A z-score tells you how many standard deviations a value is from the mean, calculated as z=σx−μ. 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.
Showing a random 20 of 50 problems.
Example 1
easy
Compute the z-score for x=15 when μ=10 and σ=5.
Example 2
easy
Compute the z-score for x=4 when μ=10 and σ=2.
Example 3
easy
If a z-score is exactly −2.5, how many standard deviations is the value from the mean, and on which side?z = −2.5 is 2.5 SDs below the mean (far left tail)
Example 4
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?x
Example 5
medium
If μ=60 and σ=5, what raw value corresponds to a z-score of +3?
Example 6
medium
A value has z-score 0.5, μ=40, σ=6. Find the raw value x.
Example 7
medium
Under the empirical (68-95-99.7) rule, approximately what percent of values in a normal distribution have ∣z∣≤2?
Example 8
challenge
For any data set with finite σ>0, prove that the mean of the z-scores is exactly 0.
Example 9
medium
Given z=−0.75, μ=200, and σ=40, find the raw value x.
Example 10
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 11
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?Top 2% begins at z ≈ 2.05; raw IQ cutoff = 100 + 2.05×15 ≈ 131
Example 12
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 13
medium
If z=2, μ=50, and σ=4, find the raw value x.
Example 14
easy
A value has z=−1. How many standard deviations from the mean is it, and in which direction?z
Example 15
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?
Example 16
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 17
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.
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
easy
A child's height has a z-score of 0 in their class. What does this tell you about the child's height relative to the class mean?
Example 20
medium
A standardized test has μ=500 and σ=100. A student scores 640. Find the z-score, and explain what it means.