Sample variance of rounded variables

TL;DR

This paper derives exact formulas for the sample variance of rounded variables, analyzing when the common approximation of variance as the sum of true variance and rounding error variance is valid, especially for large, symmetric distributions.

Contribution

It provides exact expressions for the sample variance of rounded variables and discusses the conditions under which the simple approximation holds, particularly for large, symmetric distributions.

Findings

Exact formulas for sample variance of rounded variables.

Validation of the approximation $s^2=\sigma^2 + w^2/12$ under certain conditions.

Approximate scaling behavior of the sample variance for large, symmetric distributions.

Abstract

If the rounding errors are assumed to be distributed independently from the intrinsic distribution of the random variable, the sample variance $s^2$ of the rounded variable is given by the sum of the true variance $\sigma^2$ and the variance of the rounding errors (which is equal to $w^2/12$ where $w$ is the size of the rounding window). Here the exact expressions for the sample variance of the rounded variables are examined and it is also discussed when the simple approximation $s^2=\sigma^2+w^2/12$ can be considered valid. In particular, if the underlying distribution $f$ belongs to a family of symmetric normalizable distributions such that $f(x)=\sigma^{-1}F(u)$ where $u=(x-\mu)/\sigma$, and $\mu$ and $\sigma^2$ are the mean and variance of the distribution, then the rounded sample variance scales like $s^2-(\sigma^2+w^2/12)\sim\sigma\Phi'(\sigma)$ as $\sigma\to\infty$ where $\Phi(\tau)=\int_{-\infty}^\infty{\rm d}u\,e^{iu\tau}F(u)$ is the characteristic function of $F(u)$. It follows that, roughly speaking, the approximation is valid for a slowly-varying symmetric underlying distribution with its variance sufficiently larger than the size of the rounding unit.