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The following is the formula for Pearson's coefficient of correlation :

  • A.\(\dfrac{\text{Covariance(X, Y)}}{\sigma_{X}\sigma_{Y}}\)
  • B.\(\dfrac{\text{Covariance(X, Y)}}{\sigma_{X}^{2}\sigma_{Y}^{2}}\)
  • C.\(\dfrac{\sigma_{X}^{2}}{\sigma_{Y}^{2}}\) Covariance(X, Y)
  • D.\(\sigma_{X}^{2} \times\) Covariance(X, Y) \(\times \sigma_{Y}^{2}\)

Answer: A

Pearson's coefficient is \(\dfrac{\text{Covariance(X, Y)}}{\sigma_{X}\sigma_{Y}}\).

Dividing the covariance by the product of the two standard deviations removes the units of measurement and gives a number between -1 and 1. Equivalently,

\[ \begin{aligned} r &= \frac{\sum(x - \bar{x})(y - \bar{y})}{\sqrt{D}}, \\ D &= \sum(x - \bar{x})^{2}\sum(y - \bar{y})^{2} \end{aligned} \]

Dividing by the product of the variances gives a quantity that depends on the units, and the forms that multiply the covariance by variances are not correlation coefficients at all.

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