Sheet № 66 · Higher only · AQA · Edexcel · OCR
Histograms –
Histograms are a Higher-tier topic that appears regularly on AQA, Edexcel and OCR GCSE Maths papers. Unlike a bar chart, a histogram uses frequency density on the vertical axis and has no gaps between bars, because the data is continuous. The area of each bar — not its height — represents the frequency. This distinction catches many stude
§Key definitions
Question:
The table shows the times taken by 60 runners to complete a 5K run.
Question 1:
The table shows waiting times at a surgery.
Question 2:
On a histogram, a bar covers the interval 30 ≤ x < 45 and has a frequency density of 2.4. Find the frequency.
Question 3:
A histogram shows the heights of 100 trees. The bar for 5 ≤ h < 10 has a frequency density of 8. Estimate how many trees are between 5 m and 7 m tall.
Question 4:
Explain why a histogram is more appropriate than a bar chart for displaying continuous data with unequal class widths.
§Formulas to memorise
Frequency density = Frequency ÷ Class width
Frequency = Frequency density × Class width
Calculate the frequency: frequency = frequency density × class width.
Class width = 5 − 2 = 3.
Frequency = frequency density × class width = 6 × 3 = 18 parcels.
Estimated frequency = 18 × (1 ÷ 3) = 6 parcels.
Worked example
The table shows the times taken by 60 runners to complete a 5K run. | Time (min) | 15 ≤ t < 20 | 20 ≤ t < 25 | 25 ≤ t < 35 | 35 ≤ t < 50 | |---|---|---|---|---| | Frequency | 10 | 20 | 18 | 12 | Draw a histogram for this data.
Working:
⚠ Common mistakes
- ✗Plotting frequency instead of frequency density — this is the single most common error. Always check whether class widths are equal; if they are not, you must use frequency density.
- ✗Leaving gaps between bars — a histogram represents continuous data, so bars must touch.
- ✗Incorrect class boundaries — for the interval "10 to under 20", the boundaries are 10 and 20, giving a width of 10, not 9.
- ✗Misreading the vertical axis — if the examiner gives you a histogram, read the frequency density, not the frequency, from the axis.
- ✗Forgetting to multiply back — when finding frequency from a histogram, remember frequency = frequency density × class width, not just the height.
✦ Exam tips
- →Always show the frequency density calculation — even if the question only asks you to draw the histogram, writing the FD column earns method marks.
- →Use a sharp pencil and ruler for accurate bars — examiners check boundary positions and heights.
- →Label the vertical axis "Frequency density", not "Frequency". This small detail signals that you understand the concept.
- →Check your total — add up all the frequencies (frequency density × class width for each bar) and verify it matches the total given in the question.
- →Link to cumulative frequency — some Higher questions combine histograms with cumulative frequency graphs. See cumulative frequency and box plots for more.