How does correlation standardize covariance to produce a value between −1 and +1?

Correlation standardizes covariance by removing the influence of the variables' units of measurement, which otherwise cause the magnitude of covariance to be uninformative and unbounded.1

To produce a value between −1 and +1, the correlation coefficient (r) is calculated by dividing the covariance of two variables, X and Y, by the product of their individual standard deviations. Mathematically, this is expressed as:12

$$r=\frac{\text{Cov}(X,Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}}$$

In this formula, the denominator represents the product of the standard deviations of X and Y. By dividing by these values, the resulting coefficient becomes a dimensionless quantity. This normalization ensures that the measure reflects only the strength and direction of the linear relationship, where +1 indicates a perfect positive correlation, −1 indicates a perfect negative correlation, and 0 indicates no linear association.13