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 is a valuable tool for assessing the goodness of fit and the strength of the relationship between the variables in a regression analysis.

5 Must Know Facts For Your Next Test

1. The coefficient of determination, $R^2$, is calculated as the square of the correlation coefficient, $r$, between the dependent and independent variables.
2. The value of $R^2$ ranges from 0 to 1, with 0 indicating that the independent variable(s) do not explain any of the variation in the dependent variable, and 1 indicating that the independent variable(s) explain all of the variation in the dependent variable.
3. A higher $R^2$ value indicates a stronger linear relationship between the independent and dependent variables, and a better fit of the regression model to the data.
4. The coefficient of determination is used to assess the overall fit of a regression model and to compare the explanatory power of different regression models.
5. In the context of the topics 12.2 The Regression Equation, 12.6 Regression (Distance from School), 12.7 Regression (Textbook Cost), and 12.8 Regression (Fuel Efficiency), the coefficient of determination can be used to evaluate the strength of the relationship between the independent and dependent variables in each regression model.

Review Questions

• Explain how the coefficient of determination, $R^2$, is calculated and what it represents in the context of regression analysis.
  • The coefficient of determination, $R^2$, is calculated as the square of the correlation coefficient, $r$, between the dependent and independent variables. It represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model. In other words, $R^2$ indicates the goodness of fit of the regression model, with a value closer to 1 suggesting a stronger linear relationship and a better fit of the model to the data.
• Describe how the coefficient of determination, $R^2$, can be used to compare the explanatory power of different regression models in the context of the topics covered (12.2 The Regression Equation, 12.6 Regression (Distance from School), 12.7 Regression (Textbook Cost), and 12.8 Regression (Fuel Efficiency)).
  • The coefficient of determination, $R^2$, can be used to compare the explanatory power of different regression models in the context of the topics covered. By calculating the $R^2$ value for each regression model, you can assess how much of the variation in the dependent variable (e.g., distance from school, textbook cost, fuel efficiency) is explained by the independent variable(s) in each model. A higher $R^2$ value indicates a stronger linear relationship and a better fit of the regression model, allowing you to compare the relative explanatory power of the different models and select the one that best fits the data.
• Evaluate how the coefficient of determination, $R^2$, can be used to draw conclusions about the strength of the relationship between variables in the regression models covered in the topics (12.2 The Regression Equation, 12.6 Regression (Distance from School), 12.7 Regression (Textbook Cost), and 12.8 Regression (Fuel Efficiency)).
  • The coefficient of determination, $R^2$, can be used to draw conclusions about the strength of the relationship between variables in the regression models covered in the topics. By analyzing the $R^2$ value, you can determine how much of the variation in the dependent variable is explained by the independent variable(s) in each model. A higher $R^2$ value, closer to 1, indicates a stronger linear relationship and a better fit of the regression model to the data. This allows you to assess the strength of the relationship between the variables and make informed decisions about the predictive power and reliability of the regression models in the context of the topics covered.

Related terms

Regression Analysis: Regression analysis is a statistical technique used to model the relationship between a dependent variable and one or more independent variables, allowing for the prediction of the dependent variable based on the independent variable(s).

Goodness of Fit: Goodness of fit refers to the degree to which a statistical model fits a set of observations. It is a measure of how well the regression line or curve represents the data.

Correlation Coefficient: The correlation coefficient, denoted as $r$, is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with 0 indicating no linear relationship.