- Sampling distribution
In

statistics , a**sampling distribution**is theprobability distribution , under repeated sampling of the population, of a givenstatistic (a numerical quantity calculated from thedata values in a sample).The formula for the sampling distribution depends on the distribution of the population, the statistic being considered, and the sample size used. A more precise formulation would speak of the distribution of the statistic as that for "all possible samples" of a given size, not just "under repeated sampling".

For example, consider a very large normal population (one that follows the so-called

bell curve ). Assume we repeatedly take samples of a given size from the population and calculate thesample mean ($ar\; x$, thearithmetic mean of the data values) for each sample. Different samples will lead to different sample means. The distribution of these means is the "sampling distribution of the sample mean" (for the given sample size). This distribution will be normal since the population was normal. (According to thecentral limit theorem , if the population is not normal but "sufficiently well behaved", the sampling distribution of the sample mean will still be approximately normal provided the sample size is sufficiently large.)Thus, the mean of the sampling distribution is equivalent to the expected value of any statistic. For the case where the statistic is the sample mean:

:$mu\_\{ar\; x\}\; =\; mu$

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::$sigma\_\{ar\; x\}\; =\; frac\{sigma\}\{sqrt\{n$

where $sigma$ is the standard deviation of the population distribution of that quantityand 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 designingstatistical studies where cost is a factor, this may have a factor inunderstanding 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****External links*** [

*http://www.indiana.edu/~jkkteach/ExcelSampler/ Generate sampling distributions in Excel*]

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