R-Squared Calculator

R-squared (R²) measures the proportion of variance in y that is explained by the linear regression on x. An R² of 1.0 means the regression line passes through every data point perfectly. An R² of 0 means the line explains none of the variance — knowing x tells you nothing useful about y.

The calculation uses ordinary least squares. First, fit a line y = mx + b that minimizes the sum of squared vertical distances from each point to the line. Then:

R² = 1 - SS_res / SS_tot

where SS_res is the sum of squared residuals (actual minus predicted), and SS_tot is the total sum of squares (actual minus mean).

R² is the square of the Pearson correlation coefficient r. A perfect positive correlation (r=1) gives R²=1. A perfect negative correlation (r=-1) also gives R²=1, because R² measures explanation, not direction.

Equivalently: R² = [n·Σ(xy) - Σx·Σy]² / [(n·Σx² - (Σx)²) · (n·Σy² - (Σy)²)]

What counts as a “good” R²? It depends heavily on the field. Physical experiments routinely achieve R² > 0.99. Economic and social science data may show R² of 0.3-0.5 and still be meaningful. High R² does not prove causation — a perfectly correlated spurious relationship has R² = 1.

Residuals matter as much as R². Plot your data (the chart shows the points and regression line). If the residuals form a curve rather than a random scatter, a linear model is the wrong choice regardless of what R² says.