The marks of seven students in intelligence and arithmetic tests are as follows :
`{:("Candidate",A,B,C,D,E,F,G),("Intelligence test", 30,52,60,62,45,32,41),("Arithmetic test", 41,62,70,78,53,45,57):}`
Calculate Karl Person's coefficient of correlation and interpret it .

Answer

To calculate Karl Pearson's coefficient of correlation for the given data, we will follow these steps:

### Step 1: Calculate the Mean of Intelligence and Arithmetic Test Scores

1. **Intelligence Test Scores**:
- Marks: 30, 52, 60, 62, 45, 32, 41
- Mean (X̄) = (30 + 52 + 60 + 62 + 45 + 32 + 41) / 7
- Mean (X̄) = 322 / 7 = 46

2. **Arithmetic Test Scores**:
- Marks: 41, 62, 70, 78, 53, 45, 57
- Mean (Ȳ) = (41 + 62 + 70 + 78 + 53 + 45 + 57) / 7
- Mean (Ȳ) = 406 / 7 = 58

### Step 2: Create a Table for Deviations and Products

| Candidate | X (Intelligence) | Y (Arithmetic) | X - X̄ | Y - Ȳ | (X - X̄)² | (Y - Ȳ)² | (X - X̄)(Y - Ȳ) |
|-----------|------------------|----------------|--------|--------|-----------|-----------|------------------|
| A | 30 | 41 | -16 | -17 | 256 | 289 | 272 |
| B | 52 | 62 | 6 | 4 | 36 | 16 | 24 |
| C | 60 | 70 | 14 | 12 | 196 | 144 | 168 |
| D | 62 | 78 | 16 | 20 | 256 | 400 | 320 |
| E | 45 | 53 | -1 | -5 | 1 | 25 | 5 |
| F | 32 | 45 | -14 | -13 | 196 | 169 | 182 |
| G | 41 | 57 | -5 | -1 | 25 | 1 | 5 |

### Step 3: Calculate Summations

- \( \Sigma (X - X̄)² = 256 + 36 + 196 + 256 + 1 + 196 + 25 = 966 \)
- \( \Sigma (Y - Ȳ)² = 289 + 16 + 144 + 400 + 25 + 169 + 1 = 1044 \)
- \( \Sigma (X - X̄)(Y - Ȳ) = 272 + 24 + 168 + 320 + 5 + 182 + 5 = 976 \)

### Step 4: Calculate Karl Pearson's Coefficient of Correlation (r)

Using the formula:
\[ r = \frac{\Sigma (X - X̄)(Y - Ȳ)}{\sqrt{\Sigma (X - X̄)² \cdot \Sigma (Y - Ȳ)²}} \]

Substituting the values:
\[ r = \frac{976}{\sqrt{966 \cdot 1044}} \]

Calculating the denominator:
\[ \sqrt{966 \cdot 1044} = \sqrt{1008004} \approx 1004 \]

Thus,
\[ r = \frac{976}{1004} \approx 0.971 \]

### Step 5: Interpretation

The value of Karl Pearson's coefficient of correlation \( r \approx 0.971 \) indicates a very strong positive correlation between the marks obtained in the intelligence test and the arithmetic test. This suggests that students who perform well in the intelligence test tend to also perform well in the arithmetic test.

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