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Maths · Interpreting and representing data

Statistical diagrams 1

Before data can tell you anything, it has to be sorted. Frequency tables and two-way tables organise it; bar charts and pie charts show it at a glance - as long as you choose the right one and read it carefully.

  • 6 key terms
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Before We Start

Answer each one, then check.

  1. 1

    How many degrees are there in a full turn?

    Show answerHide answer

    \(360^\circ\)

  2. 2

    What is 25% of 360?

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    \(90\)

  3. 3

    Write \(\frac{1}{4}\) as a percentage.

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    25%

  4. 4

    What is the mode of 3, 5, 5, 7?

    Show answerHide answer

    5 - the most common value.

  5. 5

    Work out \(360 \div 60\).

    Show answerHide answer

    \(6\)

Learning Objectives

  1. 1Tell the difference between types of data.
  2. 2Use frequency tables and two-way tables.
  3. 3Draw and read bar charts, including dual and composite bar charts.
  4. 4Work out the angles for a pie chart and draw it.
  5. 5Read information from a pie chart.

Types of Data

  • Qualitative (categorical)

    Described in words: favourite colour, type of pet.

  • Quantitative

    Numbers: height, number of siblings.

  • Discrete

    Can only take certain values, usually counted: number of goals, shoe size.

  • Continuous

    Can take any value in a range, usually measured: height, time, mass.

  • Primary data

    Collected by you, for example with a questionnaire.

  • Secondary data

    Collected by someone else, for example from a website.

Frequency Tables

A frequency table counts how often each value or category appears.

  • Tally

    Make a mark for each item, grouping them in fives with a line through four: it is easy to count.

  • Frequency

    The total count for each row.

  • Check the total

    The frequencies should add up to the number of items in the data.

  • Grouped data

    When there are many different values, group them into classes such as \(10 < t \le 20\) - "more than 10, up to and including 20".

Two-Way Tables

A two-way table shows two sets of information about the same group at once.

  • Rows and columns

    One piece of information across the top, the other down the side.

  • Totals

    The last row and column hold totals. Every row and every column adds up to its total.

  • Filling in gaps

    Use the totals: if you know a total and all but one of its parts, subtract.

  • Reading

    Read along a row and down a column to the cell where they meet.

A Two-Way Table

90 Year 10 students chose one school trip.

  • Boys

    Zoo: 23. Museum: 8. Total: 14. 45

  • Girls

    Zoo: 17. Museum: 15. Total: 13. 45

  • Total

    Zoo: 40. Museum: 23. Total: 27. 90

Completing a Two-Way Table

Using only these facts, find how many boys chose the museum: 90 students; 45 are boys; 40 chose the theme park, 23 of them boys; 15 girls chose the zoo; 23 students chose the zoo in total.

Show the solutionHide the solution
  1. 1 Boys who chose the zoo \(23 - 15 = 8\)
  2. 2 Boys who chose the museum = boys − theme park − zoo \(45 - 23 - 8\)
  3. 3 Work it out \(14\)

Answer14 boys chose the museum

Dual or Composite Bar Chart?

Dual bar chart

  • Two bars side by side for each category.
  • One bar for each group, e.g. boys and girls.
  • Best for comparing the groups within each category.
  • Needs a key.

Composite bar chart

  • One bar for each category, split into parts.
  • Each part shows one group, stacked on top.
  • Best for seeing the total and how it is made up.
  • Needs a key.

Drawing a Pie Chart

The whole circle, \(360^\circ\), stands for the whole of the data.

  • Degrees per item

    Divide 360 by the total frequency.

  • Each angle

    Multiply each frequency by the degrees per item.

  • Check

    The angles must add up to \(360^\circ\).

  • Draw

    Draw a radius, measure each angle with a protractor from the last line, and label every sector.

Working Out the Angles

60 students were asked their favourite sport. \(360 \div 60 = 6\), so each student is worth \(6^\circ\).

  • Football

    Frequency: 21. Angle: \(21 \times 6 = 126^\circ\)

  • Tennis

    Frequency: 9. Angle: \(9 \times 6 = 54^\circ\)

  • Swimming

    Frequency: 18. Angle: \(18 \times 6 = 108^\circ\)

  • Other

    Frequency: 12. Angle: \(12 \times 6 = 72^\circ\)

  • Total

    Frequency: 60. Angle: \(360^\circ\)

Reading a Pie Chart

A pie chart shows how 720 people travel to work. The sector for "cycle" has an angle of \(75^\circ\). How many people cycle to work?

Show the solutionHide the solution
  1. 1 The sector is a fraction of the whole circle \(\dfrac{75}{360}\)
  2. 2 Multiply by the total \(\dfrac{75}{360} \times 720\)
  3. 3 Simplify: \(720 \div 360 = 2\) \(75 \times 2\)

Answer150 people

Class Survey

Ask everyone in the class one question with four or five possible answers (for example, "How did you get to school today?"). Record the answers in a two-way table split by boys and girls. Then draw a pie chart of all the answers and a dual bar chart comparing boys and girls.

1. Collect and tally the answers.

2. Fill in the two-way table and its totals.

3. Work out the angles, then draw the pie chart.

4. Draw the dual bar chart.

A good answer shows: A completed two-way table with correct totals, a pie chart whose angles add to \(360^\circ\), and a labelled dual bar chart with a key.

Can I...?

  1. 1Say whether data is qualitative or quantitative.
  2. 2Say whether data is discrete or continuous.
  3. 3Complete a frequency table.
  4. 4Complete and read a two-way table.
  5. 5Draw and read dual and composite bar charts.
  6. 6Work out the angles for a pie chart.
  7. 7Draw a pie chart.
  8. 8Read information from a pie chart.

Summary & Exam Focus

  • Data is qualitative or quantitative; quantitative data is discrete or continuous.
  • Two-way tables: every row and column adds up to its total.
  • Pie chart angle = frequency \(\times\) (360 \(\div\) total). The angles add to \(360^\circ\).
  • Number in a sector = \(\dfrac{\text{angle}}{360} \times \text{total}\).

Exam focus

A pie chart shows how 900 people travel to work. The angle for bus is \(80^\circ\). How many people travel by bus? (2 marks) (2 marks)

For any pie chart question, first work out what \(1^\circ\) (or 1 person) is worth. Everything else follows from that one number.

Key terms

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

Frequency
How many times a value or category appears.
Discrete data
Data that can only take particular values, usually counted.
Continuous data
Data that can take any value in a range, usually measured.
Two-way table
A table showing two sets of information about the same group.
Composite bar chart
A bar chart with each bar split into parts for different groups.
Sector
A "slice" of a pie chart.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 1 mark

    Is the number of people in a car discrete data or continuous data?

    Show answerHide answer

    Model answer

    Discrete - it is counted, and can only be a whole number.

    Mark scheme

    • Discrete — B1
  2. Question 2 Non-calculator 3 marks

    80 students each study one language: French, Spanish or German. 42 of the students are girls. 17 boys study French. 21 students study Spanish, and 12 of them are girls. How many boys study German?

    Show answerHide answer

    Model answer

    Boys: \(80 - 42 = 38\). Boys studying Spanish: \(21 - 12 = 9\). Boys studying German: \(38 - 17 - 9 = 12\).

    Mark scheme

    • 38 boys, or a two-way table started with at least two correct values — P1
    • 9 boys study Spanish — P1
    • 12 — A1
  3. Question 3 Non-calculator 3 marks

    40 people were asked which pet they own. 14 own a dog, 11 own a cat, 5 own a fish and 10 own no pet. Work out the angle for each sector of a pie chart.

    Show answerHide answer

    Model answer

    \(360 \div 40 = 9^\circ\) per person. Dog \(14 \times 9 = 126^\circ\), cat \(11 \times 9 = 99^\circ\), fish \(5 \times 9 = 45^\circ\), none \(10 \times 9 = 90^\circ\).

    Mark scheme

    • \(360 \div 40 = 9\), or one correct angle — M1
    • At least two correct angles — A1
    • All four correct: \(126^\circ\), \(99^\circ\), \(45^\circ\), \(90^\circ\) — A1
  4. Question 4 Calculator 2 marks

    The pie chart shows how 900 people travel to work. The angle for bus is \(80^\circ\). How many people travel to work by bus?

    A pie chart of how 900 people travel to work: car, bus, train and walk, with the bus sector marked 80 degrees.
    Show answerHide answer

    Model answer

    \(\dfrac{80}{360} \times 900 = 200\) people

    Mark scheme

    • \(\dfrac{80}{360} \times 900\), or \(900 \div 360 = 2.5\) — M1
    • 200 — A1

Quick check

  1. Which of these is continuous data?

    1. ANumber of brothers
    2. BShoe size
    3. CTime taken to run 100 m
    4. DFavourite colour
    Show answerHide answer

    C: Time taken to run 100 m

    Time is measured and can take any value in a range.

  2. 36 people are shown on a pie chart. What angle represents one person?

    1. A\(36^\circ\)
    2. B\(10^\circ\)
    3. C\(1^\circ\)
    4. D\(12^\circ\)
    Show answerHide answer

    B: \(10^\circ\)

    \(360 \div 36 = 10\), so each person is \(10^\circ\).

  3. A pie chart shows 240 people. One sector has an angle of \(45^\circ\). How many people is that?

    1. A30
    2. B45
    3. C60
    4. D24
    Show answerHide answer

    A: 30

    \(\dfrac{45}{360} = \dfrac{1}{8}\), and \(240 \div 8 = 30\).

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