Quick answer: A score of 85 when the mean is 70 and the standard deviation is 10 has z = (85 − 70) ÷ 10 = 1.5, which is the 93.32nd percentile of a normal distribution.
How it's calculated
z = (x − μ) ÷ σ. A positive z is above the mean, a negative z below it; z = 2 means two standard deviations above. Turning it around, x = μ + z·σ.
The percentile is the area under the standard normal curve to the left of z, Φ(z). It is computed here from the error function, Φ(z) = ½·(1 + erf(z/√2)), to about 15 significant digits, so you do not need a z-table.
The tail areas are what hypothesis tests use: the two-tailed value is the p-value of a z-test. If you have raw data instead of μ and σ, find them first with the standard deviation calculator.
Worked examples
- x = 85, μ = 70, σ = 10: z = 1.5, percentile 93.32%
- x = 1.96σ above the mean: P(Z < 1.96) = 97.5%, two-tailed 5%
- z = −1, μ = 100, σ = 15: x = 85, percentile 15.87%
Common z-values
| z | P(Z < z) | Two-tailed |
|---|---|---|
| 1 | 84.13% | 31.73% |
| 1.645 | 95% | 10% |
| 1.96 | 97.5% | 5% |
| 2 | 97.72% | 4.55% |
| 2.576 | 99.5% | 1% |
| 3 | 99.87% | 0.27% |
Frequently asked questions
What is a good z-score?
A z-score isn't good or bad on its own. In a normal distribution about 68% of values have z between −1 and 1, and 95% between −1.96 and 1.96; values beyond ±3 are rare (0.27%).
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean.
Should I use the sample or population standard deviation?
Use the population σ when you know it. With only a sample, the sample standard deviation s is the usual stand-in.
What's the z-score of 85 if the mean is 70 and the SD is 10?
1.5. That puts 85 one and a half standard deviations above average, higher than about 93% of scores.
What z-score marks the top 5%?
1.645. For the top 2.5%, it's 1.96.
Can I compare marks from two different tests?
Yes, that's what z-scores are for. 80 on a test that averages 70 with an SD of 5 is z = 2, which beats 85 on a test averaging 80 with an SD of 10 (z = 0.5).