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About stdlib...

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stdevyc

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Calculate the standard deviation of a strided array using a one-pass algorithm proposed by Youngs and Cramer.

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.

Usage

import stdevyc from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-stdevyc@deno/mod.js';

stdevyc( N, correction, x, stride )

Computes the standard deviation of a strided array x using a one-pass algorithm proposed by Youngs and Cramer.

var x = [ 1.0, -2.0, 2.0 ];

var v = stdevyc( 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 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 Array or 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 of every other element in x,

import floor from 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-floor@deno/mod.js';

var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ];
var N = floor( x.length / 2 );

var v = stdevyc( N, 1, x, 2 );
// returns 2.5

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

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@deno/mod.js';
import floor from 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-floor@deno/mod.js';

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

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

Computes the standard deviation of a strided array using a one-pass algorithm proposed by Youngs and Cramer and alternative indexing semantics.

var x = [ 1.0, -2.0, 2.0 ];

var v = stdevyc.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 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

import floor from 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-floor@deno/mod.js';

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 = stdevyc.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.
  • Depending on the environment, the typed versions (dstdevyc, sstdevyc, etc.) are likely to be significantly more performant.

Examples

import randu from 'https://cdn.jsdelivr.net/gh/stdlib-js/random-base-randu@deno/mod.js';
import round from 'https://cdn.jsdelivr.net/gh/stdlib-js/math-base-special-round@deno/mod.js';
import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@deno/mod.js';
import stdevyc from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-stdevyc@deno/mod.js';

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 = stdevyc( 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.

See Also

  • @stdlib/stats-base/dstdevyc: calculate the standard deviation of a double-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.
  • @stdlib/stats-base/nanstdevyc: calculate the standard deviation of a strided array ignoring NaN values and using a one-pass algorithm proposed by Youngs and Cramer.
  • @stdlib/stats-base/sstdevyc: calculate the standard deviation of a single-precision floating-point strided array using a one-pass algorithm proposed by Youngs and Cramer.
  • @stdlib/stats-base/stdev: calculate the standard deviation of a strided array.
  • @stdlib/stats-base/varianceyc: calculate the variance of a strided array using a one-pass algorithm proposed by Youngs and Cramer.

Notice

This package is part of stdlib, a standard library 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.

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.