Correlation and Simple Linear Regression Review

Pearson's correlation coefficient (r) measures the strength and direction of the linear association between two continuous variables. r ranges from −1 (perfect negative linear relationship) through 0 (no linear relationship) to +1 (perfect positive linear relationship). Formula: r = [Σ(xᵢ − x̄)(yᵢ − ȳ)] / [(n−1) × s_x × s_y]. Interpretation: r = 0.1–0.3 (weak), 0.3–0.5 (moderate), 0.5–0.7 (strong), 0.7–0.9 (very strong), 0.9–1.0 (near perfect). These thresholds vary by field — in psychology, r = 0.3 is substantial; in physics calibration, r = 0.99 might be considered weak. r² (coefficient of determination): the proportion of variance in Y explained by X. r = 0.7 → r² = 0.49 → 49% of the variance in Y is accounted for by X. Key limitations of Pearson's r: (1) measures only linear associations — r can be near zero even when a strong nonlinear relationship exists. Always inspect a scatter plot. (2) r is sensitive to outliers — a single extreme point can dramatically change r. (3) r does not imply causation. (4) Restriction of range: if data are collected only from a narrow range of X values, r will underestimate the true association. Testing H₀: ρ = 0 (no correlation in population): t = r√(n−2) / √(1−r²), with df = n − 2.