SL 4.4—Correlation and regression

Correlation measures association, not a causal mechanism.

A correlation coefficient describes the direction and strength of a linear association, while regression predicts one variable from another under model assumptions.

Example

Ice-cream sales and sunburn may rise together because temperature affects both; the correlation does not make sales the cause.

Plot the relationship, inspect outliers and direction, then state whether the regression is for prediction or explanation.

A strong correlation can be nonlinear or confounded; it does not prove causation.

Pearson's $r$ lies between $-1$ and $1$ and measures only linear association; its sign gives direction and $|r|$ gives linear strength. For a fitted line $y=ax+b$, $a$ is the predicted change in $y$ for one unit of $x$ and $b$ is the predicted value at $x=0$ when that interpretation is meaningful. Use the $y$-on-$x$ line to predict $y$ from $x$, avoid unjustified extrapolation, and never infer causation from correlation alone.