Expected value of sample standard deviation

Solution 1:

I understand the proof where you split $(X_i-\overline{X})^2=(X_i-\mu)^2+(\overline{X}-\mu)^2-2(X_i-\mu)(\overline{X}-\mu)$ then take the expected value.

You've ommitted the sigma sign.

$=\frac{1}{n-1}E\left[\sum_{i=1}^n \left[(X_i-\mu)^2-2(\overline X-\mu)(X_i-\mu)+(\overline X-\mu)^2 \right]\right] \quad$

Writing for each summand a sigma sign

$=\frac{1}{n-1}E\left[\sum_{i=1}^n (X_i-\mu)^2-2(\overline X-\mu)\sum_{i=1}^n(X_i-\mu)+\sum_{i=1}^n(\overline X-\mu)^2 \right] \quad$

$=\frac{1}{n-1}E\left[\sum_{i=1}^n (X_i-\mu)^2-2(\overline X-\mu)\color{blue}{\sum_{i=1}^n(X_i-\mu)}+n(\overline X-\mu)^2 \right] \quad$

Transforming the blue term

$\sum_{i=1}^n(X_i-\mu)=n\cdot \overline X-n\cdot \mu$

Thus $2(\overline X-\mu)\color{blue}{\sum_{i=1}^n(X_i-\mu)}=2(\overline X-\mu)\cdot (n\cdot \overline X-n\cdot \mu)=2n( \overline X- \mu)^2$

$=\frac{1}{n-1}E\left[\sum_{i=1}^n (X_i-\mu)^2-2n( \overline X- \mu)^2+n(\overline X-\mu)^2 \right] \quad$

$=\frac{1}{n-1}E\left[\sum_{i=1}^n (X_i-\mu)^2-n( \overline X- \mu)^2\right] \quad$

I think you can finish. If not, feel free to ask.

Update

$$\sum Var(X_i-\overline{X})+\sum [E(X_i-\overline{X})]^2$$

First of all $[E(X_i-\overline{X})]^2=[E(X_i)-E(\overline{X})]^2=[\mu-\mu]^2=0$. Then $$Var(X_i-\overline{X})=Var(X_i)-2\cdot Cov(X_i,\overline X)+Var(\overline{X})$$

$$=\sigma^2-2\cdot Cov(X_i,\overline X)+\frac{\sigma^2}{n} \qquad (1)$$ Next, the covariance has to be calculated.

$$ Cov(\overline X,X_i)= Cov(\frac{1}{n}\sum_{j=1}^{n}X_j,X_i) = \frac{1}{n}Cov(\sum_{j=1}^{n}X_j,X_i) $$

The assumption is that $X_j's$ are inependent for $i\neq j$. But for i=j we get the variance. Thus $\frac{1}{n}Cov(\sum_{j=1}^{n}X_j,X_i)=\frac{\sigma^2}{n}$. Put it in (1) and we obtain

$$=\sigma^2-2\cdot \frac{\sigma^2}{n}+\frac{\sigma^2}{n} =\sigma^2-\frac{\sigma^2}{n}$$

If the sigma signs are regarded, then term has to be multiplied by n:

$$n\sigma^2-\sigma^2=\sigma^2(n-1)=(n-1)E(S^2) \quad \checkmark$$

Your approach is more interesting than it looked like at the first glance.