Correlation Calculator
Paste your X and Y values — as two lists or as one "x, y" pair per line — and get Pearson's correlation coefficient r, the coefficient of determination r², and a plain-language reading of the strength and direction. The formula is worked through with your numbers. Free, no sign-up.
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How to Use the Correlation Calculator
- 1. Choose how to enter the data. Separate X and Y lists takes two boxes of numbers in the same order. Paired takes one "x, y" pair per line; lines without two numbers are ignored.
- 2. Enter at least three pairs. The two lists must have the same length, and each variable must vary — a constant list has no correlation.
- 3. Read the result. You get r, r², the number of pairs, the strength and the direction, followed by the working: the means and the sums of squares and cross-products behind r.
- 4. Plot before you conclude. The caution note below the result is not boilerplate; r only describes a straight-line pattern.
What Pearson's r Measures
Pearson's r = Σ(x − x̄)(y − ȳ) ÷ √(Σ(x − x̄)² × Σ(y − ȳ)²). It runs from −1 (a perfect downhill line) through 0 (no straight-line pattern) to +1 (a perfect uphill line). Squaring it gives r², the share of the variance in one variable that moves with the other.
X = 1, 2, 3, 4, 5 and Y = 2, 4, 5, 4, 6
Means: x̄ = 3 and ȳ = 4.2
Σ(x − x̄)(y − ȳ) = 8, Σ(x − x̄)² = 10, Σ(y − ȳ)² = 8.8
r = 8 ÷ √(10 × 8.8) = 8 ÷ 9.3808 = 0.8528
r² = 0.7273, so about 72.7% of the variance in Y is shared with X
Reading: strong, positive
How the Calculator Labels Strength
| |r| | Label shown |
|---|---|
| 0.7 or more | strong |
| 0.4 to below 0.7 | moderate |
| 0.1 to below 0.4 | weak |
| below 0.1 | negligible |
These cut-offs are a convention, not a law. Cohen's well-known benchmarks are 0.1, 0.3 and 0.5, and what counts as large depends on the field — in physics 0.9 can be disappointing, in psychology 0.3 can be meaningful.
What r Can and Cannot Tell You
- A small sample can fool you. The example above has r = 0.85 on five pairs, yet the test of r = 0 gives t(3) = 2.83, p ≈ .066, which is not significant at .05. This calculator reports r, not a p-value, so judge significance separately.
- Curves look like zero. X = −2, −1, 0, 1, 2 with Y = 4, 1, 0, 1, 4 is a perfect U, yet r = 0, because the relationship is not a straight line.
- Outliers pull hard. One extreme pair can raise or wreck r. Look at the scatterplot and consider rank correlation if you see one.
- Correlation is not causation. A strong r may reflect a cause, a shared third factor or coincidence.
Which Correlation Tool Do You Need?
| Your data | Use |
|---|---|
| Two numeric variables, roughly linear | This Correlation Calculator (Pearson) |
| Ranks, ordinal data, or a monotonic curve or outliers | Spearman Correlation Calculator |
| Small samples with many ties, or ranks again | Kendall Tau Calculator |
| Three or more variables at once | Correlation Matrix Calculator |
| The unstandardised co-movement of two variables | Covariance Calculator |
| Whether two correlations differ | Correlation Comparison Calculator |
Frequently Asked Questions
- What is the Pearson correlation coefficient?
- Pearson's r measures the strength and direction of the linear relationship between two variables. It ranges from −1 through 0 to +1. It is the sum of the products of each pair's deviations from the means, divided by the square root of the product of the two sums of squared deviations.
- How do I interpret the strength of a correlation?
- This calculator calls |r| of 0.7 or more strong, 0.4 up to 0.7 moderate, 0.1 up to 0.4 weak and anything smaller negligible. Those bands are rules of thumb and what counts as meaningful varies by field.
- What does r² tell me?
- r², the coefficient of determination, is the proportion of the variance in one variable that is linearly associated with the other. An r of 0.8 gives r² = 0.64, so 64% of the variance is shared.
- Does correlation mean one variable causes the other?
- No. Variables can correlate because one causes the other, because a third factor drives both, or by chance. A coefficient alone cannot show cause.
- Why might Pearson's r miss a real relationship?
- It only measures straight-line association, so a U-shaped or other curved relationship can give r near zero. It is also sensitive to outliers. Plot the data, and consider Spearman's rho for a monotonic but curved pattern.
- How many pairs do I need?
- The calculator needs at least three, with equal-length lists. Statistically you want far more: with only a handful of pairs even a large r can arise by chance, as the five-pair example shows.