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≥30n \geq 30), the sampling distribution of the sample mean xˉ\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: Central Limit Theorem uses a sample result and a variation model to make a careful population statement.

Common stuck point: Students often know a procedure related to central limit theorem but skip the recognition step: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly?

Worked Examples

Example 1

medium
A population has μ=72\mu=72, σ=10\sigma=10. For n=25n=25, find P(xˉ<70)P(\bar{x}<70).

Answer

≈0.1587\approx 0.1587

First step

1
SE=10/25=2SE=10/\sqrt{25}=2; z=(70−72)/2=−1z=(70-72)/2=-1.

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Example 2

medium
A factory's bolts have weight μ=10\mu=10 g, σ=0.2\sigma=0.2 g. A box of 100 bolts is sampled. Find P(xˉ>10.03)P(\bar{x}>10.03).

Example 3

hard
Show why xˉ\bar{x} is unbiased for μ\mu regardless of population shape (CLT does not assume normality).

Example 4

challenge
XX is exponential with mean μ=10\mu=10 (so σ=10\sigma=10). For n=100n=100, approximate P(xˉ>11)P(\bar{x}>11) using the CLT.

Example 5

medium
A skewed population has μ=50\mu=50, σ=20\sigma=20. For n=100n=100, use the CLT to estimate P(xˉ>53)P(\bar{x} > 53).

Example 6

medium
A population has μ=10\mu=10 and σ=5\sigma=5. For n=100n=100, use the CLT to find the 95%95\% central range of xˉ\bar{x}.

Example 7

medium
A population of waiting times is exponential with mean μ=4\mu=4 and SD σ=4\sigma=4. For n=64n=64, estimate P(xˉ<3.5)P(\bar{x} < 3.5) via the CLT.

Example 8

medium
A population has μ=18\mu = 18, σ=9\sigma = 9. For n=81n = 81, find the value cc such that P(xˉ<c)=0.10P(\bar{x} < c) = 0.10 using the CLT.

Example 9

hard
A skewed population has μ=200\mu=200, σ=60\sigma=60. For n=144n=144, find P(195<xˉ<207)P(195 < \bar{x} < 207).

Example 10

hard
A Bernoulli population has p=0.5p=0.5. By the CLT, find the approximate probability that the sample proportion p^\hat{p} from n=100n=100 exceeds 0.550.55.

Example 11

hard
A factory's defect rate is p=0.1p=0.1. By the CLT, find the approximate probability that out of n=200n=200 items, at least 2525 are defective.

Example 12

challenge
A Cauchy distribution has no finite mean or variance. Does the CLT apply to its sample mean?

Example 13

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

Example 14

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

easy
What does the Central Limit Theorem say about the sampling distribution of xˉ\bar{x} for large nn?

Example 2

easy
Roughly what sample size is the common rule of thumb for the CLT to apply?

Example 3

easy
Does the CLT require the population to be normally distributed?

Example 4

easy
The CLT describes the distribution of which quantity?

Example 5

easy
As nn increases, the CLT approximation becomes ____.

Example 6

easy
True or false: the CLT changes the shape of the population distribution itself.

Example 7

easy
A population is heavily right-skewed. For n=50n=50, what shape is the sampling distribution of xˉ\bar{x}?

Example 8

easy
Fill in: the CLT lets us use ____-distribution methods even when the population is not normal, provided nn is large.

Example 9

medium
A population has μ=8\mu=8, σ=6\sigma=6. For n=36n=36, the CLT says xˉ\bar{x} is approximately normal. Give its mean and standard deviation.

Example 10

medium
A skewed population has μ=40\mu=40, σ=12\sigma=12. For n=144n=144, find the probability that xˉ\bar{x} exceeds 42 (use the CLT).

Example 11

medium
Why can a sample of n=100n=100 from a wildly skewed income distribution still support normal-based confidence intervals for the mean?

Example 12

medium
Distinguish the Central Limit Theorem from the Law of Large Numbers in one sentence each.

Example 13

medium
For n=4n=4 from a strongly skewed population, is it safe to assume xˉ\bar{x} is normal? Explain.

Example 14

medium
A population has μ=500\mu=500, σ=80\sigma=80, n=64n=64. Within what symmetric interval around 500 will about 95% of sample means fall?

Example 15

medium
If the population is already normal, how large must nn be for xˉ\bar{x} to be normal?

Example 16

medium
A sum of 100 independent dice rolls is recorded. Why is this sum approximately normal?

Example 17

medium
A population has μ=100\mu=100, σ=24\sigma=24, n=16n=16. By the CLT, give the mean and SD of the approximate normal distribution of xˉ\bar{x}.

Example 18

challenge
A population has μ=20\mu=20, σ=15\sigma=15. You need P(xˉ>23)≈2.5%P(\bar{x} > 23) \approx 2.5\%. What sample size nn achieves this using the CLT?

Example 19

challenge
Explain why averaging reduces both skew and spread of the sampling distribution as nn grows, referencing the CLT.

Example 20

challenge
Two independent samples of size n=36n=36 from populations with σ=12\sigma=12 each have means xˉ1,xˉ2\bar{x}_1, \bar{x}_2. By the CLT, the difference xˉ1−xˉ2\bar{x}_1-\bar{x}_2 is approximately normal with what standard deviation?

Example 21

easy
A population has mean μ=50\mu=50 and SD σ=10\sigma=10. For n=100n=100, find the standard error of xˉ\bar{x}.

Example 22

easy
If σ=20\sigma=20 and n=400n=400, what is the standard error of the sample mean?

Example 23

easy
A population has μ=100\mu=100, σ=15\sigma=15. For a sample of size n=225n=225, what is the mean of the sampling distribution of xˉ\bar{x}?

Example 24

easy
A skewed population has σ=8\sigma=8. For n=64n=64, find SExˉSE_{\bar{x}}.

Example 25

medium
XX has μ=60\mu=60, σ=12\sigma=12. For n=36n=36, find P(xˉ>62)P(\bar{x}>62) using the CLT.

Example 26

medium
A sample proportion p^\hat{p} comes from n=400n=400 with p=0.5p=0.5. Find SEp^SE_{\hat{p}}.

Example 27

medium
μ=500\mu=500, σ=50\sigma=50. For n=100n=100, find P(490<xˉ<510)P(490<\bar{x}<510).

Example 28

medium
For a sample proportion, n=100n=100, p=0.30p=0.30. What is the approximate distribution of p^\hat{p} by the CLT?

Example 29

medium
μ=250\mu=250, σ=40\sigma=40. Find sample size needed so that SExˉ≤4SE_{\bar{x}}\le 4.

Example 30

medium
A population has σ=6\sigma=6. What sample size gives SExˉ=0.5SE_{\bar{x}}=0.5?

Example 31

medium
XX is uniformly distributed on [0,1][0,1] with μ=0.5\mu=0.5, σ=1/12≈0.289\sigma=1/\sqrt{12}\approx 0.289. For n=48n=48, find SExˉSE_{\bar{x}}.

Example 32

medium
μ=70\mu=70, σ=14\sigma=14, n=49n=49. Find P(xˉ<68)P(\bar{x}<68).

Example 33

medium
n=200n=200, p=0.40p=0.40. By the CLT, approximately what is P(p^>0.45)P(\hat{p}>0.45)?

Example 34

hard
A skewed population has μ=20\mu=20, σ=8\sigma=8. For n=64n=64, find P(18<xˉ<22)P(18<\bar{x}<22).

Example 35

hard
A heavy-tailed distribution has σ=100\sigma=100. We want SExˉ≤5SE_{\bar{x}}\le 5. What nn is required?

Example 36

hard
μ=200\mu=200, σ=30\sigma=30, n=100n=100. Find the value aa such that P(xˉ>a)=0.025P(\bar{x}>a)=0.025.

Example 37

hard
An audit samples 64 invoices from a population with μ=$120\mu=\$120, σ=$24\sigma=\$24. Find P(xˉ>$123)P(\bar{x}>\$123).

Example 38

easy
State the conclusion of the CLT in one phrase.

Example 39

easy
A right-skewed population has μ=20\mu=20 and σ=8\sigma=8. For n=64n=64, by the CLT the distribution of xˉ\bar{x} is approximately ____.

Example 40

easy
Does the CLT apply equally to skewed and normal populations?

Example 41

easy
A heavy-tailed population requires what relative to the n≥30n \ge 30 rule for the CLT to look good?

Example 42

easy
For a normal population, what nn is needed for xˉ\bar{x} to be normal?

Example 43

medium
A bowl of dice rolls has μ=3.5\mu=3.5, σ≈1.71\sigma \approx 1.71. For n=36n=36 rolls, estimate P(xˉ>4)P(\bar{x} > 4).

Example 44

medium
What is the difference between the CLT and the Law of Large Numbers in one sentence?

Example 45

medium
Why is the CLT particularly useful in real-world settings where the population shape is unknown?

Example 46

medium
A population is bimodal with μ=50\mu=50. Will xˉ\bar{x} also be bimodal for n=50n = 50?

Example 47

hard
A heavily right-skewed payroll has μ=$60,000\mu=\$60{,}000 and σ=$25,000\sigma=\$25{,}000. For n=400n=400, give the approximate distribution of xˉ\bar{x}.

Example 48

hard
A study uses n=10n=10 from a heavily skewed population. Can normal-based inference for μ\mu be justified by the CLT? Explain.

Example 49

hard
A population has σ=12\sigma=12. What is the smallest nn such that the CLT-normal approximation gives an SE of at most 1.51.5?

Example 50

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

Example 51

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?
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Background Knowledge

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

sampling distributionstat normal distribution