Correlation Calculator

|r| ≈ 0.1 — weak. |r| ≈ 0.3 — moderate. |r| ≈ 0.5 — strong. |r| ≥ 0.7 — very strong. These thresholds vary by field; physics expects much higher than social science.

Sign matters: positive r means as x grows, y tends to grow; negative r means y tends to fall.

r² (coefficient of determination) is the share of variance in y explained by a linear model on x. r = 0.7 → r² = 0.49: about half of y's variation is linearly explained by x.

Even with no underlying relationship (ρ = 0), small samples produce nonzero r by chance.

Test statistic: t = r · √(n − 2) / √(1 − r²), with df = n − 2.

Two-tailed p-value tells you the probability of seeing |r| this large or larger under H₀: ρ = 0. Significant at α = 0.05 if p < 0.05.

For n = 30 with r = 0.4, t = 2.31 → p ≈ 0.028 (significant). For n = 10 with the same r, t = 1.23 → p ≈ 0.25 (not significant). Sample size matters enormously.

x = (1,2,3,4,5,6,7,8,9,10), y = (2,4,5,4,5,7,8,8,10,12).

mean(x) = 5.5, mean(y) = 6.5. Σ(xi−x̄)(yi−ȳ) = 73.5. Σ(xi−x̄)² = 82.5. Σ(yi−ȳ)² = 75.

r = 73.5 / √(82.5·75) ≈ 0.934. r² ≈ 0.873 — about 87% of y's variance is explained by a linear fit on x.

OLS slope = 73.5/82.5 ≈ 0.891; intercept = 6.5 − 0.891·5.5 ≈ 1.6. Fitted line: ŷ = 1.6 + 0.89·x.

Ice-cream sales correlate with drownings; both spike in summer (the lurking variable). r doesn't tell you which way the arrow points or whether one even exists.

Reverse causation: faster typing correlates with more typos — typing speed could cause typos, typos (frustration) could slow typing, or both depend on fatigue.

Confounders: school years and salary correlate; ability, family background, and field all confound.

Restriction of range — if you only sample the top 10% of x, r drops sharply even if the true relationship is strong.

r only catches LINEAR relationships. A perfect parabola y = x² with x spanning negative and positive gives r ≈ 0.

Outliers swing r dramatically — one extreme point can flip a weak negative to a strong positive. Always plot the scatter.

Treating r as a probability or percentage. r = 0.5 doesn't mean '50% related'; r² is the variance-explained percentage.

Computing r across heterogeneous subgroups can hide or invert the true within-group relationships (Simpson's paradox).