Coefficient of Determination (R-Squared) Calculator

The coefficient of determination, usually written R² or R-squared, describes how much variation in a dependent variable is accounted for by a fitted model. In simple linear regression with an intercept, it equals the square of the Pearson correlation coefficient between X and Y. Values normally range from zero to one: zero indicates that the fitted line explains none of the observed variation in Y, while one indicates that every point lies exactly on the line. This calculator fits an ordinary least-squares line. It finds the slope by dividing the sum of cross-products around the means by the sum of squared X deviations. The intercept positions that line through the point formed by the mean of X and mean of Y. Pearson's r measures the direction and strength of the linear relationship, and R-squared is r². Because squaring removes the sign, inspect r or the slope to distinguish positive from negative relationships. Adjusted R-squared applies a penalty for model complexity. For this simple regression with one predictor, it is calculated as 1 − (1 − R²)(n − 1)/(n − 2), where n is the number of paired observations. It can be lower than zero when the fitted model performs poorly. Adjusted R-squared is especially useful when comparing models with different numbers of predictors, although this calculator specifically fits one-predictor linear regression. A high R-squared does not establish causation, validate every model assumption, or prove that predictions will be accurate outside the observed X range. A curved relationship may have a low linear R-squared despite being highly systematic. Conversely, a strong trend caused by time, a lurking variable, or a few influential points may produce a high value without a meaningful causal connection. Always inspect a scatterplot and residuals alongside summary statistics. Least-squares inference commonly assumes independent observations, a linear mean relationship, roughly constant residual variance, and suitably distributed residuals for confidence intervals and hypothesis tests. R-squared itself can still be calculated when these conditions fail, but its interpretation may be misleading. Duplicate X values are acceptable, but X and Y must each contain variation; a constant series makes correlation and regression undefined. Enter paired values in matching order so each X corresponds to the Y at the same position. Units affect the slope and intercept but not r or R-squared. Report enough context to explain the variables, sample, and range, and avoid describing R-squared as the percentage of individual outcomes predicted correctly. It is a proportion of sample variation explained by the fitted line, not a classification accuracy score or a guarantee about future observations.