Histograms
O Level / IGCSE Mathematics – Cheat Sheet – 9.7 Histograms
Key Points
- Histogram: A diagram showing the distribution of continuous data using bars whose areas represent frequencies.
- Vertical axis: always labelled Frequency Density.
- Horizontal axis: shows the class intervals (continuous scale).
- No gaps between bars.
- Bar height is proportional to Frequency Density, not directly to frequency.
Formula
Frequency Density = Frequency ÷ Class Width
Step-by-Step Example
| Class Interval (cm) | Frequency (f) | Class Width (w) | Frequency Density (f ÷ w) |
|---|---|---|---|
| 100 – 120 | 8 | 20 | 0.4 |
| 120 – 140 | 12 | 20 | 0.6 |
| 140 – 180 | 30 | 40 | 0.75 |
| 180 – 200 | 10 | 20 | 0.5 |
Drawing the Histogram
- Draw horizontal axis (class boundaries) and vertical axis (frequency density).
- For each class:
- Calculate frequency density using formula.
- Draw a bar spanning the class width, height = frequency density.
- Make sure bars touch (continuous data).
Interpretation Tips
- Taller bars → higher frequency density, not necessarily higher frequency.
- If class widths vary, heights adjust so that area still represents frequency.
- Total frequency can be found by:
Total Frequency = Σ(Frequency Density × Class Width)