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