Pearson Correlation Coefficient of Population and Sample

We start with the Pearson correlation coefficient in the population and then show how to make inferences about it using its sample version. The correlation coefficient between trade and growth in the population, denoted by $\rho_{trade, growth}$, is defined as follows:

$$\rho_{trade, growth} = \frac{\sigma_{trade,growth}}{\sigma_{trade}\sigma_{growth}}$$

Above, $\sigma_{trade, growth}$ represents the covariance, and $\sigma_{trade}$ and $\sigma_{growth}$ represent the respective standard deviations. The population correlation coefficient is the population covariance between two variables divided by the product of their population standard deviations.

Values of Pearson’s correlation coefficient

As a result, the Pearson correlation coefficient is unit free, bounded between −1 (perfect negative linear correlation) and +1 (perfect positive linear correlation), with zero meaning no correlation, and values approaching positive or negative one indicating stronger positive or negative linear correlations. The denominator in the formula helps to standardize the covariance so the strength of correlation between any two pairs of variables can be compared.

Note: It’s also worth noting that the correlation coefficient applies to continuous random variables, and it captures only the linear, but not any non-linear, relationship between two variables.