Central Limit Theorem Examples in Math

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 Math.

Concept Recap

For sufficiently large sample size (n≥30n \geq 30 as a rule of thumb), the sampling distribution of the sample mean is approximately normal with mean μ\mu and standard deviation σn\frac{\sigma}{\sqrt{n}}, regardless of the shape of the population distribution.

Roll a single die and the outcomes are flat (uniform). But average the rolls of 30 dice and the result looks like a bell curve every time. No matter how weird the original data looks—skewed, bimodal, flat—the averages of large enough samples always settle into a normal shape. It's one of the most surprising facts in all of mathematics.

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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 CLT says sample means become approximately normal for large nn, whatever the population's shape.

Common stuck point: The procedure for central limit theorem is the easy part; the trap is claiming the raw data becomes normal. Asking "Am I claiming the distribution of a sample MEAN is approximately normal because the sample is large?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I claiming the distribution of a sample MEAN is approximately normal because the sample is large?

Worked Examples

Example 1

medium
A highly skewed population (times between bus arrivals) has μ=15\mu=15 min and σ=8\sigma=8 min. For samples of n=64n=64, describe the shape, mean, and SD of the sampling distribution of Xˉ\bar{X}, and find P(Xˉ<14)P(\bar{X} < 14).

Answer

Xˉ∼N(15,1)\bar{X} \sim N(15, 1); P(Xˉ<14)≈0.159P(\bar{X} < 14) \approx 0.159 despite non-normal population.

First step

1
CLT: despite skewed population, with n=64≥30n=64 \geq 30, Xˉ\bar{X} is approximately normally distributed

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Example 2

hard
A fair die (μ=3.5, σ=1.71) is rolled n=100n=100 times. By CLT, find the approximate probability that the sample mean is between 3.3 and 3.7.

Example 3

medium
A population has μ=70\mu = 70, σ=12\sigma = 12. For n=36n = 36, find P(Xˉ>73)P(\bar{X} > 73).

Example 4

medium
A coin is flipped n=100n = 100 times. Let Xˉ\bar{X} be the sample proportion of heads. Use CLT to find P(Xˉ>0.55)P(\bar{X} > 0.55).

Example 5

medium
A population has μ=120\mu = 120, σ=30\sigma = 30. For n=36n = 36, what is the probability Xˉ\bar{X} falls within 55 of μ\mu?

Example 6

medium
A casino's roulette bet has expected loss μ=−0.05\mu = -0.05 per dollar and σ=1\sigma = 1. After n=10000n = 10000 bets, find P(Xˉ>0)P(\bar{X} > 0) (player profits on average).

Example 7

hard
An exponential population (e.g., wait times) has μ=5\mu = 5, σ=5\sigma = 5. For n=100n = 100, find P(Xˉ>5.5)P(\bar{X} > 5.5).

Example 8

hard
A factory's items have μ=200\mu = 200g, σ=10\sigma = 10g. A box contains n=25n = 25 items. Find P(total weight>5050)P(\text{total weight} > 5050) using CLT.

Example 9

hard
A population has μ=80\mu = 80, σ=16\sigma = 16. Find the 95th percentile of Xˉ\bar{X} for n=64n = 64 (use z0.95=1.645z_{0.95} = 1.645).

Example 10

challenge
n=100n = 100 i.i.d. samples come from a Bernoulli(p=0.02p=0.02) population. Why might the CLT-based normal approximation for p^\hat{p} be poor here, and what's the alternative?

Practice Problems

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

Example 1

easy
State the Central Limit Theorem in your own words, including what conditions must be met and what it tells us about the shape of the sampling distribution.

Example 2

hard
Customers arrive at a store with mean μ=2\mu=2 per minute, σ=1.5\sigma=1.5 per minute (Poisson-like). For 36-minute observation windows, find P(total arrivals>80)P(\text{total arrivals} > 80) using CLT.

Example 3

easy
A population has standard deviation σ=20\sigma = 20. For samples of size n=100n = 100, what is the standard deviation of the sample mean (the standard error)?

Example 4

easy
A population has mean μ=50\mu = 50. What is the mean of the sampling distribution of the sample mean for samples of size n=36n = 36?

Example 5

easy
According to the common rule of thumb, what minimum sample size makes the CLT a reasonable approximation for the sample mean?

Example 6

easy
True or false: the CLT applies to the sampling distribution of the sample mean, not to individual observations.

Example 7

easy
A population has σ=12\sigma = 12. For n=9n = 9, find the standard error of the mean.

Example 8

easy
A skewed population has μ=8\mu = 8. The sampling distribution of xˉ\bar{x} for large nn is approximately what shape?

Example 9

easy
If the sample size increases from n=25n=25 to n=100n=100 (a factor of 44), the standard error of the mean changes by what factor?

Example 10

easy
A population has σ=15\sigma = 15, n=25n = 25. Express the standard error of xˉ\bar{x}.

Example 11

medium
A population has μ=70\mu = 70, σ=10\sigma = 10. For n=25n = 25, find P(xˉ>74)P(\bar{x} > 74). Use P(Z>2)≈0.0228P(Z > 2) \approx 0.0228.

Example 12

medium
A population has μ=100\mu = 100, σ=16\sigma = 16. For n=64n = 64, find P(xˉ<96)P(\bar{x} < 96). Use P(Z<−2)≈0.0228P(Z < -2) \approx 0.0228.

Example 13

medium
Heights have μ=168\mu = 168 cm, σ=8\sigma = 8 cm. For a sample of n=16n = 16, what is the SD of the sample mean?

Example 14

medium
A sample mean is needed to have standard error ≤1\le 1 when σ=6\sigma = 6. What is the smallest nn?

Example 15

medium
A die roll (uniform 11 to 66) has mean 3.53.5. The average of n=49n=49 rolls is approximately normal centered at what value?

Example 16

medium
A population has σ=25\sigma = 25. To cut the standard error of the mean from 55 to 2.52.5, by what factor must nn increase?

Example 17

medium
Package weights have μ=500\mu = 500 g, σ=30\sigma = 30 g. For n=36n = 36, find the standard error of the mean.

Example 18

medium
A population has μ=40\mu = 40, σ=9\sigma = 9. For n=81n = 81, find P(xˉ>41)P(\bar{x} > 41). Use P(Z>1)≈0.1587P(Z > 1) \approx 0.1587.

Example 19

challenge
A population has mean μ\mu and SD σ\sigma. For a sample of size nn, by what factor does the standard error of the total (sum) S=∑xiS = \sum x_i grow relative to the standard error of one observation, in terms of nn?

Example 20

challenge
A population is strongly right-skewed with μ=5\mu = 5, σ=5\sigma = 5. Explain why n=30n = 30 may be insufficient for the normal approximation of xˉ\bar{x}, and state what determines the needed nn.

Example 21

challenge
Two independent samples are averaged. Sample A: nA=100n_A = 100, σA=30\sigma_A = 30. Sample B: nB=50n_B = 50, σB=20\sigma_B = 20. Which sample mean has the smaller standard error, and what are the two values?

Example 22

medium
A population has σ=14\sigma = 14, n=49n = 49. Find the standard error of the sample mean.

Example 23

easy
A population has σ=18\sigma = 18. For samples of size n=81n = 81, what is the SE of Xˉ\bar{X}?

Example 24

easy
A population has μ=200\mu = 200, σ=50\sigma = 50. For n=100n = 100, find the SE.

Example 25

easy
For σ=6\sigma = 6 and n=36n = 36, find the SE of Xˉ\bar{X}.

Example 26

easy
A skewed population has μ=5\mu = 5, σ=4\sigma = 4. For n=100n = 100, what shape does the sampling distribution of Xˉ\bar{X} have?

Example 27

medium
A truck-weight population has μ=5000\mu = 5000 lb, σ=800\sigma = 800 lb. For n=64n = 64 trucks, find P(Xˉ<4900)P(\bar{X} < 4900).

Example 28

medium
For a population with σ=20\sigma = 20, what sample size makes the SE equal to 44?

Example 29

medium
The CLT says the SUM S=X1+X2+⋯+XnS = X_1 + X_2 + \dots + X_n is approximately normal with mean nμn\mu and SD σn\sigma\sqrt{n}. If μ=4\mu = 4, σ=2\sigma = 2, n=25n = 25, find the SD of SS.

Example 30

medium
A population has σ=10\sigma = 10. By what factor does the SE of Xˉ\bar{X} change when nn goes from 2525 to 400400?

Example 31

hard
A population has μ=50\mu = 50, σ=10\sigma = 10. To guarantee SE ≤0.5\le 0.5, what minimum nn is needed?

Example 32

hard
For μ=100\mu = 100, σ=25\sigma = 25, n=25n = 25, find P(95<Xˉ<110)P(95 < \bar{X} < 110).

Example 33

hard
A population proportion is p=0.20p = 0.20. Verify CLT applies for n=50n = 50 and find P(p^>0.25)P(\hat{p} > 0.25).

Example 34

medium
For μ=60\mu = 60, σ=8\sigma = 8, n=16n = 16, find the SE.
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Background Knowledge

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

sampling distributionnormal distribution