|
|
| Line 1: |
Line 1: |
| 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.
| | [https://www.gov.uk/search?q=Selected Selected] of the author is probably [http://browse.deviantart.com/?qh=§ion=&global=1&q=Gabrielle+Lattimer Gabrielle Lattimer]. Fish keeping is something her dad doesn't really like yet still she does. Idaho is where her home happens to be and she will never move. Software establishing is what she will do but she's always ideal her own business. She is actually running and maintaining any kind of blog here: http://prometeu.net<br><br>Feel free to visit my homepage ... [http://prometeu.net clash of clans hack apk download] |
| | |
| ==Introduction==
| |
| The '''sampling distribution''' of a statistic is the [[probability distribution|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 [[probability distribution|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 distribution|normal]] population with mean ''μ'' and variance ''σ''². Assume we repeatedly take samples of a given size from this population and calculate the [[arithmetic mean]] <math>\scriptstyle \bar x</math> 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 <math>\scriptstyle \mathcal{N}(\mu,\, \sigma^2/n)</math> (n is the size, which is number of items, in the sample) 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 population]]s. For other statistics and other populations the formulas are more complicated, and often they don't exist in [[Closed-form expression|closed-form]]. In such cases the sampling distributions may be approximated through [[Monte-Carlo simulation]]s<ref>{{cite book|last=Mooney|first=Christopher Z.|title=Monte Carlo simulation|year=1999|publisher=Sage|location=Thousand Oaks, Calif.|isbn=9780803959439|edition=|url = http://books.google.de/books?id=xQRgh4z_5acC}}</ref><sup>[p. 2]</sup>, [[Bootstrapping (statistics)|bootstrap]] methods, or [[asymptotic distribution]] theory.
| |
| | |
| ==Standard error== | |
| | |
| The [[standard deviation]] of the sampling distribution of a statistic is referred to as the
| |
| [[standard error (statistics)|standard error]] of that quantity. For the case where the statistic is the sample mean, and samples are uncorrelated, the standard error is:
| |
| | |
| :<math>\sigma_{\bar x} = \frac{\sigma}{\sqrt{n}}</math>
| |
| | |
| where <math>\sigma</math> 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==
| |
| {| class="wikitable"
| |
| |-
| |
| ! Population || Statistic || Sampling distribution
| |
| |-
| |
| | [[Normal distribution|Normal]]: <math>\mathcal{N}(\mu, \sigma^2)</math>
| |
| | Sample mean <math>\bar X</math> from samples of size ''n''
| |
| | <math>\bar X \sim \mathcal{N}\Big(\mu,\, \frac{\sigma^2}{n} \Big)</math>
| |
| |-
| |
| | [[Bernoulli distribution|Bernoulli]]: <math>\operatorname{Bernoulli}(p)</math>
| |
| | Sample proportion of "successful trials" <math>\bar X</math>
| |
| | [[Binomial distribution|<math>n \bar X \sim \operatorname{Binomial}(n, p)</math>]]
| |
| |-
| |
| | Two independent normal populations:<br>
| |
| <math>\mathcal{N}(\mu_1, \sigma_1^2)</math> and <math>\mathcal{N}(\mu_2, \sigma_2^2)</math>
| |
| | Difference between sample means, <math>\bar X_1 - \bar X_2</math>
| |
| | <math>\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)</math>
| |
| |-
| |
| | Any absolutely continuous distribution ''F'' with density ''ƒ''
| |
| | [[Median]] <math>X_{(k)}</math> from a sample of size ''n'' = 2''k'' − 1, where sample is ordered <math>X_{(1)}</math> to <math>X_{(n)}</math>
| |
| | <math>f_{X_{(k)}}(x) = \frac{(2k-1)!}{(k-1)!^2}f(x)\Big(F(x)(1-F(x))\Big)^{k-1}</math>
| |
| |-
| |
| | Any distribution with distribution function ''F''
| |
| | [[Maximum]] <math>M=\max\ X_k</math> from a random sample of size ''n''
| |
| | <math>F_M(x) = P(M\le x) = \prod P(X_k\le x)= \left(F(x)\right)^n</math>
| |
| |}
| |
| | |
| ==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 distribution]]s of any unknown quantities given the sample data, by the [[conditional distribution]]s 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.
| |
| | |
| {{unreferenced|date=June 2011}}
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| * Merberg, A. and S.J. Miller (2008). "The Sample Distribution of the Median". ''Course Notes for Math 162: Mathematical Statistics'', on the web at http://web.williams.edu/Mathematics/sjmiller/public_html/BrownClasses/162/Handouts/MedianThm04.pdf, pgs 1-9.
| |
| | |
| ==External links==
| |
| *[http://www.indiana.edu/~jkkteach/ExcelSampler/ Generate sampling distributions in Excel]
| |
| *[http://demonstrations.wolfram.com/StatisticsAssociatedWithNormalSamples/ ''Mathematica'' demonstration showing the sampling distribution of various statistics (e.g. Σ''x''²) for a normal population]
| |
| | |
| {{Statistics}}
| |
| | |
| [[Category:Statistical theory]]
| |
| | |
| [[de:Stichprobenverteilung]]
| |
Selected of the author is probably Gabrielle Lattimer. Fish keeping is something her dad doesn't really like yet still she does. Idaho is where her home happens to be and she will never move. Software establishing is what she will do but she's always ideal her own business. She is actually running and maintaining any kind of blog here: http://prometeu.net
Feel free to visit my homepage ... clash of clans hack apk download