October 7, 2025

What Is The Coefficient Of Determination R Squared

Q: Is a higher R-squared always better?

Not always. A very high R-squared in a complex model might indicate overfitting, meaning the model fits the training data too closely and may not generalize well to new data.

Q: What is the difference between R-squared and the correlation coefficient (R)?

The correlation coefficient (R) measures the strength and direction of a linear relationship between two variables. R-squared (R²) is the square of the correlation coefficient for simple linear regression and represents the proportion of variance explained by the model. R is for two variables, R² describes model fit.

Q: Can R-squared be negative?

Typically, R-squared cannot be negative in standard linear regression models as it's the square of R. However, some software might report a negative R-squared if the model performs worse than a horizontal line (the mean of the dependent variable), which usually indicates a very poor model or data issues.

Q: Does R-squared indicate causation?

No, R-squared only indicates the degree to which a model explains variance. It shows correlation and association, but it does not imply a cause-and-effect relationship between variables.

Discover R-squared, a statistical measure indicating how well a regression model predicts outcomes from input variables, explaining variance in the dependent variable.

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Definition of R-squared

The Coefficient of Determination, commonly known as R-squared (R²), is a statistical measure that represents the proportion of the variance in the dependent variable that can be explained by the independent variable(s) in a regression model. In simpler terms, it indicates how well the observed data points are replicated by the model, based on the proportion of total variation of outcomes explained by the model.

Interpreting R-squared Values

R-squared values range from 0 to 1 (or 0% to 100%). An R-squared of 1 (100%) means that the model explains all the variability of the dependent variable around its mean, suggesting a perfect fit. Conversely, an R-squared of 0 indicates that the model explains none of the variability of the dependent variable, implying that the model does not predict the outcome better than simply using the mean of the dependent variable. Higher R-squared values generally suggest a better fit for the model.

Practical Example: Predicting Test Scores

Imagine you're trying to predict a student's test score (dependent variable) based on the number of hours they studied (independent variable). After running a linear regression, you find an R-squared value of 0.75. This means that 75% of the variation in test scores can be explained by the number of hours studied. The remaining 25% of the variation is attributed to other factors not included in your model, such as prior knowledge, teaching quality, or stress levels.

Limitations and Importance

While a high R-squared is often desired, it doesn't necessarily imply that the model is correct or that the predictor variables are the true causes. It can increase with the addition of more independent variables, even if those variables are not truly related to the dependent variable. Therefore, R-squared should be interpreted in conjunction with other statistical measures and domain knowledge to assess the validity and utility of a regression model, helping researchers understand the predictive power of their analysis.

Frequently Asked Questions

Is a higher R-squared always better?

What is the difference between R-squared and the correlation coefficient (R)?

Can R-squared be negative?

Does R-squared indicate causation?