Formula for the linear Correlation Coefficient:

\[r=\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left| n\left(\sum x^2\right)-\left(\sum x\right)^2\right|\cdot\left|n\left(\sum y^2\right)-\left(\sum y\right)^2 \right|}}\]

where \(n\) is the number of data pairs.

Formula for the \(t\) test for the Correlation Coefficient

\[t=r\cdot\sqrt{\frac{n-2}{1-r^2}}\]

with degrees of freedom equal to \(n-2\), where \(n\) is the number of ordered pairs \(x,y\).

Formula for the Regression Line \(y'=\alpha+bx\):

\[\alpha=\frac{\left(\sum y\right)\left(\sum x^2\right)-\left(\sum x\right)\left(\sum xy\right)}{n\left(\sum x^2\right)-\left(\sum x\right)^2}\]

\[b=\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{n\left(\sum x^2\right)-\left(x\right)^2}\]