= Sampling Distribution = A '''sampling distribution''' describes a random variable representing a sample statistic. <> ---- == Description == For a given sample, a sample statistic can be calculated. This statistic is a fixed value but in theory it would vary if other samples were given. The sample statistic is a random variable, and the fixed value that was realized is actually the random variable conditioned on the sample. The distribution of possible values is the sampling distribution. If the population distribution is known then further statements can be made on the sampling distribution. For example, if ''X ~ N(μ,,X,,``, σ^2^,,X,,)'' then ''X̅ ~ N(μ,,X̅,,``, σ^2^,,X̅,,)'' where ''μ,,X̅,, = μ,,X,,'' and ''σ^2^,,X̅,, = σ^2^,,X,,/n''. Furthermore, a confidence interval can be constructed around the known population mean; 95% of samples will lead to a sample statistic that is within approximately 1.96 standard errors of that mean. As another example, if ''X ~ Bernoulli(p)'' then ''nX̅ ~ Binomial(n,p)''. However, by the central limit theorem, ''X̅ ~ N(μ,,X̅,,``, σ^2^,,X̅,,)'' is approximately true even for non-[[Analysis/NormalDistribution|normal]] distributions. A confidence interval constructed around a point estimate is said to contain the true population parameter with that level of confidence. ---- == Standard Error == The standard deviation of a sampling distribution is called the standard error. If the population distribution is known to follow ''X ~ N(μ,,X,,``, σ^2^,,X,,)'', then the standard error of the sample statistic ''X̅'' is given by ''σ^2^,,X̅,, = σ^2^,,X,,/n''. ---- CategoryRicottone