Correlation Coefficient Calculator
Calculate Pearson correlation from paired data and inspect strength, direction and explained variation
Pearson r
How we calculated it
About the Correlation Coefficient Calculator
Correlation Coefficient Calculator measures the strength and direction of a linear relationship between two numerical variables. Pearson’s r ranges from -1 to +1.
A strong correlation does not prove that one variable causes the other. Outliers, nonlinear patterns and shared underlying factors can all produce misleading interpretations.
How to Use the Correlation Coefficient Calculator
- Enter paired x and y observations in matching order.
- Calculate the correlation coefficient.
- Inspect a scatter plot when available to check for outliers or nonlinear patterns.
- Interpret the magnitude and sign in the context of the data rather than using one universal cutoff for strength.
How the Calculation Works
Pearson correlation r = covariance(x,y) ÷ (sx sy), where sx and sy are the standard deviations of the two variables.
Equivalently, it standardizes the joint variation of paired observations so the resulting measure is dimensionless and bounded between -1 and +1.
Worked example
Correlation Coefficient Calculator Worked Example
Perfect increasing pairs
For pairs (1,2), (2,4), (3,6) and (4,8), y is exactly 2x, so Pearson correlation is +1.
Result guide
Understanding Your Correlation Coefficient Calculator Result
A positive r indicates that larger x values tend to accompany larger y values; a negative r indicates the opposite. Values near zero indicate weak linear association, not necessarily no relationship at all. Sample size also matters when judging whether an observed correlation is stable or statistically meaningful. A large r from only a few data points can be much less convincing than a similar r supported by a large, well-sampled data set.
Before you rely on the answer
Assumptions and Limitations
- Observations are correctly paired.
- Pearson correlation is appropriate for the relationship being studied.
- Outliers and nonlinearity are examined rather than ignored.
Practical tip
Using This Result in Practice
Always inspect the data visually when possible. Two very different scatter plots can sometimes produce similar correlation coefficients.
Related calculations
Useful Calculations to Continue With
If you want the unstandardized joint variation behind correlation, use the Covariance Calculator.
If you want a fitted line as well as association strength, use the Linear Regression Calculator.
Common questions
Correlation Coefficient Calculator FAQ
What does r=1 mean?
A perfect positive linear relationship in the sample.
Does zero correlation mean no relationship?
No. It means little or no linear association; a nonlinear relationship can still exist.
Does correlation prove causation?
No. Correlation alone cannot establish a causal relationship.