Uniform quantization error (${\sigma }_{ϵ}=\Delta /\sqrt{12}$)

rounding an unclipped voltage to equally spaced levels of separation $\Delta$ leaves error in $[-\Delta /2,\Delta /2]$. If the fractional position within a conversion interval is uniformly distributed, direct integration gives zero mean and expectation value $⟨{ϵ}^2⟩={\Delta }^2/12$. This is a statistical model for the error, not a claim that the converter adds independent white noise for every possible input.

A smooth input probability density varying slowly on the scale of one conversion interval makes the fractional uniform quantization error approximately uniform. Large overall variance is not sufficient: discrete inputs aligned with conversion levels provide a counterexample. Clipping, coarse sampling and correlations between error and signal require separate analysis.