Central Limit Theorem Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Central Limit Theorem.

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 Central Limit Theorem (CLT) states that for sufficiently large sample sizes (usually n \geq 30), the sampling distribution of the sample mean \bar{x} is approximately normal, regardless of the shape of the original population distribution.

This is statistics' magic trick: no matter how weird your population looks, if you take big enough samples and average them, those averages will form a bell curve. This is why normal distribution methods work so often.

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: With a large enough sample size (usually n ≥ 30), sample means follow an approximately normal distribution regardless of the population's original shape.

Common stuck point: The CLT applies to sample means, not to individual values. Individual observations from a skewed population remain skewed no matter the sample size.

Sense of Study hint: When applying the CLT, first verify your sample size is large enough (typically n \geq 30). Then compute the mean of the sampling distribution as \mu_{\bar{x}} = \mu and the standard error as SE = \frac{\sigma}{\sqrt{n}}. Finally, use the normal distribution to find probabilities about \bar{x}.

Worked Examples

Example 1

hard
A population has a right-skewed distribution with \mu = 40 and \sigma = 10. If we take samples of size 50, describe the shape of the sampling distribution of \bar{x}.

Solution

  1. 1
    Step 1: The Central Limit Theorem states that for sufficiently large n (typically n \geq 30), the sampling distribution of \bar{x} is approximately normal.
  2. 2
    Step 2: Here n = 50 \geq 30, so the CLT applies even though the population is skewed.
  3. 3
    Step 3: The sampling distribution of \bar{x} is approximately normal with mean 40 and SE = \frac{10}{\sqrt{50}} \approx 1.41.

Answer

Approximately normal with mean 40 and SE ≈ 1.41, by the Central Limit Theorem.
The CLT is one of the most important results in statistics. It explains why normal-based inference works even for non-normal populations, provided the sample size is large enough.

Example 2

hard
Explain why the Central Limit Theorem is important for making confidence intervals.

Practice Problems

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

Example 1

hard
A uniform distribution has \mu = 5 and \sigma = 2.89. For samples of size 36, what are the mean and standard error of the sampling distribution? Is it approximately normal?

Example 2

hard
A population is strongly right-skewed. If samples of size 100 are taken repeatedly, what does the Central Limit Theorem say about the sampling distribution of the sample mean?
← Back to Central Limit Theorem Practice Mode

Background Knowledge

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

sampling distributionnormal distribution