[mturmon]

I have to agree with you here.

One problem with introducing it the way the article does is that it's hard to see why the variance is never negative, and is zero exactly when the R.V. is constant.

This is a very important property, to say the least.

It would be better to say you measure the "energy" with

  E x^2

but that this is not immune to level shifts, so you need to subtract some constant off first. And it so happens that the optimal constant to subtract off is our friend E x.

Edited to add: The notion of introducing the ideas of a sample space and a random variable (in the technical sense), as is done in the article, and at the same time being shy about calculus, is rather contradictory. That is, the intersection of

  { people who want measure-theoretic probability concepts }

and

  { people who don't know calculus }

may be empty.

[hessenwolf]

It looks like an enthusiastic newbie with some clipart and an equation. Fair play for trying.

I would suggest adding the following.

1. What the poster above said.

2. The reason for the E[(x_{bar} - x_i)^2] choice. Why not E[|x_{bar} - x_i|]? Was it a mathematical convencience? Was it, perhaps, because Gauss had the integral of e_{t^2} from -Inf to plus Inf lying around in a letter from Laplace?

3. It is an equation with a square. Use a square somewhere.

4. The square root of the variance happens to be the horizontal distance between the mean and the point of inflection in the normal distribution. How cool is that?

[mturmon]

I like (3) in particular. You could introduce, in a very simple way, the idea that the "error" (X - E X) is perpendicular to the "estimate" (E X). That's the two legs of the right triangle; the hypotenuse is "X" itself.