This tutorial provides definitions and examples of a statistic. Further, it explains the relationship between a statistic and sampling distribution.

## The Definition of a Statistic

**A statistic** is a function of observable random variables, ** T=t(X_{1}, X_{2}, …, X_{n})**, which does not depend on any unknown parameters.

script ** t** is the function that we apply to

**X**to define the statistic, which is denoted by capital

_{1}, X_{2}, …, X_{n}**.**

*T*The intent of the use of a statistic ** T** is to make inferences about the distribution of the set of random variables. Thus, if the variables are not observable or if the function

**depends on unknown parameters, T would not be useful in making such inferences.**

*t*(X_{1}, X_{2}, …, X_{n})## Example 1 of a statistic

Note that, the set of observable random variables need not be a random sample. For instance, 13 planes were in service, and the first 10 air conditioner failer times were as follows.

23,50,50,55,74,90,97,102,130,194

For this case, \( T = \sum_{i=1}^{10} y_i+2y_{10} = 1447\) is a statistic. For sure, **X _{1}, X_{2}, …, and X_{n}** can be random samples, but they need not be random samples.

## Example 2 of a statistic

Let **X _{1}, X_{2}, …, and X_{n}** represent a random sample from a population. The sample mean \( \bar{X} \) is a statistic with the function of

**. Often, the statistic of the sample mean \( \bar{X} \) is written as follows.**

*t*(X_{1}, X_{2}, …, X_{n}) = (X_{1}, X_{2}, …, X_{n})/n\( \bar{X} = \sum_{i=1}^{n} \frac{X_i}{n} \)

Note that, in the function above, \( \bar{X} \) uses the capital case of \( X \). In contrast, when a random sample is observed, the value of \( \bar{X} \) computed from the data is denoted by lowercase \( \bar{x} \).

The sample mean \( \bar{x} \) is useful because it can estimate the population mean and population variance. In particular, if **X _{1}, X_{2}, …, X_{n}** represent a random sample from a population

*with \( E(X) =\mu \) and \(Var(X) =\sigma^2 \). Then, we can get:*

**f(x)**\( E(\bar{X}) = \mu\)

\( Var(\bar{X}) = \frac{\sigma^2}{n} \)

## A statistic and sampling distribution

A statistic is also a random variable. The distribution of a statistic is referred to as a **derived distribution** or **sampling distribution**, in contrast to the **population distribution**.

Many important statistics can be expressed as a linear combination of independent normal random variables. For instance, if **X _{1}, X_{2}, …, X_{n}** denotes a random sample from \( N(\mu, \sigma^2)\), then, we can get the sampling distribution of \( \bar{X} \) as follows.

\( \bar{X} \sim N(\mu, \sigma^2/n)\)