[jmichaud]

In the chapter on variability, on page 105, in "calculating average distances", we see that the average distance comes to 0. Right below that, in "there are no dumb questions" section we have the question, "Can't we just take the positive distances and average those?" The answer seems to be:
"We could, but that's not what statisticians do, there's another way."

I appreciate that squaring the difference eliminates the negative values, but so does taking the absolute value. There must be a better mathematical or logical reason why the RMS is better (other then the handy elimination of negative signs or "that's just the way it's done" by statisticians")

Is there an inherit reason? Or is the standard deviation just an arbitrary but "well known" description of variance, perhaps the way the Dow Jones is an arbitrary but well known economic status indicator?

[RavenHawk]

I would also like an answer to this question. Having tried to find out myself from various web searches, all I'm getting is a lot of gobbly gook about it being more intuitive - not on my planet - or it being more preferred. One person said they give the same answer but they mean different things but without any futher explanation.

[jmichaud]

http://www.leeds.ac.uk/educol/documents/00003759.htm

"Revisiting a 90-year-old debate: the advantages of the mean deviation"
- Stephen Gorard, Department of Educational Studies, University of York
- presented at the British Educational Research Association Annual Conference, University of Manchester, 16-18 September 2004

This paper makes the claim that Std Deviation was chosen as the basis of classical statistics based on stability across samples of the population, and algebraic convenience. It also includes some historical background.

It seems like a valid argument to me, but it would be nice if a statistician could validate the argument and historical record (even if they don't concur with the thesis).

[RavenHawk]

Thanks for that link jmichaud, it was very interesting. Too bad this forum is quiet I would have loved to have heard from other people on this issue. Where's the author?

[jr.larsen]

Without a computer, finding the distance from the mean for a large set of numbers is a bit of a pain, particularly as the mean is often not an integer.

The formula for variance, where the distances are squared can be rearranged to give an easier method of computation. Variance = mean of the squares - mean squared.

The variance is square rooted to bring the measure of dispersion back to the same kind of scale as the initial values. That's the standard deviation. Although it is not directly the mean of the deviations, it is based on them and the alternative method of computation makes it a good choice as a measure of dispersion.

Now, I admit it, I made all that up, but it sounds like a good reason.