On the Distribution of Logarithm of Standard Deviation From a Normal Population

ABSTRACT This letter investigates the distribution of , where is the sample standard deviation from a normal population. Building on Maghsoodloo and Silva (2025), we provide a rigorous analytical validation of the probability density function (PDF) , eliminating the need for numerical verification. We derive exact closed‐form expressions for the moments , , , , , variance, skewness, and kurtosis of , using derivatives of the gamma function. Furthermore, we establish closed‐form expressions for both the cumulative distribution function (CDF) and the quantile function of , resolving an open problem from prior work. Comparisons reveal that while existing approximations for perform reasonably well for larger sample sizes, approximations for show significant discrepancies for smaller . is left skewed and leptokurtic, approaching symmetry and normal kurtosis as the sample size increases. Our results leverage gamma, incomplete gamma, and standardized incomplete gamma functions, enabling efficient computation via standard mathematical software.