6.1 Pearson’s Correlation

The Pearson product-moment correlation measures the strength and direction of a linear relationship between two continuous variables, x and y. The Pearson correlation coefficient, r, ranges from -1 (perfect negative linear relationship) to +1 (perfect positive linear relationship). A value of 0 indicates no relationship between two variables.

\[r_{x,y} = \frac{cov(x,y)}{\sigma(x) \sigma(y)}\]

The statistic can be used as an estimate of the population correlation, \(\rho\), in a test of statistical significance from 0 (H0: \(\rho\) = 0).

\[\rho_{X,Y} = \frac{cov(X,Y)}{\sigma(X) \sigma(Y)}\]

A rule of thumb interpretation is that \(|r|\) under .1 is “no correlation”, .1 - .3 is “small”, .3 - .5 “medium/moderate”, and .5 - 1.0 “large/strong”.

The test statistic below follows a t-distribution with n − 2 degrees of freedom.

\[t = r_{x,y} \sqrt{\frac{n - 2}{1 - r^2_{x,y}}}\]