Question 4 (c)
Correcting a rank correlation coefficient for a wrongly taken rank difference
4 marks
- Time to write
- About 8 minutes
- Topic
- Correlation
Question
Question 4 (c) · 4 marksThe coefficient of rank correlation between the return and the risk of 15 securities was found to be 0.3. It was later found that, for one security, the difference between the ranks in return and risk had been wrongly taken as 8 instead of 6. Find the correct value of the rank correlation coefficient.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- FindFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 ΣD² recovered from the given coefficient 2 marks 3 min
- 2 Correction of ΣD² and the correct coefficient 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 8 minutes for 4 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Formula. Spearman's rank correlation coefficient is ρ = 1 − 6ΣD² / [n(n² − 1)], where D is the difference between the two ranks of an item and n the number of items. Here n = 15 securities, so n(n² − 1) = 15 × (225 − 1) = 15 × 224 = 3,360.
The sum of squared rank differences used
The coefficient found, 0.3, was computed from the wrong ΣD²:
0.3 = 1 − 6ΣD² / 3,360 ⇒ 6ΣD² = (1 − 0.3) × 3,360 = 2,352 ⇒ ΣD² (wrong) = 2,352 / 6 = 392.
One difference was taken as 8 instead of 6, so 8² = 64 was included where 6² = 36 belongs:
ΣD² (correct) = 392 − 64 + 36 = 364.
Correct rank correlation coefficient
ρ = 1 − 6 × 364 / 3,360 = 1 − 2,184 / 3,360 = 1 − 0.65 = 0.35.
The corrected coefficient, 0.35, shows a low positive rank correlation between return and risk for these 15 securities: securities ranked higher on return tend, weakly, to be ranked higher on risk as well. The correction raises the coefficient slightly, because the wrong difference had overstated the disagreement between the two rankings.
Where the 4 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | ΣD² recovered from the given coefficient | 2 |
| 2 | Correction of ΣD² and the correct coefficient | 2 |
| Total | 4 |
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