dsemch

Calculate the standard error of the mean of a double-precision floating-point strided array using a one-pass trial mean algorithm.

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

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

where s is the sample standard deviation.

Usage

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

dsemch( N, correction, x, stride )

Computes the standard error of the mean of a double-precision floating-point strided array x using a one-pass trial mean algorithm.

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

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;

var v = dsemch( N, 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.
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 error of the mean 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 N = floor( x.length / 2 );

var v = dsemch( N, 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 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 = dsemch( N, 1, x1, 2 );
// returns 1.25

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

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

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

var x = new Float64Array( [ 1.0, -2.0, 2.0 ] );
var N = x.length;

var v = dsemch.ndarray( N, 1, x, 1, 0 );
// returns ~1.20185

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 error of the mean 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 N = floor( x.length / 2 );

var v = dsemch.ndarray( N, 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.
The underlying algorithm is a specialized case of Neely's two-pass algorithm. As the standard deviation is invariant with respect to changes in the location parameter, the underlying algorithm uses the first strided array element as a trial mean to shift subsequent data values and thus mitigate catastrophic cancellation. Accordingly, the algorithm's accuracy is best when data is unordered (i.e., the data is not sorted in either ascending or descending order such that the first value is an "extreme" value).

Examples

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

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

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.
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.
Chan, Tony F., Gene H. Golub, and Randall J. LeVeque. 1983. "Algorithms for Computing the Sample Variance: Analysis and Recommendations." The American Statistician 37 (3). American Statistical Association, Taylor & Francis, Ltd.: 242–47. doi:10.1080/00031305.1983.10483115.
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.