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https://github.com/stdlib-js/stats-base-dvarianceyc

Calculate the variance of a double-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.
https://github.com/stdlib-js/stats-base-dvarianceyc

cramer deviation dispersion javascript math mathematics node node-js nodejs sample-variance standard-deviation statistics stats stdlib strided strided-array unbiased var variance youngs

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Calculate the variance of a double-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.

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README 

        



  

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# dvarianceyc



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



> Calculate the [variance][variance] of a double-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.



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



```math

\sigma^2 = \frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2

```



where the population mean is given by



```math

\mu = \frac{1}{N} \sum_{i=0}^{N-1} x_i

```



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`,



```math

s^2 = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2

```



where the sample mean is given by



```math

\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i

```



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.



## Installation



```bash

npm install @stdlib/stats-base-dvarianceyc

```



Alternatively,



-   To load the package in a website via a `script` tag without installation and bundlers, use the [ES Module][es-module] available on the [`esm`][esm-url] branch (see [README][esm-readme]).

-   If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).

-   For use in Observable, or in browser/node environments, use the [Universal Module Definition (UMD)][umd] build available on the [`umd`][umd-url] branch (see [README][umd-readme]).



The [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.



To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.



## Usage



```javascript

var dvarianceyc = require( '@stdlib/stats-base-dvarianceyc' );

```



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



Computes the [variance][variance] of a double-precision floating-point strided array `x` using a one-pass algorithm proposed by Youngs and Cramer.



```javascript

var Float64Array = require( '@stdlib/array-float64' );



var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );

var N = x.length;



var v = dvarianceyc( 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 [`Float64Array`][@stdlib/array/float64].

-   **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 Float64Array = require( '@stdlib/array-float64' );

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



var x = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );

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



var v = dvarianceyc( 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.



```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 = dvarianceyc( N, 1, x1, 2 );

// returns 6.25

```



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



Computes the [variance][variance] of a double-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer and alternative indexing semantics.



```javascript

var Float64Array = require( '@stdlib/array-float64' );



var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );

var N = x.length;



var v = dvarianceyc.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 Float64Array = require( '@stdlib/array-float64' );

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



var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );

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



var v = dvarianceyc.ndarray( N, 1, x, 2, 1 );

// returns 6.25

```



## 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`.



## Examples



```javascript

var randu = require( '@stdlib/random-base-randu' );

var round = require( '@stdlib/math-base-special-round' );

var Float64Array = require( '@stdlib/array-float64' );

var dvarianceyc = require( '@stdlib/stats-base-dvarianceyc' );



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 = dvarianceyc( x.length, 1, x, 1 );

console.log( v );

```



* * *



## References



-   Youngs, Edward A., and Elliot M. Cramer. 1971. "Some Results Relevant to Choice of Sum and Sum-of-Product Algorithms." _Technometrics_ 13 (3): 657–65. doi:[10.1080/00401706.1971.10488826][@youngs:1971a].



* * *



## See Also



-   [`@stdlib/stats-base/dnanvarianceyc`][@stdlib/stats/base/dnanvarianceyc]: calculate the variance of a double-precision floating-point strided array ignoring NaN values and using a one-pass algorithm proposed by Youngs and Cramer.

-   [`@stdlib/stats-base/dstdevyc`][@stdlib/stats/base/dstdevyc]: calculate the standard deviation of a double-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.

-   [`@stdlib/stats-base/dvariance`][@stdlib/stats/base/dvariance]: calculate the variance of a double-precision floating-point strided array.

-   [`@stdlib/stats-base/svarianceyc`][@stdlib/stats/base/svarianceyc]: calculate the variance of a single-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.

-   [`@stdlib/stats-base/varianceyc`][@stdlib/stats/base/varianceyc]: calculate the variance of a strided array using a one-pass algorithm proposed by Youngs and Cramer.



* * *



## 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 © 2016-2024. The Stdlib [Authors][stdlib-authors].



[npm-image]: http://img.shields.io/npm/v/@stdlib/stats-base-dvarianceyc.svg

[npm-url]: https://npmjs.org/package/@stdlib/stats-base-dvarianceyc



[test-image]: https://github.com/stdlib-js/stats-base-dvarianceyc/actions/workflows/test.yml/badge.svg?branch=main

[test-url]: https://github.com/stdlib-js/stats-base-dvarianceyc/actions/workflows/test.yml?query=branch:main



[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/stats-base-dvarianceyc/main.svg

[coverage-url]: https://codecov.io/github/stdlib-js/stats-base-dvarianceyc?branch=main



[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg

[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im



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



[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors



[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-base-dvarianceyc/tree/deno

[deno-readme]: https://github.com/stdlib-js/stats-base-dvarianceyc/blob/deno/README.md

[umd-url]: https://github.com/stdlib-js/stats-base-dvarianceyc/tree/umd

[umd-readme]: https://github.com/stdlib-js/stats-base-dvarianceyc/blob/umd/README.md

[esm-url]: https://github.com/stdlib-js/stats-base-dvarianceyc/tree/esm

[esm-readme]: https://github.com/stdlib-js/stats-base-dvarianceyc/blob/esm/README.md

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



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



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



[@stdlib/array/float64]: https://github.com/stdlib-js/array-float64



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



[@youngs:1971a]: https://doi.org/10.1080/00401706.1971.10488826



[@stdlib/stats/base/dnanvarianceyc]: https://github.com/stdlib-js/stats-base-dnanvarianceyc



[@stdlib/stats/base/dstdevyc]: https://github.com/stdlib-js/stats-base-dstdevyc



[@stdlib/stats/base/dvariance]: https://github.com/stdlib-js/stats-base-dvariance



[@stdlib/stats/base/svarianceyc]: https://github.com/stdlib-js/stats-base-svarianceyc



[@stdlib/stats/base/varianceyc]: https://github.com/stdlib-js/stats-base-varianceyc