# Sampling distribution

﻿
Sampling distribution

In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data 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 the sample mean (, the arithmetic 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 the central 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:

:

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:

:

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&times;) 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

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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Sampling Distribution — A probability distribution of a statistic obtained through a large number of samples drawn from a specific population. The sampling distribution of a given population is the distribution of frequencies of a range of different outcomes that could… …   Investment dictionary

• sampling distribution — noun : the distribution of a statistic (as a sample mean) …   Useful english dictionary

• Distribution GNU/Linux — Distribution Linux Pour les articles homonymes, voir Distribution et Linux (homonymie). Une distribution Linux (ou distro), appelée aussi distribution GNU/Linux pour faire référence aux bibliothèques et logiciels du projet GNU, est un ensemble… …   Wikipédia en Français

• Distribution linux — Pour les articles homonymes, voir Distribution et Linux (homonymie). Une distribution Linux (ou distro), appelée aussi distribution GNU/Linux pour faire référence aux bibliothèques et logiciels du projet GNU, est un ensemble cohérent de logiciels …   Wikipédia en Français

• sampling — noun 1. (statistics) the selection of a suitable sample for study (Freq. 4) • Topics: ↑statistics • Hypernyms: ↑choice, ↑selection, ↑option, ↑pick …   Useful english dictionary

• Sampling bias — In statistics, sampling bias is when a sample is collected in such a way that some members of the intended population are less likely to be included than others. It results in a biased sample, a non random sample[1] of a population (or non human… …   Wikipedia

• sampling — A method for collecting information and drawing inferences about a larger population or universe, from the analysis of only part thereof, the sample. Censuses of the population are an expensive way of monitoring social and economic change, and… …   Dictionary of sociology

• Distribution Linux — Une distribution Linux, appelée aussi distribution GNU/Linux pour faire référence aux logiciels du projet GNU, est un ensemble cohérent de logiciels, la plupart étant logiciels libres, assemblés autour du noyau Linux. Il existe une très grande… …   Wikipédia en Français

• distribution curve — noun : a graph of the frequencies of different values of a variable in a statistical distribution * * * Statistics. the curve or line of a graph in which cumulative frequencies are plotted as ordinates and values of the variate as abscissas. * *… …   Useful english dictionary

• distribution — 1. The passage of the branches of arteries or nerves to the tissues and organs. 2. The area in which the branches of an artery or a nerve terminate, or the area supplied by such an artery or nerve. 3. The relative numbers of individuals in each… …   Medical dictionary