z = (x − μ) / σ. A positive z is above the mean; negative is below.
Calculation details
Planning estimate only, not financial, tax, or legal advice. Verify assumptions and current rules before making decisions.
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Use this result
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What does this calculator estimate?
A test score of 88 with a mean of 80 and a standard deviation of 5 has a z-score of (88 - 80) / 5 = 1.6, meaning it sits 1.6 standard deviations above the average. Positive z-scores are above the mean; negative ones are below.
- A z-score of 1.6 corresponds to roughly the 94.5th percentile of a normal distribution.
- Z-scores have no units, which lets you compare values across different scales.
- The z-score formula subtracts the mean and divides by the standard deviation (NIST).
What z-scores are for
Z-scores put values from different distributions on a common scale: how far above or below the mean, in standard deviation units. They power percentile lookup, anomaly detection, and comparison across tests with different units.
Limitations to watch for
Z-scores assume the data is roughly normal for percentile interpretation — skewed data misleads. The sign matters: positive is above the mean, negative below. The tool computes the z; probability tables convert it to percentiles.
How to use it in practice
Use z-scores to compare performance across different tests or metrics (a 3200 SAT vs a 31 ACT). For outlier detection, |z| > 3 is commonly flagged. Convert to percentiles with a standard normal table.
['Enter the value.', 'Enter the mean and standard deviation.', 'Read the z-score and its direction.']
How this calculator works
Formula
Z-score = (x − μ) ÷ σ, where x is the value, μ the mean, and σ the standard deviation. A z-score states how many standard deviations a value is from the mean.
Worked example
A score of 85 with mean 70 and SD 10: z = (85 − 70) ÷ 10 = 1.5 — 1.5 standard deviations above the mean.
Assumptions to verify
- The distribution is approximately normal.
- Mean and standard deviation are known.
- The value is a member of that distribution.
Frequently asked questions
What is a z-score?
How many standard deviations a value is from the mean: (x − μ) ÷ σ. 85 with mean 70, SD 10 → z = 1.5.
How do I interpret it?
Z = 0 is average; positive is above, negative below. ±1 is common, ±2 is unusual, ±3 is very rare in a normal distribution.
What's the z-score for 95% coverage?
±1.96 — the range containing 95% of values in a normal distribution.
Can I use it for grades?
Yes — standardizing test scores lets you compare performance across different exams.
What if my data isn't normal?
Z-scores still describe distance from the mean, but percentile interpretation needs normality.
How do I find the percentile?
Look up the z in a standard normal table: z = 1.5 corresponds to about the 93rd percentile.
Is |z| > 3 an outlier?
Commonly treated as one — it falls outside 99.7% of a normal distribution.
Cite this tool
BoringToolsKit. “Z-Score Calculator.” boringtoolskit.com/z-score-calculator/ (reviewed 2026-08-25). Free to reference in articles, syllabi, and answer posts with a link.
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