# Thread: Variance Formula: why not absolute values?

1. ## Variance Formula: why not absolute values?

This might be a basic question, or it might be very complex. I asked my professor but he sort of shrugged and said it complicates probability theory later on down the road.

Question: Why does the variance formula square the difference between each data point and the mean, rather than simply use the absolute value of the difference?

2. ## Re: Variance Formula: why not absolute values?

Originally Posted by jsndacruz
This might be a basic question, or it might be very complex. I asked my professor but he sort of shrugged and said it complicates probability theory later on down the road.

Question: Why does the variance formula square the difference between each data point and the mean, rather than simply use the absolute value of the difference?
The absolute value is perfectly usable, it is just not called "variance". The standard deviation (square root of the variance) is based on the "standard" definition of distance in n dimensions, $\displaystyle \sqrt{\sum_{i=1}^n (x_i- x_{0i})^2}$. But other definitions of "distance" do use $\displaystyle \sum_{i=1}^n |x_i- x_{0i}|$ and even $\displaystyle max{|x_i- x_{0i}|$.

3. ## Re: Variance Formula: why not absolute values?

Originally Posted by jsndacruz
This might be a basic question, or it might be very complex. I asked my professor but he sort of shrugged and said it complicates probability theory later on down the road.

Question: Why does the variance formula square the difference between each data point and the mean, rather than simply use the absolute value of the difference?
What Hall's said, and being a smooth function of the deviations means that calculus can more easily be used on it which is a very great advantage down the line.

CB