Intuition behind Variance formula [duplicate]

Variance is given as: $\operatorname{Var}(X) = \mathbb{E}[(X-\mathbb{E}(X))^2]$.

Is there an intuition behind this and can you find this formula starting from the second generating moment ?

Solution 1:

We want to measure the expected “deviation” of a random variable from it's expected value. What comes first in mind is the expected difference between the r.v. and it's expected value, that is $E\bigl(X-E(X)\bigr)$. But this number is always zero. --

Next, as drhab pointed out, is the average distance, i.e., the absolute value of the difference, between the r.v. and it's expected value, that is $E\bigl(|X-E(X)|\bigr)$. Now that appears to be a “natural” measure of the deviation, but there is a big drawback: it's not everywhere differentiable, so we can't apply Analysis to it.

Now we're stuck; what to do know? Well, let's follow an idea of a great mathematician, in our case Gauss. He introduced in his early years the expected value of the squared difference between the r.v. and it's expected value, that is $E\bigl((X-E(X))^2\bigr)$, aka variance of $X$. Its drawback: it isn't intuitive.

At the other hand, as Stefanos wrote: “The moment of the most use is when k=2.” That's right, but why is it the moment of most use? The main reason for this is doubtlessly Chebyshev's inequality: http://en.wikipedia.org/wiki/Chebyshev%27s_inequality And this inequality is very intuitive.

Solution 2:

If I want to 'measure' how much random variable $X$ is expected to differ from its expected value than intuitively I think of things like $\mathbb E(|X-\mathbb E(X)|)$ or $\mathbb E(X-\mathbb E(X))^2$. Another possibility in the same line is the root of the variance, named deviation. Have a look at the anwer of Stefanos when it concerns second generating moment.

Solution 3:

The idea one can say, is that the variance is a measure of how the values of a random variables are spread in the space. So if you have a random variable that takes many values identical to the mean value and few values far away from the mean then that means that distribution is more focused in a smaller space, hence the variance is smaller. In other words the variance of a random variable ($X$) measures the amount by which $X$ tends to deviate from the average value.

If you want to have a formula for the variance having the second generating moment: $$Var(X)=E[(X-E(X))^2]=E(X^2)-[E(X)]^2$$

In general we define $\sigma_k=E[(X-\mu_k)^k]$ as the $k_{th}$ central moment of $X$. The moment of the most use is when $k=2$