Instatistics, asampling distribution orfinite-sample distribution is theprobability distribution of a givenrandom-sample-basedstatistic. For an arbitrarily large number of samples where each sample, involving multiple observations (data points), is separately used to compute one value of a statistic (for example, thesample mean or samplevariance) per sample, the sampling distribution is the probability distribution of the values that the statistic takes on. In many contexts, only one sample (i.e., a set of observations) is observed, but the sampling distribution can be found theoretically.

Sampling distributions are important in statistics because they provide a major simplification en route tostatistical inference. More specifically, they allow analytical considerations to be based on the probability distribution of a statistic, rather than on thejoint probability distribution of all the individual sample values.

Thesampling distribution of a statistic is thedistribution of that statistic, considered as arandom variable, when derived from arandom sample of size${\displaystyle n}$. It may be considered as the distribution of the statistic forall possible samples from the same population of a given sample size. The sampling distribution depends on the underlyingdistribution 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 anasymptotic distribution, which corresponds to the limiting case either as the number of random samples of finite size, taken from an infinite population and used to produce the distribution, tends to infinity, or when just one equally-infinite-size "sample" is taken of that same population.

For example, consider anormal population with mean${\displaystyle \mu }$ and variance${\displaystyle {\sigma }^2}$. Assume we repeatedly take samples of a given size from this population and calculate thearithmetic mean${\displaystyle \overline{x}}$ for each sample – this statistic is called thesample mean. The distribution of these means, or averages, is called the "sampling distribution of the sample mean". This distribution is normal${\displaystyle \mathcal{N}(\mu ,{\sigma }^2/n)}$ (n is the sample size) since the underlying population is normal, although sampling distributions may be close to normal even when the population distribution is not (seecentral limit theorem). An alternative to the sample mean is the samplemedian. 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 simpleststatistical populations. For other statistics and other populations the formulas are more complicated, and often they do not exist inclosed-form. In such cases the sampling distributions may be approximated throughMonte-Carlo simulations,^[1]bootstrap methods, orasymptotic distribution theory.

Thestandard deviation of the sampling distribution of astatistic is referred to as thestandard error of the statistic. For the case where the statistic is the sample mean, and samples are uncorrelated, the standard error is:$${\displaystyle {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}}$$where${\displaystyle \sigma }$ is the standard deviation of the population distribution of that quantity and${\displaystyle n}$ is the sample 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.

For the case where the statistic is the sample total, and samples are uncorrelated, the standard error is:$${\displaystyle {\sigma }_{\mathrm{\Sigma }x}=\sigma \sqrt{n}}$$where, again,${\displaystyle \sigma }$ is the standard deviation of the population distribution of that quantity and${\displaystyle n}$ is the sample size (number of items in the sample).

Population	Statistic	Sampling distribution
Normal:${\displaystyle \mathcal{N}(\mu ,{\sigma }^2)}$	Sample mean${\displaystyle \overline{X}}$ from samples of sizen	${\displaystyle \overline{X}\sim \mathcal{N}(\mu ,\frac{{\sigma }^2}{n})}$. If the standard deviation${\displaystyle \sigma }$ is not known, one can consider${\displaystyle T=\left(\overline{X}-\mu \right)\frac{\sqrt{n}}{S}}$, which follows theStudent's t-distribution with${\displaystyle \nu =n-1}$ degrees of freedom. Here${\displaystyle S^2}$ is the sample variance, and${\displaystyle T}$ is apivotal quantity, whose distribution does not depend on${\displaystyle \sigma }$.
Bernoulli:${\displaystyle \mathrm{Bernoulli}(p)}$	Sample proportion of "successful trials"${\displaystyle \overline{X}}$	${\displaystyle n\overline{X}\sim \mathrm{Binomial}(n,p)}$
Two independent normal populations: ${\displaystyle \mathcal{N}({\mu }_1,{\sigma }_1^2)}$  and ${\displaystyle \mathcal{N}({\mu }_2,{\sigma }_2^2)}$	Difference between sample means,${\displaystyle {\overline{X}}_1-{\overline{X}}_2}$	${\displaystyle {\overline{X}}_1-{\overline{X}}_2\sim \mathcal{N}\left({\mu }_1-{\mu }_2,\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}\right)}$
Any absolutely continuous distributionF with densityf	Median${\displaystyle X_{(k)}}$ from a sample of sizen = 2k − 1, where sample is ordered${\displaystyle X_{(1)}}$ to${\displaystyle X_{(n)}}$	${\displaystyle f_{X_{(k)}}(x)=\frac{(2k-1)!}{(k-1){!}^2}f(x)(F(x)(1-F(x)){)}^{k-1}}$
Any distribution with distribution functionF	Maximum${\displaystyle M=max{X}_k}$ from a random sample of sizen	${\displaystyle F_M(x)=P(M\le x)=\prod P(X_k\le x)={\left(F(x)\right)}^n}$

• Merberg, A. and S.J. Miller (2008)."The Sample Distribution of the Median".Course Notes for Math 162: Mathematical Statistics, pgs 1–9.

• Mathematica demonstration showing the sampling distribution of various statistics (e.g. Σx²) for a normal population

• Statistical inference
• Sampling (statistics)

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