Histograms and Frequency Density: GCSE Maths Guide

A GCSE histogram uses frequency density on the vertical axis, not frequency. This is essential when class widths are unequal — it ensures the area of each bar represents the frequency, not the height. Frequency density = frequency ÷ class width. This single formula unlocks both drawing and reading histograms.

How is a GCSE histogram different from a bar chart?

At KS3, students often draw bar charts where the height of each bar shows the frequency. A GCSE histogram is different in two key ways:

The vertical axis shows frequency density, not frequency. This means tall bars don't automatically represent more data — a tall, narrow bar can represent the same frequency as a short, wide bar.

The area of each bar represents the frequency. Area = frequency density × class width = (frequency ÷ class width) × class width = frequency.

There are no gaps between bars, because the data is continuous.

This design ensures that bars with different widths are fairly compared — a class interval of width 20 would otherwise look twice as prominent as one of width 10 even with the same number of data values.

What is the frequency density formula?

Frequency density = frequency ÷ class width

Rearranging this gives:

Frequency = frequency density × class width (to recover frequency from a histogram)

Class width = frequency ÷ frequency density (to find the class width if unknown)

These three versions of the same formula are all you need.

How do you draw a histogram?

Worked example: Draw a histogram for this grouped data about the lengths (in cm) of 80 leaves.

Length (cm) | Frequency | Class width | Frequency density
0 ≤ l < 5 | 10 | 5 | 10 ÷ 5 = 2.0
5 ≤ l < 10 | 24 | 5 | 24 ÷ 5 = 4.8
10 ≤ l < 20 | 30 | 10 | 30 ÷ 10 = 3.0
20 ≤ l < 30 | 12 | 10 | 12 ÷ 10 = 1.2
30 ≤ l < 50 | 4 | 20 | 4 ÷ 20 = 0.2

Step 1 — Calculate frequency density for each row (shown above).

Step 2 — Draw axes. Horizontal axis: the continuous variable (length in cm), from 0 to 50. Vertical axis: frequency density (no units — it's a derived measure), from 0 to at least 4.8.

Step 3 — Draw bars. Each bar runs from the lower to upper class boundary on the horizontal axis, and its height equals the frequency density. No gaps between bars.

Step 4 — Label axes clearly: "Length (cm)" and "Frequency density."

Check: total frequency = sum of (frequency density × class width) = (2×5) + (4.8×5) + (3×10) + (1.2×10) + (0.2×20) = 10 + 24 + 30 + 12 + 4 = 80 ✓

How do you read frequencies from a histogram?

If a histogram is given and you need to find frequencies, multiply each bar's height (frequency density) by its class width.

Worked example: A histogram bar for the class 15 ≤ t < 25 has frequency density 3.5. What is the frequency?

Frequency = 3.5 × (25 − 15) = 3.5 × 10 = 35

How do you find a missing frequency from a histogram?

Exam questions often give a partial histogram and ask you to complete it — or give the histogram and ask for the missing frequency in the table.

Worked example: A histogram for grouped data shows the class 0 ≤ w < 4 has frequency density 5, and 4 ≤ w < 6 has frequency density 8. Find both frequencies.

Frequency for 0 ≤ w < 4: 5 × 4 = 20

Frequency for 4 ≤ w < 6: 8 × 2 = 16

What common mistakes should you avoid?

Using frequency as the bar height instead of frequency density. This error makes bars with wider class intervals look disproportionately large.

Leaving gaps between bars. Histograms display continuous data — bars must touch.

Forgetting that "class width" is the difference between the boundaries, not the number of values. For 10 ≤ l < 30, class width = 30 − 10 = 20, not 10 or 30.

Mislabelling the vertical axis as "frequency." The vertical axis must be labelled "frequency density."

Frequently asked questions

Why do we use frequency density instead of frequency?

When class widths are equal, frequency and frequency density tell the same story (the bars differ only by a constant multiplier). But when class widths differ, plotting raw frequency makes wider intervals look artificially important. Frequency density corrects this by standardising each bar to "frequency per unit of class width," so the area — not the height — carries the meaning.

How does a histogram relate to a frequency polygon?

A frequency polygon is drawn by connecting the midpoints of the tops of histogram bars with straight lines. At GCSE, frequency polygons are usually drawn from equal-width class intervals and plot frequency directly; histograms handle unequal widths using frequency density. The two representations complement each other.

Can I estimate the median from a histogram?

Yes — add up areas (frequencies) starting from the left until you reach the halfway point of the total. The value at which you reach this halfway point is the estimated median. This is equivalent to finding the median from a cumulative frequency graph and is a standard Higher GCSE skill.

Are histograms only on the Higher GCSE paper?

Histograms with unequal class widths and frequency density calculations are Higher tier only. Foundation tier students may encounter simple bar charts for continuous data with equal class widths, which look like histograms but do not require frequency density.

For Socratic GCSE statistics practice including histograms, see aitutors.me.