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How to use
Pearson’s r measures the direction and strength of a linear relationship between paired numeric series. The tool returns r, r², a simple regression line and basic statistics, but does not interpret correlation as causation. For X=[1,2,3] and Y=[1,3,2], both means are 2, the cross-deviation sum is 1 and each squared-deviation sum is 2. Thus r=0.5 and r²=0.25. In simple linear regression with an intercept, r² here corresponds to 25% of the observed Y variation explained by the fitted model; it is not a 25% prediction accuracy or a causal claim.
- Enter X values separated by spaces or commas.
- Enter Y values with the same count and pairing order.
- Calculate.
- Review r, r², regression line and pair count.
How it is calculated
Example
For X=[1,2,3,4] and Y=[2,4,6,8], r=1, a perfect positive linear relationship, and the regression line is y=2x.
Important notes
Pearson measures linear association and is sensitive to outliers. r near zero does not rule out a nonlinear relationship, and correlation does not prove causation.
Worked examples and interpreting results
If the first list is 1, 2, 3 and the second is 2, 4, 6, their relationship is perfectly positive and linear. Replacing the second list with 6, 4, 2 makes it perfectly negative and linear.
| Case | Calculation | Result |
|---|---|---|
| Both lists increase together | X = [1,2,3]; Y = [2,4,6] → r | 1 |
| One decreases while the other increases | X = [1,2,3]; Y = [6,4,2] → r | -1 |
| Pearson r: X=[1,2,3], Y=[1,3,2] | 1 ÷ √(2 × 2) | 0.5 |
| Squared correlation for the same data | 0.5² | 0.25 |
How to check the result
Matching positions must represent the same observation. Never reorder just one list. Pearson correlation is undefined if either list is constant, and a strong correlation does not establish causation.
Pearson correlation coefficient
Pearson's r describes the strength and direction of a linear relationship in paired observations. It ranges from minus one to plus one, with zero indicating no linear correlation.
- Perfect positive correlation: r([1,2,3],[2,4,6]) = +1
- Perfect negative correlation: r([1,2,3],[6,4,2]) = −1
- Zero linear correlation: r([1,2,3],[1,0,1]) = 0
Important note
Correlation does not establish causation. Pearson's r is undefined when either series is constant.
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Frequently asked questions
What is the range of r?
From -1 to +1.
What does r² mean?
The squared correlation; in simple linear regression it is the coefficient of determination.
Does strong correlation prove causation?
No. Correlation measures the strength and direction of a linear association, but it does not prove that one variable causes the other. A third variable, bias or chance may explain the pattern.
Can correlation be negative while r² is positive?
Yes. If r=−0.5, r²=0.25. The sign of r gives the direction of the linear association; squaring removes that direction.
Why is r undefined for a constant data series?
Its squared-deviation sum is zero, making the Pearson denominator zero. Both series need nonzero variation.