Histograms
Histograms are used for grouped continuous data. They look similar to bar charts, but they are different because the area of each bar represents the frequency.
Frequency density = frequency ÷ class width
Frequency = frequency density × class width
Why do histograms use frequency density?
In many histogram questions, the class intervals are not all the same width. This means we cannot use frequency directly as the height of the bars. Instead, we use frequency density.
Example 1: Find frequency density
A class interval is 10 < x ≤ 20 and its frequency is 30.
The class width is:
20 − 10 = 10
Now calculate frequency density:
30 ÷ 10 = 3
The frequency density is 3.
Completing a histogram table
GCSE histogram questions often give you a grouped frequency table and ask you to calculate frequency density before drawing the histogram.
Example 2: Complete a frequency density table
Complete the frequency density column:
0 < x ≤ 10, frequency 20
10 < x ≤ 30, frequency 50
30 < x ≤ 40, frequency 25
Calculate each class width and divide frequency by class width:
20 ÷ 10 = 2
50 ÷ 20 = 2.5
25 ÷ 10 = 2.5
The frequency densities are 2, 2.5 and 2.5.
Drawing a histogram
To draw a histogram, put the class intervals on the horizontal axis and frequency density on the vertical axis.
The bars should touch.
The height of each bar is the frequency density.
The width of each bar is the class width.
The area of each bar represents the frequency.
Finding frequency from a histogram
If a histogram is already drawn, you can find the frequency by multiplying the class width by the frequency density.
Example 3: Find frequency from a histogram bar
A histogram bar has class width 15 and frequency density 4.
Frequency = 15 × 4 = 60
The frequency is 60.
Using area in histogram questions
Because frequency is represented by area, two bars can have different widths and heights but still represent the same frequency.
Example 4: Compare two histogram bars
Bar A has width 10 and frequency density 6.
Bar B has width 20 and frequency density 3.
Bar A frequency = 10 × 6 = 60
Bar B frequency = 20 × 3 = 60
Both bars represent the same frequency.
A common mistake is using frequency as the bar height. In a histogram, the bar height is frequency density, not frequency.
If the class widths are unequal, immediately think “frequency density”. Check the axis label carefully before reading values from the graph.
Video explanation
A short Worthing Maths Tutor video explanation for histograms can be embedded here later to improve student engagement and time on page.
Practice questions
- A class interval has frequency 40 and class width 8. Find the frequency density.
- A class interval is 20 < x ≤ 50. Find the class width.
- A histogram bar has width 12 and frequency density 5. Find the frequency.
- A class interval has frequency 72 and class width 18. Find the frequency density.
- Explain why bars in a histogram touch.
Answers
- 5
- 30
- 60
- 4
- Because histograms show continuous grouped data.
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