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2.5 Correlation

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Pearson's Correlation Coefficient

The correlation coefficient (r) — denoted ${r}$

r = \frac{1}{n-1} \sum\left(\frac{x_i - \bar{x}}{s_x}\right)\left(\frac{y_i - \bar{y}}{s_y}\right)

Where ${s_x}$ and ${s_y}$ are the standard deviations of ${x}$ and ${y}$ .

Properties of ${r}$ :

Always between ${-1}$ and ${+1}$ : ${-1 \leq r \leq 1}$
${r > 0}$ → positive association; ${r < 0}$ → negative association
${r = 0}$ → no linear association (but could be nonlinear!)
${r = 1}$ or ${r = -1}$ → perfect linear association
r close to 1 → strong linear; r close to 0 → weak linear

Rough guidelines:

${|r| \geq 0.8}$ → strong linear association
${0.5 \leq |r| < 0.8}$ → moderate
${|r| < 0.5}$ → weak

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