Two numbers within the bracket denote the rank of 10 students of a class in two subjects : `(1,10),(2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2),(10,1)`. Then rank correlation coefficient is

Options

A. 0
B. `-1`
C. 1
D. `0.5`

Answer

To find the rank correlation coefficient for the given ranks of 10 students in two subjects, we will use Spearman's rank correlation formula. Here’s a step-by-step solution:

### Step 1: List the Ranks
We have the ranks of 10 students in two subjects:
- Subject 1 ranks: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
- Subject 2 ranks: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

### Step 2: Create a Difference Table
We will calculate the difference (D) between the ranks in the two subjects for each student.

| Student | Rank in Subject 1 (X) | Rank in Subject 2 (Y) | D = X - Y |
|---------|------------------------|------------------------|-----------|
| 1 | 1 | 10 | -9 |
| 2 | 2 | 9 | -7 |
| 3 | 3 | 8 | -5 |
| 4 | 4 | 7 | -3 |
| 5 | 5 | 6 | -1 |
| 6 | 6 | 5 | 1 |
| 7 | 7 | 4 | 3 |
| 8 | 8 | 3 | 5 |
| 9 | 9 | 2 | 7 |
| 10 | 10 | 1 | 9 |

### Step 3: Calculate D²
Next, we will square the differences (D²).

| Student | D | D² |
|---------|----|-----|
| 1 | -9 | 81 |
| 2 | -7 | 49 |
| 3 | -5 | 25 |
| 4 | -3 | 9 |
| 5 | -1 | 1 |
| 6 | 1 | 1 |
| 7 | 3 | 9 |
| 8 | 5 | 25 |
| 9 | 7 | 49 |
| 10 | 9 | 81 |

### Step 4: Sum of D²
Now, we will calculate the sum of D².

\[ \sum D² = 81 + 49 + 25 + 9 + 1 + 1 + 9 + 25 + 49 + 81 = 330 \]

### Step 5: Apply the Spearman's Rank Correlation Formula
The formula for Spearman's rank correlation coefficient (R) is:

\[ R = 1 - \frac{6 \sum D²}{N(N^2 - 1)} \]

Where:
- \(N\) = number of students = 10
- \(\sum D² = 330\)

Substituting the values:

\[ R = 1 - \frac{6 \times 330}{10(10^2 - 1)} \]
\[ R = 1 - \frac{1980}{10(100 - 1)} \]
\[ R = 1 - \frac{1980}{10 \times 99} \]
\[ R = 1 - \frac{1980}{990} \]
\[ R = 1 - 2 = -1 \]

### Final Result
The rank correlation coefficient \(R\) is \(-1\).