# Pearson's Product Moment Correlation

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

$r = \frac{\sum ^n _{i=1}w_i(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}w_i(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}w_i(y_i - \bar{y})^2}}$

where:

$w_i$ is the Calibrated Weight for the $i$th of $n$ observations
$p \approx \Pr(t_{\sum^n_{i=1}w_i-2} \ge r\sqrt{\frac{\sum^n_{i=1}w_i-2}{1 - r^2}})$