A histogram shows grouped continuous data, often with unequal class widths. The height of each bar is the frequency density, which is frequency ÷ class width, so the area of each bar represents the frequency. To find a frequency from a histogram, multiply the frequency density by the class width. To estimate part of a class, assume the values are spread evenly across it.
Histograms
- Frequency density = frequency ÷ class width
- Frequency = frequency density × class width (the bar's area)
- Modal class: the bar with the greatest frequency density
- Part of a class: frequency × (part width ÷ class width)
Why frequency density?
When classes have different widths, using frequency as the height would make wide classes look too important. Dividing by the class width gives frequency density, which puts every class on a fair scale. The area of each bar then equals its frequency.
Drawing a histogram
Work out the class width and frequency density for each class, then draw bars with no gaps between them, each spanning its class on a continuous horizontal scale. For a class 0 < w ≤ 20 with frequency 18, the width is 20 and the frequency density is 0.9. Label the vertical axis “frequency density”.
Reading a histogram
To find the frequency of a class, multiply its frequency density by its width. A bar of height 2.4 over a class of width 15 represents 36 values. If the vertical scale is not labelled with numbers, use a bar whose frequency is known to work out the scale.
The modal class
On a histogram with unequal widths, the modal class is the class with the greatest frequency density, the tallest bar, not necessarily the one with the greatest frequency. A wide class can contain many values while being less concentrated.
Estimating within a class
To estimate how many values lie in part of a class, assume they are spread evenly. If 30 values lie in 20 < w ≤ 40, the estimate for 20 < w ≤ 25 is 30 × 5/20 = 7.5. Add the parts of other classes as needed.
Worked examples
Example 1. A class 0 < w ≤ 20 has frequency 18. Find its frequency density.
- Class width: 20.
- 18 ÷ 20.
Answer: 0.9.
Example 2. Complete the frequency density for the class 20 < w ≤ 30 with frequency 18, and for 30 < w ≤ 60 with frequency 27.
- Widths: 10 and 30.
- 18 ÷ 10 and 27 ÷ 30.
Answer: 1.8 and 0.9.
Example 3. A histogram bar has frequency density 2.4 and covers a class of width 15. How many values does it represent?
- Frequency = 2.4 × 15.
Answer: 36.
Example 4. There are 30 values in the class 20 < w ≤ 40. Estimate how many lie between 20 and 25.
- Assume an even spread: 5 out of the 20 units.
- 30 × 5/20.
Answer: 7.5, so about 7 or 8 values.
Example 5. Classes 0–10, 10–20, 20–40 and 40–70 have frequency densities 0.6, 2.4, 1.5 and 0.4. Write down the modal class.
- The modal class has the greatest frequency density.
Answer: 10 < w ≤ 20.
Common mistakes
- Using frequency as the bar height. The height is frequency density.
- Reading a frequency straight off the axis. Multiply the frequency density by the class width.
- Choosing the widest bar as the modal class. The modal class has the greatest frequency density.
- Leaving gaps between bars. A histogram's bars touch, because the data is continuous.
- Wrong class width. Width = upper boundary − lower boundary.
How histograms are examined
Unequal-width histograms are a Higher-only topic for UK GCSE and International GCSE candidates, and they often appear late in the paper. Questions ask you to:
- complete a frequency density column and draw a histogram
- find frequencies from a given histogram
- find the total number of values, or the number in part of a class
- identify the modal class
Practise with the histograms worksheet, which includes answers, and the GCSE data revision engine.
Histograms display grouped data, and they are often paired with cumulative frequency questions.
Frequently asked questions
What is frequency density?
Frequency divided by class width.
How do you find a frequency from a histogram?
Multiply the bar's frequency density by its class width; this is the bar's area.
Why do histograms use frequency density?
So that classes of different widths are compared fairly, with area representing frequency.