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README.md

stdevch

Calculate the standard deviation of an array using a one-pass trial mean algorithm.

The population standard deviation of a finite size population of size N is given by

$$\sigma = \sqrt{\frac{1}{N} \sum_{i=0}^{N-1} (x_i - \mu)^2}$$

where the population mean is given by

$$\mu = \frac{1}{N} \sum_{i=0}^{N-1} x_i$$

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,

$$s = \sqrt{\frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x})^2}$$

where the sample mean is given by

$$\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i$$

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 stdevch = require( '@stdlib/stats/array/stdevch' );

stdevch( x[, correction] )

Computes the standard deviation of an array using a one-pass trial mean algorithm.

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

var v = stdevch( x );
// returns ~2.0817

The function has the following parameters:

  • x: input array.
  • 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 N corresponds to the number of array elements and 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 unbiased 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). Default: 1.0.

By default, the function computes the sample standard deviation. To adjust the degrees of freedom when computing the standard deviation, provide a correction argument.

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

var v = stdevch( x, 0.0 );
// returns ~1.6997

Notes

  • If provided an empty array, the function returns NaN.
  • If N - c is less than or equal to 0 (where c corresponds to the provided degrees of freedom adjustment), the function returns NaN.
  • The function supports array-like objects having getter and setter accessors for array element access (e.g., @stdlib/array/base/accessor).

Examples

var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
var stdevch = require( '@stdlib/stats/array/stdevch' );

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

var v = stdevch( x );
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.