Sampling Variability Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sampling Variability.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

Sampling variability is the natural sample-to-sample difference that appears when we take repeated random samples from the same population. Even good random samples do not all produce identical statistics.

If you take two honest random samples, they can still disagree a little. That disagreement is not necessarily bias or a mistake; it is part of how sampling works.

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 Variability 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 variability 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?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Sampling Bias Mistakes

Worked Examples

Example 1

easy

A population has

\mu=200

\sigma=20

. For samples of size

n=100

, what is the standard error of the sample mean?

Answer

2

First step

Standard error of

\bar{x}

\sigma/\sqrt{n}

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Example 2

medium

A teacher takes many random samples of

n=36

test scores from a population with

\mu=75

and

\sigma=18

. Sketch in words the sampling distribution of

\bar{x}

Example 3

medium

A factory claim says mean bolt length is

\mu = 50

mm with

\sigma = 0.4

mm. A sample of

n=64

gives

\bar{x} = 50.1

mm. Is

0.1

mm unusual?

Example 4

medium

In one random sample of

n=50

, the sample mean is

\bar{x}=12

. Why is it wrong to claim 'the population mean is exactly

12

Example 5

hard

A population has

\sigma = 15

. You want the standard error of

\bar{x}

to be at most

1.5

. What is the minimum

n

Example 6

hard

A sample of

n=36

from a population with

\mu=120

\sigma=24

gives

\bar{x}=132

. How many standard errors above

\mu

\bar{x}

Example 7

hard

The standard error of

\bar{x}

3

for

n=25

. What is it for

n=225

, assuming the same population?

Example 8

hard

Population standard deviation is

\sigma = 6

. For random samples of

n=9

, find a typical range that should contain about

95\%

of sample means around

\mu = 50

Example 9

challenge

A factory's bolts have

\mu = 5.00

cm,

\sigma = 0.10

cm. A quality engineer rejects a shift if

\bar{x}

from

n=25

bolts differs from

5.00

by more than

0.04

cm. What is the rejection probability for an in-control shift?

Example 10

challenge

A population has

\sigma

unknown. From

n=100

data values, the sample SD is

s=8

and

\bar{x}=52

. Estimate the standard error of

\bar{x}

and a

95\%

interval for

\mu

Practice Problems

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

Example 1

easy

Two honest random samples of 50 voters give 48% and 52% support for a measure. Does this difference prove one sample was biased?

Example 2

easy

A population has mean

\mu = 100

. Three random samples give means

98

101

, and

103

. What explains the spread of these sample means?

Example 3

easy

True or false: if a random sample's statistic does not equal the population parameter, the sample must be biased.

Example 4

easy

Polling 30 students gives a mean study time of 4.2 hours; another 30 gives 3.9 hours. What kind of difference is this most likely?

Example 5

easy

Which decreases sampling variability of the sample mean: a larger sample or a smaller sample?

Example 6

easy

A scientist remeasures the same object 5 times with a scale and gets slightly different readings. Is this sampling variability?

Example 7

easy

Fill in: sampling variability is the natural ____ difference seen when taking repeated random samples.

Example 8

easy

A claim states: 'Our one sample of 1000 gives the exact population average.' What is wrong with this?

Example 9

medium

A population of test scores has

\mu=70

\sigma=10

. You take many samples of size

n=25

. Roughly how much should sample means typically vary around 70?

Example 10

medium

A survey of 100 people estimates 60% approval. A repeat survey of 100 gets 56%. A third gets 63%. What single concept best describes this pattern of differing results?

Example 11

medium

If you quadruple the sample size, by what factor does the typical sampling variability of the mean change?

Example 12

medium

A teacher says: 'Class A averaged 82 and Class B averaged 80 on a random quiz sample, so A is the stronger class.' Why is caution warranted?

Example 13

medium

Population proportion is

p=0.5

. For samples of

n=100

, the standard deviation of the sample proportion is

\sqrt{p(1-p)/n}

. Compute it.

Example 14

medium

Why do two honest pollsters using proper random sampling sometimes report different percentages?

Example 15

medium

A sample of 36 items from a population with

\sigma=18

gives mean 50. What is the typical sampling variability of this mean, and is a value of 53 surprising?

Example 16

medium

Among these, which is NOT a source of sampling variability: (a) which people land in the sample, (b) random selection, (c) a miscalibrated scale used on every unit?

Example 17

medium

You take 1000 random samples of size 40 and plot all 1000 sample means. The plot's spread reflects what?

Example 18

challenge

A population has

\sigma=20

. A researcher wants the typical sampling variability of the mean to be at most 2. What minimum sample size is needed?

Example 19

challenge

Sample A (n=25) and sample B (n=100) come from the same population with

\sigma=15

. By what factor is sample A's typical variability larger than sample B's?

Example 20

challenge

A pollster reports a result with no margin of error, claiming their single random sample of 500 gives the precise population value. Explain the conceptual error and what they should report.

Example 21

easy

Two random samples of size

40

from the same school give mean GPAs of

3.10

and

3.18

. What concept best explains the gap?

Example 22

easy

A population proportion is

p = 0.5

. A random sample of

100

gives

\hat{p} = 0.46

. Is the difference

0.04

surprising?

Example 23

easy

Which sample produces less sampling variability for the sample mean:

n = 25

n = 400

Example 24

easy

A radio host claims one sample of

500

adults gives the 'exact' national approval rating. What is the flaw?

Example 25

medium

A population has

\sigma=12

. To cut the standard error of the sample mean from

4

1

, by what factor must the sample size grow?

Example 26

medium

A poll of

400

voters shows

52\%

support. The standard error of

\hat{p}

is about

0.025

. Roughly how far can

\hat{p}

stray from

p

Example 27

medium

Why is sampling variability not a 'mistake' that good statisticians can avoid?

Example 28

medium

A population has

\mu=60

\sigma=10

. Compare the standard error of

\bar{x}

for

n=25

n=100

Example 29

medium

A coin is fair (

p=0.5

). Out of

100

flips, you get

58

heads. Is this strong evidence the coin is biased?

Example 30

hard

Two polls each survey

1000

adults randomly. Poll A reports

42\%

approval, Poll B reports

46\%

. Is the

4

-point gap likely real?

Example 31

hard

A sociologist insists 'my one sample shows kids today watch more TV than last year.' What flaw of reasoning does this overlook?

Example 32

hard

A class simulates taking

1000

random samples of

n=40

and records each sample's mean. What does the spread of those

1000

means estimate?

Example 33

hard

A scientist wants to compare two diets using random samples of

n=10

each, then

n=200

each. Which design will show smaller differences as significant?

Example 34

challenge

A poll of

n=1000

adults estimates support at

51\%

. The margin of error (at

95\%

) is about

\pm 3.1\%

. Why can't the poll conclude that 'a majority supports' the measure?

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Related Concepts

Random Sampling Population vs Sample Sampling Distribution Standard Error Margin of Error

Background Knowledge

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

random samplingpopulation vs sample