Correlation Coefficient Calculator

Pearson's r measures the strength and direction of the linear relationship between two variables. It ranges from −1 (perfect negative) to +1 (perfect positive). A value of 0 means no linear relationship. The sign tells you direction; the magnitude tells you strength. r only captures linear association; it can be near 0 for relationships that are strong but curved.

For each pair, compute the deviation from the mean of x and the mean of y. Multiply those deviations and sum the products. Divide by the square root of the product of the sum of squared x-deviations and the sum of squared y-deviations. The formula is r = Σ((x − x̄)(y − ȳ)) ÷ √(Σ(x − x̄)² × Σ(y − ȳ)²).

r squared is the proportion of variance in y that is explained by a linear relationship with x. An r of 0.8 means r squared = 0.64, so a linear fit to x explains 64 percent of the variance in y. It is also the same as the R² that comes out of a simple linear regression on the same data.

Common rules of thumb: |r| above 0.7 is strong, 0.3 to 0.7 is moderate, 0.1 to 0.3 is weak, and below 0.1 is little to no linear association. These bands are field-specific in practice. A physics experiment expects much higher correlations than a social science study, and the same value of r can mean different things depending on the data type.

No. r measures association only. Two variables can be highly correlated because one causes the other, both are caused by a third variable, the data is selected in a biased way, or pure coincidence. Causal claims require study design (experiments, controlled conditions) that goes beyond correlation.

r is undefined when either x or y has zero variance (every value is the same). The formula divides by the spread of each variable, and that spread is zero in this case. Add some variation to the data and try again.