Histograms are a Higher-tier topic that consistently appears on AQA, Edexcel and OCR papers. The key difference between a histogram and a bar chart is that a histogram uses frequency density on the vertical axis, and the area of each bar — not its height — represents the frequency. This guide focuses on the practical skills of drawing histograms from tables and reading values from given histograms.
What Is a Histogram?
A histogram displays the distribution of continuous grouped data. Unlike a bar chart, the bars have no gaps and can have unequal widths. The vertical axis shows frequency density rather than frequency, which ensures a fair comparison between classes of different widths.
The area of each bar equals the frequency for that class interval. This means a wide bar does not appear misleadingly large just because it covers a broader range — its height is adjusted downward by dividing by the class width.
To draw a histogram, you calculate the frequency density for each class. To read a histogram, you reverse the process: read the frequency density from the height and multiply by the class width to recover the frequency.
Key Formulas
Step-by-Step Method
- Calculate the class width for each interval (upper boundary minus lower boundary).
- Calculate the frequency density for each class: frequency ÷ class width.
- Draw the horizontal axis with a continuous scale showing class boundaries (no category labels).
- Draw each bar from the lower to the upper boundary with height equal to the frequency density.
- Label the vertical axis "Frequency density" — never "Frequency".
Worked Example 1 — Foundation Level
Question: The table shows the ages of 80 members of a gym. Draw a histogram.
| Age (years) | 15-24 | 25-34 | 35-44 | 45-64 |
|---|---|---|---|---|
| Frequency | 20 | 24 | 16 | 20 |
Working: Class widths: 10, 10, 10, 20. Frequency densities: 20÷10=2.0, 24÷10=2.4, 16÷10=1.6, 20÷20=1.0. Draw the horizontal axis from 15 to 65. Draw bars at heights 2.0, 2.4, 1.6, and 1.0 with no gaps.
Answer: The histogram has four bars with frequency densities 2.0, 2.4, 1.6, and 1.0 respectively.
Worked Example 2 — Higher Level
Question: A histogram shows journey times. The bar for 10 ≤ t < 25 has a frequency density of 3.2 and the bar for 25 ≤ t < 30 has a frequency density of 5.0. Find the frequency for each class and the total number of journeys in these two classes.
Working: Class 10 ≤ t < 25: width = 15. Frequency = 3.2 × 15 = 48. Class 25 ≤ t < 30: width = 5. Frequency = 5.0 × 5 = 25. Total = 48 + 25 = 73 journeys.
Answer: 48 journeys in the first class, 25 in the second, 73 total.
Worked Example 3 — Exam Style
Question: From the histogram in Worked Example 2, estimate how many journeys took between 10 and 15 minutes.
Working: The range 10 ≤ t < 15 covers 5 minutes of the class 10 ≤ t < 25 (total width 15). Frequency for full class = 48. Assuming even distribution: estimated frequency = 48 × (5/15) = 48 × 1/3 = 16.
Answer: An estimated 16 journeys took between 10 and 15 minutes.
Common Mistakes
- Plotting frequency instead of frequency density. If class widths are unequal, this distorts the diagram. Always calculate frequency density first.
- Leaving gaps between bars. Histograms represent continuous data, so bars must touch with no gaps.
- Misreading the axis when recovering frequency. To find frequency from a histogram, multiply the height (frequency density) by the class width — do not just read the height as the frequency.
Exam Tips
- Always show the frequency density calculations in a table alongside your diagram. Examiners award marks for this working.
- Use a ruler and sharp pencil for accurate bars. Check that the boundaries align exactly with the scale.
- When estimating a frequency within part of a class, assume the data is evenly distributed and use the proportion of the class width.
Practice Questions
Q1 (Foundation): Calculate the frequency densities for these reaction times.
| Time (s) | 0 ≤ t < 0.2 | 0.2 ≤ t < 0.4 | 0.4 ≤ t < 0.8 | 0.8 ≤ t < 1.0 |
|---|---|---|---|---|
| Frequency | 6 | 14 | 12 | 8 |
Q2 (Foundation): A histogram bar covers the interval 50 ≤ x < 80 and has a frequency density of 1.5. Find the frequency.
Q3 (Higher): A histogram bar covers 20 ≤ w < 35 with a frequency density of 4. Estimate how many data values fall between 20 and 25.
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Related Topics
Summary
- Frequency density = Frequency ÷ Class width. The area of each bar represents the frequency.
- To draw a histogram, calculate frequency densities and plot bars with no gaps on a continuous scale.
- To read a histogram, multiply frequency density by class width to recover the frequency.
- To estimate within part of a class, assume even distribution and use the proportional width.
- Always label the vertical axis "Frequency density" and show your calculation table.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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