Coefficient of Determination

Definition

The coefficient of determination, denoted as $R^2$, is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model. It provides an indication of the goodness of fit of the regression line to the observed data.

5 Must Know Facts For Your Next Test

1. The coefficient of determination, $R^2$, ranges from 0 to 1, with 0 indicating no linear relationship and 1 indicating a perfect linear relationship.
2. A higher $R^2$ value indicates that a larger proportion of the variability in the dependent variable is explained by the independent variable(s) in the regression model.
3. The coefficient of determination is calculated as the square of the correlation coefficient, $r$, between the observed and predicted values of the dependent variable.
4. The coefficient of determination is used to assess the overall fit of the regression model and the strength of the relationship between the independent and dependent variables.
5. In the context of 13.4 The Regression Equation, $R^2$ is used to evaluate the goodness of fit of the regression line, while in 13.6 Predicting with a Regression Equation, it is used to quantify the reliability of predictions made using the regression model.

Review Questions

• Explain how the coefficient of determination, $R^2$, is calculated and its interpretation in the context of a regression analysis.
  • The coefficient of determination, $R^2$, is calculated as the square of the correlation coefficient, $r$, between the observed and predicted values of the dependent variable. It represents the proportion of the variance in the dependent variable that is explained by the independent variable(s) in the regression model. An $R^2$ value of 0 indicates no linear relationship, while an $R^2$ value of 1 indicates a perfect linear relationship. The higher the $R^2$ value, the better the regression model fits the observed data.
• Describe the role of the coefficient of determination, $R^2$, in evaluating the goodness of fit of a regression equation (as discussed in 13.4 The Regression Equation).
  • In the context of 13.4 The Regression Equation, the coefficient of determination, $R^2$, is used to assess the overall fit of the regression model. A higher $R^2$ value indicates that a larger proportion of the variability in the dependent variable is explained by the independent variable(s) in the regression equation. This provides an indication of how well the regression line fits the observed data points. The $R^2$ value is a key statistic in evaluating the predictive power and reliability of the regression model.
• Explain how the coefficient of determination, $R^2$, can be used to quantify the reliability of predictions made using a regression equation (as discussed in 13.6 Predicting with a Regression Equation).
  • In the context of 13.6 Predicting with a Regression Equation, the coefficient of determination, $R^2$, can be used to quantify the reliability of predictions made using the regression model. A higher $R^2$ value indicates that a larger proportion of the variability in the dependent variable is explained by the independent variable(s) in the regression equation. This means that the regression model is better able to predict the values of the dependent variable based on the independent variable(s). The $R^2$ value provides an estimate of the accuracy and precision of the predictions made using the regression equation.

Related terms

Regression Analysis: A statistical process for estimating the relationships between a dependent variable and one or more independent variables.

Correlation Coefficient: A measure of the strength and direction of the linear relationship between two variables, denoted as $r$.

Variance: A measure of the spread or dispersion of a set of data, calculated as the average squared deviation from the mean.