Calculate Pearson's correlation coefficient (r), coefficient of determination (R²), covariance, and correlation strength for paired datasets.
Click "Calculate Correlation" to evaluate relationship strength.
A marketing analyst examining digital ad spend versus monthly revenue records paired observation pairs across 12 months: calculating Pearson's correlation coefficient yields r = +0.92, indicating a strong positive linear relationship, with an R² = 0.846 (84.6%) meaning 84.6% of monthly revenue variation is directly explained by ad budget variance. An educational researcher analyzing study hours versus exam scores evaluates paired student outcomes to determine linear predictive strength.
Pearson's product-moment correlation coefficient (r) is a standardized statistical metric measuring the strength and direction of a linear relationship between two continuous quantitative variables.
The value of r always ranges from -1.0 to +1.0:
This calculator evaluates Pearson's r, covariance, R² variance explained, and qualitative strength descriptions. The following sections explain covariance formulas, non-linear limitation caveats, and practical applications in finance, marketing, and biostatistics.
When paired X and Y datasets are submitted, the engine evaluates sample means x̄ and ȳ, computes cross-product sum of deviations SP_xy, and calculates r.
1. Pearson's Correlation Coefficient Formula (r):
r = SP_xy / √( SS_xx × SS_yy )
Where:
SP_xy = ∑ (xᵢ - x̄)(yᵢ - ȳ)
SS_xx = ∑ (xᵢ - x̄)²
SS_yy = ∑ (yᵢ - ȳ)²
2. Sample Covariance Cov(X, Y):
Cov(X, Y) = SP_xy / (n - 1)
3. Coefficient of Determination (R²):
R² = r²
Represents the percentage of total variance in Y explained by X.
Financial portfolio management and asset diversification. Portfolio managers pair asset returns to find non-correlated (r ≈ 0) or negatively correlated assets to hedge market risk.
Marketing attribution and sales forecasting. Digital marketers measure linear correlation between social media impression counts and website lead generation.
Epidemiology and public health. Medical researchers evaluate correlation between dietary habits and disease incidence rates across population cohorts.
Biometrics and educational research. Psychometricians assess test-retest reliability by correlating student scores across repeated exam administrations.
Ensure your X and Y input lists have the exact same number of data points (paired observations).
Remember: Correlation does not imply causation. A high r value demonstrates a statistical association, not a direct causal mechanism.
For finding slope and y-intercept line of best fit parameters, pair this tool with our Linear Regression Calculator.
The calculation engine operates client-side in JavaScript using IEEE 754 double-precision floats. Paired datasets with up to 20,000 observations process in under 10 milliseconds.
| r Value Range | Direction | Strength Interpretation |
|---|---|---|
| 0.80 to 1.00 (-0.80 to -1.00) | Positive (Negative) | Very Strong linear relationship |
| 0.60 to 0.79 (-0.60 to -0.79) | Positive (Negative) | Strong linear relationship |
| 0.40 to 0.59 (-0.40 to -0.59) | Positive (Negative) | Moderate linear relationship |
| 0.20 to 0.39 (-0.20 to -0.39) | Positive (Negative) | Weak linear relationship |
| 0.00 to 0.19 (-0.00 to -0.19) | Neutral | Very Weak or No linear correlation |
No. Pearson's r specifically measures linear relationships. Quadratic, exponential, or U-shaped relationships can have r ≈ 0 even when a strong non-linear pattern exists.
Covariance indicates direction but its magnitude depends on unit scale. Correlation standardizes covariance onto a unitless scale from -1 to +1.
r indicates direction (+ or -) and linear strength. R² (the square of r) indicates the proportion of variance (from 0% to 100%) in Y explained by X.
Linear Regression Calculator — Calculates line of best fit slope, intercept, and prediction equation.
Standard Deviation Calculator — Computes sample and population standard deviations.
P-Value Calculator — Evaluates statistical significance p-values.