OxfordAQA International GCSE Mathematics: Statistical charts and diagrams (9260)

This guide is written for OxfordAQA International GCSE Mathematics (9260) and follows Version 5.1 of the specification, covering exams from May/June 2018. It covers reference S4 in section 3.4.1, Presentation and analysis, and teaches how to produce and read each chart the specification names. Extension papers can also assess the Core content. Three items are Extension only: histograms with unequal class intervals, cumulative frequency diagrams and box plots.

Pair this guide with the short revision notes and a set of practice questions. To see what else the course needs, try the checklist, the diagnostics or the course hub.

What this unit covers

Spec reference	You must be able to produce and read	Tier
S4	tally charts, pictograms, bar charts, dual bar charts, composite bar charts	Core
S4	pie charts, line graphs, stem-and-leaf diagrams, scatter graphs	Core
S4	frequency polygons, histograms with equal class intervals	Core
S4	histograms with unequal class intervals	Extension only
S4	cumulative frequency diagrams	Extension only
S4	box plots	Extension only

1. Tally charts and frequency tables

A tally chart counts raw data one value at a time. Strike through four strokes with a fifth to make a group of five. Then convert each tally to a number to give the frequency.

Worked example. Twenty-four pupils each report how many messages they sent before school: 3, 1, 0, 2, 4, 1, 2, 2, 3, 0, 1, 5, 2, 3, 1, 2, 4, 2, 0, 3, 1, 2, 5, 2.

Messages  Tally      Frequency
0         |||        3
1         ||||/      5
2         ||||/ |||  8
3         ||||       4
4         ||         2
5         ||         2
                     total 24

Check that the frequencies total the number of values you started with. Here 3 + 5 + 8 + 4 + 2 + 2 = 24.

2. Pictograms

A pictogram uses symbols, and the key says what one symbol stands for. Divide by the key to find how many symbols to draw. Multiply to read a value. Use half symbols when the key allows.

Worked example. Key: one symbol = 8 loaves. A bakery sells 36 loaves on Monday and 20 on Tuesday. Monday: 36 ÷ 8 = 4.5, so four whole symbols and one half. Tuesday: 20 ÷ 8 = 2.5, so two whole symbols and one half. Reading back: a Thursday row with 4 symbols shows 4 × 8 = 32 loaves.

3. Bar charts: dual and composite

In a bar chart the bars have equal widths with equal gaps, and the height shows the frequency. Label both axes.

A dual bar chart pairs two bars at each category, which lets you compare two data sets directly. Add a key to say which bar is which.

A composite bar chart stacks the parts of each total in one bar. Each segment starts where the one below ends. It shows the total well but makes it harder to compare the upper segments.

Worked example. A club has these members.

Year	Boys	Girls
9	12	8
10	10	14
11	6	9

Dual chart: two bars per year, of heights 12 and 8 for Year 9, and so on. Composite chart: Year 9 has a boys segment from 0 to 12 and a girls segment from 12 to 20. Year 10 runs 0 to 10, then 10 to 24. Year 11 runs 0 to 6, then 6 to 15.

4. Pie charts

A pie chart splits a whole into sectors. Frequencies and angles are in the same ratio.

angle = (frequency ÷ total) × 360°

Worked example. 72 pupils are asked how they travel to school: walk 18, bus 27, car 15, cycle 12. One pupil is 360° ÷ 72 = 5°. Walk 18 × 5 = 90°. Bus 27 × 5 = 135°. Car 15 × 5 = 75°. Cycle 12 × 5 = 60°. Check: 90 + 135 + 75 + 60 = 360°.

Reading a pie chart works backwards. A sector of 42° in a survey of 240 households stands for (42 ÷ 360) × 240 = 28 households.

5. Line graphs

A line graph plots a quantity against time, with points joined by straight lines. Use it to show a trend.

Worked example. Greenhouse temperatures (°C) were 12, 15, 19, 25, 27, 22, 16 at 2-hour intervals from 06:00 to 18:00. The changes between readings are +3, +4, +6, +2, −5, −6. The largest rise is 6 °C, between 10:00 and 12:00. The largest fall is 6 °C, between 16:00 and 18:00. Halfway between the 12:00 and 14:00 readings the line gives (25 + 27) ÷ 2 = 26 °C. This is only an estimate, because nothing was measured at 13:00.

6. Stem-and-leaf diagrams

The stem holds the leading digits and each leaf is the last digit. Order the leaves from smallest to largest, keep the columns aligned and always give a key.

Worked example. Quiz scores out of 50: 23, 31, 18, 42, 35, 27, 31, 44, 19, 38, 25, 31, 47, 29, 36, 22, 41, 33, 28.

1 | 8 9
2 | 2 3 5 7 8 9
3 | 1 1 1 3 5 6 8
4 | 1 2 4 7
Key: 2 | 5 means 25

The diagram holds 19 values. The range is 47 − 18 = 29. Three scores of 31 make it the most common. Six scores are over 35 (36, 38, 41, 42, 44, 47). The middle value is the 10th, which is 31.

7. Scatter graphs

A scatter graph plots paired data, one point per pair. Put the variable you control or that comes first on the horizontal axis. Choose scales that spread the points over most of the grid.

Worked example. Hours of revision against test score for eight pupils: (1, 34), (2, 41), (3, 45), (4, 58), (5, 60), (6, 71), (7, 69), (8, 80). Horizontal axis 0 to 8 hours, one square per hour. Vertical axis 30 to 90 marks, steps of 10. Mark each point with a small cross. The points climb from left to right, so more revision goes with higher scores. A ninth pupil, (6, 35), lies far below the other points. Say so, and say whether you would check that point.

8. Frequency polygons and histograms with equal class intervals

For grouped continuous data, a frequency polygon plots each frequency against the class midpoint. Join the points with straight lines.

A histogram with equal class intervals is a bar chart for continuous data. There are no gaps, the horizontal axis is a continuous scale, and the bar height is the frequency.

Worked example. Journey times of 50 employees.

Time t (minutes)	0 < t ≤ 10	10 < t ≤ 20	20 < t ≤ 30	30 < t ≤ 40	40 < t ≤ 50
Frequency	5	12	18	10	5

Midpoints are 5, 15, 25, 35, 45. The polygon points are (5, 5), (15, 12), (25, 18), (35, 10), (45, 5). The histogram has five touching bars, each 10 wide, with heights 5, 12, 18, 10 and 5.

9. Histograms with unequal class intervals (Extension only)

With classes of different widths, a bar’s area stands for frequency. Its height is therefore the frequency density.

frequency density = frequency ÷ class width, so frequency = frequency density × class width

Worked example. Ages of 205 visitors.

Age a (years)	Frequency	Width	Frequency density
0 < a ≤ 10	30	10	3
10 < a ≤ 30	70	20	3.5
30 < a ≤ 40	45	10	4.5
40 < a ≤ 70	60	30	2

Estimate how many visitors were aged 20 to 30. Assume the class is evenly spread. The bar height is 3.5, so the frequency is 3.5 × 10 = 35.

To read a bar, multiply height by width. A bar 1.6 high and 20 wide stands for 32 people.

10. Cumulative frequency diagrams (Extension only)

Cumulative frequency is a running total. Mark each total above the upper class boundary, with the curve starting at (lower end, 0), and join the points with a smooth curve.

Worked example. Minutes 80 callers waited in a phone queue.

Time t (minutes)	0 < t ≤ 10	10 < t ≤ 20	20 < t ≤ 30	30 < t ≤ 40	40 < t ≤ 50	50 < t ≤ 60
Frequency	3	12	25	20	12	8
Cumulative frequency	3	15	40	60	72	80

The points are (10, 3), (20, 15), (30, 40), (40, 60), (50, 72), (60, 80), plus (0, 0).

For the median, go up to n ÷ 2 = 40 on the vertical axis, across to the curve, then down. The curve passes through (30, 40), so the median is 30 minutes. Lower quartile: the 20th value. Between (20, 15) and (30, 40) this is about 22 minutes. Upper quartile: the 60th value is at 40 minutes. Interquartile range: 40 − 22 = 18 minutes.

To find how many waited more than 45 minutes, read the curve at 45. It is about 66, so 80 − 66 = 14 callers.

11. Box plots (Extension only)

A box plot shows five values on one scale: the minimum, lower quartile, median, upper quartile and maximum. Draw a box between the two quartiles and mark the median across it. Lines called whiskers then reach out to the smallest and largest values.

Worked example. Use the queue above, with a shortest wait of 3 minutes and a longest of 58 minutes. The five values are 3, 22, 30, 40, 58.

  |--------[---|----]--------|
0    10   20   30   40   50   60  minutes

The interquartile range is 18 minutes. To compare two box plots, quote a middle value (the median) and a spread (the interquartile range or the range). Suppose a second queue has quartiles 15 and 30 and a median of 22. Its median is lower and its interquartile range is 15, so its waits are shorter and less varied.

Common errors

• Missing a key on a pictogram, stem-and-leaf diagram or dual bar chart.
• Drawing gaps between histogram bars, or plotting a frequency polygon at class edges instead of midpoints.
• Plotting cumulative frequency at the midpoint or lower boundary instead of the upper boundary.
• Using frequency as bar height when the class widths are unequal.
• Reading the median at the 50% mark of the horizontal axis instead of the n ÷ 2 position on the vertical axis.
• Leaving leaves in the order they were collected.
• Joining scatter graph points with a line.

Official syllabus

Specification used: Version 5.1 of the OxfordAQA specification for International GCSE Mathematics (9260), which is for exams May/June 2018 onwards. Section 3.4.1, Presentation and analysis, reference S4.