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About

The coefficient of determination R² quantifies the proportion of variance in a dependent variable y that is predictable from an independent variable x. A value of 0.85 means 85% of the observed variation in y is explained by the linear model. Misinterpreting R² leads to overfitting, false confidence in weak models, or rejection of adequate ones. This calculator performs Ordinary Least Squares regression, computes R² via 1 − SSE ÷ SST, and reports adjusted R² which penalizes model complexity. It assumes a linear relationship and homoscedastic residuals.

Note: R² alone does not confirm causation and can be misleading with nonlinear data. Always inspect residual patterns. For datasets with fewer than 5 observations, adjusted R² becomes unreliable due to small-sample bias. The F-statistic reported here tests the null hypothesis that the slope equals zero. A high R² with a non-significant F-statistic signals insufficient data.

Formulas

The coefficient of determination is defined as the complement of the ratio of residual variance to total variance:

R² = 1 − SSESST

Where the component sums of squares are computed as:

SST = n∑i=1 (y_i − y)²

SSE = n∑i=1 (y_i − y_i)²

SSR = n∑i=1 (y_i − y)²

The OLS regression line y = b₀ + b₁x has coefficients:

b₁ = n∑x_iy_i − ∑x_i ∑y_in∑x_i² − (∑x_i)²

b₀ = y − b₁ ⋅ x

Adjusted R-squared corrects for the number of predictors k:

R²_adj = 1 − (1 − R²)(n − 1)n − k − 1

The Pearson correlation coefficient r satisfies R² = r² for simple linear regression. The F-statistic is:

F = SSR ÷ kSSE ÷ (n − k − 1)

Where n = number of observations, k = number of predictors (1 for simple linear regression), y_i = observed value, y_i = predicted value, y = mean of observed values, b₀ = y-intercept, b₁ = slope.

Reference Data

R² Range	Interpretation	Typical Domain	Action Guidance
0.95 - 1.00	Excellent fit	Physics, engineering calibration	Verify not overfitting; check for data leakage
0.85 - 0.95	Strong fit	Chemistry, controlled experiments	Model reliable for prediction within range
0.70 - 0.85	Moderate fit	Biology, agriculture	Consider additional predictors or transformations
0.50 - 0.70	Weak - moderate fit	Social sciences, psychology	Model captures trend but high residual noise
0.30 - 0.50	Weak fit	Economics, marketing	Useful for directional insight only
0.00 - 0.30	Poor fit	Behavioral data, stock returns	Linear model inadequate; try nonlinear or add variables
< 0.00	Worse than mean model	Misspecified models	Model is harmful; discard and re-specify
Key Statistics Reference
SST	Total Sum of Squares - total variance of y around its mean y
SSR	Regression Sum of Squares - variance explained by the regression line
SSE	Error (Residual) Sum of Squares - unexplained variance; SST = SSR + SSE
r	Pearson correlation coefficient; R² = r² for simple linear regression
SEE	Standard Error of Estimate - average distance of data from regression line in y units
F	F-statistic - ratio of explained to unexplained variance per degree of freedom
Adj. R²	Adjusted R-squared - penalizes adding predictors that do not improve fit
n	Sample size - minimum 3 for regression, 10+ recommended
k	Number of independent predictors - 1 in simple linear regression

Frequently Asked Questions

A negative R² means the fitted model performs worse than a horizontal line at the mean y. This occurs when the model is fundamentally misspecified - for example, fitting a positive-slope line to data with a negative trend, or applying linear regression to strongly nonlinear data. A negative value is a signal to discard the current model and re-examine the relationship between variables.

Standard R² never decreases when a predictor is added, even if that predictor is noise. Adjusted R² applies a penalty proportional to n − 1n − k − 1, so it can decrease if a new variable does not improve the model enough to justify the lost degree of freedom. Use adjusted R² whenever you compare models with different numbers of predictors. For simple linear regression with k = 1, the difference is small when n is large but significant when n < 10.

No. R² is scale-dependent on SST, which is unique to each dependent variable's distribution. Comparing R² = 0.90 from a model predicting temperature with R² = 0.70 from a model predicting stock price is meaningless. For cross-model comparison with different response variables, use information criteria (AIC, BIC) or out-of-sample prediction error (RMSE on test data).

For simple linear regression, the F-statistic is a monotonic transformation of R²: F = R² ÷ k(1 − R²) ÷ (n − k − 1). A high R² with a low F-statistic (below approximately 4.0 for common significance levels) means the sample is too small to confirm the relationship is not due to chance. Always check F alongside R².

Statistically, you need at least n ≥ k + 2 to compute regression at all (for k = 1, minimum 3 points). However, with very small samples, R² is inflated - two points always yield R² = 1.00 trivially. A practical guideline is n ≥ 10 per predictor for exploratory work and n ≥ 20 per predictor for confirmatory analysis.

No. R² measures linear association, not causation. Two variables may correlate due to a shared confounding variable. For example, ice cream sales and drowning rates both correlate with temperature, producing a high R² between them despite no causal link. Establishing causation requires controlled experiments, instrumental variables, or quasi-experimental designs - none of which are captured by R² alone.