Final answer:
The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables, with a range between -1 and +1. It is calculated using a specific formula and does not depend on which variable is designated as x or y. The coefficient of determination (r²) is the percentage of variation explained by the linear relationship.
Step-by-step explanation:
The correlation coefficient, represented by r, is used to determine the strength and direction of a linear relationship between two variables, which are represented by the x-values and y-values in a dataset. The coefficient ranges from -1 to +1, with positive values indicating a direct correlation and negative values indicating an inverse correlation.
To calculate r, use the formula: r = [n(∑xy) - (∑x)(∑y)] / √{[n∑x² - (∑x)²][n∑y² - (∑y)²]}, where n is the number of data points, ∑xy is the sum of the product of x and y pairings, ∑x is the sum of the x-values, and ∑y is the sum of the y-values. If you switch x and y, calculating r again but reversing their roles, you will find that the value of r remains the same, as correlation does not depend on which variable is considered x or y.
The coefficient of determination, r², represents the percentage of variation in the dependent variable that can be explained by the independent variable using the regression line. For example, if r = .6631, then r² = .4397, which means approximately 44 percent of the variation can be explained by the linear relationship.