Covariance vs Correlation

Covariance provides a basic measure of how two variables move together, correlation offers a more interpretable and standardized way to understand their relationship.

Definition:

• Covariance measures the degree to which two variables change together. It can take any value, which makes it difficult to interpret the strength of the relationship.
• Correlation, on the other hand, is a standardized measure of the relationship between two variables, ranging from -1 to 1. This standardization allows for easier interpretation of the strength and direction of the relationship.

Formula:

• Correlation is derived from covariance. The formula for the correlation coefficient $r$between two variables $X$ and $Y$ is:

$$r=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$$

where ${\sigma }_X$and ${\sigma }_Y$ are the standard deviations of $X$and $Y$, respectively. This formula shows that correlation is essentially the covariance normalized by the product of the standard deviations of the two variables.

Interpretation:

• While covariance can indicate the direction of the relationship (positive or negative), it does not provide information about the strength of that relationship.
• Correlation, being bounded between -1 and 1, allows for a clearer understanding of how strongly the two variables are related.