Correlation Coefficient Calculator
Analyze the linear relationship between two variables with high precision. Our premium studio calculates Pearson's correlation coefficient (r), coefficient of determination (r²), covariance, and least-squares regression lines with custom labels and vector scatter plots.
Variable Datasets
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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 + cm = r(sy / sx) and intercept c = ȳ - mx̄Linear 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.
r = Σ((x - x̄)(y - ȳ)) / √[Σ(x - x̄)² × Σ(y - ȳ)²]Coefficient of Determination (R-Squared)
The proportion of variance in the dependent variable (Y) that is explained by the independent variable (X).
R² = r² (Calculated by squaring Pearson's r value) 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.
y = mx + c where slope m = r(sy / sx) and intercept c = ȳ - mx̄Covariance
Indicates the directional relationship between X and Y. A positive covariance implies variables increase together.
Cov(X,Y) = Σ((xᵢ - x̄)(yᵢ - ȳ)) / nCorrelation 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. |
Overview & Capabilities
Analyze the linear relationship between two variables with high precision. Our premium studio calculates Pearson's correlation coefficient (r), coefficient of determination (r²), covariance, and least-squares regression lines with custom labels and vector scatter plots.
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Key Features
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Frequently Asked Questions
Q How is Pearson's correlation coefficient (r) calculated and interpreted?
Pearson's r is calculated by dividing the covariance of two variables by the product of their standard deviations. Its value ranges from -1.0 to +1.0. A value of +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative relationship, and 0 indicates no linear correlation whatsoever.
Q What is the difference between correlation and causation?
Correlation measures the statistical association or relationship between two variables—showing that they change together. Causation proves that a change in one variable directly causes the change in the other. Association does not prove cause, as a third hidden variable (confounding factor) could be driving both changes.
Q How do I interpret the Coefficient of Determination (r²)?
The Coefficient of Determination (r² or R-squared) represents the proportion of variance in the dependent variable (Y) that is predictable from the independent variable (X). For example, an r-squared value of 0.81 means that 81% of the variation in Y is explained by the variation in X, with the remaining 19% driven by other factors.
Q What is the linear regression line and how is it calculated?
The linear regression line (y = mx + c) represents the straight line of 'best fit' that minimizes the sum of squared vertical distances from the scattered data points. The slope (m) is calculated as r * (sy / sx), representing the change in Y per unit change in X, and the intercept (c) represents the value of Y when X is zero.
Q What is covariance and how does it differ from correlation?
Both measure linear association, but covariance is scale-dependent and expressed in the multiplied units of X and Y, making its value difficult to interpret directly. Correlation normalizes covariance by dividing it by the standard deviations of both variables, producing a scale-free value between -1.0 and +1.0.