npm package: @stdlib/stats-base-dists-normal-cdf
Description: Normal distribution cumulative distribution function (CDF).
Author: The Stdlib Authors
Latest version: 0.3.1

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# Cumulative Distribution Function

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> [Normal][normal-distribution] distribution [cumulative distribution function][cdf].

<section class="intro">

The [cumulative distribution function][cdf] for a [normal][normal-distribution] random variable is

<!-- <equation class="equation" label="eq:normal_cdf" align="center" raw="F(x;\mu,\sigma) = \frac{1}{2} \left[ 1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right]" alt="Cumulative distribution function for a normal distribution."> -->

<div class="equation" align="center" data-raw-text="F(x;\mu,\sigma) = \frac{1}{2} \left[ 1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right]" data-equation="eq:normal_cdf">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/normal/cdf/docs/img/equation_normal_cdf.svg" alt="Cumulative distribution function for a normal distribution.">
    <br>
</div>

<!-- </equation> -->

where `µ` is the mean and `σ` is the standard deviation.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-normal-cdf
```

</section>

<section class="usage">

## Usage

```javascript
var cdf = require( '@stdlib/stats-base-dists-normal-cdf' );
```

#### cdf( x, mu, sigma )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [normal][normal-distribution] distribution with parameters `mu` (mean) and `sigma` (standard deviation).

```javascript
var y = cdf( 2.0, 0.0, 1.0 );
// returns ~0.977

y = cdf( 0.0, 0.0, 1.0 );
// returns 0.5

y = cdf( -1.0, 4.0, 2.0 );
// returns ~0.006
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var y = cdf( NaN, 0.0, 1.0 );
// returns NaN

y = cdf( 0.0, NaN, 1.0 );
// returns NaN

y = cdf( 0.0, 0.0, NaN );
// returns NaN
```

If provided `sigma < 0`, the function returns `NaN`.

```javascript
var y = cdf( 2.0, 0.0, -1.0 );
// returns NaN
```

If provided `sigma = 0`, the function evaluates the [CDF][cdf] of a [degenerate distribution][degenerate-distribution] centered at `mu`.

```javascript
var y = cdf( 2.0, 8.0, 0.0 );
// returns 0.0

y = cdf( 8.0, 8.0, 0.0 );
// returns 1.0

y = cdf( 10.0, 8.0, 0.0 );
// returns 1.0
```

#### cdf.factory( mu, sigma )

Returns a function for evaluating the [cumulative distribution function][cdf] of a normal distribution with parameters `mu` and `sigma`.

```javascript
var mycdf = cdf.factory( 10.0, 2.0 );

var y = mycdf( 10.0 );
// returns 0.5

y = mycdf( 8.0 );
// returns ~0.159
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var cdf = require( '@stdlib/stats-base-dists-normal-cdf' );

var opts = {
    'dtype': 'float64'
};
var sigma = uniform( 10, 0.0, 20.0, opts );
var mu = uniform( 10, -5.0, 5.0, opts );
var x = uniform( 10, 0.0, 10.0, opts );

logEachMap( 'x: %0.4f, µ: %0.4f, σ: %0.4f, F(x;µ,σ): %0.4f', x, mu, sigma, cdf );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/normal/cdf.h"
```

#### stdlib_base_dists_normal_cdf( x, a, b )

Evaluates the [cumulative distribution function][cdf] (CDF) for a [normal][normal-distribution] distribution with parameters `mu` (mean) and `sigma` (standard deviation).

```c
double out = stdlib_base_dists_normal_cdf( 2.0, 0.0, 1.0 );
// returns ~0.977
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.
-   **mu**: `[in] double` mean.
-   **sigma**: `[in] double` standard deviation.

```c
double stdlib_base_dists_normal_cdf( cconst double x, const double mu, const double sigma );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/stats/base/dists/normal/cdf.h"
#include <stdlib.h>
#include <stdio.h>

static double random_uniform( const double min, const double max ) {
    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
    return min + ( v*(max-min) );
}

int main( void ) {
    double sigma;
    double mu;
    double x;
    double y;
    int i;

    for ( i = 0; i < 10; i++ ) {
        x = random_uniform( 0.0, 10.0 );
        mu = random_uniform( -5.0, 5.0 );
        sigma = random_uniform( 0.1, 20.0 );
        y = stdlib_base_dists_normal_cdf( x, mu, sigma );
        printf( "x: %lf, µ: %lf, σ: %lf, F(x;µ,σ): %lf\n", x, mu, sigma, y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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<section class="related">

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-base-dists-normal-cdf/main/LICENSE

[cdf]: https://en.wikipedia.org/wiki/Cumulative_distribution_function

[normal-distribution]: https://en.wikipedia.org/wiki/Normal_distribution

[degenerate-distribution]: https://en.wikipedia.org/wiki/Degenerate_distribution

</section>

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