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 equals StartRoot StartFraction 1 Over upper N EndFraction sigma-summation Underscript i equals 0 Overscript upper N minus 1 Endscripts left-parenthesis x Subscript i Baseline minus mu right-parenthesis squared EndRoot

where the population mean is given by

mu equals StartFraction 1 Over upper N EndFraction sigma-summation Underscript i equals 0 Overscript upper N minus 1 Endscripts x Subscript 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 equals StartRoot StartFraction 1 Over n minus 1 EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts left-parenthesis x Subscript i Baseline minus x overbar right-parenthesis squared EndRoot

where the sample mean is given by

x overbar equals StartFraction 1 Over n EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts x Subscript 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

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

dstdevwd( N, correction, x, strideX )

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 v = dstdevwd( x.length, 1.0, 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.
  • strideX: stride length for x.

The N and stride parameters determine which elements in the strided array are accessed at runtime. For example, to compute the standard deviation of every other element in x,

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

var x = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );

var v = dstdevwd( 4, 1.0, 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 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 v = dstdevwd( 4, 1.0, x1, 2 );
// returns 2.5

dstdevwd.ndarray( N, correction, x, strideX, offsetX )

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 v = dstdevwd.ndarray( x.length, 1.0, x, 1, 0 );
// returns ~2.0817

The function has the following additional parameters:

  • offsetX: 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 element in x starting from the second element

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

var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );

var v = dstdevwd.ndarray( 4, 1.0, 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 discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
var dstdevwd = require( '@stdlib/stats/base/dstdevwd' );

var x = discreteUniform( 10, -50, 50, {
    'dtype': 'float64'
});
console.log( x );

var v = dstdevwd( x.length, 1.0, x, 1 );
console.log( v );

C APIs

Usage

#include "stdlib/stats/base/dstdevwd.h"

stdlib_strided_dstdevwd( N, correction, *X, strideX )

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

const double x[] = { 1.0, -2.0, 2.0 };

double v = stdlib_strided_dstdevwd( 3, 1.0, x, 1 );
// returns ~2.0817

The function accepts the following arguments:

  • N: [in] CBLAS_INT number of indexed elements.
  • correction: [in] double 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: [in] double* input array.
  • strideX: [in] CBLAS_INT stride length for X.
double stdlib_strided_dstdevwd( const CBLAS_INT N, const double correction, const double *X, const CBLAS_INT strideX );

stdlib_strided_dstdevwd_ndarray( N, correction, *X, strideX, offsetX )

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

const double x[] = { 1.0, -2.0, 2.0 };

double v = stdlib_strided_dstdevwd_ndarray( 3, 1.0, x, 1, 0 );
// returns ~2.0817

The function accepts the following arguments:

  • N: [in] CBLAS_INT number of indexed elements.
  • correction: [in] double 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: [in] double* input array.
  • strideX: [in] CBLAS_INT stride length for X.
  • offsetX: [in] CBLAS_INT starting index for X.
double stdlib_strided_dstdevwd_ndarray( const CBLAS_INT N, const double correction, const double *X, const CBLAS_INT strideX, const CBLAS_INT offsetX );

Examples

#include "stdlib/stats/base/dstdevwd.h"
#include <stdio.h>

int main( void ) {
    // Create a strided array:
    const double x[] = { 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0 };

    // Specify the number of elements:
    const int N = 4;

    // Specify the stride length:
    const int strideX = 2;

    // Compute the variance:
    double v = stdlib_strided_dstdevwd( N, 1.0, x, strideX );

    // Print the result:
    printf( "sample standard deviation: %lf\n", 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

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