Histograms
A histogram looks like a bar chart but works differently: with unequal class widths, the area of each bar represents the frequency, not its height. Every question in this topic is an application of one formula.
1. The formula
Frequency density = Frequency ÷ Class width
And rearranged, for reading a histogram:
Frequency = Frequency density × Class width — i.e. the AREA of the bar
The y-axis is FREQUENCY DENSITY, never frequency. Label it that way — it is a mark, and mislabelling it undermines every reading taken from the graph.
Area represents frequency, not height. This is the defining property of a histogram and the source of most errors. A wide, short bar can represent more data than a narrow, tall one.
2. Why frequency density is needed
With unequal class widths, plotting raw frequency would be misleading: a class twice as wide naturally collects more values, so its bar would look more important than it is. Dividing by the class width corrects for the width, making the bars fairly comparable.
Histograms vs bar charts:
| Histogram | Bar chart | |
|---|---|---|
| Data | continuous | discrete or categorical |
| y-axis | frequency density | frequency |
| Bars | no gaps | gaps between bars |
| Widths | may differ | equal |
| Represents frequency | area | height |
Histograms have no gaps, because the data is continuous and the classes run into each other.
3. Class width
Class width = upper boundary − lower boundary
| Class | Width |
|---|---|
| 0 < t ≤ 10 | 10 |
| 10 < t ≤ 20 | 10 |
| 20 < t ≤ 50 | 30 |
| 50 < t ≤ 60 | 10 |
Work out the class width first, in its own column. Ignoring unequal widths, or confusing the class width with a frequency, were both recorded errors — and the whole calculation depends on getting it right.
4. Drawing a histogram
Add two columns to the table:
| Time (min) | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0 < t ≤ 10 | 8 | 10 | 8 ÷ 10 = 0.8 |
| 10 < t ≤ 20 | 15 | 10 | 15 ÷ 10 = 1.5 |
| 20 < t ≤ 50 | 24 | 30 | 24 ÷ 30 = 0.8 |
| 50 < t ≤ 60 | 5 | 10 | 5 ÷ 10 = 0.5 |
Then plot frequency density on the y-axis, with bars spanning each class and no gaps.
Always add both columns. Tutors flagged this specifically — missing the class width column is where students lose marks before any plotting happens.
Notice that the first and third classes have the same frequency density (0.8) despite very different frequencies (8 and 24) — because the third class is three times as wide. That is exactly the correction frequency density performs.
5. Reading a histogram
Frequency = frequency density × class width — the area of the bar.
Example: a bar spanning 20 to 50 with a height of 0.8:
- frequency = 0.8 × 30 = 24
To find a total frequency, calculate the area of every bar and add them.
Never read the frequency off the y-axis. The y-axis gives frequency density; the frequency is the area. This was the most common misunderstanding recorded.
Part of a class
If a question asks for the frequency between values that don’t line up with the class boundaries, take the proportion of that bar.
Example: for the bar from 20 to 50 with frequency 24, how many fall between 20 and 30?
- That is 10 of the 30 units of width, so 10/30 × 24 = 8
This assumes the data is evenly spread within the class — which is why the answer is an estimate.
6. Method
- Work out class widths
- Calculate frequency density for each class
- Plot with frequency density on the y-axis, labelled
- Draw bars with no gaps, spanning the full class width
- To read: area = frequency
Check your scale before plotting, and use the full grid.
7. Mistakes that cost marks
Plotting frequency instead of frequency density.
Failing to label the y-axis as frequency density.
Reading frequency off the y-axis rather than calculating the area.
Ignoring unequal class widths.
Calculating the class width wrongly.
Multiplying when you should divide, or vice versa.
Leaving gaps between bars.
Missing out the class width column in the table.
Forgetting to add all the bars for a total frequency.
Frequently asked questions
What is frequency density? Frequency ÷ class width — it goes on the y-axis.
Why do we use frequency density? Because the class widths are unequal, so plotting raw frequency would exaggerate the wider classes.
What does the area of a bar represent? The frequency of that class.
How do I find the frequency from a histogram? Frequency density × class width — the area of the bar.
How is a histogram different from a bar chart? Histograms show continuous data with no gaps, may have unequal widths, and use area rather than height to represent frequency.
What goes on the y-axis? Frequency density — always.
How do I find the frequency for part of a class? Take the proportion of the bar’s area, assuming the data is evenly spread.
How do I find the total frequency? Add the areas of all the bars.
Quick revision checklist
- I know frequency density = frequency ÷ class width
- I can calculate class widths from the boundaries
- I always add class width and frequency density columns
- I plot frequency density on the y-axis and label it
- I draw bars with no gaps
- I know area = frequency
- I can find a frequency from a bar’s dimensions
- I can find a total frequency by adding all the areas
- I can find the frequency for part of a class by proportion
- I know how a histogram differs from a bar chart
- I understand why frequency density is used
These notes cover histograms in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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