Sample variance of rounded variables
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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 σ^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=σ^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)=σ^-1F(u) where u=(x-μ)/σ, and μ and σ^2 are the mean and variance of the distribution, then the rounded sample variance scales like s^2-(σ^2+w^2/12)∼σΦ'(σ) as σ→∞ where Φ(τ)=∫_-∞^∞ du e^iuτF(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.
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