Z-Score (Standard Score) Examples: 47 Problems with Answers

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

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

Concept Recap

A z-score tells you how many standard deviations a value is from the mean, calculated as z=x−μσ. Positive z-scores are above the mean; negative z-scores are below. Z-scores allow comparison of values from different distributions.

Z-scores put everything on the same scale. A z-score of +2 means 'two standard deviations above average' - unusually high. A z-score of -1 means 'one SD below average' - somewhat low but normal.

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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: Z-Score (Standard Score) asks how a value or feature behaves inside the full distribution.

Common stuck point: Students often know a procedure related to z-score (standard score) but skip the recognition step: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?

Worked Examples

Example 1

medium
A standardized test has μ=500 and σ=100. A student scores 640. Find the z-score, and explain what it means.

Answer

z=1.4

First step

1
z=640−500100=140100=1.4.

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Example 2

medium
A baby weighs 4.2 kg at birth. Birth weights have μ=3.4 kg and σ=0.5 kg. Find the z-score.

Example 3

medium
A factory's bolts have mean length 50 mm with SD 0.4 mm. A bolt measures 49.2 mm. Compute the z-score and decide whether it lies within ±2 SD of the mean.

Example 4

hard
A normally distributed exam has μ=72 and σ=6. The top 2.5% correspond approximately to z≥1.96. What raw score is the cutoff?

Example 5

challenge
Adult-male IQ scores are modeled as normal with μ=100,σ=15. Mensa requires roughly the top 2%, corresponding to about z≥2.05. Approximately what IQ qualifies?

Example 6

medium
A student scores 78 on a test where μ=70 and σ=4. Calculate her z-score and interpret it.

Example 7

hard
Alice scores 85 in Maths (μ=75,σ=5) and 90 in English (μ=80,σ=10). In which subject did she perform better relative to her class?

Practice Problems

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

Example 1

easy
Write the formula for a z-score.

Example 2

easy
Compute the z-score for x=15 when μ=10 and σ=5.

Example 3

easy
Compute the z-score for x=4 when μ=10 and σ=2.

Example 4

easy
A z-score of 0 means the value equals what?

Example 5

easy
Does a positive z-score mean the value is above or below the mean?

Example 6

easy
Which is more unusual under a normal model: a z-score of +2.5 or +0.5?

Example 7

easy
A value has z=−1. How many standard deviations from the mean is it, and in which direction?

Example 8

easy
Can z-scores be used to compare a test score and a height measurement?

Example 9

medium
Student A scored 85 on a test with μ=70,σ=10. Student B scored 90 on a test with μ=80,σ=20. Who did relatively better?

Example 10

medium
If z=2, μ=50, and σ=4, find the raw value x.

Example 11

medium
A value has z=−1.5, μ=200, σ=20. Find x.

Example 12

medium
In a normal distribution, a z-score of +1 corresponds to roughly what percentile (using the empirical rule)?

Example 13

medium
Why does scaling all data by a constant leave each value's z-score unchanged (assume also scaling shifts cancel)? Give the key reason.

Example 14

medium
A data point has z=3.2. Under the common ∣z∣>3 rule, is it an outlier?

Example 15

medium
Two values from the SAME distribution have z-scores +1 and −1. What is true about their distances from the mean?

Example 16

medium
A z-score is computed for non-normal, heavily skewed data. Is the ∣z∣>3 outlier rule reliable here?

Example 17

medium
A value has z-score 0.5, μ=40, σ=6. Find the raw value x.

Example 18

challenge
On test X (μ=75,σ=5) Maria scored 82. On test Y (μ=88,σ=4) she scored 94. On which test was her standing higher?

Example 19

challenge
A value's z-score is 2. The data is then transformed by y=3x+7. What is the z-score of the transformed value?

Example 20

challenge
A normal data set has μ=100. A value at the 97.5th percentile has what approximate z-score (empirical rule)?

Example 21

easy
Compute the z-score for x=22 when μ=18 and σ=2.

Example 22

easy
Find the z-score for x=7 when μ=10 and σ=1.5.

Example 23

easy
True or false: standardizing every value in a data set by computing z-scores produces a new data set with mean 0 and standard deviation 1.

Example 24

easy
Compute the z-score for x=100 when μ=80 and σ=25.

Example 25

easy
Find the z-score for x=50 when μ=50 and σ=7.

Example 26

medium
Given z=−0.75, μ=200, and σ=40, find the raw value x.

Example 27

medium
An NBA player is 213 cm tall. Adult-male heights have μ=175 cm and σ=8 cm. Find his z-score (to two decimal places).

Example 28

medium
On Math, Anya scored 88 with μ=70,σ=12. On English, she scored 82 with μ=65,σ=10. On which test did she perform relatively better?

Example 29

medium
If z-scores of −1.2 and +1.8 are computed for the same value under two different (μ,σ) models, are these compatible interpretations of the same observation?

Example 30

medium
Find the z-score for x=12.5 when μ=10 and σ=2.5.

Example 31

medium
If μ=60 and σ=5, what raw value corresponds to a z-score of +3?

Example 32

hard
Two friends compare resting heart rates. Min: 58 bpm (group μ=72,σ=8). Sam: 66 bpm (group μ=78,σ=6). Whose heart rate is more unusually low for their group?

Example 33

hard
An observation has z=2.4 in a distribution with μ=30. The raw value is 42. Find σ.

Example 34

hard
A standardized variable is rescaled by y=a+bx where b>0. If the original value x has z-score zx, what is the z-score of the new value y in the new distribution?

Example 35

hard
A data set has values {4,8,8,12}. Find the z-score of the value 12 using the population SD σ.

Example 36

hard
Under a normal model, an observation with ∣z∣≥3 is often flagged as an outlier. A reading is x=200 in a process with μ=170,σ=8. Is it flagged?

Example 37

challenge
For any data set with finite σ>0, prove that the mean of the z-scores is exactly 0.

Example 38

challenge
Two values from different distributions have z-scores z1 and z2 with z1<z2<0. Which value is closer to its distribution's mean (in standard-deviation units)?

Example 39

medium
A data value has a z-score of −1.5. If the distribution has μ=50 and σ=8, find the original value.

Example 40

medium
In a class, test scores have mean μ=65 and standard deviation σ=5. A student scores 55. Find the z-score and interpret it.
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Background Knowledge

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

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