Histograms and Frequency Density
A histogram is a chart for continuous grouped data with unequal class widths, in which the area of each bar, rather than its height, represents the frequency of that class. The height of each bar is called the frequency density, found by dividing the frequency of a class by its class width, and this Higher-tier topic extends the equal-width bar charts met earlier in the course.
Before you start
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Method
- Find the class width of each group by subtracting the lower class boundary from the upper class boundary.
- Calculate the frequency density for each group using frequency density = frequency / class width.
- Draw bars with no gaps between them, since the data is continuous, with class boundaries on the horizontal axis and frequency density on the vertical axis; the height of each bar equals its frequency density.
- To find a frequency from a histogram, read the frequency density (the height) and multiply by the class width, since frequency = frequency density x class width, which equals the area of the bar.
- For a class split by an extra vertical line, calculate the area of only the relevant part rather than assuming an even spread across the whole interval.
- Check that summing frequency density x class width over every group gives the total frequency stated in the question.
Worked example
The table shows the time, t minutes, taken by 60 people to complete a puzzle: 0<=t<10 (frequency 12), 10<=t<20 (frequency 24), 20<=t<40 (frequency 18), 40<=t<60 (frequency 6). Find the frequency density for each class.
- Class widths: 10, 10, 20, 20.
- Frequency density = frequency / class width for each class: 12/10 = 1.2, 24/10 = 2.4, 18/20 = 0.9, 6/20 = 0.3.
- Check: total frequency = 12 + 24 + 18 + 6 = 60, which matches the 60 people given.
- Final answer: frequency densities are 1.2, 2.4, 0.9 and 0.3.
Practice questions
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Q1A class interval has width 5 and frequency 20. Find the frequency density.Show answer
Answer: 4 (20/5)
Q2A class interval 10-30 (width 20) has frequency density 3. Find the frequency.Show answer
Answer: 60 (3 x 20)
Q3A histogram bar has height (frequency density) 2.5 and covers the interval 0-8. Find the frequency it represents.Show answer
Answer: 20 (2.5 x 8)
Q4A frequency table has intervals 0-10 (frequency 15) and 10-15 (frequency 20). Calculate the frequency density for each.Show answer
Answer: 1.5 for 0-10 (15/10), 4 for 10-15 (20/5)
Q5Two intervals both have frequency 30: one has width 5 and one has width 15. Explain which interval has the taller histogram bar, and why.Show answer
Answer: The interval with width 5, because frequency density = frequency/width is larger for the narrower interval (30/5=6 compared with 30/15=2), and the bar height equals the frequency density
Q6A histogram has classes 20-25 (frequency density 4), 25-35 (frequency density 2.4), 35-50 (frequency density 1.6). Find the total number of data values represented.Show answer
Answer: 68 ((4x5)+(2.4x10)+(1.6x15) = 20+24+24)
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
The table shows the masses, m kg, of 50 parcels: 0<=m<5 (frequency 10), 5<=m<15 (frequency 30), 15<=m<25 (frequency 10). Calculate the frequency density for each class interval.
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A histogram represents the ages, in years, of 90 members of a gym. The class 10<=a<20 has frequency density 1.5, and the class 20<=a<50 has frequency density 2. There is one more class, 50<=a<70, with frequency density d. (a) Find the frequency for each of the first two classes. (b) Given that the total number of members is 90, find d.
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Explain why the vertical axis of a histogram is labelled frequency density, rather than frequency, when the class widths are not all equal.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
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This topic is chapter 13 of GCSE Statistics Higher Workbook 1, the whole course as one free printable PDF.
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