When we take a sample and calculate its mean and standard deviation, this is treated as a random variable for the population mean/standard deviation.

We treat *each* observation as a random variable.

The random variables \(X_{1},\dots,X_{n}\) form a **random sample**
of size \(n\) if:

- The \(X_{i}\) are independent.
- Every \(X_{i}\) has the same probability distribution.

Note that the second is not true when sampling without replacement. We treat it as a good approximation, though, if \(n/N\le0.05\)

The above two conditions are the same as saying the \(X_{i}\) are
**independent and identically distributed** (iid).

## Normal Distribution

For a normal distribution, the expected value of the sample mean is:

\begin{equation*}
E(S)=\sqrt{\frac{2}{n-1}}\Gamma\left(\frac{n}{2}\right)\frac{\sigma}{\Gamma\left(\frac{1}{2}(n-1)\right)}
\end{equation*}