Pearson Correlation Coefficient (r

Statistical Theory of Pearson's Correlation Coefficient (r)

Derived by Karl Pearson in the 1890s based on Francis Galton's work, Pearson's product-moment correlation coefficient r measures the strength and direction of linear association between two continuous variables:

• The Normalized Scale:

• Perfect positive linear association (as X increases, Y increases along a straight line).

• Zero linear association (variables have no straight-line dependency).

• Perfect negative linear association (as X increases, Y decreases along a straight line).

• Coefficient of Determination (R^2): r^2 measures the percentage of shared variance between the two variables. If \ (64% shared variance, with 36% unexplained).

• Fisher's z-Transformation: Because the sampling distribution of r is highly skewed when |r| is close to 1, Fisher's hyperbolic tangent transformation \ normalizes the distribution to construct accurate 95% Confidence Intervals.

How to Use This Calculator

Enter your Variable X Values (comma or space separated) (e.g. 4, 6, 8, 11, 13, 16, 18, 20), Variable Y Values (comma or space separated) (e.g. 12, 16, 19, 25, 29, 36, 41, 45), and Significance Level (α) — choose from α = 0.05 (Standard 95% Confidence), α = 0.01 (Strict 99% Confidence), α = 0.10 (Exploratory 90% Confidence). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Pearson Product-Moment Correlation & Fisher z-Transform equations:

Sample Covariance: Cov(X, Y) = Σ_i=1^n (X_i - X)(Y_i - Y)n - 1

Pearson Correlation: r = (Cov(X, Y)) / (s_x s_y) = Σ (X_i - X)(Y_i - Y)√(Σ (X_i - X))^2 Σ (Y_i - Y)^2

Coefficient of Determination: R^2 = r^2

T-Statistic for Significance: t = r √((n - 2) / (1 - r^2)), df = n - 2

Fisher's z-Transformation: z_r = (1) / (2) ((1 + r) / (1 - r)), SE_z = (1) / (√(n - 3))

Worked Example

For example, evaluating 8 paired observations \ and With \ and and . Pearson . The t-statistic is \ (p < 0.0001), confirming a very strong, statistically significant positive linear correlation.

Understanding Your Result

Statistical results require context: a mean alone is misleading without knowing the standard deviation (spread). Normal distribution reference: 68% of values fall within ±1 SD of the mean, 95% within ±2 SD, 99.7% within ±3 SD. For small sample sizes (n < 30), use t-distributions rather than z-scores. Statistical significance (p < 0.05) does not imply practical significance — effect size (Cohen's d, R², odds ratio) measures real-world importance. Correlation coefficient r: |r| < 0.3 weak, 0.3–0.7 moderate, > 0.7 strong.