Histograms

In a histogram, the area of each bar shows the frequency. When classes have different widths, the height is the frequency density: frequency ÷ class width. To read a frequency back, multiply density by width.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Calculate frequency density for each class.
Draw a histogram with unequal class widths.
Read frequencies from a histogram.
Estimate the number of values in part of a class.

Pause and think

$20$ people are aged $0$ – $10$ and $20$ are aged $10$ – $50$ . On a bar chart both bars are the same height. Is that a fair picture?

Section 2 of 14

Core idea

Start hereFair bars

Group ages as 0–10, 10–20, then 50–80. The 30-year-wide class collects more people just because it's wider. Plotting frequency density makes each bar's area its frequency, so wide classes don't mislead.

The big idea

Frequency density $=\frac{\text{frequency}}{\text{class width}}$ ; area of a bar = frequency.

1. Drawing

Ages of 100 visitors (unequal classes)

The 50–80 class has 15 people but is 30 years wide, so its height is only

0.5

. The numbers are the frequencies (the areas).

\text{fd}=\frac{f}{w}\iff f=\text{fd}\times w

$w$: class width.

Worked exampleDensities

0–10 (12), 10–20 (25), 50–80 (15).

Goal: Heights.
Strategy: Divide by width.

$12\div10=1.2$
Width 10.
$25\div10=2.5$
Width 10.
$15\div30=0.5$
Width 30.

Answer

1.2

2.5

0.5

✓ CheckWidest class has the lowest bar despite

f=15

✓.

Worked exampleReading back

A bar of width 5, height 3.2.

Goal: Frequency.
Strategy: Area.

$3.2\times5$
fd × width.
$=16$
Evaluate.

Answer

16

✓ CheckArea units = number of items ✓.

Pause and think

Why not plot frequency?

Summary

fd = f ÷ width.
Area = frequency.

Frequency density: Frequency per unit width.

HistoryNaming the histogram. Karl Pearson introduced the word "histogram" in his 1891 lectures, probably from the Greek histos (something set upright, like a mast) and gramma (drawing).

Fun factYour phone camera has one. Many camera apps can show a histogram of the brightness of every pixel. Photographers use it to spot photos that are too dark (bars piled on the left) or blown out (bars piled on the right).

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