Pearson Correlation Coefficient (r) Calculator
Compute Pearson’s r between two paired lists, with r², the covariance and a plain-language reading of how strongly the variables move together.
How it works
r = Σ(x − x̄)(y − ȳ) ÷ √(Σ(x − x̄)² × Σ(y − ȳ)²)
Pearson’s r rescales the covariance of two variables by the product of their standard deviations, giving a unitless number from −1 to +1. Positive values mean the variables rise together, negative values mean one falls as the other rises, and zero means no straight-line link. Because it divides out the scales, r is comparable across data sets, but it only sees linear patterns and never proves causation.
Worked example
For X = 10, 20, 30, 40, 50 and Y = 12, 24, 33, 46, 55, the cross-deviation sum is 1030 while Σ(x − x̄)² = 1000 and Σ(y − ȳ)² = 1069.2. So r = 1030 ÷ √(1000 × 1069.2) ≈ 0.996 — a very strong positive relationship with r² ≈ 0.99.
Frequently asked questions
What counts as a strong correlation?
Rough conventions call |r| above 0.7 strong, 0.5–0.7 moderate and 0.3–0.5 weak, but context matters. In physics an r of 0.95 may be disappointing, while in social science 0.4 can be a notable signal.
Does r change if I swap X and Y?
No. Pearson’s r is symmetric — correlating X with Y gives exactly the same number as correlating Y with X. That is different from regression, where swapping the roles changes the slope.
Why can a strong pattern still give r near zero?
Because r measures only straight-line association. A clear curved relationship, such as a U-shape, can average out to an r close to zero, so you should always plot the points as well.
Related calculators
For education — check the assumptions (sample size, independence, distribution) before using these statistics in real decisions.