Differing variance equations

Question

I see both of these equations on Wikipedia and Cross Validated being referred to as "variance":

$\mathrm{Var}(x) = \mathrm{E}\left[ \left( x - \mathrm{E}[x] \right)^2 \right]$

$\frac{1}{n} \sum_i \left(x_i - \bar{x} \right)^2 \quad \text{ where } \quad \bar{x} = \frac{1}{n} \sum x_i$

Does the $\mathrm{E}$ simply imply that the equation in brackets needs to be averaged over all iterations, hence why we place the $\frac{1}{n}$ at the front of the equation?

I just need to be walked through how the top equation allows us to arrive at the bottom equation and both of these are referred to as variance.

Also, is there a term you could use to refer specifically to $\sum_{i=1}^n(x_i-\bar{x})^2$ when not multiplied by $\frac{1}{n}$? Just sum of squares?

Answer 1

The top equation is theoretical, you cannot use it directly with the data you have. You should have the PDF/CDF to calculate theoretical expected values (i.e. population values). And you should think of population as infinite samples, not 100 as you say.

The second equation is an estimation procedure, so it is an estimation of the variance, i.e. $\hat{var(X)}$. And, in practice, it is usually not equal the theoretical value calculated from the first equation. Moreover, another estimation of variance uses $\frac{1}{n-1}$ instead of $\frac{1}{n}$ because it is unbiased. Please check calculation of $E[S^2]$ in the link.

Finally, yes, the $E$ sign implies taking mean of the inner expression; but they're not equal. For example, approximating $E[g(X)]$ via $\frac{1}{n}\sum{g(x_i)}$ from data is not a rare situation in statistics.