EDEXCEL GCSE MATHS · HIGHER
Drawing histograms
Further statistics · Lesson 4 of 6
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
1. What is the width of the class 10 < t ≤ 30?
20
2. Work out 30 ÷ 20.
1.5
3. What is a frequency table?
A table of how often values occur
4. How does a bar chart differ from a histogram?
A histogram's bar area shows frequency; classes are continuous
5. Write 12 ÷ 10 as a decimal.
1.2
Learning Objectives
1. Explain why a histogram uses frequency density.
2. Work out the frequency density for each class.
3. Draw a histogram with unequal class widths.
4. Label the axes correctly.
Histogram
In a histogram the area of each bar shows the frequency. The height is the frequency density.
frequency density = frequency/(class width). The bars touch and there are no gaps.
A Histogram
|
Wider bars are shorter for the same frequency. |
A histogram of times with unequal class widths, with bar heights of frequency density and frequencies written inside the bars.
Working Out Frequency Density
Times taken (t minutes) by 94 customers.
|
Time |
Frequency |
Class width |
Frequency density |
|---|---|---|---|
|
0 < t ≤ 10 |
12 |
10 |
1.2 |
|
10 < t ≤ 30 |
30 |
20 |
1.5 |
|
30 < t ≤ 40 |
26 |
10 |
2.6 |
|
40 < t ≤ 60 |
18 |
20 |
0.9 |
|
60 < t ≤ 100 |
8 |
40 |
0.2 |
Drawing a Histogram
Follow the steps.
|
1 Work out the class widths Upper boundary minus lower boundary |
2 Work out the frequency density Frequency divided by class width |
3 Draw the axes Data on the horizontal axis, frequency density on the vertical axis |
4 Draw the bars Bars touch; the height is the frequency density |
5 Label Add a scale and a label for each axis |
Frequency Density
|
A class 20 < x ≤ 50 has frequency 60. Work out the frequency density. |
1. Class width
50 − 20 = 30
2. Divide
60/30 = 2
Answer: The frequency density is 2, so the bar has height 2.
Completing a Histogram
|
Age (a years): 0 < a ≤ 10: 8, 10 < a ≤ 20: 20, 20 < a ≤ 40: 36, 40 < a ≤ 60: 30. The first three bars are drawn. Work out the height of the last bar. |
1. Class width
60 − 40 = 20
2. Frequency density
30/20 = 1.5
Answer: Draw 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".
Key Terms
|
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. |
Your Task: Find the Density
10 minutes
|
Work out the frequency density for each class. (a) 0 < w ≤ 5, frequency 15 (b) 5 < w ≤ 15, frequency 40 (c) 15 < w ≤ 35, frequency 30 (d) 35 < w ≤ 60, frequency 10. 1. Class width first. 2. Then divide. |
A good answer shows: (a) 3 (b) 4 (c) 1.5 (d) 0.4
Note: Ask which bar will be tallest and why.
Can I...?
☐ Find the class widths.
☐ Work out frequency density.
☐ Draw and label axes.
☐ Draw bars that touch.
☐ Use unequal class widths.
☐ Complete a partly drawn histogram.
☐ Explain why frequency density is used.
☐ Avoid common mistakes.
Summary
✓ Frequency density = frequency/(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) Add a frequency density column to the table first. |
Exam Practice: Drawing histograms
Answer all questions. Show your working. · 30 minutes
▸ Question 1 · 3 marks · Complete. The table shows the times, t minutes, taken by 94 customers. Frequencies: 0 < t ≤ 10: 12, 10 < t ≤ 30: 30, 30 < t ≤ 40: 26, 40 < t ≤ 60…
▸ Question 2 · 3 marks · Complete the histogram. The table shows information about the ages, a years, of 94 people: 0 < a ≤ 10: 8, 10 < a ≤ 20: 20, 20 < a ≤ 40: 36, 40 < a ≤ 60: 30…
▸ Question 3 · 2 marks · Work out. A class 5 < x ≤ 25 has frequency 40. Work out the frequency density.
▸ Question 4 · 2 marks · Work out. A histogram bar for the class 10 < x ≤ 14 has a frequency density of 3. Work out the frequency of the class.
▸ Question 5 · 2 marks · Explain. Explain why the height of a bar in a histogram is not the frequency when the class widths are unequal.
▸ Question 6 · 3 marks · Draw. Describe how to draw a histogram from a grouped frequency table with unequal class widths.
Question 1 · 3 marks · Complete
|
“The table shows the times, t minutes, taken by 94 customers. Frequencies: 0 < t ≤ 10: 12, 10 < t ≤ 30: 30, 30 < t ≤ 40: 26, 40 < t ≤ 60: 18, 60 < t ≤ 100: 8. Work out the frequency density for each class.” |
HOW TO ANSWER IT Command word: Complete. Worth 3 marks, so plan before writing.
Question 1 · mark scheme
3 marks available. Award a mark for each point made.
▸ Class widths. M1
▸ At least three correct. M1
▸ All correct. A1
▸ Model answer. 1.2, 1.5, 2.6, 0.9, 0.2
Question 2 · 3 marks · Complete the histogram
|
The table shows information about the ages, a years, of 94 people: 0 < a ≤ 10: 8, 10 < a ≤ 20: 20, 20 < a ≤ 40: 36, 40 < a ≤ 60: 30. Complete the histogram. (3 marks) |
|
Question 2 · mark scheme
3 marks available. Award a mark for each point made.
▸ Frequency density 1.5. M1
▸ Bar from 40 to 60. A1
▸ Height 1.5. A1
▸ Model answer. Frequency density = 30 ÷ 20 = 1.5; a bar drawn from 40 to 60 of height 1.5.
Question 3 · 2 marks · Work out
|
“A class 5 < x ≤ 25 has frequency 40. Work out the frequency density.” |
HOW TO ANSWER IT Command word: Work out. Worth 2 marks, so plan before writing.
Question 3 · mark scheme
2 marks available. Award a mark for each point made.
▸ Class width 20. M1
▸ 2. A1
▸ Model answer. 40 ÷ 20 = 2
Question 4 · 2 marks · Work out
|
“A histogram bar for the class 10 < x ≤ 14 has a frequency density of 3. Work out the frequency of the class.” |
HOW TO ANSWER IT Command word: Work out. Worth 2 marks, so plan before writing.
Question 4 · mark scheme
2 marks available. Award a mark for each point made.
▸ Uses 4 × 3. M1
▸ 12. A1
▸ Model answer. Width 4; frequency = 4 × 3 = 12
Question 5 · 2 marks · Explain
|
“Explain why the height of a bar in a histogram is not the frequency when the class widths are unequal.” |
HOW TO ANSWER IT Command word: Explain. Worth 2 marks, so plan before writing.
Question 5 · mark scheme
2 marks available. Award a mark for each point made.
▸ Area shows frequency. M1
▸ Frequency density used for the height. C1
▸ 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.
Question 6 · 3 marks · Draw
|
“Describe how to draw a histogram from a grouped frequency table with unequal class widths.” |
HOW TO ANSWER IT Command word: Draw. Worth 3 marks, so plan before writing.
Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ Frequency density. B1
▸ Axes correctly labelled. B1
▸ Touching bars with correct heights. B1
▸ 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.