IB Mathematics SL 4.10 Spearman’s rank correlation coefficient AI SL Paper 1- Exam Style Questions- New Syllabus
Question
| Rank | \( 1 \) | \( 2 \) | \( 3 \) | \( 4 \) | \( 5 \) | \( 6 \) | \( 7 \) | \( 8 \) |
|---|---|---|---|---|---|---|---|---|
| Brett | \( \text{A} \) | \( \text{C} \) | \( \text{D} \) | \( \text{B} \) | \( \text{E} \) | \( \text{F} \) | \( \text{G} \) | \( \text{H} \) |
| Santa | \( \text{A} \) | \( \text{B} \) | \( \text{D} \) | \( \text{C} \) | \( \text{E} \) | \( \text{G} \) | \( \text{F} \) | \( \text{H} \) |
Most-appropriate topic codes (IB Math AI 2025):
• SL 4.10: Calculation and interpretation of Spearman’s rank correlation coefficient — parts (b), (c)
▶️ Answer/Explanation
(a)
From Brett’s row, dog B is in rank position 4.
\(\boxed{4}\)
(b)
First, assign numerical ranks to each dog for both judges:
| Dog | A | B | C | D | E | F | G | H |
|---|---|---|---|---|---|---|---|---|
| Brett \(R_B\) | 1 | 4 | 2 | 3 | 5 | 6 | 7 | 8 |
| Santa \(R_C\) | 1 | 2 | 4 | 3 | 5 | 7 | 6 | 8 |
| Diff. \(d\) | 0 | 2 | -2 | 0 | 0 | -1 | 1 | 0 |
| \(d^2\) | 0 | 4 | 4 | 0 | 0 | 1 | 1 | 0 |
Sum of \(d^2 = 0 + 4 + 4 + 0 + 0 + 1 + 1 + 0 = 10\)
Spearman’s rank correlation coefficient:
\( r_s = 1 – \frac{6 \sum d^2}{n(n^2 – 1)} = 1 – \frac{6 \times 10}{8 \times (64 – 1)} \)
\( = 1 – \frac{60}{8 \times 63} = 1 – \frac{60}{504} \)
\( = 1 – 0.1190476\ldots = 0.880952\ldots \approx 0.881 \)
\(\boxed{0.881}\) (to 3 s.f.)
(c)
Since \( r_s \) is close to +1, there is a strong positive correlation between Brett’s and Santa’s rankings. This means the judges largely agree on the ordering of the dogs’ skill levels, though not perfectly (since \( r_s \neq 1 \)).
Comment: There is a strong positive association between the rankings given by Brett and Santa. They largely agree but not completely.