In statistics, we work with samples. For example, the sample mean is easily calculated. However, it is the population mean that is more valuable.

Suppose we have one sample $S_i$, which is used to calculate the mean of the sample $\mu_i$. We have two key problems to solve at this moment.

  1. Can we use this sample mean $\mu_i$ to represent the population mean $\mu_p$?
  2. How good is our estimations?

To answer these questions, we need to work out the properties of the samples themselves and work out a theory to instruct us to infer population statistics from sample statistics.