Pearson Correlation Coefficient

r = 0 means there is no linear association between the two variables — knowing X tells you nothing about Y in a linear sense. However, r = 0 does not mean the variables are unrelated. They could have a strong nonlinear relationship (e.g., Y = X² gives r = 0 for symmetric X) or a non-monotonic relationship (e.g., a U-shape). Always plot the data before concluding the variables are independent. Additionally, r = 0 does not mean the variables are statistically independent — independence is stronger than zero correlation. Two variables can be dependent with r = 0, but if they are independent, r must be 0.

It depends entirely on context. In physics, correlations above 0.99 are common for well-controlled measurements. In psychology and social science, correlations of 0.3–0.5 are considered meaningful; 0.5–0.7 is strong. In medicine, even 0.2 between a biomarker and disease risk can be clinically important at population scale. In finance, correlations between 0.3 and 0.7 between asset returns are typical. A useful guideline (Cohen's conventions): |r| < 0.1 negligible, 0.1–0.3 weak, 0.3–0.5 moderate, 0.5–0.7 strong, > 0.7 very strong. But these thresholds are conventions, not laws — always interpret r alongside the scatter plot, sample size, and domain knowledge.

No. Correlation measures statistical association, not causal effect. A high |r| between X and Y could mean X causes Y, Y causes X, or a third variable Z causes both. Classic example: ice cream sales and drowning deaths are positively correlated because both are driven by warm weather — but ice cream doesn't cause drowning. Establishing causation requires experimental design (randomized controlled trials), instrumental variables, natural experiments, or other causal-inference methods. Pearson's r alone is a descriptive statistic, not evidence of causation. The famous phrase "correlation does not imply causation" is one of the most important caveats in all of statistics.

Small sample sizes produce unstable r estimates with wide confidence intervals. A correlation of 0.30 based on n = 5 is not significantly different from zero, whereas the same r = 0.30 with n = 100 is highly significant (p < 0.01). The t-statistic for testing r = 0 is t = r·√(n−2)/√(1−r²), with df = n − 2. For r = 0.30 and n = 5, t = 0.55 (p ≈ 0.62). For r = 0.30 and n = 100, t = 3.08 (p ≈ 0.003). Always report both r and n (or the confidence interval), never r alone. A very high r from a tiny sample should not be treated as reliable evidence of a strong association.

Spearman's ρ (rho) is a rank-based correlation that measures monotonic association — whether Y tends to increase as X increases (not necessarily linearly). Use Spearman's ρ when: (1) the relationship is monotonic but nonlinear (e.g., logarithmic); (2) the data are ordinal (ranks, categories with order); (3) the data contain outliers that could distort Pearson's r; (4) the distributions are heavily skewed; (5) both variables are continuous but you want a robust measure. For roughly linear relationships with no strong outliers, Pearson's r and Spearman's ρ give similar answers. When they disagree substantially, it usually means the relationship is monotonic but nonlinear, and Spearman's ρ is more appropriate.

R² (the coefficient of determination) is simply r squared. It represents the fraction of variance in Y that is explained by the linear relationship with X. For example, r = 0.80 gives R² = 0.64, meaning 64% of the variance in Y is explained by X, and 36% is unexplained. Note that R² is always between 0 and 1 (because r² is), so it loses the sign information — a strong negative correlation and a strong positive correlation give the same R². When interpreting, remember that "explained variance" is a statistical decomposition, not a causal claim. Even a high R² can arise from confounding, and low R² does not necessarily mean a weak relationship if the relationship is nonlinear.