Sampling Distribution Examples in Math

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 Math.

Concept Recap

The probability distribution of a statistic (such as the sample mean) computed from all possible random samples of the same size drawn from a population.

Imagine you survey 50 random people about their height, compute the average, then repeat with a different group of 50, again and again. Each group gives a slightly different average. The pattern of all those averages forms the sampling distribution. It's like taking the temperature of a city by sending out 100 different thermometers—each reads slightly differently, but together they cluster around the truth.

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: Even if the population isn't normal, the distribution of sample means approaches a normal shape as sample size grows—this is the bridge from raw data to statistical inference.

Common stuck point: The sampling distribution is NOT the distribution of the raw data—it's the distribution of a statistic (like \bar{x}) computed from many samples.

Worked Examples

Example 1

medium
A population has \mu=50 and \sigma=12. For random samples of size n=36, describe the sampling distribution of \bar{X}: find its mean, standard error, and shape.

Solution

  1. 1
    Mean of sampling distribution: \mu_{\bar{X}} = \mu = 50
  2. 2
    Standard error: \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{36}} = \frac{12}{6} = 2
  3. 3
    Shape: by CLT (n=36 ≥ 30), approximately Normal even if population is not Normal
  4. 4
    Full description: \bar{X} \sim N(50, 2)

Answer

\bar{X} \sim N(\mu=50,\ SE=2). Sample means are normally distributed around 50 with SD=2.
The sampling distribution describes the distribution of sample means across all possible samples of size n. Its mean equals the population mean (unbiased); its SD (standard error) = σ/√n. The CLT guarantees approximate normality for large n.

Example 2

hard
For a population with \mu=100, \sigma=20, and n=25: (a) find P(\bar{X} > 104), (b) find the value c such that P(\bar{X} < c) = 0.90.

Practice Problems

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

Example 1

easy
The standard error of a sample mean is \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}. If \sigma=10 and n=100, find the standard error. What happens to the SE if n is quadrupled to 400?

Example 2

hard
A population proportion is p=0.40. For samples of size n=100, describe the sampling distribution of \hat{p} and find P(\hat{p} > 0.45).
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Background Knowledge

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

normal distributionmeanstandard deviation