For the following data :
`sum (x-overset(-)(x))^(2)=25, sum (y-overset(y))^(2)=256, sum (x-overset(-)(x)) (y-overset(y))=80`, find the Karl Pearson's coefficeint of correlation.

Answer

To find Karl Pearson's coefficient of correlation (denoted as \( r \)), we can use the formula:

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

Given data:
- \( \sum (x - \overline{x})^2 = 25 \)
- \( \sum (y - \overline{y})^2 = 256 \)
- \( \sum (x - \overline{x})(y - \overline{y}) = 80 \)

### Step 1: Substitute the values into the formula

Now, we can substitute the given values into the formula for \( r \):

\[ r = \frac{80}{\sqrt{25} \cdot \sqrt{256}} \]

### Step 2: Calculate the square roots

Next, we calculate the square roots:

\[ \sqrt{25} = 5 \]
\[ \sqrt{256} = 16 \]

### Step 3: Substitute the square root values back into the formula

Now substitute these values back into the formula:

\[ r = \frac{80}{5 \cdot 16} \]

### Step 4: Calculate the denominator

Now calculate the denominator:

\[ 5 \cdot 16 = 80 \]

### Step 5: Calculate the final value of \( r \)

Now substitute this back into the equation:

\[ r = \frac{80}{80} = 1 \]

### Conclusion

Thus, the Karl Pearson's coefficient of correlation \( r \) is:

\[ \boxed{1} \]