<P> 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 en 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 . </P> <P> The sampling distribution of a statistic is the distribution of that statistic, considered as a random variable, when derived from a random sample of size n (\ displaystyle n). It may be considered as the distribution of the statistic for all possible samples from the same population of a given sample size . The sampling distribution depends on the underlying 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 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 . </P> <P> For example, consider a normal population with mean μ (\ displaystyle \ mu) and variance σ 2 (\ displaystyle \ sigma ^ (2)). Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean x _̄ (\ displaystyle \ scriptstyle (\ bar (x))) for each sample--this statistic is called the sample mean . The distribution of these means, or averages, is called the "sampling distribution of the sample mean". This distribution is normal N (μ, σ 2 / n) (\ displaystyle \ scriptstyle (\ mathcal (N)) (\ mu, \, \ sigma ^ (2) / n)) (n is the sample size) 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). </P>

The distribution of means uses many samples of
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