Pearson's Product Moment Correlation

The correlation between two variables, $x$ and $y$, with weighted means of ${\bar {x}}$ and ${\bar {y}}$ respectively, is:

$r={\frac {\sum _{i=1}^{n}w_{i}(x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{{\sqrt {\sum _{i=1}^{n}w_{i}(x_{i}-{\bar {x}})^{2}}}{\sqrt {\sum _{i=1}^{n}w_{i}(y_{i}-{\bar {y}})^{2}}}}}$

where:

$w_{i}$ is the Calibrated Weight for the $i$th of $n$ observations

$p\approx \Pr(t_{\sum _{i=1}^{n}w_{i}-2}\geq r{\sqrt {\frac {\sum _{i=1}^{n}w_{i}-2}{1-r^{2}}}})$

See also

Correlations - Comparing Two Numeric Variables