Differences between revisions 1 and 3 (spanning 2 versions)
⇤ ← Revision 1 as of 2026-07-31 16:51:29 →
Size: 1003
Editor: DominicRicottone
Comment: Initial commit
← Revision 3 as of 2026-08-02 06:31:51 → ⇥
Size: 1273
Editor: DominicRicottone
Comment: Note
Deletions are marked like this. Additions are marked like this.
Line 17: Line 17:
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''. Furthermore, a 95% confidence interval can be constructed around the known population mean; 95% of samples will lead to a sample statistic that is within 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.

Sampling Distribution

A sampling distribution describes a random variable representing a sample statistic.

Contents

  1. Sampling Distribution
    1. Description

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. Furthermore, a 95% confidence interval can be constructed around the known population mean; 95% of samples will lead to a sample statistic that is within 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(μX̅, σ2X̅) is approximately true even for non-normal distributions.

CategoryRicottone

Statistics/SamplingDistribution (last edited 2026-08-02 17:13:49 by DominicRicottone)