R-Squared CalculatorCalculator
Calculate R² (Coefficient of Determination), Adjusted R², SST, SSR, and SSE to evaluate regression goodness of fit and explained variance.
Dataset Input & Model Configuration
Goodness-of-Fit Summary
R-Squared Calculator - Coefficient of Determination & Model Fit: Calculate coefficient of determination (R²), Adjusted R², SST, SSR, and SSE variance components. Fast, free, and secure client-side execution in your browser.
How to Use the R
Input Observed Y Data
Enter empirical, recorded values for the dependent variable.
Input Predicted Ŷ Values
Enter corresponding fitted values generated by your regression model.
Specify Predictor Count (p)
Enter the number of independent variables in the model to compute Adjusted R².
Evaluate Sum of Squares
Inspect SST (total variance), SSR (explained variance), and SSE (residual error).
Statistical Modeling & Mathematical Analysis
The Coefficient of Determination (R²) is the preeminent metric used to assess the explanatory power of a regression model. Formally, R² represents the proportion of the total variance in the dependent variable (Y) that is predictable from or explained by the independent variables: R² = 1 - (SSE / SST) = SSR / SST.
Total Sum of Squares (SST = Σ(Y_i - ȳ)²) measures the dispersion of observed data points around their simple sample mean. The Residual Sum of Squares (SSE = Σ(Y_i - ŷ_i)²) measures the unexplained variation remaining after applying the regression equation. An R² of 0.85 indicates that 85% of total variance is successfully explained by the model, leaving only 15% to unexplained noise.
A notorious pitfall of standard R² is that it mechanically increases whenever new predictor variables are added to a model, even if those variables are completely irrelevant random numbers. To combat overfitting, statisticians rely on Adjusted R²: 1 - [(1 - R²) · (n - 1) / (n - p - 1)]. Adjusted R² penalizes model complexity and only increases if a new predictor improves model fit more than expected by random chance.
What constitutes an "acceptable" R² depends entirely on the scientific domain. In physics and laboratory chemistry, instrumentation regressions typically demand R² > 0.99. In contrast, in psychology, sociology, and stock market return analysis, an R² of 0.20 to 0.40 is considered informative due to the inherent complexity and randomness of human behavior.
Key Analytical Advantages
Total Variance Decomposition
Deconstructs total sum of squares into explained regression and residual error.
Adjusted R² Penalty
Protects against multi-collinearity and overfitting by factoring in predictor degrees of freedom.
Model Comparison Metric
Provides a normalized, unitless scale from 0% to 100% to rank competitive algorithms.
Unexplained Variance Highlight
Explicitly calculates 1 - R² to identify remaining unmodeled residual noise.
High-Precision Client Computation
Computes multi-point sum of squares sums in microseconds on your device.
Absolute Data Confidentiality
Confidential corporate forecasting models are processed locally without cloud leakage.
Frequently Asked Questions
An R-squared of 0.75 means that 75% of the variance in the dependent variable is explained by the independent variables in the model, while 25% remains unexplained.
Adjusted R-squared penalizes the model for each additional predictor variable added. It is always slightly lower than or equal to standard R-squared.
In standard ordinary least squares linear regression with an intercept, R² is mathematically constrained between 0 and 1. However, if a regression model is forced through the origin without an intercept, or if an external model performs worse than a horizontal mean line, R² can mathematically be negative.
In simple linear regression, R is Pearson's correlation coefficient (-1 to +1 indicating direction and strength), while R-squared is the square of R (0 to 1 indicating percentage of explained variance).
Not necessarily. A very high R² (e.g., 0.999) on noisy data can indicate severe overfitting. Additionally, an R² can be high even if the model violates essential regression assumptions like homoscedasticity or normality of errors.