What does the Pearson Correlation Coefficient (r) measure?

Pearson's r measures the strength and direction of a linear relationship between two continuous variables, ranging from -1.0 (perfect negative linear correlation) to +1.0 (perfect positive linear correlation), with 0 indicating no linear relationship.

Why does statistical correlation not establish causality?

Correlation indicates that two variables fluctuate together, but does not prove that changes in one variable cause changes in the other (which may be driven by confounding third variables or coincidence).

How is the Coefficient of Determination (R²) interpreted?

R² is the square of Pearson's r. An r of 0.80 yields an R² of 0.64, meaning 64% of the variance in variable Y is explained by its linear relationship with variable X.

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Pearson Correlation Coefficient (r) Calculator

Evaluate linear relationships between two variables with our Pearson Correlation Coefficient Calculator! Enter paired $(X, Y)$ datasets to calculate Pearson's $r$, Coefficient of Determination ($R^2$), Covariance ($\text{Cov}(X,Y)$), and linear regression slope ($y = mx + b$) with interactive scatter plots.

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Variable Datasets

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X-Axis Variable Title

Y-Axis Variable Title

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Drop Dataset X .txt file, or select manually

Values X (Comma, space, or newline separated)

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Drop Dataset Y .txt file, or select manually

Values Y (Comma, space, or newline separated)

Popular Correlation Scenarios

Correlation Formula Guide

Pearson's correlation coefficient ($r$) measures the strength and direction of a linear relationship between two variables:

r = Σ((x - x̄)(y - ȳ)) / √[Σ(x - x̄)² × Σ(y - ȳ)²]

The least squares regression line maps a straight trend path through scattered data points:

y = mx + c

where slope m = r(sy / sx) and intercept c = ȳ - mx̄

Linear Correlation & Regression: Concepts & Formulas

Explore the mathematical measures used to trace strength, direction, and predictive patterns between numerical variables.

Pearson Correlation Coefficient (r)

Measures the strength and direction of a linear relationship between two continuous variables, ranging from -1.0 to +1.0.

Formula:r = Σ((x - x̄)(y - ȳ)) / √[Σ(x - x̄)² × Σ(y - ȳ)²]

Standard Use: Determining if two factors change together (e.g., student study hours vs exam performance).

R²

Coefficient of Determination (R-Squared)

The proportion of variance in the dependent variable (Y) that is explained by the independent variable (X).

Formula:R² = r² (Calculated by squaring Pearson's r value)

Standard Use: Evaluating model fitting accuracy—for instance, an R² = 0.81 implies that 81% of Y's variation is predictable from X.

y=mx+c

Least-Squares Linear Regression

The straight line of best fit that minimizes the sum of squared vertical residuals (distances) between the data points and the trend line.

Formula:y = mx + c where slope m = r(sy / sx) and intercept c = ȳ - mx̄

Standard Use: Forecasting future values based on historic trends (e.g. predicting sales revenues based on targeted advertising budgets).

Cov

Covariance

Indicates the directional relationship between X and Y. A positive covariance implies variables increase together.

Formula:Cov(X,Y) = Σ((xᵢ - x̄)(yᵢ - ȳ)) / n

Standard Use: Portfolio risk calculations to determine asset movement directions relative to each other.

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Correlation Strength & Significance Scale

How to interpret the magnitude and direction of the Pearson coefficient (r) value:

Coefficient Range (r)	Relationship Strength	Direction & Practical Meaning
+1.0 / -1.0	Perfect Correlation	Complete linear matching. All points sit exactly on the trend line.
±0.7 to ±0.9	Strong Correlation	Highly reliable relationship. Significant predictive capabilities.
±0.3 to ±0.6	Moderate Correlation	Clear observable pattern, but with substantial individual variance.
±0.1 to ±0.2	Weak / Negligible	Minimal linear connection. Very poor predictive fitting.
0.0	No Correlation	Completely random scattered points. Variables are independent.