Inverse Normal Distribution Calculator
Go backwards from a probability to a value — the quantile function, which answers "what score puts me in the top 5%?".
The normal CDF Φ turns a value into a probability; the inverse turns a probability back into a value.
How the inverse normal distribution calculator works
The normal CDF Φ turns a value into a probability; the inverse turns a probability back into a value. There is no closed form, so it is found here by bisection on Φ to full double precision.
Given a mean and standard deviation, the answer is μ + zσ where z is the standard normal quantile — which is how percentile cut-offs, control limits and value-at-risk are all computed.
Formula: x = μ + σ Φ⁻¹(p)
Worked examples
| Inputs | Value at that percentile | Note |
|---|---|---|
| 95th percentile of IQ | 124.672802 | 124.7 — z = 1.645 |
| The median | 100 | z = 0, value = the mean |
| The top 1% | 134.895205 | 134.9 |
FAQFrequently asked questions
What is the inverse normal used for?
Turning a percentile into a score: exam cut-offs, control chart limits, value-at-risk, sample-size formulas.
What is the z for the 95th percentile?
1.645. For the 97.5th it is 1.960, which is where the familiar ±1.96 of a 95% interval comes from.
Why is there no formula?
The normal CDF has no elementary antiderivative, so its inverse cannot be written in closed form. Everything computes it numerically.
What is the difference between one- and two-sided?
The 95th percentile leaves 5% above; a two-sided 95% interval leaves 2.5% in each tail and runs from the 2.5th to the 97.5th percentile.
Is this the same as the probit function?
Yes — probit is another name for the standard normal quantile function.
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.