How it works
A z-score (standard score) tells you how many standard deviations a value sits from the mean. Zero is exactly average, +2 is two standard deviations above, −1 is one below. Z-scores let you compare results from different tests, spot outliers, and turn any value into a percentile under the normal curve — no z-table needed.
z = (x − μ) ÷ σ P(X < x) = Φ(z) · P(X > x) = 1 − Φ(z)Worked example. A student scores 85 on a test with mean 70 and standard deviation 10. z = (85 − 70) ÷ 10 = 1.5. If scores are normally distributed, 93.32% of students scored lower and 6.68% higher — roughly the 93rd percentile. 86.64% of scores fall between z = −1.5 and +1.5.
Reading a z-score
| z | Share below | Meaning |
|---|---|---|
| −2 | 2.28% | Very low |
| −1 | 15.87% | Below average |
| 0 | 50% | Exactly average |
| +1 | 84.13% | Above average |
| +1.645 | 95% | One-tailed 95% cut-off |
| +1.96 | 97.5% | Two-tailed 95% confidence bound |
| +3 | 99.87% | Exceptional, or a possible outlier |
Comparing scores from different tests
You scored 80 in maths (mean 65, SD 10) and 85 in science (mean 80, SD 5). Which was better? Maths z = 1.5; science z = 1.0. Relative to classmates, the maths result is stronger, even though the raw score is lower. Standardised tests use the same idea to put results from different forms on one scale.
When the percentages apply
The z-score itself is valid for any data, but turning it into a probability assumes the data are roughly normal (bell-shaped). For strongly skewed data, like incomes or house prices, the percentages are only approximate. The calculator evaluates the normal CDF with the Abramowitz–Stegun approximation (error below 0.0000001), as accurate as a printed z-table.
Outliers
Values with |z| above 3 are commonly flagged as outliers; some analysts use 2.5. Check them before deleting — they may be data-entry errors or genuinely important cases.
Finding a value from a percentile
Rearranged, the formula gives x = μ + z × σ. To find the score at the 90th percentile (z ≈ 1.2816) with mean 70 and SD 10: 70 + 12.8 = 82.8.
Don’t have the standard deviation yet? Calculate it from your data with the standard deviation calculator. Planning a study? Use the sample size calculator; for probabilities from counting, the combination calculator.