Sampling Distribution Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sampling Distribution.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

The sampling distribution is the probability distribution of a statistic (such as the sample mean \bar{x}) computed from all possible random samples of a given size n drawn from a population. It describes how that statistic varies from sample to sample.

If you took 1000 different random samples and calculated the mean of each, those 1000 means would form a distribution. That's the sampling distribution - it shows how sample statistics vary.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The sampling distribution describes how a sample statistic (like the mean) varies across all possible samples of the same size from the same population.

Common stuck point: Students confuse the sampling distribution with the population distribution. The sampling distribution of means is narrower and becomes more normal as sample size grows.

Sense of Study hint: When working with a sampling distribution, first identify the statistic of interest (usually \bar{x}). Then find its center, which equals the population parameter (\mu_{\bar{x}} = \mu). Finally, calculate the standard error SE = \frac{\sigma}{\sqrt{n}} to determine how spread out the distribution is.

Worked Examples

Example 1

hard
A population has mean \mu = 60 and standard deviation \sigma = 12. If we take samples of size n = 36, what is the standard error of the sample mean?

Solution

  1. 1
    Step 1: The standard error is \text{SE} = \frac{\sigma}{\sqrt{n}}.
  2. 2
    Step 2: \text{SE} = \frac{12}{\sqrt{36}} = \frac{12}{6} = 2.
  3. 3
    Step 3: The sampling distribution of \bar{x} has mean 60 and standard error 2.

Answer

\text{SE} = 2
The standard error measures the variability of sample means. As sample size increases, the standard error decreases, meaning sample means cluster more tightly around the population mean.

Example 2

hard
Why does increasing the sample size from 25 to 100 improve the precision of a sample mean?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

hard
A population has \sigma = 20. Find the standard error for sample sizes n = 16 and n = 64.

Example 2

hard
A population has mean \mu = 120 and standard deviation \sigma = 24. For samples of size n = 36, what are the mean and standard error of the sampling distribution of \bar{x}?
← Back to Sampling Distribution Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

population vs samplemeanstandard deviation intro