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 xˉ\bar{x}) computed from all possible random samples of a given size nn 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.

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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: Sampling Distribution uses a sample result and a variation model to make a careful population statement.

Common stuck point: Students often know a procedure related to sampling distribution 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=75\mu = 75, σ=15\sigma = 15. For n=25n = 25, find the center and standard error of xˉ\bar{x}.

Answer

Center =75= 75; SE =3= 3.

First step

1
μxˉ=μ=75\mu_{\bar{x}} = \mu = 75.

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Example 2

medium
What three things does the sampling distribution of xˉ\bar{x} describe?

Example 3

hard
A skewed population has μ=12,σ=4\mu = 12, \sigma = 4. For n=64n = 64, is the sampling distribution of xˉ\bar{x} approximately normal? Why?

Example 4

hard
Population is uniform on [0,10][0, 10] (μ=5,σ=10/12≈2.89\mu = 5, \sigma = 10/\sqrt{12} \approx 2.89). Describe the sampling distribution of xˉ\bar{x} for n=50n = 50.

Example 5

challenge
Population σ\sigma is unknown; we estimate it with sample SD ss. Why does the sampling distribution of xˉ\bar{x} then follow a tt-distribution rather than zz?

Example 6

medium
A population has μ=150\mu=150 and σ=30\sigma=30. For n=100n=100, find the probability that xˉ\bar{x} lies between 147147 and 153153.

Example 7

medium
A population has μ=50\mu=50, σ=10\sigma=10. For n=25n=25, find the probability that xˉ\bar{x} exceeds 5252.

Example 8

medium
From a normal population with μ=70\mu=70, σ=15\sigma=15, samples of size n=9n=9 are drawn. Find the probability that xˉ<65\bar{x} < 65.

Example 9

medium
A normal population has μ=500\mu=500, σ=100\sigma=100. For n=25n=25, find the value cc such that P(xˉ>c)=0.05P(\bar{x} > c) = 0.05.

Example 10

hard
A normal population has μ=80\mu=80 and σ=12\sigma=12. A sample of n=36n=36 is drawn. Find the probability that xˉ\bar{x} lies more than 33 units away from μ\mu.

Example 11

hard
A normal population has μ=20\mu=20, σ=4\sigma=4. For n=16n=16, find the 90%90\% central interval for xˉ\bar{x}.

Example 12

hard
A factory's tubes have μ=500\mu=500 ml and σ=10\sigma=10 ml. Quality inspectors test n=25n=25 tubes. What is the probability the sample mean lies outside the warning band 498498 to 502502?

Example 13

challenge
A uniform population on [0,10][0, 10] has μ=5\mu=5 and σ=102/12≈2.887\sigma=\sqrt{10^2/12} \approx 2.887. Approximate P(xˉ>5.5)P(\bar{x} > 5.5) for n=100n=100 using the CLT.

Example 14

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

Example 15

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

easy
What is the sampling distribution of a statistic?

Example 2

easy
A population has mean μ=50\mu=50. What is the mean (center) of the sampling distribution of the sample mean xˉ\bar{x}?

Example 3

easy
A population has σ=12\sigma=12. For samples of size n=9n=9, what is the standard deviation of the sampling distribution of xˉ\bar{x}?

Example 4

easy
As sample size increases, does the sampling distribution of xˉ\bar{x} get wider or narrower?

Example 5

easy
True or false: the sampling distribution and the population distribution are the same thing.

Example 6

easy
Which statistic's sampling distribution does SE=σ/nSE=\sigma/\sqrt{n} describe the spread of?

Example 7

easy
If you took 1000 random samples and recorded each sample's mean, the histogram of those 1000 means approximates what?

Example 8

easy
Fill in: the sampling distribution of xˉ\bar{x} has mean equal to ____ and standard deviation equal to ____.

Example 9

medium
A population has μ=200\mu=200, σ=30\sigma=30. For n=36n=36, give the mean and standard deviation of the sampling distribution of xˉ\bar{x}.

Example 10

medium
For σ=20\sigma=20, the sampling distribution of xˉ\bar{x} should have SD =4=4. What sample size achieves this?

Example 11

medium
Two sampling distributions of xˉ\bar{x} from the same population use n=16n=16 and n=64n=64. Which is narrower and by what factor?

Example 12

medium
A skewed population has μ=10\mu=10, σ=4\sigma=4. For n=64n=64, what is the approximate shape, center, and spread of the sampling distribution of xˉ\bar{x}?

Example 13

medium
Using μ=200\mu=200, SD of xˉ\bar{x} =5=5 (from n=36n=36, σ=30\sigma=30), what fraction of sample means fall above 205?

Example 14

medium
Why is the sampling distribution of xˉ\bar{x} narrower than the population distribution?

Example 15

medium
A sampling distribution of xˉ\bar{x} is centered at 75 with SD 3. A particular sample gives xˉ=81\bar{x}=81. How many standard errors from the center is this?

Example 16

medium
Which is required for the sampling distribution of xˉ\bar{x} to be exactly normal (not just approximately) for any nn?

Example 17

medium
A sampling distribution of xˉ\bar{x} has mean 60 and SD 4. A sample gives xˉ=68\bar{x}=68. How many standard errors above the center is this?

Example 18

challenge
A population has σ=10\sigma=10. You want 95% of sample means to lie within 1 unit of μ\mu. Using ±2\pm 2 SE for 95%, find the needed sample size.

Example 19

challenge
Sampling distribution of xˉ\bar{x}: μ=500\mu=500, σ=60\sigma=60, n=144n=144. What is the probability a sample mean exceeds 510? (Use SE and z.)

Example 20

challenge
Explain why, for a fixed population, the sampling distribution of xˉ\bar{x} becomes both narrower and more bell-shaped as nn grows.

Example 21

easy
A population has mean μ=100\mu = 100. For samples of size n=25n = 25, what is the mean of the sampling distribution of xˉ\bar{x}?

Example 22

easy
Population σ=20\sigma = 20. For samples of size n=100n = 100, find the standard deviation of xˉ\bar{x}.

Example 23

easy
σ=30\sigma = 30, n=36n = 36. Find the standard error of xˉ\bar{x}.

Example 24

medium
σ=12\sigma = 12, n=4n = 4. Find the standard error of xˉ\bar{x}.

Example 25

medium
If nn is multiplied by 44, the standard error of xˉ\bar{x} is multiplied by what factor?

Example 26

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

Example 27

medium
μ=60\mu = 60, σ=10\sigma = 10, n=25n = 25. Use the normal approximation to find P(xˉ>62)P(\bar{x} > 62).

Example 28

medium
A population proportion is p=0.4p = 0.4. For n=100n = 100, find the standard error of p^\hat{p}.

Example 29

medium
σ=8\sigma = 8. What sample size nn makes the standard error of xˉ\bar{x} equal to 22?

Example 30

medium
Distinguish: population SD vs sample SD vs SD of the sampling distribution. Which is largest, for n>1n > 1?

Example 31

medium
Population has mean μ=50,σ=6\mu = 50, \sigma = 6. Find SE for n=9,36,144n = 9, 36, 144.

Example 32

hard
μ=70,σ=8,n=16\mu = 70, \sigma = 8, n = 16. Use normal approximation to find P(xˉ<68)P(\bar{x} < 68).

Example 33

hard
p=0.5,n=400p = 0.5, n = 400. Find the SE of p^\hat{p}.

Example 34

hard
We want SE of p^\hat{p} to be at most 0.020.02 when p=0.5p = 0.5. What sample size nn is needed?

Example 35

medium
Population μ=0,σ=1\mu = 0, \sigma = 1 (standard normal). For n=100n = 100, find SE of xˉ\bar{x}.

Example 36

medium
The sampling distribution of xˉ\bar{x} has SE =2.5= 2.5. If n=16n = 16, what is the population standard deviation σ\sigma?

Example 37

hard
μ=100,σ=16,n=64\mu = 100, \sigma = 16, n = 64. Find the probability xˉ\bar{x} is within 11 unit of μ\mu.

Example 38

easy
A population has μ=100\mu=100 and σ=20\sigma=20. What is the mean of the sampling distribution of xˉ\bar{x} for n=25n=25?

Example 39

easy
A population has μ=100\mu=100 and σ=20\sigma=20. What is the standard deviation of the sampling distribution of xˉ\bar{x} for n=25n=25?

Example 40

easy
A population has σ=24\sigma=24. For n=16n=16, find σxˉ\sigma_{\bar{x}}.

Example 41

easy
A population has μ=75\mu=75 and σ=10\sigma=10. For n=100n=100, the sampling distribution of xˉ\bar{x} has mean and SD equal to what?

Example 42

easy
For a normal population with σ=8\sigma=8 and n=64n=64, find the standard deviation of xˉ\bar{x}.

Example 43

medium
To cut the SD of the sampling distribution of xˉ\bar{x} in thirds, by what factor must nn increase?

Example 44

medium
A normal population has μ=200\mu=200, σ=40\sigma=40. For n=16n=16, find the probability that xˉ\bar{x} lies between 190190 and 210210.

Example 45

medium
A population has μ=12\mu=12 and σ=6\sigma=6. What sample size makes the standard deviation of the sampling distribution of xˉ\bar{x} equal to 0.50.5?

Example 46

medium
True or false: when the population is normal, the sampling distribution of xˉ\bar{x} is exactly normal for any sample size.

Example 47

medium
In what sense does the sampling distribution describe 'sample-to-sample variability'?

Example 48

hard
A skewed population has σ=30\sigma=30. We want σxˉ\sigma_{\bar{x}} at most 22 and the sampling distribution to be approximately normal. What nn achieves both?

Example 49

hard
Why is the sampling distribution of xˉ\bar{x} used to construct confidence intervals for μ\mu?

Example 50

hard
A normal population has μ=60\mu=60, σ=18\sigma=18. What sample size guarantees that xˉ\bar{x} falls within 33 units of μ\mu at least 95%95\% of the time?

Example 51

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

Example 52

hard
A population has mean μ=120\mu = 120 and standard deviation σ=24\sigma = 24. For samples of size n=36n = 36, what are the mean and standard error of the sampling distribution of xˉ\bar{x}?
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Background Knowledge

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

population vs samplemean fair sharestandard deviation intro