What this calculator does
A standard z-score calculation goes from a value to a percentile: given a score, mean and standard deviation, it tells you where that score ranks. The inverse normal distribution calculation runs the other way. Given a target percentile or probability, such as the 90th percentile, it works out the value on the underlying normal distribution that sits exactly there.
This is the calculation behind statements like "what score puts someone in the top 10%" or "what cut-off value corresponds to the 95th percentile of this distribution". It needs the inverse of the standard normal cumulative distribution function, sometimes called the probit function, which has no simple closed-form formula and is evaluated here with a well-established numerical approximation, accurate to better than one part in a billion across the full range of probabilities.
The formula
Convert the target percentile to a probability between 0 and 1, find the z-score that produces that probability under the standard normal distribution (the inverse normal CDF), then convert that z-score back to the original scale using x = mean + z × standard deviation.
| Term | Meaning |
|---|---|
| Percentile / probability | The target proportion of the distribution that falls below the value being solved for, entered as a percentage. |
| Z-score | The number of standard deviations above or below the mean that corresponds to the target percentile on a standard normal distribution. |
| Inverse normal CDF (probit) | The function that converts a probability back into a z-score; the mathematical inverse of the standard normal cumulative distribution function Φ. |
The inputs explained
| Field | What to enter |
|---|---|
| Target percentile / probability (%) | The target percentile or probability, as a percentage between 0 and 100 (exclusive). |
| Mean (μ) | The mean of the normal distribution being used. |
| Standard deviation (σ) | The standard deviation of the normal distribution. Must be greater than zero. |
When to use it
Setting a cut-off score
A test designer who wants the top 5% of scorers to pass a threshold can use the 95th percentile to find the exact score that cut-off represents, given the distribution's mean and standard deviation.
Working out a specification limit
A manufacturing process with a known mean and standard deviation can use a target percentile to set an upper or lower specification limit that only a chosen small fraction of output is expected to fall outside.
Checking a common statistical constant
The well-known z-values of roughly 1.645 for the 95th percentile and 1.96 for the 97.5th percentile, familiar from one-tailed and two-tailed confidence intervals, both fall directly out of this calculation.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the z-score and value change with the target percentile
A fixed mean and standard deviation, across a range of target percentiles.
| Target percentile | Z-score | Value at this percentile |
|---|---|---|
| 50% | 0 | 100.000 |
| 75% | 0.6745 | 110.117 |
| 90% | 1.282 | 119.223 |
| 95% | 1.645 | 124.673 |
| 97.5% | 1.960 | 129.399 |
| 99% | 2.326 | 134.895 |
How the value at a fixed percentile changes with the standard deviation
A fixed mean and target percentile, across a range of standard deviations.
| Standard deviation | Z-score | Value at this percentile |
|---|---|---|
| 5 | 1.282 | 106.408 |
| 10 | 1.282 | 112.816 |
| 15 | 1.282 | 119.223 |
| 20 | 1.282 | 125.631 |
| 25 | 1.282 | 132.039 |
Questions
What is the inverse normal distribution calculation actually used for?
Anywhere a target proportion needs converting into an actual cut-off value: exam pass marks, manufacturing tolerance limits, statistical confidence interval multipliers, or working out what score a given percentile rank actually represents on a known distribution.
How accurate is the underlying approximation?
The rational approximation used here is accurate to within about 1.15 × 10⁻⁹ across the full range of probabilities, verified against known reference values such as z ≈ 1.645 at the 95th percentile and z ≈ 1.96 at the 97.5th percentile, which is more than precise enough for any practical statistical use.
What happens at exactly the 0th or 100th percentile?
Those are undefined: the standard normal distribution never quite reaches a probability of 0 or 1 at any finite z-score, so this calculator requires a percentile strictly between 0 and 100.
How does this differ from a regular z-score or percentile calculator?
A regular z-score calculator starts from a value and works out its percentile. This calculator runs the calculation backwards, starting from a target percentile and working out the value that produces it, which needs the inverse of the cumulative distribution function rather than the function itself.
To go the other direction, from a value to its percentile, use the z-score and percentile calculator. For the share of a normal distribution within a set number of standard deviations, see the empirical rule calculator.