Histograms, explained with examples

By The STEM-Tutor team · Updated October 2026

A histogram shows grouped continuous data. In GCSE Maths Higher, histograms use frequency density, so the area of each bar, not its height, shows the frequency. Once that idea clicks, histogram questions become reliable marks.

Histograms versus bar charts

A bar chart displays categories, such as favourite colours, with gaps between the bars. A histogram displays continuous data grouped into classes, such as times or heights, with no gaps between bars, because one class runs straight into the next. In GCSE Higher histograms, the classes can have different widths, and that is why we need frequency density.

Frequency density

Frequency density = frequency ÷ class width

The vertical axis of a histogram shows frequency density, not frequency. Rearranging gives the key fact: frequency = frequency density × class width, which is the area of the bar. Using area means a wide class with the same number of items does not look misleadingly large.

Why area, not height?

Imagine the 30–50 class drawn with its frequency of 8 as the height. It would be twice as tall as it is now and four times as wide as some other bars, so it would look like the biggest group, even though only 8 people are in it. Using frequency density keeps the picture honest: the space a bar takes up on the page matches how many people it represents.

Drawing a histogram

1. Find the width of each class.
2. Calculate frequency density for each class.
3. Draw a continuous horizontal scale for the data and a frequency density scale up the side.
4. Draw each bar from the start to the end of its class, with height equal to its frequency density.
5. Label both axes.

Example 1: building the table

Journey times for 69 people are: 0–10 minutes, 10 people; 10–15, 15; 15–20, 20; 20–30, 16; 30–50, 8. Find the frequency densities.

• Widths: 10, 5, 5, 10, 20.
• Frequency densities: 10/10 = 1, 15/5 = 3, 20/5 = 4, 16/10 = 1.6, 8/20 = 0.4.
• These are the heights of the bars in the diagram above.

Reading frequencies from a histogram

To find how many items lie in a class, multiply the bar's height (frequency density) by its width. Read the scale carefully: the frequency density axis often goes up in steps like 0.2 or 0.5, not 1.

Example 2: reading a bar

From the histogram, how many people took between 20 and 30 minutes?

• Height 1.6, width 10.
• Frequency = 1.6 × 10 = 16 people.

Example 3: part of a class

Estimate how many people took between 12 and 25 minutes.

• From 12 to 15: 3 minutes of the 10–15 bar, so 3 × 3 = 9 people.
• From 15 to 20: the whole bar, 20 people.
• From 20 to 25: 5 minutes of the 20–30 bar, so 1.6 × 5 = 8 people.
• Total ≈ 37 people. This assumes values are spread evenly within each class, so the answer is an estimate.

Example 4: working out the scale

A histogram has no numbers on the frequency density axis. The 10–20 bar is 3 cm tall and represents 15 people. The 20–25 bar is 4 cm tall. How many people does it represent?

• First bar: area = 10 × 3 = 30 units for 15 people, so each unit of area is 0.5 people.
• Second bar: area = 5 × 4 = 20 units, so it represents 20 × 0.5 = 10 people.
• The method always works: find how many items one unit of area stands for, then apply it to the other bars.

Estimating the mean

You cannot find the exact mean from grouped data, because you do not know each value. Instead, assume every value in a class sits at the class midpoint, multiply each midpoint by its frequency, add up, and divide by the total frequency.

Example 5: an estimated mean

Estimate the mean journey time from the histogram data.

• Midpoints: 5, 12.5, 17.5, 25 and 40. Frequencies: 10, 15, 20, 16 and 8.
• Σfx = 50 + 187.5 + 350 + 400 + 320 = 1307.5.
• Mean ≈ 1307.5 ÷ 69 ≈ 18.9 minutes. It is higher than the median of 17.5 minutes because the long tail of slow journeys pulls the mean up.

Comparing two histograms

When asked to compare two distributions, make two comparisons, each in context: one about a typical value (the median or mean, or where the peak is) and one about spread (the range, interquartile range, or how wide the bars spread). For example: “Journeys by bus were typically longer than by car, and bus times were more spread out.” Comparing the shapes, such as one being more skewed, can earn an extra mark.

Estimating the median from a histogram

The median is the middle value. With 69 people, it is the 35th value. Count through the classes: 10 in the first class, 25 after the second, 45 after the third. So the 35th value is in the 15–20 class, 10 people into its 20. That is halfway through the class, so the median is about 15 + 2.5 = 17.5 minutes. This uses the same even-spread assumption as Example 3.

Interpreting the shape

The shape of a histogram tells a story. Most journeys here take 10 to 20 minutes, with a long tail of slower journeys up to 50 minutes. That tail makes the distribution positively skewed. In exam questions, you may be asked to compare two histograms: compare where most values lie (the peak), the spread, and any skew.

Common mistakes

• Plotting frequency instead of frequency density on the vertical axis.
• Reading frequency as the height of the bar instead of height × width.
• Leaving gaps between bars.
• Misreading the frequency density scale.
• Forgetting to label the axes.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are histograms on the Foundation paper?

Simple histograms with equal widths can appear on Foundation papers. Frequency density with unequal class widths is a Higher tier topic.

Do I need to know histograms for A level Maths?

Yes. A level Maths statistics uses histograms with frequency density, and questions often involve interpolation and comparing distributions.

Key takeaways

• Frequency density = frequency ÷ class width.
• The area of a bar equals its frequency.
• Bars touch, because the data are continuous.
• For part of a class, use density × the part of the width.
• Estimates assume values spread evenly within each class.

Related guides

• Box plots
• Cumulative frequency
• Venn diagrams