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R-Squared

General

Coefficient of Determination

The proportion of variance in a dependent variable that is explained by a regression model, ranging from 0 (no explanatory power) to 1 (perfect fit).

Definition

R-squared, also called the coefficient of determination, is the proportion of variance in a dependent variable that is explained by a regression model, expressed on a scale from 0 to 1 (or 0% to 100%). An R-squared of 0.75, for example, means the model explains 75% of the variation seen in the outcome variable, while the remaining 25% is due to factors outside the model or random noise.

R-squared is the standard headline statistic reported alongside any regression model to communicate how well the model fits the data. The Linear Regression Calculator reports R-squared directly as part of its output, letting users judge at a glance how reliable the fitted line is likely to be for prediction.

Because R-squared is derived from correlation, it is closely linked to the Correlation Coefficient computed by the Correlation Coefficient Calculator — in simple linear regression, squaring Pearson's r gives R-squared directly.

Formula

R² = 1 − (SSres ÷ SStot)

Where SSres is the sum of squared residuals (actual minus predicted values) and SStot is the total sum of squares (actual values minus their mean). For simple linear regression with one predictor, this simplifies to:

R² = r²

where r is the Pearson correlation coefficient between the predictor and outcome.

Worked Example

A simple linear regression of advertising spend against sales produces a Pearson correlation coefficient of r = 0.85.

R² = 0.85² = 0.7225

This means the regression model explains approximately 72.25% of the variance in sales, while the remaining 27.75% is attributable to other factors, such as seasonality, pricing, or competitor activity, not captured by the model.

Key Things to Know

  • Bounded between 0 and 1: an R-squared of 1 means the model explains all variance in the outcome; 0 means it explains none.
  • Equals r² in simple regression: in a single-predictor model, R-squared is exactly the square of the Correlation Coefficient.
  • Always rises with more predictors: adding variables to a model can only increase or maintain R-squared, even if those variables are meaningless — adjusted R-squared corrects for this.
  • Doesn't confirm causation or model validity: a high R-squared reflects fit to the sample data, not proof that the model is correctly specified or will generalize well.

Related Calculators

Linear Regression Calculator →Calculate the simple linear regression line for paired X, Y data in seconds. Get the slope, intercept, and R² so you can predict Y from any new X value instantly.
Correlation Coefficient Calculator →Calculate the Pearson correlation coefficient (r) between two variables in seconds. Enter paired X, Y data points to measure the strength and direction of their relationship.

Related Terms

Correlation CoefficientPearson Correlation Coefficient (r)

Frequently Asked Questions

What counts as 'good' depends heavily on the field — in controlled physical sciences an R-squared above 0.9 might be expected, while in social science or behavioral research an R-squared of 0.3 to 0.5 can be considered meaningful given the inherent noise in human data. There is no universal cutoff, so R-squared should always be interpreted relative to the field and the goal of the model.
Not necessarily — a high R-squared shows the model explains a lot of variance in the sample data, but it doesn't guarantee the model is correctly specified, free of overfitting, or useful for prediction on new data. A model can have a high R-squared and still produce biased or misleading predictions if key assumptions are violated.
R-squared always increases (or stays the same) as more predictor variables are added to a model, even if those variables are irrelevant, while adjusted R-squared penalizes the addition of variables that don't meaningfully improve the fit. Adjusted R-squared is generally preferred when comparing models with different numbers of predictors.
In simple linear regression with a single predictor, R-squared is exactly the square of the Pearson correlation coefficient (r), so R² = r². This means a correlation coefficient of 0.8 corresponds to an R-squared of 0.64, meaning 64% of the variance is explained by the model.
In standard linear regression, R-squared ranges from 0 to 1, but it is possible to see a negative value if a model performs worse than simply predicting the mean of the data — a sign of a poorly fitted or misspecified model. This typically only occurs with certain nonlinear models or when a fitted model is evaluated on entirely new data.