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Pie Charts, Bar Charts and Stem-and-Leaf Diagrams

Drawing and reading pie charts, bar charts and stem-and-leaf diagrams, and spotting misleading charts.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Read, draw and interpret pie charts, including finding angles and frequencies.
  2. 2Read bar charts and compare data, including dual bar charts.
  3. 3Construct and interpret stem-and-leaf diagrams, and find the median, mode and range from one.
  4. 4Spot misleading features in a chart.

Showing data clearly

A good chart lets the reader see a pattern at a glance, and examiners test whether you can both read a chart accurately and decide what it shows. Pie charts are about angles and fractions, bar charts are about heights, and stem-and-leaf diagrams keep every value while sorting them into order. All three need careful reading of the key and scale, and all the numbers are chosen to be easy: a pie chart of 60 people has 6 degrees for each person, and 360 divides neatly by 36, 40, 60, 72 and 120.

Pie charts

A pie chart shows how a whole is shared out, with the angle of each sector proportional to its frequency.

  • Angle for a sector

    \(\dfrac{\text{frequency}}{\text{total}} \times 360^\circ\).

  • Degrees for each item

    Work out \(\dfrac{360}{\text{total}}\) first. For 60 students, each is \(6^\circ\).

  • Frequency from an angle

    \(\dfrac{\text{angle}}{360} \times \text{total}\). In a pie chart of 36 people, \(100^\circ\) is \(\dfrac{100}{360} \times 36 = 10\) people.

  • Check

    The angles add up to \(360^\circ\), and so do the frequencies for the total.

Drawing a pie chart

40 students choose their favourite fruit: apple 16, banana 12, pear 8, grape 4. Work out the angle for each sector.

Show the solutionHide the solution
  1. 1 Degrees per student \(\dfrac{360}{40} = 9^\circ\).
  2. 2 Multiply Apple \(16 \times 9 = 144\), banana \(12 \times 9 = 108\), pear \(8 \times 9 = 72\), grape \(4 \times 9 = 36\).
  3. 3 Check \(144 + 108 + 72 + 36 = 360\).
  4. 4 Draw Use a protractor and label each sector.

AnswerApple \(144^\circ\), banana \(108^\circ\), pear \(72^\circ\), grape \(36^\circ\)

Bar charts

A bar chart compares categories, with the height of each bar showing the frequency.

  • Gaps between bars

    For data in categories, such as colours or sports.

  • Dual bar charts

    Two sets of bars side by side to compare groups, such as boys and girls.

  • Read the scale

    Check what each gridline is worth, as some scales go up in 2s or 5s.

  • Compare totals

    To compare groups of different sizes, use fractions or percentages, not just heights.

Stem-and-leaf diagrams

A stem-and-leaf diagram keeps the exact data. The stem is the first digit or digits, and each leaf is the last digit.

  • A key

    It says what a value means, such as \(2 \mid 3\) means 23. A diagram without a key loses a mark.

  • Ordered

    The leaves go in order, smallest nearest the stem.

  • Median

    Count to the middle value, or the average of the middle two.

  • Back to back

    Two sets of data can share the same stem, with leaves going left and right.

Finding the median

A stem-and-leaf diagram shows the masses of 13 parcels in kilograms. Stem 1 has leaves 3 and 7. Stem 2 has leaves 0, 4, 4, 5 and 8. Stem 3 has leaves 1, 3, 6 and 9. Stem 4 has leaves 2 and 5. The key says that stem 2 with leaf 4 means 24 kg. Find the median mass.

Show the solutionHide the solution
  1. 1 Count the values \(2 + 5 + 4 + 2 = 13\).
  2. 2 Find the position The median is the \(\dfrac{13 + 1}{2} = 7\)th value.
  3. 3 Count along the rows Stem 1 gives the 1st and 2nd values, and stem 2 gives the 3rd to 7th values: 20, 24, 24, 25, 28.
  4. 4 Read the 7th value The 7th value is 28, so the median is 28 kg.

Answer28 kg

Misleading charts

Examiners like to ask what is wrong with a chart.

  • A scale that does not start at zero

    It makes small differences look large.

  • Uneven scales or bars

    Bars with different widths, or scales that jump, are misleading.

  • A 3D effect

    It distorts the sizes of bars and pie sectors.

  • Missing labels or a missing key

    The reader cannot tell what the chart shows.

Test yourself

  1. 1

    How do you find the angle for a sector?

    Show answerHide answer

    Frequency divided by total, times 360.

  2. 2

    For 40 people, how many degrees represent each person?

    Show answerHide answer

    \(\dfrac{360}{40} = 9^\circ\).

  3. 3

    What must every stem-and-leaf diagram have?

    Show answerHide answer

    A key.

  4. 4

    How do you find the median from a stem-and-leaf diagram with 11 values?

    Show answerHide answer

    The 6th value.

  5. 5

    Why might a bar chart be misleading?

    Show answerHide answer

    The scale may not start at zero, or the bars may be uneven.

Exam technique: charts

Be exact with angles and keys.

  • Measure angles accurately

    Within 2 degrees is usually accepted.

  • Show your working

    Write \(\dfrac{360}{40} = 9\) before multiplying.

  • Write the key

    Stem and leaf diagrams must show it.

  • Say "compare" correctly

    Use both averages and spread, and give figures.

Summary and exam focus

  • A sector's angle is \(\dfrac{\text{frequency}}{\text{total}} \times 360^\circ\), and a frequency is \(\dfrac{\text{angle}}{360} \times \text{total}\).
  • Bar charts compare categories, and a dual bar chart compares two groups.
  • A stem-and-leaf diagram has ordered leaves and a key, and gives the median, mode and range.
  • Check scales and labels for misleading features.

Exam focus

A pie chart shows the favourite drinks of 40 pupils. The angle for tea is \(135^\circ\). How many pupils chose tea? (2 marks) (2 marks)

Each pupil is \(\dfrac{360}{40} = 9^\circ\), so tea is \(\dfrac{135}{9} = 15\) pupils. Alternatively, \(\dfrac{135}{360} \times 40 = 15\). Either way, show the method.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Pie chart
A circular chart where the angle of each sector represents its share of the total.
Sector
A slice of a pie chart.
Bar chart
A chart that uses bars of different heights to show frequencies.
Dual bar chart
A bar chart with two sets of bars side by side.
Stem-and-leaf diagram
A diagram that shows data by splitting each value into a stem and a leaf.
Stem
The first digit or digits of each value in a stem-and-leaf diagram.
Leaf
The last digit of each value in a stem-and-leaf diagram.
Key
A note that explains what the values in a chart or diagram mean.
Frequency
The number of times a value or category occurs.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Work out 3 marks Easier

Some children are asked their favourite colour. 9 choose blue, 6 choose red, 5 choose green and 4 choose yellow. The results are shown in a pie chart. Work out the angle for each colour. [3 marks]

Mark scheme — 3 marks available

  • \(24\) children and \(360 \div 24 = 15\) — M1
  • At least two angles correct — A1
  • \(135^\circ, 90^\circ, 75^\circ, 60^\circ\) — A1

Model answer

There are 24 children, so each child is \(\dfrac{360}{24} = 15^\circ\). Blue: \(9 \times 15 = 135^\circ\). Red: \(6 \times 15 = 90^\circ\). Green: \(5 \times 15 = 75^\circ\). Yellow: \(4 \times 15 = 60^\circ\).

2. Exam question Work out 5 marks Core

The pie chart shows the favourite subjects of 40 pupils. The angle for dance is not shown. (a) How many pupils chose music? [2 marks] (b) Work out the angle for dance. [1 mark] (c) How many pupils chose dance? [1 mark] (d) What fraction of the pupils chose art? Give your answer in its simplest form. [1 mark]

A pie chart of the favourite subject of 40 pupils with angles of 135 degrees for music, 90 degrees for art, 81 degrees for drama, and dance unlabelled.

Mark scheme — 5 marks available

  • (a) \(\dfrac{135}{360} \times 40\) or \(135 \div 9\) — M1
  • (a) 15 — A1
  • (b) \(54^\circ\) — B1
  • (c) 6 — B1
  • (d) \(\dfrac{1}{4}\) — B1

Model answer

(a) Each pupil is \(\dfrac{360}{40} = 9^\circ\), so \(\dfrac{135}{9} = 15\) pupils. (b) \(360 - 135 - 90 - 81 = 54^\circ\). (c) \(\dfrac{54}{9} = 6\). (d) \(\dfrac{90}{360} = \dfrac{1}{4}\).

3. Exam question Work out 4 marks Core

The stem-and-leaf diagram shows the masses of 13 parcels. (a) Write down the mode. [1 mark] (b) Work out the median. [2 marks] (c) Work out the range. [1 mark]

A stem-and-leaf diagram of the masses of 13 parcels with a key where 4 bar 2 means 42 kg.

Mark scheme — 4 marks available

  • (a) 42 — B1
  • (b) The 7th value identified — M1
  • (b) 49 — A1
  • (c) 30 — B1

Model answer

(a) 42 occurs twice. (b) There are 13 values, so the median is the 7th value, which is 49 kg. (c) \(64 - 34 = 30\) kg.

4. Exam question Explain 2 marks Core

Give a reason why each of these charts may be misleading. (a) A pie chart with a 3D effect. [1 mark] (b) A bar chart where the vertical scale starts at 50 instead of 0. [1 mark]

Mark scheme — 2 marks available

  • (a) Distorts the sizes of the sectors — B1
  • (b) Exaggerates the differences between the bars — B1

Model answer

(a) The 3D effect distorts the sizes of the sectors, so they are hard to compare. (b) A small difference between the bars looks much bigger than it really is.

5. Exam question Work out 3 marks Core

The stem-and-leaf diagram shows the heights of some plants. The stem 4 has leaves 1, 3 and 7. The stem 5 has leaves 0, 2, 2 and 8. The stem 6 has leaves 3 and 5. The key says that 4 bar 1 means 41 cm. Work out (a) the mode, (b) the median, (c) the range. [3 marks]

Mark scheme — 3 marks available

  • (a) 52 — B1
  • (b) 52 — B1
  • (c) 24 — B1

Model answer

(a) 52 occurs twice. (b) There are 9 values, so the median is the 5th value, 52 cm. (c) \(65 - 41 = 24\) cm.

6. Exam question Work out 3 marks Stretch

A pie chart shows the favourite pets of 72 people. The angle for cats is \(160^\circ\) and the angle for dogs is \(120^\circ\). (a) Work out the number of people who chose cats. [2 marks] (b) How many more people chose cats than dogs? [1 mark]

Mark scheme — 3 marks available

  • (a) \(360 \div 72 = 5\) or \(\dfrac{160}{360} \times 72\) — M1
  • (a) 32 — A1
  • (b) 8 — B1

Model answer

(a) Each person is \(\dfrac{360}{72} = 5^\circ\), so cats is \(\dfrac{160}{5} = 32\). (b) Dogs is \(\dfrac{120}{5} = 24\), so \(32 - 24 = 8\) more chose cats.

7. Multiple choice 1 mark Easier

60 students choose a sport. 24 choose football. What is the angle for football on a pie chart?

  1. A \(24^\circ\)
  2. B \(120^\circ\)
  3. C \(144^\circ\) Correct
  4. D \(60^\circ\)

Why: Each student is \(\dfrac{360}{60} = 6^\circ\), so \(24 \times 6 = 144^\circ\).

8. Multiple choice 1 mark Easier

A pie chart shows 40 people. How many degrees represent each person?

  1. A \(40^\circ\)
  2. B \(9^\circ\) Correct
  3. C \(4^\circ\)
  4. D \(90^\circ\)

Why: \(\dfrac{360}{40} = 9\).

9. Multiple choice 1 mark Core

A pie chart shows 36 people. A sector is \(100^\circ\). How many people does it represent?

  1. A \(10\) Correct
  2. B \(100\)
  3. C \(36\)
  4. D \(3.6\)

Why: \(\dfrac{100}{360} \times 36 = 10\).

10. Multiple choice 1 mark Easier

What fraction of a pie chart is a \(90^\circ\) sector?

  1. A \(\dfrac{1}{3}\)
  2. B \(\dfrac{1}{2}\)
  3. C \(\dfrac{9}{10}\)
  4. D \(\dfrac{1}{4}\) Correct

Why: \(\dfrac{90}{360} = \dfrac{1}{4}\).

11. Multiple choice 1 mark Easier

What must every stem-and-leaf diagram have?

  1. A A title only
  2. B Gaps between the rows
  3. C A key Correct
  4. D Leaves in a random order

Why: Without a key the values cannot be read.

12. Multiple choice 1 mark Core

A stem-and-leaf diagram has 14 ordered values. The 7th is 26 and the 8th is 29. What is the median?

  1. A \(26\)
  2. B \(27.5\) Correct
  3. C \(29\)
  4. D \(28\)

Why: \(\dfrac{26 + 29}{2} = 27.5\).

13. Multiple choice 1 mark Easier

The values in a stem-and-leaf diagram run from 12 to 45. What is the range?

  1. A \(33\) Correct
  2. B \(57\)
  3. C \(28.5\)
  4. D \(45\)

Why: \(45 - 12 = 33\).

14. Multiple choice 1 mark Core

Which of these makes a bar chart misleading?

  1. A The bars have labels
  2. B The chart has a title
  3. C The bars are the same width
  4. D The vertical scale does not start at zero Correct

Why: A scale that does not start at zero exaggerates differences.

15. Multiple choice 1 mark Stretch

On a pie chart for 60 people, a sector is \(54^\circ\). What percentage is that?

  1. A \(54\%\)
  2. B \(9\%\)
  3. C \(15\%\) Correct
  4. D \(30\%\)

Why: \(\dfrac{54}{360} = 0.15\).