Question 3 (a)
Spearman's rank correlation of performance and customer satisfaction, with tied ranks
9 marks
- Time to write
- About 18 minutes
- Topic
- Correlation
Question
Question 3 (a) · 9 marksA company that installs water purifiers is studying how employee performance relates to customer satisfaction in its sales department. The performance scores and the consumer satisfaction scores of 10 sales representatives are shown below:
| Sales Representatives | Performance Scores | Customer Satisfaction Scores |
|---|---|---|
| A | 47 | 44 |
| B | 52 | 49 |
| C | 43 | 40 |
| D | 44 | 47 |
| E | 40 | 43 |
| F | 49 | 46 |
| G | 42 | 40 |
| H | 33 | 36 |
| I | 37 | 39 |
| J | 44 | 42 |
Determine, using Spearman's Rank Correlation Coefficient, whether there is a significant correlation between the employees' performance scores and the customer satisfaction scores. Interpret the result.
Reworded from the university paper; 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
- Determine…Interpret…Two asks in one question: answer both.
Write it in this order
- Read; underline the asks; jot your points 1 min
- 1 Ranking, including tied ranks 3 marks 5 min
- 2 ΣD² and the tie correction 2 marks 4 min
- 3 Rank correlation coefficient 2 marks 4 min
- 4 Significance and interpretation 2 marks 3 min
- Check: every ask answered, key terms underlined 1 min
Short on time? Trim steps 1–4 evenly rather than skip one: each is capped at its own marks.
Try it first. About 18 minutes for 9 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.
Ranking. Rank 1 is given to the highest score in each series. Where scores are equal, each receives the average of the ranks they would occupy: performance 44 (D and J) share ranks 4 and 5, so each gets 4.5; satisfaction 40 (C and G) share ranks 7 and 8, so each gets 7.5. With tied ranks, Spearman's formula adds a correction of (m³ − m) / 12 to ΣD² for every group of m tied values:
ρ = 1 − 6[ΣD² + Σ(m³ − m)/12] / [n(n² − 1)], with D the difference in ranks.
| Representative | Performance | Satisfaction | Rank R₁ | Rank R₂ | D = R₁ − R₂ | D² |
|---|---|---|---|---|---|---|
| A | 47 | 44 | 3 | 4 | −1 | 1 |
| B | 52 | 49 | 1 | 1 | 0 | 0 |
| C | 43 | 40 | 6 | 7.5 | −1.5 | 2.25 |
| D | 44 | 47 | 4.5 | 2 | 2.5 | 6.25 |
| E | 40 | 43 | 8 | 5 | 3 | 9 |
| F | 49 | 46 | 2 | 3 | −1 | 1 |
| G | 42 | 40 | 7 | 7.5 | −0.5 | 0.25 |
| H | 33 | 36 | 10 | 10 | 0 | 0 |
| I | 37 | 39 | 9 | 9 | 0 | 0 |
| J | 44 | 42 | 4.5 | 6 | −1.5 | 2.25 |
| Total | 0 | 22 |
Two ties of m = 2 (one in each series): correction = (2³ − 2)/12 + (2³ − 2)/12 = 0.5 + 0.5 = 1.
ρ = 1 − 6(22 + 1) / [10(10² − 1)] = 1 − 138 / 990 = 1 − 0.1394 = 0.8606.
Is the correlation significant? Judged by its probable error (Unit 3 of the syllabus,): P.E. = 0.6745 (1 − ρ²) / √n = 0.6745 × (1 − 0.7406) / √10 = 0.0553, and 6 × P.E. = 0.3319. Since ρ = 0.8606 is far larger than 6 × P.E., the correlation is significant: it is not a chance result of these ten representatives.
Interpretation. There is a high positive rank correlation (0.86) between the representatives' performance scores and the satisfaction of their customers: representatives who perform better generally leave customers more satisfied. The company can use performance reviews and training as a lever on customer satisfaction, keeping in mind that correlation alone does not prove which one causes the other.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Ranking, including tied ranks | 3 |
| 2 | ΣD² and the tie correction | 2 |
| 3 | Rank correlation coefficient | 2 |
| 4 | Significance and interpretation | 2 |
| Total | 9 |
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