Equivalently, about variance?
I realize it measures the spread of a distribution, but many other metrics could do the same (e.g., the average absolute deviation). What is its deeper significance? Does it have
- a particular geometric interpretation (in the sense, e.g., that the mean is the balancing point of a distribution)?
- any other intuitive interpretation that differentiates it from other possible measures of spread?
What's so special about it that makes it act as a normalizing factor in all sorts of situations (for example, convert covariance to correlation)?