Mathematical statistics uses probability theory and other branches of mathematics to study statistics from a purely mathematical standpoint.

Thus, for example, a random sample is denoted by a sequence of random variables,

each following a certain probability distribution (usually they are considered independent and identically distributed). The sample mean is expressed as a function of those variables,

and is therefore itself a random variable with a distribution.

By proceeding in this fashion, mathematical statistics attempts to provide a theoretical basis for many practices in applied statistics.