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Calculator Answers

Correlation Coefficient Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A correlation coefficient calculator computes Pearson's r, which measures the strength and direction of the linear relationship between two variables. The calculator processes coordinate pairs (X, Y) to yield a value between negative one and positive one. A value near positive one indicates a strong positive linear correlation, a value near negative one indicates a strong negative linear correlation, and a value near zero indicates no linear correlation. Statisticians and researchers use this tool to identify trends and analyze relationships within experimental datasets.

Add paired x and y values, one row at a time. The calculator returns the Pearson correlation coefficient (r), r squared, and a plain-English strength label.

See also: P Value Calculator, Variance Calculator, and Standard Deviation Calculator – Variance & SD.

Quick Answer

Compute Pearson's correlation coefficient for two variables. Enter your data pairs to find the strength and direction of their linear relationship.

xy

Enter at least two paired (x, y) values. Empty rows are ignored. r is undefined if either x or y has zero variance (all the same value).

Pearson correlation

5 pairs

0.8704

Strong positive correlation

Pearson r0.8704
r squared0.7576
Mean x3
Mean y4.4
Pairs counted5
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Examples

Pairs (1,2), (2,4), (3,5), (4,4), (5,7)

r ≈ 0.872 (strong positive)

Pairs (1,5), (2,4), (3,3), (4,2), (5,1)

r = −1.000 (perfect negative)

Pairs (1,2), (2,1), (3,3), (4,2), (5,1)

r ≈ −0.408 (moderate negative)

How it works

Subtract the mean from each x and each y. Multiply the deviations pairwise and sum. Divide by the square root of the product of squared deviations to standardize. The result is between −1 and +1.

r · Σ((x − x̄)(y − ȳ)) ÷ √(Σ(x − x̄)² × Σ(y − ȳ)²)

Strength bands: |r| ≥ 0.7 strong · 0.3 to 0.7 moderate · 0.1 to 0.3 weak · below 0.1 little/no linear relationship.

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Frequently asked questions

Pearson's r measures the strength and direction of the linear relationship between two variables. It ranges from −1 (perfect negative) to +1 (perfect positive). A value of 0 means no linear relationship. The sign tells you direction; the magnitude tells you strength. r only captures linear association; it can be near 0 for relationships that are strong but curved.

For each pair, compute the deviation from the mean of x and the mean of y. Multiply those deviations and sum the products. Divide by the square root of the product of the sum of squared x-deviations and the sum of squared y-deviations. The formula is r = Σ((x − x̄)(y − ȳ)) ÷ √(Σ(x − x̄)² × Σ(y − ȳ)²).

r squared is the proportion of variance in y that is explained by a linear relationship with x. An r of 0.8 means r squared = 0.64, so a linear fit to x explains 64 percent of the variance in y. It is also the same as the R² that comes out of a simple linear regression on the same data.

Common rules of thumb: |r| above 0.7 is strong, 0.3 to 0.7 is moderate, 0.1 to 0.3 is weak, and below 0.1 is little to no linear association. These bands are field-specific in practice. A physics experiment expects much higher correlations than a social science study, and the same value of r can mean different things depending on the data type.

No. r measures association only. Two variables can be highly correlated because one causes the other, both are caused by a third variable, the data is selected in a biased way, or pure coincidence. Causal claims require study design (experiments, controlled conditions) that goes beyond correlation.

r is undefined when either x or y has zero variance (every value is the same). The formula divides by the spread of each variable, and that spread is zero in this case. Add some variation to the data and try again.