The formula to calculate the Sample Correlation Coefficient \( R \) is:
\[ R = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \sum{(y_i - \bar{y})^2}}} \]
The Sample Correlation Coefficient \( R \), also known as Pearson’s correlation coefficient, is a statistical measure that calculates the strength of the relationship between the relative movements of two variables. The values range between -1.0 and 1.0. A calculated number greater than 1.0 or less than -1.0 means that there was an error in the correlation measurement. A correlation of -1.0 shows a perfect negative correlation, while a correlation of 1.0 shows a perfect positive correlation. A correlation of 0.0 shows no linear relationship between the movement of the two variables.
Let's assume the following data points:
Step 1: Calculate the means:
\[ \bar{x} = \frac{2 + 4 + 6 + 8}{4} = 5 \]
\[ \bar{y} = \frac{1 + 3 + 5 + 7}{4} = 4 \]
Step 2: Calculate the numerator:
\[ \sum{(x_i - \bar{x})(y_i - \bar{y})} = (2-5)(1-4) + (4-5)(3-4) + (6-5)(5-4) + (8-5)(7-4) = 20 \]
Step 3: Calculate the denominator:
\[ \sqrt{\sum{(x_i - \bar{x})^2} \sum{(y_i - \bar{y})^2}} = \sqrt{20 \times 20} = 20 \]
Therefore, the Sample Correlation Coefficient \( R \) is:
\[ R = \frac{20}{20} = 1 \]
