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A Bayesian solution to identify mean and standard deviation of a distribution when only minimum and maximum are known: moving further from the rule of thumb

Marc Girondot et al.

Environmental and Ecological Statistics2026https://doi.org/10.1007/s10651-026-00707-8article
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Abstract

Many historical datasets report only summary statistics such as minimum and maximum values, limiting their utility in meta-analyses due to the absence of detailed distributional information. Traditional methods to estimate standard deviation from such limited data make strong assumptions, provide point estimates without uncertainty, and fail when the range is zero. This paper introduces a novel Bayesian approach to infer the underlying distribution from observed minima and maxima, using the generalized extreme value (GEV) distribution. By treating the problem as an inverse problem, the method simulates data from candidate distributions, fits GEV parameters to the extremes, and combines information from other available statistics (e.g., mean, median, quantiles) to compute a posterior distribution of the parameters. Implemented in the R package HelpersMG, the method allows estimation of the mean and standard deviation—even when Max = Min —along with associated uncertainty. Applied to incubation duration data for chelonians, the method successfully reconstructs plausible distributions and outperforms previous approaches in robustness and flexibility. Despite its computational intensity for non-exponential family distributions, this method enhances the value of sparse historical data in ecological and biological research.

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https://doi.org/https://doi.org/10.1007/s10651-026-00707-8

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@article{marc2026,
  title        = {{A Bayesian solution to identify mean and standard deviation of a distribution when only minimum and maximum are known: moving further from the rule of thumb}},
  author       = {Marc Girondot et al.},
  journal      = {Environmental and Ecological Statistics},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1007/s10651-026-00707-8},
}

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