Coefficient of Determination Calculator

The coefficient of determination, often denoted as R², plays a crucial role in statistical models, especially in linear regression. It measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

Historical Background

Originally developed in the early 20th century, the concept of R² has been pivotal in regression analysis, allowing researchers to quantify the strength of a model's predictive capability.

Calculation Formula

To compute the coefficient of determination, the formula used is:

\[ R^2 = 1 - \frac{RSS}{TSS} \]

where:

• \(R^2\) is the coefficient of determination,
• \(RSS\) is the sum of squares of residuals,
• \(TSS\) is the total sum of squares.

Example Calculation

For instance, if the sum of squares of residuals (RSS) is 50 and the total sum of squares (TSS) is 200, then:

\[ R^2 = 1 - \frac{50}{200} = 0.75 \]

This means that 75% of the variance in the dependent variable can be predicted from the independent variable.

Importance and Usage Scenarios

The coefficient of determination is essential for assessing the quality of a regression model. A higher R² value indicates a model that better fits the data, whereas a lower R² suggests a less accurate model. It's particularly useful in comparing the explanatory power of models.

Common FAQs

1. What does an R² value of 1 signify?

   • An R² value of 1 indicates that the regression predictions perfectly fit the data.

2. Can R² be negative?

   • Yes, R² can be negative when the chosen model fits the data worse than a horizontal line representing the mean of the dependent variable.

3. How does R² relate to correlation?

   • R² is the square of the correlation coefficient, reflecting the degree of linear correlation between variables squared.

This calculator streamlines the process of computing the coefficient of determination, making it accessible for students, researchers, and professionals involved in statistical analysis and modeling.