Standard Deviation

[cpkz]

If I did not know any statistics and I heard the term "standard deviation," I would think "average distance from the mean," rather than the way it's actually defined. Why is it defined as it is, and not intuitively? Or is my intuition off?

Thanks!

[alekhine4149]

No, you are asking a very good question, and an important one underlying statistics. The measure you describe has a name - the average absolute deviation = E(|x-mu|).

Standard deviation = sqrt(E((x-mu)^2)) isn't so terribly different. Squaring the differences serves the purpose of making all differences non-negative, and taking the square root puts it back in the unit's terms. The real difference is in the effect: standard deviation is more sensitive to outliers. For instance, given the data set: 1,1,1,1,6:

AAD = 1.6000
STD = 2.2361

The standard deviation from the mean is higher; it's more responsive to the 6 in the dataset. This is possibly why it has become the more favored tool in statistical analysis; I do not know the history behind it. But by no means is using the average absolute deviation incorrect. Sometimes an analyst prefers to use measurements that are less sensitive to outliers. This would be an example of a way you could.

Not sure if that's what you meant, but I hope this helps some...

[cpkz]

Yes, that is precisely what I was getting at. Your explanation is very helpful.

So, just so I am thinking about this correctly.. the use of standard deviation as opposed to average absolute deviation is entirely arbitrary, right? Like, we could redo everything in terms of AAD and we would not be going against something fundamental to the nature of mathematics or reality?

[alekhine4149]

We would not be going against something fundamental to the nature of mathematics or reality, that's definitely right. But entirely arbitary is much too strong a phrase. I'd say standard deviation is the favored tool or the default tool. I'd say that there are cases when one is either more or less concerned about outlier statistics, and that I can say I've deliberately selected AAD for a work application before. And I'd say it's good that you're giving it thought. Hopefully an actual professor can come on here and give you a history of the use of standard deviation. lol