npm package: @stdlib/stats-array-variancewd
Description: Calculate the variance of an array using Welford's algorithm.
Author: The Stdlib Authors
Latest version: 0.1.1

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# variancewd

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> Calculate the [variance][variance] of an array using Welford's algorithm.

<section class="intro">

The population [variance][variance] of a finite size population of size `N` is given by

<!-- <equation class="equation" label="eq:population_variance" align="center" raw="\sigma^2 = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2" alt="Equation for the population variance."> -->

<!-- </equation> -->

where the population mean is given by

<!-- <equation class="equation" label="eq:population_mean" align="center" raw="\mu = \frac{1}{N} \sum_{i=0}^{N-1} x_i" alt="Equation for the population mean."> -->

<!-- </equation> -->

Often in the analysis of data, the true population [variance][variance] is not known _a priori_ and must be estimated from a sample drawn from the population distribution. If one attempts to use the formula for the population [variance][variance], the result is biased and yields a **biased sample variance**. To compute an **unbiased sample variance** for a sample of size `n`,

<!-- <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 computing an unbiased sample variance."> -->

<!-- </equation> -->

where the sample mean is given by

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

<!-- </equation> -->

The use of the term `n-1` is commonly referred to as Bessel's correction. Note, however, that applying Bessel's correction can increase the mean squared error between the sample variance and population variance. Depending on the characteristics of the population distribution, other correction factors (e.g., `n-1.5`, `n+1`, etc) can yield better estimators.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-array-variancewd
```

</section>

<section class="usage">

## Usage

```javascript
var variancewd = require( '@stdlib/stats-array-variancewd' );
```

#### variancewd( x\[, correction] )

Computes the variance of an array.

```javascript
var x = [ 1.0, -2.0, 2.0 ];

var v = variancewd( x );
// returns ~4.3333
```

The function has the following parameters:

-   **x**: input array.
-   **correction**: degrees of freedom adjustment. Setting this parameter to a value other than `0` has the effect of adjusting the divisor during the calculation of the [variance][variance] according to `N-c` where  `N` corresponds to the number of array elements and `c` corresponds to the provided degrees of freedom adjustment. When computing the [variance][variance] of a population, setting this parameter to `0` is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the unbiased sample [variance][variance], setting this parameter to `1` is the standard choice (i.e., the provided array contains data sampled from a larger population; this is commonly referred to as Bessel's correction). Default: `1.0`.

By default, the function computes the sample [variance][variance]. To adjust the degrees of freedom when computing the [variance][variance], provide a `correction` argument.

```javascript
var x = [ 1.0, -2.0, 2.0 ];

var v = variancewd( x, 0.0 );
// returns ~2.8889
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   If provided an empty array, the function returns `NaN`.
-   If provided a `correction` argument which is greater than or equal to the number of elements in a provided input array, the function returns `NaN`.
-   The function supports array-like objects having getter and setter accessors for array element access (e.g., [`@stdlib/array-base/accessor`][@stdlib/array/base/accessor]).

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```javascript
var discreteUniform = require( '@stdlib/random-array-discrete-uniform' );
var variancewd = require( '@stdlib/stats-array-variancewd' );

var x = discreteUniform( 10, -50, 50, {
    'dtype': 'float64'
});
console.log( x );

var v = variancewd( x );
console.log( v );
```

</section>

<!-- /.examples -->

* * *

<section class="references">

## References

-   Welford, B. P. 1962. "Note on a Method for Calculating Corrected Sums of Squares and Products." _Technometrics_ 4 (3). Taylor & Francis: 419–20. doi:[10.1080/00401706.1962.10490022][@welford:1962a].
-   van Reeken, A. J. 1968. "Letters to the Editor: Dealing with Neely's Algorithms." _Communications of the ACM_ 11 (3): 149–50. doi:[10.1145/362929.362961][@vanreeken:1968a].

</section>

<!-- /.references -->

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

<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]: https://github.com/stdlib-js/stdlib

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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-array-variancewd/tree/deno
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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-array-variancewd/main/LICENSE

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

[@welford:1962a]: https://doi.org/10.1080/00401706.1962.10490022

[@vanreeken:1968a]: https://doi.org/10.1145/362929.362961

[@stdlib/array/base/accessor]: https://www.npmjs.com/package/@stdlib/array-base-accessor

</section>

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