Bias Correction Constant for Sample Standard Deviation

Compute the constant \(c_4\), which corrects the bias of the sample standard deviation \(S\) as an estimator of the process standard deviation \(\sigma\) under normality.

Usage

c4(n)

Arguments

n

Integer sample size, at least 2. Can be a vector.

Value

Numeric vector of \(c_4\) values, the same length as n.

Details

The constant is given by $$c_4 = \sqrt{\frac{2}{n-1}} \,\frac{\Gamma(n/2)}{\Gamma\{(n-1)/2\}},$$ so that \(E[S] = c_4 \, \sigma\). Multiplying \(S\) by \(1 / c_4\) produces an unbiased estimator of \(\sigma\).

References

Montgomery, D. C. (2009). Introduction to Statistical Quality Control, 6th ed. Wiley.

See also

d2, d3, table.const

Author

Daniela R. Recchia, Emanuel P. Barbosa

Examples

c4(5)
#> [1] 0.9399856
c4(2:15)
#>  [1] 0.7978846 0.8862269 0.9213177 0.9399856 0.9515329 0.9593688 0.9650305
#>  [8] 0.9693107 0.9726593 0.9753501 0.9775594 0.9794056 0.9809714 0.9823162