A Positive Dispersion Optimum for the Rounded Sample Mean: Sufficiency and Necessity of Finite Variance

The problem considered in this paper has its origin in Lumsden's~\cite{Lumsden1976} ``flogging wall'' thought experiment, first encountered by MRD as an undergraduate student in 1979. Lumsden used the flogging wall to test the counterintuitive possibility that, in some circumstances, an unreliable psychometric instrument might outperform a reliable one. Between 1989 and 2014, Daniel D. Reidpath (ORCID: \href{https://orcid.org/0000-0002-8796-0420}{0000-0002-8796-0420}) and MRD intermittently explored a variety of formalized simulation models of the flogging wall which strongly suggested that instrument unreliability could be advantageous. Two unpublished papers were developed on the basis of the simulation results. In 2023 MRD reformulated the problem in terms of a weighing scale that \emph{truncates} to finite precision, as opposed to the simpler rounding formulation used here. AOBM observed that this finite precision formulation could be expressed naturally as a question about quantization; see for example \cite[p.~1]{WidrowKollar2008}. Our first formalization was restricted to Gaussian scatter using, as a starting point, the treatment of quantized distributions given by Schwartz \cite{Schwartz1980}, and drawing also on the quantization framework of Widrow~\cite{Widrow1961}. This led to a moment-based derivation of a formula for the mean-squared error \(\MSE_n(u,s)\) around a specific $u$, followed by integration over \(u\in[-\tfrac12,\tfrac12)\), differentiation with respect to \(s\), and a Jacobi theta-function expression for the minimizing scale. That work was developed into a short unpublished manuscript. The present paper arose from the attempt to identify which features of the Gaussian calculation were essential. In particular, MRD conjectured from the moment-based model that positive finite variance might be the necessary and sufficient condition for the existence of a strictly positive minimizer of the cell-averaged mean-squared error, for \(n>1\). The central simplification in the present paper is the replacement of the Gaussian moment calculation by the autocorrelation formulation used above. This change separates the existence argument from the special analytic form of the Gaussian density, and is what allows the result to be stated for the more general class of distributions considered here. Figures were initially developed in Wolfram Mathematica and then translated into TikZ for inclusion in the manuscript. The mathematical arguments, exposition, and final form of the paper are the responsibility of the authors. \section*{Acknowledgements and AI tool disclosure} The authors used several generative AI tools during the development of this paper, including OpenAI ChatGPT, Anthropic Claude, Google Gemini, and Moonshot AI Kimi. The tools were used to check algebraic derivations from the quantization literature, to test and expand justifications involving Fourier series, convergence, and exchanges of limits, sums, and integrals, and to help identify simpler proof strategies from earlier moment-based formulations. In particular, ChatGPT and Claude independently suggested that it might be possible to simplify the existence argument by working with an autocorrelation function rather than with the moment formula derived from the Gaussian model. AI tools were also used for proofreading and for checks of notational consistency. Google Gemini and Moonshot Kimi were particularly helpful with LaTeX assistance, and the translation of Wolfram Mathematica plots into TikZ. All AI-assisted mathematical suggestions were reviewed, revised, and checked by the authors. The text, mathematical claims, and final editorial decisions are those of the authors, who take responsibility for any remaining errors.