Correlation

Correlation measures the strength and direction of the linear relationship between two variables $X$ and $Y$. The Pearson correlation coefficient:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \in [-1, 1]$

Interpretation:

• $r = 1$: perfect positive linear relationship.
• $r = -1$: perfect negative linear relationship.
• $r = 0$: no linear relationship (but possibly a non-linear one!).
• $|r| > 0.7$: strong; $0.3 < |r| < 0.7$: moderate; $|r| < 0.3$: weak.

Crucial caveats:

• Correlation is not causation. Ice cream sales correlate with drowning deaths — both driven by hot weather.
• Sensitive to outliers. A single extreme point can flip $r$.
• Linear only. A perfect quadratic relationship $y = x^2$ has $r \approx 0$ around symmetric data.

For ranked / non-linear monotonic relationships, use Spearman's $\rho$. For categorical association, use chi-square or Cramér's V.