README.md

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dsemtk

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

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

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

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

$$\sigma_{\bar{x}} \approx \frac{s}{\sqrt{n}}$$

where s is the sample standard deviation.

Usage

import dsemtk from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dsemtk@esm/index.mjs';

dsemtk( N, correction, x, strideX )

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

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';

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

var v = dsemtk( 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,

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';

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

var v = dsemtk( 4, 1, x, 2 );
// returns 1.25

Note that indexing is relative to the first index. To introduce an offset, use typed array views.

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';

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

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

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

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';

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

var v = dsemtk.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

import Float64Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-float64@esm/index.mjs';

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

var v = dsemtk.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.
• Some caution should be exercised when using the one-pass textbook algorithm. Literature overwhelmingly discourages the algorithm's use for two reasons: 1) the lack of safeguards against underflow and overflow and 2) the risk of catastrophic cancellation. These concerns have merit; however, the one-pass textbook algorithm should not be dismissed outright. For data distributions with a moderately large standard deviation to mean ratio (i.e., coefficient of variation), the one-pass textbook algorithm may be acceptable, especially when performance is paramount and some precision loss is acceptable (including a risk of computing a negative variance due to floating-point rounding errors!). In short, no single "best" algorithm for computing the standard error of the mean exists. The "best" algorithm depends on the underlying data distribution, your performance requirements, and your minimum precision requirements. When evaluating which algorithm to use, consider the relative pros and cons, and choose the algorithm which best serves your needs.

Examples

<!DOCTYPE html>
<html lang="en">
<body>
<script type="module">

import discreteUniform from 'https://cdn.jsdelivr.net/gh/stdlib-js/random-array-discrete-uniform@esm/index.mjs';
import dsemtk from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-base-dsemtk@esm/index.mjs';

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

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

</script>
</body>
</html>

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References

• 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.

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See Also

• @stdlib/stats-base/dsem: calculate the standard error of the mean for a double-precision floating-point strided array.
• @stdlib/stats-base/dstdevtk: calculate the standard deviation of a double-precision floating-point strided array using a one-pass textbook algorithm.

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Notice

This package is part of stdlib, a standard library with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2025. The Stdlib Authors.