Pearson's Product Moment Correlation

The correlation between two variables, [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], with weighted means of [math]\displaystyle{ \bar{x} }[/math] and [math]\displaystyle{ \bar{y} }[/math] respectively, is:

[math]\displaystyle{ r = \frac{\sum ^n _{i=1}w_i(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}w_i(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}w_i(y_i - \bar{y})^2}} }[/math]

where:

[math]\displaystyle{ w_i }[/math] is the Calibrated Weight for the [math]\displaystyle{ i }[/math]th of [math]\displaystyle{ n }[/math] observations

[math]\displaystyle{ p \approx \Pr(t_{\sum^n_{i=1}w_i-2} \ge r\sqrt{\frac{\sum^n_{i=1}w_i-2}{1 - r^2}}) }[/math]

See also

Correlations - Comparing Two Numeric Variables