npm package: @stdlib/stats-incr-vmr
Description: Compute a variance-to-mean ratio (VMR) incrementally.
Author: The Stdlib Authors
Latest version: 0.2.3

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# incrvmr

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

> Compute a [variance-to-mean ratio][variance-to-mean-ratio] (VMR) incrementally.

<section class="intro">

The [unbiased sample variance][sample-variance] is defined as

<!-- <equation class="equation" label="eq:unbiased_sample_variance" align="center" raw="s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2" alt="Equation for the unbiased sample variance."> -->

<div class="equation" align="center" data-raw-text="s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2" data-equation="eq:unbiased_sample_variance">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7fe559e94716008fb414ec7c6b3d0e3e1194f2ba/lib/node_modules/@stdlib/stats/incr/vmr/docs/img/equation_unbiased_sample_variance.svg" alt="Equation for the unbiased sample variance.">
    <br>
</div> -->

<!-- </equation> -->

and the [arithmetic mean][arithmetic-mean] is defined as

<!-- <equation class="equation" label="eq:arithmetic_mean" align="center" raw="\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i" alt="Equation for the arithmetic mean."> -->

<!-- <div class="equation" align="center" data-raw-text="\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i" data-equation="eq:arithmetic_mean">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@86f8c49b0e95ee794f0b098b8d17444c0cbeea0a/lib/node_modules/@stdlib/stats/incr/vmr/docs/img/equation_arithmetic_mean.svg" alt="Equation for the arithmetic mean.">
    <br>
</div> -->

<!-- </equation> -->

The [variance-to-mean ratio][variance-to-mean-ratio] (VMR) is thus defined as

<!-- <equation class="equation" label="eq:variance_to_mean_ratio" align="center" raw="D = \frac{s^2}{\bar{x}}" alt="Equation for the variance-to-mean ratio (VMR)."> -->

<!-- <div class="equation" align="center" data-raw-text="D = \frac{s^2}{\bar{x}}" data-equation="eq:variance_to_mean_ratio">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@86f8c49b0e95ee794f0b098b8d17444c0cbeea0a/lib/node_modules/@stdlib/stats/incr/vmr/docs/img/equation_variance_to_mean_ratio.svg" alt="Equation for the variance-to-mean ratio (VMR).">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-incr-vmr
```

</section>

<section class="usage">

## Usage

```javascript
var incrvmr = require( '@stdlib/stats-incr-vmr' );
```

#### incrvmr( \[mean] )

Returns an accumulator `function` which incrementally computes a [variance-to-mean ratio][variance-to-mean-ratio].

```javascript
var accumulator = incrvmr();
```

If the mean is already known, provide a `mean` argument.

```javascript
var accumulator = incrvmr( 3.0 );
```

#### accumulator( \[x] )

If provided an input value `x`, the accumulator function returns an updated accumulated value. If not provided an input value `x`, the accumulator function returns the current accumulated value.

```javascript
var accumulator = incrvmr();

var D = accumulator( 2.0 );
// returns 0.0

D = accumulator( 1.0 ); // => s^2 = ((2-1.5)^2+(1-1.5)^2) / (2-1)
// returns ~0.33

D = accumulator( 3.0 ); // => s^2 = ((2-2)^2+(1-2)^2+(3-2)^2) / (3-1)
// returns 0.5

D = accumulator();
// returns 0.5
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   Input values are **not** type checked. If provided `NaN` or a value which, when used in computations, results in `NaN`, the accumulated value is `NaN` for **all** future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly **before** passing the value to the accumulator function.

-   The following table summarizes how to interpret the [variance-to-mean ratio][variance-to-mean-ratio]:

    |        VMR        |   Description   |     Example Distribution     |
    | :---------------: | :-------------: | :--------------------------: |
    |         0         |  not dispersed  |           constant           |
    | 0 &lt; VMR &lt; 1 | under-dispersed |           binomial           |
    |         1         |        --       |            Poisson           |
    |         >1        |  over-dispersed | geometric, negative-binomial |

    Accordingly, one can use the [variance-to-mean ratio][variance-to-mean-ratio] to assess whether observed data can be modeled as a Poisson process. When observed data is "under-dispersed", observed data may be more regular than as would be the case for a Poisson process. When observed data is "over-dispersed", observed data may contain clusters (i.e., clumped, concentrated data).

-   The [variance-to-mean ratio][variance-to-mean-ratio] is typically computed on nonnegative values. The measure may lack meaning for data which can assume both positive and negative values.

-   The [variance-to-mean ratio][variance-to-mean-ratio] is also known as the **index of dispersion**, **dispersion index**, **coefficient of dispersion**, and **relative variance**.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random-base-randu' );
var incrvmr = require( '@stdlib/stats-incr-vmr' );

var accumulator;
var v;
var i;

// Initialize an accumulator:
accumulator = incrvmr();

// For each simulated datum, update the variance-to-mean ratio...
for ( i = 0; i < 100; i++ ) {
    v = randu() * 100.0;
    accumulator( v );
}
console.log( accumulator() );
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/stats-incr/mean`][@stdlib/stats/incr/mean]</span><span class="delimiter">: </span><span class="description">compute an arithmetic mean incrementally.</span>
-   <span class="package-name">[`@stdlib/stats-incr/mvmr`][@stdlib/stats/incr/mvmr]</span><span class="delimiter">: </span><span class="description">compute a moving variance-to-mean ratio (VMR) incrementally.</span>
-   <span class="package-name">[`@stdlib/stats-incr/variance`][@stdlib/stats/incr/variance]</span><span class="delimiter">: </span><span class="description">compute an unbiased sample variance incrementally.</span>

</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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[umd]: https://github.com/umdjs/umd
[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules

[deno-url]: https://github.com/stdlib-js/stats-incr-vmr/tree/deno
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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-incr-vmr/main/LICENSE

[variance-to-mean-ratio]: https://en.wikipedia.org/wiki/Index_of_dispersion

[arithmetic-mean]: https://en.wikipedia.org/wiki/Arithmetic_mean

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

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[@stdlib/stats/incr/mean]: https://www.npmjs.com/package/@stdlib/stats-incr-mean

[@stdlib/stats/incr/mvmr]: https://www.npmjs.com/package/@stdlib/stats-incr-mvmr

[@stdlib/stats/incr/variance]: https://www.npmjs.com/package/@stdlib/stats-incr-variance

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