Histograms and Cumulative Frequency
Histograms
A histogram looks like a bar chart, but it is used for continuous data grouped into classes. The key difference:
In a histogram, frequency density is plotted on the y-axis, not frequency.
$$\text{Frequency density} = \frac{\text{Frequency}}{\text{Class width}}$$
$$\text{Frequency} = \text{Frequency density} \times \text{Class width}$$
The area of each bar represents the frequency.
Why? When classes have unequal widths, a wider class would appear to have more data just because it is wider — frequency density corrects for this.
Reading a Histogram
Example: A bar has frequency density 4 and class width 5. $$\text{Frequency} = 4 \times 5 = 20$$
Drawing a Histogram
Example Data:
| Height (cm) | Frequency | Class width | Freq. density |
|---|---|---|---|
| 150–160 | 15 | 10 | 1.5 |
| 160–165 | 20 | 5 | 4.0 |
| 165–170 | 30 | 5 | 6.0 |
| 170–180 | 24 | 10 | 2.4 |
Note: the 160–165 and 165–170 bars are narrower but taller; their areas (20, 30) correctly represent their frequencies.
Cumulative Frequency
Cumulative frequency is a running total of frequencies. It shows how many data values are at or below a given value.
Constructing a Cumulative Frequency Table
Add a running total to your frequency table:
| Score | Freq | Cumulative Freq |
|---|---|---|
| 0–20 | 5 | 5 |
| 20–40 | 12 | 17 |
| 40–60 | 18 | 35 |
| 60–80 | 10 | 45 |
| 80–100 | 5 | 50 |
Drawing the Graph
Plot cumulative frequency against the upper class boundary of each interval. Join points with a smooth S-shaped curve (ogive).
Reading Off Statistics
From a cumulative frequency graph of $n$ values:
| Statistic | Position on y-axis |
|---|---|
| Median | $n/2$ |
| Lower quartile (Q1) | $n/4$ |
| Upper quartile (Q3) | $3n/4$ |
| Interquartile range | Q3 − Q1 |
IQR (interquartile range) measures the spread of the middle 50% of data.
Box Plots
Cumulative frequency graphs are often paired with box plots (box-and-whisker plots), which show: minimum, Q1, median, Q3, maximum.
WJEC Exam Tips
- Histogram: never plot raw frequency on the y-axis for unequal class widths.
- Always calculate frequency density = frequency ÷ class width.
- Cumulative frequency: plot at the end (upper boundary) of each class.
- Read off values carefully — use a ruler and draw a horizontal/vertical line on your graph.
- IQR is the preferred measure of spread for skewed data.
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