{"id":23840328,"url":"https://github.com/stdlib-js/stats-ttest2","last_synced_at":"2025-09-07T17:31:25.836Z","repository":{"id":41443260,"uuid":"377247308","full_name":"stdlib-js/stats-ttest2","owner":"stdlib-js","description":"Two-sample Student's t-Test.","archived":false,"fork":false,"pushed_at":"2024-12-23T01:54:42.000Z","size":4700,"stargazers_count":4,"open_issues_count":0,"forks_count":0,"subscribers_count":3,"default_branch":"main","last_synced_at":"2025-01-02T00:33:11.128Z","etag":null,"topics":["hypothesis","javascript","math","mathematics","mean","node","node-js","nodejs","normal","statistics","stats","stdlib","summary","test","two-sample","welch"],"latest_commit_sha":null,"homepage":"https://github.com/stdlib-js/stdlib","language":"JavaScript","has_issues":false,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"apache-2.0","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/stdlib-js.png","metadata":{"files":{"readme":"README.md","changelog":"CHANGELOG.md","contributing":"CONTRIBUTING.md","funding":null,"license":"LICENSE","code_of_conduct":"CODE_OF_CONDUCT.md","threat_model":null,"audit":null,"citation":"CITATION.cff","codeowners":null,"security":"SECURITY.md","support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null},"funding":{"github":["stdlib-js"],"open_collective":"stdlib","tidelift":"npm/@stdlib/stdlib"}},"created_at":"2021-06-15T17:42:45.000Z","updated_at":"2024-12-23T01:39:25.000Z","dependencies_parsed_at":"2023-10-03T06:19:18.092Z","dependency_job_id":"2c05df0c-d6de-4f19-b777-a898b895d6b6","html_url":"https://github.com/stdlib-js/stats-ttest2","commit_stats":{"total_commits":50,"total_committers":1,"mean_commits":50.0,"dds":0.0,"last_synced_commit":"6bc7273aa1f66c9afbfd54a624df56dc20090b95"},"previous_names":[],"tags_count":23,"template":false,"template_full_name":null,"repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fstats-ttest2","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fstats-ttest2/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fstats-ttest2/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/stdlib-js%2Fstats-ttest2/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/stdlib-js","download_url":"https://codeload.github.com/stdlib-js/stats-ttest2/tar.gz/refs/heads/main","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":232230606,"owners_count":18492090,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["hypothesis","javascript","math","mathematics","mean","node","node-js","nodejs","normal","statistics","stats","stdlib","summary","test","two-sample","welch"],"created_at":"2025-01-02T17:31:23.153Z","updated_at":"2025-09-07T17:31:25.666Z","avatar_url":"https://github.com/stdlib-js.png","language":"JavaScript","readme":"\u003c!--\n\n@license Apache-2.0\n\nCopyright (c) 2018 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# Student's t-Test\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 Two-sample Student's t-Test.\n\n\u003csection class=\"installation\"\u003e\n\n## Installation\n\n```bash\nnpm install @stdlib/stats-ttest2\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 ttest2 = require( '@stdlib/stats-ttest2' );\n```\n\n#### ttest2( x, y\\[, opts] )\n\nBy default, the function performs a two-sample t-test for the null hypothesis that the data in [arrays][mdn-array] or [typed arrays][mdn-typed-array] `x` and `y` is independently drawn from normal distributions with _equal_ means.\n\n```javascript\n// Student's sleep data:\nvar x = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];\nvar y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];\n\nvar out = ttest2( x, y );\n/* e.g., returns\n    {\n        'rejected': false,\n        'pValue': ~0.079,\n        'statistic': ~-1.861,\n        'ci': [ ~-3.365, ~0.205 ],\n        // ...\n    }\n*/\n```\n\nThe returned object comes with a `.print()` method which when invoked will print a formatted output of the results of the hypothesis test. `print` accepts a `digits` option that controls the number of decimal digits displayed for the outputs and a `decision` option, which when set to `false` will hide the test decision.\n\n\u003c!-- run-disable --\u003e\n\n```javascript\nconsole.log( out.print() );\n/* e.g., =\u003e\n    Welch two-sample t-test\n\n    Alternative hypothesis: True difference in means is not equal to 0\n\n        pValue: 0.0794\n        statistic: -1.8608\n        95% confidence interval: [-3.3655,0.2055]\n\n    Test Decision: Fail to reject null in favor of alternative at 5% significance level\n*/\n```\n\nThe function accepts the following `options`:\n\n-   **alpha**: `number` in the interval `[0,1]` giving the significance level of the hypothesis test. Default: `0.05`.\n-   **alternative**: Either `two-sided`, `less` or `greater`. Indicates whether the alternative hypothesis is that `x` has a larger mean than `y` (`greater`), `x` has a smaller mean than `y` (`less`) or the means are the same (`two-sided`). Default: `two-sided`.\n-   **difference**: `number` denoting the difference in means under the null hypothesis. Default: `0`.\n-   **variance**: `string` indicating if the test should be conducted under the assumption that the unknown variances of the normal distributions are `equal` or `unequal`. Default: `unequal`.\n\nBy default, the hypothesis test is carried out at a significance level of `0.05`. To choose a different significance level, set the `alpha` option.\n\n```javascript\nvar x = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];\nvar y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];\n\nvar out = ttest2( x, y, {\n    'alpha': 0.1\n});\nvar table = out.print();\n/* e.g., returns\n    Welch two-sample t-test\n\n    Alternative hypothesis: True difference in means is not equal to 0\n\n        pValue: 0.0794\n        statistic: -1.8608\n        90% confidence interval: [-3.0534,-0.1066]\n\n    Test Decision: Reject null in favor of alternative at 10% significance level\n*/\n```\n\nBy default, a two-sided test is performed. To perform either of the one-sided tests, set the `alternative` option to `less` or `greater`.\n\n```javascript\n// Student's sleep data:\nvar x = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];\nvar y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];\n\nvar out = ttest2( x, y, {\n    'alternative': 'less'\n});\nvar table = out.print();\n/* e.g., returns\n    Welch two-sample t-test\n\n    Alternative hypothesis: True difference in means is less than 0\n\n        pValue: 0.0397\n        statistic: -1.8608\n        df: 17.7765\n        95% confidence interval: [-Infinity,-0.1066]\n\n    Test Decision: Reject null in favor of alternative at 5% significance level\n*/\n\nout = ttest2( x, y, {\n    'alternative': 'greater'\n});\ntable = out.print();\n/* e.g., returns\n    Welch two-sample t-test\n\n    Alternative hypothesis: True difference in means is greater than 0\n\n        pValue: 0.9603\n        statistic: -1.8608\n        df: 17.7765\n        95% confidence interval: [-3.0534,Infinity]\n\n    Test Decision: Fail to reject null in favor of alternative at 5% significance level\n*/\n```\n\nAs a default choice, the `ttest2` function carries out the Welch test (using the Satterthwaite approximation for the degrees of freedom), which does not have the requirement that the variances of the underlying distributions are equal. If the equal variances assumption seems warranted, set the `variance` option to `equal`.\n\n```javascript\nvar x = [ 2, 3, 1, 4 ];\nvar y = [ 1, 2, 3, 1, 2, 5, 3, 4 ];\n\nvar out = ttest2( x, y, {\n    'variance': 'equal'\n});\nvar table = out.print();\n/* e.g., returns\n    Two-sample t-test\n\n    Alternative hypothesis: True difference in means is not equal to 0\n\n        pValue: 0.8848\n        statistic: -0.1486\n        df: 10\n        95% confidence interval: [-1.9996,1.7496]\n\n    Test Decision: Fail to reject null in favor of alternative at 5% significance level\n*/\n```\n\nTo test whether the difference in the population means is equal to some other value than `0`, set the `difference` option.\n\n```javascript\nvar normal = require( '@stdlib/random-base-normal' ).factory;\n\nvar rnorm = normal({\n    'seed': 372\n});\n\nvar x = [];\nvar i;\nfor ( i = 0; i \u003c x.length; i++ ) {\n    x.push( rnorm( 2.0, 3.0 ) );\n}\n\nvar y = [];\nfor ( i = 0; i \u003c x.length; i++ ) {\n    y.push( rnorm( 1.0, 3.0 ) );\n}\n\nvar out = ttest2( x, y, {\n    'difference': 1.0,\n    'variance': 'equal'\n});\n/* e.g., returns\n    {\n        'rejected': false,\n        'pValue': ~0.642,\n        'statistic': ~-0.466,\n        'ci': [ ~-0.0455, ~1.646 ],\n        // ...\n    }\n*/\n\nvar table = out.print();\n/* e.g., returns\n    Two-sample t-test\n\n    Alternative hypothesis: True difference in means is not equal to 1\n\n        pValue: 0.6419\n        statistic: -0.4657\n        df: 198\n        95% confidence interval: [-0.0455,1.646]\n\n    Test Decision: Fail to reject null in favor of alternative at 5% significance level\n*/\n```\n\n\u003c/section\u003e\n\n\u003c!-- /.usage --\u003e\n\n\u003csection class=\"examples\"\u003e\n\n## Examples\n\n\u003c!-- eslint no-undef: \"error\" --\u003e\n\n```javascript\nvar incrspace = require( '@stdlib/array-base-incrspace' );\nvar ttest2 = require( '@stdlib/stats-ttest2' );\n\nvar a = incrspace( 1, 11, 1 );\nvar b = incrspace( 7, 21, 1 );\n\nvar out = ttest2( a, b );\nvar table = out.print();\n/* e.g., returns\n    Welch two-sample t-test\n\n    Alternative hypothesis: True difference in means is not equal to 0\n\n        pValue: 0\n        statistic: -5.4349\n        95% confidence interval: [-11.0528,-4.9472]\n\n    Test Decision: Reject null in favor of alternative at 5% significance level\n*/\n```\n\n\u003c/section\u003e\n\n\u003c!-- /.examples --\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* * *\n\n## See Also\n\n-   \u003cspan class=\"package-name\"\u003e[`@stdlib/stats-ttest`][@stdlib/stats/ttest]\u003c/span\u003e\u003cspan class=\"delimiter\"\u003e: \u003c/span\u003e\u003cspan class=\"description\"\u003eone-sample and paired Student's t-Test.\u003c/span\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; 2016-2025. 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-ttest2.svg\n[npm-url]: https://npmjs.org/package/@stdlib/stats-ttest2\n\n[test-image]: https://github.com/stdlib-js/stats-ttest2/actions/workflows/test.yml/badge.svg?branch=main\n[test-url]: https://github.com/stdlib-js/stats-ttest2/actions/workflows/test.yml?query=branch:main\n\n[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/stats-ttest2/main.svg\n[coverage-url]: https://codecov.io/github/stdlib-js/stats-ttest2?branch=main\n\n\u003c!--\n\n[dependencies-image]: https://img.shields.io/david/stdlib-js/stats-ttest2.svg\n[dependencies-url]: https://david-dm.org/stdlib-js/stats-ttest2/main\n\n--\u003e\n\n[chat-image]: https://img.shields.io/gitter/room/stdlib-js/stdlib.svg\n[chat-url]: https://app.gitter.im/#/room/#stdlib-js_stdlib:gitter.im\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-ttest2/tree/deno\n[deno-readme]: https://github.com/stdlib-js/stats-ttest2/blob/deno/README.md\n[umd-url]: https://github.com/stdlib-js/stats-ttest2/tree/umd\n[umd-readme]: https://github.com/stdlib-js/stats-ttest2/blob/umd/README.md\n[esm-url]: https://github.com/stdlib-js/stats-ttest2/tree/esm\n[esm-readme]: https://github.com/stdlib-js/stats-ttest2/blob/esm/README.md\n[branches-url]: https://github.com/stdlib-js/stats-ttest2/blob/main/branches.md\n\n[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/stats-ttest2/main/LICENSE\n\n[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array\n\n[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Typed_arrays\n\n\u003c!-- \u003crelated-links\u003e --\u003e\n\n[@stdlib/stats/ttest]: https://github.com/stdlib-js/stats-ttest\n\n\u003c!-- \u003c/related-links\u003e --\u003e\n\n\u003c/section\u003e\n\n\u003c!-- /.links --\u003e\n","funding_links":["https://github.com/sponsors/stdlib-js","https://opencollective.com/stdlib","https://tidelift.com/funding/github/npm/@stdlib/stdlib"],"categories":[],"sub_categories":[],"project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fstdlib-js%2Fstats-ttest2","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fstdlib-js%2Fstats-ttest2","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fstdlib-js%2Fstats-ttest2/lists"}