# Thread: Expected Value of nS^2

1. ## Expected Value of nS^2

I am curious if any expert knows what the expected value is of SUM (Xi - X-bar)^2 from 1 to n. Where Xi are a random sample from a distribution?

We have been showing that the expected value of S^2 is sigma squared. But I am wondering how to do this without the n or (n-1) divisor.

We were able to do the formers by re-writing S^2 as SUM(Xi^2)^2 /n - x-bar^2

but this does not work for SUM (Xi - X-bar)^2

Thanks!

2. Originally Posted by B_Miner
I am curious if any expert knows what the expected value is of SUM (Xi - X-bar)^2 from 1 to n. Where Xi are a random sample from a distribution?

We have been showing that the expected value of S^2 is sigma squared. But I am wondering how to do this without the n or (n-1) divisor.

We were able to do the formers by re-writing S^2 as SUM(Xi^2)^2 /n - x-bar^2

but this does not work for SUM (Xi - X-bar)^2

Thanks!
You know that $S^2 = \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n-1}$ is an unbaised estimator of $\sigma^2$. Therefore:

$E\left( \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n-1} \right) = \sigma^2$

$\Rightarrow E\left(\sum_{i=1}^n (x_i - \overline{x})^2 \right) = (n - 1) \sigma^2$.