dmeanstdevpn

Calculate the mean and standard deviation of a double-precision floating-point strided array using a two-pass algorithm.

The population standard deviation 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 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,

where the sample mean is given by

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

dmeanstdevpn( N, correction, x, strideX, out, strideOut )

Computes the mean and standard deviation of a double-precision floating-point strided array x using a two-pass algorithm.

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

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var out = new Float64Array( 2 );

var v = dmeanstdevpn( x.length, 1, x, 1, out, 1 );
// returns <Float64Array>[ ~0.3333, ~2.0817 ]

var bool = ( v === out );
// returns true

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: index increment for x.
out: output Float64Array for storing results.
strideOut: index increment for out.

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

var Float64Array = require( '@stdlib/array/float64' );
var floor = require( '@stdlib/math/base/special/floor' );

var x = new Float64Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ] );
var out = new Float64Array( 2 );
var N = floor( x.length / 2 );

var v = dmeanstdevpn( N, 1, x, 2, out, 1 );
// returns <Float64Array>[ 1.25, 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 floor = require( '@stdlib/math/base/special/floor' );

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 out0 = new Float64Array( 4 );
var out1 = new Float64Array( out0.buffer, out0.BYTES_PER_ELEMENT*2 ); // start at 3rd element

var N = floor( x0.length / 2 );

var v = dmeanstdevpn( N, 1, x1, 2, out1, 1 );
// returns <Float64Array>[ 1.25, 2.5 ]

dmeanstdevpn.ndarray( N, correction, x, strideX, offsetX, out, strideOut, offsetOut )

Computes the mean and standard deviation of a double-precision floating-point strided array using a two-pass algorithm and alternative indexing semantics.

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

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var out = new Float64Array( 2 );

var v = dmeanstdevpn.ndarray( x.length, 1, x, 1, 0, out, 1, 0 );
// returns <Float64Array>[ ~0.3333, ~2.0817 ]

The function has the following additional parameters:

offsetX: starting index for x.
offsetOut: starting index for out.

While typed array views mandate a view offset based on the underlying buffer, the offset parameters support indexing semantics based on a starting index. For example, to calculate the mean and standard deviation for every other value in x starting from the second value

var Float64Array = require( '@stdlib/array/float64' );
var floor = require( '@stdlib/math/base/special/floor' );

var x = new Float64Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] );
var out = new Float64Array( 4 );
var N = floor( x.length / 2 );

var v = dmeanstdevpn.ndarray( N, 1, x, 2, 1, out, 2, 1 );
// returns <Float64Array>[ 0.0, 1.25, 0.0, 2.5 ]

Notes

If N <= 0, both functions return a mean and standard deviation equal to NaN.
If N - c is less than or equal to 0 (where c corresponds to the provided degrees of freedom adjustment), both functions return a standard deviation equal to NaN.

Examples

var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var Float64Array = require( '@stdlib/array/float64' );
var dmeanstdevpn = require( '@stdlib/stats/base/dmeanstdevpn' );

var out;
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 );

out = new Float64Array( 2 );
dmeanstdevpn( x.length, 1, x, 1, out, 1 );
console.log( out );

References

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.
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.