Histograms trip up more students than almost any other statistics topic — not because the maths is hard, but because students confuse them with bar charts. Once you understand what a histogram actually shows and why frequency density exists, the questions become very manageable. This topic appears on all three main exam boards: AQA (8300), Edexcel (1MA1), and OCR (J560). Expect it in the statistics section, usually on Paper 2 or Paper 3, often worth 4–6 marks.
What You Need to Know First
Before tackling histogram questions, make sure you're confident with:
- The difference between discrete and continuous data (histograms are always for continuous data grouped in class intervals)
- Calculating with fractions and decimals — you'll be dividing and multiplying throughout
- Reading scales carefully on axes — histogram scales can be tricky and are often where marks are lost
- The meaning of class width — the size of each group (e.g. 10 ≤ x < 20 has class width 10)
- Basic area calculations — in a histogram, area = frequency
Frequency Density: The Key Formula
In a bar chart, the height of each bar shows frequency. In a histogram, the height shows frequency density — and this distinction is everything.
The reason: when class widths are unequal (which they often are in exam questions), taller bars would be misleading if they simply showed frequency. A group covering a 20-year range shouldn't look the same as one covering a 5-year range with identical frequencies. Frequency density corrects for this by spreading each group's frequency across its width.
The formula:
Frequency density = Frequency ÷ Class width
Rearranged:
Frequency = Frequency density × Class width Class width = Frequency ÷ Frequency density
These three versions are equally important. In an exam you might need any one of them.
Worked Example 1: Drawing a Histogram
The table below shows the time (in minutes) 60 students spent on homework one evening.
| Time (t minutes) | Frequency |
|---|---|
| 0 ≤ t < 20 | 8 |
| 20 ≤ t < 30 | 15 |
| 30 ≤ t < 40 | 18 |
| 40 ≤ t < 60 | 12 |
| 60 ≤ t < 100 | 7 |
Step 1: Find the class width for each group.
- 0 ≤ t < 20: width = 20
- 20 ≤ t < 30: width = 10
- 30 ≤ t < 40: width = 10
- 40 ≤ t < 60: width = 20
- 60 ≤ t < 100: width = 40
Step 2: Calculate frequency density = frequency ÷ class width.
- 0 ≤ t < 20: 8 ÷ 20 = 0.4
- 20 ≤ t < 30: 15 ÷ 10 = 1.5
- 30 ≤ t < 40: 18 ÷ 10 = 1.8
- 40 ≤ t < 60: 12 ÷ 20 = 0.6
- 60 ≤ t < 100: 7 ÷ 40 = 0.175
Step 3: Draw the histogram. The x-axis shows time (0 to 100). The y-axis shows frequency density. Each bar spans its class interval and has height equal to its frequency density. Bars touch each other — there are no gaps in a histogram.
Check: total frequency = (0.4 × 20) + (1.5 × 10) + (1.8 × 10) + (0.6 × 20) + (0.175 × 40) = 8 + 15 + 18 + 12 + 7 = 60 ✓
Interpreting a Histogram
Just as often, you'll be given a completed histogram and asked to find frequencies or estimate the number of people in a particular range. This is where the area formula is your best friend.
Frequency = Frequency density × Class width = Area of bar
Worked Example 2: Reading from a Histogram
A histogram shows the ages of people at a sports centre. Use it to complete the frequency table and find the total number of people surveyed.
The histogram shows the following bars (reading off the frequency densities):
- 0 ≤ age < 10: frequency density = 2.4
- 10 ≤ age < 20: frequency density = 3.6
- 20 ≤ age < 40: frequency density = 2.1
- 40 ≤ age < 60: frequency density = 1.8
- 60 ≤ age < 80: frequency density = 0.9
Step 1: Write down the class width for each bar.
- 0–10: width = 10
- 10–20: width = 10
- 20–40: width = 20
- 40–60: width = 20
- 60–80: width = 20
Step 2: Calculate frequency = frequency density × class width.
- 0–10: 2.4 × 10 = 24
- 10–20: 3.6 × 10 = 36
- 20–40: 2.1 × 20 = 42
- 40–60: 1.8 × 20 = 36
- 60–80: 0.9 × 20 = 18
Step 3: Total = 24 + 36 + 42 + 36 + 18 = 156 people
Worked Example 3: Finding a Missing Frequency
Sometimes a table is partially completed and you're given the histogram to fill in one missing value.
Suppose you know that 24 people are in the group 0 ≤ age < 10, and the frequency density for that bar reads 2.4 on the histogram.
You can verify: 2.4 × 10 = 24 ✓
Now suppose the bar for 20 ≤ age < 40 has a frequency density of 2.1 and the class width is 20.
Frequency = 2.1 × 20 = 42 people in the 20–40 age group.
The key skill is spotting which version of the formula you need. Read the question carefully — if you're given frequency density and class width, multiply. If you're given frequency and class width, divide.
Common Mistakes Students Make
- Drawing a bar chart instead of a histogram. The most common error. If you plot raw frequency on the y-axis, you lose all the marks. Always label the y-axis "Frequency Density" and calculate it correctly.
- Confusing class width with class boundaries. The class width of 30 ≤ t < 50 is 20, not 30 or 50. It's the upper boundary minus the lower boundary: 50 − 30 = 20.
- Leaving gaps between bars. Histograms represent continuous data — the bars must touch. Gaps suggest discrete categories, which is a bar chart.
- Not checking total frequency. After reading a histogram, add up all your calculated frequencies. If the question says 80 people were surveyed and your total is 92, you've misread a scale. Always cross-check.
- Misreading the frequency density scale. Histogram y-axes often use unusual intervals (0.2, 0.5, etc.). Count the gridlines carefully before reading off values.
Exam Tips for Histograms
- Always write down frequency density = frequency ÷ class width before you start. Stating the formula shows the examiner you know what you're doing and earns method marks even if you make an arithmetic error.
- Label your axes correctly. The x-axis must show the measured quantity with units. The y-axis must say "Frequency Density" — not "Frequency". Incorrect labelling loses marks.
- Use a ruler for the bars. Marks can be lost for bars that aren't at the exact correct height. Precision matters. Plot each bar carefully, particularly when frequency densities are awkward decimals like 0.175.
- AQA often asks for an estimate of the number in a specific sub-range. For example: "estimate how many people took between 25 and 30 minutes." Find what fraction of the bar falls in that range and multiply. The bar for 20–30 has class width 10; the portion 25–30 is width 5, so half the bar's area: frequency = 0.5 × 15 = 7.5, which you'd round to 8 people.
- If you're given a partially completed histogram and table, use the given row to find the frequency density scale. One row where you know both frequency and class width gives you frequency density — from that, you can often read off the scale and find all missing values.
Practise Now
The best way to get confident with histograms is to practise exam-style questions with instant feedback. Try histograms questions now on GCSEMathsAI — AI-generated questions matched to your board and tier, with detailed marking and personalised feedback.