npm package: @stdlib/stats-array-stdevyc
Description: Calculate the standard deviation of an array using a one-pass algorithm proposed by Youngs and Cramer.
Author: The Stdlib Authors
Latest version: 0.1.1

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# stdevyc

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> Calculate the [standard deviation][standard-deviation] of an array using a one-pass algorithm proposed by Youngs and Cramer.

<section class="intro">

The population [standard deviation][standard-deviation] of a finite size population of size `N` is given by

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

<div class="equation" align="center" data-raw-text="\sigma = \sqrt{\frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2}" data-equation="eq:population_standard_deviation">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@425217464d5d132b77bbfeea67dc2d8f8c58139f/lib/node_modules/@stdlib/stats/strided/stdevyc/docs/img/equation_population_standard_deviation.svg" alt="Equation for the population standard deviation.">
    <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@425217464d5d132b77bbfeea67dc2d8f8c58139f/lib/node_modules/@stdlib/stats/strided/stdevyc/docs/img/equation_population_mean.svg" alt="Equation for the population mean.">
    <br>
</div> -->

<!-- </equation> -->

Often in the analysis of data, the true population [standard deviation][standard-deviation] 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 [standard deviation][standard-deviation], the result is biased and yields an **uncorrected sample standard deviation**. To compute a **corrected sample standard deviation** for a sample of size `n`,

<!-- <equation class="equation" label="eq:corrected_sample_standard_deviation" align="center" raw="s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2}" alt="Equation for computing a corrected sample standard deviation."> -->

<!-- <div class="equation" align="center" data-raw-text="s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2}" data-equation="eq:corrected_sample_standard_deviation">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@425217464d5d132b77bbfeea67dc2d8f8c58139f/lib/node_modules/@stdlib/stats/strided/stdevyc/docs/img/equation_corrected_sample_standard_deviation.svg" alt="Equation for computing a corrected sample standard deviation.">
    <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@425217464d5d132b77bbfeea67dc2d8f8c58139f/lib/node_modules/@stdlib/stats/strided/stdevyc/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 standard deviation and population standard deviation. 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-stdevyc
```

</section>

<section class="usage">

## Usage

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

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

Computes the [standard deviation][standard-deviation] of an array using a one-pass algorithm proposed by Youngs and Cramer.

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

var v = stdevyc( x );
// returns ~2.0817
```

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 [standard deviation][standard-deviation] 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 [standard deviation][standard-deviation] 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 [standard deviation][standard-deviation], 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 [standard deviation][standard-deviation]. To adjust the degrees of freedom when computing the [standard deviation][standard-deviation], provide a `correction` argument.

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

var v = stdevyc( x, 0.0 );
// returns ~1.6997
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   If provided an empty array, the function returns `NaN`.
-   If `N - c` is less than or equal to `0` (where `c` corresponds to the provided degrees of freedom adjustment), 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 stdevyc = require( '@stdlib/stats-array-stdevyc' );

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

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

</section>

<!-- /.examples -->

* * *

<section class="references">

## 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].

</section>

<!-- /.references -->

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<section class="related">

</section>

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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].

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[standard-deviation]: https://en.wikipedia.org/wiki/Standard_deviation

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

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

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