Pearson Correlation Coefficient (r) Calculator
Enter paired values for X and Y in the same order. Use commas, spaces, semicolons, or line breaks as separators.
What is the correlation coefficient r?
The correlation coefficient r (usually called Pearson’s r) measures the strength and direction of a linear relationship between two variables. It ranges from -1 to +1.
- r = +1: perfect positive linear relationship
- r = 0: no linear relationship
- r = -1: perfect negative linear relationship
How to use this calculator
- Enter your X data points in the first field.
- Enter the corresponding Y data points in the second field.
- Make sure both lists have the same number of values (paired observations).
- Click Calculate r to get the result instantly.
Example pair structure: if your first X value is 10 and your first Y value is 25, those two values are treated as one pair. Keep all pairs aligned in order.
The formula behind Pearson’s r
This calculator uses the standard computational formula:
r = [nΣxy - (Σx)(Σy)] / √{[nΣx² - (Σx)²][nΣy² - (Σy)²]}
where n is the number of paired observations.
Interpreting correlation strength (rule of thumb)
- 0.00 to 0.19: very weak
- 0.20 to 0.39: weak
- 0.40 to 0.59: moderate
- 0.60 to 0.79: strong
- 0.80 to 1.00: very strong
The sign (+ or -) tells direction. The absolute value tells strength.
Why this matters
Correlation is used in business analytics, social science, health research, machine learning, and finance. It helps answer questions like:
- Do study hours and exam scores move together?
- Is advertising spend associated with revenue?
- Do sleep duration and mood ratings show a pattern?
Important cautions
1) Correlation is not causation
A high r does not prove one variable causes the other. Hidden factors or coincidence may explain the relationship.
2) Outliers can distort results
One extreme value can inflate or deflate correlation dramatically. Always inspect your data visually when possible.
3) Pearson r captures linear association
If the relationship is curved (nonlinear), Pearson r may be low even if variables are clearly related.
Quick example
Suppose X is weekly exercise hours and Y is resting heart rate improvement score. If r = -0.72, that indicates a strong negative linear relationship: as exercise hours increase, heart rate score tends to decrease (improve).
Final thoughts
Use this correlation coefficient r calculator for quick, accurate Pearson correlation results. For complete analysis, combine r with data visualization, context knowledge, and statistical significance testing.