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Mathematics of Statistics

Definition[edit]

Pearson’s correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a “product moment”, that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.

For a population[edit]

(X,Y)

Pearson’s correlation coefficient when applied to a population is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient. Given a pair of random variables {\displaystyle (X,Y)}, the formula for ρ[7] is:

{\displaystyle \rho _{X,Y}={\frac {\operatorname {cov} (X,Y)}{\sigma _{X}\sigma _{Y}}}}    (Eq.1)

where:

  • {\displaystyle \operatorname {cov} } is the covariance
  • {\displaystyle \sigma _{X}} is the standard deviation of {\displaystyle X}
  • {\displaystyle \sigma _{Y}} is the standard deviation of {\displaystyle Y}
\rho

The formula for {\displaystyle \rho } can be expressed in terms of mean and expectation. Since{\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})],}

{\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})],}

[7]

\rho

the formula for {\displaystyle \rho } can also be written as

{\displaystyle \rho _{X,Y}={\frac {\operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]}{\sigma _{X}\sigma _{Y}}}}    (Eq.2)

where:

  • {\displaystyle {\sigma _{Y}}} and {\displaystyle \sigma _{X}} are defined as above
  • {\displaystyle \mu _{X}} is the mean of {\displaystyle X}
  • {\displaystyle \mu _{Y}} is the mean of {\displaystyle Y}
  • {\displaystyle \operatorname {E} } is the expectation.
\rho

The formula for {\displaystyle \rho } can be expressed in terms of uncentered moments. Since

  • {\displaystyle \mu _{X}=\operatorname {E} [X]}
  • {\displaystyle \mu _{Y}=\operatorname {E} [Y]}
  • {\displaystyle \sigma _{X}^{2}=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\operatorname {E} [X^{2}]-[\operatorname {E} [X]]^{2}}
  • {\displaystyle \sigma _{Y}^{2}=\operatorname {E} [(Y-\operatorname {E} [Y])^{2}]=\operatorname {E} [Y^{2}]-[\operatorname {E} [Y]]^{2}}
  • {\displaystyle \operatorname {E} [(X-\mu _{X})(Y-\mu _{Y})]=\operatorname {E} [(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])]=\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y],\,}
\rho

the formula for {\displaystyle \rho } can also be written as{\displaystyle \rho _{X,Y}={\frac {\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}{{\sqrt {\operatorname {E} [X^{2}]-[\operatorname {E} [X]]^{2}}}~{\sqrt {\operatorname {E} [Y^{2}]-[\operatorname {E} [Y]]^{2}}}}}.}

{\displaystyle \rho _{X,Y}={\frac {\operatorname {E} [XY]-\operatorname {E} [X]\operatorname {E} [Y]}{{\sqrt {\operatorname {E} [X^{2}]-[\operatorname {E} [X]]^{2}}}~{\sqrt {\operatorname {E} [Y^{2}]-[\operatorname {E} [Y]]^{2}}}}}.}