# nanstdevyc
> Calculate the [standard deviation][standard-deviation] of a strided array ignoring `NaN` values and using a one-pass algorithm proposed by Youngs and Cramer.
The population [standard deviation][standard-deviation] of a finite size population of size `N` is given by
where the population mean is given by
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. 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`,
where the sample mean is given by
The 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.
## Usage
```javascript
var nanstdevyc = require( '@stdlib/stats/base/nanstdevyc' );
```
#### nanstdevyc( N, correction, x, stride )
Computes the [standard deviation][standard-deviation] of a strided array `x` ignoring `NaN` values and using a one-pass algorithm proposed by Youngs and Cramer.
```javascript
var x = [ 1.0, -2.0, NaN, 2.0 ];
var v = nanstdevyc( x.length, 1, x, 1 );
// returns ~2.0817
```
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 [`Array`][mdn-array] or [`typed array`][mdn-typed-array].
- **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 deviation][standard-deviation] of every other element in `x`,
```javascript
var floor = require( '@stdlib/math/base/special/floor' );
var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0, NaN ];
var N = floor( x.length / 2 );
var v = nanstdevyc( N, 1, x, 2 );
// returns 2.5
```
Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views.
```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, NaN ] );
var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var N = floor( x0.length / 2 );
var v = nanstdevyc( N, 1, x1, 2 );
// returns 2.5
```
#### nanstdevyc.ndarray( N, correction, x, stride, offset )
Computes the [standard deviation][standard-deviation] of a strided array ignoring `NaN` values and using a one-pass algorithm proposed by Youngs and Cramer and alternative indexing semantics.
```javascript
var x = [ 1.0, -2.0, NaN, 2.0 ];
var v = nanstdevyc.ndarray( x.length, 1, x, 1, 0 );
// returns ~2.0817
```
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 deviation][standard-deviation] for every other value in `x` starting from the second value
```javascript
var floor = require( '@stdlib/math/base/special/floor' );
var x = [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ];
var N = floor( x.length / 2 );
var v = nanstdevyc.ndarray( N, 1, x, 2, 1 );
// returns 2.5
```
## 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 and `n` corresponds to the number of non-`NaN` indexed elements), both functions return `NaN`.
- Depending on the environment, the typed versions ([`dnanstdevyc`][@stdlib/stats/base/dnanstdevyc], [`snanstdevyc`][@stdlib/stats/base/snanstdevyc], etc.) are likely to be significantly more performant.
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float64Array = require( '@stdlib/array/float64' );
var nanstdevyc = require( '@stdlib/stats/base/nanstdevyc' );
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 = nanstdevyc( x.length, 1, x, 1 );
console.log( v );
```
* * *
## References
- Youngs, Edward A., and Elliot M. Cramer. 1971. "Some Results Relevant to Choice of Sum and Sum-of-Product Algorithms." _Technometrics_ 13 (3): 657–65. doi:[10.1080/00401706.1971.10488826][@youngs:1971a].
[standard-deviation]: https://en.wikipedia.org/wiki/Standard_deviation
[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array
[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
[@stdlib/stats/base/dnanstdevyc]: https://www.npmjs.com/package/@stdlib/stats/tree/main/base/dnanstdevyc
[@stdlib/stats/base/snanstdevyc]: https://www.npmjs.com/package/@stdlib/stats/tree/main/base/snanstdevyc
[@youngs:1971a]: https://doi.org/10.1080/00401706.1971.10488826
