S2, S4StatisticsFoundationGrades 1–5

Charts & Graphs

Charts turn a table of numbers into a picture that can be read at a glance. You need to draw and interpret bar charts, pie charts, time series graphs, stem-and-leaf diagrams and frequency polygons — and to criticise graphs that are drawn to mislead.

40 min AQA GCSE Maths

The big picture

Each chart suits a different job. Bar charts compare categories. Pie charts show how a whole is shared out. Time series show change over time. Stem-and-leaf diagrams keep every data value while showing the shape. Frequency polygons compare distributions of grouped data.

Most chart questions are really number questions in disguise: a pie chart is a proportion of $360^\circ$ , a stem-and-leaf gives an ordered list ready for the median, and a frequency polygon plots midpoints.

What you'll learn

Draw and interpret bar charts, pictograms and time series graphs
Draw and interpret pie charts
Draw and use stem-and-leaf diagrams
Draw frequency polygons and identify misleading graphs

Pie charts

The whole circle, $360^\circ$ , represents the total. Angle for a category $= \dfrac{\text{frequency}}{\text{total}} \times 360^\circ$ . Work out the angle per item first: $360 \div \text{total}$ .

To read a pie chart, reverse it: frequency $= \dfrac{\text{angle}}{360} \times \text{total}$ .

48 students: each student is

360 \div 48 = 7.5^\circ

. Football:

16 \times 7.5 = 120^\circ

Example 1In a survey of 60 people, 25 chose tea, 20 coffee and 15 juice. Find the angle for each sector.

1Each person:

360 \div 60 = 6^\circ

2Tea

150^\circ

, coffee

120^\circ

, juice

90^\circ

Answer

150^\circ

120^\circ

90^\circ

Your turnSame method, new numbers — try it before moving on.

A pie chart shows how 90 people travel to work. The "bus" sector has an angle of $84^\circ$ . How many people travel by bus?

Your answer

Show worked solution

Solution

1Each person: $360 \div 90 = 4^\circ$ .
2 $84 \div 4 = 21$ people.

Answer: $21$

Tip — Check your angles add up to $360^\circ$ before drawing.

Stem-and-leaf diagrams

Split each value into a stem (e.g. tens) and a leaf (e.g. units). Leaves are written in order, and a key is essential: $3 \mid 7$ means 37.

Because the data is already ordered, the median, mode and range can be read off directly.

Example 1Find the median and range from this stem-and-leaf diagram.

\begin{array}{r|l} 1 & 2\ 5\ 8 \\ 2 & 0\ 3\ 3\ 7 \\ 3 & 1\ 4 \end{array}

Key:

2 \mid 3 = 23

1There are 9 values; the median is the 5th: 12, 15, 18, 20, then 23.

2Range:

34 - 12 = 22

Answer

Median 23, range 22

Your turnSame method, new numbers — try it before moving on.

Using the same diagram, what is the mode?

Your answer

Show worked solution

Solution

1The leaf 3 appears twice on stem 2.
2Mode $= 23$ .

Answer: $23$

Time series and frequency polygons

A $\textbf{time series}$ plots values against time, joined with straight lines. Describe the trend (overall rise or fall) and any seasonal pattern.

A $\textbf{frequency polygon}$ plots frequency against the $\textbf{midpoint}$ of each class and joins the points with straight lines. Two polygons on one grid make comparing distributions easy.

A frequency polygon for classes 0–10, 10–20, …, 40–50: points are plotted at the midpoints 5, 15, 25, 35, 45.

Misleading graphs

Look for: a vertical axis that does not start at zero, uneven scales, missing labels, 3D effects that exaggerate size, and pictograms where the symbol size changes instead of the number of symbols.

Example 1A bar chart of sales has its vertical axis starting at 900. The bars for 2024 and 2025 are 1 cm and 3 cm tall, representing 950 and 1050 sales. Explain why it is misleading.

1The 2025 bar looks three times as tall.

2But sales only rose from 950 to 1050 — about 10.5%.

Answer

The axis does not start at zero, so the increase looks far larger than it is.

Context matters

Truncated axes are not always dishonest — they can reveal small but important changes. What matters is that the axis is clearly labelled, so readers can judge the real size of the difference.

Think like an examiner

Pie chart angles are marked within $\pm 2^\circ$ — use a protractor carefully and label each sector.
Stem-and-leaf diagrams need a key and ordered leaves for full marks.
When criticising a graph, give a specific reason referring to the diagram, not "it is misleading".

Chart calculations

\text{angle} = \dfrac{f}{\text{total}} \times 360^\circ

Pie chart sectors.

f = \dfrac{\text{angle}}{360} \times \text{total}

Reading a pie chart.

\text{plot at midpoints}

Frequency polygons.

Watch out for these

Pie chart angle

=

frequency (e.g. drawing

16^\circ

for 16 people).

Multiply by the angle per person:

16 \times 7.5 = 120^\circ

Frequency polygon plotted at the class ends.

Plot at the class midpoints.

Stem-and-leaf diagram without a key.

Always include a key such as

2 \mid 3 = 23

Comparing pie charts of different totals by angle alone.

A bigger angle does not mean more people if the totals differ.

Stretch yourself

Two pie charts show the favourite fruit of pupils in two schools. In school A (360 pupils) apples have a $90^\circ$ sector. In school B (240 pupils) apples have a $120^\circ$ sector. Which school has more pupils who prefer apples?

Hint — Convert each angle to a number of pupils.

Reveal solution

SolutionSchool A:

\frac{90}{360} \times 360 = 90

pupils. School B:

\frac{120}{360} \times 240 = 80

pupils. School A has more, even though its sector is smaller — pie charts show proportions, not amounts.

\blacksquare

Questions students ask

When should I use a pie chart rather than a bar chart?

Use a pie chart to show how a total is divided into parts. Use a bar chart to compare the sizes of categories, especially across different groups.

What is a dual bar chart?

A bar chart with two bars per category (for example, boys and girls), side by side, so two groups can be compared.

Can frequency polygons be used for discrete data?

They are mainly used for grouped continuous data. For discrete data, a vertical line chart or bar chart is usually clearer.

Key takeaways

Pie charts: angle per item $= 360 \div$ total.
Stem-and-leaf: ordered leaves and a key; read off median and range.
Time series: describe the trend and seasonal pattern.
Frequency polygons: plot frequency at class midpoints.
Misleading graphs: check axes, scales, labels and symbol sizes.

Exam-style questions

Practise it the way it is examined

2 original questions · 5 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 23 marks

72 people were asked their favourite season: spring 14, summer 30, autumn 10, winter 18.

Work out the angle for each season on a pie chart.[3]
Show mark scheme
- M1 $360 \div 72 = 5^\circ$ per person
- A1Spring $70^\circ$ , summer $150^\circ$
- A1Autumn $50^\circ$ , winter $90^\circ$

Question 2Grade 42 marks

The ages of 15 people at a party: $\begin{array}{r|l} 1 & 4\ 6\ 9 \\ 2 & 1\ 2\ 2\ 5\ 8 \\ 3 & 0\ 3\ 5\ 7 \\ 4 & 1\ 5\ 6 \end{array}$ Key: $2 \mid 5 = 25$ years

(a)Find the median age.[1]
Show mark scheme (a)
- B1 $28$ (the 8th value)
Answer: $28$
(b)Find the range.[1]
Show mark scheme (b)
- B1 $46 - 14 = 32$
Answer: $32$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

A pie chart represents 120 people. Angle per person?
- $3^\circ$
- $120^\circ$
- $0.33^\circ$
- $36^\circ$
Show answer and explanation
Answer: $3^\circ$
$360 \div 120$ .
Key $4 \mid 7 = 47$ . What does $5 \mid 2$ mean?
- $52$
- $5.2$
- $25$
- $7$
Show answer and explanation
Answer: $52$
Stem 5, leaf 2.
A frequency polygon plots points at…
- class midpoints
- class upper limits
- class lower limits
- the mean
Show answer and explanation
Answer: class midpoints
Midpoints represent each class.
30 of 180 people chose red. Angle for red?
- $60^\circ$
- $30^\circ$
- $90^\circ$
- $16.7^\circ$
Show answer and explanation
Answer: $60^\circ$
$\frac{30}{180} \times 360$ .
A $45^\circ$ sector in a pie chart of 400 people represents…
- 50 people
- 45 people
- 90 people
- 8 people
Show answer and explanation
Answer: 50 people
$\frac{45}{360} \times 400$ .
Which chart best shows temperature change across a year?
- time series graph
- pie chart
- stem-and-leaf
- pictogram
Show answer and explanation
Answer: time series graph
Change over time.
A graph's vertical axis starts at 80 instead of 0. This…
- exaggerates differences
- hides differences
- makes it more accurate
- has no effect
Show answer and explanation
Answer: exaggerates differences
Small changes look large.
Stem-and-leaf with 20 values: the median is…
- halfway between the 10th and 11th
- the 10th value
- the 11th value
- the middle stem
Show answer and explanation
Answer: halfway between the 10th and 11th
Position $\frac{20 + 1}{2} = 10.5$ .

Next step

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