# Normal Distribution Calculator

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

> The normal distribution calculator finds the probability that a normally distributed variable falls below, above or between values, given the mean and standard deviation.

## Inputs

- Mean (μ)
- Standard deviation (σ)
- Lower value
- Upper value

## Results

- P(a < X < b)
- P(X < a)
- P(X > b)

## Formula

```
z = (x − μ) ÷ σ
P(X < x) = Φ(z)
```

## Explanation

Many natural measurements, such as heights and test scores, are approximately normal.

### How to use the Normal Distribution Calculator

- **Mean (μ)**
- **Standard deviation (σ)**
- **Lower value**
- **Upper value**

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

### How it works

Convert each value to a z-score and use the standard normal cumulative distribution.

### Inputs explained

FieldWhat to enterMean (μ)A numberStandard deviation (σ)A numberLower valueA numberUpper valueA number

### What each result means

- **P(a < X < b)** = (ncdf(([Upper value] - [Mean (μ)])÷[Standard deviation (σ)]) - ncdf(([Lower value] - [Mean (μ)])÷[Standard deviation (σ)]))×100
- **P(X < a)** = ncdf(([Lower value] - [Mean (μ)])÷[Standard deviation (σ)])×100
- **P(X > b)** = (1 - ncdf(([Upper value] - [Mean (μ)])÷[Standard deviation (σ)]))×100

### Example values

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

Mean (μ)P(a < X < b) (%)500.98087524.866210068.268912524.86621500.98082007.2801e-7

## Worked example

Inputs — Mean (μ): 100; Standard deviation (σ): 15; Lower value: 85; Upper value: 115.

Results — **P(a < X < b) = 68.2689 %**; P(X < a) = 15.8655 %; P(X > b) = 15.8655 %.

## FAQ

**What is the 68-95-99.7 rule?**
About 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.

**What is an IQ of 130?**
About the 98th percentile, two SDs above the mean.

**What formula does the Normal Distribution Calculator use?**
z = (x − μ) ÷ σ.


## Sources

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