npm package: @stdlib/stats-base-dists-hypergeometric-variance
Description: Hypergeometric distribution variance.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Variance

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

> [Hypergeometric][hypergeometric-distribution] distribution [variance][variance].

<!-- 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">

Imagine a scenario with a population of size `N`, of which a subpopulation of size `K` can be considered successes. We draw `n` observations from the total population. Defining the random variable `X` as the number of successes in the `n` draws, `X` is said to follow a [hypergeometric distribution][hypergeometric-distribution]. The [variance][variance] for a [hypergeometric][hypergeometric-distribution] random variable is

<!-- <equation class="equation" label="eq:hypergeometric_variance" align="center" raw="\operatorname{Var}\left( X \right) = n{K \over N}{(N-K) \over N}{N-n \over N-1}" alt="Variance for a hypergeometric distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Var}\left( X \right) = n{K \over N}{(N-K) \over N}{N-n \over N-1}" data-equation="eq:hypergeometric_variance">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/hypergeometric/variance/docs/img/equation_hypergeometric_variance.svg" alt="Variance for a hypergeometric distribution.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<!-- Package usage documentation. -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-base-dists-hypergeometric-variance
```

</section>

<section class="usage">

## Usage

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

#### variance( N, K, n )

Returns the [variance][variance] of a [hypergeometric][hypergeometric-distribution] distribution with parameters `N` (population size), `K` (subpopulation size), and `n` (number of draws).

```javascript
var v = variance( 16, 11, 4 );
// returns ~0.688

v = variance( 2, 1, 1 );
// returns 0.25
```

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

```javascript
var v = variance( NaN, 10, 4 );
// returns NaN

v = variance( 20, NaN, 4 );
// returns NaN

v = variance( 20, 10, NaN );
// returns NaN
```

If provided a population size `N`, subpopulation size `K`, or draws `n` which is not a nonnegative integer, the function returns `NaN`.

```javascript
var v = variance( 10.5, 5, 2 );
// returns NaN

v = variance( 10, 1.5, 2 );
// returns NaN

v = variance( 10, 5, -2.0 );
// returns NaN
```

If the number of draws `n` or the subpopulation size `K` exceed population size `N`, the function returns `NaN`.

```javascript
var v = variance( 10, 5, 12 );
// returns NaN

v = variance( 10, 12, 5 );
// returns NaN
```

</section>

<!-- /.usage -->

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

<section class="notes">

</section>

<!-- /.notes -->

<!-- Package usage examples. -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random-base-randu' );
var round = require( '@stdlib/math-base-special-round' );
var variance = require( '@stdlib/stats-base-dists-hypergeometric-variance' );

var v;
var i;
var N;
var K;
var n;

for ( i = 0; i < 10; i++ ) {
    N = round( randu() * 20 );
    K = round( randu() * N );
    n = round( randu() * K );
    v = variance( N, K, n );
    console.log( 'N: %d, K: %d, n: %d, Var(X;N,K,n): %d', N, K, n, v.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

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<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/stats/base/dists/hypergeometric/variance.h"
```

#### stdlib_base_dists_hypergeometric_variance( N, K, n )

Returns the [variance][variance] of a [hypergeometric][hypergeometric-distribution] distribution with parameters `N` (population size), `K` (subpopulation size), and `n` (number of draws).

```c
double out = stdlib_base_dists_hypergeometric_variance( 16, 11, 4 );
// returns ~0.688
```

The function accepts the following arguments:

-   **N**: `[in] int32_t` population size.
-   **K**: `[in] int32_t` subpopulation size.
-   **n**: `[in] int32_t` number of draws.

```c
double stdlib_base_dists_hypergeometric_variance( const int32_t N, const int32_t K, const int32_t n );
```

</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/hypergeometric/variance.h"
#include "stdlib/math/base/special/ceil.h"
#include <stdlib.h>
#include <stdint.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 ) {
    int32_t N;
    int32_t K;
    int32_t n;
    double v;
    int i;

    for ( i = 0; i < 10; i++ ) {
        N = stdlib_base_ceil( random_uniform( 2.0, 100.0 ) );
        K = stdlib_base_ceil( random_uniform( 0.0, N ) );
        n = stdlib_base_ceil( random_uniform( 0.0, N ) );
        v = stdlib_base_dists_hypergeometric_variance( N, K, n );
        printf( "N: %d, K: %d, n: %d, Var(X;N,K,n): %lf\n", N, K, n, v );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

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</section>

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</section>

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* * *

## 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].

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## License

See [LICENSE][stdlib-license].


## Copyright

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

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[hypergeometric-distribution]: https://en.wikipedia.org/wiki/Hypergeometric_distribution

[variance]: https://en.wikipedia.org/wiki/Variance

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