Question: QUESTION 2: BOX PLOT (DOK 3 - Strategic Thinking)
A mathematics teacher recorded the test scores (out of 100) of 15 students:
65, 72, 78, 81, 68, 85, 90, 76, 88, 92, 70, 84, 79, 86, 95
a) Arrange the data in ascending order and find the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum).
b) Draw a box plot to represent this data.
c) Identify any potential outliers using the Interquartile Range (IQR) method and explain your reasoning.
d) Interpret the box plot: What does the position of the median line within the box tell you about the distribution of test scores?
1. **State the problem:** We have 15 test scores and need to find the five-number summary, draw a box plot, identify outliers using IQR, and interpret the median's position.
2. **Arrange data in ascending order:**
$$65, 68, 70, 72, 76, 78, 79, 81, 84, 85, 86, 88, 90, 92, 95$$
3. **Find the five-number summary:**
- Minimum = $65$
- Maximum = $95$
- Median (middle value for 15 data points) is the 8th value: $81$
4. **Find $Q_1$ (lower quartile):** median of lower half (first 7 values): $65, 68, 70, 72, 76, 78, 79$
- Median of these 7 values is the 4th value: $72$
5. **Find $Q_3$ (upper quartile):** median of upper half (last 7 values): $84, 85, 86, 88, 90, 92, 95$
- Median of these 7 values is the 4th value: $88$
6. **Five-number summary:**
$$\text{Minimum} = 65, \quad Q_1 = 72, \quad \text{Median} = 81, \quad Q_3 = 88, \quad \text{Maximum} = 95$$
7. **Calculate Interquartile Range (IQR):**
$$IQR = Q_3 - Q_1 = 88 - 72 = 16$$
8. **Identify potential outliers:**
- Lower bound = $Q_1 - 1.5 \times IQR = 72 - 1.5 \times 16 = 72 - 24 = 48$
- Upper bound = $Q_3 + 1.5 \times IQR = 88 + 1.5 \times 16 = 88 + 24 = 112$
9. **Check for outliers:**
- Minimum $65$ is above lower bound $48$
- Maximum $95$ is below upper bound $112$
- Therefore, **no outliers**
10. **Interpret the box plot median position:**
- Median $81$ is closer to $Q_1=72$ than to $Q_3=88$
- This means the lower half of scores is more spread out, and the distribution is slightly skewed towards higher scores.
**Final answers:**
- Five-number summary: $65, 72, 81, 88, 95$
- No outliers detected by IQR method
- Median position indicates slight skewness towards higher scores