Mathematics.

inferential statistics

Sampling Distributions

Statistics25 minDifficulty4 out of 10

You should know: normal distribution, descriptive statistics

Overview

A sampling distribution is the probability distribution of a statistic (such as the sample mean or sample proportion) obtained by repeatedly drawing random samples of a fixed size n from a population. Rather than describing individual data points, it describes how the statistic itself would vary across many hypothetical samples. The sampling distribution of the sample mean has the same mean as the population, μ, but a smaller standard deviation, σ/√n, called the standard error, which shrinks as sample size grows. Sampling distributions are the theoretical foundation for confidence intervals and hypothesis tests, since they tell us how much sample statistics fluctuate due to random sampling alone.

Intuition

Imagine repeatedly drawing samples of size n from a population, computing the mean of each sample, and plotting a histogram of all those sample means. That histogram is the sampling distribution. Individual data points can vary wildly, but averages of many points tend to cluster more tightly around the true population mean — the larger the sample, the tighter the cluster, which is why the standard error shrinks as 1/√n.

Formal Definition

Definition

If X₁, ..., Xₙ are independent random samples from a population with mean μ and standard deviation σ, and X̄ is the sample mean:

E[Xˉ]=μE[\bar{X}] = \mu
Mean of the sampling distribution of the sample mean
σXˉ=σn\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}
Standard error of the mean
Var⁡(Xˉ)=σ2n\operatorname{Var}(\bar{X}) = \frac{\sigma^2}{n}
Variance of the sampling distribution

Worked Examples

  1. Apply the standard error formula.

    σXˉ=2025=205=4\sigma_{\bar{X}} = \frac{20}{\sqrt{25}} = \frac{20}{5} = 4

Answer: The standard error is 4.

Practice Problems

Difficulty 3/10

A population has σ = 30. Find the standard error of the mean for a sample of size n = 9.

Difficulty 4/10

If the standard error is 5 when n = 16, what is the population standard deviation σ?

Difficulty 5/10

Quadrupling the sample size (from n to 4n) has what effect on the standard error of the sample mean?

Quiz

The sampling distribution of the sample mean describes:
As sample size n increases, the standard error of the mean:
The mean of the sampling distribution of the sample mean equals:

Summary

  • A sampling distribution describes how a statistic (like the sample mean) varies across repeated random samples of fixed size.
  • The sampling distribution of the sample mean has mean μ (same as the population) and standard error σ/√n.
  • Standard error shrinks as sample size grows, which is why larger samples give more precise estimates of the population mean.

References

  1. WebsiteWikipedia — Sampling distribution