Correlation Calculator
Measure the strength and direction of a linear relationship.
How Pearson correlation works
Pearson's correlation coefficient r measures the strength and direction of a linear relationship between two variables, from −1 (perfect negative) through 0 (no linear relationship) to +1 (perfect positive).
The calculator requires paired X and Y values of equal length, at least two pairs, and at least one variable must vary (a constant column has zero variance and no defined correlation).
Correlation describes association, not causation: two variables can be strongly correlated without one causing the other.
The formula
r = Σ((x−x̄)(y−ȳ)) ÷ √(Σ(x−x̄)² · Σ(y−ȳ)²)
For each pair, multiply the deviations of x and y from their means, sum them, and divide by the square root of the product of the summed squared deviations of x and y.
Worked example
Perfect positive relationship
For X = 1,2,3,4,5 and Y = 2,4,6,8,10 every point lies exactly on a rising line, so r = 1. If Y were 10,8,6,4,2, r would be −1.
Frequently asked questions
How do I interpret the strength of r?
As a rough guide, |r| below 0.3 is weak, 0.3–0.7 is moderate, and above 0.7 is strong. Context and sample size matter — always consider both.
Why do I get an error for constant data?
If all X values are identical there is no variance in X, so the correlation is undefined. Add a dataset with some variation.
Does correlation imply causation?
No. Correlation alone cannot establish cause and effect — a third variable may drive both, or the relationship may be coincidental.
Related research methods
- Correlation Analysis
- Regression Analysis
When to use this tool
Use this to quantify the strength and direction of a linear relationship between two continuous variables.
Assumptions
- Approximately linear relationship.
- Approximate bivariate normality is helpful for inference; r is otherwise a descriptive measure.
- Independent observations.
Limitations
- Correlation is not causation.
- r is sensitive to outliers and only captures linear relationships.
- Restricted range in either variable can deflate r.
How to interpret the result
r ranges from −1 to +1. Values near ±1 indicate a strong linear relationship; the sign indicates direction.
How to report the result
Report r, the sample size, and a confidence interval — for example, 'r(48) = 0.62, 95% CI 0.40–0.78'.
Next steps in your analysis
Related tools
Related guides
- How to Choose the Right Statistical TestA practical framework for picking the right statistical test: start from your research question, data type and study design, then check the assumptions.
- Correlation vs RegressionThe difference between measuring a linear association (correlation) and modelling a relationship to make predictions (regression), and how R² connects them.
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