dvariancewd
Calculate the variance of a doubleprecision floatingpoint strided array using Welford's algorithm.
The population variance 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 variance 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 variance, the result is biased and yields a biased sample variance. To compute an unbiased sample variance for a sample of size n
,
where the sample mean is given by
The use of the term n1
is commonly referred to as Bessel's correction. Note, however, that applying Bessel's correction can increase the mean squared error between the sample variance and population variance. Depending on the characteristics of the population distribution, other correction factors (e.g., n1.5
, n+1
, etc) can yield better estimators.
Installation
npm install @stdlib/statsbasedvariancewd
Usage
var dvariancewd = require( '@stdlib/statsbasedvariancewd' );
dvariancewd( N, correction, x, stride )
Computes the variance of a doubleprecision floatingpoint strided array x
using Welford's algorithm.
var Float64Array = require( '@stdlib/arrayfloat64' );
var x = new Float64Array( [ 1.0, 2.0, 2.0 ] );
var N = x.length;
var v = dvariancewd( N, 1, x, 1 );
// returns ~4.3333
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 variance according toNc
wherec
corresponds to the provided degrees of freedom adjustment. When computing the variance of a population, setting this parameter to0
is the standard choice (i.e., the provided array contains data constituting an entire population). When computing the unbiased sample variance, setting this parameter to1
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 variance of every other element in x
,
var Float64Array = require( '@stdlib/arrayfloat64' );
var floor = require( '@stdlib/mathbasespecialfloor' );
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 = dvariancewd( N, 1, x, 2 );
// returns 6.25
Note that indexing is relative to the first index. To introduce an offset, use typed array
views.
var Float64Array = require( '@stdlib/arrayfloat64' );
var floor = require( '@stdlib/mathbasespecialfloor' );
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 = dvariancewd( N, 1, x1, 2 );
// returns 6.25
dvariancewd.ndarray( N, correction, x, stride, offset )
Computes the variance of a doubleprecision floatingpoint strided array using Welford's algorithm and alternative indexing semantics.
var Float64Array = require( '@stdlib/arrayfloat64' );
var x = new Float64Array( [ 1.0, 2.0, 2.0 ] );
var N = x.length;
var v = dvariancewd.ndarray( N, 1, x, 1, 0 );
// returns ~4.33333
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 variance for every other value in x
starting from the second value
var Float64Array = require( '@stdlib/arrayfloat64' );
var floor = require( '@stdlib/mathbasespecialfloor' );
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 = dvariancewd.ndarray( N, 1, x, 2, 1 );
// returns 6.25
Notes
 If
N <= 0
, both functions returnNaN
.  If
N  c
is less than or equal to0
(wherec
corresponds to the provided degrees of freedom adjustment), both functions returnNaN
.
Examples
var randu = require( '@stdlib/randombaserandu' );
var round = require( '@stdlib/mathbasespecialround' );
var Float64Array = require( '@stdlib/arrayfloat64' );
var dvariancewd = require( '@stdlib/statsbasedvariancewd' );
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 = dvariancewd( 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.
 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.
See Also

@stdlib/stats/base/dnanvariancewd
: calculate the variance of a doubleprecision floatingpoint strided array ignoring NaN values and using Welford's algorithm. 
@stdlib/stats/base/dstdevwd
: calculate the standard deviation of a doubleprecision floatingpoint strided array using Welford's algorithm. 
@stdlib/stats/base/dvariance
: calculate the variance of a doubleprecision floatingpoint strided array. 
@stdlib/stats/base/svariancewd
: calculate the variance of a singleprecision floatingpoint strided array using Welford's algorithm. 
@stdlib/stats/base/variancewd
: calculate the variance of a strided array using Welford's algorithm.
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.
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License
See LICENSE.
Copyright
Copyright © 20162022. The Stdlib Authors.