time-to-botec/js/node_modules/@stdlib/stats/base/dsemtk/README.md

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# dsemtk

> Calculate the [standard error of the mean][standard-error] of a double-precision floating-point strided array using a one-pass textbook algorithm.

<section class="intro">

The [standard error of the mean][standard-error] of a finite size sample of size `n` is given by

<!-- <equation class="equation" label="eq:standard_error_of_the_mean" align="center" raw="\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}" alt="Equation for the standard error of the mean."> -->

<div class="equation" align="center" data-raw-text="\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}" data-equation="eq:standard_error_of_the_mean">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@5801695664293426455789e96b013ef4320d0569/lib/node_modules/@stdlib/stats/base/dsemtk/docs/img/equation_standard_error_of_the_mean.svg" alt="Equation for the standard error of the mean.">
    <br>
</div>

<!-- </equation> -->

where `σ` is the population [standard deviation][standard-deviation].

Often 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. In this scenario, one must use a sample [standard deviation][standard-deviation] to compute an estimate for the [standard error of the mean][standard-error]

<!-- <equation class="equation" label="eq:standard_error_of_the_mean_estimate" align="center" raw="\sigma_{\bar{x}} \approx \frac{s}{\sqrt{n}}" alt="Equation for estimating the standard error of the mean."> -->

<div class="equation" align="center" data-raw-text="\sigma_{\bar{x}} \approx \frac{s}{\sqrt{n}}" data-equation="eq:standard_error_of_the_mean_estimate">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@5801695664293426455789e96b013ef4320d0569/lib/node_modules/@stdlib/stats/base/dsemtk/docs/img/equation_standard_error_of_the_mean_estimate.svg" alt="Equation for estimating the standard error of the mean.">
    <br>
</div>

<!-- </equation> -->

where `s` is the sample [standard deviation][standard-deviation].

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

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

Computes the [standard error of the mean][standard-error] 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 = dsemtk( N, 1, x, 1 );
// returns ~1.20185
```

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 [standard deviation][standard-deviation] according to `N-c` where `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 corrected 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).
-   **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 [standard error of the mean][standard-error] 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 = dsemtk( N, 1, x, 2 );
// returns 1.25
```

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 = dsemtk( N, 1, x1, 2 );
// returns 1.25
```

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

Computes the [standard error of the mean][standard-error] 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 = dsemtk.ndarray( N, 1, x, 1, 0 );
// returns ~1.20185
```

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 [standard error of the mean][standard-error] 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 = dsemtk.ndarray( N, 1, x, 2, 1 );
// returns 1.25
```

</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. 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 error of the mean 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 dsemtk = require( '@stdlib/stats/base/dsemtk' );

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 = dsemtk( 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 class="links">

[standard-error]: https://en.wikipedia.org/wiki/Standard_error

[standard-deviation]: https://en.wikipedia.org/wiki/Standard_deviation

[@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

</section>

<!-- /.links -->