Percentile to Z-Score Calculator
Turn a percentile into a z-score — the inverse normal, which tells you how many standard deviations a given rank sits from the mean.
This inverts the normal CDF numerically.
How the percentile to z-score calculator works
This inverts the normal CDF numerically. The 50th percentile is z = 0, the 84.1st is z = 1, and the 97.5th is z = 1.96.
The function grows without bound as the percentile approaches 0 or 100, which is why extreme percentiles are so sensitive: moving from the 99th to the 99.9th is a jump of 0.76 standard deviations, and 99.9th to 99.99th is another 0.63.
Formula: z = Φ⁻¹(p)
Worked examples
| Inputs | Z-score | Note |
|---|---|---|
| 90th percentile | 1.281552 | z = 1.282 |
| The median | 3.9273e-89 | z = 0 |
| The top 0.1% | 3.090253 | z = 3.09 |
FAQFrequently asked questions
What z-score is the 90th percentile?
1.282 — nine in ten fall below it.
Why can I not enter 0 or 100?
Because the normal distribution has no finite endpoints. The z-score would be infinite.
What is the top 1%?
z = 2.326.
Why does the top 0.01% need only z = 3.72?
Because the normal tail is very thin. Every extra tenth of a standard deviation removes most of what is left.
Does this apply to my exam score?
Only if the scores are roughly normal. Many are deliberately scaled to be, which is what makes the conversion routine.
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.