Z-Score Examples in Math

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−μ)/σz = (x - \mu)/\sigma.

A universal measuring stick—z=2z = 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 σ\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 8282 on an exam where the mean was 7474 and the standard deviation was 88. What is the student's z-score?

Answer

z=1.0z = 1.0

First step

1
Recall the z-score formula: z=x−μσz = \frac{x - \mu}{\sigma}, which measures how many standard deviations xx is from the mean.

Full solution

  1. 2
    Identify given values: x=82x = 82, μ=74\mu = 74, σ=8\sigma = 8.
  2. 3
    Substitute and calculate: z=82−748=88=1.0z = \frac{82 - 74}{8} = \frac{8}{8} = 1.0
A z-score of 1.01.0 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 7878 (μ=70\mu = 70, σ=5\sigma = 5). On Test B, she scored 8585 (μ=80\mu = 80, σ=10\sigma = 10). On which test did she perform relatively better?

Example 3

easy
A car gets 3232 mpg. The fleet has μ=28\mu = 28 mpg and σ=4\sigma = 4 mpg. Compute the z-score and describe what it means.

Example 4

medium
Heights are normal with μ=170\mu = 170 cm and σ=8\sigma = 8 cm. A person is 182182 cm tall. What is the z-score?

Example 5

hard
SAT and ACT scores are normalized. A student gets SAT =1300= 1300 (μ=1050\mu = 1050, σ=200\sigma = 200) and ACT =28= 28 (μ=21\mu = 21, σ=5\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 4545 in a distribution with μ=50\mu = 50 and σ=4\sigma = 4. Find its z-score.

Example 2

medium
In a class, test scores have mean 7070 and standard deviation 88. What raw score corresponds to a z-score of 1.251.25?

Example 3

easy
A value is x=15x=15, mean μ=10\mu=10, SD σ=5\sigma=5. Find its z-score.

Example 4

easy
Find the z-score for x=4x=4, μ=10\mu=10, σ=2\sigma=2.

Example 5

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

Example 6

easy
Find the z-score for x=85x=85, μ=70\mu=70, σ=15\sigma=15.

Example 7

easy
A z-score is 22 with μ=50\mu=50, σ=10\sigma=10. What is the raw value xx?

Example 8

easy
Find the z-score for x=12x=12, μ=12\mu=12, σ=3\sigma=3.

Example 9

easy
A z-score is −1.5-1.5 with μ=20\mu=20, σ=4\sigma=4. Find xx.

Example 10

easy
Find the z-score for x=8x=8, μ=5\mu=5, σ=1.5\sigma=1.5.

Example 11

medium
Student A scored 8080 on a test with μ=70\mu=70, σ=5\sigma=5. Student B scored 9090 with μ=85\mu=85, σ=10\sigma=10. Who did relatively better?

Example 12

medium
A value has z-score 1.21.2. After every data value (including this one) is doubled, what is its new z-score?

Example 13

medium
A test has μ=60\mu=60, σ=12\sigma=12. What raw score corresponds to the 84th percentile (normal data)?

Example 14

medium
A value's deviation from the mean is x−μ=−9x-\mu=-9 and σ=6\sigma=6. Find the z-score.

Example 15

medium
On a normal distribution, a z-score of +2+2 means the value is in roughly what top percent?

Example 16

medium
Two values from the same distribution have z-scores −1-1 and +1+1. If μ=50\mu=50 and σ=8\sigma=8, find both raw values.

Example 17

medium
A normal distribution has μ=100\mu=100. A value of 115115 has z-score 1.51.5. Find σ\sigma.

Example 18

medium
A value has z-score 0.50.5. After adding 2020 to every data value, what is its new z-score?

Example 19

medium
Find the z-score for x=2x=2, μ=8\mu=8, σ=3\sigma=3.

Example 20

challenge
Test scores are normal, μ=500\mu=500, σ=100\sigma=100. A scholarship requires the top 16%. What minimum score qualifies?

Example 21

challenge
Distribution X has μ=70,σ=10\mu=70,\sigma=10; distribution Y has μ=70,σ=5\mu=70,\sigma=5. A raw value of 8080 appears in both. In which is it more extreme, and by how many SDs more?

Example 22

challenge
A value xx satisfies z=2z=2 in a distribution with σ=4\sigma=4, and the same xx gives z=4z=4 in another distribution with the same mean. Find that second SD.

Example 23

easy
A value x=22x = 22 comes from a distribution with μ=18\mu = 18 and σ=2\sigma = 2. Find the z-score.

Example 24

easy
Find the z-score for x=30x = 30 when μ=40\mu = 40 and σ=5\sigma = 5.

Example 25

easy
Compute the z-score for x=6x = 6, μ=4\mu = 4, σ=0.5\sigma = 0.5.

Example 26

easy
Find the z-score of x=17x = 17 in a distribution with μ=20\mu = 20, σ=6\sigma = 6.

Example 27

medium
Quiz A: Jay scored 8888 with μ=80\mu = 80, σ=4\sigma = 4. Quiz B: Jay scored 9292 with μ=80\mu = 80, σ=8\sigma = 8. On which did Jay do relatively better?

Example 28

medium
A normal distribution has μ=200\mu = 200 and σ=25\sigma = 25. What raw value has z-score −1.6-1.6?

Example 29

medium
A test has μ=75\mu = 75, σ=5\sigma = 5. Two students score 8080 and 6565. Find both z-scores.

Example 30

medium
Two raw scores have z-scores −0.5-0.5 and +2+2. Their difference is 2020. Find σ\sigma.

Example 31

medium
A value with z-score 1.21.2 has every value in the dataset shifted up by 77. What is the value's new z-score?

Example 32

medium
If every value in a dataset is multiplied by 55, what happens to a z-score of 0.70.7?

Example 33

medium
On a normal distribution with μ=0\mu = 0, σ=1\sigma = 1, what is the z-score of x=1.96x = 1.96?

Example 34

medium
On a normal distribution with μ=50\mu = 50, σ=10\sigma = 10, approximately what percent of values lie above z=1z = 1?

Example 35

hard
A value xx has z-score 1.51.5 in distribution AA with σA=8\sigma_A = 8. The same xx has z-score 33 in distribution BB with the same mean. Find σB\sigma_B.

Example 36

hard
On a normally distributed test with μ=500\mu = 500, σ=100\sigma = 100, what minimum score corresponds to the top 2.5%2.5\%?

Example 37

hard
A dataset has μ=50\mu = 50 and σ=5\sigma = 5. Two values have z-scores −2-2 and +1.4+1.4. What is the difference between the raw values?

Example 38

hard
In a normal distribution, approximately what percent of values have ∣z∣≤2|z| \leq 2?

Example 39

hard
A value at the 2525th percentile of a normal distribution corresponds to which approximate z-score?

Example 40

challenge
A scholarship requires the top 5%5\% on a normal test with μ=60\mu = 60 and σ=8\sigma = 8. What is the minimum qualifying raw score?

Example 41

challenge
In a normal distribution, what is the approximate z-score of the 9090th percentile?
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Background Knowledge

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

meanstandard deviation