DMAR (version 1.0.0)

sd_unbiased: Unbiased Estimate of the Population Standard Deviation Under Normality

Description

Returns the unbiased estimate of the population standard deviation under normality. Although the sample variance \(s^2\) is unbiased for \(\sigma^2\) when computed with \(N - 1\) in the denominator, its square root \(s\) is biased downward for \(\sigma\) because the square root function is concave (Jensen's inequality). sd_unbiased applies the classical Holtzman (1950) correction factor that exactly removes that bias under normality.

Usage

sd_unbiased(s = NULL, N = NULL, X = NULL)

Value

A 1-row data.frame with columns term and value. The term value is "sd" and value is the unbiased estimate of \(\sigma\).

Arguments

s

The usual estimate of the standard deviation (the square root of the unbiased variance \(s^2\)).

N

The sample size on which s is based.

X

Optional vector of raw scores from which s and N are inferred (s = sd(X), N = length(X)). Mutually exclusive with s and N.

Author

Ken Kelley [email protected]

Details

The sample variance computed with \(N - 1\) in the denominator is unbiased for the population variance \(\sigma^2\), but its square root \(s\) is biased downward for \(\sigma\). Under normality, the multiplicative bias is $$E[s] \;=\; \sigma \cdot c_N^{-1}, \qquad c_N \;=\; \sqrt{(N - 1)/2}\, \cdot \Gamma((N - 1)/2)\,/\,\Gamma(N/2),$$ (Holtzman, 1950). Multiplying \(s\) by \(c_N\) therefore yields an unbiased estimator of \(\sigma\). The correction is non-trivial in small samples: \(c_N\) is about 1.064 at \(N = 5\), 1.028 at \(N = 10\), 1.009 at \(N = 30\), and is essentially 1 by \(N = 100\) (about 1.003). For most applied work the bias of \(s\) is small enough to ignore, but it matters when \(s\) feeds into downstream quantities (variance components, standardizers, planning calculations) at small \(N\).

Implementation note: the factor \(c_N\) is computed via lgamma to avoid the overflow of gamma above \(N \approx 340\), so the function is numerically stable for arbitrarily large \(N\).

References

Holtzman, W. H. (1950). The unbiased estimate of the population variance and standard deviation. American Journal of Psychology, 63, 615--617.

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

See Also

sd, cv

Examples

set.seed(113)
X <- rnorm(10, 100, 15)

# The plug-in estimate sqrt(s^2) is biased downward for sigma.
var(X)^.5

# Holtzman (1950) bias-corrected estimate, supplying s and N.
sd_unbiased(s = var(X)^.5, N = length(X))

# Equivalent call from the raw vector.
sd_unbiased(X = X)