R-Squared Calculator

Historical Background

R-squared, also known as the coefficient of determination, is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. The concept of R-squared has been widely used in statistical modeling and regression analysis since the 20th century.

Calculation Formula

R-squared is calculated using the formula:

\[ R^2 = 1 - \frac{\text{Sum of Squares of the Residuals (SSR)}}{\text{Total Sum of Squares (SST)}} \]

Where:

• SSR (Sum of Squares of the Residuals) is the sum of the squares of the model's residuals.
• SST (Total Sum of Squares) is the total sum of the squares of differences from the mean.

Example Calculation

For instance, if the SSR is 20 and the SST is 100, the R-squared value would be:

\[ R^2 = 1 - \frac{20}{100} = 0.8 \]

Importance and Usage Scenarios

R-squared is important for:

1. Model Evaluation: It helps in assessing the fit of a regression model.
2. Predictive Analysis: In forecasting, R-squared indicates how well future outcomes are likely to be predicted by the model.
3. Statistical Analysis: It's used in various fields like economics, engineering, and social sciences for data analysis.

Common FAQs

1. What does an R-squared value of 0.8 mean?

   • It means that 80% of the variance in the dependent variable is predictable from the independent variable(s).

2. Is a higher R-squared always better?

   • Not necessarily. A high R-squared doesn’t imply that the model is good. Other factors like the nature of data and the purpose of the model should be considered.

3. Can R-squared be used for non-linear models?

   • R-squared is most commonly used for linear regression models. For non-linear models, other goodness-of-fit measures might be more appropriate.