The population [variance][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][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][variance], the result is biased and yields a **biased sample variance**. To compute an **unbiased sample variance** for a sample of size `n`,

Equation for computing an unbiased sample variance.

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.

## Usage ```javascript var variancetk = require( '@stdlib/stats/base/variancetk' ); ``` #### variancetk( N, correction, x, stride ) Computes the [variance][variance] of a strided array `x` using a one-pass textbook algorithm. ```javascript var x = [ 1.0, -2.0, 2.0 ]; var v = variancetk( x.length, 1, x, 1 ); // returns ~4.3333 ``` 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 [variance][variance] according to `N-c` where `c` corresponds to the provided degrees of freedom adjustment. When computing the [variance][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][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). - **x**: input [`Array`][mdn-array] or [`typed array`][mdn-typed-array]. - **stride**: index increment for `x`. The `N` and `stride` parameters determine which elements in `x` are accessed at runtime. For example, to compute the [variance][variance] of every other element in `x`, ```javascript var floor = require( '@stdlib/math/base/special/floor' ); var x = [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0 ]; var N = floor( x.length / 2 ); var v = variancetk( N, 1, x, 2 ); // returns 6.25 ``` Note that indexing is relative to the first index. To introduce an offset, use [`typed array`][mdn-typed-array] views. ```javascript var Float64Array = require( '@stdlib/array/float64' ); var floor = require( '@stdlib/math/base/special/floor' ); 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 N = floor( x0.length / 2 ); var v = variancetk( N, 1, x1, 2 ); // returns 6.25 ``` #### variancetk.ndarray( N, correction, x, stride, offset ) Computes the [variance][variance] of a strided array using a one-pass textbook algorithm and alternative indexing semantics. ```javascript var x = [ 1.0, -2.0, 2.0 ]; var v = variancetk.ndarray( x.length, 1, x, 1, 0 ); // returns ~4.33333 ``` The function has the following additional parameters: - **offset**: starting index for `x`. While [`typed array`][mdn-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 [variance][variance] for every other value in `x` starting from the second value ```javascript var floor = require( '@stdlib/math/base/special/floor' ); var x = [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ]; var N = floor( x.length / 2 ); var v = variancetk.ndarray( N, 1, x, 2, 1 ); // returns 6.25 ```

## 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 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. - Depending on the environment, the typed versions ([`dvariancetk`][@stdlib/stats/base/dvariancetk], [`svariancetk`][@stdlib/stats/base/svariancetk], etc.) are likely to be significantly more performant.