The population [standard deviation][standard-deviation] of a finite size population of size `N` is given by

Equation for the population standard deviation.

where the population mean is given by

Often in the analysis of data, the true population [standard deviation][standard-deviation] 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 [standard deviation][standard-deviation], the result is biased and yields an **uncorrected sample standard deviation**. To compute a **corrected sample standard deviation** for a sample of size `n`,

Equation for computing a corrected sample standard deviation.

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 standard deviation and population standard deviation. 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 snanstdevtk = require( '@stdlib/stats/base/snanstdevtk' ); ``` #### snanstdevtk( N, correction, x, stride ) Computes the [standard deviation][standard-deviation] of a single-precision floating-point strided array `x` ignoring `NaN` values and using a one-pass textbook algorithm. ```javascript var Float32Array = require( '@stdlib/array/float32' ); var x = new Float32Array( [ 1.0, -2.0, NaN, 2.0 ] ); var v = snanstdevtk( x.length, 1, x, 1 ); // returns ~2.0817 ``` 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 [standard deviation][standard-deviation] according to `N-c` where `c` corresponds to the provided degrees of freedom adjustment. When computing the [standard deviation][standard-deviation] 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 [standard deviation][standard-deviation], 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 [standard deviation][standard-deviation] 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 = snanstdevtk( N, 1, x, 2 ); // returns 2.5 ``` 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 = snanstdevtk( N, 1, x1, 2 ); // returns 2.5 ``` #### snanstdevtk.ndarray( N, correction, x, stride, offset ) Computes the [standard deviation][standard-deviation] of a single-precision floating-point strided array ignoring `NaN` values and using a one-pass textbook 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 = snanstdevtk.ndarray( x.length, 1, x, 1, 0 ); // returns ~2.0817 ``` 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 [standard deviation][standard-deviation] 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 = snanstdevtk.ndarray( N, 1, x, 2, 1 ); // returns 2.5 ```

## 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`. - 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 computing a negative variance due to floating-point rounding errors!). In short, no single "best" algorithm for computing the standard deviation 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.