The **coefficient of determination** (often denoted as \( R^2 \)) is a statistical measure used to assess the goodness of fit of a model, specifically in regression analysis. It represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). In other words, it tells you how well the independent variable(s) explain the variation in the dependent variable.

### Formula
In a simple linear regression model, the coefficient of determination is calculated as:
\[
R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2}
\]
Where:
- \( y_i \) are the actual observed values.
- \( \hat{y}_i \) are the predicted values from the regression model.
- \( \bar{y} \) is the mean of the observed values.

### Interpretation
- \( R^2 = 0 \) means the model explains none of the variance of the dependent variable.
- \( R^2 = 1 \) means the model explains all of the variance of the dependent variable.
- A value between 0 and 1 indicates the proportion of variance explained by the model.

### Key Points:
- A higher \( R^2 \) value indicates a better fit of the model to the data.
- However, a high \( R^2 \) does not necessarily imply causality or that the model is the best. It's only one indicator of model quality.

Do you want more details about how it's used in different regression models? Or do you have a specific example in mind?

### Related Questions:
1. What is the difference between \( R^2 \) and adjusted \( R^2 \)?
2. How does the coefficient of determination relate to correlation?
3. How do outliers affect the \( R^2 \) value?
4. What are the limitations of using \( R^2 \) as a sole measure of model performance?
5. How can \( R^2 \) be interpreted in multiple regression models?

### Tip:
Be cautious when comparing \( R^2 \) values between models with different numbers of predictors. A higher \( R^2 \) might simply indicate overfitting, especially if the model includes too many variables.

Learn about the coefficient of determination (R^2), a key statistical measure in regression analysis that quantifies how well the independent variables explain the variance in the dependent variable. The page covers the formula, interpretation, and real-world applications of R^2. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation