Spearman’s rank correlation
Answers are for parents and teachers.
1
Give the ranks (smallest = 1) of the weights 15, 20, 15, 10, 25, 15 kg.
In order: 10, 15, 15, 15, 20, 25. Ranks: 3, 5, 3, 1, 6, 3 (the three 15’s share places 2–4; sum 21).
2
Six pupils: x = hours of practice and y = score (table). Find Spearman’s r.
| x | 8 | 5 | 9 | 6 | 5 | 7 |
|---|---|---|---|---|---|---|
| y | 14 | 9 | 16 | 12 | 10 | 12 |
Ranks of x: 5, 1.5, 6, 3, 1.5, 4. Ranks of y: 5, 1, 6, 3.5, 2, 3.5. D = 0, 0.5, 0, −0.5, −0.5, 0.5 (sum 0); ΣD² = 1. r = 1 − 6 ÷ 210 = 0.9714.
3
Yeast cells in a school experiment are counted each day: x = 1, 2, 3, 4, 5 and y = 1, 3, 9, 27, 81 (in thousands). Which coefficient is larger, Spearman’s or Pearson’s? Give both values.
Spearman’s r = 1 (y always rises with x); Pearson’s r = 0.8720 (the dots curve upward steeply).
4
Two teachers give six pupils levels. Teacher 1: pass, good, excellent, pass, weak, good. Teacher 2: good, good, very good, weak, weak, pass. Use weak < pass < good < very good < excellent. Find Spearman’s r.
Ranks 1: 2.5, 4.5, 6, 2.5, 1, 4.5. Ranks 2: 4.5, 4.5, 6, 1.5, 1.5, 3. D = −2, 0, 0, 1, −0.5, 1.5 (sum 0); ΣD² = 4 + 0 + 0 + 1 + 0.25 + 2.25 = 7.5. r = 1 − 45 ÷ 210 = 0.7857.
5
A student calculates ΣD² = −4. Explain why this must be wrong.
D² is a square, so it can never be negative, and neither can the sum of squares.