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In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given statistic based on a random sample. Sampling distributions are important in statistics because they provide a major simplification on the route to statistical inference. More specifically, they allow analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution of all the individual sample values.

## Introduction

The sampling distribution of a statistic is the distribution of that statistic, considered as a random variable, when derived from a random sample of size n. It may be considered as the distribution of the statistic for all possible samples from the same population of a given size. The sampling distribution depends on the underlying distribution of the population, the statistic being considered, the sampling procedure employed and the sample size used. There is often considerable interest in whether the sampling distribution can be approximated by an asymptotic distribution, which corresponds to the limiting case as n → ∞.

For example, consider a normal population with mean μ and variance σ². Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean $\scriptstyle \bar x$ for each sample — this statistic is called the sample mean. Each sample has its own average value, and the distribution of these averages is called the "sampling distribution of the sample mean". This distribution is normal $\scriptstyle \mathcal{N}(\mu,\, \sigma^2/n)$ since the underlying population is normal, although sampling distributions may also often be close to normal even when the population distribution is not (see central limit theorem). An alternative to the sample mean is the sample median. When calculated from the same population, it has a different sampling distribution to that of the mean and is generally not normal (but it may be close for large sample sizes).

The mean of a sample from a population having a normal distribution is an example of a simple statistic taken from one of the simplest statistical populations. For other statistics and other populations the formulas are more complicated, and often they don't exist in closed-form. In such cases the sampling distributions may be approximated through Monte-Carlo simulations, bootstrap methods, or asymptotic distribution theory.

## Standard error

The standard deviation of the sampling distribution of the statistic is referred to as the standard error of that quantity. For the case where the statistic is the sample mean, and samples are uncorrelated, the standard error is:

$\sigma_{\bar x} = \frac{\sigma}{\sqrt{n}}$

where $\sigma$ is the standard deviation of the population distribution of that quantity and n is the size (number of items) in the sample.

An important implication of this formula is that the sample size must be quadrupled (multiplied by 4) to achieve half (1/2) the measurement error. When designing statistical studies where cost is a factor, this may have a role in understanding cost-benefit tradeoffs.

## Examples

Population Statistic Sampling distribution
Normal: $\mathcal{N}(\mu, \sigma^2)$ Sample mean $\bar X$ from samples of size n $\bar X \sim \mathcal{N}\Big(\mu,\, \frac{\sigma^2}{n} \Big)$
Bernoulli: $\operatorname{Bernoulli}(p)$ Sample proportion of "successful trials" $\bar X$ $n \bar X \sim \operatorname{Binomial}(n, p)$
Two independent normal populations:

$\mathcal{N}(\mu_1, \sigma_1^2)$  and  $\mathcal{N}(\mu_2, \sigma_2^2)$

Difference between sample means, $\bar X_1 - \bar X_2$ $\bar X_1 - \bar X_2 \sim \mathcal{N}\! \left(\mu_1 - \mu_2,\, \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} \right)$
Any absolutely continuous distribution F with density ƒ Median $X_{(k)}$ from a sample of size n = 2k − 1, where sample is ordered $X_{(1)}$ to $X_{(n)}$ $f_{X_{(k)}}(x) = \frac{(2k-1)!}{(k-1)!^2}f(x)\Big(F(x)(1-F(x))\Big)^{k-1}$
Any distribution with distribution function F Maximum $M=\max\ X_k$ from a random sample of size n $F_M(x) = P(M\le x) = \prod P(X_k\le x)= \left(F(x)\right)^n$

## Statistical inference

In the theory of statistical inference, the idea of a sufficient statistic provides the basis of choosing a statistic (as a function of the sample data points) in such a way that no information is lost by replacing the full probabilistic description of the sample with the sampling distribution of the selected statistic.

In frequentist inference, for example in the development of a statistical hypothesis test or a confidence interval, the availability of the sampling distribution of a statistic (or an approximation to this in the form of an asymptotic distribution) can allow the ready formulation of such procedures, whereas the development of procedures starting from the joint distribution of the sample would be less straightforward.

In Bayesian inference, when the sampling distribution of a statistic is available, one can consider replacing the final outcome of such procedures, specifically the conditional distributions of any unknown quantities given the sample data, by the conditional distributions of any unknown quantities given selected sample statistics. Such a procedure would involve the sampling distribution of the statistics. The results would be identical provided the statistics chosen are jointly sufficient statistics.

## References

 140835 videos foundNext >
 INTRO TO SAMPLING DISTRIBUTIONSWhat is a sampLING distribution? It is a distribution of a bunch of p-hats or x-bars... watch and see. Sampling distributionA summary of the logic behind the sampling distribution. Sampling Distribution Example ProblemLearn more: http://www.khanacademy.org/video?v=0ZstEh_8bYc Figuring out the probability of running out of water on a camping trip. Sampling Distributions: Introduction to the ConceptI discuss the concept of sampling distributions (an important concept that underlies much of statistical inference), and illustrate the sampling distribution... The Sampling Distribution of the Sample MeanI discuss the sampling distribution of the sample mean, and work through an example of a probability calculation. (I only briefly mention the central limit t... Introductory Statistics - Chapter 6: Sampling distributionsA video summary of chapter 6 in Perdisco's Introductory Statistics 360Textbook. To find out more, visit www.perdisco.com/introstats. Sampling Distribution of the Sample MeanLearn more: http://www.khanacademy.org/video?v=FXZ2O1Lv-KE The central limit theorem and the sampling distribution of the sample mean. Example of Sampling Distribution of the MeanStatistics: A sample of 169 fish is randomly selected from a large fish population. Fish length X is distributed with a mean of 50 cm and a standard deviatio... How likely to get lots of cheese lovers?? Sampling Distribution for p-hatOften you are asked the likelihood of getting some proportion of your sample to have some characteristic when you know the proportion of the population who h... Sample problem using a sampling distribution 7.1This is a sample problem using the normal sampling distribution.
 140835 videos foundNext >

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