Standard Error Examples: 44 Problems with Answers

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

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 standard error (SE) is the standard deviation of a sampling distribution, measuring how much a sample statistic (like the sample mean) typically varies from the true population parameter across repeated samples. It decreases as sample size increases.

Standard error tells you how much your sample estimate might be 'off' from the true value. Larger samples have smaller SE because they're more precise - like asking 1000 people vs 10.

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: Standard Error uses a sample result and a variation model to make a careful population statement.

Common stuck point: Students often know a procedure related to standard error 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

You want to shrink the SE of the mean from

5

1

keeping the same population. By what factor must

n

grow?

Answer

25 times

First step

Ratio of SEs:

5/1 = 5

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

medium

A pollster wants a margin of error no larger than

3

percentage points at

95\%

confidence for a proportion close to

0.5

. Find the minimum sample size.

Example 3

medium

For two studies on the same trait, study A has

n=100

and study B has

n=400

. Both use the same

\sigma

. How do their SEs compare?

Example 4

medium

A sample of

n=64

has mean

\bar{x}=120

and SD

s=16

. Construct an approximate

95\%

confidence interval for

\mu

using

\pm 2 \cdot SE

Example 5

hard

A study of

n=100

has mean

\bar{x}=75

and SD

s=10

. The researcher proposes 'the SE of a single observation is

1

.' Is this right?

Example 6

hard

Two independent samples have means

\bar{x}_1

and

\bar{x}_2

with SEs

1.2

and

0.9

. Find the SE of the difference

\bar{x}_1 - \bar{x}_2

Example 7

hard

In a two-sample t-test with

\bar{x}_1=50

\bar{x}_2=46

s_1=8

s_2=6

n_1=n_2=64

. Find the SE of the mean difference.

Example 8

challenge

A stratified sample draws

n_1=40

from a stratum with

\sigma_1=8

and

n_2=60

from a stratum with

\sigma_2=12

, with stratum weights

w_1=0.4

w_2=0.6

. Find the SE of the overall stratified mean.

Example 9

easy

A population has a standard deviation of

\sigma = 20

. If you take a random sample of

n = 100

, what is the standard error of the sample mean?

Example 10

medium

How does the standard error change when you quadruple the sample size from

n = 25

n = 100

? Assume

\sigma = 30

Practice Problems

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

Example 1

easy

Write the formula for the standard error of the sample mean.

Example 2

easy

A population has

\sigma=20

and

n=4

. Compute the standard error of the mean.

Example 3

easy

Standard error measures the standard deviation of what?

Example 4

easy

As sample size increases, the standard error ____.

Example 5

easy

Is the standard error usually larger or smaller than the population standard deviation

\sigma

Example 6

easy

A population SD is 15 and

n=9

. What is the standard error of the mean?

Example 7

easy

Which symbol denotes population standard deviation in the SE formula

\sigma/\sqrt{n}

Example 8

easy

n=1

, what does the standard error equal?

Example 9

medium

To halve the standard error of the mean, by what factor must the sample size increase?

Example 10

medium

A study reports sample SD

s=24

from

n=36

observations. Estimate the standard error of the mean.

Example 11

medium

A 95% confidence interval uses

\pm 2\,SE

. If SE

=3

, what is the half-width (margin of error)?

Example 12

medium

Population SD is 50. What sample size makes the standard error of the mean equal to 5?

Example 13

medium

Why use the population SD

\sigma

(not sample-to-sample range) in the standard error formula?

Example 14

medium

Two studies estimate the same mean: study A has SE

=2

, study B has SE

=5

. Which estimate is more precise?

Example 15

medium

A sample mean is 80 with SE

=4

. How many standard errors away is a hypothesized mean of 72?

Example 16

medium

A proportion

\hat{p}=0.4

comes from

n=100

. The SE of a proportion is

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

. Compute it.

Example 17

medium

A sample SD is

s=30

from

n=25

. Estimate the standard error of the mean.

Example 18

challenge

Population SD is 40. A researcher needs a 95% margin of error (

2\,SE

) of at most 4. Find the minimum sample size.

Example 19

challenge

Combine two independent sample means each with SE

=3

into their difference. What is the standard error of the difference?

Example 20

challenge

A sample of

n=100

gives SE

=2

. A colleague claims increasing to

n=110

will noticeably shrink the SE. Evaluate this quantitatively.

Example 21

easy

Population SD is

30

and

n=25

. Find the SE of the mean.

Example 22

easy

Population SD is

14

and

n=49

. Find the SE of the mean.

Example 23

easy

\sigma=10

stays fixed, what happens to SE when

n

doubles?

Example 24

easy

A sample of

n=36

has sample SD

s=24

. Estimate the SE of the mean.

Example 25

easy

Compute the SE of the mean when

\sigma=18

and

n=81

Example 26

medium

95\%

confidence interval for

\mu

uses

\bar{x} \pm 1.96 \cdot SE

. If

SE=2.5

, what is the margin of error?

Example 27

medium

A sample of

n=16

has SD

s=12

. Estimate the SE of the mean.

Example 28

medium

A normal population has

\sigma=50

. Find the sample size needed so that the SE of the mean is at most

5

Example 29

medium

In a poll of

n=1600

voters,

52\%

support a candidate. Find the SE of the proportion.

Example 30

hard

In linear regression, the SE of the slope decreases when which two things happen?

Example 31

hard

A bootstrap procedure resamples

1000

samples and computes

\bar{x}^*

each time. The SD of those

1000

means is the bootstrap estimate of what?

Example 32

hard

A study reports

\bar{x}=68

with SE

=1.5

n=36

. What is the implied sample SD?

Example 33

medium

A researcher measures the reaction time of 64 participants and finds a sample standard deviation of

s = 40

ms. Calculate the standard error and construct an approximate 95% confidence interval if the sample mean is 250 ms.

Example 34

hard

A polling company wants the standard error of a proportion to be no more than 0.02 (2%). If a preliminary estimate suggests

\hat{p} \approx 0.5

, what minimum sample size is needed? Use

SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

← Back to Standard Error Formula Explained Practice Mode

Related Concepts

Standard Deviation Sampling Distribution Confidence Interval Margin of Error

Background Knowledge

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

standard deviation introsampling distribution