sampling distribution: Definition from Answers.com

In statistics, a sampling distribution is the probability distribution of a given statistic based on a random sample of size n. It may be considered as the distribution of the statistic for all possible samples of a given size. The sampling distribution depends on the underlying distribution of the population, the statistic being considered, and the sample size used. The sampling distribution is frequently opposed to the asymptotic distribution, which corresponds to the limit case 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 will have its own average value, and the distribution of these averages will be called the “sampling distribution of the sample mean”. This distribution will be normal $\scriptstyle \mathcal{N}(\mu,\, \sigma^2/n)$ since the underlying population is normal.

This was an example of a simple statistic taken from one of the simplest statistical populations. For other statistics and other populations the formulas are frequently more complicated, and oftentimes they don’t even exist in closed-form. In such cases the sampling distributions may be approximated through Monte-Carlo simulations, bootstrap method, or asymptotic distribution theory.

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, the standard error is:

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

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

A very important implication of this formula is that you must quadruple the sample size (4×) to achieve half (1/2) the measurement error. When designing statistical studies where cost is a factor, this may have a factor in understanding cost-benefit tradeoffs.

Alternatively, consider the sample median from the same population. It has a different sampling distribution which is generally not normal (but may be close under certain circumstances).

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” $\bar X$	$\bar X \sim \operatorname{Binomial}(n, p)$
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 $X (k - 1)$ from a sample of size n = 2k − 1	$f_{X_{(k-1)}}(x) = \frac{(2k-1)!}{(k-1)!^2}f(x)\Big(F(x)(1-F(x))\Big)^{k-1}$

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Inference	Confidence interval (Frequentist inference) · Credible interval (Bayesian inference) · Significance · Meta-analysis

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