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Calculate the variance of a doubleprecision floatingpoint strided array using a onepass trial mean 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.
npm install @stdlib/statsbasedvariancech
var dvariancech = require( '@stdlib/statsbasedvariancech' );
Computes the variance of a doubleprecision floatingpoint strided array x
using a onepass trial mean algorithm.
var Float64Array = require( '@stdlib/arrayfloat64' );
var x = new Float64Array( [ 1.0, 2.0, 2.0 ] );
var N = x.length;
var v = dvariancech( 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 = dvariancech( 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 = dvariancech( N, 1, x1, 2 );
// returns 6.25
Computes the variance of a doubleprecision floatingpoint strided array using a onepass trial mean 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 = dvariancech.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 = dvariancech.ndarray( N, 1, x, 2, 1 );
// returns 6.25
 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
.  The underlying algorithm is a specialized case of Neely's twopass algorithm. As the variance 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).
var randu = require( '@stdlib/randombaserandu' );
var round = require( '@stdlib/mathbasespecialround' );
var Float64Array = require( '@stdlib/arrayfloat64' );
var dvariancech = require( '@stdlib/statsbasedvariancech' );
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 = dvariancech( x.length, 1, x, 1 );
console.log( v );
 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.
 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.
 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.
 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.

@stdlib/statsbase/dnanvariancech
: calculate the variance of a doubleprecision floatingpoint strided array ignoring NaN values and using a onepass trial mean algorithm. 
@stdlib/statsbase/dstdevch
: calculate the standard deviation of a doubleprecision floatingpoint strided array using a onepass trial mean algorithm. 
@stdlib/statsbase/dvariance
: calculate the variance of a doubleprecision floatingpoint strided array. 
@stdlib/statsbase/svariancech
: calculate the variance of a singleprecision floatingpoint strided array using a onepass trial mean algorithm. 
@stdlib/statsbase/variancech
: calculate the variance of a strided array using a onepass trial mean algorithm.
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For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.
See LICENSE.
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