Correlation vs Regression: Statistics for CA Foundation
Correlation tells you IF two variables are related and HOW STRONGLY. Regression tells you HOW EXACTLY one variable changes when the other changes — and lets you predict.
head-to-Head Comparison
| Basis | Correlation | Regression |
|---|---|---|
| Purpose | Measures the degree and direction of linear relationship between two variables | Estimates the value of one variable (dependent) from another (independent) |
| Output | Correlation Coefficient (r) ranging from −1 to +1 | Regression Equation: Y = a + bX (Line of Best Fit) |
| Cause-Effect | Does NOT imply causation — only association | Implies a functional/predictive relationship between variables |
| Symmetry | Correlation of X on Y = Correlation of Y on X (symmetric) | Regression of Y on X ≠ Regression of X on Y (asymmetric) |
| Key Formula | r = Σ(dx × dy) / √[Σdx² × Σdy²] (Karl Pearson's Coefficient) | b (regression coefficient) = Σ(dx × dy) / Σdx²; a = Ȳ − bX̄ |
The 'r² = b_yx × b_xy' Trap
The key relationship: r² = byx × bxy, so r = √(byx × bxy). Importantly, byx and bxy must have the same sign as r. If both regression coefficients are negative, r is negative. If one is positive and one negative, something is wrong — check your calculations.
Common Ground (Similarities)
- Both study the relationship between two (or more) quantitative variables.
- Both require the same raw data (paired observations of X and Y).
- The regression coefficient (b) and correlation coefficient (r) are related: r = b_yx × b_xy (geometric mean relation: r² = b_yx × b_xy).
Test Your Understanding
Q1: If correlation coefficient r = 0, it means:
Perfect positive correlation
No linear correlation between the variables ✅
Perfect negative correlation
Variables are identical
Explanation: r = 0 indicates no linear relationship between the two variables. However, they may still have a non-linear relationship.
Q2: The regression line of Y on X passes through the point:
(0, 0)
(X̄, Ȳ) ✅
(σx, σy)
(1, 1)
Explanation: Both regression lines always pass through the point of means (X̄, Ȳ). This is a key property used in regression calculations.