including \(3\). The modal class has the largest frequency.
Sector graph (pie chart): sector angle \(= \dfrac{\text{frequency}}{\text{total}} \times 360^\circ\), and the angles add to \(360^\circ\). For percentages, angle \(=\) percentage \(\div 100 \times 360^\circ\). Read it backwards to get a count or a percentage from an angle.
Divided bar graph. One bar stands for the whole. A part’s length is \(\dfrac{\text{frequency}}{\text{total}} \times\) the bar length (or angle \(\div 360^\circ \times\) bar length).
Column graph and histogram. A column graph shows categories, with gaps between the columns. A histogram shows grouped numbers, and its columns touch because the classes join.
Dot plot and stem-and-leaf plot. A dot plot shows one dot per value; the total of the data is each value times its number of dots, added. A stem-and-leaf plot splits each value into a stem and a leaf, with the leaves in order and a key such as \(4 \mid 3 = 4.3\) kg.
Label it. Every display needs a title, axis labels with units, a scale in equal steps, and a key for a stem-and-leaf plot or a picture graph.
Which display?
| Display | Best for |
|---|---|
| Column graph | categories (with gaps) |
| Sector graph | parts of a whole |
| Divided bar graph | parts of a whole |
| Dot plot | small sets of numbers |
| Stem-and-leaf plot | numbers, keeping every value |
| Histogram | grouped numbers (no gaps) |
| Line graph | change over time |
Organising data
- Tally each value or category in a frequency table (group into classes if needed).
- Count the tallies and check the frequencies add to the total.
Sector graphs and divided bars
- Find each fraction: frequency \(\div\) total.
- Multiply by \(360^\circ\) for an angle, or by the bar length for a bar.
- Check the angles add to \(360^\circ\) (or the lengths to the bar).
- Label every part and give the graph a title.
There are \(10 + 6 + 8 + 12 + 4 = 40\) jelly beans. As decimals: white \(0.25\), black \(0.15\), green \(0.2\), blue \(0.3\) and red \(0.1\) of the bar.
Blue occurs most often (\(30\%\) of the bar).
| \(\text{White}\) | \(=\) | \(0.25 \times 360^\circ = 90^\circ\) |
| \(\text{Black}\) | \(=\) | \(0.15 \times 360^\circ = 54^\circ\) |
| \(\text{Green}\) | \(=\) | \(0.20 \times 360^\circ = 72^\circ\) |
| \(\text{Blue}\) | \(=\) | \(0.30 \times 360^\circ = 108^\circ\) |
| \(\text{Red}\) | \(=\) | \(0.10 \times 360^\circ = 36^\circ\) |
The angles add to \(360^\circ\). Red occurs least.
Tally each answer in turn, then count the tallies.
| \(\text{Beach}\) | \(=\) | \(6\) |
| \(\text{Home}\) | \(=\) | \(5\) |
| \(\text{Camping}\) | \(=\) | \(3\) |
| \(\text{Family}\) | \(=\) | \(2\) |
| \(6 + 5 + 3 + 2\) | \(=\) | \(16\) |
The frequencies add to \(16\). The most common answer is beach.
(i) Find the angle of the music sector.
(ii) On a divided bar graph \(12\) cm long, how long is the coding part?
| \(\dfrac{13}{40} \times 360^\circ\) | \(=\) | \(13 \times 9^\circ = 117^\circ\) |
| \(\dfrac{9}{40} \times 12\) | \(=\) | \(9 \times 0.3 = 2.7\) |
(i) \(117^\circ\). All four angles: \(117 + 99 + 81 + 63 = 360\).
(ii) \(2.7\) cm
(i) Name the modal class.
(ii) How many sent \(20\) or more?
| \(0\text{ to }9\) | \(\) | \(4,\ 7,\ 9,\ 2 \to 4\) |
| \(10\text{ to }19\) | \(\) | \(13,\ 15,\ 17,\ 14,\ 19 \to 5\) |
| \(20\text{ to }29\) | \(\) | \(23,\ 26 \to 2\) |
| \(30\text{ to }39\) | \(\) | \(31 \to 1\) |
The frequencies add to \(12\).
(i) \(10\)–\(19\)
(ii) \(2 + 1 = 3\) students
(i) Find the U17 angle.
(ii) How many members does the club have?
(iii) What percentage are U15?
| \(108 + 72 + 96\) | \(=\) | \(276\) |
| \(360 - 276\) | \(=\) | \(84\) |
| \(108 \div 27\) | \(=\) | \(4^\circ \text{ per member}\) |
| \(360 \div 4\) | \(=\) | \(90\) |
| \(\dfrac{72}{360} \times 100\%\) | \(=\) | \(20\%\) |
(i) \(84^\circ\)
(ii) \(90\) members
(iii) \(20\%\)
The tens digit is the stem and the units digit is the leaf. Put the leaves of each stem in order.
| Stem | Leaf |
|---|---|
| 2 | 3 6 8 |
| 3 | 0 5 9 |
| 4 | 1 4 |
Key: \(2 \mid 3 = 23\). There are \(8\) leaves, one for each score.
Common pitfalls
Frequently asked questions
How do you work out the angle for a sector graph?
Divide the frequency by the total and multiply by \(360^\circ\): \(15\) out of \(60\) is \(\dfrac{15}{60} \times 360^\circ = 90^\circ\).
What is the difference between a column graph and a histogram?
A column graph shows categories and has gaps; a histogram shows grouped numerical data and its columns touch.
What is a class interval?
A group of values of equal width, such as 10 to 19, used when there are many different numbers.
What is the modal class?
The class interval with the largest frequency.
What should every graph have?
A title, labelled axes with units, an even scale, and a key where one is needed.