Cumulative Frequency Diagrams
O Level / IGCSE Mathematics – Cheat Sheet – 9.6 Cumulative Frequency Diagrams
Key Points
- Cumulative Frequency (CF): running total of frequencies in a dataset.
- Cumulative Frequency Table: shows each data value/class and the total frequency up to that value.
- Cumulative Frequency Diagram: plotted points show the cumulative total against the upper boundary of each class interval. Points are joined by a smooth curve (not straight lines).
- Interpretation:
- Median: value at CF = Total CF ÷ 2
- Lower quartile (Q₁): value at CF = Total CF ÷ 4
- Upper quartile (Q₃): value at CF = (3 × Total CF) ÷ 4
- Interquartile range (IQR) = Q₃ − Q₁
- Percentiles: value at CF = (percent ÷ 100) × Total CF
Step-by-Step Example
| Class Interval (cm) | Frequency (f) | Cumulative Frequency (CF) |
|---|---|---|
| 120 – 130 | 4 | 4 |
| 130 – 140 | 6 | 10 |
| 140 – 150 | 10 | 20 |
| 150 – 160 | 8 | 28 |
| 160 – 170 | 2 | 30 |
How to Find Key Values
- Total CF = 30
- Median position = Total CF ÷ 2 = 30 ÷ 2 = 15 → read value from graph at CF = 15
- Q₁ position = Total CF ÷ 4 = 30 ÷ 4 = 7.5 → read value from graph at CF = 7.5
- Q₃ position = (3 × Total CF) ÷ 4 = (3 × 30) ÷ 4 = 22.5 → read value from graph at CF = 22.5
- IQR = Q₃ − Q₁
Tips for Drawing
- Mark cumulative frequencies against the upper class boundaries.
- Plot points clearly as × marks.
- Use a smooth curve, not a polygon.
- Extend curve only within the data range.