npm package: @stdlib/stats-base-variancewd
Description: Calculate the variance of a strided array using Welford's algorithm.
Author: The Stdlib Authors
Latest version: 0.2.2

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# variancewd

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

> Calculate the [variance][variance] of a strided 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."> -->

<div class="equation" align="center" data-raw-text="\sigma^2 = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2" data-equation="eq:population_variance">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@00dc7e1a49e6c42611726e55478a569d55020169/lib/node_modules/@stdlib/stats/base/variancewd/docs/img/equation_population_variance.svg" alt="Equation for the population variance.">
    <br>
</div> -->

<!-- </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."> -->

<!-- <div class="equation" align="center" data-raw-text="\mu = \frac{1}{N} \sum_{i=0}^{N-1} x_i" data-equation="eq:population_mean">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@00dc7e1a49e6c42611726e55478a569d55020169/lib/node_modules/@stdlib/stats/base/variancewd/docs/img/equation_population_mean.svg" alt="Equation for the population mean.">
    <br>
</div> -->

<!-- </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."> -->

<!-- <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@00dc7e1a49e6c42611726e55478a569d55020169/lib/node_modules/@stdlib/stats/base/variancewd/docs/img/equation_unbiased_sample_variance.svg" alt="Equation for computing an unbiased sample variance.">
    <br>
</div> -->

<!-- </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."> -->

<!-- <div class="equation" align="center" data-raw-text="\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i" data-equation="eq:sample_mean">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@00dc7e1a49e6c42611726e55478a569d55020169/lib/node_modules/@stdlib/stats/base/variancewd/docs/img/equation_sample_mean.svg" alt="Equation for the sample mean.">
    <br>
</div>

<!-- </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-base-variancewd
```

</section>

<section class="usage">

## Usage

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

#### variancewd( N, correction, x, stride )

Computes the [variance][variance] of a strided array `x` using Welford's algorithm.

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

var v = variancewd( x.length, 1, x, 1 );
// returns ~4.3333
```

The function has the following parameters:

-   **N**: number of indexed elements.
-   **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 `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).
-   **x**: input [`Array`][mdn-array] or [`typed array`][mdn-typed-array].
-   **stride**: index increment for `x`.

The `N` and `stride` parameters determine which elements in `x` are accessed at runtime. For example, to compute the [variance][variance] of every other element in `x`,

```javascript
var floor = require( '@stdlib/math-base-special-floor' );

var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ];
var N = floor( x.length / 2 );

var v = variancewd( N, 1, x, 2 );
// returns 6.25
```

Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.

<!-- eslint-disable stdlib/capitalized-comments -->

```javascript
var Float64Array = require( '@stdlib/array-float64' );
var floor = require( '@stdlib/math-base-special-floor' );

var x0 = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

var N = floor( x0.length / 2 );

var v = variancewd( N, 1, x1, 2 );
// returns 6.25
```

#### variancewd.ndarray( N, correction, x, stride, offset )

Computes the [variance][variance] of a strided array using Welford's algorithm and alternative indexing semantics.

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

var v = variancewd.ndarray( x.length, 1, x, 1, 0 );
// returns ~4.33333
```

The function has the following additional parameters:

-   **offset**: starting index for `x`.

While [`typed array`][mdn-typed-array] views mandate a view offset based on the underlying `buffer`, the `offset` parameter supports indexing semantics based on a starting index. For example, to calculate the [variance][variance] for every other value in `x` starting from the second value

```javascript
var floor = require( '@stdlib/math-base-special-floor' );

var x = [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ];
var N = floor( x.length / 2 );

var v = variancewd.ndarray( N, 1, x, 2, 1 );
// returns 6.25
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   If `N <= 0`, both functions return `NaN`.
-   If `N - c` is less than or equal to `0` (where `c` corresponds to the provided degrees of freedom adjustment), both functions return `NaN`.
-   Depending on the environment, the typed versions ([`dvariancewd`][@stdlib/stats/base/dvariancewd], [`svariancewd`][@stdlib/stats/base/svariancewd], etc.) are likely to be significantly more performant.

</section>

<!-- /.notes -->

<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 Float64Array = require( '@stdlib/array-float64' );
var variancewd = require( '@stdlib/stats-base-variancewd' );

var x;
var i;

x = new Float64Array( 10 );
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = round( (randu()*100.0) - 50.0 );
}
console.log( x );

var v = variancewd( x.length, 1, x, 1 );
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">

* * *

## See Also

-   <span class="package-name">[`@stdlib/stats-base/dvariancewd`][@stdlib/stats/base/dvariancewd]</span><span class="delimiter">: </span><span class="description">calculate the variance of a double-precision floating-point strided array using Welford's algorithm.</span>
-   <span class="package-name">[`@stdlib/stats-base/nanvariancewd`][@stdlib/stats/base/nanvariancewd]</span><span class="delimiter">: </span><span class="description">calculate the variance of a strided array ignoring NaN values and using Welford's algorithm.</span>
-   <span class="package-name">[`@stdlib/stats-base/stdevwd`][@stdlib/stats/base/stdevwd]</span><span class="delimiter">: </span><span class="description">calculate the standard deviation of a strided array using Welford's algorithm.</span>
-   <span class="package-name">[`@stdlib/stats-base/variance`][@stdlib/stats/base/variance]</span><span class="delimiter">: </span><span class="description">calculate the variance of a strided array.</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; 2016-2024. The Stdlib [Authors][stdlib-authors].

</section>

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[branches-url]: https://github.com/stdlib-js/stats-base-variancewd/blob/main/branches.md

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

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

[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array

[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray

[@stdlib/stats/base/svariancewd]: https://www.npmjs.com/package/@stdlib/stats-base-svariancewd

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

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

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[@stdlib/stats/base/stdevwd]: https://www.npmjs.com/package/@stdlib/stats-base-stdevwd

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

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</section>

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