{"id":"hexagon/2610.00230","pdf":"https://hexagonmath.org/pdf/2610.00230v1","url":"https://hexagonmath.org/abs/2610.00230v1","tags":["msc:62F10","msc:42A16","Fourier series","characteristic functions","mean-squared error","quantization error","rounded sample mean","rounding error"],"score":0,"title":"A Positive Dispersion Optimum for the Rounded Sample Mean: Sufficiency and Necessity of Finite Variance","domain":"math.ST","origin":"ai-generated","source":"hexagon","authors":["Mark R. Diamond","Angela O'Brien-Malone"],"reviews":0,"subject":"math.ST","summary":"A continuous quantity, observable only after nearest-lattice rounding, is estimated by the mean of $n$ rounded readings of an imperfect instrument. Averaged over the position of the true value within a rounding cell, the mean-squared error of this estimator is a function of one dimensionless parameter: the pre-rounding standard deviation $s$, measured in cell widths. A perfectly repeatable instrument ($s=0$) returns the same rounded value on every reading, so averaging provides no additional information; its cell-averaged error is the rounding floor $1/12$. We show that this floor is never…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","Gemini","ChatGPT","Kimi","Mathematica"],"discussed":0,"published":"2026-10-14","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}