# What is a Statistic

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(X1, X2, …, Xn), which does not depend on any unknown parameters.

script t is the function that we apply to X1, X2, …, Xn to define the statistic, which is denoted by capital 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 t(X1, X2, …, Xn) depends on unknown parameters, T would not be useful in making such inferences.

## 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, X1, X2, …, and Xn can be random samples, but they need not be random samples.

## Example 2 of a statistic

Let X1, X2, …, and Xn represent a random sample from a population. The sample mean $$\bar{X}$$ is a statistic with the function of t(X1, X2, …, Xn) = (X1, X2, …, Xn)/n. Often, the statistic of the sample mean $$\bar{X}$$ is written as follows.

$$\bar{X} = \sum_{i=1}^{n} \frac{X_i}{n}$$

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 X1, X2, …, Xn represent a random sample from a population f(x) with $$E(X) =\mu$$ and $$Var(X) =\sigma^2$$. Then, we can get:

$$E(\bar{X}) = \mu$$

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

## 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 X1, X2, …, Xn denotes a random sample from $$N(\mu, \sigma^2)$$, then, we can get the sampling distribution of $$\bar{X}$$ as follows.

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