Pearson Correlation Coefficient

Pearson Correlation Coefficient

• $r=\frac{\sum (X_i-\overline{X})(Y_i-\overline{Y})}{\sqrt{\sum (X_i-\overline{X}{)}^2\cdot \sum (Y_i-\overline{Y}{)}^2}}$
  • or $r=\frac{1}{n-1}\sum (\frac{X_i-\overline{X}}{s_X})(\frac{Y_i-\overline{Y}}{s_Y})={\hat{\beta }}_1\frac{s_X}{s_Y}$
  • function of $\widehat{{\beta }_1}$ with the standard deviations of X and Y
    • sd is always positive, so r will always have the same sign as $\widehat{{\beta }_1}$ (slope)
• properties of the correlation coefficient
  • r is unitless
  • $-1\le r\le 1$
  • r has the same sign as $\widehat{{\beta }_1}$
  • if r = 1 or -1, all points fall directly on a line
    • values close to 1 or -1 indicate a strong linear relationship
  • for simple linear regression models, $r^2=R^2$ (Coefficient of Determination)