The marks in Physics and Biology of 12 students in a public examination are as follows :
`{:("Physics",69,36,39,71,67,76,40,20,85,65,55,34),("Biology",33,52,71,25,79,22,83,81,24,35,46,64):}`
Calculate the coefficient of rank correlation. What conclusion can be made from the result ?

Answer

To calculate the coefficient of rank correlation for the given marks in Physics and Biology of 12 students, we will follow these steps:

### Step 1: Organize the Data
We will create a table to organize the marks of each student in Physics and Biology, along with their ranks.

| Student | Physics Marks | Biology Marks |
|---------|---------------|---------------|
| A | 69 | 33 |
| B | 36 | 52 |
| C | 39 | 71 |
| D | 71 | 25 |
| E | 67 | 79 |
| F | 76 | 22 |
| G | 40 | 83 |
| H | 20 | 81 |
| I | 85 | 24 |
| J | 65 | 35 |
| K | 55 | 46 |
| L | 34 | 64 |

### Step 2: Assign Ranks
Next, we will assign ranks to the marks in Physics and Biology. The highest mark gets the rank of 1, the second highest gets 2, and so on. If there are ties, we assign the average rank.

**Ranks for Physics:**
- 69 (4)
- 36 (10)
- 39 (9)
- 71 (3)
- 67 (5)
- 76 (2)
- 40 (8)
- 20 (12)
- 85 (1)
- 65 (6)
- 55 (7)
- 34 (11)

**Ranks for Biology:**
- 33 (9)
- 52 (6)
- 71 (4)
- 25 (10)
- 79 (3)
- 22 (12)
- 83 (1)
- 81 (2)
- 24 (11)
- 35 (8)
- 46 (7)
- 64 (5)

### Step 3: Create a Rank Difference Table
Now, we will create a table to find the difference in ranks (d) and the square of the differences (d²).

| Student | Rank in Physics | Rank in Biology | d (Rank Difference) | d² (d Squared) |
|---------|------------------|------------------|---------------------|-----------------|
| A | 4 | 9 | 5 | 25 |
| B | 10 | 6 | 4 | 16 |
| C | 9 | 4 | 5 | 25 |
| D | 3 | 10 | 7 | 49 |
| E | 5 | 3 | 2 | 4 |
| F | 2 | 12 | 10 | 100 |
| G | 8 | 1 | 7 | 49 |
| H | 12 | 2 | 10 | 100 |
| I | 1 | 11 | 10 | 100 |
| J | 6 | 8 | 2 | 4 |
| K | 7 | 7 | 0 | 0 |
| L | 11 | 5 | 6 | 36 |

### Step 4: Calculate the Sum of d²
Now, we will calculate the sum of d²:
\[ \sum d² = 25 + 16 + 25 + 49 + 4 + 100 + 49 + 100 + 100 + 4 + 0 + 36 = 508 \]

### Step 5: Use the Rank Correlation Formula
The formula for the coefficient of rank correlation (r) is:
\[ r = 1 - \frac{6 \sum d²}{n(n² - 1)} \]
Where \( n \) is the number of students.

Substituting the values:
- \( n = 12 \)
- \( \sum d² = 508 \)

Calculating:
\[ r = 1 - \frac{6 \times 508}{12(12² - 1)} = 1 - \frac{3048}{12 \times 143} = 1 - \frac{3048}{1716} \]
Calculating \( 12 \times 143 = 1716 \):
\[ r = 1 - 1.77 = -0.77 \]

### Conclusion
Since \( r = -0.77 \), which is less than 0, we can conclude that there is a strong negative correlation between the marks in Physics and Biology. This implies that students who performed well in Physics tended to perform poorly in Biology, and vice versa.