# dsemwd
> Calculate the [standard error of the mean][standard-error] of a double-precision floating-point strided array using Welford's algorithm.
The [standard error of the mean][standard-error] of a finite size sample of size `n` is given by
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]
where `s` is the sample [standard deviation][standard-deviation].
## Usage
```javascript
var dsemwd = require( '@stdlib/stats/base/dsemwd' );
```
#### dsemwd( N, correction, x, stride )
Computes the [standard error of the mean][standard-error] of a double-precision floating-point strided array `x` using Welford's algorithm.
```javascript
var Float64Array = require( '@stdlib/array/float64' );
var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;
var v = dsemwd( 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 = dsemwd( 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.
```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 = dsemwd( N, 1, x1, 2 );
// returns 1.25
```
#### dsemwd.ndarray( N, correction, x, stride, offset )
Computes the [standard error of the mean][standard-error] of a double-precision floating-point strided array using Welford's 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 = dsemwd.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 = dsemwd.ndarray( N, 1, x, 2, 1 );
// returns 1.25
```
## 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 Float64Array = require( '@stdlib/array/float64' );
var dsemwd = require( '@stdlib/stats/base/dsemwd' );
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 = dsemwd( 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-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
[@welford:1962a]: https://doi.org/10.1080/00401706.1962.10490022
[@vanreeken:1968a]: https://doi.org/10.1145/362929.362961
