npm package: @stdlib/stats-base-dsvariancepn
Description: Calculate the variance of a single-precision floating-point strided array using a two-pass algorithm with extended accumulation and returning an extended precision result.
Author: The Stdlib Authors
Latest version: 0.2.2

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# dsvariancepn

[![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 single-precision floating-point strided array using a two-pass algorithm with extended accumulation and returning an extended precision result.

<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@9aa0053c951406de8e19e50168cb8e4b76af2548/lib/node_modules/@stdlib/stats/base/dsvariancepn/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@9aa0053c951406de8e19e50168cb8e4b76af2548/lib/node_modules/@stdlib/stats/base/dsvariancepn/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@9aa0053c951406de8e19e50168cb8e4b76af2548/lib/node_modules/@stdlib/stats/base/dsvariancepn/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@9aa0053c951406de8e19e50168cb8e4b76af2548/lib/node_modules/@stdlib/stats/base/dsvariancepn/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-dsvariancepn
```

</section>

<section class="usage">

## Usage

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

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

Computes the [variance][variance] of a single-precision floating-point strided array `x` using a two-pass algorithm with extended accumulation and returning an extended precision result.

```javascript
var Float32Array = require( '@stdlib/array-float32' );

var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;

var v = dsvariancepn( N, 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 [`Float32Array`][@stdlib/array/float32].
-   **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 Float32Array = require( '@stdlib/array-float32' );
var floor = require( '@stdlib/math-base-special-floor' );

var x = new Float32Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
var N = floor( x.length / 2 );

var v = dsvariancepn( 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 Float32Array = require( '@stdlib/array-float32' );
var floor = require( '@stdlib/math-base-special-floor' );

var x0 = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var x1 = new Float32Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element

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

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

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

Computes the [variance][variance] of a single-precision floating-point strided array using a two-pass algorithm with extended accumulation and alternative indexing semantics and returning an extended precision result.

```javascript
var Float32Array = require( '@stdlib/array-float32' );

var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;

var v = dsvariancepn.ndarray( N, 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 Float32Array = require( '@stdlib/array-float32' );
var floor = require( '@stdlib/math-base-special-floor' );

var x = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var N = floor( x.length / 2 );

var v = dsvariancepn.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`.
-   Accumulated intermediate values are stored as double-precision floating-point numbers. 

</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 Float32Array = require( '@stdlib/array-float32' );
var dsvariancepn = require( '@stdlib/stats-base-dsvariancepn' );

var x;
var i;

x = new Float32Array( 10 );
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = round( (randu()*100.0) - 50.0 );
}
console.log( x );

var v = dsvariancepn( x.length, 1, x, 1 );
console.log( v );
```

</section>

<!-- /.examples -->

* * *

<section class="references">

## References

-   Neely, Peter M. 1966. "Comparison of Several Algorithms for Computation of Means, Standard Deviations and Correlation Coefficients." _Communications of the ACM_ 9 (7). Association for Computing Machinery: 496–99. doi:[10.1145/365719.365958][@neely:1966a].
-   Schubert, Erich, and Michael Gertz. 2018. "Numerically Stable Parallel Computation of (Co-)Variance." In _Proceedings of the 30th International Conference on Scientific and Statistical Database Management_. New York, NY, USA: Association for Computing Machinery. doi:[10.1145/3221269.3223036][@schubert:2018a].

</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/dvariancepn`][@stdlib/stats/base/dvariancepn]</span><span class="delimiter">: </span><span class="description">calculate the variance of a double-precision floating-point strided array using a two-pass algorithm.</span>
-   <span class="package-name">[`@stdlib/stats-base/dsvariance`][@stdlib/stats/base/dsvariance]</span><span class="delimiter">: </span><span class="description">calculate the variance of a single-precision floating-point strided array using extended accumulation and returning an extended precision result.</span>
-   <span class="package-name">[`@stdlib/stats-base/variancepn`][@stdlib/stats/base/variancepn]</span><span class="delimiter">: </span><span class="description">calculate the variance of a strided array using a two-pass algorithm.</span>
-   <span class="package-name">[`@stdlib/stats-base/svariancepn`][@stdlib/stats/base/svariancepn]</span><span class="delimiter">: </span><span class="description">calculate the variance of a single-precision floating-point strided array using a two-pass algorithm.</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

[![Chat][chat-image]][chat-url]

---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2024. The Stdlib [Authors][stdlib-authors].

</section>

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

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

[@stdlib/array/float32]: https://www.npmjs.com/package/@stdlib/array-float32

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

[@neely:1966a]: https://doi.org/10.1145/365719.365958

[@schubert:2018a]: https://doi.org/10.1145/3221269.3223036

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

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

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