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Calculate the variance of a doubleprecision floatingpoint strided array using a onepass textbook algorithm.
The population variance of a finite size population of size N
is given by
where the population mean is given by
After rearranging terms, the population variance can be equivalently expressed as
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
Similar to the population variance, after rearranging terms, the unbiased sample variance can be equivalently expressed as
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/statsbasedvariancetk
var dvariancetk = require( '@stdlib/statsbasedvariancetk' );
Computes the variance of a doubleprecision floatingpoint strided array x
using a onepass textbook algorithm.
var Float64Array = require( '@stdlib/arrayfloat64' );
var x = new Float64Array( [ 1.0, 2.0, 2.0 ] );
var N = x.length;
var v = dvariancetk( 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 = dvariancetk( 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 = dvariancetk( N, 1, x1, 2 );
// returns 6.25
Computes the variance of a doubleprecision floatingpoint strided array using a onepass textbook 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 = dvariancetk.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 = dvariancetk.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
.  Some caution should be exercised when using the onepass textbook algorithm. Literature overwhelmingly discourages the algorithm's use for two reasons: 1) the lack of safeguards against underflow and overflow and 2) the risk of catastrophic cancellation when subtracting the two sums if the sums are large and the variance small. These concerns have merit; however, the onepass textbook algorithm should not be dismissed outright. For data distributions with a moderately large standard deviation to mean ratio (i.e., coefficient of variation), the onepass textbook algorithm may be acceptable, especially when performance is paramount and some precision loss is acceptable (including a risk of returning a negative variance due to floatingpoint rounding errors!). In short, no single "best" algorithm for computing the variance exists. The "best" algorithm depends on the underlying data distribution, your performance requirements, and your minimum precision requirements. When evaluating which algorithm to use, consider the relative pros and cons, and choose the algorithm which best serves your needs.
var randu = require( '@stdlib/randombaserandu' );
var round = require( '@stdlib/mathbasespecialround' );
var Float64Array = require( '@stdlib/arrayfloat64' );
var dvariancetk = require( '@stdlib/statsbasedvariancetk' );
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 = dvariancetk( x.length, 1, x, 1 );
console.log( v );
 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.

@stdlib/statsbase/dnanvariancetk
: calculate the variance of a doubleprecision floatingpoint strided array ignoring NaN values and using a onepass textbook algorithm. 
@stdlib/statsbase/dstdevtk
: calculate the standard deviation of a doubleprecision floatingpoint strided array using a onepass textbook algorithm. 
@stdlib/statsbase/dvariance
: calculate the variance of a doubleprecision floatingpoint strided array. 
@stdlib/statsbase/svariancetk
: calculate the variance of a singleprecision floatingpoint strided array using a onepass textbook algorithm. 
@stdlib/statsbase/variancetk
: calculate the variance of a strided array using a onepass textbook 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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