Sampling Distributions and the Central Limit Theorem

Manual: General · Subject: Statistics

Learn how statistics vary from sample to sample and why that variability can be predicted.

Why Sample Statistics Vary

From data to sampling distributions

A sampling distribution is the distribution of a statistic, such as $\bar{x}$ , across many hypothetical samples of the same size. It describes how much a statistic typically changes from sample to sample.

Key ideas

Sampling distribution

The distribution of a statistic over repeated samples.

Standard error

The standard deviation of a sampling distribution.

Central limit theorem

For large

n

, the sampling distribution of many statistics becomes approximately normal under broad conditions.

Mean and proportion sampling behavior

Statistic	Center	Spread
Sample mean $\bar{x}$	Approximately $\mu$	$\sigma/\sqrt{n}$
Sample proportion $\hat{p}$	Approximately $p$	$\sqrt{\frac{p(1-p)}{n}}$

⚠️

Sample size matters

As $n$ increases, the standard error typically decreases like $1/\sqrt{n}$ , so estimates become more stable.

Using the CLT correctly

1

Step 1: Identify the statistic being studied, such as $\bar{x}$ or $\hat{p}$ .
2

Step 2: Check the conditions that justify an approximate normal sampling distribution.
3

Step 3: Compute the standard error.
4

Step 4: Use the approximate normal model to estimate probabilities or margins of error.

What is the standard error of the sample mean when the population standard deviation is $\sigma$ and sample size is $n$ ?

\sigma n

\sigma/\sqrt{n}

\sqrt{n}/\sigma

\sigma^2/n

What does the central limit theorem help justify?

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Sampling Distributions and the Central Limit Theorem

Why Sample Statistics Vary

From data to sampling distributions

Using the CLT correctly