Editing Pearson correlation coefficient (section)

===For a sample===
Pearson's correlation coefficient, when applied to a [[sample (statistics)|sample]], is commonly represented by <math>r_{xy}</math> and may be referred to as the ''sample correlation coefficient'' or the ''sample Pearson correlation coefficient''. We can obtain a formula for <math>r_{xy}</math> by substituting estimates of the covariances and variances based on a sample into the formula above. Given paired data <math>\left\{ (x_1,y_1),\ldots,(x_n,y_n) \right\}</math> consisting of <math>n</math> pairs, <math>r_{xy}</math> is defined as

<math display=block>r_{xy} =\frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}(y_i - \bar{y})^2}}</math>

where
*<math>n</math> is sample size
*<math>x_i, y_i</math> are the individual sample points indexed with ''i''
*<math display="inline">\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i</math> (the sample mean); and analogously for <math>\bar{y}</math>.

Rearranging gives us this<ref name="RealCorBasic"/> formula for <math>r_{xy}</math>:

:<math>r_{xy} = \frac{\sum_i x_i y_i-n\bar{x}\bar{y}}
{\sqrt{\sum_i x_i^2-n\bar{x}^2}~\sqrt{\sum_i y_i^2-n\bar{y}^2}},</math>

where <math>n, x_i, y_i, \bar{x}, \bar{y}</math> are defined as above.

Rearranging again gives us this formula for <math>r_{xy}</math>:

:<math>r_{xy} = \frac{n\sum x_i y_i - \sum x_i\sum y_i}
{\sqrt{n\sum x_i^2-\left(\sum x_i\right)^2}~\sqrt{n\sum y_i^2-\left(\sum y_i\right)^2}},</math>

where <math>n, x_i, y_i</math> are defined as above.

This formula suggests a convenient single-pass algorithm for calculating sample correlations, though depending on the numbers involved, it can sometimes be [[numerical stability|numerically unstable]].

An equivalent expression gives the formula for <math>r_{xy}</math> as the mean of the products of the [[standard score]]s as follows:

:<math>r_{xy} = \frac{1}{n-1} \sum ^n _{i=1} \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right)</math>

where
*<math>n, x_i, y_i, \bar{x}, \bar{y}</math> are defined as above, and <math>s_x, s_y</math> are defined below
*<math display="inline">\left( \frac{x_i - \bar{x}}{s_x} \right)</math> is the standard score (and analogously for the standard score of <math>y</math>).

Alternative formulae for <math>r_{xy}</math> are also available. For example, one can use the following formula for <math>r_{xy}</math>:

:<math>r_{xy} =\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}</math>
where
*<math>n, x_i, y_i, \bar{x}, \bar{y}</math> are defined as above and:
*<math display="inline">s_x = \sqrt{\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2}</math> (the [[sample standard deviation]]); and analogously for <math>s_y</math>.