Pearson Correlation Calculator — r, r² and Significance

Formula

How this is calculated

Pearson's r measures the strength and direction of the linear relationship between two continuous variables. It is computed from paired observations (x₁,y₁), …, (xₙ,yₙ): subtract the mean from each value to get deviations, multiply paired deviations and sum them (the covariance numerator Sxy), then divide by the geometric mean of the two variance sums √(Sxx · Syy). The result is dimensionless and lies in [−1, +1]: +1 is a perfect positive line, −1 is a perfect negative line, 0 means no linear trend. r² (the coefficient of determination) gives the proportion of variance in Y that is linearly explained by X.

To test whether the correlation is statistically significant, r is converted to a t-statistic: t = r × √(n − 2) / √(1 − r²), which follows a t-distribution with n − 2 degrees of freedom under the null hypothesis that the true population correlation ρ = 0. This calculator reports the t-statistic and whether |t| exceeds the two-tailed critical value at α = 0.05 (the precise critical value depends on df: 12.706 for df = 1, 2.571 for df = 5, 2.042 for df = 30+).

Pearson's r captures only monotone linear patterns and is sensitive to outliers. It assumes the variables are measured on a continuous scale, the observations are independent, and ideally that the data are bivariate-normal for the significance test to be exact. At least 3 pairs are needed (the t-distribution has df = n − 2 ≥ 1). Correlation does not imply causation.