R-Squared Calculator
Calculate Coefficient of Determination ($R^2 = \frac{\text{SSR}}{\text{SST}} = 1 - \frac{\text{SSE}}{\text{SST}}$), explained variance %, and unexplained residual error.
Calculate Coefficient of Determination R²
Enter Sum of Squares Regression (SSR) and Sum of Squares Error (SSE).
Calculation Results
Calculated using coefficient of determination equations: R^2 = \frac{\text{SSR}}{\text{SST}} = 1 - \frac{\text{SSE}}{\text{SST}}
Quick Summary
The R-Squared Calculator evaluates the coefficient of determination ($R^2 = \frac{\text{SSR}}{\text{SST}}$), percentage explained variance, and unexplained residual error percentage.
Formula Explanation
R^2 = \frac{\text{SSR}}{\text{SST}} = 1 - \frac{\text{SSE}}{\text{SST}}
\text{SST} = \text{SSR} + \text{SSE}How It Works
R-squared ($R^2$) measures goodness-of-fit in regression analysis. The R-Squared Calculator divides explained Sum of Squares Regression (SSR) by Total Sum of Squares (SST = SSR + SSE).
Step-by-Step Worked Example
Practical Problem: A statistical model has $\text{SSR} = 75$ and $\text{SSE} = 25$. Calculate $R^2$ coefficient of determination.
- Step 1: Calculate Total Sum of Squares (SST): $\text{SST} = \text{SSR} + \text{SSE} = 75 + 25 = \mathbf{100.0}$.
- Step 2: Divide SSR by SST: $R^2 = 75 / 100 = \mathbf{0.7500}$.
- Step 3: Convert to percentage: $0.7500 \times 100 = \mathbf{75.00\%}$.
- Step 4: Calculate unexplained variance percentage: $100\% - 75\% = \mathbf{25.00\%}$.
- Step 5: Interpretation: The model explains 75.00% of the total variance in the outcome variable $Y$.
Real-World Calculation Examples
Scenario 1: Model SSR = 75, SSE = 25
Parameters: SSR = 75, SSE = 25
Result: R² = 0.7500 (75.00% variance explained).
Scenario 2: High Precision Physics Experiment
Parameters: SSR = 99, SSE = 1
Result: R² = 0.9900 (99% precision fit).
Scenario 3: Social Science Survey Behavior
Parameters: SSR = 20, SSE = 80
Result: R² = 0.2000 (20% variance explained).
Scenario 4: Stock Market Beta CAPM Model
Parameters: Market variance fit
Result: R-squared benchmark evaluation.
Key Benefits of Using This Calculator
Goodness-of-Fit Quantification
Measures model explanatory power on a standardized 0% to 100% scale.
Explained vs Unexplained Breakdown
Outputs both explained model percentage and residual error percentage.
Sum of Squares Partitioning
Displays Total Sum of Squares (SST = SSR + SSE).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is R-squared ($R^2$)?
R-squared ($R^2$) is the coefficient of determination measuring the proportion of variance in dependent outcome $Y$ explained by regression inputs.
What is range of R-squared?
In OLS linear regression, $R^2$ ranges strictly from $0.0$ (0%) to $1.0$ (100%).
Can R-squared be negative?
In standard OLS without forced zero-intercept, $R^2 \ge 0$. In non-linear regression or forced zero-intercept models, $R^2$ can technically be negative.
How do I calculate R-squared in Excel?
Use formula =RSQ(known_y, known_x).
What is a "good" R-squared value?
In physics, $R^2 > 0.90$ is common; in social sciences and economics, $R^2 \ge 0.30$ or $0.50$ is often considered strong.
What is difference between R-squared and Pearson $r$?
In simple linear regression, $R^2 = r^2$ (the square of Pearson correlation coefficient $r$).
Does a high R-squared prove causation?
No — a high $R^2$ indicates high predictive accuracy, but does NOT prove causal direction.
What is Total Sum of Squares (SST)?
$\text{SST} = \sum (y_i - \bar{y})^2$, representing total variation in outcome $Y$ around its mean.
What is Residual Sum of Squares (SSE)?
$\text{SSE} = \sum (y_i - \hat{y}_i)^2$, representing unexplained model residual error.
Why is Adjusted R-squared preferred for multiple regression?
Adjusted $R^2$ adjusts for number of predictors $k$, preventing artificial inflation when adding extra variables.