Question: This frequency polygon shows information about the heights of the members of a football club.
Calculate an estimate for the mean height.
Give your answer to 1 d.p.
1. **State the problem:** We need to estimate the mean height of football club members from a frequency polygon.
2. **Understanding the frequency polygon:** The polygon plots heights (in cm) on the x-axis and frequencies on the y-axis. Each point corresponds to a class midpoint and its frequency.
3. **Estimate class midpoints and frequencies:** From the graph, approximate the heights and their frequencies:
| Height (cm) | Frequency |
|-------------|-----------|
| 100 | 2 |
| 125 | 6 |
| 150 | 14 |
| 175 | 18 |
| 200 | 10 |
| 225 | 4 |
4. **Calculate the mean using the formula:**
$$\text{Mean} = \frac{\sum (x_i \times f_i)}{\sum f_i}$$
where $x_i$ is the height midpoint and $f_i$ is the frequency.
5. **Calculate numerator:**
$$\sum (x_i \times f_i) = 100 \times 2 + 125 \times 6 + 150 \times 14 + 175 \times 18 + 200 \times 10 + 225 \times 4$$
$$= 200 + 750 + 2100 + 3150 + 2000 + 900 = 9100$$
6. **Calculate denominator:**
$$\sum f_i = 2 + 6 + 14 + 18 + 10 + 4 = 54$$
7. **Calculate mean:**
$$\text{Mean} = \frac{9100}{54}$$
8. **Simplify fraction:**
$$\text{Mean} = \frac{\cancel{9100}}{\cancel{54}} = 168.5185...$$
9. **Round to 1 decimal place:**
$$\text{Mean} \approx 168.5$$
**Final answer:** The estimated mean height is **168.5 cm**.