To calculate the Correlation Factor (r):
\[ r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2}} \cdot \sqrt{\sum{(y_i - \bar{y})^2}}} \]
Where:
A correlation factor, also known as the Pearson correlation coefficient, measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no linear correlation.
Let's assume the following values:
Using the formula:
\[ \bar{x} = \frac{2 + 4 + 6 + 8}{4} = 5 \]
\[ \bar{y} = \frac{1 + 3 + 5 + 7}{4} = 4 \]
\[ r = \frac{(2-5)(1-4) + (4-5)(3-4) + (6-5)(5-4) + (8-5)(7-4)}{\sqrt{(2-5)^2 + (4-5)^2 + (6-5)^2 + (8-5)^2} \cdot \sqrt{(1-4)^2 + (3-4)^2 + (5-4)^2 + (7-4)^2}} = 1 \]
The Correlation Factor (r) is 1.
Let's assume the following values:
Using the formula:
\[ \bar{x} = \frac{1 + 2 + 3 + 4}{4} = 2.5 \]
\[ \bar{y} = \frac{4 + 3 + 2 + 1}{4} = 2.5 \]
\[ r = \frac{(1-2.5)(4-2.5) + (2-2.5)(3-2.5) + (3-2.5)(2-2.5) + (4-2.5)(1-2.5)}{\sqrt{(1-2.5)^2 + (2-2.5)^2 + (3-2.5)^2 + (4-2.5)^2} \cdot \sqrt{(4-2.5)^2 + (3-2.5)^2 + (2-2.5)^2 + (1-2.5)^2}} = -1 \]
The Correlation Factor (r) is -1.
