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, σ²_X) then X̅ ~ N(μ_X̅, σ²_X̅) where μ_X̅ = μ_X and σ²_X̅ = σ²_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, σ²_X̅) is approximately true even for non-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, σ²_X), then the standard error of the sample statistic X̅ is given by σ²_X̅ = σ²_X/n.

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