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Apache-2.0\n\nCopyright (c) 2025 The Stdlib Authors.\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n   http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n\n--\u003e\n\n\n\u003cdetails\u003e\n  \u003csummary\u003e\n    About stdlib...\n  \u003c/summary\u003e\n  \u003cp\u003eWe believe in a future in which the web is a preferred environment for numerical computation. To help realize this future, we've built stdlib. stdlib is a standard library, with an emphasis on numerical and scientific computation, written in JavaScript (and C) for execution in browsers and in Node.js.\u003c/p\u003e\n  \u003cp\u003eThe library is fully decomposable, being architected in such a way that you can swap out and mix and match APIs and functionality to cater to your exact preferences and use cases.\u003c/p\u003e\n  \u003cp\u003eWhen you use stdlib, you can be absolutely certain that you are using the most thorough, rigorous, well-written, studied, documented, tested, measured, and high-quality code out there.\u003c/p\u003e\n  \u003cp\u003eTo join us in bringing numerical computing to the web, get started by checking us out on \u003ca href=\"https://github.com/stdlib-js/stdlib\"\u003eGitHub\u003c/a\u003e, and please consider \u003ca href=\"https://opencollective.com/stdlib\"\u003efinancially supporting stdlib\u003c/a\u003e. We greatly appreciate your continued support!\u003c/p\u003e\n\u003c/details\u003e\n\n# stdevtk\n\n[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] \u003c!-- [![dependencies][dependencies-image]][dependencies-url] --\u003e\n\n\u003e Calculate the [standard deviation][standard-deviation] of an array using a one-pass textbook algorithm.\n\n\u003csection class=\"intro\"\u003e\n\nThe population [standard deviation][standard-deviation] of a finite size population of size `N` is given by\n\n\u003c!-- \u003cequation 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.\"\u003e --\u003e\n\n```math\n\\sigma = \\sqrt{\\frac{1}{N} \\sum_{i=0}^{N-1} (x_i - \\mu)^2}\n```\n\n\u003c!-- \u003cdiv 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\"\u003e\n    \u003cimg src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@1685f915feee8c7d26b90643d00105b4b6803eb4/lib/node_modules/@stdlib/stats/strided/stdevtk/docs/img/equation_population_standard_deviation.svg\" alt=\"Equation for the population standard deviation.\"\u003e\n    \u003cbr\u003e\n\u003c/div\u003e --\u003e\n\n\u003c!-- \u003c/equation\u003e --\u003e\n\nwhere the population mean is given by\n\n\u003c!-- \u003cequation 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.\"\u003e --\u003e\n\n```math\n\\mu = \\frac{1}{N} \\sum_{i=0}^{N-1} x_i\n```\n\n\u003c!-- \u003cdiv class=\"equation\" align=\"center\" data-raw-text=\"\\mu = \\frac{1}{N} \\sum_{i=0}^{N-1} x_i\" data-equation=\"eq:population_mean\"\u003e\n    \u003cimg src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@1685f915feee8c7d26b90643d00105b4b6803eb4/lib/node_modules/@stdlib/stats/strided/stdevtk/docs/img/equation_population_mean.svg\" alt=\"Equation for the population mean.\"\u003e\n    \u003cbr\u003e\n\u003c/div\u003e --\u003e\n\n\u003c!-- \u003c/equation\u003e --\u003e\n\nOften 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`,\n\n\u003c!-- \u003cequation 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.\"\u003e --\u003e\n\n```math\ns = \\sqrt{\\frac{1}{n-1} \\sum_{i=0}^{n-1} (x_i - \\bar{x})^2}\n```\n\n\u003c!-- \u003cdiv 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\"\u003e\n    \u003cimg src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@1685f915feee8c7d26b90643d00105b4b6803eb4/lib/node_modules/@stdlib/stats/strided/stdevtk/docs/img/equation_corrected_sample_standard_deviation.svg\" alt=\"Equation for computing a corrected sample standard deviation.\"\u003e\n    \u003cbr\u003e\n\u003c/div\u003e --\u003e\n\n\u003c!-- \u003c/equation\u003e --\u003e\n\nwhere the sample mean is given by\n\n\u003c!-- \u003cequation 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.\"\u003e --\u003e\n\n```math\n\\bar{x} = \\frac{1}{n} \\sum_{i=0}^{n-1} x_i\n```\n\n\u003c!-- \u003cdiv class=\"equation\" align=\"center\" data-raw-text=\"\\bar{x} = \\frac{1}{n} \\sum_{i=0}^{n-1} x_i\" data-equation=\"eq:sample_mean\"\u003e\n    \u003cimg src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@1685f915feee8c7d26b90643d00105b4b6803eb4/lib/node_modules/@stdlib/stats/strided/stdevtk/docs/img/equation_sample_mean.svg\" alt=\"Equation for the sample mean.\"\u003e\n    \u003cbr\u003e\n\u003c/div\u003e --\u003e\n\n\u003c!-- \u003c/equation\u003e --\u003e\n\nThe 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.\n\n\u003c/section\u003e\n\n\u003c!-- /.intro --\u003e\n\n\u003csection class=\"installation\"\u003e\n\n## Installation\n\n```bash\nnpm install @stdlib/stats-array-stdevtk\n```\n\nAlternatively,\n\n-   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]).\n-   If you are using Deno, visit the [`deno`][deno-url] branch (see [README][deno-readme] for usage intructions).\n-   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]).\n\nThe [branches.md][branches-url] file summarizes the available branches and displays a diagram illustrating their relationships.\n\nTo 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.\n\n\u003c/section\u003e\n\n\u003csection class=\"usage\"\u003e\n\n## Usage\n\n```javascript\nvar stdevtk = require( '@stdlib/stats-array-stdevtk' );\n```\n\n#### stdevtk( x\\[, correction] )\n\nComputes the [standard deviation][standard-deviation] of an array using a one-pass textbook algorithm.\n\n```javascript\nvar x = [ 1.0, -2.0, 2.0 ];\n\nvar v = stdevtk( x );\n// returns ~2.0817\n```\n\nThe function has the following parameters:\n\n-   **x**: input array.\n-   **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`.\n\nBy 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.\n\n```javascript\nvar x = [ 1.0, -2.0, 2.0 ];\n\nvar v = stdevtk( x, 0.0 );\n// returns ~1.6997\n```\n\n\u003c/section\u003e\n\n\u003c!-- /.usage --\u003e\n\n\u003csection class=\"notes\"\u003e\n\n## Notes\n\n-   If provided an empty array, the function returns `NaN`.\n-   If `N - c` is less than or equal to `0` (where `c` corresponds to the provided degrees of freedom adjustment), the function returns `NaN`.\n-   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]).\n-   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 computing a negative variance due to floating-point rounding errors!). In short, no single \"best\" algorithm for computing the standard deviation 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.\n\n\u003c/section\u003e\n\n\u003c!-- /.notes --\u003e\n\n\u003csection class=\"examples\"\u003e\n\n## Examples\n\n\u003c!-- eslint no-undef: \"error\" --\u003e\n\n```javascript\nvar discreteUniform = require( '@stdlib/random-array-discrete-uniform' );\nvar stdevtk = require( '@stdlib/stats-array-stdevtk' );\n\nvar x = discreteUniform( 10, -50, 50, {\n    'dtype': 'float64'\n});\nconsole.log( x );\n\nvar v = stdevtk( x );\nconsole.log( v );\n```\n\n\u003c/section\u003e\n\n\u003c!-- /.examples --\u003e\n\n* * *\n\n\u003csection class=\"references\"\u003e\n\n## References\n\n-   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 \u0026 Francis, Ltd.: 859–66. doi:[10.2307/2286154][@ling:1974a].\n\n\u003c/section\u003e\n\n\u003c!-- /.references --\u003e\n\n\u003c!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. --\u003e\n\n\u003csection class=\"related\"\u003e\n\n\u003c/section\u003e\n\n\u003c!-- /.related --\u003e\n\n\u003c!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --\u003e\n\n\n\u003csection class=\"main-repo\" \u003e\n\n* * *\n\n## Notice\n\nThis 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.\n\nFor 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].\n\n#### Community\n\n[![Chat][chat-image]][chat-url]\n\n---\n\n## License\n\nSee [LICENSE][stdlib-license].\n\n\n## Copyright\n\nCopyright \u0026copy; 2016-2026. The Stdlib [Authors][stdlib-authors].\n\n\u003c/section\u003e\n\n\u003c!-- /.stdlib --\u003e\n\n\u003c!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. --\u003e\n\n\u003csection class=\"links\"\u003e\n\n[npm-image]: http://img.shields.io/npm/v/@stdlib/stats-array-stdevtk.svg\n[npm-url]: https://npmjs.org/package/@stdlib/stats-array-stdevtk\n\n[test-image]: https://github.com/stdlib-js/stats-array-stdevtk/actions/workflows/test.yml/badge.svg?branch=main\n[test-url]: https://github.com/stdlib-js/stats-array-stdevtk/actions/workflows/test.yml?query=branch:main\n\n[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/stats-array-stdevtk/main.svg\n[coverage-url]: https://codecov.io/github/stdlib-js/stats-array-stdevtk?branch=main\n\n\u003c!--\n\n[dependencies-image]: https://img.shields.io/david/stdlib-js/stats-array-stdevtk.svg\n[dependencies-url]: https://david-dm.org/stdlib-js/stats-array-stdevtk/main\n\n--\u003e\n\n[chat-image]: https://img.shields.io/badge/zulip-join_chat-brightgreen.svg\n[chat-url]: https://stdlib.zulipchat.com\n\n[stdlib]: https://github.com/stdlib-js/stdlib\n\n[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors\n\n[umd]: https://github.com/umdjs/umd\n[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules\n\n[deno-url]: https://github.com/stdlib-js/stats-array-stdevtk/tree/deno\n[deno-readme]: https://github.com/stdlib-js/stats-array-stdevtk/blob/deno/README.md\n[umd-url]: https://github.com/stdlib-js/stats-array-stdevtk/tree/umd\n[umd-readme]: https://github.com/stdlib-js/stats-array-stdevtk/blob/umd/README.md\n[esm-url]: https://github.com/stdlib-js/stats-array-stdevtk/tree/esm\n[esm-readme]: https://github.com/stdlib-js/stats-array-stdevtk/blob/esm/README.md\n[branches-url]: https://github.com/stdlib-js/stats-array-stdevtk/blob/main/branches.md\n\n[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-array-stdevtk/main/LICENSE\n\n[standard-deviation]: https://en.wikipedia.org/wiki/Standard_deviation\n\n[@stdlib/array/base/accessor]: https://github.com/stdlib-js/array-base-accessor\n\n[@ling:1974a]: https://doi.org/10.2307/2286154\n\n\u003c/section\u003e\n\n\u003c!-- /.links --\u003e\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fstdlib-js%2Fstats-array-stdevtk","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fstdlib-js%2Fstats-array-stdevtk","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fstdlib-js%2Fstats-array-stdevtk/lists"}