Key takeaway: A z-score counts the standard deviations separating a data point from its average, so results measured on unrelated scales can be compared fairly.
Grade 11 – Algebra 2S-ID.A.4Statistics and Data
Definition of Z-Score
A z-score converts a raw measurement into a count of standard deviations from the mean. Positive means above average, negative means below, and zero means right on the average.
Raw numbers from different tests are hard to compare because the scales differ. A 41 on one quiz and a 640 on another say little by themselves. After conversion, both live on one common scale where 0 is typical and each unit is one spread. A z-score beyond 2 or below -2 marks a value that is unusual for bell-shaped data.
Formula and notation
Take a measurement \(x\), subtract the average \(\mu\), then divide by the spread \(\sigma\):
\[z=\frac{x-\mu}{\sigma}.\]
To go backward, \(x=\mu+z\sigma\). For a sample, the mean \(\,\overline{x}\) and the sample standard deviation take their places.
Key facts
- The sign shows direction: positive above the mean, negative below it.
- Standardized values always have mean 0 and standard deviation 1.
- A z-score has no units; the units cancel in the division.
- Larger absolute values mean the data point is rarer.
Z-Score example
Maya scored 640 on a test with mean 520 and standard deviation 80. Leo scored 41 on a different test with mean 33 and standard deviation 5. Who did better relative to their own test?
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Maya: \(z=\frac{640-520}{80}=\frac{120}{80}=1.5\).
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Leo: \(z=\frac{41-33}{5}=\frac{8}{5}=1.6\).
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Compare: \(1.6>1.5\).
Leo did slightly better relative to his group: z = 1.6 versus 1.5.
Common mistake
Watch out
Priya computes Maya’s value as \(\frac{520-640}{80}=-1.5\) and says Maya scored below average.
Why it is wrong
The order of subtraction matters: the mean is subtracted from the value, not the reverse. The correct result is \(\frac{640-520}{80}=+1.5\). A score above the mean must have a positive z-score.
Related terms and lessons
- Algebra 2 · Data, Normal Distributions and Statistical InferenceLessonWorksheetQuiz
- AP Statistics · Binomial and Normal DistributionsLessonWorksheetQuiz
Browse the glossary: A–Z list · Algebra 2 vocabulary · Statistics and Data