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Maths · Further statistics

Drawing histograms

Calculate frequency density for grouped data with unequal class widths and draw a histogram.

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  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What is the width of the class \(10 < t \le 30\)?

    Show answerHide answer

    \(20\)

  2. 2

    Work out \(30 \div 20\).

    Show answerHide answer

    \(1.5\)

  3. 3

    What is a frequency table?

    Show answerHide answer

    A table of how often values occur

  4. 4

    How does a bar chart differ from a histogram?

    Show answerHide answer

    A histogram's bar area shows frequency; classes are continuous

  5. 5

    Write \(12 \div 10\) as a decimal.

    Show answerHide answer

    \(1.2\)

Learning Objectives

  1. 1Explain why a histogram uses frequency density.
  2. 2Work out the frequency density for each class.
  3. 3Draw a histogram with unequal class widths.
  4. 4Label the axes correctly.

HISTOGRAM

In a histogram the area of each bar shows the frequency. The height is the frequency density.

\(\text{frequency density} = \dfrac{\text{frequency}}{\text{class width}}\). The bars touch and there are no gaps.

Working Out Frequency Density

Times taken (t minutes) by 94 customers.

  • \(0 < t \le 10\)

    Frequency: \(12\). Class width: \(10\). Frequency density: \(1.2\)

  • \(10 < t \le 30\)

    Frequency: \(30\). Class width: \(20\). Frequency density: \(1.5\)

  • \(30 < t \le 40\)

    Frequency: \(26\). Class width: \(10\). Frequency density: \(2.6\)

  • \(40 < t \le 60\)

    Frequency: \(18\). Class width: \(20\). Frequency density: \(0.9\)

  • \(60 < t \le 100\)

    Frequency: \(8\). Class width: \(40\). Frequency density: \(0.2\)

Drawing a Histogram

Follow the steps.

  1. 1 Work out the class widths

    Upper boundary minus lower boundary
  2. 2 Work out the frequency density

    Frequency divided by class width
  3. 3 Draw the axes

    Data on the horizontal axis, frequency density on the vertical axis
  4. 4 Draw the bars

    Bars touch; the height is the frequency density
  5. 5 Label

    Add a scale and a label for each axis

Frequency Density

A class \(20 < x \le 50\) has frequency 60. Work out the frequency density.

Show the solutionHide the solution
  1. 1 Class width \(50 - 20 = 30\)
  2. 2 Divide \(\dfrac{60}{30} = 2\)

AnswerThe frequency density is 2, so the bar has height 2.

Completing a Histogram

Age (a years): \(0 < a \le 10\): 8, \(10 < a \le 20\): 20, \(20 < a \le 40\): 36, \(40 < a \le 60\): 30. The first three bars are drawn. Work out the height of the last bar.

Show the solutionHide the solution
  1. 1 Class width \(60 - 40 = 20\)
  2. 2 Frequency density \(\dfrac{30}{20} = 1.5\)

AnswerDraw a bar from 40 to 60 with height 1.5.

Common Mistakes

Watch for these.

  • Frequency as the height

    Use frequency density.

  • Gaps between bars

    The bars must touch.

  • Widths

    Use the actual class width, not the number of classes.

  • Labels

    The vertical axis is "Frequency density", not "Frequency".

Find the Density

Work out the frequency density for each class. (a) \(0 < w \le 5\), frequency 15 (b) \(5 < w \le 15\), frequency 40 (c) \(15 < w \le 35\), frequency 30 (d) \(35 < w \le 60\), frequency 10.

1. Class width first.

2. Then divide.

A good answer shows: (a) 3 (b) 4 (c) 1.5 (d) 0.4

Can I...?

  1. 1Find the class widths.
  2. 2Work out frequency density.
  3. 3Draw and label axes.
  4. 4Draw bars that touch.
  5. 5Use unequal class widths.
  6. 6Complete a partly drawn histogram.
  7. 7Explain why frequency density is used.
  8. 8Avoid common mistakes.

Summary & Exam Focus

  • Frequency density \(= \dfrac{\text{frequency}}{\text{class width}}\).
  • Area of bar = frequency.
  • Bars touch; classes may have different widths.
  • The vertical axis is frequency density.

Exam focus

A table shows the ages of 94 people in classes with different widths. Work out the frequency density for each class and draw the histogram. (4 marks) (4 marks)

Add a frequency density column to the table first.

Key terms

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

Histogram
A chart where bar area shows frequency.
Frequency density
Frequency divided by class width.
Class width
The upper boundary minus the lower boundary.
Continuous data
Data measured, not counted.
Unequal class widths
Classes of different sizes.
Area
Height multiplied by width.

Practice questions

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

  1. Question 1 Complete 3 marks

    The table shows the times, t minutes, taken by 94 customers. Frequencies: \(0 < t \le 10\): 12, \(10 < t \le 30\): 30, \(30 < t \le 40\): 26, \(40 < t \le 60\): 18, \(60 < t \le 100\): 8. Work out the frequency density for each class.

    Show answerHide answer

    Model answer

    1.2, 1.5, 2.6, 0.9, 0.2

    Mark scheme

    • Class widths — M1
    • At least three correct — M1
    • All correct — A1
  2. Question 2 Complete the histogram 3 marks

    The table shows information about the ages, a years, of 94 people: \(0 < a \le 10\): 8, \(10 < a \le 20\): 20, \(20 < a \le 40\): 36, \(40 < a \le 60\): 30. Complete the histogram.

    A histogram of ages with three bars drawn and the last class from 40 to 60 missing.
    Show answerHide answer

    Model answer

    Frequency density \(= 30 \div 20 = 1.5\); a bar drawn from 40 to 60 of height 1.5.

    Mark scheme

    • Frequency density 1.5 — M1
    • Bar from 40 to 60 — A1
    • Height 1.5 — A1
  3. Question 3 Work out 2 marks

    A class \(5 < x \le 25\) has frequency 40. Work out the frequency density.

    Show answerHide answer

    Model answer

    \(40 \div 20 = 2\)

    Mark scheme

    • Class width 20 — M1
    • 2 — A1
  4. Question 4 Work out 2 marks

    A histogram bar for the class \(10 < x \le 14\) has a frequency density of 3. Work out the frequency of the class.

    Show answerHide answer

    Model answer

    Width \(4\); frequency \(= 4 \times 3 = 12\)

    Mark scheme

    • Uses \(4 \times 3\) — M1
    • 12 — A1
  5. Question 5 Explain 2 marks

    Explain why the height of a bar in a histogram is not the frequency when the class widths are unequal.

    Show answerHide answer

    Model answer

    The area of a bar shows the frequency, so a wider class needs a lower bar for the same frequency. The height is the frequency density.

    Mark scheme

    • Area shows frequency — M1
    • Frequency density used for the height — C1
  6. Question 6 Draw 3 marks

    Describe how to draw a histogram from a grouped frequency table with unequal class widths.

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    Model answer

    Work out the class width and then the frequency density for each class. Draw a frequency density scale on the vertical axis and the data on the horizontal axis. Draw touching bars with heights equal to the frequency densities.

    Mark scheme

    • Frequency density — B1
    • Axes correctly labelled — B1
    • Touching bars with correct heights — B1

Quick check

  1. Frequency density is...

    1. AFrequency times class width
    2. BFrequency divided by class width
    3. CClass width divided by frequency
    4. DFrequency plus class width
    Show answerHide answer

    B: Frequency divided by class width

    Frequency divided by class width.

  2. What is shown by the area of a bar in a histogram?

    1. AThe class width
    2. BThe frequency density
    3. CThe frequency
    4. DThe mean
    Show answerHide answer

    C: The frequency

    The frequency.

  3. A class of width 20 has frequency 30. Its frequency density is...

    1. A1.5
    2. B0.67
    3. C50
    4. D600
    Show answerHide answer

    A: 1.5

    \(30 \div 20 = 1.5\).

  4. Bars in a histogram...

    1. AHave gaps
    2. BAre all equal in height
    3. CAre always tall
    4. DTouch each other
    Show answerHide answer

    D: Touch each other

    Touch, because the data is continuous.

  5. The vertical axis of a histogram is labelled...

    1. AFrequency
    2. BFrequency density
    3. CClass width
    4. DCumulative frequency
    Show answerHide answer

    B: Frequency density

    Frequency density.

  6. Class width 5 and frequency density 4 give a frequency of...

    1. A0.8
    2. B9
    3. C20
    4. D1.25
    Show answerHide answer

    C: 20

    \(5 \times 4 = 20\).

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