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

where the population mean is given by

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,

where the sample mean is given by

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.

var variancetk = require( '@stdlib/stats/array/variancetk' );

Computes the variance of an array.

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

var v = variancetk( 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 adjust the degrees of freedom when computing the variance, provide a correction argument.

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

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

• 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).
• 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 when subtracting the two sums if the sums are large and the variance small. 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 returning a negative variance due to floating-point rounding errors!). In short, no single "best" algorithm for computing the variance 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.

• 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.