SL 4.4—Correlation and regression

A correlation coefficient summarises the direction and strength of an association between paired variables. A regression model uses an explanatory variable to estimate a response, with residuals showing what the model misses.

A positive correlation means larger values tend to occur together; a negative one means one tends to fall as the other rises. Strength concerns consistency around a pattern, not the slope's units or a causal mechanism.

If study time and score have r=0.82, a fitted line may predict scores within the observed range. It cannot show that study time alone caused the result: prior attainment, teaching and selection may also matter.

r=0 does not prove independence, and a strong r does not prove causation. Check outliers, range restriction, residual pattern and whether extrapolation leaves the observed data.

A by-eye best-fit line should pass through the mean point $(\bar x,\bar y)$. For $y=ax+b$, $a$ is the predicted change in $y$ per unit increase in $x$ and $b$ is predicted $y$ at $x=0$ when meaningful. A y-on-x regression predicts $y$ from $x$; rearranging it to predict $x$ is not generally valid. Pearson's $r$ measures only linear association.