The marks of ten intelligent students in two subjects as given below.
`{:("Students","English","Physics"),(A,88,74),(B,74,66),(C,80,72),(D,76,90),(E,40,54),(F,65,50),(G,68,54),(H,80,43),(I,40,30),(J,43,38):}`
Calculate the rank correlation coefficient and interpret your result.

Answer

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

### Step 1: Organize the Data
We will create a table to list the students along with their marks in English and Physics.

| Students | English | Physics |
|----------|---------|---------|
| A | 88 | 74 |
| B | 74 | 66 |
| C | 80 | 72 |
| D | 76 | 90 |
| E | 40 | 54 |
| F | 65 | 50 |
| G | 68 | 54 |
| H | 80 | 43 |
| I | 40 | 30 |
| J | 43 | 38 |

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

#### Ranks for English:
- A: 1 (88)
- C: 2 (80)
- H: 2 (80) (tie)
- D: 4 (76)
- B: 5 (74)
- F: 6 (65)
- G: 7 (68)
- E: 8 (40)
- I: 9 (40) (tie)
- J: 10 (43)

#### Ranks for Physics:
- D: 1 (90)
- A: 2 (74)
- B: 3 (66)
- C: 4 (72)
- E: 5 (54)
- G: 5 (54) (tie)
- F: 7 (50)
- H: 8 (43)
- J: 9 (38)
- I: 10 (30)

### Step 3: Create a Rank Table
Now we will create a table with the ranks and calculate the difference (d) between the ranks.

| Students | English Rank (R1) | Physics Rank (R2) | d = R1 - R2 | d² |
|----------|--------------------|-------------------|-------------|------|
| A | 1 | 2 | -1 | 1 |
| B | 5 | 3 | 2 | 4 |
| C | 2 | 4 | -2 | 4 |
| D | 4 | 1 | 3 | 9 |
| E | 8 | 5 | 3 | 9 |
| F | 6 | 7 | -1 | 1 |
| G | 7 | 5 | 2 | 4 |
| H | 2 | 8 | -6 | 36 |
| I | 9 | 10 | -1 | 1 |
| J | 10 | 9 | 1 | 1 |

### Step 4: Calculate Summation of d²
Now we will sum up the d² values.

\[ \sum d² = 1 + 4 + 4 + 9 + 9 + 1 + 4 + 36 + 1 + 1 = 70 \]

### Step 5: Calculate the Rank Correlation Coefficient (r)
We will use the formula for the rank correlation coefficient:

\[ r = 1 - \frac{6 \sum d²}{n(n² - 1)} \]

Where:
- \( n = 10 \) (number of students)

Substituting the values:

\[ r = 1 - \frac{6 \times 70}{10(10² - 1)} = 1 - \frac{420}{10 \times 99} = 1 - \frac{420}{990} = 1 - 0.4242 = 0.5758 \]

### Step 6: Interpretation
The rank correlation coefficient \( r = 0.5758 \) indicates a moderate positive correlation between the marks in English and Physics. This suggests that students who perform well in English tend to also perform reasonably well in Physics, although the correlation is not very strong.