Question: Estimate the interquartile range of the data shown below.
Circumference of tree trunks
Cumulative frequency
80
60
40
20
0
0
50
100
150
200
250
300
Circumference (cm)
Graph: a cumulative frequency curve labeled "Circumference of tree trunks" is centered on the page; the curve rises from top-left to top-right and flattens near a cumulative frequency of 60 at around 300 cm.
1. **State the problem:** We need to estimate the interquartile range (IQR) of the tree trunk circumferences from the cumulative frequency graph.
2. **Recall the formula:** The interquartile range is given by $$\text{IQR} = Q_3 - Q_1$$ where $Q_1$ is the first quartile (25th percentile) and $Q_3$ is the third quartile (75th percentile).
3. **Find total frequency:** From the graph, the maximum cumulative frequency is approximately $60$.
4. **Calculate positions of quartiles:**
- $Q_1$ position = $\frac{1}{4} \times 60 = 15$
- $Q_3$ position = $\frac{3}{4} \times 60 = 45$
5. **Estimate $Q_1$ from the graph:** Locate cumulative frequency $15$ on the vertical axis and find the corresponding circumference on the horizontal axis. From the graph, this is approximately $100$ cm.
6. **Estimate $Q_3$ from the graph:** Locate cumulative frequency $45$ on the vertical axis and find the corresponding circumference on the horizontal axis. From the graph, this is approximately $250$ cm.
7. **Calculate the interquartile range:**
$$\text{IQR} = Q_3 - Q_1 = 250 - 100 = 150$$
**Final answer:** The estimated interquartile range of the tree trunk circumferences is $150$ cm.