Correlation Coefficient Calculator

The formula

r = Σ((xᵢ - x̄)(yᵢ - ȳ)) / √(Σ(xᵢ - x̄)² × Σ(yᵢ - ȳ)²)

Equivalently: r = cov(X, Y) / (σ_X × σ_Y). It is the covariance normalised by the product of the standard deviations, which forces the result into [-1, +1].

Interpreting r

r² (coefficient of determination) is the fraction of variance in Y explained by a linear model in X. r = 0.7 means r² = 0.49, about half the variance is linear, half is not.

When Pearson is the wrong tool

Pearson assumes a linear relationship with roughly normal data and no big outliers. It fails badly on:

• Curved relationships: y = x² over the range [-5, 5] has r = 0 despite being perfectly predictable.
• Outliers: a single point 5 standard deviations out can double or halve r.
• Rank data: use Spearman’s ρ (correlation of ranks) instead.
• Binary or categorical: use point-biserial, phi, or Cramer’s V depending on types.

Correlation is not causation: the classic

Two variables can correlate strongly because:

1. X causes Y.
2. Y causes X.
3. A third variable Z causes both (confounding).
4. Pure chance (in small samples).
5. Selection bias in how the data was collected.

Ice cream sales and drowning deaths correlate +0.8 over summer months. Ice cream does not cause drowning, hot weather causes both.

Sample size and significance

A small-sample r can be misleading. For n = 5, r = 0.5 is not statistically distinguishable from zero (p ≈ 0.4). For n = 100, r = 0.2 is clearly non-zero (p < 0.05). Always report the sample size alongside r. A p-value for r tests the null hypothesis that the true correlation is zero.