Pearson Correlation (r) Calculator
Enter two equal-length data series (X and Y) to calculate the Pearson correlation coefficient r. Separate values by commas, spaces, or new lines.
What is a correlation r value?
The correlation coefficient r (usually called Pearson’s r) measures the strength and direction of a linear relationship between two numeric variables. In plain language, it tells you whether two things tend to move together, and how tightly they move together.
- r = +1: perfect positive linear relationship
- r = 0: no linear relationship
- r = -1: perfect negative linear relationship
How to use this correlation calculator
Step-by-step
- Paste your first variable into the X values box.
- Paste your second variable into the Y values box.
- Make sure both lists contain the same number of observations.
- Click Calculate r.
The calculator returns r, r², sample size, and a quick interpretation of relationship strength.
Pearson correlation formula
This page uses the standard Pearson product-moment correlation formula:
Where n is the number of paired observations. The formula normalizes covariance by each variable’s spread, which keeps r in the range from -1 to +1.
Interpreting r values (rule-of-thumb)
Interpretation depends on context, but this quick guide is commonly used:
- |r| < 0.10: negligible linear relationship
- 0.10 to 0.29: weak
- 0.30 to 0.49: moderate
- 0.50 to 0.69: moderately strong
- 0.70 to 0.89: strong
- 0.90 to 1.00: very strong
When should you use Pearson’s r?
Pearson correlation works best when:
- Both variables are continuous numerical data.
- The relationship is approximately linear.
- Outliers are limited (or handled carefully).
- Observations are paired correctly and independent.
Common mistakes to avoid
- Mismatched rows: each X value must correspond to the correct Y value.
- Using r on ranked or ordinal data: Spearman’s rho may be better in that case.
- Ignoring outliers: a single extreme value can distort correlation.
- Assuming causal effects: a high r alone is not proof of cause-and-effect.
Quick example
Suppose you track weekly study hours and exam score. If higher study hours generally come with higher scores, you may see a positive r (e.g., 0.78). That indicates a strong positive linear relationship, but still does not prove study hours are the only reason scores increased.
FAQ
Can r be greater than 1?
No. By definition, Pearson’s r is always between -1 and +1.
What if one variable never changes?
If all X values or all Y values are identical, correlation is undefined because standard deviation is zero. This calculator reports that as an error.
What is r²?
r² is the coefficient of determination. It is the proportion of variance in one variable that is linearly associated with the other. For example, r = 0.60 gives r² = 0.36, meaning 36% shared linear variance.
Is a negative correlation bad?
Not necessarily. A negative r simply means one variable tends to increase while the other decreases.