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

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Title: On the Distribution of Logarithm of Standard Deviation From a Normal Population.
Authors: Nadarajah, Saralees1 (AUTHOR) mbbsssn2@manchester.ac.uk, Kurdi, Talal1 (AUTHOR)
Source: Quality & Reliability Engineering International. Apr2026, Vol. 42 Issue 3, p1399-1403. 5p.
Subjects: Standard deviations, Logarithms, Probability density function, Quantiles, Cumulative distribution function, Statistics, Gaussian distribution, Distribution (Probability theory)
Abstract: This letter investigates the distribution of Y=lnS$Y = \ln S$, where S$S$ 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) fY(y)=Cexpay−a2exp(2y)$f_Y (y) = C \exp \left[ a y - \frac{a}{2} \exp (2 y) \right]$, eliminating the need for numerical verification. We derive exact closed‐form expressions for the moments EYm$E \left(Y^m\right)$, E(Y)$E (Y)$, EY2$E \left(Y^2\right)$, EY3$E \left(Y^3\right)$, EY4$E \left(Y^4\right)$, variance, skewness, and kurtosis of Y$Y$, using derivatives of the gamma function. Furthermore, we establish closed‐form expressions for both the cumulative distribution function (CDF) and the quantile function of Y$Y$, resolving an open problem from prior work. Comparisons reveal that while existing approximations for E(Y)$E(Y)$ perform reasonably well for larger sample sizes, approximations for Var(Y)$Var(Y)$ show significant discrepancies for smaller n$n$. Y$Y$ 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. [ABSTRACT FROM AUTHOR]
Copyright of Quality & Reliability Engineering International is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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  Data: On the Distribution of Logarithm of Standard Deviation From a Normal Population.
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  Data: <searchLink fieldCode="AR" term="%22Nadarajah%2C+Saralees%22">Nadarajah, Saralees</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> mbbsssn2@manchester.ac.uk</i><br /><searchLink fieldCode="AR" term="%22Kurdi%2C+Talal%22">Kurdi, Talal</searchLink><relatesTo>1</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Quality+%26+Reliability+Engineering+International%22">Quality & Reliability Engineering International</searchLink>. Apr2026, Vol. 42 Issue 3, p1399-1403. 5p.
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  Data: <searchLink fieldCode="DE" term="%22Standard+deviations%22">Standard deviations</searchLink><br /><searchLink fieldCode="DE" term="%22Logarithms%22">Logarithms</searchLink><br /><searchLink fieldCode="DE" term="%22Probability+density+function%22">Probability density function</searchLink><br /><searchLink fieldCode="DE" term="%22Quantiles%22">Quantiles</searchLink><br /><searchLink fieldCode="DE" term="%22Cumulative+distribution+function%22">Cumulative distribution function</searchLink><br /><searchLink fieldCode="DE" term="%22Statistics%22">Statistics</searchLink><br /><searchLink fieldCode="DE" term="%22Gaussian+distribution%22">Gaussian distribution</searchLink><br /><searchLink fieldCode="DE" term="%22Distribution+%28Probability+theory%29%22">Distribution (Probability theory)</searchLink>
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  Data: This letter investigates the distribution of Y=lnS$Y = \ln S$, where S$S$ 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) fY(y)=Cexpay−a2exp(2y)$f_Y (y) = C \exp \left[ a y - \frac{a}{2} \exp (2 y) \right]$, eliminating the need for numerical verification. We derive exact closed‐form expressions for the moments EYm$E \left(Y^m\right)$, E(Y)$E (Y)$, EY2$E \left(Y^2\right)$, EY3$E \left(Y^3\right)$, EY4$E \left(Y^4\right)$, variance, skewness, and kurtosis of Y$Y$, using derivatives of the gamma function. Furthermore, we establish closed‐form expressions for both the cumulative distribution function (CDF) and the quantile function of Y$Y$, resolving an open problem from prior work. Comparisons reveal that while existing approximations for E(Y)$E(Y)$ perform reasonably well for larger sample sizes, approximations for Var(Y)$Var(Y)$ show significant discrepancies for smaller n$n$. Y$Y$ 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. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Quality & Reliability Engineering International is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1002/qre.70133
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        Text: English
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        PageCount: 5
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      – SubjectFull: Standard deviations
        Type: general
      – SubjectFull: Logarithms
        Type: general
      – SubjectFull: Probability density function
        Type: general
      – SubjectFull: Quantiles
        Type: general
      – SubjectFull: Cumulative distribution function
        Type: general
      – SubjectFull: Statistics
        Type: general
      – SubjectFull: Gaussian distribution
        Type: general
      – SubjectFull: Distribution (Probability theory)
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      – TitleFull: On the Distribution of Logarithm of Standard Deviation From a Normal Population.
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            NameFull: Nadarajah, Saralees
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            – D: 01
              M: 04
              Text: Apr2026
              Type: published
              Y: 2026
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