Editing Pearson correlation coefficient (section)

===For a population===
Pearson's correlation coefficient, when applied to a [[statistical population|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 <math>(X,Y)</math> (for example, Height and Weight), the formula for ''ρ''<ref name="RealCorBasic">Real Statistics Using Excel, "[http://www.real-statistics.com/correlation/basic-concepts-correlation/ Basic Concepts of Correlation]", retrieved 22 February 2015.</ref> is<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Statistical Correlation|url=https://mathworld.wolfram.com/StatisticalCorrelation.html|access-date=2020-08-22|website=Wolfram MathWorld|language=en}}</ref>

<math display=block> \rho_{X,Y}= \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y}</math>

where
*<math> \operatorname{cov} </math> is the [[covariance]]
*<math> \sigma_X </math> is the [[standard deviation]] of  <math> X </math>
*<math> \sigma_Y </math> is the standard deviation of  <math> Y </math>.

The formula for <math>\operatorname{cov}(X,Y)</math> can be expressed in terms of [[mean]] and [[Expected Value|expectation]]. Since<ref name="RealCorBasic"/>

:<math>\operatorname{cov}(X,Y) = \operatorname\mathbb{E}[(X-\mu_X)(Y-\mu_Y)],</math>

the formula for <math>\rho</math> can also be written as

<math display=block> \rho_{X,Y} = \frac{\operatorname\mathbb{E}[(X - \mu_X)(Y - \mu_Y)]}{\sigma_X\sigma_Y}</math>

where
*<math> \sigma_Y </math> and <math> \sigma_X </math> are defined as above
*<math> \mu_X </math> is the mean of <math> X </math>
*<math> \mu_Y </math> is the mean of <math> Y </math>
*<math> \operatorname\mathbb{E} </math> is the expectation.

The formula for <math>\rho</math> can be expressed in terms of uncentered moments.  Since

:<math>\begin{align}
       \mu_X ={} &\operatorname\mathbb{E}[X] \\
       \mu_Y ={} &\operatorname\mathbb{E}[Y] \\
  \sigma_X^2 ={} &\operatorname\mathbb{E}\left[\left(X - \operatorname\mathbb{E}[X]\right)^2\right] = \operatorname\mathbb{E}\left[X^2\right] - \left(\operatorname\mathbb{E}[X]\right)^2 \\
   \sigma_Y^2 ={} &\operatorname\mathbb{E}\left[\left(Y - \operatorname\mathbb{E}[Y]\right)^2\right] = \operatorname\mathbb{E}\left[Y^2\right] - \left(\operatorname\mathbb{E}[Y]\right)^2 \\
\operatorname{cov}(X,Y) ={} &\operatorname\mathbb{E}[\left(X - \mu_X\right)\left(Y - \mu_Y\right)] = \operatorname\mathbb{E}[\left(X - \operatorname\mathbb{E}[X]\right)\left(Y - \operatorname\mathbb{E}[Y]\right)] = \operatorname\mathbb{E}[XY] - \operatorname\mathbb{E}[X]\operatorname\mathbb{E}[Y] ,
\end{align}</math>

the formula for <math>\rho</math> can also be written as
<math display="block">\rho_{X,Y} =
  \frac{\operatorname\mathbb{E}[XY] - \operatorname\mathbb{E}[X]\operatorname\mathbb{E}[Y]}{\sqrt{\operatorname\mathbb{E}\left[X^2\right] - \left(\operatorname\mathbb{E}[X] \right)^2} ~ \sqrt{\operatorname\mathbb{E}\left[Y^2\right] - \left(\operatorname\mathbb{E}[Y] \right)^2}}.</math>