Correlation Coefficient Calculator
Paste paired x and y values to get the Pearson correlation coefficient r — the strength and direction of the straight-line relationship between them.
Quick answer: Pearson's r = Σ(x−x̄)(y−ȳ) ÷ √(Σ(x−x̄)²·Σ(y−ȳ)²), ranging from −1 to +1. For x = 1,2,3,4,5 and y = 2,4,5,4,5, r ≈ 0.775 — a fairly strong positive relationship.
The Pearson correlation coefficient measures how closely two variables move together in a straight line. It runs from −1 (perfect inverse) through 0 (no linear link) to +1 (perfect direct). Its square, r², is the share of variation in one variable explained by the other under a linear fit.
The formula
r = Σ(x − x̄)(y − ȳ) ÷ √( Σ(x − x̄)² · Σ(y − ȳ)² )
Worked example
For x = 1,2,3,4,5 (x̄ = 3) and y = 2,4,5,4,5 (ȳ = 4): Σ(x−x̄)(y−ȳ) = 6, Σ(x−x̄)² = 10, Σ(y−ȳ)² = 6. r = 6 ÷ √(10 × 6) = 6 ÷ √60 ≈ 0.775.
Frequently asked questions
What does the correlation coefficient tell me?
The strength and direction of the linear relationship between two variables, from −1 to +1.
What counts as a strong correlation?
As a rough guide, |r| above 0.7 is strong, 0.3–0.7 moderate, and below 0.3 weak — but context matters.
Does correlation prove causation?
No. A high r shows association, not that one variable causes the other; a third factor may drive both.
What is r²?
The coefficient of determination — the fraction of variance in one variable explained by a linear fit to the other.
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