Z-Score to Percentile Calculator
Turn a z-score into a percentile — the share of a normal distribution below it, with both tails and the two-sided probability.
The percentile is Φ(z), the normal cumulative distribution evaluated at z.
How the z-score to percentile calculator works
The percentile is Φ(z), the normal cumulative distribution evaluated at z. Because the distribution is symmetric, Φ(−z) = 1 − Φ(z), so the lower tail at −1 is the same as the upper tail at +1.
The familiar landmarks: z = 0 is the 50th percentile, z = 1 the 84.1st, z = 2 the 97.7th, and z = 1.96 the 97.5th — the last being where the 95% confidence interval comes from.
Formula: percentile = Φ(z)
Worked examples
| Inputs | Percentile | Note |
|---|---|---|
| z = 1.5 | 93.319277% | 93.3rd percentile |
| z = 0 | 50% | exactly the 50th |
| z = −2 | 2.275006% | 2.3rd percentile |
FAQFrequently asked questions
What percentile is a z-score of 1?
The 84.1st — about five in six of the distribution sits below it.
What is the 68-95-99.7 rule?
The share of a normal distribution within one, two and three standard deviations of the mean.
Can a z-score be negative?
Yes — it simply means below the mean, and the percentile falls below 50.
Does this need the data to be normal?
Yes, for the percentile. The z-score itself is meaningful for any distribution; the percentile conversion is not.
Why is z = 1.96 special?
It cuts off 2.5% in each tail, which is what makes ±1.96 the 95% interval.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.