What's so special about standard deviation?

Question

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)?

Answer 1

Patrick provided an excellent example of how standard deviations work, but since you asked for a broader interpretation of it, I opted to chime in with additional perspective.

Central tendency (which can be mean, median, mode, and sometimes others you likely haven't heard of) try to tell you where the center of the data rests. Each has strengths and weaknesses, such as vulnerability to outliers or a narrow view of the data. In your case, we use the mean, because it is the only one that lets you calculate the standard deviation.

What is the standard deviation? Well, the mean is where you expect me to be. You could say it is in the middle of all the places I might have been. What if you don't find me there? How far could I have moved? Well, if mean is the center, the standard deviation is the size of my steps. Every time I walk away from the center, my stride is one standard deviation. With each step, I move that many standard deviations further away.

As a corollary to this, you can now understand Z-scores easily: they are the number of steps I took.

Answer 2

It allows you to break the data into classes and categories, which in turn is used to determine if

a particular score is good or bad.

For example, the consider an exam which has a mean/average of 250.

Your score is 300.

Is your score good or bad?

Without the standard deviation, it is hard to tell.

Now, let's suppose the standard deviation is 50. THen your score is 1 standard deviation above

the mean average which makes it a grade of B.

On the other hand, if the standard deviation is 100 then your score is just average, exactly 1/2 standard deviation above the mean, which is a C.

By sliding the window for a grade of C between 1 to 1.5 standard deviations above the mean,

the teacher can curve the grades, so that the scores are normally distributed, which in turn benefits you.