0.997359
0.998678
2.01
-0.01
40.508
0.107
40.401
6.02
0.997359
0.998678
2.01
-0.01
40.508
0.107
40.401
6.02
The Coefficient of Determination (R²) Calculator measures how well a linear regression model fits observed data. R², often called R-squared, is the proportion of variance in the dependent variable that is explained by the independent variable in a linear regression model. It is the single most commonly reported statistic for assessing the goodness of fit of a regression model and is used across virtually every quantitative discipline including economics, psychology, biology, engineering, and data science.
R² ranges from 0 to 1 for standard linear regression. A value of 0 means the model explains none of the variability in the response data, while a value of 1 means the model explains all the variability. In practice, R² values between 0.7 and 1.0 are generally considered strong in the natural sciences and engineering, while values as low as 0.3 may be considered meaningful in social science research where human behavior introduces inherent variability.
This calculator accepts up to five paired data points (x, y), fits a simple linear regression line using the ordinary least squares (OLS) method, and computes R² along with related statistics including the Pearson correlation coefficient r, the total sum of squares (SS Total), the residual sum of squares (SS Residual), and the regression sum of squares (SS Regression). These components satisfy the fundamental decomposition: SS Total = SS Regression + SS Residual.
Understanding R² is essential for anyone working with regression models. A high R² does not necessarily mean the model is good — overfitting, omitted variable bias, and spurious correlations can all produce misleadingly high R² values. Conversely, a low R² does not mean the model is useless; in some fields, even modest predictive ability provides valuable insight. This calculator helps you decompose the variance in your data to understand how much of it your linear model captures.
The closely related Pearson correlation coefficient (r) is simply the signed square root of R². While R² tells you the proportion of variance explained, r tells you the strength and direction of the linear association. A positive r indicates that y increases as x increases; a negative r indicates that y decreases as x increases. Both statistics are reported by this calculator, giving you a complete picture of the linear relationship between your variables.
The calculator first fits a simple linear regression model to the provided data points using the ordinary least squares method. The regression line takes the form: $$\hat{y} = b_0 + b_1 x$$
The slope b1 and intercept b0 are computed as: $$b_1 = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2}$$ $$b_0 = \bar{y} - b_1 \bar{x}$$
Next, the calculator computes the three key sums of squares. The Total Sum of Squares (SS Total) measures the total variability in y: $$SS_{\text{total}} = \sum_{i=1}^{n}(y_i - \bar{y})^2$$
The Residual Sum of Squares (SS Residual) measures the unexplained variability: $$SS_{\text{res}} = \sum_{i=1}^{n}(y_i - \hat{y}_i)^2$$
The Regression Sum of Squares (SS Regression) measures the explained variability: $$SS_{\text{reg}} = SS_{\text{total}} - SS_{\text{res}}$$
Finally, R² is computed as: $$R^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{total}}}$$
And the correlation coefficient r is: $$r = \text{sign}(b_1) \cdot \sqrt{R^2}$$
An R² of 1.0 means all data points fall exactly on the regression line (perfect fit). An R² of 0 means the regression line explains none of the variance (the model is no better than simply predicting the mean for every observation). Values between 0 and 1 indicate partial explanatory power. The correlation coefficient r carries a sign: positive r means y tends to increase with x, negative r means y tends to decrease with x. SS Total decomposes into SS Regression (explained) plus SS Residual (unexplained), providing a complete partition of variance.
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With these nearly linear data points, R² = 0.997, indicating that 99.7% of the variance in y is explained by x. The correlation r = 0.999 confirms a very strong positive linear relationship.
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With some scatter in the data, R² = 0.771, meaning 77.1% of variance is explained. This is a moderately strong fit typical of many real-world datasets.
There is no universal threshold for a 'good' R² value — it depends entirely on the field and context. In physics and engineering, R² above 0.95 is common because physical systems follow predictable laws. In social sciences and psychology, R² of 0.3-0.5 may be considered strong because human behavior is inherently variable. In financial forecasting, even R² of 0.1 can be economically significant. Always compare R² to benchmarks in your specific field.
For simple linear regression with an intercept, R² is always between 0 and 1. However, R² can be negative in special cases: (1) when using a model without an intercept, (2) when applying a model's R² to out-of-sample test data where the model fits worse than the mean, or (3) when computing R² for nonlinear models. A negative R² means the model is worse than simply predicting the mean for all observations.
R² always increases (or stays the same) as you add more predictors to a model, even if those predictors are meaningless noise. Adjusted R² penalizes for the number of predictors, decreasing if a new variable does not improve the model enough to justify its inclusion. Use adjusted R² when comparing models with different numbers of predictors. For simple regression with one predictor, the difference is usually small.
For simple linear regression with one predictor, R² is exactly equal to the square of the Pearson correlation coefficient: $$R^2 = r^2$$. The correlation r ranges from -1 to +1 and indicates both strength and direction, while R² ranges from 0 to 1 and indicates only the proportion of explained variance. For multiple regression with several predictors, R² is the square of the multiple correlation coefficient R.
No. A high R² only means the model fits the observed data well, not that the model is correctly specified. Common pitfalls include: overfitting (fitting noise rather than signal), spurious correlation (two variables moving together by coincidence), omitted variable bias (missing an important confounding variable), and nonlinearity (forcing a linear model on curved data). Always examine residual plots and use domain knowledge alongside R².
These three statistics decompose the total variance in your data into explained and unexplained components: $$SS_{\text{total}} = SS_{\text{reg}} + SS_{\text{res}}$$. This decomposition is the foundation of analysis of variance (ANOVA) for regression. SS Regression tells you how much variability the model captures; SS Residual tells you how much remains unexplained. The F-statistic for testing overall model significance is based on the ratio of these components.
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