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

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Τίτλος: On the Distribution of Logarithm of Standard Deviation From a Normal Population.
Συγγραφείς: Nadarajah, Saralees1 (AUTHOR) mbbsssn2@manchester.ac.uk, Kurdi, Talal1 (AUTHOR)
Πηγή: Quality & Reliability Engineering International. Apr2026, Vol. 42 Issue 3, p1399-1403. 5p.
Θεματικοί όροι: *Standard deviations, *Logarithms, *Probability density function, *Quantiles, *Cumulative distribution function, *Statistics, *Gaussian distribution, *Distribution (Probability theory)
Περίληψη: 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]
Βάση Δεδομένων: Academic Search Index
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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.
– Name: Subject
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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>
– Name: Abstract
  Label: Abstract
  Group: Ab
  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]
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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1002/qre.70133
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 5
        StartPage: 1399
    Subjects:
      – 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)
        Type: general
    Titles:
      – TitleFull: On the Distribution of Logarithm of Standard Deviation From a Normal Population.
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          Name:
            NameFull: Nadarajah, Saralees
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            NameFull: Kurdi, Talal
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          Dates:
            – D: 01
              M: 04
              Text: Apr2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 07488017
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            – Type: volume
              Value: 42
            – Type: issue
              Value: 3
          Titles:
            – TitleFull: Quality & Reliability Engineering International
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