Question: QUESTION 2: BOX PLOT (DOK 3 - Strategic Thinking)
Total Marks: 13
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, Q1, median, Q3, 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 box plot.
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) = 8th value = $81$
4. **Find Q1 (first 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 Q3 (third 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:**
$$\text{Minimum} = 65, \quad Q1 = 72, \quad \text{Median} = 81, \quad Q3 = 88, \quad \text{Maximum} = 95$$
7. **Draw box plot:**
- Draw a number line covering $65$ to $95$
- Draw a box from $Q1=72$ to $Q3=88$
- Draw a line inside the box at the median $81$
- Draw whiskers from minimum $65$ to maximum $95$
8. **Identify outliers using IQR method:**
- Calculate IQR: $$IQR = Q3 - Q1 = 88 - 72 = 16$$
- Lower bound for outliers: $$Q1 - 1.5 \times IQR = 72 - 1.5 \times 16 = 72 - 24 = 48$$
- Upper bound for outliers: $$Q3 + 1.5 \times IQR = 88 + 1.5 \times 16 = 88 + 24 = 112$$
- Any data point below $48$ or above $112$ is an outlier.
- Since all scores are between $65$ and $95$, there are **no outliers**.
9. **Interpret the box plot:**
- The median line at $81$ is closer to $Q1=72$ than to $Q3=88$.
- This indicates the distribution is slightly skewed to the higher scores (right skewed).
- More scores are concentrated between $72$ and $81$ than between $81$ and $88$.
**Final answers:**
- Five-number summary: $65, 72, 81, 88, 95$
- No outliers detected
- Median position suggests slight right skew in scores distribution