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# Variance

> [Discrete uniform][discrete-uniform-distribution] distribution [variance][variance].

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<section class="intro">

The [variance][variance] for a [discrete uniform][discrete-uniform-distribution] random variable is

<!-- <equation class="equation" label="eq:discrete_uniform_variance" align="center" raw="\operatorname{Var}\left( X \right) = \frac{\left( b - a + 1 \right)^2 - 1}{12}" alt="Variance for a discrete uniform distribution."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Var}\left( X \right) = \frac{\left( b - a + 1 \right)^2 - 1}{12}" data-equation="eq:discrete_uniform_variance">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/7e0a95722efd9c771b129597380c63dc6715508b/lib/node_modules/@stdlib/stats/base/dists/discrete-uniform/variance/docs/img/equation_discrete_uniform_variance.svg" alt="Variance for a discrete uniform distribution.">
    <br>
</div>

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where `a` is the minimum support and `b` the maximum support of the distribution.

</section>

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<section class="usage">

## Usage

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

#### variance( a, b )

Returns the [variance][variance] of a [discrete uniform][discrete-uniform-distribution] distribution with parameters `a` (minimum support) and `b` (maximum support).

```javascript
var v = variance( 0, 1 );
// returns ~0.25

v = variance( 4, 12 );
// returns ~6.667

v = variance( -4, 4 );
// returns ~6.667
```

If `a` or `b` is not an integer value, the function returns `NaN`.

```javascript
var v = variance( 0.1, 2 );
// returns NaN

v = variance( 0, 2.2 );
// returns NaN

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

v = variance( 2, NaN );
// returns NaN
```

If provided `a > b`, the function returns `NaN`.

```javascript
var v = variance( 3, 2 );
// returns NaN

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

</section>

<!-- /.usage -->

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<section class="notes">

</section>

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<section class="examples">

## Examples

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

```javascript
var randint = require( '@stdlib/random/base/discrete-uniform' );
var variance = require( '@stdlib/stats/base/dists/discrete-uniform/variance' );

var randa = randint.factory( 0, 10 );
var randb = randint.factory();
var a;
var b;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    a = randa();
    b = randb( a, a+randa() );
    v = variance( a, b );
    console.log( 'a: %d, b: %d, Var(X;a,b): %d', a.toFixed( 4 ), b.toFixed( 4 ), v.toFixed( 4 ) );
}
```

</section>

<!-- /.examples -->

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<section class="references">

</section>

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<section class="links">

[discrete-uniform-distribution]: https://en.wikipedia.org/wiki/Discrete_uniform_distribution

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

</section>

<!-- /.links -->
