@stdlib/stats/incr/skewness/README.md

Standard library statistical functions.

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# incrskewness

> Compute a [corrected sample skewness][sample-skewness] incrementally.

<section class="intro">

The [skewness][sample-skewness] for a random variable `X` is defined as

<!-- <equation class="equation" label="eq:skewness" align="center" raw="\operatorname{Skewness}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^3 \biggr]" alt="Equation for skewness."> -->

<div class="equation" align="center" data-raw-text="\operatorname{Skewness}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^3 \biggr]" data-equation="eq:skewness">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@49d8cabda84033d55d7b8069f19ee3dd8b8d1496/lib/node_modules/@stdlib/stats/incr/skewness/docs/img/equation_skewness.svg" alt="Equation for skewness.">
    <br>
</div> -->

<!-- </equation> -->

For a sample of `n` values, the [sample skewness][sample-skewness] is

<!-- <equation class="equation" label="eq:sample_skewness" align="center" raw="b_1 = \frac{m_3}{s^3} = \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^3}{\biggl( \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2 \biggr)^{3/2}}" alt="Equation for the sample skewness."> -->

<!-- <div class="equation" align="center" data-raw-text="b_1 = \frac{m_3}{s^3} = \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^3}{\biggl( \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2 \biggr)^{3/2}}" data-equation="eq:sample_skewness">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@49d8cabda84033d55d7b8069f19ee3dd8b8d1496/lib/node_modules/@stdlib/stats/incr/skewness/docs/img/equation_sample_skewness.svg" alt="Equation for the sample skewness.">
    <br>
</div> -->

<!-- </equation> -->

where `m_3` is the sample third central moment and `s` is the sample standard deviation.

An alternative definition for the [sample skewness][sample-skewness] which includes an adjustment factor (and is the implemented definition) is

<!-- <equation class="equation" label="eq:adjusted_sample_skewness" align="center" raw="G_1 = \frac{n^2}{(n-1)(n-2)} \frac{m_3}{s^3} = \frac{\sqrt{n(n-1)}}{n-2} \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^3}{\biggl( \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^2 \biggr)^{3/2}}" alt="Equation for the adjusted sample skewness."> -->

<!-- <div class="equation" align="center" data-raw-text="G_1 = \frac{n^2}{(n-1)(n-2)} \frac{m_3}{s^3} = \frac{\sqrt{n(n-1)}}{n-2} \frac{\frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^3}{\biggl( \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \bar{x})^2 \biggr)^{3/2}}" data-equation="eq:adjusted_sample_skewness">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@49d8cabda84033d55d7b8069f19ee3dd8b8d1496/lib/node_modules/@stdlib/stats/incr/skewness/docs/img/equation_adjusted_sample_skewness.svg" alt="Equation for the adjusted sample skewness.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var incrskewness = require( '@stdlib/stats/incr/skewness' );
```

#### incrskewness()

Returns an accumulator `function` which incrementally computes a [corrected sample skewness][sample-skewness].

```javascript
var accumulator = incrskewness();
```

#### accumulator( \[x] )

If provided an input value `x`, the accumulator function returns an updated [corrected sample skewness][sample-skewness]. If not provided an input value `x`, the accumulator function returns the current [corrected sample skewness][sample-skewness].

```javascript
var accumulator = incrskewness();

var skewness = accumulator();
// returns null

skewness = accumulator( 2.0 );
// returns null

skewness = accumulator( -5.0 );
// returns null

skewness = accumulator( -10.0 );
// returns ~0.492

skewness = accumulator();
// returns ~0.492
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   Input values are **not** type checked. If provided `NaN` or a value which, when used in computations, results in `NaN`, the accumulated value is `NaN` for **all** future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly **before** passing the value to the accumulator function.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var incrskewness = require( '@stdlib/stats/incr/skewness' );

var accumulator;
var v;
var i;

// Initialize an accumulator:
accumulator = incrskewness();

// For each simulated datum, update the corrected sample skewness...
for ( i = 0; i < 100; i++ ) {
    v = randu() * 100.0;
    accumulator( v );
}
console.log( accumulator() );
```

</section>

<!-- /.examples -->

* * *

<section class="references">

## References

-   Joanes, D. N., and C. A. Gill. 1998. "Comparing measures of sample skewness and kurtosis." _Journal of the Royal Statistical Society: Series D (The Statistician)_ 47 (1). Blackwell Publishers Ltd: 183–89. doi:[10.1111/1467-9884.00122][@joanes:1998].

</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/incr/kurtosis`][@stdlib/stats/incr/kurtosis]</span><span class="delimiter">: </span><span class="description">compute a corrected sample excess kurtosis incrementally.</span>
-   <span class="package-name">[`@stdlib/stats/incr/mean`][@stdlib/stats/incr/mean]</span><span class="delimiter">: </span><span class="description">compute an arithmetic mean incrementally.</span>
-   <span class="package-name">[`@stdlib/stats/incr/stdev`][@stdlib/stats/incr/stdev]</span><span class="delimiter">: </span><span class="description">compute a corrected sample standard deviation incrementally.</span>
-   <span class="package-name">[`@stdlib/stats/incr/summary`][@stdlib/stats/incr/summary]</span><span class="delimiter">: </span><span class="description">compute a statistical summary incrementally.</span>
-   <span class="package-name">[`@stdlib/stats/incr/variance`][@stdlib/stats/incr/variance]</span><span class="delimiter">: </span><span class="description">compute an unbiased sample variance incrementally.</span>

</section>

<!-- /.related -->

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<section class="links">

[sample-skewness]: https://en.wikipedia.org/wiki/Skewness

[@joanes:1998]: http://onlinelibrary.wiley.com/doi/10.1111/1467-9884.00122/

<!-- <related-links> -->

[@stdlib/stats/incr/kurtosis]: https://github.com/stdlib-js/stats/tree/main/incr/kurtosis

[@stdlib/stats/incr/mean]: https://github.com/stdlib-js/stats/tree/main/incr/mean

[@stdlib/stats/incr/stdev]: https://github.com/stdlib-js/stats/tree/main/incr/stdev

[@stdlib/stats/incr/summary]: https://github.com/stdlib-js/stats/tree/main/incr/summary

[@stdlib/stats/incr/variance]: https://github.com/stdlib-js/stats/tree/main/incr/variance

<!-- </related-links> -->

</section>

<!-- /.links -->