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dstdevwd

Calculate the standard deviation of a double-precision floating-point strided array using Welford's algorithm.

The population standard deviation of a finite size population of size N is given by

$$\sigma = \sqrt{\frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2}$$

where the population mean is given by

$$\mu = \frac{1}{N} \sum_{i=0}^{N-1} x_i$$

Often in the analysis of data, the true population 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, the result is biased and yields an uncorrected sample standard deviation. To compute a corrected sample standard deviation for a sample of size n,

$$s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2}$$

where the sample mean is given by

$$\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i$$

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.

Installation

npm install @stdlib/stats-base-dstdevwd

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var dstdevwd = require( '@stdlib/stats-base-dstdevwd' );

dstdevwd( N, correction, x, stride )

Computes the standard deviation of a double-precision floating-point strided array x using Welford's algorithm.

var Float64Array = require( '@stdlib/array-float64' );

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;

var v = dstdevwd( 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 according to N-c where c corresponds to the provided degrees of freedom adjustment. When computing the 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, 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.
• 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 of every other element in x,

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 = dstdevwd( N, 1, x, 2 );
// returns 2.5

Note that indexing is relative to the first index. To introduce an offset, use typed array views.

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 = dstdevwd( N, 1, x1, 2 );
// returns 2.5

dstdevwd.ndarray( N, correction, x, stride, offset )

Computes the standard deviation of a double-precision floating-point strided array using Welford's algorithm and alternative indexing semantics.

var Float64Array = require( '@stdlib/array-float64' );

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;

var v = dstdevwd.ndarray( N, 1, x, 1, 0 );
// returns ~2.0817

The function has the following additional parameters:

• offset: starting index for x.

While 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 for every other value in x starting from the second value

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 = dstdevwd.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

var randu = require( '@stdlib/random-base-randu' );
var round = require( '@stdlib/math-base-special-round' );
var Float64Array = require( '@stdlib/array-float64' );
var dstdevwd = require( '@stdlib/stats-base-dstdevwd' );

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 = dstdevwd( x.length, 1, x, 1 );
console.log( v );

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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.
• 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.

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See Also

• @stdlib/stats-base/dnanstdevwd: calculate the standard deviation of a double-precision floating-point strided array ignoring NaN values and using Welford's algorithm.
• @stdlib/stats-base/dstdev: calculate the standard deviation of a double-precision floating-point strided array.
• @stdlib/stats-base/dvariancewd: calculate the variance of a double-precision floating-point strided array using Welford's algorithm.
• @stdlib/stats-base/sstdevwd: calculate the standard deviation of a single-precision floating-point strided array using Welford's algorithm.
• @stdlib/stats-base/stdevwd: calculate the standard deviation of a strided array using Welford's algorithm.

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

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