To compute Pearson's $r$ manually, start by transforming the raw scores of two quantitative variables into $z$ scores. For example, consider an $X$ variable with a mean of $4.00$ and a standard deviation of $1.90$, and a $Y$ variable with a mean of $40.00$ and a standard deviation of $11.78$. If an individual's $X$ score is $4$ (a $z$ score of $0.00$) and their $Y$ score is $30$ (a $z$ score of $-0.85$), their cross-product is $0.00$. Another individual might have an $X$ score of $7$ ($z = 1.58$) and a $Y$ score of $54$ ($z = 1.19$), resulting in a cross-product of $1.88$. By taking the mean of these cross-products for the entire sample, we arrive at the value for Pearson's $r$. In this example, the mean of the cross-products is $+0.53$, indicating a positive relationship.