Sample Variance

Why do we divide by n-1 when calculating the sample standard deviation?

For a shorter proof, here are a few things we need to know: $x_{1}, x_{2},x_{3}, \cdots, x_{n}$ are independent observations from a population with mean $\mu$ and variance $\sigma^{2}$. $E[x_{i}]=\mu, V(x_{i})=\sigma^2.$ $E[x_{i}^2]=\sigma^{2}+\mu^{2}.$ $V(X)=E[X^{2}]-E[X]^{2}$ $E \left[ \bar{X}^{2} \right] =\mu^{2} + \frac{\sigma^{2}}{n}$

The sample variance is defined as follows: $S^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\bar{X})^2,$ where $\bar{X}$ is the sample mean.

On the right hand side of equation,

If we take the expectation value on the right hand side,