# Inverse Normal Distribution Calculator

URL: https://speedycalculator.com/calculators/statistics/inverse-normal/
Last updated: 2026-10-25

> The inverse normal calculator finds the value x below which a given percentage of a normal distribution lies, for a chosen mean and standard deviation.

## Inputs

- Mean (μ)
- Standard deviation (σ)
- Area to the left (%)

## Results

- z-score
- Value x

## Formula

```
x = μ + σ × Φ⁻¹(p)
```

## Explanation

Use it to set cut-offs, for example the score needed to be in the top 10%.

### How to use the Inverse Normal Distribution Calculator

- **Mean (μ)**
- **Standard deviation (σ)**
- **Area to the left**

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Find the z-score for the probability with the inverse normal function, then convert to the original scale.

### Inputs explained

FieldWhat to enterMean (μ)A numberStandard deviation (σ)A numberArea to the leftIn %

### What each result means

- **z-score** = ninv([Area to the left]÷100)
- **Value x** = [Mean (μ)] + [Standard deviation (σ)]×[z-score]

### Example values

How the main result changes when **Mean** changes and everything else stays at the example values above:

Mean (μ)Value x5069.22337594.2233100119.2233125144.2233150169.2233200219.2233

## Worked example

Inputs — Mean (μ): 100; Standard deviation (σ): 15; Area to the left: 90 %.

Results — z-score = 1.2816; **Value x = 119.2233**.

## FAQ

**What is z for the 95th percentile?**
About 1.645.

**How do I get a two-sided cut-off?**
Use (1 + confidence)/2, e.g. 97.5% for a 95% interval.

**What formula does the Inverse Normal Distribution Calculator use?**
x = μ + σ × Φ⁻¹(p).


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)