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Encyclopedia > 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). This article is about the field of statistics. ... In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ... Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ... A statistic (singular) is the result of applying a statistical algorithm to a set of data. ... For other uses, see Data (disambiguation). ... Sampling is that part of statistical practice concerned with the selection of individual observations intended to yield some knowledge about a population of concern, especially for the purposes of statistical inference. ...


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". In probability theory, every random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset (more precisely every measurable subset) of its state space in such a way that the probability axioms are satisfied. ...


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 (bar x, 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.) The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The graph of the probability density function of the normal distribution is sometimes called the bell curve or the bell-shaped curve; see normal distribution. ... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ...


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: In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

mu_{bar 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: The standard error of a method of measurement or estimate is the estimated standard deviation of the error in that method. ...

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). This article is about the statistical concept. ...


Examples

Population Sample statistic Sampling distribution
Infinite, X sim N(mu, sigma^2) Sample mean, bar X bar X sim N left (mu, frac{sigma^2}{n} right )
Finite (size N), X sim N(mu, sigma^2) Sample mean, bar X bar X sim N left (mu, frac{N - n}{N - 1} times frac{sigma^2}{n} right )
Infinite, X sim operatorname{Binomial}(p) Sample proportion, bar p bar p sim operatorname{Binomial}(p)
Infinite, X_1 sim N(mu_1, sigma_1^2), X_2 sim N(mu_2, sigma_2^2) Sample difference between means, bar X_1 - bar X_2 bar X_1 - bar X_2 sim N left (mu_1 - mu_2, frac{sigma_1^2}{n_1} + frac{sigma_2^2}{n_2}right )

See also

In statistics, when analyzing collected data, the samples observed differ in such things as means and standard deviations from the population from which the sample is taken. ... The normal distribution, also called Gaussian distribution (named after Carl Friedrich Gauss, a German mathematician, although Gauss was not the first to work with it), is an extremely important probability distribution in many fields. ...

External links

  • Generate sampling distributions in Excel
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  Results from FactBites:
 
Statistical Terms in Sampling (1621 words)
The distribution of an infinite number of samples of the same size as the sample in your study is known as the sampling distribution.
When we keep the sampling distribution in mind, we realize that while the statistic we got from our sample is probably near the center of the sampling distribution (because most of the samples would be there) we could have gotten one of the extreme samples just by the luck of the draw.
For starters, we assume that the mean of the sampling distribution is the mean of the sample, which is 3.75.
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