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 snanvariancech = require( '@stdlib/stats/base/snanvariancech' ); ``` #### snanvariancech( N, correction, x, stride ) Computes the [variance][variance] of a single-precision floating-point strided array `x` ignoring `NaN` values and using a one-pass trial mean algorithm. ```javascript var Float32Array = require( '@stdlib/array/float32' ); var x = new Float32Array( [ 1.0, -2.0, NaN, 2.0 ] ); var v = snanvariancech( 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 and `n` corresponds to the number of non-`NaN` indexed elements. 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 [`Float32Array`][@stdlib/array/float32]. - **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 Float32Array = require( '@stdlib/array/float32' ); var floor = require( '@stdlib/math/base/special/floor' ); var x = new Float32Array( [ 1.0, 2.0, 2.0, -7.0, -2.0, 3.0, 4.0, 2.0, NaN ] ); var N = floor( x.length / 2 ); var v = snanvariancech( 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 Float32Array = require( '@stdlib/array/float32' ); var floor = require( '@stdlib/math/base/special/floor' ); var x0 = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0, NaN ] ); var x1 = new Float32Array( x0.buffer, x0.BYTES_PER_ELEMENT*1 ); // start at 2nd element var N = floor( x0.length / 2 ); var v = snanvariancech( N, 1, x1, 2 ); // returns 6.25 ``` #### snanvariancech.ndarray( N, correction, x, stride, offset ) Computes the [variance][variance] of a single-precision floating-point strided array ignoring `NaN` values and using a one-pass trial mean algorithm and alternative indexing semantics. ```javascript var Float32Array = require( '@stdlib/array/float32' ); var x = new Float32Array( [ 1.0, -2.0, NaN, 2.0 ] ); var v = snanvariancech.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 Float32Array = require( '@stdlib/array/float32' ); var floor = require( '@stdlib/math/base/special/floor' ); var x = new Float32Array( [ 2.0, 1.0, 2.0, -2.0, -2.0, 2.0, 3.0, 4.0 ] ); var N = floor( x.length / 2 ); var v = snanvariancech.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 and `n` corresponds to the number of non-`NaN` indexed elements), both functions return `NaN`. - The underlying algorithm is a specialized case of Neely's two-pass algorithm. As the variance is invariant with respect to changes in the location parameter, the underlying algorithm uses the first non-`NaN` strided array element as a trial mean to shift subsequent data values and thus mitigate catastrophic cancellation. Accordingly, the algorithm's accuracy is best when data is **unordered** (i.e., the data is **not** sorted in either ascending or descending order such that the first value is an "extreme" value).

## 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][@neely:1966a]. - 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][@ling:1974a]. - 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][@chan:1983a]. - 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][@schubert:2018a].