Histograms and Frequency Density

Representing grouped data when class widths are unequal

A histogram represents grouped continuous data. Unlike a bar chart, its bars touch and the area of each bar represents frequency. Students usually meet this at GCSE Higher and IGCSE level.

Key rule and step-by-step method

Frequency density = frequency ÷ class width. First calculate each class width, then divide the frequency by that width, choose a suitable vertical scale and draw touching bars. To recover a frequency, multiply frequency density by class width. Review the glossary definition of a histogram

Worked examples

Example 1: Class 0–5 has frequency 20. Width = 5, so density = 20 ÷ 5 = 4. Example 2: Class 5–15 has frequency 30. Width = 10, so density = 3. Example 3: A bar has density 2.5 and width 8. Frequency = 2.5 × 8 = 20.

Common mistakes and exam advice

Do not use frequency as the bar height when widths differ. Check that bars touch, calculate width by subtracting boundaries, and label the vertical axis frequency density. Use exact values until the final stage.

Practice exercises and answers

Exercise 1: Frequency 18, class 0–6. Answer: density = 18 ÷ 6 = 3. Exercise 2: Frequency 28, class 6–20. Answer: width 14, density = 2. Exercise 3: Density 3.5, class width 12. Answer: frequency = 3.5 × 12 = 42.

Related learning across the website

Continue with Statistics Lessons, frequency tables, statistics, mean, median, mode, range, scatter graphs, probability lessons and fractions lessons

Benefits of one-to-one maths tutoring

One-to-one lessons with maths tutor Ryan Harvey can target errors with class width, scale selection and graph interpretation through personalised questions and immediate feedback.

Contact and lesson enquiry

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