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Coefficient of Determination Calculator

R-squared (R²) quantifies the proportion of variance in your dependent variable explained by independent variables in a regression model. Researchers, data analysts, and statisticians use this metric to assess goodness of fit—whether a linear model captures the underlying relationship or misses important patterns. Values range from 0 (no explanatory power) to 1 (perfect prediction). Enter your paired observations and the calculator returns both the coefficient and its interpretation.

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Creators Anna Szczepanek, PhD Statistics
7,135 people find this calculator helpful

What is R-Squared?

The coefficient of determination, denoted R², measures the fraction of total variation in a dependent variable that a regression model successfully explains. In simple linear regression Y ~ aX + b, it answers the question: what percentage of Y's fluctuations are driven by X?

An R² of 0.75 means the model accounts for 75% of the variance; the remaining 25% stems from unmeasured factors, measurement error, or genuine randomness. Unlike correlation (which ranges from −1 to +1), R² is always non-negative and bounded at 1. It represents the square of the Pearson correlation coefficient in bivariate settings.

R² is fundamental to:

  • Model selection—comparing competing regression specifications
  • Predictive inference—determining whether forecasts are reliable
  • Publishing—journals often require R² as evidence of model fit
  • Quality assurance—identifying when a fitted line is misleading

The R-Squared Formula

R² is computed from three sums of squares. First, calculate the mean of your y-values (ȳ), then fit your regression line to obtain predicted values (ŷᵢ). The formula then becomes:

R² = 1 − (SSE / SST)

or equivalently

R² = SSR / SST

where:

SSE = Σ(yᵢ − ŷᵢ)²

SST = Σ(yᵢ − ȳ)²

SSR = Σ(ŷᵢ − ȳ)²

  • SSE — Sum of squared errors (residual sum of squares); measures divergence between observed and predicted y-values
  • SST — Total sum of squares; measures total variance in the y-variable around its mean
  • SSR — Sum of squares due to regression; captures variance explained by the fitted model
  • ȳ — Mean of all observed y-values
  • ŷᵢ — Fitted (predicted) y-value for the i-th observation

Interpreting R² Values

R² = 1.0: Perfect fit. All observations lie exactly on the regression line; prediction is error-free.

R² = 0.9 to 0.99: Excellent fit. The model explains 90–99% of variation. Rare in real-world data; suggests strong causal or functional relationships.

R² = 0.7 to 0.9: Strong fit. Common in physics, engineering, and controlled experiments. The model is predictively useful, though unexplained variation remains.

R² = 0.5 to 0.7: Moderate fit. Typical in social sciences and observational studies. The model captures meaningful patterns but substantial noise persists.

R² = 0.3 to 0.5: Weak fit. The model has limited predictive power; consider alternative specifications or additional variables.

R² < 0.3: Poor fit. The independent variable(s) explain less than 30% of variation. Re-examine model assumptions and data quality.

Context matters: a low R² does not disqualify a model if its coefficients are statistically significant and theoretically sound.

Worked Example

Suppose you have three data points: (0, 1), (2, 4), (4, 4).

Step 1: Calculate ȳ = (1 + 4 + 4) / 3 = 3

Step 2: Fit the line Y ~ 0.75X + 1.5 using least squares regression.

Step 3: Compute predicted values:

  • ŷ₁ = 0.75(0) + 1.5 = 1.5
  • ŷ₂ = 0.75(2) + 1.5 = 3.0
  • ŷ₃ = 0.75(4) + 1.5 = 4.5

Step 4: Calculate SST = (1 − 3)² + (4 − 3)² + (4 − 3)² = 4 + 1 + 1 = 6

Step 5: Calculate SSE = (1 − 1.5)² + (4 − 3)² + (4 − 4.5)² = 0.25 + 1 + 0.25 = 1.5

Step 6: R² = 1 − (1.5 / 6) = 1 − 0.25 = 0.75

The model explains 75% of the variance in y.

Common Pitfalls and Caveats

R² is a powerful diagnostic, but several limitations deserve attention.

  1. R² always rises with more variables — Adding predictors mechanically inflates R², even if they have no real relationship to the outcome. Use adjusted R² or information criteria (AIC, BIC) when comparing models with different numbers of variables. Adjusted R² penalises complexity and provides a fairer comparison.
  2. High R² does not imply causation — A strong fit indicates predictive association, not causality. Two variables may both be driven by a hidden confounder. Always ground your interpretation in theory and experimental design, not statistics alone.
  3. R² is scale-sensitive in weighted regression — When observations have different reliabilities or sample sizes, unweighted R² can mislead. Use weighted least squares and report both weighted and unweighted R² if heteroscedasticity is suspected.
  4. Outliers and leverage can distort R² — A single extreme point can dramatically shift the regression line and inflate or deflate R². Always inspect residual plots and identify influential observations using leverage and Cook's distance diagnostics.

Frequently Asked Questions

What does an R² of 0.85 mean?

An R² of 0.85 means your regression model accounts for 85% of the variation in the dependent variable. The remaining 15% is due to factors not captured by your independent variable(s), measurement error, or random fluctuation. In most applied contexts, 0.85 indicates a strong fit, especially if your field is biology, economics, or social science where multiple unmeasured influences are typical. In physics or engineering, you might demand higher R² values.

Is correlation the same as R-squared?

No. Correlation (r) ranges from −1 to +1 and indicates both direction and strength of linear association. R² is the square of the correlation coefficient and ranges from 0 to 1, capturing only strength. If r = 0.9, then R² = 0.81, meaning 81% of variance is explained. In simple linear regression, R² = r². For multiple regression with multiple predictors, correlation and R² are distinct: you cannot recover R² from pairwise correlations alone.

Can R² ever be negative?

In ordinary least-squares regression, R² is mathematically constrained between 0 and 1. However, if you fit a regression model without an intercept term, or use prediction on data far outside your training range, reported R² values can dip below zero. A negative R² signals that your model performs worse than simply predicting the mean of your data—a red flag to reconsider your specification.

How much R² is 'good enough'?

Acceptability depends on your field and purpose. In physics, R² > 0.95 is often expected. In psychology or medicine, R² = 0.5–0.7 is respectable because human behaviour and biological systems are inherently variable. For forecasting business metrics, stakeholders often demand R² > 0.8. Start by benchmarking against published studies in your discipline and ensure your model makes theoretical sense, not just achieving a numerical threshold.

Should I report R² or adjusted R²?

Always report both if you are comparing models with different numbers of predictors. Adjusted R² penalises model complexity and is more honest when variables are added. The adjustment shrinks R² by a factor depending on sample size (n) and number of predictors (k): Adjusted R² = 1 − [(1 − R²)(n − 1)/(n − k − 1)]. For a single model, report the unadjusted R²; for model selection, lean on adjusted R² or information criteria.

What if my R² is very low but my p-value is significant?

This signals that your independent variable has a statistically detectable linear relationship with the outcome (p < 0.05), but explains little of its overall variation (low R²). This often occurs in large samples where even weak effects become statistically significant. It suggests the relationship is real but modest in practical importance. Consider whether the effect size is large enough for your application and whether additional variables or non-linear models might improve fit.

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