# dstdevch
> Calculate the [standard deviation][standard-deviation] of a double-precision floating-point strided array using a one-pass trial mean 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 dstdevch = require( '@stdlib/stats/base/dstdevch' );
```
#### dstdevch( N, correction, x, stride )
Computes the [standard deviation][standard-deviation] of a double-precision floating-point strided array `x` using a one-pass trial mean algorithm.
```javascript
var Float64Array = require( '@stdlib/array/float64' );
var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;
var v = dstdevch( 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 [`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 deviation][standard-deviation] 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 = dstdevch( 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 ] );
var x1 = new Float64Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element
var N = floor( x0.length / 2 );
var v = dstdevch( N, 1, x1, 2 );
// returns 2.5
```
#### dstdevch.ndarray( N, correction, x, stride, offset )
Computes the [standard deviation][standard-deviation] of a double-precision floating-point strided array using a one-pass trial mean 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 = dstdevch.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 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 = dstdevch.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`.
- The underlying algorithm is a specialized case of Neely's two-pass algorithm. As the standard deviation is invariant with respect to changes in the location parameter, the underlying algorithm uses the first strided array element as a trial mean to shift subsequent data values and thus mitigate catastrophic cancellation. Accordingly, the algorithm's accuracy is best when data is **unordered** (i.e., the data is **not** sorted in either ascending or descending order such that the first value is an "extreme" value).
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float64Array = require( '@stdlib/array/float64' );
var dstdevch = require( '@stdlib/stats/base/dstdevch' );
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 = dstdevch( x.length, 1, x, 1 );
console.log( v );
```
* * *
## References
- Neely, Peter M. 1966. "Comparison of Several Algorithms for Computation of Means, Standard Deviations and Correlation Coefficients." _Communications of the ACM_ 9 (7). Association for Computing Machinery: 496–99. doi:[10.1145/365719.365958][@neely:1966a].
- 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].
- Chan, Tony F., Gene H. Golub, and Randall J. LeVeque. 1983. "Algorithms for Computing the Sample Variance: Analysis and Recommendations." _The American Statistician_ 37 (3). American Statistical Association, Taylor & Francis, Ltd.: 242–47. doi:[10.1080/00031305.1983.10483115][@chan:1983a].
- Schubert, Erich, and Michael Gertz. 2018. "Numerically Stable Parallel Computation of (Co-)Variance." In _Proceedings of the 30th International Conference on Scientific and Statistical Database Management_. New York, NY, USA: Association for Computing Machinery. doi:[10.1145/3221269.3223036][@schubert:2018a].
[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
[@neely:1966a]: https://doi.org/10.1145/365719.365958
[@ling:1974a]: https://doi.org/10.2307/2286154
[@chan:1983a]: https://doi.org/10.1080/00031305.1983.10483115
[@schubert:2018a]: https://doi.org/10.1145/3221269.3223036
