varianceyc

Calculate the variance of an array using a one-pass algorithm proposed by Youngs and Cramer.

The population variance of a finite size population of size N is given by

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

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 variance 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 variance, the result is biased and yields a biased sample variance. To compute an unbiased sample variance for a sample of size n,

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

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 variance and population variance. 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 varianceyc = require( '@stdlib/stats/array/varianceyc' );

varianceyc( x[, correction] )

Computes the variance of an array.

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

var v = varianceyc( x );
// returns ~4.3333

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 variance 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 variance 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 variance, 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 variance. To use adjust the degrees of freedom when computing the variance, provide a correction argument.

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

var v = varianceyc( x, 0.0 );
// returns ~2.8889

Notes

• If provided an empty array, the function returns NaN.
• If provided a correction argument which is greater than or equal to the number of elements in a provided input array, 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 varianceyc = require( '@stdlib/stats/array/varianceyc' );

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

var v = varianceyc( x );
console.log( v );

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References

• Youngs, Edward A., and Elliot M. Cramer. 1971. "Some Results Relevant to Choice of Sum and Sum-of-Product Algorithms." Technometrics 13 (3): 657–65. doi:10.1080/00401706.1971.10488826.