Correlation Coefficient Calculator
Pearson's correlation coefficient between two sets of paired values — how strongly they move together, and whether the relationship is statistically significant.
Pearson's r measures how closely two variables move together on a straight line, from −1 (perfectly opposed) through 0 (no linear relationship) to +1 (perfectly together).
How the correlation coefficient calculator works
Pearson's r measures how closely two variables move together on a straight line, from −1 (perfectly opposed) through 0 (no linear relationship) to +1 (perfectly together). It is the covariance divided by the product of the two standard deviations, which strips out the units so r is comparable across any pair of measurements.
Two cautions come with it. r² is the share of variance explained, and it is always smaller than r — a correlation of 0.7 explains only 49% of the variation. And r measures *linear* association only: a perfect parabola has an r near zero, and correlation never establishes cause.
Formula: r = Σ(x−x̄)(y−ȳ) ÷ √(Σ(x−x̄)² Σ(y−ȳ)²)
Worked examples
| Inputs | Correlation (r) | Note |
|---|---|---|
| A strong positive pair | 0.95119 | r ≈ 0.91 |
| A perfect straight line | 1 | r = 1 |
| Perfectly opposed | -1 | r = −1 |
FAQFrequently asked questions
What does the correlation coefficient measure?
How closely two variables move together on a straight line, from −1 through 0 to +1.
What is a strong correlation?
Above about 0.7 in absolute value is usually called strong, 0.5 moderate and below 0.3 weak — though what counts as strong varies a great deal by field.
What is r²?
The share of variance in one variable explained by the other. It is always smaller than r: a correlation of 0.7 explains only 49%.
Does correlation prove causation?
No. It measures association. A third variable, coincidence or reverse causation can all produce a high r.
Can r miss a real relationship?
Yes — it only sees straight lines. A perfect parabola gives an r of zero despite a perfect relationship.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.