Ad Banner Placeholder

Cambridge IGCSE Mathematics — 0580 Extended

Topic 9.8: Statistics — Histograms

On a histogram the vertical axis is labelled Frequency density, the bars touch, and area equals frequency. Unequal class widths are the Extended skill: a wide class is drawn short so that its area still matches \(f\).

\[ \text{frequency density} = \dfrac{\text{frequency}}{\text{class width}} \]

Area is frequency

Two bars with the same area represent the same frequency, even if one is tall and thin and the other is short and wide.

A tall thin histogram bar and a short wide bar with the same area 12
1.2 × 10 = 12 and 0.6 × 20 = 12. Equal area ⇒ equal frequency.

Frequency density table

Add a width column, then divide. The last class \(30 \le t < 50\) has width 20, so \(fd = 6 \div 20 = 0.3\), not \(6 \div 10\).

Method

  1. Class width \(=\) upper end \(-\) lower end.
  2. \(fd = f \div\) width, for every row.
  3. Draw bars that touch. Height \(= fd\). Width \(=\) class width on the scale.
  4. Label the vertical axis “Frequency density”.
Frequency density table including a width-20 class with fd 0.3
Last class: 6 ÷ 20 = 0.3. Never leave that row as frequency 6 on the vertical scale.

Drawing the histogram

Draw the histogram of the journey times.

Unequal-width histogram of journey times with frequency density on the vertical axis
The 30–50 bar is twice as wide as the others, and short. Bars touch because the data is continuous.

Frequency from a bar

A missing frequency is the area you can see: \(fd \times\) width.

Unknown last histogram bar recovered as 0.3 times 20 equals 6
Read fd from the scale, multiply by the class width on the horizontal axis.
Three contrasts between a bar chart and a histogram
Bar chart: frequency, equal width, gaps. Histogram: frequency density, class width, bars touch.

Paper 2 (non-calculator)

\(6 \div 20 = 0.3\). \(12 \div 10 = 1.2\). Keep the decimal; do not “simplify” \(0.3\) to \(1/3\) unless the question asks for a fraction — \(0.3 = 3/10\), not \(1/3\).

Try this

A histogram bar has fd 0.8 and class width 10. What is the frequency?

Show answer
Answer

  1. Area = frequency.

    \[ 0.8 \times 10 = 8 \]

Exam Traps

  • Labelling the vertical axis “Frequency” on an unequal-width histogram is a lost mark even if every bar height is numerically right — the heights are densities, not frequencies.

0/10

Ad Banner Placeholder