Editing Correlation (section)

===Sample correlation coefficient===
Given a series of <math>n</math> measurements of the pair <math>(X_i,Y_i)</math> indexed by <math>i=1,\ldots,n</math>, the ''sample correlation coefficient'' can be used to estimate the population Pearson correlation <math>\rho_{X,Y}</math> between <math>X</math> and <math>Y</math>.  The sample correlation coefficient is defined as

:<math>
r_{xy} \quad \overset{\underset{\mathrm{def}}{}}{=} \quad \frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{(n-1)s_x s_y}
      =\frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}
            {\sqrt{\sum\limits_{i=1}^n (x_i-\bar{x})^2 \sum\limits_{i=1}^n (y_i-\bar{y})^2}},
</math>

where <math>\overline{x}</math> and <math>\overline{y}</math> are the sample [[arithmetic mean|means]] of <math>X</math> and <math>Y</math>, and <math>s_x</math> and <math>s_y</math> are the [[Standard deviation#Corrected sample standard deviation|corrected sample standard deviations]] of <math>X</math> and <math>Y</math>.

Equivalent expressions for <math>r_{xy}</math> are
:<math>
\begin{align}
r_{xy} &=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{n s'_x s'_y} \\[5pt]
       &=\frac{n\sum x_iy_i-\sum x_i\sum y_i}{\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.
\end{align}
</math>
where <math>s'_x</math> and <math>s'_y</math> are the [[Standard deviation#Uncorrected sample standard deviation|''uncorrected'' sample standard deviations]] of <math>X</math> and <math>Y</math>.

If <math>x</math> and <math>y</math> are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range.<ref>{{cite journal|last=Francis|first=DP|author2=Coats AJ|author3=Gibson D|title=How high can a correlation coefficient be?|journal=Int J Cardiol|year=1999|volume=69|pages=185–199|doi=10.1016/S0167-5273(99)00028-5|issue=2|pmid=10549842}}</ref> For the case of a linear model with a single independent variable, the [[Coefficient of determination|coefficient of determination (R squared)]] is the square of <math>r_{xy}</math>, Pearson's product-moment coefficient.