stdevwd

Calculate the standard deviation of a strided array using Welford's 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 stdevwd = require( '@stdlib/stats/base/stdevwd' );

stdevwd( N, correction, x, stride )

Computes the standard deviation of a strided array x using Welford's algorithm.

var x = [ 1.0, -2.0, 2.0 ];

var v = stdevwd( x.length, 1, 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 Array or typed array.
stride: index increment for x.

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

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

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

var v = stdevwd( N, 1, 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 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 N = floor( x0.length / 2 );

var v = stdevwd( N, 1, x1, 2 );
// returns 2.5

stdevwd.ndarray( N, correction, x, stride, offset )

Computes the standard deviation of a strided array using Welford's algorithm and alternative indexing semantics.

var x = [ 1.0, -2.0, 2.0 ];

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

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 standard deviation for every other value in x starting from the second value

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

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

var v = stdevwd.ndarray( N, 1, 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.
Depending on the environment, the typed versions (dstdevwd, sstdevwd, etc.) are likely to be significantly more performant.

Examples

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

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 = stdevwd( x.length, 1, x, 1 );
console.log( 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.