Coefficient of Determination (R²)

Square the correlation and you get the fraction of the variance in y that the regression accounts for. A correlation of 0.9 means R² = 0.81, so 81% of the variation is explained by the line and 19% remains as residual scatter. The squaring is deflating on purpose: a correlation of 0.5, which sounds like a solid relationship, explains only a quarter of the variation. Karl Pearson formalised the correlation coefficient in 1896, building on Galton's earlier work, and the squared version quickly became the standard summary of how well a straight line fits.

Two traps. First, R² says nothing about causation or about whether a line is the right shape — Anscombe's famous 1973 quartet contains four data sets with identical R² of 0.67, one a clean linear trend, one a perfect parabola, one a straight line ruined by a single outlier. Always plot the data. Second, this calculator returns the positive root when you invert it, because squaring destroys the sign: R² = 0.64 implies |r| = 0.8, but only the scatterplot or the slope tells you whether the relationship runs up or down.