Z-Score Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Z-Score.

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

Concept Recap

A z-score measures how many standard deviations a data value is above or below the mean: $z = (x - \mu)/\sigma$ .

A universal measuring stick— $z = 2$ means '2 SDs above average.'

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: A z-score restates a value as the number of standard deviations it sits above or below the average.

Common stuck point: The procedure for z-score is the easy part; the trap is forgetting to divide by $\sigma$ . Asking "Am I expressing this value as a number of standard deviations from its mean?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I expressing this value as a number of standard deviations from its mean?

Worked Examples

Example 1

easy

A student scored

82

on an exam where the mean was

74

and the standard deviation was

8

. What is the student's z-score?

Answer

z = 1.0

First step

Recall the z-score formula:

z = \frac{x - \mu}{\sigma}

, which measures how many standard deviations

x

is from the mean.

Full solution

2
Identify given values: $x = 82$ , $\mu = 74$ , $\sigma = 8$ .
3
Substitute and calculate: $z = \frac{82 - 74}{8} = \frac{8}{8} = 1.0$

A z-score of

1.0

means the student scored exactly one standard deviation above the mean. Z-scores allow comparison across different scales.

Example 2

medium

On Test A, Maria scored

78

(

\mu = 70

\sigma = 5

). On Test B, she scored

85

(

\mu = 80

\sigma = 10

). On which test did she perform relatively better?

Example 3

easy

A car gets

32

mpg. The fleet has

\mu = 28

mpg and

\sigma = 4

mpg. Compute the z-score and describe what it means.

Example 4

medium

Heights are normal with

\mu = 170

cm and

\sigma = 8

cm. A person is

182

cm tall. What is the z-score?

Example 5

hard

SAT and ACT scores are normalized. A student gets SAT

= 1300

(

\mu = 1050

\sigma = 200

) and ACT

= 28

(

\mu = 21

\sigma = 5

). Which is the stronger performance?

Practice Problems

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

Example 1

easy

A data point has value

45

in a distribution with

\mu = 50

and

\sigma = 4

. Find its z-score.

Example 2

medium

In a class, test scores have mean

70

and standard deviation

8

. What raw score corresponds to a z-score of

1.25

Example 3

easy

A value is

x=15

, mean

\mu=10

, SD

\sigma=5

. Find its z-score.

Example 4

easy

Find the z-score for

x=4

\mu=10

\sigma=2

Example 5

easy

A value equals the mean. What is its z-score?

Example 6

easy

Find the z-score for

x=85

\mu=70

\sigma=15

Example 7

easy

A z-score is

2

with

\mu=50

\sigma=10

. What is the raw value

x

Example 8

easy

Find the z-score for

x=12

\mu=12

\sigma=3

Example 9

easy

A z-score is

-1.5

with

\mu=20

\sigma=4

. Find

x

Example 10

easy

Find the z-score for

x=8

\mu=5

\sigma=1.5

Example 11

medium

Student A scored

80

on a test with

\mu=70

\sigma=5

. Student B scored

90

with

\mu=85

\sigma=10

. Who did relatively better?

Example 12

medium

A value has z-score

1.2

. After every data value (including this one) is doubled, what is its new z-score?

Example 13

medium

A test has

\mu=60

\sigma=12

. What raw score corresponds to the 84th percentile (normal data)?

Example 14

medium

A value's deviation from the mean is

x-\mu=-9

and

\sigma=6

. Find the z-score.

Example 15

medium

On a normal distribution, a z-score of

+2

means the value is in roughly what top percent?

Example 16

medium

Two values from the same distribution have z-scores

-1

and

+1

. If

\mu=50

and

\sigma=8

, find both raw values.

Example 17

medium

A normal distribution has

\mu=100

. A value of

115

has z-score

1.5

. Find

\sigma

Example 18

medium

A value has z-score

0.5

. After adding

20

to every data value, what is its new z-score?

Example 19

medium

Find the z-score for

x=2

\mu=8

\sigma=3

Example 20

challenge

Test scores are normal,

\mu=500

\sigma=100

. A scholarship requires the top 16%. What minimum score qualifies?

Example 21

challenge

Distribution X has

\mu=70,\sigma=10

; distribution Y has

\mu=70,\sigma=5

. A raw value of

80

appears in both. In which is it more extreme, and by how many SDs more?

Example 22

challenge

A value

x

satisfies

z=2

in a distribution with

\sigma=4

, and the same

x

gives

z=4

in another distribution with the same mean. Find that second SD.

Example 23

easy

A value

x = 22

comes from a distribution with

\mu = 18

and

\sigma = 2

. Find the z-score.

Example 24

easy

Find the z-score for

x = 30

when

\mu = 40

and

\sigma = 5

Example 25

easy

Compute the z-score for

x = 6

\mu = 4

\sigma = 0.5

Example 26

easy

Find the z-score of

x = 17

in a distribution with

\mu = 20

\sigma = 6

Example 27

medium

Quiz A: Jay scored

88

with

\mu = 80

\sigma = 4

. Quiz B: Jay scored

92

with

\mu = 80

\sigma = 8

. On which did Jay do relatively better?

Example 28

medium

A normal distribution has

\mu = 200

and

\sigma = 25

. What raw value has z-score

-1.6

Example 29

medium

A test has

\mu = 75

\sigma = 5

. Two students score

80

and

65

. Find both z-scores.

Example 30

medium

Two raw scores have z-scores

-0.5

and

+2

. Their difference is

20

. Find

\sigma

Example 31

medium

A value with z-score

1.2

has every value in the dataset shifted up by

7

. What is the value's new z-score?

Example 32

medium

If every value in a dataset is multiplied by

5

, what happens to a z-score of

0.7

Example 33

medium

On a normal distribution with

\mu = 0

\sigma = 1

, what is the z-score of

x = 1.96

Example 34

medium

On a normal distribution with

\mu = 50

\sigma = 10

, approximately what percent of values lie above

z = 1

Example 35

hard

A value

x

has z-score

1.5

in distribution

A

with

\sigma_A = 8

. The same

x

has z-score

3

in distribution

B

with the same mean. Find

\sigma_B

Example 36

hard

On a normally distributed test with

\mu = 500

\sigma = 100

, what minimum score corresponds to the top

2.5\%

Example 37

hard

A dataset has

\mu = 50

and

\sigma = 5

. Two values have z-scores

-2

and

+1.4

. What is the difference between the raw values?

Example 38

hard

In a normal distribution, approximately what percent of values have

|z| \leq 2

Example 39

hard

A value at the

25

th percentile of a normal distribution corresponds to which approximate z-score?

Example 40

challenge

A scholarship requires the top

5\%

on a normal test with

\mu = 60

and

\sigma = 8

. What is the minimum qualifying raw score?

Example 41

challenge

In a normal distribution, what is the approximate z-score of the

90

th percentile?

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Related Concepts

Mean Standard Deviation Normal Distribution

Background Knowledge

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

meanstandard deviation