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

These notes are a short recall sheet for OxfordAQA International GCSE Mathematics (9260), based on Version 5.1 of the specification (exams from May/June 2018), section 3.4.1, reference S4. The study guide has the full explanations. Extension papers can also assess Core content. Items marked Extension only are for Extension candidates alone.

Put it into practice with the practice set. For the wider course, see the checklist, the diagnostics and the course hub.

Chart choice at a glance

Diagram	Use it for	Key rule	Tier
Tally chart	counting raw data	groups of five, then a frequency column	Core
Pictogram	simple comparisons	a key; half symbols allowed	Core
Bar chart	discrete or categorical data	equal bar widths, gaps between bars	Core
Dual bar chart	comparing two data sets	bars side by side, with a key	Core
Composite bar chart	totals and their parts	segments stacked from the baseline	Core
Pie chart	share of a total	angle = (frequency ÷ total) × 360°	Core
Line graph	change over time	plot points, join with straight lines	Core
Stem-and-leaf	small data sets	ordered leaves, a key	Core
Scatter graph	paired data	one point per pair	Core
Frequency polygon	grouped continuous data	plot at class midpoints	Core
Histogram, equal widths	grouped continuous data	no gaps, height = frequency	Core
Histogram, unequal widths	grouped continuous data	height = frequency density	Extension only
Cumulative frequency	median, quartiles, counts	plot at upper boundaries	Extension only
Box plot	five-number summary	box from lower to upper quartile	Extension only

Formulas

Result	Formula
Pie chart angle	(frequency ÷ total) × 360°
Frequency from a sector	(angle ÷ 360°) × total
Class midpoint	(lower boundary + upper boundary) ÷ 2
Frequency density (Extension only)	frequency ÷ class width
Frequency from a bar (Extension only)	frequency density × class width
Interquartile range (Extension only)	upper quartile − lower quartile

Methods in steps

Pie chart

1. Total all the frequencies.
2. Divide 360° by that total to get the angle for a single item.
3. Scale each frequency by this angle.
4. Add the angles to confirm 360°, then draw and label the sectors.

Histogram with unequal intervals (Extension only)

1. Subtract to get each class width.
2. Frequency ÷ width gives the height of the bar.
3. Name the vertical axis “frequency density”.
4. Draw touching bars as wide as their classes.
5. Going backwards, height × width recovers the frequency.

Cumulative frequency diagram (Extension only)

1. Add a running total column to the table.
2. Place each total above the upper end of its class, and add the point (start, 0).
3. Draw one smooth curve through the points.
4. Read across from n ÷ 2 for the median, and from n ÷ 4 and 3n ÷ 4 for the quartiles.

Box plot (Extension only)

1. List the smallest value, three quartile values (lower, median, upper) and the largest value.
2. Draw one scale and mark all five.
3. Box the two quartiles and cut the box at the median.
4. Extend whiskers out to the smallest and largest values.

Worked reminders

Pie chart. 45 pupils choose a club: art 9, drama 14, music 22. One pupil is 360° ÷ 45 = 8°. Angles: 72°, 112°, 176°. Check: 72 + 112 + 176 = 360°.

Unequal histogram (Extension only).

Class x	0 < x ≤ 5	5 < x ≤ 10	10 < x ≤ 20	20 < x ≤ 40
Frequency	15	30	44	60
Width	5	5	10	20
Frequency density	3	6	4.4	3

The total frequency is 149.

Quartile positions (Extension only). For 60 values the curve is read at 30 for the median, 15 for the lower quartile and 45 for the upper quartile.

Cumulative frequency table (Extension only). Scores x for 60 pupils.

Class x	0 < x ≤ 20	20 < x ≤ 40	40 < x ≤ 60	60 < x ≤ 80	80 < x ≤ 100
Frequency	3	12	30	9	6
Cumulative frequency	3	15	45	54	60

The points are (20, 3), (40, 15), (60, 45), (80, 54), (100, 60), plus (0, 0). The curve reads 40 at height 15, 50 at height 30 and 60 at height 45. So the quartiles are 40 and 60, the median is 50 and the interquartile range is 20.

Composite bar chart. A cafe sells 18 teas, 12 coffees and 9 juices in one hour. The bar has a teas segment from 0 to 18, a coffees segment from 18 to 30 and a juices segment from 30 to 39.

Frequency polygon. For classes 0 < x ≤ 4, 4 < x ≤ 8 and 8 < x ≤ 12 the midpoints are 2, 6 and 10. Plot each frequency above its midpoint and join the points with straight lines.

Comparing box plots (Extension only). Delivery times in minutes.

Shop	Minimum	Lower quartile	Median	Upper quartile	Maximum
A	4	10	15	22	35
B	6	12	14	17	24

Shop B has the lower median (14 against 15) and the smaller interquartile range (5 against 12). Its deliveries are slightly quicker and much more consistent.

Reading and describing a diagram

• Start with the title, axis labels, units and key. Only then take values.
• For a line graph, say what happens, when it happens and by how much. A rise of 6 °C between 10:00 and 12:00 is a fuller answer than “it goes up”.
• For a scatter graph, describe the direction of the points, any point that sits apart from the rest, and what the pattern would mean in context.
• For a pie chart, remember that equal angles mean equal frequencies. A pie chart gives shares, so you cannot tell the totals of two pie charts apart unless the question says so.
• When a value is read between two plotted points, call it an estimate.

Must-know distinctions

• Bar chart or histogram. A bar chart has gaps and handles categories or discrete values. A histogram has no gaps and a continuous scale.
• Frequency polygon or line graph. A polygon plots frequency at class midpoints. A line graph plots a value against time.
• Dual or composite bar chart. Dual bars sit beside each other. Composite bars stack to show a total.
• Frequency or frequency density (Extension only). Frequency is the count. Frequency density is the bar height, and area stands for frequency.
• Histogram or cumulative frequency (Extension only). A histogram shows how many are in each class. A cumulative frequency diagram shows how many are up to each boundary.

Quick self-test

1. In a pictogram one symbol stands for 6 books. How many books do 4.5 symbols show?
2. In a survey of 48 pupils, 30 chose football. What sector angle shows this on a pie chart?
3. A sector of 40° represents a survey of 135 people. How many people does it stand for?
4. (Extension only) A class 10 < x ≤ 25 has frequency 54. Find the frequency density.
5. (Extension only) A histogram bar is 12 wide and 2.5 high. What frequency does it show?
6. State the midpoint of the class 15 < x ≤ 25 for a frequency polygon.
7. (Extension only) At which part of each class do you plot a cumulative frequency point?
8. (Extension only) A cumulative frequency curve is drawn for 84 values. At what heights do you read the median and the two quartiles?
9. (Extension only) A lower quartile is 14 and an upper quartile is 31. Find the interquartile range.
10. A composite bar has a bottom segment of 15 and a middle segment of 11. At what height does the middle segment end?
11. A line graph shows 14 at 3 pm and 20 at 5 pm. Estimate the value at 4 pm from the joining line.
12. A stem-and-leaf key says 3 | 8 means 3.8 kg. What does 6 | 1 mean?

Answers

1. 4.5 × 6 = 27 books.
2. (30 ÷ 48) × 360° = 225°.
3. (40 ÷ 360) × 135 = 15 people.
4. 54 ÷ 15 = 3.6.
5. 2.5 × 12 = 30.
6. (15 + 25) ÷ 2 = 20.
7. The upper class boundary.
8. The median at 42, the lower quartile at 21 and the upper quartile at 63.
9. 31 − 14 = 17.
10. 15 + 11 = 26.
11. Halfway between 14 and 20, so 17.
12. 6.1 kg.

Where marks are usually lost

• Leaving out the key on a pictogram, a dual bar chart or a stem-and-leaf diagram.
• Pie chart angles that sum to something other than 360°, from an early division slip.
• Plotting a frequency polygon at the class edges instead of the midpoints.
• Using frequency as the bar height for unequal classes. Show the division by width.
• Plotting cumulative frequency against the lower or middle value of a class.
• Reading the median from the horizontal axis at its halfway mark instead of at n ÷ 2 on the vertical axis.
• Writing the answer without showing the reading lines or the working. Drawn lines help if the reading is off.
• Comparing box plots by describing the shape only. Quote a median and a spread, with the figures.
• Putting axis scales on a graph that do not use most of the grid, or have uneven steps.

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.