EDEXCEL GCSE MATHS · HIGHER
Drawing histograms
Further statistics · Lesson 4 of 6
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
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. |