# sstdevwd
> Calculate the [standard deviation][standard-deviation] of a single-precision floating-point strided array using Welford's algorithm.
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 sstdevwd = require( '@stdlib/stats/base/sstdevwd' );
```
#### sstdevwd( N, correction, x, stride )
Computes the [standard deviation][standard-deviation] of a single-precision floating-point strided array `x` using Welford's algorithm.
```javascript
var Float32Array = require( '@stdlib/array/float32' );
var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;
var v = sstdevwd( N, 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 [`Float32Array`][@stdlib/array/float32].
- **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 Float32Array = require( '@stdlib/array/float32' );
var floor = require( '@stdlib/math/base/special/floor' );
var x = new Float32Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
var N = floor( x.length / 2 );
var v = sstdevwd( 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 Float32Array = require( '@stdlib/array/float32' );
var floor = require( '@stdlib/math/base/special/floor' );
var x0 = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var x1 = new Float32Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var N = floor( x0.length / 2 );
var v = sstdevwd( N, 1, x1, 2 );
// returns 2.5
```
#### sstdevwd.ndarray( N, correction, x, stride, offset )
Computes the [standard deviation][standard-deviation] of a single-precision floating-point strided array using Welford's algorithm and alternative indexing semantics.
```javascript
var Float32Array = require( '@stdlib/array/float32' );
var x = new Float32Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;
var v = sstdevwd.ndarray( N, 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 Float32Array = require( '@stdlib/array/float32' );
var floor = require( '@stdlib/math/base/special/floor' );
var x = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var N = floor( x.length / 2 );
var v = sstdevwd.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), both functions return `NaN`.
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float32Array = require( '@stdlib/array/float32' );
var sstdevwd = require( '@stdlib/stats/base/sstdevwd' );
var x;
var i;
x = new Float32Array( 10 );
for ( i = 0; i < x.length; i++ ) {
x[ i ] = round( (randu()*100.0) - 50.0 );
}
console.log( x );
var v = sstdevwd( x.length, 1, x, 1 );
console.log( v );
```
* * *
## References
- Welford, B. P. 1962. "Note on a Method for Calculating Corrected Sums of Squares and Products." _Technometrics_ 4 (3). Taylor & Francis: 419–20. doi:[10.1080/00401706.1962.10490022][@welford:1962a].
- van Reeken, A. J. 1968. "Letters to the Editor: Dealing with Neely's Algorithms." _Communications of the ACM_ 11 (3): 149–50. doi:[10.1145/362929.362961][@vanreeken:1968a].
[standard-deviation]: https://en.wikipedia.org/wiki/Standard_deviation
[@stdlib/array/float32]: https://www.npmjs.com/package/@stdlib/array-float32
[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypedArray
[@welford:1962a]: https://doi.org/10.1080/00401706.1962.10490022
[@vanreeken:1968a]: https://doi.org/10.1145/362929.362961
