npm package: @stdlib/stats-base-dvariancetk
Description: Calculate the variance of a double-precision floating-point strided array using a one-pass textbook algorithm.
Author: The Stdlib Authors
Latest version: 0.2.2

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# dvariancetk

[![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 double-precision floating-point strided array using a one-pass textbook 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@14f73db6358fc1febe42f8220332bf3af188416c/lib/node_modules/@stdlib/stats/base/dvariancetk/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@14f73db6358fc1febe42f8220332bf3af188416c/lib/node_modules/@stdlib/stats/base/dvariancetk/docs/img/equation_population_mean.svg" alt="Equation for the population mean.">
    <br>
</div> -->

<!-- </equation> -->

After rearranging terms, the population [variance][variance] can be equivalently expressed as

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

<!-- <div class="equation" align="center" data-raw-text="\sigma^2 = \frac{1}{N}\biggl(\ \sum_{i=0}^{N-1} x_i^2 - \frac{1}{N}\biggl(\ \sum_{i=0}^{N-1} x_i \ \biggr)^2\ \biggr)" data-equation="eq:population_variance_textbook">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@225f0ea70463a8ba0111dd791d7274ad81ad6c3c/lib/node_modules/@stdlib/stats/base/dvariancetk/docs/img/equation_population_variance_textbook.svg" alt="Equation for the population variance (one-pass textbook formula).">
    <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@14f73db6358fc1febe42f8220332bf3af188416c/lib/node_modules/@stdlib/stats/base/dvariancetk/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@14f73db6358fc1febe42f8220332bf3af188416c/lib/node_modules/@stdlib/stats/base/dvariancetk/docs/img/equation_sample_mean.svg" alt="Equation for the sample mean.">
    <br>
</div> -->

<!-- </equation> -->

Similar to the population [variance][variance], after rearranging terms, the **unbiased sample variance** can be equivalently expressed as

<!-- <equation class="equation" label="eq:unbiased_sample_variance_textbook" align="center" raw="s^2 = \frac{1}{n-1}\biggl(\ \sum_{i=0}^{n-1} x_i^2 - \frac{1}{n}\biggl(\ \sum_{i=0}^{n-1} x_i \ \biggr)^2\ \biggr)" alt="Equation for the unbiased sample variance (one-pass textbook formula)."> -->

<!-- <div class="equation" align="center" data-raw-text="s^2 = \frac{1}{n-1}\biggl(\ \sum_{i=0}^{n-1} x_i^2 - \frac{1}{n}\biggl(\ \sum_{i=0}^{n-1} x_i \ \biggr)^2\ \biggr)" data-equation="eq:unbiased_sample_variance_textbook">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@225f0ea70463a8ba0111dd791d7274ad81ad6c3c/lib/node_modules/@stdlib/stats/base/dvariancetk/docs/img/equation_unbiased_sample_variance_textbook.svg" alt="Equation for the unbiased sample variance (one-pass textbook formula).">
    <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-dvariancetk
```

</section>

<section class="usage">

## Usage

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

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

Computes the [variance][variance] of a double-precision floating-point strided array `x` using a one-pass textbook algorithm.

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

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

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

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

Computes the [variance][variance] of a double-precision floating-point strided array using a one-pass textbook algorithm 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 = dvariancetk.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 = dvariancetk.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`.
-   Some caution should be exercised when using the one-pass textbook algorithm. Literature overwhelmingly discourages the algorithm's use for two reasons: 1) the lack of safeguards against underflow and overflow and 2) the risk of catastrophic cancellation when subtracting the two sums if the sums are large and the variance small. These concerns have merit; however, the one-pass textbook algorithm should not be dismissed outright. For data distributions with a moderately large standard deviation to mean ratio (i.e., **coefficient of variation**), the one-pass textbook algorithm may be acceptable, especially when performance is paramount and some precision loss is acceptable (including a risk of returning a negative variance due to floating-point rounding errors!). In short, no single "best" algorithm for computing the variance exists. The "best" algorithm depends on the underlying data distribution, your performance requirements, and your minimum precision requirements. When evaluating which algorithm to use, consider the relative pros and cons, and choose the algorithm which best serves your needs.

</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 dvariancetk = require( '@stdlib/stats-base-dvariancetk' );

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

</section>

<!-- /.examples -->

* * *

<section class="references">

## References

-   Ling, Robert F. 1974. "Comparison of Several Algorithms for Computing Sample Means and Variances." _Journal of the American Statistical Association_ 69 (348). American Statistical Association, Taylor & Francis, Ltd.: 859–66. doi:[10.2307/2286154][@ling:1974a].

</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/dnanvariancetk`][@stdlib/stats/base/dnanvariancetk]</span><span class="delimiter">: </span><span class="description">calculate the variance of a double-precision floating-point strided array ignoring NaN values and using a one-pass textbook algorithm.</span>
-   <span class="package-name">[`@stdlib/stats-base/dstdevtk`][@stdlib/stats/base/dstdevtk]</span><span class="delimiter">: </span><span class="description">calculate the standard deviation of a double-precision floating-point strided array using a one-pass textbook algorithm.</span>
-   <span class="package-name">[`@stdlib/stats-base/dvariance`][@stdlib/stats/base/dvariance]</span><span class="delimiter">: </span><span class="description">calculate the variance of a double-precision floating-point strided array.</span>
-   <span class="package-name">[`@stdlib/stats-base/svariancetk`][@stdlib/stats/base/svariancetk]</span><span class="delimiter">: </span><span class="description">calculate the variance of a single-precision floating-point strided array using a one-pass textbook algorithm.</span>
-   <span class="package-name">[`@stdlib/stats-base/variancetk`][@stdlib/stats/base/variancetk]</span><span class="delimiter">: </span><span class="description">calculate the variance of a strided array using a one-pass textbook 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>

<!-- /.stdlib -->

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

[@stdlib/array/float64]: https://www.npmjs.com/package/@stdlib/array-float64

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

[@ling:1974a]: https://doi.org/10.2307/2286154

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