Standard Deviation Examples in Math

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 Math.

Concept Recap

The standard deviation measures the average distance of data values from the mean, giving a typical spread around the center.

The typical distance from the average. Low SD = clustered. High SD = spread out.

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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: Standard deviation is roughly how far a normal data value sits from the mean, on average.

Common stuck point: The procedure for standard deviation is the easy part; the trap is forgetting the square root. Asking "Am I measuring how far values typically fall from the mean, in the data's own units?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I measuring how far values typically fall from the mean, in the data's own units?

Worked Examples

Example 1

medium

Find the population standard deviation of

\{2, 4, 4, 4, 5, 5, 7, 9\}

Answer

\sigma = 2

First step

Compute the mean:

\bar{x} = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5

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Example 2

hard

Find the sample standard deviation of

\{10, 12, 23, 23, 16, 23, 21, 16\}

Example 3

medium

Find the population standard deviation of

\{3, 7, 7, 19\}

Example 4

medium

Find the population SD of

\{1, 5, 9\}

Example 5

medium

Find the sample SD of

\{2, 4, 6, 8\}

Example 6

medium

A population has SD

\sigma

. Apply

y = 4x + 7

. Find SD of

y

Example 7

hard

Find the sample SD of

\{6, 8, 10, 12, 14\}

Practice Problems

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

Example 1

medium

Find the population standard deviation of

\{6, 2, 3, 1\}

Example 2

medium

Compare the population standard deviations of

A = \{4, 4, 4, 4\}

and

B = \{2, 4, 4, 6\}

Example 3

easy

All five values in a data set equal

8

. What is the standard deviation?

Example 4

easy

Data

2, 4, 6

has mean

4

. Find the population standard deviation.

Example 5

easy

A data set has variance

25

. What is its standard deviation?

Example 6

easy

Which data set has larger spread:

A=\{5,5,5\}

B=\{1,5,9\}

Example 7

easy

Population data

10, 12

has mean

11

. Find the SD.

Example 8

easy

Units check: a data set of weights in kilograms has SD

3

. What are the units of the SD?

Example 9

easy

Data

3, 3, 3, 3, 7

— without computing, will the SD be small or large relative to a tightly clustered set?

Example 10

easy

If every value in a data set is multiplied by

3

, what happens to the SD?

Example 11

medium

Find the population standard deviation of

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

Example 12

medium

Find the sample standard deviation of

4, 8, 12

(mean

8

Example 13

medium

Two classes have the same mean test score, but class A's SD is

4

and class B's SD is

10

. Which class is more consistent?

Example 14

medium

A data set has SD

6

. Every value is increased by

20

. What is the new SD?

Example 15

medium

Population data

1, 3, 5, 7, 9

has mean

5

. Find the SD.

Example 16

medium

Compare the SDs of

\{10,20,30\}

and

\{110,120,130\}

(both population).

Example 17

medium

A population SD is

4

. What is the variance?

Example 18

medium

Data

6, 6, 6, 6, 6, 6, 6, 14

(population, mean

=7

). Find the SD.

Example 19

medium

Population data

4, 4, 4, 4

— find the SD and explain in one phrase.

Example 20

challenge

A population of

n

values has SD

\sigma

. Each value is transformed by

y=3x-5

. Find the new SD.

Example 21

challenge

Two data points

a

and

b

form a population with SD

5

. Find

|a-b|

Example 22

challenge

A data set has mean

50

and SD

0

. What must be true, and what is its variance?

Example 23

easy

A data set has variance

49

. Find the standard deviation.

Example 24

easy

A data set has SD

9

. What is the variance?

Example 25

easy

Population data

4, 6

has mean

5

. Find the SD.

Example 26

easy

Which data set has larger SD:

A=\{7,7,7,7\}

B=\{3,7,7,11\}

Example 27

easy

A data set has SD

5

. Every value is decreased by

12

. What is the new SD?

Example 28

medium

Find the sample standard deviation of

\{5, 7, 9\}

Example 29

medium

A population SD is

4

. If each value is multiplied by

-2

, what is the new SD?

Example 30

medium

Two stores' weekly sales have the same mean of

500

. Store A has SD

20

; store B has SD

80

. Which store has more variable sales?

Example 31

medium

A population has

n=4

, sum

\sum(x-\mu)^2 = 100

. Find the SD.

Example 32

medium

A data set has mean

20

and SD

3

. Roughly what range contains values within one SD of the mean?

Example 33

medium

A population has values

a, a, a, a, b

with mean

\mu = a + 1

. Find the SD in terms of

b

and

a

given

b = a+5

Example 34

hard

A population

\{x_1,\dots,x_5\}

has mean

10

and

\sum x_i^2 = 530

. Find the population SD.

Example 35

hard

Three population values

a-d, a, a+d

have mean

a

and SD

\sigma

. Find

d

in terms of

\sigma

Example 36

hard

A student reports an SD of

0.45

for a list of test scores between

0

and

100

. Does this look plausible?

Example 37

hard

A population

\{2, 4, 4, 6\}

has mean

4

and SD

\sigma_1

. The set

\{2, 4, 4, 6, 4\}

has SD

\sigma_2

. Which is larger and why?

Example 38

challenge

A population of size

4

has values

0, 0, 0, k

with

k>0

. Find the SD as a function of

k

, and the value of

k

that makes SD

= 3

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Related Concepts

Mean Square Roots Variance Normal Distribution

Background Knowledge

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

meansquare roots