dsemwd

Calculate the standard error of the mean of a double-precision floating-point strided array using Welford's algorithm.

The standard error of the mean of a finite size sample of size n is given by

sigma Subscript x overbar Baseline equals StartFraction sigma Over StartRoot n EndRoot EndFraction

where σ is the population standard deviation.

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. In this scenario, one must use a sample standard deviation to compute an estimate for the standard error of the mean

sigma Subscript x overbar Baseline almost-equals StartFraction s Over StartRoot n EndRoot EndFraction

where s is the sample standard deviation.

Usage

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

dsemwd( N, correction, x, strideX )

Computes the standard error of the mean 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 = dsemwd( x.length, 1, x, 1 );
// returns ~1.20185

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 error of the mean 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 = dsemwd( 4, 1, x, 2 );
// returns 1.25

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 = dsemwd( 4, 1, x1, 2 );
// returns 1.25

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

Computes the standard error of the mean 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 = dsemwd.ndarray( x.length, 1, x, 1, 0 );
// returns ~1.20185

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 error of the mean 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 = dsemwd.ndarray( 4, 1, x, 2, 1 );
// returns 1.25

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 dsemwd = require( '@stdlib/stats/base/dsemwd' );

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

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

C APIs

Usage

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

stdlib_strided_dsemwd( N, correction, *X, strideX )

Computes the standard error of the mean 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_dsemwd( 3, 1.0, x, 1 );
// returns ~1.20185

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_dsemwd( const CBLAS_INT N, const double correction, const double *X, const CBLAS_INT strideX );

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

Computes the standard error of the mean 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_dsemwd_ndarray( 3, 1.0, x, 1, 0 );
// returns ~1.20185

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_dsemwd_ndarray( const CBLAS_INT N, const double correction, const double *X, const CBLAS_INT strideX, const CBLAS_INT offsetX );

Examples

#include "stdlib/stats/base/dsemwd.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 standard error of the mean:
    double v = stdlib_strided_dsemwd( N, 1.0, x, strideX );

    // Print the result:
    printf( "standard error of the mean: %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

  • @stdlib/stats/base/dsem: calculate the standard error of the mean for a double-precision floating-point strided array.
  • @stdlib/stats/base/dstdevwd: calculate the standard deviation of a double-precision floating-point strided array using Welford's algorithm.
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