Math · Introductory statistics · Worked example
Compare scores on different scales with z-scores
One student scores 82 on a test with mean 70 and standard deviation 8. Another scores 27 on a test with mean 21 and standard deviation 5. Which score is more unusual?
Standardize each score
Compare
The first score is 1.5 standard deviations above its mean and the second 1.2, so the first is further above average for its test.
If both tests are normal
Φ(1.5) ≈ 0.933 and Φ(1.2) ≈ 0.885: the first score beats about 93% of its group, the second about 89% of its group.
Result
The 82 is more unusual: z = 1.5 against z = 1.2.
Your turn
A value has z = −2 on a scale with mean 50 and standard deviation 4. What is the value?
Show the answer and explanation
42.
x = μ + zσ = 50 + (−2)(4) = 42.
Keep exploring
In Statistics, Distributions with the left tail gives P(X ≤ 1.5) ≈ 0.9332 and P(X ≤ 1.2) ≈ 0.8849.
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