How does a sample correlation coefficient r relate to a population correlation coefficient?

In statistics, the sample correlation coefficient $r$ serves as an estimate of the population correlation coefficient. While $r$ is calculated from a specific data set to quantify the strength and direction of a linear association between two variables, it represents only the observed relationship within that sample.1

The population correlation coefficient (often denoted by the Greek letter $\rho$) represents the true, underlying linear relationship between the variables for the entire population. Because $r$ is derived from a sample, it is subject to sampling variability; therefore, researchers use statistical methods, such as a t-test, to determine the significance of the observed $r$ and to infer whether the relationship exists within the broader population. It is essential to remember that neither the sample nor the population coefficient establishes a cause-and-effect relationship, as they only measure the degree of linear association.1