Correlation Coefficient and Regression Lines


When analyzing the relationship between two variables, one commonly used statistical measure is the correlation coefficient. The correlation coefficient, denoted by r, is a numerical value that indicates the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with -1 representing a perfect negative correlation, 1 representing a perfect positive correlation, and 0 indicating no correlation.


A correlation coefficient of -1 indicates a perfect negative correlation between two variables. In this case, as one variable increases, the other variable decreases in a perfectly linear manner. When the correlation coefficient is -1, the two regression lines resulting from the relationship between the variables have the following properties:

1. Slope: The slope of the regression line for the independent variable is negative, indicating a downward trend.


2. Intercept: The intercept of the regression line for the independent variable is positive, indicating that it intersects the y-axis above zero.


3. Scatterplot: The scatterplot of the data points will show a clear downward trend.


4. Prediction: The regression line can be used to predict the value of the dependent variable from the given value of the independent variable.

A correlation coefficient of 1 indicates a perfect positive correlation between two variables. In this case, as one variable increases, the other variable also increases in a perfectly linear manner. When the correlation coefficient is 1, the two regression lines resulting from the relationship between the variables have the following properties:

1. Slope: The slope of the regression line for the independent variable is positive, indicating an upward trend.


2. Intercept: The intercept of the regression line for the independent variable is positive, indicating that it intersects the y-axis above zero.


3. Scatterplot: The scatterplot of the data points will show a clear upward trend.


4. Prediction: The regression line can be used to predict the value of the dependent variable from the given value of the independent variable.

In summary, when the correlation coefficient is -1 or 1, it indicates a perfect negative or positive correlation, respectively, between two variables. The resulting regression lines have specific characteristics, including the slope, intercept, scatterplot appearance, and prediction capabilities. Understanding these properties helps in interpreting the relationship between variables and making predictions based on the regression lines.