Enter paired x and y values (comma-separated) to get Pearson’s correlation coefficient — how strongly they move together.
What Is a Correlation Coefficient?
A correlation calculator measures how two variables move together: Pearson’s r ranges from −1 to +1. For x = 1,2,3,4,5 and y = 2,4,5,4,5: r = 0.7746 — a strong positive correlation. r near +1: they rise together; near −1: one rises as the other falls; near 0: no linear relationship. Correlation is not causation — the classic warning, because it never is.
How to Use
- Enter the x values, comma-separated.
- Enter the y values, comma-separated (same count).
- Click “Calculate r”.
Formula
r = Σ((x−x̄)(y−ȳ)) ÷ √(Σ(x−x̄)² × Σ(y−ȳ)²).
Worked Example
x = 1,2,3,4,5; y = 2,4,5,4,5 → r = 0.7746 (strong positive).
How It Works, In Plain Words
Pearson’s r answers one question: when x goes up, does y tend to go up too — or down? The two inputs are lists of paired measurements: the first x belongs with the first y, the second with the second, and so on, which is why both lists must have the same count. The tool first works out the mean of the x values (x̄) and the mean of the y values (ȳ) — for the worked example those means are 3 and 4.
Then it measures every point against its own mean. For each pair it multiplies the two gaps, (x−x̄) × (y−ȳ): positive when both sit on the same side of their means, negative when they sit on opposite sides. Adding those products gives the top of the formula. The bottom is built from the squared gaps of each list on its own, multiplied and square-rooted, which scales the answer so it always lands between −1 and +1. Finally the tool labels the result using the bands on this page: above 0.7 in size is strong, 0.3 to 0.7 moderate, below 0.3 weak.
Real-World Uses
A student revising for an exam pairs hours of study with practice-test scores to check whether the two really move together before changing their routine. A shop owner pairs weekly advertising spend with weekly sales to see whether the adverts track revenue at all — remembering that a strong r still does not prove the adverts caused the sales. A runner pairs training distance with resting heart rate over a season, expecting a negative r as fitness improves. A teacher might pair attendance with grades across a class to spot whether the pattern is strong enough to act on. In every case the pairing is the point: each y must belong to the x beside it.
Benefits
- Pearson r plus a plain-English strength label;
- Input validation (equal counts, numeric);
- The correlation ≠ causation warning;
- Useful for research and business analysis.
Features of Our Tool
- Instant results in your browser;
- No sign-up, no download, 100% free;
- Works on mobile, tablet and desktop;
- Clear step-by-step explanation below the tool;
- Private — your entries never leave your device.
Common Mistakes
- Claiming causation from correlation;
- Missing non-linear relationships (r only sees straight lines);
- Letting one outlier drive the whole r.
Common Mistakes to Avoid
The deepest mistake is reading causation into the number — the page repeats the warning because r cannot show it: two variables may share a hidden cause, or move together by chance. Second, trusting r on curved data: Pearson only sees straight-line relationships, so a strong curve can score near zero; plot the points first, as the FAQ advises. Third, letting one outlier dominate: a single wild pair can drag r up or down, which is why the mistakes list flags it. Fourth, sloppy pairing — lists of different lengths, or a y typed against the wrong x — the tool rejects mismatched counts, but only you can check the pairs line up.
FAQ
What is a good correlation?
|r| > 0.7 is strong, 0.3–0.7 moderate, < 0.3 weak — context decides.
Does correlation imply causation?
No — correlated variables may share a cause, or it may be coincidence.
What is r = 0.7746?
A strong positive correlation — they tend to rise together.
Can r be negative?
Yes — negative r means one rises as the other falls.
What if my data curves?
Pearson r only measures linear relationships — plot the data or use rank correlation.
How many pairs do I need for a meaningful r?
The tool accepts as few as two pairs, but a correlation from very few points is fragile — one pair can swing it. Treat small samples as a hint, not evidence, and prefer more pairs before drawing conclusions; the strength bands on this page assume a reasonable amount of data behind them.
What does r squared tell me?
Squaring r gives the share of one variable’s variation that lines up with the other’s, as a proportion. For the worked example, 0.7746 × 0.7746 ≈ 0.60, so about 60% of the movement in y tracks the movement in x — the rest comes from other factors and noise.