$$ \mathrm{Y}: \mid $$ Arithmetic means of X and Y series are 6 and 8 respectively. In order to find the correlation coefficient between the two variables X and Y from 12 pair of observations, the following calculations were made: $$ \Sigma \mathrm{X}=30, \quad \Sigma \mathrm{Y}=5, \quad \Sigma \mathrm{X}^{2}=670, \quad \Sigma \mathrm{Y}^{2}=285, \quad \Sigma \mathrm{XY}=334 $$ On subsequent verification, it was found that the pair $$ (\mathrm{X}=11, \mathrm{Y}=4) $$ was copied wrong, the correct value being $$ (\mathrm{X}=10, \mathrm{Y}=14) $$. Find the correct value of correlation coefficient.

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To find the correct value of the correlation coefficient, we need to recalculate the values of $$ \Sigma \mathrm{X} $$, $$ \Sigma \mathrm{Y} $$, $$ \Sigma \mathrm{X}^{2} $$, $$ \Sigma \mathrm{Y}^{2} $$, and $$ \Sigma \mathrm{XY} $$ after correcting the pair $$ (X=11, Y=4) $$ to $$ (X=10, Y=14) $$.

Given:
- $$ \Sigma \mathrm{X} = 30 $$
- $$ \Sigma \mathrm{Y} = 5 $$
- $$ \Sigma \mathrm{X}^{2} = 670 $$
- $$ \Sigma \mathrm{Y}^{2} = 285 $$
- $$ \Sigma \mathrm{XY} = 334 $$

Corrected values:
- $$ \Sigma \mathrm{X} = 30 - 11 + 10 = 29 $$
- $$ \Sigma \mathrm{Y} = 5 - 4 + 14 = 15 $$
- $$ \Sigma \mathrm{X}^{2} = 670 - 121 + 100 = 649 $$
- $$ \Sigma \mathrm{Y}^{2} = 285 - 16 + 196 = 465 $$
- $$ \Sigma \mathrm{XY} = 334 - 44 + 140 = 430 $$

Now, we can calculate the correlation coefficient using the formula:
$$ \text{Correlation Coefficient} = \frac{\Sigma \mathrm{XY} - \frac{\Sigma \mathrm{X} \times \Sigma \mathrm{Y}}{n}}{\sqrt{(\Sigma \mathrm{X}^{2} - \frac{(\Sigma \mathrm{X})^{2}}{n}) \times (\Sigma \mathrm{Y}^{2} - \frac{(\Sigma \mathrm{Y})^{2}}{n})}} $$

Substitute the corrected values into the formula to find the correct correlation coefficient.
Calculate the value by following steps:
- step0: Calculate:
  $$430-\frac{\left(29\times 15\right)}{12}$$
- step1: Remove the parentheses:
  $$430-\frac{29\times 15}{12}$$
- step2: Reduce the fraction:
  $$430-\frac{145}{4}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{430\times 4}{4}-\frac{145}{4}$$
- step4: Transform the expression:
  $$\frac{430\times 4-145}{4}$$
- step5: Multiply the numbers:
  $$\frac{1720-145}{4}$$
- step6: Subtract the numbers:
  $$\frac{1575}{4}$$
The correct value of the correlation coefficient after correcting the pair $$ (X=11, Y=4) $$ to $$ (X=10, Y=14) $$ is $$ 393.75 $$.
The correct correlation coefficient is 393.75.

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Bonus Knowledge

Step 1: Adjust the summations

Step 2: Calculate the correlation coefficient

Final calculation of :

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