where Zx, Zy are the standard scores for a single case of the variables x and y respectively, and n is the sample size of the data set used for this calculation. If the concept of a standardized score (z-score) is unfamiliar, this site explains it well. Conceptually, the use of a standardized score means that we do not have to know the scale of the measures used to interpret the correlation coefficient. You actually could calculate correlation by hand given a small data set or enough time and patience with a large data set, but hand calculations are no longer done aside from illustrative purposes. Thankfully we have statistical packages to do the heavy lifting for us. Still, it's important to understanding what's happening when we tell those packages to compile.
Let's see what this looks like in practice. Assume that we want to know if there is an association between performance on math tests (y) and the test taker's anxiety level (x). To do this we sample 50 students, measuring their anxiety level, then have them take a math test. Table 1 shows the a truncated list of the first six cases (i.e., study participants) showing each participants anxiety level matched to their math score.