Correlation Coefficient Calculator
Pearson’s r for two data sets, with the least-squares line and how much variance it explains.
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Your details
Result
- Correlation (r)
- 0.529809
- R² — variance explained
- 28.0697%
- Regression line
- y = 0.3852x + 65.1416
- Covariance
- 95.6
- p-value for r ≠ 0
- 0.279645
- Pairs
- 6
Moderate positive relationship
The line accounts for 28.0697% of the variation in Y
Not significant at the 5% level
SD of X 15.753, SD of Y 11.454
6 paired observations…
r = Σ((x−x̄)(y−ȳ)) ÷ √(Σ(x−x̄)² × Σ(y−ȳ)²)
Covariance = 95.6
SD(x) × SD(y) = 15.7533 × 11.4543 = 180.4424
95.6 ÷ 180.4424
r = 0.529809
Formula — written step by step
Six paired observations…
r = Σ((x−x̄)(y−ȳ)) ÷ √(Σ(x−x̄)² × Σ(y−ȳ)²)
Covariance = 95.6
SD(x) = 15.7533, SD(y) = 11.4543
95.6 ÷ (15.7533 × 11.4543)
r = 0.529809
r runs from −1 to +1. Zero means no linear relationship; the sign gives the direction and the magnitude the strength. Squaring it gives the share of variance the straight line accounts for.
A worked example
What r² actually tells you
- Correlation
- r = 0.53
- Squared
- r² = 0.28
- r² = 0.53² = 0.2809
- 28.09% of the variation in Y is explained
- The other 71.91% is not
- A "moderate" correlation explains under a third
Just 28% of the variance is accounted for
This is why r flatters relationships. A correlation of 0.5 sounds like half a relationship; it explains a quarter of the variation. Even r = 0.7, usually called strong, leaves more than half the variance unexplained.
How to use this calculator
- Paste your X values and your Y values, one list in each box, in matching order.
- Each X needs a Y — the calculator says so if the counts differ.
- Read r for direction and strength, and r² for how much of the variation the line accounts for.
- The p-value tests whether the correlation differs from zero. With a small sample even a large r can fail to reach significance.
What Pearson’s r measures
Pearson’s correlation coefficient measures the strength and direction of a straight-line relationship between two variables. It runs from −1, a perfect inverse relationship, through 0, no linear relationship at all, to +1, a perfect direct one.
It is scale-free. Converting heights from centimetres to inches, or prices from one currency to another, leaves r unchanged — which is what makes it comparable across studies measuring very different things.
Correlation is not causation, and r is not r²
That correlation does not imply causation is the most repeated warning in statistics, and still the most ignored. Two variables can move together because one causes the other, because both are driven by something else, or by pure coincidence in a small sample.
The less-repeated warning is that r overstates the relationship to the eye. The share of variance explained is r², not r. A correlation of 0.5 explains 25% of the variation, not half of it, and 0.3 explains under a tenth.
What r cannot see
Pearson’s r detects linear relationships only. A perfect parabola — y = x² over a symmetric range — has a correlation of exactly zero, despite y being completely determined by x. A low r rules out a straight line, not a relationship.
It is also acutely sensitive to outliers. A single extreme point can drag r from near zero to near one, or the reverse. Anscombe’s quartet is the classic demonstration: four data sets with identical correlations, means and regression lines, that look nothing alike when plotted. Always plot the data.
Formula & methodology
What this calculator assumes
- Detects linear relationships only. A perfect parabola has a correlation near zero, so a low r rules out a straight line rather than a relationship.
- Highly sensitive to outliers — a single extreme point can move r from near zero to near one. Plot the data.
- Computed from deviations about the mean rather than by the raw sum-of-squares shortcut, which loses catastrophic precision when values are large and their variance small.
- Correlation does not establish causation, and r overstates the relationship to the eye: the variance explained is r², not r.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods, 2012 — measures of association.
- Anscombe, "Graphs in Statistical Analysis", The American Statistician 27(1), 1973 — four data sets with identical correlations that look nothing alike.
Interpreting r, and what it explains
| |r| | Usually called | r² — variance explained |
|---|---|---|
| 0.9 to 1.0 | Very strong | 81% to 100% |
| 0.7 to 0.9 | Strong | 49% to 81% |
| 0.5 to 0.7 | Moderate | 25% to 49% |
| 0.3 to 0.5 | Weak | 9% to 25% |
| 0.0 to 0.3 | Very weak or none | Under 9% |
These labels are conventions, not thresholds, and they vary by field. In physics r = 0.9 may indicate a problem with the apparatus; in psychology it would be extraordinary.
Frequently asked questions
What is the difference between r and r²?
r measures strength and direction, from −1 to +1. r² is the share of variance in Y explained by the line, from 0 to 1. r = 0.7 sounds strong but explains only 49%.
Can a correlation of zero mean no relationship?
No — it means no linear relationship. A perfect U-shaped relationship has r near zero, which is why plotting the data matters more than the coefficient.
How many points do I need?
At least three, since two always fall exactly on a line. Correlations from fewer than about ten pairs are unstable, which is what the p-value here is testing.
Should I use Pearson or Spearman?
Pearson for linear relationships between roughly normal variables. Spearman, which correlates ranks instead, is better for ordinal data, monotonic-but-curved relationships, or data with outliers.