Pearson Correlation Coefficient (PCC) is a statistic method of measuring linear correlation between two sets of data.

Definition

Say \(r_{xy}\) is representing a PCC between two variables, \(x\) and \(y\). Then, \(r_{xy}\) is defined as:

\[r_{xy} = \frac{\sum_{i=1}^{n}{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum_{i=1}^{n}{(x_i - \bar{x})^2}}\sqrt{\sum_{i=1}^{n}{(y_i - \bar{y})^2}}}\]

where $n$ is sample size, \(x_i, y_i\) are the samples with the index number of \(i\), \(\bar{x}\) is the sample mean and analogously for \(\bar{y}\). The \(r_{xy}\) is in the range of \([-1, 1]\). As the value gets closer to 1, it represents a strong linear correlation. And conversely when it gets near to -1, it means the two are inversely proportional. Finally, if it is 0, then there is no correlation.

The above equation can also be expressed as follows:

\[r_{xy} = \frac{cov(x, y)}{\sigma_x \sigma_y}\]

where \(cov\) is covariance, and \(\sigma\) is standard deviation.