Calculate Kari Pearson's coefficient of correlation between the values of x and y for the following data :
`n = 10, sum x = 55, sum y = 40, sum x^(2) = 385, sum y^(2) = 192 and sum (x + y)^(2) = 947`

Answer

To calculate the Karl Pearson's coefficient of correlation (r) between the values of x and y, we will use the following formula:

\[ r = \frac{n \sum xy - \sum x \sum y}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}} \]

### Step 1: Identify the given values
From the problem statement, we have:
- \( n = 10 \)
- \( \sum x = 55 \)
- \( \sum y = 40 \)
- \( \sum x^2 = 385 \)
- \( \sum y^2 = 192 \)
- \( \sum (x + y)^2 = 947 \)

### Step 2: Find \( \sum xy \)
We know that:

\[ \sum (x + y)^2 = \sum x^2 + \sum y^2 + 2 \sum xy \]

Substituting the known values:

\[ 947 = 385 + 192 + 2 \sum xy \]

This simplifies to:

\[ 947 = 577 + 2 \sum xy \]

Now, isolating \( \sum xy \):

\[ 2 \sum xy = 947 - 577 \]
\[ 2 \sum xy = 370 \]
\[ \sum xy = \frac{370}{2} = 185 \]

### Step 3: Substitute values into the formula
Now that we have \( \sum xy = 185 \), we can substitute all values into the formula for \( r \):

\[ r = \frac{10 \cdot 185 - 55 \cdot 40}{\sqrt{(10 \cdot 385 - 55^2)(10 \cdot 192 - 40^2)}} \]

### Step 4: Calculate the numerator
Calculating the numerator:

\[ 10 \cdot 185 = 1850 \]
\[ 55 \cdot 40 = 2200 \]
\[ \text{Numerator} = 1850 - 2200 = -350 \]

### Step 5: Calculate the denominator
Calculating the denominator:

1. Calculate \( n \sum x^2 - (\sum x)^2 \):

\[ 10 \cdot 385 - 55^2 = 3850 - 3025 = 1825 \]

2. Calculate \( n \sum y^2 - (\sum y)^2 \):

\[ 10 \cdot 192 - 40^2 = 1920 - 1600 = 320 \]

3. Now, calculate the denominator:

\[ \text{Denominator} = \sqrt{1825 \cdot 320} \]

Calculating \( 1825 \cdot 320 \):

\[ 1825 \cdot 320 = 584000 \]
\[ \sqrt{584000} \approx 764.42 \]

### Step 6: Final Calculation of \( r \)
Now substituting back into the formula:

\[ r = \frac{-350}{764.42} \approx -0.458 \]

### Final Answer
Thus, the Karl Pearson's coefficient of correlation \( r \approx -0.458 \).