README.md

variancetk

Calculate the variance of a one-dimensional ndarray using a one-pass textbook algorithm.

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 an uncorrected sample variance. To compute a corrected 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 variancetk = require( '@stdlib/stats/base/ndarray/variancetk' );

variancetk( arrays )

Computes the variance of a one-dimensional ndarray using a one-pass textbook algorithm.

var ndarray = require( '@stdlib/ndarray/base/ctor' );
var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );

var opts = {
    'dtype': 'generic'
};

var xbuf = [ 1.0, -2.0, 2.0 ];
var x = new ndarray( opts.dtype, xbuf, [ 3 ], [ 1 ], 0, 'row-major' );
var correction = scalar2ndarray( 1.0, opts );

var v = variancetk( [ x, correction ] );
// returns ~4.3333

The function has the following parameters:

• arrays: array-like object containing two elements: a one-dimensional input ndarray and a zero-dimensional ndarray specifying the degrees of freedom adjustment. Providing a non-zero degrees of freedom adjustment has the effect of adjusting the divisor during the calculation of the variance according to N-c where N is the number of elements in the input ndarray 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 corrected 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).

Notes

• If provided an empty one-dimensional ndarray, the function returns NaN.
• If N - c is less than or equal to 0 (where N corresponds to the number of elements in the input ndarray and c corresponds to the provided degrees of freedom adjustment), the function returns NaN.

Examples

var discreteUniform = require( '@stdlib/random/array/discrete-uniform' );
var ndarray = require( '@stdlib/ndarray/base/ctor' );
var scalar2ndarray = require( '@stdlib/ndarray/from-scalar' );
var ndarray2array = require( '@stdlib/ndarray/to-array' );
var variancetk = require( '@stdlib/stats/base/ndarray/variancetk' );

var opts = {
    'dtype': 'float64'
};

var xbuf = discreteUniform( 10, -50, 50, opts );
var x = new ndarray( opts.dtype, xbuf, [ xbuf.length ], [ 1 ], 0, 'row-major' );
console.log( ndarray2array( x ) );

var correction = scalar2ndarray( 1.0, opts );
var v = variancetk( [ x, correction ] );
console.log( v );

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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.