Unbiased Estimate of the Population Standard Deviation Under Normality

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)

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.

Value

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

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.

Author

Ken Kelley kkelley@nd.edu

Examples

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

# The plug-in estimate sqrt(s^2) is biased downward for sigma.
var(X)^.5
#> [1] 14.31461

# Holtzman (1950) bias-corrected estimate, supplying s and N.
sd_unbiased(s = var(X)^.5, N = length(X))
#>  term value
#>  sd   14.7 

# Equivalent call from the raw vector.
sd_unbiased(X = X)
#>  term value
#>  sd   14.7