Histograms and cumulative frequency
These are Paper 3 (calculator) topics on OCR J560. Histograms with unequal class widths are a common higher-tier question (frequency density). Cumulative frequency graphs enable reading off estimates for medians and quartiles.
Histograms
Equal class widths
Frequency on y-axis (same as a bar chart). Bars touch (continuous data, no gaps).
Unequal class widths — Frequency Density
When class widths differ, you CANNOT use frequency on the y-axis (bars would be misleading). Use frequency density:
Frequency density = Frequency ÷ Class width
To find frequency from a histogram: Frequency = Frequency density × Class width (area of bar).
Example:
| Class | Width | Freq density | Frequency |
|---|---|---|---|
| 0–10 | 10 | 3 | 30 |
| 10–15 | 5 | 6 | 30 |
| 15–25 | 10 | 2 | 20 |
| 25–40 | 15 | 1 | 15 |
The bar for 10–15 is twice as tall as 0–10 even though they have the same frequency — because the class is half as wide.
Cumulative frequency
Building a table
Add up frequencies as you go through the classes:
| Height (cm) | Freq | Cumulative freq |
|---|---|---|
| 140–150 | 8 | 8 |
| 150–160 | 15 | 23 |
| 160–170 | 20 | 43 |
| 170–180 | 7 | 50 |
Plotting the graph
Plot the upper class boundary against the cumulative frequency. Connect points with a smooth curve. Start at (140, 0) — lower boundary of first class.
Reading off estimates
For n data points:
- Median: n/2 th value → read off x from the curve.
- Lower quartile (Q1): n/4 th value.
- Upper quartile (Q3): 3n/4 th value.
- Interquartile range (IQR) = Q3 − Q1.
Example (n=50): Median at 25th value; Q1 at 12.5th (≈13th); Q3 at 37.5th (≈38th).
Box plots from cumulative frequency
From the cumulative frequency graph, read off: minimum, Q1, median, Q3, maximum to draw a box plot.
Common OCR exam mistakes
- Using frequency instead of frequency density for unequal class-width histograms.
- Plotting cumulative frequency at the class MID-POINT instead of the UPPER boundary.
- Reading off Q1 as n/4 but forgetting it's the value at the n/4 th count, then reading across to the x-axis.
- Forgetting to include the (lower boundary, 0) starting point on the cumulative frequency graph.
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