npm package: @stdlib/stats-bartlett-test
Description: Bartlett’s test for equal variances.
Author: The Stdlib Authors
Latest version: 0.2.3

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# bartlettTest

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> Compute Bartlett’s test for equal variances.

<section class="intro">

Bartlett's test is used to test the null hypothesis that the variances of k groups are equal against the alternative that at least two of them are different.

For `k` groups each with `n_i` observations, the test statistic is

<!-- <equation class="equation" label="eq:bartlett-test-statistic" align="center" raw="\chi^2 = \frac{\displaystyle N\ln(S^2) - \sum_{i=0}^{k-1} n_i \ln(S_i^2)}{\displaystyle 1 + \frac{1}{3(k-1)}\left(\sum_{i=0}^{k-1} \frac{1}{n_i} - \frac{1}{N}\right)}" alt="Equation for Bartlett's test statistic."> -->

<div class="equation" align="center" data-raw-text="\chi^2 = \frac{\displaystyle N\ln(S^2) - \sum_{i=0}^{k-1} n_i \ln(S_i^2)}{\displaystyle 1 + \frac{1}{3(k-1)}\left(\sum_{i=0}^{k-1} \frac{1}{n_i} - \frac{1}{N}\right)}" data-equation="eq:bartlett-test-statistic">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@4b1db4ebd815eb54bf53a3fa132b992604743d9c/lib/node_modules/@stdlib/stats/bartlett-test/docs/img/equation_bartlett-test-statistic.svg" alt="Equation for Bartlett's test statistic.">
    <br>
</div>

<!-- </equation> -->

where `N` is the total number of observations, `S_i` are the biased group-level variances and `S^2` is a (biased) pooled estimate for the variance. Under the null hypothesis, the test statistic follows a _chi-square_ distribution with `df = k - 1` degrees of freedom.

</section>

<!-- /.intro -->

<section class="installation">

## Installation

```bash
npm install @stdlib/stats-bartlett-test
```

</section>

<section class="usage">

## Usage

```javascript
var bartlettTest = require( '@stdlib/stats-bartlett-test' );
```

#### bartlettTest( a\[,b,...,k]\[, opts] )

For input arrays `a`, `b`, ... holding numeric observations, this function calculates Bartlett’s test, which tests the null hypothesis that the variances in all `k` groups are the same.

```javascript
// Data from Hollander & Wolfe (1973), p. 116:
var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = bartlettTest( x, y, z );
/* returns
    {
        'rejected': false,
        'alpha': 0.05,
        'df': 2,
        'pValue': ~0.573,
        'statistic': ~1.112,
        ...
    }
*/
```

The function accepts the following `options`:

-   **alpha**: `number` in the interval `[0,1]` giving the significance level of the hypothesis test. Default: `0.05`.
-   **groups**: an `array` of group indicators. If set, the function assumes that only a single numeric array is provided holding all observations.

By default, the test is carried out at a significance level of `0.05`. To choose a custom significance level, set the `alpha` option.

```javascript
var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = bartlettTest( x, y, z, {
    'alpha': 0.01
});
/* returns
    {
        'rejected': false,
        'alpha': 0.01,
        'df': 2,
        'pValue': ~0.573,
        'statistic': ~1.112,
        ...
    }
*/
```

The function provides an alternate interface by supplying an array of group indicators to the `groups` option. In this case, it is assumed that only a single numeric array holding all observations is provided to the function.

<!-- eslint-disable array-element-newline -->

```javascript
var arr = [
    2.9, 3.0, 2.5, 2.6, 3.2,
    3.8, 2.7, 4.0, 2.4,
    2.8, 3.4, 3.7, 2.2, 2.0
];
var groups = [
    'a', 'a', 'a', 'a', 'a',
    'b', 'b', 'b', 'b',
    'c', 'c', 'c', 'c', 'c'
];
var out = bartlettTest( arr, {
    'groups': groups
});
```

The 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.

```javascript
var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = bartlettTest( x, y, z );
console.log( out.print() );
/* =>
    Bartlett's test of equal variances

    Null hypothesis: The variances in all groups are the same.

        pValue: 0.5735
        statistic: 1.1122
        df: 2

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var bartlettTest = require( '@stdlib/stats-bartlett-test' );

// Data from Hollander & Wolfe (1973), p. 116:
var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
var y = [ 3.8, 2.7, 4.0, 2.4 ];
var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];

var out = bartlettTest( x, y, z );
/* returns
    {
        'rejected': false,
        'alpha': 0.05,
        'df': 2,
        'pValue': ~0.573,
        'statistic': ~1.112,
        ...
    }
*/

var table = out.print();
/* returns
    Bartlett's test of equal variances

    Null hypothesis: The variances in all groups are the same.

        pValue: 0.5735
        statistic: 1.1122
        df: 2

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/
```

</section>

<!-- /.examples -->

<section class="references">

</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-vartest`][@stdlib/stats/vartest]</span><span class="delimiter">: </span><span class="description">two-sample F-test for equal variances.</span>
-   <span class="package-name">[`@stdlib/stats-levene-test`][@stdlib/stats/levene-test]</span><span class="delimiter">: </span><span class="description">Levene's test for equal variances.</span>

</section>

<!-- /.related -->

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* * *

## Notice

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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

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## Copyright

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

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