Standard Deviation Examples in Statistics

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

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

Concept Recap

Standard deviation is a measure of how spread out data values are from the mean, representing the typical distance of data points from the average. A small standard deviation means data clusters tightly around the mean; a large one means data is widely spread.

If the mean is 'home base,' standard deviation tells you how far data points typically wander from home. Small SD = data clusters close to the mean (like a tight group of friends). Large SD = data is scattered (friends spread all over town).

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 Deviation asks how tightly or loosely the values sit around the data set, not just where the middle is.

Common stuck point: Students often know a procedure related to standard deviation but skip the recognition step: Do I need to describe how far the data values extend or vary, rather than where the middle is? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Do I need to describe how far the data values extend or vary, rather than where the middle is?

Worked Examples

Example 1

medium

Compute the population SD of

1, 3, 5, 7, 9

Answer

\sigma = 2\sqrt{2}

First step

Mean

= 25/5 = 5

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Example 2

medium

Compute the population SD of

2, 4, 4, 4, 5, 5, 7, 9

Example 3

hard

For data

\{1, 2, 3, 4, 5\}

, compute the population SD.

Example 4

medium

Calculate the population standard deviation of: 4, 8, 6, 2, 10.

Example 5

hard

Dataset A: {5, 5, 5, 5}. Dataset B: {2, 4, 6, 8}. Without full calculation, which has a larger standard deviation and why?

Practice Problems

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

Example 1

easy

The data set

4, 4, 4, 4

has what standard deviation?

Example 2

easy

Can a standard deviation be

-3

Example 3

easy

Find the mean of

2, 4, 6

Example 4

easy

A data set with values tightly clustered near the mean has a (small / large) standard deviation?

Example 5

easy

Which data set is more spread out:

A=\{5,5,5\}

B=\{1,5,9\}

Set B has values 1, 5, 9 — spread widely from low to high. Set A = {5, 5, 5} has no spread at all.

Example 6

easy

Compute the variance of

2, 4

(population).

Example 7

easy

If the variance is

25

, what is the standard deviation?

Example 8

easy

Two classes have the same mean test score, but class X has SD

2

and class Y has SD

10

. Which class is more consistent?

Class X: a narrow bell curve (σ = 2). Most scores fall within 2 points of the mean — highly consistent.

Example 9

medium

Compute the population standard deviation of

2, 4, 4, 4, 5, 5, 7, 9

The 8 data values {2, 4, 4, 4, 5, 5, 7, 9} plotted in order. The mean is 5 and σ = 2.

Example 10

medium

A set has mean

10

and population variance

16

. A new value equal to

10

is added. Does the SD increase, decrease, or stay roughly the same direction?

Example 11

medium

Each value in a data set is multiplied by

3

. The original SD was

4

. What is the new SD?

Example 12

medium

Each value in a data set has

7

added to it. The original SD was

5

. What is the new SD?

Example 13

medium

Find the population SD of

1, 3, 5, 7

Example 14

medium

A sample

3, 5, 7

has mean

5

. Compute the SAMPLE standard deviation (divide by

n-1

Example 15

medium

Data set

A

ranges from

0

100

; data set

B

ranges from

40

60

. Both have

5

values. Which likely has the larger SD?

Set A spans 0 to 100 — five values spread across the full range. Set B spans only 40 to 60.

Example 16

medium

If a population SD is

0

, what must be true of the data?

Example 17

medium

The population variance of

a, b

equals

\left(\frac{a-b}{2}\right)^2

. Use this to find the SD of

6, 14

The two data points 6 and 14. Their mean is 10; each sits exactly 4 units away, so σ = 4.

Example 18

challenge

Two data sets each have

4

values and the same mean. Set

P=\{2,2,8,8\}

and set

Q=\{4,4,6,6\}

. By how much larger is

\sigma_P

than

\sigma_Q

Example 19

challenge

A set of

n

values has SD

\sigma

. You transform each value by

y = 2x - 5

. Express the new SD in terms of

\sigma

Example 20

challenge

Show why adding a value far above the mean increases the SD more than adding a value near the mean (one sentence reason).

Example 21

easy

A set has population variance

36

. What is its standard deviation?

Example 22

easy

Compute the population SD of

4, 8

Example 23

easy

Can SD ever be negative?

Example 24

easy

A constant data set has SD equal to what?

Example 25

medium

Compute the population SD of

0, 0, 6, 6

Example 26

medium

A set has population SD

4

. Each value is divided by

2

. What is the new SD?

Example 27

medium

A data set has SD

6

. Each value gets

10

added. What is the new SD?

Example 28

medium

A set has population SD

\sigma

and MAD

d

. If all values are the same, what are

\sigma

and

d

Example 29

medium

A new value equal to the mean is added to a data set. Does the population SD increase, decrease, or stay the same direction?

Example 30

medium

Compute the population variance of

5, 5, 5, 5, 25

Example 31

hard

Find the population SD of

5, 5, 5, 5, 25

Example 32

hard

Two data sets have the same range but different SDs. Which one is necessarily true?

Example 33

hard

What is the SD of

a, a

(only two equal values)?

Example 34

hard

Set

A=\{1,2,3,4,5\}

has population SD

\sqrt 2

. Multiply each value by

10

then add

7

. What is the new SD?

Example 35

hard

A set has population SD

0

. Is it possible for the values to differ? Explain.

Example 36

medium

For population variance

\sigma^2

and sample variance

s^2

, which divides by

n-1

Example 37

hard

A class of

20

students has mean test score

80

and population SD

5

. About what range covers most of the scores by the 68-95-99.7 rule, assuming roughly normal?

Example 38

challenge

Two data sets each have

n=4

values with mean

5

and population SD

2

. Must the two sets have the same values?

Example 39

medium

Calculate the population standard deviation of: 10, 12, 14.

Example 40

medium

Calculate the population standard deviation of the data set: 3, 3, 7, 7.

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Related Concepts

Mean as Fair Share Data Variability Z-Score (Standard Score)

Background Knowledge

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

mean fair sharevariability intro