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

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: 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

\mu = 75

\sigma = 15

. For

n = 25

, find the center and standard error of

\bar{x}

Sampling distribution of x̄: center = μ = 75, SE = σ/√n = 3

Answer

Center

= 75

; SE

= 3

First step

\mu_{\bar{x}} = \mu = 75

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Example 2

medium

What three things does the sampling distribution of

\bar{x}

describe?

Example 3

hard

A skewed population has

\mu = 12, \sigma = 4

. For

n = 64

, is the sampling distribution of

\bar{x}

approximately normal? Why?

Despite a skewed population, CLT gives x̄ ≈ N(12, 0.5) for n = 64

Example 4

hard

Population is uniform on

[0, 10]

(

\mu = 5, \sigma = 10/\sqrt{12} \approx 2.89

). Describe the sampling distribution of

\bar{x}

for

n = 50

Example 5

challenge

Population

\sigma

is unknown; we estimate it with sample SD

s

. Why does the sampling distribution of

\bar{x}

then follow a

t

-distribution rather than

z

Example 6

medium

A population has

\mu=150

and

\sigma=30

. For

n=100

, find the probability that

\bar{x}

lies between

147

and

153

Example 7

medium

A population has

\mu=50

\sigma=10

. For

n=25

, find the probability that

\bar{x}

exceeds

52

Example 8

medium

From a normal population with

\mu=70

\sigma=15

, samples of size

n=9

are drawn. Find the probability that

\bar{x} < 65

Example 9

medium

A normal population has

\mu=500

\sigma=100

. For

n=25

, find the value

c

such that

P(\bar{x} > c) = 0.05

Example 10

hard

A normal population has

\mu=80

and

\sigma=12

. A sample of

n=36

is drawn. Find the probability that

\bar{x}

lies more than

3

units away from

\mu

Example 11

hard

A normal population has

\mu=20

\sigma=4

. For

n=16

, find the

90\%

central interval for

\bar{x}

Example 12

hard

A factory's tubes have

\mu=500

ml and

\sigma=10

ml. Quality inspectors test

n=25

tubes. What is the probability the sample mean lies outside the warning band

498

502

Example 13

challenge

A uniform population on

[0, 10]

has

\mu=5

and

\sigma=\sqrt{10^2/12} \approx 2.887

. Approximate

P(\bar{x} > 5.5)

for

n=100

using the CLT.

Example 14

hard

A population has mean

μ = 60

and standard deviation

σ = 12

. If we take samples of size

n = 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

\mu=50

. What is the mean (center) of the sampling distribution of the sample mean

\bar{x}

Example 3

easy

A population has

\sigma=12

. For samples of size

n=9

, what is the standard deviation of the sampling distribution of

\bar{x}

Example 4

easy

As sample size increases, does the sampling distribution of

\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=\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

\bar{x}

has mean equal to ____ and standard deviation equal to ____.

Example 9

medium

A population has

\mu=200

\sigma=30

. For

n=36

, give the mean and standard deviation of the sampling distribution of

\bar{x}

Sampling distribution of x̄: centered at μ = 200, spread SE = σ/√n

Example 10

medium

For

\sigma=20

, the sampling distribution of

\bar{x}

should have SD

=4

. What sample size achieves this?

Example 11

medium

Two sampling distributions of

\bar{x}

from the same population use

n=16

and

n=64

. Which is narrower and by what factor?

Example 12

medium

A skewed population has

\mu=10

\sigma=4

. For

n=64

, what is the approximate shape, center, and spread of the sampling distribution of

\bar{x}

CLT in action: even with a skewed population, the sampling distribution of x̄ for n

Example 13

medium

Using

\mu=200

, SD of

\bar{x}

=5

(from

n=36

\sigma=30

), what fraction of sample means fall above 205?

What fraction of sample means fall above 205? (205 is one SE above μ

Example 14

medium

Why is the sampling distribution of

\bar{x}

narrower than the population distribution?

Example 15

medium

A sampling distribution of

\bar{x}

is centered at 75 with SD 3. A particular sample gives

\bar{x}=81

. How many standard errors from the center is this?

Example 16

medium

Which is required for the sampling distribution of

\bar{x}

to be exactly normal (not just approximately) for any

n

Example 17

medium

A sampling distribution of

\bar{x}

has mean 60 and SD 4. A sample gives

\bar{x}=68

. How many standard errors above the center is this?

Example 18

challenge

A population has

\sigma=10

. You want 95% of sample means to lie within 1 unit of

\mu

. Using

\pm 2

SE for 95%, find the needed sample size.

Example 19

challenge

Sampling distribution of

\bar{x}

\mu=500

\sigma=60

n=144

. What is the probability a sample mean exceeds 510? (Use SE and z.)

Sampling distribution of x̄: μ

Example 20

challenge

Explain why, for a fixed population, the sampling distribution of

\bar{x}

becomes both narrower and more bell-shaped as

n

grows.

Example 21

easy

A population has mean

\mu = 100

. For samples of size

n = 25

, what is the mean of the sampling distribution of

\bar{x}

Example 22

easy

Population

\sigma = 20

. For samples of size

n = 100

, find the standard deviation of

\bar{x}

Example 23

easy

\sigma = 30

n = 36

. Find the standard error of

\bar{x}

Example 24

medium

\sigma = 12

n = 4

. Find the standard error of

\bar{x}

Example 25

medium

n

is multiplied by

4

, the standard error of

\bar{x}

is multiplied by what factor?

Example 26

medium

What does the Central Limit Theorem say about the sampling distribution of

\bar{x}

when

n

is large?

Example 27

medium

\mu = 60

\sigma = 10

n = 25

. Use the normal approximation to find

P(\bar{x} > 62)

Sampling distribution with μ

Example 28

medium

A population proportion is

p = 0.4

. For

n = 100

, find the standard error of

\hat{p}

Example 29

medium

\sigma = 8

. What sample size

n

makes the standard error of

\bar{x}

equal to

2

Example 30

medium

Distinguish: population SD vs sample SD vs SD of the sampling distribution. Which is largest, for

n > 1

Example 31

medium

Population has mean

\mu = 50, \sigma = 6

. Find SE for

n = 9, 36, 144

Example 32

hard

\mu = 70, \sigma = 8, n = 16

. Use normal approximation to find

P(\bar{x} < 68)

Sampling distribution with μ

Example 33

hard

p = 0.5, n = 400

. Find the SE of

\hat{p}

Example 34

hard

We want SE of

\hat{p}

to be at most

0.02

when

p = 0.5

. What sample size

n

is needed?

Example 35

medium

Population

\mu = 0, \sigma = 1

(standard normal). For

n = 100

, find SE of

\bar{x}

Example 36

medium

The sampling distribution of

\bar{x}

has SE

= 2.5

. If

n = 16

, what is the population standard deviation

\sigma

Example 37

hard

\mu = 100, \sigma = 16, n = 64

. Find the probability

\bar{x}

is within

1

unit of

\mu

Example 38

easy

A population has

\mu=100

and

\sigma=20

. What is the mean of the sampling distribution of

\bar{x}

for

n=25

Example 39

easy

A population has

\mu=100

and

\sigma=20

. What is the standard deviation of the sampling distribution of

\bar{x}

for

n=25

Example 40

easy

A population has

\sigma=24

. For

n=16

, find

\sigma_{\bar{x}}

Example 41

easy

A population has

\mu=75

and

\sigma=10

. For

n=100

, the sampling distribution of

\bar{x}

has mean and SD equal to what?

Example 42

easy

For a normal population with

\sigma=8

and

n=64

, find the standard deviation of

\bar{x}

Example 43

medium

To cut the SD of the sampling distribution of

\bar{x}

in thirds, by what factor must

n

increase?

Example 44

medium

A normal population has

\mu=200

\sigma=40

. For

n=16

, find the probability that

\bar{x}

lies between

190

and

210

Example 45

medium

A population has

\mu=12

and

\sigma=6

. What sample size makes the standard deviation of the sampling distribution of

\bar{x}

equal to

0.5

Example 46

medium

True or false: when the population is normal, the sampling distribution of

\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

\sigma=30

. We want

\sigma_{\bar{x}}

at most

2

and the sampling distribution to be approximately normal. What

n

achieves both?

Example 49

hard

Why is the sampling distribution of

\bar{x}

used to construct confidence intervals for

\mu

Example 50

hard

A normal population has

\mu=60

\sigma=18

. What sample size guarantees that

\bar{x}

falls within

3

units of

\mu

at least

95\%

of the time?

Example 51

hard

A population has

σ = 20

. Find the standard error for sample sizes

n = 16

and

n = 64

Example 52

hard

A population has mean

μ = 120

and standard deviation

σ = 24

. For samples of size

n = 36

, what are the mean and standard error of the sampling distribution of

\overset{ˉ}{x}

← Back to Sampling Distribution Practice Mode

Related Concepts

Population vs Sample Mean as Fair Share Standard Deviation Central Limit Theorem Standard Error

Background Knowledge

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

population vs samplemean fair sharestandard deviation intro