What is the formula for Pearson correlation?

The Pearson correlation is denoted with r and calculated as follows:

$$r = \frac{covariance\:of\:x\:and\:y}{variability\:of\:x\:and\:y\:seperately}$$

or:

$$r = \frac{\sum{(x-\bar{x})(y-\bar{y})}}{\sqrt{\sum{(x-\bar{x})^2}\sum{(y-\bar{y})^2}}}$$

which is the same as:

$$r = \frac{N \sum{xy}-(\sum{x})(\sum{y})}{\sqrt{[N\sum{x^2}-(\sum{x})^2] [N\sum{y^2}-(\sum{y})^2]}}$$

• N: number of pairs of scores
• x: x scores
• y: y scores
• x̄: mean of x scores
• ȳ: mean of y scores