Coefficient of Determination
Compute the coefficient of determination (R²) for X-Y data. See adjusted R², correlation r, equation, sum of squares breakdown, and interactive charts.
About This Calculator
The coefficient of determination (R²) calculator measures how well a linear regression model fits your paired X-Y data. R² represents the proportion of variance in the dependent variable (Y) that can be explained by the independent variable (X). This calculator is essential for students learning regression analysis, researchers evaluating model fit, data scientists performing exploratory analysis, and anyone working with bivariate data.
The calculator uses ordinary least squares (OLS) linear regression to fit the model Y = aX + b, where a is the slope and b is the intercept. It computes R² as the ratio of explained variance (SSR) to total variance (SST). The calculation also provides the Pearson correlation coefficient (r), adjusted R² that penalizes for sample size, and a complete sum of squares breakdown including SSR, SSE, and SST. The interactive scatter chart plots actual data points against predicted values from the regression line, while the residual bar chart helps identify outliers and assess model fit.
To use the calculator, enter your X values (independent variable) and Y values (dependent variable) as comma-separated numbers in the input fields. The calculator requires at least 3 paired data points and supports up to 30 points. After clicking Calculate R², the results display the coefficient of determination, correlation coefficient, adjusted R², and regression equation. The sum of squares breakdown table shows SSR, SSE, and SST values for deeper analysis. The actual-vs-predicted scatter chart visualizes how well the regression line fits your data, and the residual bar chart helps identify patterns in prediction errors. All results can be shared via URL parameters for easy collaboration with colleagues or classmates.
Regional Notes
India (IN): R² is widely used in Indian academic research, economics, and data science programs. The interpretation follows standard statistical conventions.
United States (US): R² is a standard output in regression analysis taught in AP Statistics and university-level econometrics courses. US researchers commonly report both R² and adjusted R² in publications.
United Kingdom (UK): UK statistics curricula cover R² at GCSE and A-level mathematics. In UK business and economics research, R² is reported alongside other goodness-of-fit measures like the F-statistic.
Frequently Asked Questions
What is the coefficient of determination (R²)?
The coefficient of determination, denoted R², is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It ranges from 0 to 1, where 1 indicates a perfect linear relationship and 0 indicates no linear relationship. R² is commonly used to assess the goodness of fit of a regression model.
How is R² calculated?
R² is calculated as SSR/SST, where SSR is the regression sum of squares (explained variance) and SST is the total sum of squares (total variance). It is also equal to the square of the Pearson correlation coefficient (r) between the observed and predicted values. An equivalent formula is R² = 1 - SSE/SST, where SSE is the sum of squared errors (residuals).
What is the difference between R² and adjusted R²?
Adjusted R² modifies the R² value to account for the number of independent variables in the model. Unlike R² which always increases when you add more predictors, adjusted R² penalizes unnecessary variables. It is calculated as adjusted R² = 1 - (1 - R²)(n - 1)/(n - k - 1), where n is sample size and k is the number of predictors. For simple linear regression with one predictor, k = 1.
How many data points do I need for R² calculation?
You need at least 3 paired X-Y data points to calculate the coefficient of determination. More data points generally provide a more reliable R² estimate. The calculator supports up to 30 data points. Each pair must have both an X and Y value, and the number of X and Y values must be equal.
What does an R² value of 0.75 mean?
An R² value of 0.75 means that 75% of the variance in the dependent variable (Y) can be explained by the independent variable (X) using the linear regression model. The remaining 25% of the variance is unexplained and attributed to residuals or other factors. In simple linear regression, this also means the Pearson correlation coefficient r = √0.75 ≈ 0.866, indicating a strong positive correlation.
Can R² be negative?
In the context of linear regression with an intercept term, R² is always between 0 and 1. However, R² can be negative in two scenarios: when the regression model does not include an intercept (forced through zero) and the model fits worse than the mean model, or when evaluating a model on out-of-sample data. Our calculator uses ordinary least squares with an intercept, so R² will always be between 0 and 1.
What is the relationship between R² and correlation coefficient r?
In simple linear regression with one predictor variable, the coefficient of determination R² is exactly the square of the Pearson correlation coefficient r. For example, if r = 0.9, then R² = 0.81, meaning 81% of the variance in Y is explained by X. This relationship only holds for simple linear regression; in multiple regression, R² is not simply the square of a single correlation.
How do I interpret the residual plot?
The residual plot shows the difference between actual Y values and predicted Y values (Ŷ) for each data point. If the residuals are randomly scattered around zero with no clear pattern, it suggests the linear model is appropriate. A funnel-shaped pattern or curvature in the residuals may indicate heteroscedasticity or non-linearity. Large individual residuals indicate potential outliers that may influence the regression results.