Histograms and cumulative frequency
Histograms
Histograms display continuous grouped data. Unlike bar charts, it is the area of each bar (not height) that represents frequency.
Frequency density = Frequency ÷ Class width.
The y-axis of a histogram is always labelled "Frequency density" — not "Frequency".
Reading a histogram: Frequency = Frequency density × Class width.
Example: a bar with fd = 3 and class width 10 → frequency = 30.
When class widths are unequal, taller bars do NOT necessarily have the highest frequency — you must calculate area.
Constructing a histogram
| Class | Frequency | Class width | Frequency density |
|---|---|---|---|
| 0–10 | 20 | 10 | 2 |
| 10–15 | 30 | 5 | 6 |
| 15–25 | 40 | 10 | 4 |
Cumulative frequency graphs
Plot cumulative (running total) frequency against the upper class boundary.
- Join points with a smooth curve (not straight lines between points, ideally).
- The curve always starts at (lower boundary of first class, 0).
Reading off estimates from the curve:
- Median: read off at cumulative frequency = n/2.
- Lower quartile (Q1): read off at n/4.
- Upper quartile (Q3): read off at 3n/4.
- IQR = Q3 − Q1.
- Percentiles: p-th percentile → cumulative frequency = pn/100.
Box plots can be drawn from Q1, median, Q3, min, max.
⚠Common mistakes
- Plotting cumulative frequency against midpoints instead of upper class boundaries.
- Using bar heights as frequencies in a histogram — always multiply by class width.
- Not starting the cumulative frequency curve at zero.
- Reading off the wrong value: for the median, read at n/2 on the y-axis, then across to the curve, then down to the x-axis.
AI-generated · claude-opus-4-7 · v3-edexcel-maths