Histograms and Cumulative Frequency [H]
Higher Tier — This topic is assessed at Higher tier only (grades 4–9).
Histograms and Cumulative Frequency
Histograms with Unequal Class Widths
In a histogram, the area of each bar (not the height) represents the frequency. When class widths are unequal, frequency density is used on the vertical axis:
\[ \text{Frequency density} = \frac{\text{frequency}}{\text{class width}} \] \[ \text{Frequency} = \text{frequency density} \times \text{class width} \]There are no gaps between bars, and all bars are adjacent (continuous data).
Example — Reading a Histogram
| Class | Width | Freq. density | Frequency |
|---|---|---|---|
| 0 ≤ t < 10 | 10 | 2.5 | 25 |
| 10 ≤ t < 20 | 10 | 4.0 | 40 |
| 20 ≤ t < 30 | 10 | 3.0 | 30 |
| 30 ≤ t < 50 | 20 | 1.5 | 30 |
Cumulative Frequency
A cumulative frequency table records the running total of frequencies up to and including each class. The cumulative frequency graph (ogive) plots cumulative frequency against the upper class boundary. It is an S-shaped (ogive) curve.
From the curve, read off:
- Median: value at cumulative frequency = \( n/2 \).
- Lower quartile (Q1): value at \( n/4 \).
- Upper quartile (Q3): value at \( 3n/4 \).
- Interquartile range (IQR): \( Q_3-Q_1 \).
Key Takeaways
- Histogram: area = frequency. Use frequency density (freq ÷ class width) on y-axis when class widths differ.
- No gaps between histogram bars — continuous data.
- Cumulative frequency: running total plotted at upper class boundaries.
- Read median, Q1, Q3 at \( n/2 \), \( n/4 \), \( 3n/4 \) on the cumulative frequency axis.
- IQR = Q3 − Q1; estimates the spread of the middle 50% of the data.