Turn a value into a z-score and percentile, find the value at any percentile, or find the probability between two values on a normal curve.
How a z-score is calculated
A z-score says how many standard deviations a value is from the mean. Positive is above the mean, negative is below.
z = (x − μ) ÷ σ x = μ + z × σ
The percentile is the area under the standard normal curve to the left of z, written Φ(z). This calculator works out Φ(z) from the error function with a series and a continued fraction, accurate to about 15 significant digits, so you don't need a z-table. It finds a value from a percentile by inverting Φ the same way.
Worked example
Test scores have a mean of 70 and a standard deviation of 10. A score of 85 gives z = (85 − 70) ÷ 10 = 1.5. The area to the left of z = 1.5 is 0.9332, so 85 is at about the 93rd percentile. About 6.68% of scores are higher.
The chance of a score between 60 and 80 is the area between z = −1 and z = 1: 0.8413 − 0.1587 = 0.6827, the "68%" in the 68–95–99.7 rule.
Common z-scores
| Percentile | z | Used for |
|---|
| 90th | 1.2816 | 80% two-sided interval |
| 95th | 1.6449 | 90% two-sided interval, 5% one-sided test |
| 97.5th | 1.9600 | 95% two-sided interval |
| 99.5th | 2.5758 | 99% two-sided interval |
When it doesn't apply
- Percentiles and probabilities assume the data follows a normal (bell-shaped) distribution. For skewed data, like incomes or wait times, they can be far off. The z-score itself is still a valid measure of distance from the mean.
- If you estimated the mean and standard deviation from a small sample, use the t distribution instead; the confidence interval calculator does this.
Sources
How we checked this
Before every update, this calculator is run in a real browser with these known answers, and it must match all of them:
- 99.99999999th percentile: z = 6.361340902 (scipy isf(1e-10)) ✓
- x = 85, μ = 70, σ = 10 gives z = 1.5 ✓
- z = 1.5 is the 93.32nd percentile ✓
- 97.5th percentile of the standard normal = 1.959963985 ✓
- P(−1.96 < Z < 1.96) = 0.95000421 ✓
- μ = 100, σ = 15: P(X > 130) = 0.022750132 ✓
- A standard deviation of 0 is rejected ✓
Last checked October 26, 2026. Spotted something wrong? The math is done in your browser with the formula shown on this page, so you can check any result by hand.
Questions people ask
What does a z-score of 2 mean?
The value is 2 standard deviations above the mean. In a normal distribution, that's higher than about 97.7% of values.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. z = −1 is lower than about 84% of values, so it's around the 16th percentile.
How do I convert a z-score to a percentile?
Find the area to the left of z under the standard normal curve and multiply by 100. Choose Find the z-score above and the percentile is shown with it.
What z-score is the 95th percentile?
z ≈ 1.6449. The 97.5th percentile, z ≈ 1.96, is the one used for 95% confidence intervals because it leaves 2.5% in each tail.
What is the two-tailed p-value?
The chance of a result at least as far from the mean as yours, in either direction: 2 × the area beyond |z|. It's what a two-sided z-test reports.