Vol. 8, 2023-02 Cite as: Hernandez, H. (2023). Probability Distribution and Bias of the Sample Standard Deviation. ForsChem Research Reports, 8, 2023-02, 1 - 26. Publication Date: 25/01/2023. Probability Distribution and Bias of the Sample Standard Deviation Hugo Hernandez ForsChem Research, 050030 Medellin, Colombia hugo.hernandez@forschem.org doi: 10.13140/RG.2.2.22144.51205 Abstract Since the 19 th century, it has been known that the estimation of the population variance from a sample needs to be corrected to remove bias (Bessel’s correction), and that even when the estimation of the variance can be unbiased, the corresponding estimation of the standard deviation is still biased. Unfortunately, the probability distribution of the standard deviation of a sample strongly depends on the particular distribution of the elements in the population, and only for very well-known distributions, such as the normal distribution, a distribution model of the sample standard deviation is possible (i.e. Helmert or Chi distribution). In this report, the general formulation of the probability density of the sample standard deviation is presented, along with different approximations for reducing its mathematical complexity. These approximations are also particularly considered for normal populations. In addition, a Monte Carlo simulation was performed to obtain a general empirical approximation of estimation bias for arbitrary distributions, based on the kurtosis of the population (if known) or alternatively, on the kurtosis of the sample. While normal populations can be satisfactorily estimated using Helmert’s distribution, non-normal distributions require the proposed empirical correction. Keywords Bessel’s Correction, Change of Variable, Chi Distribution, Helmert Distribution, Kurtosis, Normal Distribution, Randomistics, Sampling, Standard Deviation, Unbiased Estimators, Variance 1. Introduction The variance   of a population of  elements is defined as the mean square deviation from the mean  (also known as the second central moment of the distribution):    ∑ (   )     (1.1)