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| 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''. However, by the central limit theorem, ''X̅ ~ N(μ,,X̅,,``, σ^2^,,X̅,,)'' is approximately true even for non-[[Analysis/NormalDistribution|normal]] distributions. | 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''. 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. |
Sampling Distribution
A sampling distribution describes a random variable representing a sample statistic.
Contents
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(μX, σ2X) then X̅ ~ N(μX̅, σ2X̅) where μX̅ = μX and σ2X̅ = σ2X/n. As another example, if X ~ Bernoulli(p) then nX̅ ~ Binomial(n,p). However, by the central limit theorem, X̅ ~ N(μX̅, σ2X̅) is approximately true even for non-normal distributions.