Histograms

What you will learn

• Calculate class width from grouped intervals.
• Use frequency density = frequency ÷ class width.
• Draw histogram bars so area represents frequency.
• Find frequency from density and width.
• Read and estimate frequencies from a histogram scale.
• Explain why histograms are different from bar charts.

Why this matters

Histograms let you display grouped continuous data when class widths are not all equal. The key GCSE idea is that frequency is shown by area, so a wider bar can represent many values even when it is not very tall.

Prior knowledge

You should already be comfortable with:

• Area of rectangles.
• Grouped frequency tables.
• Decimals and division.
• Graph scales.

Clear explanation

Main idea

In a histogram, the area of each bar represents the frequency. The height is called frequency density.

• Frequency density: frequency ÷ class width.
• Frequency: frequency density × class width.
• Class width: upper boundary − lower boundary.

Method

Find each class width, calculate frequency density for each row, choose a sensible vertical scale, then draw each bar across its whole class interval with no gaps between neighbouring intervals.

When you read from a completed histogram, check the horizontal width of the interval and the vertical density scale separately. If only part of a bar is needed, use the same area idea for that part of the interval, so an estimate from 20 to 25 uses width 5 multiplied by the bar's density.

Exam tip

Do not draw ordinary frequency bars when the widths are unequal. If the question asks for a missing frequency, use the rectangle area: width × density.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For histograms, marks usually come from calculating class widths accurately, using frequency density rather than raw frequency for bar heights, drawing bars to the correct scale, and using rectangle area to recover missing frequencies. In explanations, state clearly that area represents frequency.

Common mistakes

• Using frequency as height: this only works like a bar chart when all widths are equal, and GCSE histogram questions often use unequal widths.
• Forgetting class width: always subtract lower boundary from upper boundary before finding density.
• Leaving gaps between touching intervals: histograms for continuous grouped data usually have adjacent bars touching.
• Reading height as frequency: multiply density by width when the question asks how many values are in a class.

Extension challenge

Create a grouped frequency table with unequal class widths, draw the matching histogram, then write one sentence explaining how a reader can check one bar's frequency from its area.

Exam-board guidance

Histograms appears within GCSE Maths statistics where pupils draw and interpret grouped-data diagrams. The shared skill is to use frequency density for unequal class widths and to remember that bar area, not height alone, represents frequency.

Next lesson

This completes the currently planned GCSE Maths lesson roadmap.