Can someone help me understand what the difference is—intuitively—between these concepts? Here's how I currently think of these things.
The covariance, represented by:
... is essentially a mean area, in standard units, representing the average "size" of the joint variation from the variables' respective means.
But the product of the standard deviations ($\sigma_x\cdot\sigma_y$) seems to encapsulate this exact same idea—the average area of the joint variation from the variables' respective means.
How do I distinguish these ideas conceptually?
Edit to add:
If the correlation coefficient (r) is the ratio $\frac{cov_{xy}}{\sigma_x\cdot\sigma_y}$, then what does this ratio embody, i.e., how many whats of what?